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Department of Civil Engineering
Highly Non-linear Post-buckling
Analysis of Shell Structures
A thesis in partial fulfilment of the requirements
for a Bachelor of Sciences degree in the field of
Civil Engineering
Prepared for: University of Cape Town
Author:
Mr. Gary Hopkins
Supervisor:
Dr. Sebastian Skatulla
November, 2014
Plagiarism Declaration
ˆ I know that plagiarism is wrong. Plagiarism is to use anothers work and to pretend that
it is ones own.
ˆ I have used the Harvard Convention for citation and referencing. Each significant contri-
bution to and quotation in this report from the work or works of other people has been
attributed and has been cited and referenced.
ˆ This thesis is my own work.
ˆ I have not allowed and will not allow anyone to copy my work with the intention of passing
it as his or her own work.
Name Student No Date Signed
Gary Hopkins HPKGAR001 10 November 2014
i
Abstract
Shell structures have been widely used in many engineering applications such as pipelines,
liquid-retaining structures and aerospace structures. As shell structures are commonly con-
structed with thin walls, buckling is generally the primary failure mechanism of concern. Many
non-linear studies concerned with buckling have been conducted with regards to shell struc-
tures. However, limited studies have been conducted in which the post-buckling behaviour of
such structures has been investigated, and for this reason the post-buckling behaviour of shell
structures remains misunderstood.
In the post-buckling regime of ductile metallic structures, large deformations are often ex-
perienced with corresponding plastic deformation. Therefore, it is required that analyses are
conducted in a non-linear manner with respect to deformation and stress-strain relationships.
The Finite Element Method is commonly employed as a tool to model such behaviour, how-
ever, problems such as shear locking result in computational difficulties. Such difficulties are
often avoided with the use of specialised shell elements making simplifications in terms of
geometry and behaviour.
This study aims to implement an elasto-plastic constitutive law based on the logarithmic
right Cauchy-Green deformation tensor, in order to make possible computational analysis
of shell structures in the post-buckling regime using three-dimensional continuum mechanics
and finite strain theory, avoiding as far as possible the simplification of shell geometry and
behaviour. The constitutive law is implemented within the existing framework of SESKA,
a C++ code developed by Doctor Sebastian Skatulla with the purpose of solving equation
systems as required within this study. Implementation within the current framework allows
for the use of highly smooth meshfree approximations of the Element-free Galerkin Method,
avoiding the usual limitations of the Finite Element Method.
Following implementation, simple shell structures are modelled and analysed in order to gain
a preliminary understanding of the behaviour of such structures in the post-buckling regime.
Additionally, analysis is conducted in order to ascertain whether use of the methods employed
within this study are feasible and suitable for further use in non-linear post-buckling analyses
of shell structures.
This study is unique in that it appears to be the first known attempt to implement an elasto-
plastic post-buckling analysis tool for shell structures making use of three-dimensional con-
tinuum mechanics and meshfree methods, as opposed to specialised shell elements. Therefore,
to conclude this study, results are benchmarked against verified analyses of similar structures
which have made use of specialised shell elements and a visco-plastic material law.
ii
Acknowledgements
Firstly, I would like to thank my supervisor, Dr Sebastian Skatulla, for his guidance and sup-
port throughout this study, and for providing me with such an interesting topic to undertake.
While I have been overwhelmed with an abundance of new information and skills required, Dr
Skatulla has continuously put in great effort to guide me through this study. Additionally, his
patience and enthusiasm while explaining countless topics is highly appreciated and I would
have not been able to undertake this study without his expert guidance.
Secondly, my parents, for funding me through my schooling and most of my university career;
providing me with the opportunity to excel academically and as a person, by setting a flawless
example for me to aspire to. I extend many thanks to my brother Ryan, girlfriend Michaela
and all other friends who have supported and guided me.
The funding of Stefanutti Stocks Marine for the past two years is acknowledged and highly
appreciated, relieving my parents of financial pressure.
This thesis project has been supported by the Centre for High Performance Computing,
Mowbray.
iii
Table of Contents
Plagiarism Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Notation and list of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background to study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research project motivation, aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Motivation for research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Aim of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.3 Objectives of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Scope and limitations of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Expected results of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1 Benchmarking of constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.2 Geometric results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.3 Load carrying capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Layout of this document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
iv
2.2 Shell structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Methods of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Geometry of deformed structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Post-buckling strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Continuum mechanics theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 Elastic and plastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.2 Balance law of continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Stress measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 The constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.2 Elastic constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5.3 Plastic constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Computational analysis principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 General analysis considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 Buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Linear and non-linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.3 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Modelling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 Gauss quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.3 Element-free Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
4.4 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4.1 Time integration and local iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4.2 Stress and algorithmic tangent operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Assembly of equation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.2 Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2.3 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Simply supported plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4 Pinched cylinder with rigid diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.5 Axially compressed cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.5.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.5.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 Conclusions and recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 Implementation of constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.3 Benchmarking of constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4 Pinched cylinder with rigid diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.5 Axially compressed cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
vi
6.6 Further use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Appendix A Source code of implemented
constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
vii
List of Figures
2.1 Simple geometry of a shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 A buckled cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Typical computational buckling dimples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Stress versus strain of a ductile material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Deformation of a solid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.1 Cantilever beam - undeformed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Cantilever beam - static load deformation graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Cantilever beam - deformed configuration displacement contour plot . . . . . . . . . . 35
5.4 Cantilever beam - deformed configuration plastic strain contour plot . . . . . . . . . . 35
5.5 Cantilever beam - time deformation graph during dynamic analysis. . . . . . . . . . . 36
5.6 Simply supported plates - Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.7 Simply supported plates - Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.8 Simply supported plates - undeformed configuration . . . . . . . . . . . . . . . . . . . . . . . . 38
5.9 Simply supported plate - Case I load deformation graph . . . . . . . . . . . . . . . . . . . . 38
5.10 Simply supported plate - Case II load deformation graph. . . . . . . . . . . . . . . . . . . . 39
5.11 Simply supported plate - case I deformed configuration . . . . . . . . . . . . . . . . . . . . . 39
5.12 Simply supported plate - case I plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.13 Simply supported plate - case II deformed configuration . . . . . . . . . . . . . . . . . . . . 40
5.14 Simply supported plate - case II plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.15 Pinched cylinder - problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.16 Pinched cylinder - undeformed configuration with coarse mesh . . . . . . . . . . . . . . . 42
5.17 Pinched cylinder - undeformed configuration with fine mesh . . . . . . . . . . . . . . . . . 42
5.18 Pinched cylinder (coarse mesh) - load deformation graph . . . . . . . . . . . . . . . . . . . . 43
viii
5.19 Pinched cylinder - maximum deformation and plastic strain at end of first
loading cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.20 Pinched cylinder - final deformation and plastic strain . . . . . . . . . . . . . . . . . . . . . . 45
5.21 Pinched cylinder (fine mesh) - load deformation graph . . . . . . . . . . . . . . . . . . . . . . 46
5.22 Pinched cylinder - deformed configuration at 5mm displacement . . . . . . . . . . . . . 46
5.23 Pinched cylinder - deformed configuration at 15mm displacement . . . . . . . . . . . . 47
5.24 Pinched cylinder - effective plastic strain at 15mm displacement . . . . . . . . . . . . . 47
5.25 Pinched cylinder - deformed configuration at 20mm displacement . . . . . . . . . . . . 47
5.26 Compressed cylinder - undeformed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.27 Compressed cylinder - load deformation graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.28 Compressed cylinder - deformed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.29 Compressed cylinder - final plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.30 Compressed cylinder - experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
ix
List of Tables
5.1 Cantilever beam elastic and plastic material parameters. . . . . . . . . . . . . . . . . . . . . 33
5.2 Cantilever beam dynamic material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Simply supported plates - elastic and plastic material parameters . . . . . . . . . . . . 37
x
Notation and list of symbols
Symbols used are defined as they appear throughout the text. In the following, the general
scheme of notation and list of symbols is assembled:
a roman lower-case letters denote scalars
A roman upper-case bold-face letters denote tensors
a roman lower-case bold-face letters denote vectors
A−1
inverse of a tensor
AT
transpose of a tensor
A calligraphic upper-case letters denote sets
:= definition of equivalence
Subscripts
e elastic
p plastic
ref in the reference configuration
xi
Chapter 1
Introduction
This thesis serves the purpose to document the process followed in order to implement a non-
linear elasto-plastic material law to conduct qualitative computational investigations into the
post-buckling behaviour of shell structures.
1.1 Background to study
Shell structures are commonly used in engineering practice due to their many advantages over
other engineering structures. When compared to other engineering structures, such advantages
include high stiffness, efficiency and strength to weight ratio (Fung and Sechler, 1974). Due to
such properties, shell structures are able to serve the purpose of other engineering structures
while requiring less material. Contemporary examples of shell structures include arch dams,
domed roofs, liquid-retaining structures, aircraft and carbon nanotubes.
Shell structures have been extensively studied over the past century. With increasing com-
puting power, many experimental and theoretical investigations have been conducted into the
strength and stability of these structures. According to Mandal and Callandine (2000), when
shell structures are loaded in compression, buckling constitutes the most common form of
failure. Furthermore, as elastic buckling is often catastrophic, many studies concerned with
failure of shell structures have been aimed at predicting initial buckling loads. For this reason,
few studies have been undertaken to explore the post-buckling behaviour of shell structures
deep into the non-linear deformation and plastic regimes. (Vaziri, 2009)
Additionally, as mentioned by Vaziri (2009), recent advancements in nanotechnology have
increased the potential use of shell structures. However, understanding of the post-buckling
behaviour of shells remains ambiguous and requires further investigation. Chapelle and Bathe
(1998) refer to shells as the prima donnas of structures, and raise the problem that analysis
of such structures is difficult due to their unpredictable behaviour with small changes in
geometry. According to Vaziri (2009), “Our understanding of shell and membrane structures
is still in its infancy”, and many further studies are required to fully understand the mechanics
of these highly significant structures.
1
1.2 Research project motivation, aims and objectives
In order to successfully further the understanding of the post-buckling behaviour of shell
structures, the following research motivation, aims and objectives are specified.
1.2.1 Motivation for research
While shell structures have been extensively studied, few studies have been concerned with
plastic post-buckling behaviour. This is partly due to failure concerns related to initial buck-
ling, and partly due to the difficulty of conducting such experiments caused by shear locking
and excessive mesh distortion when making use of the Finite Element Method. Additionally,
studies which have investigated the post-buckling regime have predominantly investigated
elastic buckling. However, during buckling of metallic structures, plastic deformation is expe-
rienced and it is required that plasticity is accounted for.
As structures require larger and more efficient spans, are required to contain more water as
population increases, or as nanotechnology advances, a further understanding of shell struc-
tures will be required. Therefore, it is critical that tools be developed in order to obtain a
better knowledge of the post-buckling behaviour of shell structures in terms of stability, buck-
ling mechanisms and load carrying capacity to allow for more efficient and safe use of such
structures.
1.2.2 Aim of research
This study aims to implement and utilise an elasto-plastic constitutive law based on the loga-
rithmic right Cauchy-Green deformation tensor, in order to make possible the computational
analysis of shell structures in the post-buckling regime using three-dimensional continuum
mechanics, avoiding as far as possible the simplification of shell geometry and behaviour. It is
aimed that the constitutive law be programmed within the existing framework of the in-house
structural analysis software SESKA.
Following implementation, this study aims to ascertain the feasibility of using the material law
in further studies in order to gain an improved understanding of the post-buckling behaviour
of shell structures. Therefore, it is aimed that the implemented material law is benchmarked
against verified results of analysis conducted with the use of specialised shell elements.
Subsequently, it is aimed that preliminary qualitative analyses concerning the post-buckling
behaviour of shell structures be conducted taking into consideration stability, geometric
changes during deformation, and load carrying capacity.
2
1.2.3 Objectives of study
The objectives of this research project are:
1. Familiarisation with the in-house structural analysis software SESKA;
2. Presentation of background knowledge required for the analysis of such structures;
3. Implementation of an elasto-plastic constitutive law within the existing framework of
SESKA;
4. Creation of simple models of shell structures using the commercial pre-processing software
GiD;
5. Computational analysis of models created using the implemented material law;
6. Benchmarking against verified results of similar shell structures in order to determine the
feasibility of further use of the implemented law; and
7. Post-processing of results using GiD in order to gain a preliminary qualitative understand-
ing into the post-buckling regime of shell structures with respect to stability, deformation
behaviour and load carrying capacity.
1.3 Scope and limitations of investigation
Prior to the implementation of the constitutive law used in this study, a von Mises elasto-
plastic constitutive law based on the logarithmic unsymmetric stretch tensor had been imple-
mented. However, it was required that a constitutive law making use of the right Cauchy-Green
deformation tensor be implemented due to the unsymmetric nature of the previously imple-
mented law; causing increased computational time. Due to the many similarities between the
two laws, the existing law was modified in order to implement the law used in this study.
The law used in this study makes use of the existing computational continuum mechanics
framework contained within SESKA, and therefore the documentation of implementation is
primarily concerned with detailing the additional methods followed during implementation.
Subsequently, simple models of thin-walled and specifically shell structures are created and
analysed. This study is aimed at qualitative analysis of such structures in the post-buckling
and plastic regimes, and therefore aspects such as initial material imperfections are not ac-
counted for as initial buckling loads are not of high importance within this study. However,
plasticity of the materials analysed is taken into account with the inclusion of plastic history
variables, and strain hardening with the inclusion of linear and non-linear hardening terms.
This study does not attempt to conduct a full analysis of such structures, but rather to
gain a preliminary understanding of behaviour of such structures when analysed with the
use of elasto-plasticity and three dimensional continuum mechanics. Additionally, included
within the scope of this study is the use of the implemented constitutive law in order to
3
determine feasibility for use in further, more detailed investigations. This investigation and
the constitutive law implemented are limited to the use of metallic structures only.
1.4 Expected results of investigation
The following results in terms implementation, benchmarking and analysis are expected:
1.4.1 Benchmarking of constitutive law
It is expected that in comparison to the visco-plastic shell element analysis against which
the implemented law is to be benchmarked, similar behaviour within the elastic regime is
to be experienced. However, it is not known how the two methods of analysis will compare
in the plastic regime; elasto-plasticity is rate-independent, whereas visco-plasticity is rate-
dependant. Despite this difference, similar qualitative behaviour of the two laws is expected.
1.4.2 Geometric results
During post-buckling analysis of cylindrical shells, the formation of inward ellipsoidal de-
flections in the metal known as buckling “dimples” is expected. However, the locations and
number of such dimples is not known. It is expected that the greatest deformation is to be
expected at such dimples, and dimple behaviour is to govern overall behaviour of the struc-
tures in terms of structural integrity. Additionally, it is expected that buckling is to occur in
the locations of these so-called dimples.
1.4.3 Load carrying capacity
Following initial buckling and movement into the plastic regime, a decrease in load resisted is
expected. However, the amount by which the load is expected to decrease is not known and
will most likely be influenced by initial material parameters assigned to the materials tested,
and the behaviour of buckling dimples. It is expected that in the post-buckling and plastic
regimes, a portion of the load carrying capacity will be retained.
It is expected that the formation of the previously mentioned dimples is related to the loss
of load carrying capacity, however, it is not known whether these dimples form as a result
of decreased load carrying capacity or vice-versa. The presence of a post-buckling plateau,
during which large deformations are experienced with small increases in stress; similar to
the onset of plastic deformation, is expected. It is expected that this plateau will occur once
structures have reached a stable mode of deformation.
4
1.5 Layout of this document
The following breakdown contains the primary information which will be addressed within
each chapter:
Chapter one provides an introduction to the topic of this document by providing the reader
with relevant background information. The motivation, aims and objectives of this research
project are subsequently addressed. Furthermore, this chapter contains the scope and limita-
tions of this document.
Chapter two contains a review of the available literature concerning buckling and post-
buckling studies related to shell structures. The chapter begins with an introduction into
shell structures gleaned from the review of various studies and texts. Subsequently, methods
of analysis and results of previously conducted studies are presented. Within this chapter,
conclusions are made concerning the need for further study in the field.
Chapter three is concerned with the continuum mechanics theory used to implement the mate-
rial law. General considerations when dealing with elasto-plasticity are illustrated, following
which the kinematics of an elastic-plastic body is described. Subsequently, the continuum
mechanics of the constitutive law is presented.
Chapter four deals with computational aspects of implementation, such as types of analy-
sis and modelling methods. Additionally, the computational methods followed in order to
implement the constitutive law are described.
Chapter five describes all preliminary models created and analysed, and provides the problem
definitions for each model respectively in terms of geometry, loading conditions and type
of analysis conducted. Initial configurations of all models are presented, along with material
parameters used. Additionally, this chapter illustrates the results of analysis of all preliminary
models and comments on relevant aspects of these results.
Chapter six makes conclusions and recommendations with regards to the investigation con-
ducted and whether the implemented law is feasible for further use in the post-buckling
analysis of shell structures. Additionally, conclusions are made with respect to analysis of
models created.
Appendix A contains the source code of the implemented material law as compiled.
5
Chapter 2
Literature Review
2.1 Introduction
This chapter contains a review of available literature concerned with shell structures and
previously conducted investigations into the buckling and post-buckling behaviour of such
structures. Initially, an introduction to shell structures is presented with the intention to fa-
miliarise the reader with such structures. Subsequently, previously conducted investigations
are reviewed with attention being given to methods of investigation, geometry and load bear-
ing capacity in the post-buckling regime with the aim of illustrating that further research in
the field is required.
2.2 Shell structures
As explained by Lowe (1970), shells may be defined as a body bound by two curved surfaces,
where the distance between the two surfaces is small in comparison with the other dimensions.
The line which is equidistant from both curved surfaces is defined as the middle surface of a
shell, as illustrated overleaf in Figure 2.1.
Shell structures are commonly used in engineering applications due to the many advantages
they possess over alternative types of structures. Some advantages, as listed by Venstel and
Krauthammer (2001), are efficiency, high strength to weight ratio and a very high stiffness.
Contemporary examples of shell structures are liquid-retaining structures, arch dams, aircraft
and nano structures.
Shells are particularly similar to plates; however, they differ in that shells have a curved
surface (Venstel and Krauthammer, 2001). Various classifications of shells exist, based upon
surface curvature, and shells are generally classified in such a manner as the behaviour of a
shell is governed primarily by curvature. Additionally, according to Lowe (1970), the middle
surface, thickness and edges define the geometry of a shell.
As shells gain their unique properties largely from geometric configuration (Venstel and
Krauthammer, 2001), multiple theories have been developed in order to predict the be-
haviour of shell structures of varying geometry. Shells may be divided into varying forms
of geometry according to surface curvature; such as cylindrical, conical, spherical, ellipsoidal
or paraboloidal. Furthermore, shells may be classified as either thick or thin.
6
Figure 2.1. Simple geometry of a shell (Venstel and Krauthammer, 2001)
According to Lowe (1970), in engineering applications, a shell is regarded as thin if
max
h
R
<
1
20
(2.1)
where h is the thickness of the shell, and R is the radius of curvature of the middle surface.
Otherwise the shell is defined as thick.
As a result of the wide use of shells, many experimental and theoretical studies have been
conducted in order to gain insight into the behaviour of such structures, taking into account
varying geometry, initial material imperfections and loading conditions. Mandal and Callan-
dine (2000) state that when shells are loaded in compression, buckling constitutes the most
common form of failure. Therefore, many of the studies mentioned have been concerned with
investigating buckling of shells.
Zhu, Mandal and Callandine (2002) mention that often failure of shell structures is catas-
trophic. As shell structures are commonly used in industries where risks of failure are high,
many studies concerned with buckling of shells have focused on predicting the load at which
certain shells fail due to buckling. Consequently, there has been limited investigation into the
post-buckling behaviour of shell structures.
2.3 Methods of investigation
During investigation into the post-buckling behaviour of shell structures, various aspects
which influence this behaviour are of interest. Such aspects are material used, loading and
boundary conditions, methods of analysis, geometry of the structure and presence of initial
imperfections.
The post-buckling behaviour of steel cylindrical shells was investigated by Aghajari, Abedi
and Showkati (2006), and the behaviour analysed with respect to pressure versus displacement
and the number of inward indentations, referred to buckling waves or dimples as seen in Figure
2.2 overleaf. Analysis of the post-buckling regime is generally investigated with reference to
7
these criteria, and has been made use of by various other authors. While investigating metallic
cylindrical shells, Mandal and Callandine (2000) also used the presence of such dimples in an
attempt to explain the loss of strength of cylindrical shells following initial buckling. Fakhim,
Showkati and Abedi (2009) did not record the presence of such dimples, but made use of
pressure versus displacement as an analysis tool. The presence of buckling dimples has not
been limited to the analysis of metallic shells, however.
Figure 2.2. A buckled cylindrical shell (Fakhim, Showkati and Abedi, 2009)
Zhu, Mandal and Callandine (2002) investigated the post-buckling behaviour of cylindrical
silicone rubber shells under axial compression, in an attempt to explain observations not pre-
dicted by classical shell theory. Similar to this, Mandal and Callandine (2000) also investigated
the behaviour of silicone rubber shells. Both authors observed the post-buckling behaviour
making notes of the deformed geometry with reference to buckling waves. While the stud-
ies conducted on silicone-rubber shells were experimental and lab-based, computational and
theoretical analyses may also be conducted, employing various forms of numerical analysis.
In order to conduct analyses on post-buckling behaviour, many authors have conducted non-
linear computational analyses, as buckling of shells is a non-linear problem due to large
changes in geometry and loading conditions. Aghajari, Abedi and Showkati (2006) conducted
a non-linear analysis of cylindrical shells, and made use of the Arc-Length-Type method, as
according to Memon and Su (2004), ordinary solution techniques such as the Newton-Raphson
technique are not able to limit instability near limit points and fail to accurately predict the
load versus displacement response. According to Chatzi (2014), the Newton-Raphson tech-
nique performs poorly in buckling analysis as the slope at limit points is equal to zero. As
8
described by Memon and Su (2004), the Arc-Length-Type method is conducted by modifying
the load factor at each iteration step. While conducting a non-linear analysis of shells, Zhu,
Mandal and Callandine (2002) made use of the commercial analysis software ABAQUS, solv-
ing non-linear equations with the Arc-Length-Type method. In addition to various methods
of analysis, different types of mechanical behaviour may be analysed.
The previously mentioned investigations were concerned with elastic buckling. However,
Cortivo, et al (2009) investigated the plastic buckling and collapse of thin shell structures.
In order to carry out the non-linear investigation, the Assumed Natural Deviatoric Strain
(ANDES) finite element was utilised. While this appears to have been successful, during this
current investigation, it will not be required that a shell element be used to model shell struc-
tures, as the analysis will be conducted making use of three dimensional continuum mechanics,
thus reducing simplification of shell behaviour. In order to better predict the plastic behaviour
of shells the ANDES shell element was modified to be elasto-plastic with the introduction of
the von Mises yield criterion, while taking into consideration linear and isotropic hardening
in a rate independent analysis. The introduction of the yield criterion allowed for the identi-
fication of the point at which plastic deformation occurs. Furthermore, the Arc-Length-Type
method was utilised. Additionally, this study made use of the stress resultant approach to
predict non-linear behaviour of the structures. The use of the stress resultant approach incor-
porates the change in stress in the material due to plastic deformation. Skallerud and Haugen
(1999) conducted a similar study in which the same method of analysis was used; however, the
effectiveness of such an analysis was investigated. It was concluded that the use of a modified
ANDES element yielded results with low error.
While some of the previously mentioned investigations conducted laboratory experiments, all
made use of the Finite Element Method in order to test theoretical ideas against observed
data. Two interesting studies were conducted by Mandal and Callandine (2000) and Zhu,
Mandal and Callandine (2002) in which silicone-rubber was used to investigate elastic buck-
ling of cylindrical and domed shells, as previously discussed. The use of silicone-rubber as a
material was interesting as it allowed one to easily observe typical post-buckling behaviour.
Cylindrical shells were tested under self weight, and domed shells under a point load. Mandal
and Callandine (2000) concluded that buckling heights under self weight were proportional to
the thickness of the shell, and noted the presence of dimples during buckling. Under a point
load, it was observed that domed shells develop various buckling patterns with varying thick-
ness. It was noted that as thickness is varied, the number of vertices formed during buckling
varies.
While conducting similar experiments, Zhu, Mandal and Callandine (2002) concluded that the
method conducted illustrated well-characterised post-buckling behaviour of shell structures.
While the materials and methods outlined will not be utilised in this research project, the
outcomes are of interest as it is suspected that a plastic analysis of metallic shells will yield
similar qualitative results characteristic of the post-buckling regime of shell structures. During
these studies, initial material imperfection was taken into account.
Both studies incorporated such imperfections during finite element analysis. It was not investi-
gated whether these initial imperfections are influential in the post-buckling regime. However,
it was concluded by studies such as Fakhim, Showkati and Abedi (2009), and various other
9
authors that initial geometric imperfections in shell structures are instrumental in decreasing
the initial buckling load.
During their investigations, Mandal and Callandine (2000) and Zhu, Mandal and Callandine
(2002) did take initial imperfections into account, but concluded that there was no correlation
between initial geometric imperfections and buckling load. Mandal and Callandine (2000)
suggested that during their study, the presence of initial stress was minimised reducing scatter
of results and increasing initial buckling load. Buckling loads observed to be lower than
that of predicted theoretical loads were explained in terms of static determinacy. Mandal
and Callandine (2000) suggested that because their experimental structures were statically
determinate, residual stresses were lower; the cause of scatter in other studies. It is the view of
the author that the material choice may have impacted these results, as there may have been
lower residual stresses in the silicone-rubber as opposed to metallic structures investigated,
increasing initial buckling loads. It was noted by Fung and Sechler (1974) that initial geometric
imperfections are thought to be the underlying cause of scatter in experimental results.
In order to account for initial imperfections, specimen thickness at strategic points was mea-
sured and included in models to be computationally analysed. Alternatively, initial imperfec-
tions were included making use of trigonometric functions with dampening effects applied.
In the latter case, imperfections introduced by Mandal and Callandine (2000) took a circular
form and were deflected inwards, in order to ascertain whether the initial imperfection affected
buckling behaviour. It was noted that a dimple did form near the initial imperfection, but
subsequently migrated upwards after forming. However, other theories have been put forward
concerning the differences between experimental and theoretical results.
Shamass, Alfano and Guarracino (2014) conducted a non-linear plastic investigation into the
post-buckling behaviour of cylindrical shells loaded under axial compression, making use of
the deformation and flow theories of plasticity. In this study, it was concluded that the dis-
crepancy between theoretical and experimental results may be addressed through careful and
appropriate use of plastic theory. They did, however, take initial imperfections into account.
Additionally, it was hypothesised that plastic buckling of shell structures generally occurs
in thicker shells. The study tested cylinders made of an aluminium alloy. Various boundary
conditions were tested, such as fully fixed, clamped and hinged. Cylindrical specimens were
modelled using ABAQUS and a general purpose 4-noded shell element was used, based upon
thick shell theory. Additionally, a structured mesh was used. In order to trace the non-linear
response, the modified Arc-Length-Type approach was used. While this study is similar to
the proposed study of this document, the use of shell elements differs to the proposed use of
a three dimensional continuum approach.
2.4 Geometry of deformed structures
During the analysis of shell structures, it has been noted that such structures primarily fail due
to buckling (Mandal and Callandine, 2000). Therefore, loading which places these structures
in compression is often investigated. While investigating buckling of shell structures, axial and
pressure loading are common. During various studies making use of such loading, it appears
that there are common post-buckling deformation patterns which occur.
10
Indentations which form on shell structures in the post-buckling regime, often referred to as
dimples or buckles, have been used in some studies to analyse the nature of buckling. While
dimples of various quantities and size were noted in many studies mentioned in the previous
section, all studies did note the presence of such dimples. An example of buckling dimples
is illustrated in Figure 2.3 as red and yellow indentations on the wall of the cylindrical shell
under axial compression.
Figure 2.3. Typical computational buckling dimples (Skatulla, Sansour and Hijiaj, 2014)
While conducting experimental tests on cylindrical shells, thickness was varied by Fakhim,
Showkati and Abedi (2009). Each shell tested had a constant thickness throughout the length.
However, it was concluded that the height of cylindrical shells influenced the location and size
of the dimples, and thickness had no effect. In comparison, Aghajari, Abedi and Showkati
(2006) investigated cylindrical shells which varied in thickness throughout the length of the
cylinder. It was concluded that the variation in thickness had a direct influence on final
buckling waves, and subsequently recommended that cylindrical shells with less thickness
variation be used in construction making use of shells.
Due to the presence of dimples, shells often lose the property of symmetry in the post-
buckling regime. When loaded axially, eccentricities may form causing the structure to bend
perpendicular to the loading axis.
11
2.5 Post-buckling strength
As discussed by Mandal and Callandine (2000), experimental studies conducted in the 1930’s
with connection to the aircraft industry revealed that buckling loads were lower than that of
loads predicted by classical theory. Furthermore, there was a wide scatter in results and failure
of shell structures was catastrophic. This, therefore, led to the buckling load of shell structures
being studied in great detail. However, a significant amount of studies were concerned with
predicting buckling loads of shell structures, and not with post-buckling strength.
Studies which have investigated the post-buckling strength of shell structures, such as Mandal
and Callandine (2000) Zhu, Mandal and Callandine (2002) and Fakhim, Showkati and Abedi
(2009) have noted an interesting phenomenon. Whether loaded axially or by pressure, it has
been observed that at a certain point during loading, shell structures exhibit large deflections
with small corresponding changes in load - referred to in these studies as the post-buckling
plateau.
Mandal and Callandine (2000) noted that during the formation of dimples, in the region of
the dimple formed, the structure is no longer able to support axial loading due to an inward
tensile force experienced at the centre of the dimple. The formation of such dimples may
therefore be an explanation for the post-buckling plateau, along with plastic deformation.
While testing cylindrical shells, Fakhim, Showkati and Abedi (2009) noted that all specimens
exhibited load resistance in the post-buckling regime. Therefore, during this current study, it
is expected that a post-buckling plateau will be encountered, and that the structures will still
exhibit some structural integrity following buckling.
2.6 Concluding remarks
Some of the methods of analysis available for shell structures have been illustrated in this
chapter. Through review of available literature, it has been concluded that some studies have
been conducted investigating the post-buckling behaviour of shell structures while most stud-
ies have been concerned with events leading up to buckling. It appears as though the causes
of the observed post-buckling behaviour are not entirely known; but may be accurately pre-
dicted taking imperfections into account and a careful use of constitutive laws. Additionally,
all computational studies have been conducted with the use of a shell element in one form
or another, thus simplifying analysis of the structures. It appears that a three dimensional
continuum post-buckling plastic buckling analysis has not been conducted to date.
12
Chapter 3
Continuum mechanics theory
3.1 Introduction
During engineering analysis, it is common to analyse bodies which are far larger than the
matter from which the body is composed. Therefore, it is valid to assume that the body is
continuous, meaning that elements contained within the body share equal properties with the
entire body. This implies that if the body is analysed as a whole, or an element within the
body is analysed under identical loading and boundary conditions, both analyses will yield
identical results; based on the assumption that the underlying molecular structure of the
material analysed is generalised by the overlying material properties. This assumption that a
body is made up of elements of matter which behave identically to the entire body gives way
to the definition of a continuum. Following this, continuum mechanics is concerned with the
analysis of such bodies, making use of discretisation techniques and the tools of algebra and
calculus of vectors and tensors.
This chapter begins with a description of general theory to be taken into account when mod-
elling such bodies. Subsequently, relevant non-linear equations of continuum mechanics are
presented, and their relation to stress and strain illustrated to finally present the methodol-
ogy with respect to continuum mechanics followed in order to implement the elasto-plastic
constitutive law.
3.2 General considerations
3.2.1 Elastic and plastic behaviour
Consider a ductile, metallic material loaded under simple tension by a force F as illustrated
in Figure 3.1 with a length l and area A. Stress and strain within the material may be defined
respectively as
σtrue =
F
A
and (3.1)
true =
l
l0
dl
l
(3.2)
for small changes in l. Initially, as σ is increased, small changes in l occur resulting in a small
value of . During this initial stage, it is assumed that if unloading were to occur, the material
13
will deform back to the original length l0. Such a situation describes elastic behaviour and
gives way to the principle of elasticity, stating that deformations are reversible and occur at
small strains. When in the elastic regime of deformation, the relation between σ and is of a
linear manner and related by the scalar Young’s Modulus E. However, if further loading were
to occur resulting in larger strains, the relation between σ and becomes non-linear and the
elastic relation assumed is no longer valid.
Figure 3.1. Stress versus strain of a ductile material (Knutsen, 2012)
Assume the material under consideration is further deformed. As l increases, deformation
is resisted by internal forces within the crystal lattice constituting the underlying material
structure of the metal. However, a point is reached where some metallic bonds within the
lattice are no longer able to resist further loading. At this point, these bonds are broken
causing movement within the lattice. If unloading were to occur at this point, the structure
of the crystal lattice would not return to the original configuration. Such movement within
the lattice is termed dislocation, occurring along a so-called slip plane within the lattice.
Following dislocation, the material is said to have yielded and the relation between stress and
strain has changed, and further increase in l is termed plastic deformation. (Hibbeler, 2011)
Disque (1971) explains that there is a small increase in strength during plastic deformation
due to strain hardening, as illustrated in Figure 3.1. Following yielding, a process of partly
irreversible deformation takes place as a result of further formation and movement of dislo-
cations. Due to dislocations being formed, and slip of the crystal structure along dislocation
planes caused by shear stress placed on the crystal lattice, deformation of this nature is partly
irreversible.
When modelling plastic deformation due to large strains, it is no longer permissible to make
use of the theory of elasticity due to non-linear behaviour, nor is it acceptable to make use of
14
theory derived for the use of small strains as detailed for computation of σ and . Therefore,
the use of finite strain theory is required.
3.2.2 Balance law of continuum mechanics
Preceding the descriptions of methods used to model finite strain behaviour, a general descrip-
tion of the fundamental laws underlying such methods is required. When subjected to any
form of loading (such as force or temperature), the behaviour of the body may be described
by the following laws fundamental to continuum mechanics (Mase, 1999):
ˆ The law of Conservation of Mass ensures that there is no mass loss or gain when the
body deforms;
ˆ The law of Linear and Angular Momentum Conservation which is related to New-
ton’s Second Law which defines the rate of change of linear and angular momentum over
time, resulting in force and momentum; and
ˆ The law of Energy Conservation which is related to the Principle law of Energy which
states that energy can neither be created nor destroyed but can only change forms.
3.3 Kinematics
Consider a body B, which is to be analysed during deformation. During analysis, two states
or configurations are commonly referred to: the reference (undeformed) configuration and the
current (deformed) configuration. The state of the material in these two configurations is
illustrated overleaf in Figure 3.2.
In order to characterise the deformation of B, a second order tensor referred to as the de-
formation gradient F is utilised. F characterises the local deformation of point x contained
within B, and thus the change in position of a particle with respect to time and may be
expressed as
F = F i
j ei ⊗ Ij
=
∂xi
∂Xj
⊗ Ij
(3.3)
where dX and dx represent material line elements in the reference and current configurations
respectively, and ⊗ represents the dyadic product of terms. Defining the change in position
of a particle is important, as it has been assumed that the body being analysed is constituted
of many infinitesimally small particles. Therefore, by characterising the deformation of all
discrete particles within B, it is possible to characterise the deformation of B itself (Mase,
1999).
As discussed in Section 3.2, deformation may be of an elastic or plastic nature. During defor-
mation of B, before the yield stress of the material is reached, deformation is purely elastic,
and is described by F e. However, following yielding of the material, deformation consists of
elastic and plastic components, with the plastic component being denoted F p (Haupt, 2000).
15
Figure 3.2. Deformation of a solid body (Sanpaz, 2011)
In such a situation, the final deformation gradient is expressed with the use of multiplicative
decomposition as by Sansour and Kollmann (1997) as
F = F eF p. (3.4)
Next, the change in length of a line during deformation is to be accounted for. Two Cauchy-
Green deformation tensors are available, namely the left and right, and are defined so by
their relation to the left and right stretch tensors. The deformation tensors may be thought
of as the squared length of a deformed fibre dx in a solid body (Bower, 2009). The total
right Cauchy-Green deformation tensor C, and its inverse C−1
are expressed respectively as
(Capaldi, 2012)
C = F T
F and (3.5)
C−1
= F −1
F −T
. (3.6)
Similar to F , C is decomposed into its elastic and plastic components with
Ce = F T
e F e and (3.7)
Cp = F T
p F p. (3.8)
as detailed by Sansour and Kollmann (1997).
16
3.4 Stress measures
In Section 3.2, stress was defined with the use of Eq. (3.1). However, as mentioned, an alter-
nate means of describing the stress within the material is required during this investigation.
The stress tensor to be used is the second Piola-Kirchoff stress tensor, S. Capaldi (2012)
explains S as a measure of the transformed force per unit area of an element in the reference
configuration. S is expressed as
S = C−1
Ξ (3.9)
and is symmetric (Chandrasekharaiah & Debnath, 1994). Ξ describes the elastic material
stress as is addressed later in Section 3.5.2.
3.5 The constitutive model
With the requirement to make use of an alternate method to linear elasticity, and the kine-
matics of B defined, the following sections detail the theory and implementation of the chosen
constitutive model, following the methods described by Sansour and Kollmann (1997).
3.5.1 Background and motivation
Capaldi (2012) explains that the model developed and used for linear elasticity reasonably
describes the behaviour of metals at small values of strain. However, this model is no longer
able to accurately predict the response of the material to prescribed boundary and loading
conditions at larger values of strain due to non-linear behaviour. Ramos (1992) states that
constitutive laws may be defined as a set of equations applied to a specific material which
mathematically represent experimental and empirical observations concerning responses of
particular materials to applied loads and boundary conditions, and may be applied during
analysis of finite strain.
Elasto-plasticity is a theory which attempts to predict the effect of plastic deformation on a
material. During plastic behaviour, the metal is subject to a stress which causes a deformation
which permanently influences the geometry of the metal, as a portion of the deformation is not
recovered elastically. Therefore, it is useful to discern between the elastic (recoverable) and
plastic (permanent) components of strain with the use of elasto-plastic strain decomposition.
History of previous deformation is accounted for in order to account for unloading behaviour
and strain hardening. Elasto-plastic constitutive laws are rate independent; the rate at which
the material is loaded or unloaded is inconsequential. (Bower, 2009)
Classical elasto-plasticity is based upon the assumption that when a material is loaded such
that the stress experienced is significantly lower than the yield stress of the material, elastic
behaviour will be experienced. However, when loaded such that stress within the material
reaches a critical value (generally the yield stress denoted σy), plastic flow occurs leaving the
material permanently deformed. (Haupt, 2000)
Haupt (2000) explains that elasto-plastic models generally describe material behaviour realis-
tically, and apply particularly to metals; characterised by a crystalline structure. As described
17
in Section 3.2.1, plastic flow may be described by the formation and destruction of dislocations
within the crystal lattice, for which complicated dynamic processes are responsible. However,
such models do not deal with these processes, and rather make use of a phenomenological
representation of the material under study. For this reason, only the relevant macroscopic
effects are described with the use of an elasto-plastic model, even though the effects are
fundamentally caused by dislocation behaviour.
In the phenomenological representation, a critical quantitative load is defined - above which
dislocations are predicted to occur and below which the effect of dislocations is ignored. To
this end, elasto-plastic models are not able to model plastic deformation such as creep whereby
low loads over sustained time periods are able to cause plastic deformation. The critical load
used to discern the onset of dislocation, and hence plastic deformation, may be represented
by means of the yield surface in the space of the relevant strain tensors of the constitutive
law. (Haupt, 2000)
Macroscopic effects of dislocations depend on the previous history of loading and unloading,
and are dealt with by a variable Y which accounts for an increase in the yield stress following
plastic deformation. Additionally, kinematic or isotropic hardening is taken into account and
may be interpreted as the process-dependant obstacle impeding the motion of further dislo-
cation. Following initial plastic deformation, further dislocation movement and formation is
hindered due to an increased energy required for further dislocation formation and movement
within the crystal lattice structure of the metal. (Haupt, 2000)
Another fundamental aspect of the phenomenological representation is the concept of a yield
function, which is a scalar-valued function depending on stresses, strains and internal vari-
ables. The yield function determines the yield surface and consequently the nature of defor-
mation of the material. The yield function defines whether deformation is of an elastic or
plastic nature. Furthermore, an associated plastic flow rule is determined with the use of the
yield function. (Haupt, 2000)
With the use of the above assumptions and the yield surface, any state of stress which lies
outside the yield surface is excluded, with the state of stress moving exclusively within the
yield surface in the elastic state. However, during plastic deformation, it is required that the
stress tensor used stay on the yield surface. In order to achieve this, a plastic multiplier is
introduced. Therefore, hardening can be easily interpreted as the change in location, size or
shape of the yield surface. (Haupt, 2000)
An elasto-plastic constitutive model generally contains the following components listed below
(de Souza Neto, Peric and Owen, 2008):
ˆ elasto-plastic strain decomposition;
ˆ an elastic law;
ˆ a yield criterion, with the use of a yield function;
ˆ a plastic flow rule, defining the evolution of plastic strain; and
ˆ a hardening law, characterising the change of the yield limit.
18
3.5.2 Elastic constitutive model
Due to the linear nature of elastic deformation and the non-linear nature of plastic deforma-
tion, different methods were required to compute stress and strain of the material depending
on the nature of deformation occurring at a particular point in time. The multiplicative
decomposition of the deformation gradient tensor and the right Cauchy-Green deformation
tensor allowed two models to be developed; each to capture the elastic and plastic effects of
deformation respectively. The methodology presented in this section follows that of Sansour
and Kollmann (1997), and was not developed by the author of this paper.
Firstly, consider the elastic material stress tensor Ξ used in this study expressed as
Ξ = 2ρref F T
p Ce
∂ψ (Ce, Z)
∂Ce
F −T
p . (3.10)
In the above expression, for a unified inelastic constitutive model, a phenomenological internal
variable Z is taken into account (corresponding to the strength increase due to plastic hard-
ening) and the existence of a stored free energy function ψ is assumed such that ψ (Ce, Z).
ρref is the density of the material before deformation has taken place i.e. in the reference
configuration.
When concerned with the elastic constitutive model, ψ is decomposed into two parts via
additive decomposition; the first depending on Ce and the second on Z:
ψ = ψe (Ce) + ψZ (Z) (3.11)
Following the definition of Ce in Eq. (3.7), the logarithmic measure of strain α may be
introduced:
α = ln Ce (3.12)
Making use of the assumption that an elastically isotropic material is being dealt with, the
following holds true:
Ce
∂ψe (Ce, Z)
∂Ce
=
∂ψe (α)
∂α
(3.13)
Thus, making use of Eq. (3.11) and Eq. (3.13), Eq. (3.10) is expressed alternatively as
Ξ = 2ρref F T
p
∂ψe (α)
∂α
F −1
p . (3.14)
At this point, it is required that α be modified in order to simplify the computation of the
material stress Ξ. Therefore, the modified logarithmic measure of strain α is introduced as
α = F T
p αF −1
p . (3.15)
19
During computation, however, it is preferable to make us of an alternative definition of α in
order to increase computational efficiency. Therefore, α is alternatively expressed as
α = ln C−1
p C or (3.16)
αT
= ln CC−1
p . (3.17)
In application, it is useful to make use of the logarithmic strain measure, as further strain
is taken into account via an additive process, as opposed to a multiplicative process. The
additive process is illustrated as follows:
αt+1
= αt
+ ∆α (3.18)
Following the introduction of α in Eq. (3.15), Eq. (3.14) is further simplified and expressed
as
Ξ = 2ρref
∂ψe (α)
∂α
. (3.19)
A linear constitutive model is chosen such that Ξ is finally expressed as
Ξ = KtrαT
1 + µ devαT
(3.20)
where K and µ are elastic material parameters denoting the bulk modulus and shear modulus
respectively.
3.5.3 Plastic constitutive model
The inelastic constitutive model was used in order to capture the effects of plastic deformation.
As this investigation was concerned with the post-buckling behaviour of shell structures, it was
expected that plastic deformation of the metal tested would occur at larger, finite strains.
Therefore, it was required that a method be employed allowing plastic deformation to be
accounted for, making it possible to compute stress and strain due to plastic deformation.
The methodology presented in this section follows that of Sansour and Kollmann (1997), and
was not developed by the author of this paper.
As a distinction was made between the elastic and plastic deformation measures, it was
required that the type of deformation at each time step of the calculation be discerned. To
this end, a second internal variable is required; namely Lp, referred to as the plastic rate.
In order to derive the evolution equations for the internal variables Lp and Z an elastic region
is considered which is confined by the yield function f and is assumed to depend on the
thermodynamical quantities Ξ and Y such that the following relation holds:
E := {(Ξ, Y ) : f (Ξ, Y ) 0} (3.21)
20
The often used von Mises yield function is defined by Sansour et al. (2006) as
f (λ, Lp) = ||devΞ|| −
2
3
(σy − Y ) (3.22)
where ||devΞ|| denotes the Frobenius norm defined as tr (devΞ)2
, and Y is expressed as
Y (λ) = −H Zn +
2
3
∆tλ − (σ∞ − σy) 1 − exp −η Zn +
2
3
∆tλ . (3.23)
In Eq. (3.23) above, H is a plastic material parameter controlling linear plastic hardening
which is a special case of a work hardening material as the yield surface expands uniformly,
and may be thought of as the slope of the stress-strain curve in the plastic regime, analogous
to E in the elastic regime. η is a second plastic material parameter which serves a similar
purpose to H, with the addition of a non-linear hardening component. For a work hardening
material the plastic multiplier λ is inversely related to the hardening parameter η.
It should be noted in Equation (3.22) that Y is subtracted from the initial yield stress σy.
This is due to Y (λ) being computed as a negative value. Initially, the term was expressed as
(σy + Y ) in the C++ code implemented by the author as detailed an Appendix A and was
incorrect causing computational issues during analysis. The expression as originally imple-
mented is incorrect due to the implication that yield stress decreases during strain hardening.
Therefore, Eq. (3.22), and subsequently Eqs. (4.11) and (4.12) were additionally corrected ac-
cordingly in order to allow for an increase in yield stress during plastic deformation attributed
to the processes within the crystal lattice of the metal impeding further dislocation.
The yield function f may be viewed as a surface in the three-dimensional stress space which
encompasses all possible stress states for which elastic deformation occurs. f is used to de-
termine whether deformation is elastic or plastic in the following manner: for f < 0 elastic
deformation occurs, for f = 0 and λ = 0 neutral loading occurs. Plastic deformation occurs
in the case of f = 0 and λ > 0. For f > 0, the function is outside the yield surface; such
a situation is not possible and the calculation step is re-estimated and the yield function
recalculated.
With the introduction of λ and f, the associated flow rule Lp and Z are expressed as
Lp = λ
∂f
∂Ξ
= λ
devΞT
||devΞ||
= λνT
and (3.24)
Z = λ
∂f
∂Y
=
2
3
λ (3.25)
where ν is termed the yield-locus normal, and may be thought of as the normal vector to the
yield surface.
21
3.6 Concluding remarks
In this section, the kinematics required in order to describe the deformation of a body were
presented, along with general considerations to be taken into account, such as elastic and plas-
tic material behaviour. Furthermore, the continuum mechanics of the elasto-plastic material
law was illustrated. However, it is apparent that making use of the theory presented in this
chapter to analyse shell structures analytically would not be practically possible. Therefore,
it was required that computational methods be employed for efficient and accurate analysis.
22
Chapter 4
Computational analysis principles
4.1 Introduction
In Chapter 3, the continuum mechanics principles required in order to implement the elasto-
plastic material law were addressed. It was concluded that the use of computational techniques
were required in order to apply the law to shell structures. Therefore, this chapter begins with
a description of relevant considerations to be made when conducting computational analysis of
such structures. Subsequently, computational modelling techniques which were employed are
discussed, with methods of analysis being illustrated. Furthermore, detailed computational
methodology with respect to the constitutive law is presented, which follows the work of
Sansour and Kollmann (1997). Additionally, techniques used to set-up and solve for the global
equation system are addressed.
4.2 General analysis considerations
Various types of analysis are available, depending on the problem at hand. Therefore, distinc-
tion is made between buckling and post-buckling analysis and linear and non-linear analysis,
in order to motivate the choices of each technique for use is this study. Additionally, general
theory of dynamic behaviour and analysis is addressed.
4.2.1 Buckling analysis
Members which are considered long or slender often exhibit a form of failure referred to
as buckling when loaded axially under compression. Global buckling failure is observed when
slender members subject to axial compression deflect laterally or side-sway, changing from one
configuration to another over a short period of time, referred to as bifurcation. This concept
may be further applied to members such as cylinders, where loading causes a sudden increase
in strain within a local area of the cylinder, while stress remains constant or decreases. While
the cylinder has not deflected laterally or side-swayed, indentations within the cylinder walls
are said to have buckled locally. In both conditions, the load at which this initial sudden
deformation occurs is termed the critical buckling load, and is the maximum load which may
be supported when the structure is on the verge of buckling. During a critical buckling load
analysis, experiments are conducted and theory developed in order to predict the load at
which a structure will begin to buckle. During such an analysis, the primary concerns are
factors which influence initial buckling load. (Hibbeler, 2011)
23
In contrast to initial buckling, a post-buckling analysis may be conducted. During such an
analysis, aspects following initial buckling of a structure are investigated. Typical aspects
which are investigated are structural integrity with increased loading and structural strength
in the post-buckling regime. In the analysis of shell structures, other post-buckling aspects
of interest are those as investigated by Mandal and Callandine (2000) Zhu, Mandal and
Callandine (2002) and Fakhim, Showkati and Abedi (2009) such as the formation of the post-
buckling plateau and dimples. In order to observe such post-buckling behaviour, structures
are further loaded after the onset of initial buckling. During post-buckling deformation, large
deformations are often experienced, and specific analysis is required in order to account for
corresponding non-linear behaviour.
4.2.2 Linear and non-linear analysis
As explained by Bower (2009), linear analysis of models is only suited for analyses in which
specimens exhibit purely reversible deformation with small displacements; if a specimen de-
formed during some loading condition, upon removal of the load, returns back to its original
shape and state of stress. Furthermore, linear analysis is only appropriate to specimens with
which strain does not depend on the rate of loading applied, and history of loading and de-
formation. Additionally, in the linear regime of material behaviour, stress and strain of the
material are related in a linear manner. In the case of metallic materials, this relationship
only holds true if the strain is small.
As is the case with buckling analyses as described in Section 4.2.1, a non-linear analysis is
required (Cortivo et al, 2009). This is due to a non-linear stress-strain relationship, and large
deformations of the material being investigated. As explained in Chapter 3, at strains within
the strain hardening and necking regions or deformation of a material, the relationship between
stress and strain is no longer of a linear relationship. As there is an increase in strain with
a lower increase in stress, the stiffness of the material has decreased, decreasing the Young’s
modulus of elasticity E. It is at this point that stress is now considered a function of strain
and no longer linearly related to E. When there is a decrease in stiffness of a material altering
the relationship between stress and strain, behaviour becomes non-linear and a non-linear
analysis of the material is required. (Solidworks, 2008)
Various types of non-linearity during analysis may occur. The first type of non-linearity is as
previously discussed, where the material being deformed exhibits varying properties (stiffness)
under altered loading. A second type of non-linearity which occurs at large deformations is
geometrically non-linear behaviour. Global non-linear geometry may be observed in the case
of a slender cylindrical shell under axial compression. As deformation and load increase, the
shell begins to deform and exhibits bending about the horizontal axis. Initially, it was assumed
that the axial load was only a cause of compressive forces. However, due to imperfections in
the material and eccentricity of loading, the shell begins to deform under flexure causing
bending of the element. As bending occurs, the loading is no longer purely axial, thus altering
the loading conditions. At extreme deformations, the load may no longer be axial at all.
24
4.2.3 Dynamic analysis
In order to conduct a post-buckling analysis, it is often required that dynamics of the structure
be accounted for. This is the case as buckling is often sudden and results in an increased
velocity of portions undergoing buckling behaviour. In order to account for dynamic effects,
material density ρ is required to be specified, as inertia and thus acceleration are affected.
Additionally, a damping parameter c is required to specify the rate at which the material
returns to a point of stable equilibrium. Furthermore, the choice of time increment per loading
step is required in spite of the use of a rate-independent material law, as large loading rates
may nullify the effects of dynamics being accounted for, and small loading rates may cause
computational difficulties and unrealistic material behaviour.
4.3 Modelling methods
As stated by Rock, Zhang and Wilkinson (2008), numerical modelling is a tool that is often
employed and extremely useful for engineering design and analysis. Numerical modelling may
be broadly defined as solving physical problems making use of appropriate simplification of
reality. In order to simplify problems in engineering analysis, it is required that key variables
be identified or introduced and relations between such variables established. Once variables
and relations have been established, it is common to construct a set of differential equations
to be solved allowing engineers to predict the outcomes of complex models and scenarios
without conducting physical testing.
4.3.1 Finite Element Method
The continuous development of new structural materials leads to ever increasingly complex
structural designs that require careful analysis. Although analytical techniques are important,
the use of numerical methods to solve mathematical models of complex structures has become
an essential tool in the design process. The Finite Element Method (FEM) has been the
fundamental numerical procedure for the analysis of shells. (Bucalem and Bathe, 1997)
During this study, shell structures are subjected to large deformations in the post-buckling
and plastic regimes. According to Bower (2009), FEM is the most widely used and appropriate
method available to account for large shape changes and non-linear material behaviour. FEM
is a technique applied to computational analysis used to solve partial differential equations.
An application of solving such equations is to predict the deformation and stress fields within
solid bodies subjected to external forces such as concentrated and distributed loads.
According to Roylance (2001), the finite element method is usually carried out making use of
three steps: pre-processing, analysis and post-processing. During pre-processing, a model of
the material to be analysed is created. The geometry of the model is discretised into so-called
elements, which is commonly assisted by various computer software programs. In order to
carry out a finite element analysis, a finite element mesh is required, as mentioned by Bower
(2009). The finite element mesh is used to specify the geometry of the solid to be modelled,
25
and is additionally used to describe the displacement field within the solid. With the use of
later described meshfree methods, a mesh is also required in order to discretise the geometry
of the structure to be analysed.
During the pre-processing phase, a mesh is created and is defined by a set of nodes together
with a set of finite elements, which as a whole represent the model to be analysed. A node
is described by Bower (2009) as a discrete point within the solid body. The boundaries of
elements within the body are represented by such nodes. Furthermore, support and loading
conditions are applied to nodes. With the input geometry and boundary conditions, a dataset
to be analysed is created.
Following pre-processing, analysis of the model is conducted. During analysis, the dataset
created during pre-processing is used as the input into analysis. With this input, a set of
linear or non-linear algebraic equations is constructed and solved based on the fundamental
equations of physics governing the problem. The general system of equations is defined as
Ku = F (4.1)
where u is the displacement matrix of the nodes, and F is the force matrix applied at the
nodes or upon elements bound by nodes in the case of line, traction, pressure and volume
loads. The K matrix is dependent on the type of problem to be analysed. During analysis, it is
possible to either solve for a matrix of unknown forces, or a matrix of unknown displacements.
(Roylance, 2001)
Once the system of equations has been solved using numerical methods, the results are inter-
preted during the post-processing phase. Generally, results are visualised using post-processing
software, which creates, for example, a coloured contour image of the initial or deformed body
illustrating stress, strain, displacement, rotation and temperature. (Roylance, 2001)
4.3.2 Gauss quadrature
During computational analysis of continuum mechanics problems, if a function to be solved is
not simple or given in a tabulated form, analytical analysis of its corresponding integral can be
difficult or impossible. In such cases, the use of numerical methods is required, and subsequent
formulas used for numerical integration are named quadratures. If functions used to approxi-
mate such integrals are evaluated using unequally spaced base points, the Gauss quadrature
formulas are obtained. In such formulas, integration points are strategically selected in order
to obtain the best possible accuracy. (Shabana, 2012)
Shabana (2012) states that in Gauss quadrature, an integral to be solved is evaluated by
approximation of a function f (x) by a polynomial Pn (x). When using such an approximation,
an error may be realised. Therefore, base points are selected in order to make the integral of
the error function equal to zero.
26
4.3.3 Element-free Galerkin Method
As previously mentioned in Section 4.3.1, FEM is a tool used to calculate approximate so-
lutions to a set of differential equations in order to analyse the behaviour of materials. As
described by Belytschko, Lu and Gu (1994), the Element-free Galerkin Method (EFGM) is a
method used to solve such equations by converting the governing set of equations to the weak
form, known as the principle of virtual work making use of moving least-squares interpolants.
The EFGM was developed in order to reduce computational time required for mesh generation.
When utilising the EFGM, Belytschko, Lu and Gu (1994) explain that models to be analysed
are free of any mesh, and only an array of nodes in the domain of consideration is required.
Advantages of the EFGM which aid in the post-buckling analysis of shell structures are that
the EFGM does not seem to exhibit volumetric and shear locking and the rate of convergence
is quicker than that of a conventional finite element (Belytschko, Lu and Gu, 1994). Therefore,
the use of the EFGM is essential in this study in order to conduct analysis of three dimensional
shells avoiding volumetric and shear locking.
4.4 Implementational aspects
Certain aspects of the constitutive law presented in Chapter 3 require specific computational
techniques and modification in order to be used in computational analysis. Therefore, the fol-
lowing computational issues were dealt with following the methods described by Sansour and
Kollmann (1997). The corresponding C++ code used in order to implement the constitutive
law within the framework of SESKA may be viewed in Appendix A.
4.4.1 Time integration and local iteration
During computation, the state of deformation was computed over an interval of time. The
time interval was incremented with a number of calculation steps, such that the sum of time
of all calculation steps was equal to the total calculation time. Therefore, it was required
that various parameters be updated during each respective time step, taking into account
the history of variables in the previous calculation step. The incorporation of history into
calculation steps was crucial in order to track plastic history variables in the case of strain
hardening.
For discrete times tn and tn+1 over an increment ∆t, the following updating algorithms are
applied to the plastic deformation gradient F p and the strain measure CC−1
p :
F p|n+1 = F p|n exp (∆tLp) and (4.2)
CC−1
p |n+1 = C|n+1 exp (−∆tLp) C−1
p |n exp −∆tLT
p . (4.3)
27
Furthermore, it is required that the increment of the plastic multiplier ∆λ be computed; which
is determined with the use of the von Mises yield function - described by Eq. (3.22). It may
be noted that Lp and λ depend on Ξ. However, with CC−1
p depending also on Lp, and Ξ
depending on CC−1
p in the computation of α, an alternative method is required to compute
the terms in consideration. Therefore, the stress and plastic deformation state variables are
computed via the so-called corrector-predictor method making use of the Newton-Raphson
method.
With the use of the plastic corrector-predictor method, a separate sequence of plastic itera-
tion steps j is applied within each iteration step i of the time step ∆t, during which global
equilibrium of the system is sought. Therefore, within each iteration conducted to solve the
global equation system, a second Newton-Raphson iteration is conducted.
The plastic-corrector method is carried out over two steps during each Newton-Raphson
iteration. During the first step, referred to as the trial step with j = 0, the elastic constitutive
model is used in order to determine a trial stress Ξtrial
, with Ce|i
= Ce|i
n+1C−1
p |n. During
this step, C is kept constant, while λ is updated via
λ|j+1 = λ|j + ∆λ. (4.4)
Updating λ as above has the result of C−1
p being updated as
C−1
p |n+1 = exp (−∆tLp) C−1
p |n exp −∆tLT
p . (4.5)
Additionally, the state variable Z is updated as
Z|n+1 = Z|n +
2
3
∆tλ|j+1. (4.6)
In order to compute the increment of the plastic multiplier ∆λ, firstly an expression for Ξ is
required. Due to the modified logarithmic strain measure being used, Eq. (4.3) is modified in
order to compute the expression of the logarithmic strain as
ln CC−1
p |n+1 = ln C|n+1 exp (−∆tLp) C−1
p |n exp −∆tLT
p
= ln C|n+1C−1
p |n exp −2∆tLT
p or
αT
|n+1 = αtrial T
− 2∆tLT
p . (4.7)
Therefore, with the substitution of Eq. (4.7) into Eq. (3.20), the elastic material stress Ξ is
updated as
Ξ|n+1 = Ξtrial
− 2µ∆tλ
devΞtrial
||devΞtrial
||
. (4.8)
Secondly, Yn+1 is substituted into the yield function resulting in the expression used to com-
pute f:
f (λ, Lp) = ||devΞtrial
|| − 2µ∆tλ −
2
3
(σy − Y |n+1) = 0 (4.9)
28
With the use of Eq. (4.9), a non-linear equation for λ is obtained. However, it is still required
that the increment ∆λ be computed. In order to compute this increment, Eq. (4.9) is linearised
for each plastic step j where:
∆λ = −f
∂λ
∂f
(4.10)
In order to implement Eq. (4.10), it is required that the yield function tangent ∂f
∂λ
be computed
as
∂f
∂λ
= −2µ∆t +
2
3
∂Y |n+1
∂λ
(4.11)
where ∂Y |n+1
∂λ
is referred to as the Y tangent. Finally, the Y tangent is computed as
∂Y |n+1
∂λ
= −
2
3
∆t H + η (σ∞ − σy) exp −η Zn +
2
3
∆tλ . (4.12)
4.4.2 Stress and algorithmic tangent operator
In order to compute the second Piola-Kirchoff stress tensor S over each iteration, the following
expression is used:
S = C−1
Ξtrial
− 2µ∆tλν (4.13)
Additionally, the algorithmic tangent operator H is obtained by linearising S with respect
to C:
H =
∂S
∂C
=
∂C−1
∂C
Ξtrial
− 2µ∆tλν + C−1 ∂Ξtrial
∂C
− 2µ∆t
∂λ
∂Ξtrial
∂Ξtrial
∂C
C−1
ν
− 2µ∆tλC−1 ∂ν
∂Ξtrial
∂Ξtrial
∂C
(4.14)
However, it is further required that the derivatives in Eq. (4.14) be evaluated in order to make
use of the expression. Therefore, the final equation conveniently expressed in index notation
which is used in order to compute the algorithmic tangent operator is as follows:
∂Sij
∂Crs
= −C−1
ir C−1
sk Ξtrial
kj − 2µ∆tλνkj + C−1
ik K −
1
3
µ δkjC−1
rs + µδksC−1
rj
− 2µ∆t
νba
2µ∆t 2
3
∂Y |n+1
∂λ
K −
1
3
µ δabC−1
rs + µδasC−1
rb C−1
ik νkj
− 2µ∆tC−1
ik
1
||devΞtrial||
δkaδjb −
1
3
δkjδab − νkjνba K −
1
3
µ δabC−1
rs + µδasC−1
rb
where δ represents the Kronecker delta.
29
4.5 Variational formulation
Following the computation of S and subsequently H, a set of equation systems may finally
be set up and solved. In the analysis of non-linear initial value boundary problems, a coupled
system of partial differential equations has to be considered. During the analysis of shell
structures, such relations are governed by equilibrium equations, and the complex geometry
to which they are applied requires computational approximation methods to be employed.
The previously mentioned Element Free Galerkin Method (the so-called meshfree method)
was therefore utilised and is based on a variational formulation of the governing equations,
also known as the principle of virtual work or the weak form. (Sack, 2013)
The variational formulation of continuum mechanics states that the external potential, Wext,
is equal to the internal potential, Wint. The external potential corresponds to the work done
by the external forces such as traction forces acting on B.
Taking the above into consideration, the principle of virtual work is expressed in the following
manner:
δΨ = Wint − Wext = 0 (4.15)
The relevance of the computation of S in apparent with the definition of Wint as
Wint =
B
S : δEdV =
B
1
2
S : δCdV (4.16)
where E is the Green strain tensor and : signifies the multiplication of two tensors. The
relevance in computation of F is apparent with the definition of E as
E =
1
2
F T
F − 1 =
1
2
(C − 1) . (4.17)
The final variational formulation is defined as
δΨ =
B
1
2
S : δCdV − Wext = 0 (4.18)
and is governed by the set of binding equilibrium equations. The linearisation of the variational
formulation making use of a Taylor expansion is expressed as
δΨ =
v
1
2
S : δCdV +
v
1
2
H∆C : δC +
1
2
S : ∆δCdV − Wext = 0. (4.19)
30
4.6 Assembly of equation system
At the assembly stage, the objective is to obtain a discrete set of equations that can be
subsequently solved for the unknown field, such as displacement in the case of this study.
With Eq. (4.1) in mind, the equation system to be solved takes the form of
K∆u − [fext − fint] = 0 (4.20)
where K is the stiffness or tangent matrix, ∆u is the unknown displacement field increment,
and fext and fint are the external and internal loading and reaction vectors respectively.
However, for non-linear problems such as in this study, no equality is present in Eq. (4.20),
and hence an iterative procedure is required to find the unknown displacement field. To this
end, the Newton-Raphson method is employed to finally solve for the unknown displacement
field.
4.7 Concluding remarks
This chapter has introduced general considerations to be taken into account during the com-
putational analysis of structures with specific relevance to shell structures. Methods of com-
putational modelling were dealt with, and subsequently the specific methods employed in
order to computationally make use of the constitutive law implemented such as the use of
the plastic corrector-predictor method. Additionally, the variational formulation and assem-
bly of the final equation system were addressed in order to illustrate the method used to
finally set-up and solve for the global equation system of structures analysed. Following the
implementational aspects of this study, attention is now paid to the analysis of simplified
structures.
31
Chapter 5
Numerical examples
5.1 Introduction
With the computational analysis tools described in Chapters 3 and 4, analysis was possible
of structures making use of the implemented constitutive law. Various types of simplified
structures were modelled with increasing complexity in order to meet the modelling objec-
tives of this study. This chapter contains the problem set-up for structures analysed, with
corresponding results and discussions being presented.
Initially, a simple cantilever beam was modelled for benchmarking purposes, in order to de-
termine whether performance of the implemented material law was similar to that of the
previously implemented stretch based material law. Subsequently, two plates were modelled
in order to make comparisons against the verified results of Sansour and Wagner (2003)
who analysed identical structures making use of a visco-plastic material law. Furthermore,
two cylindrical shell structures were modelled with varying loading and boundary conditions.
While results of these analyses were also benchmarked against verified results, these models
served an additional purpose of providing a preliminary insight into the post-buckling regime
of shell structures with the use of an elasto-plastic material law, three dimensional continuum
mechanics and meshfree methods.
5.2 Cantilever beam
5.2.1 Problem set-up
The simple cantilever beam modelled was defined geometrically by a length L = 10 m and a
square cross-section of h = d = 1 m. One end surface of the beam was prescribed displacement
boundary conditions such that no displacement in all three degrees of freedom occurred,
emulating a built-in boundary constraint. The initial configuration of the cantilever beam is
illustrated overleaf in Figure 5.1.
The free end of the beam was assigned a traction load in the vertical direction, being controlled
by Dirichlet loading conditions; such that the end of the beam was displaced by a user defined
increment during each loading step. With the use of such loading, following the displacement
32
Figure 5.1. Cantilever beam - undeformed configuration (Essack, 2014)
Table 5.1. Cantilever beam elastic and plastic material parameters
Parameter Value (kN/mm2
)
µ 88.0
K 238.3
σy 0.450
σ∞ 0.715
H 0.129
η 16.29
Table 5.2. Cantilever beam dynamic material parameters
Parameter Value
ρ 4.5 x 10−9
kg/mm3
c 10−6
N · s/mm
increment being applied, the force required for displacement is computed. Material parameters
used were those for titanium, as detailed in Tables 5.1 and 5.2.
Two types of analyses were conducted on the cantilever beam; namely static and dynamic. The
static analysis was conducted in order to make comparisons with the previously mentioned
stretch based material law, and therefore two simulations on the beam were run, making use
of the respective constitutive laws. Subsequently, a dynamic analysis was conducted in order
to determine the initial expected behaviour with the use of an elasto-plastic material law
taking dynamic effects into consideration.
5.2.2 Static analysis
During analysis, a load deformation graph was generated for the tip of the beam undergoing
displacement. It may be observed overleaf in Figure 5.2 that both material laws predict very
similar behaviour of the beam in the elastic region and early stages of plastic deformation.
Elastic deformation is apparent from 0 m to 0.15 m displacement preceding a change in
gradient of the curve. Following a displacement of 0.15 m, yielding of the material is evident
33
with a change in the slope of the load deformation graph. Following initial yielding, a slight
increase in load during further deformation is required, due to to the effects of strain hardening
which have been incorporated into both material laws.
0 1 2 3 4 5 6 7 8 9
deformation [m]
0
50
100
150
200
250
loadfactor[kN]
Cauchy-Green
Stretch
Figure 5.2. Cantilever beam - static load deformation graph
However, it may be noted that the stretch based law extends only to 2.8 m of displacement;
due to analysis ceasing to converge. Additionally, analysis time of this material law was
significantly slower than that of the material law implemented in this study. The discrepancy
in analysis time may be attributed to the theory of the material laws; as the stretch based
law takes into account six degrees of freedom with the inclusion of rotation about all three
axes.
Following 2.8 m of displacement, the implemented material law continues calculation as fur-
ther strain hardening occurs over a significant period of displacement in the plastic region;
indicated by the increase in the load factor during further displacement. Within this region,
effective plastic strain of the material continues to increase as stress further increases. A steep
increase in the load factor following 6 m of deflection is attributed to tension forces within
the beam. A second period of yielding begins at approximately 8 m where the material begins
to yield under tensile forces as opposed to flexure. The deformed configuration at 4 m of
displacement is illustrated in Figure 5.3 with a contour fill depicting effective plastic strain in
Figure 5.4.
34
Figure 5.3. Cantilever beam - deformed configuration displacement contour plot
Figure 5.4. Cantilever beam - deformed configuration plastic strain contour plot
5.2.3 Dynamic analysis
Following static analysis, a dynamic analysis was conducted in order to determine feasible
dynamic material parameters for further testing, and secondly to gain an understanding of
the behaviour of the elasto-plastic material law under dynamic loading. The time deformation
graph produced is illustrated by Figure 5.5 overleaf.
Figure 5.5 depicts displacement up to -0.23 m. Following the prescribed displacement being
reached, the load applied was maintained while calculation steps continued. The change in
deformation occurring over subsequent time steps is attributed to oscillation of the beam, and
is observed to decrease over the remaining time period of analysis indicating that material
parameters chosen were feasible with no under damping or over damping observed.
35
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
time [seconds]
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
deformation[m]
Dynamic Cauchy-Green
Figure 5.5. Cantilever beam - time deformation graph during dynamic analysis
5.3 Simply supported plates
5.3.1 Problem set-up
The first thin structures modelled were those of two simply supported plates; serving the
purpose of further calibration against verified results. Sansour and Wagner (2003) previously
modelled two plates with the use of a visco-plastic material law, and specialised shell elements.
Therefore, modelling of the two plates served the purpose of benchmarking the implemented
law against verified results produced with the use of an alternate material law and analysis
techniques. Additionally, a preliminary insight was to be gained into the analysis of thin
walled structures with the use of the newly implemented material law.
Geometrically, the two square plates modelled were identical, with lengths and breadths of
500 mm and a thickness of 1 mm. The first plate modelled (case I) was fixed in all three
directions of displacement along the edges, while edges of the second plate (case II) were
fixed against displacement in the vertical direction only, allowing inward movement during
deformation as detailed overleaf by Figures 5.6 and 5.7 respectively. A uniformly distributed
dead load was applied over the plate surfaces, with the centre particle of the plates being
incremented at 0.1 mm per loading step.
Material parameters were chosen in accordance with Sansour and Wagner (2003) described in
Table 5.3 overleaf, for that of titanium facilitating comparison of results, and were adjusted
to compensate for differences in elastic material law equations. The non-linear hardening
parameter η was not used in order to allow for less stiff behaviour of the plates.
36
As buckling during deformation is unlikely to occur in such structures, dynamics of the struc-
tures was not accounted for, resulting in relatively simple initial models in order to illustrate
differences between the elastic and plastic regimes. Due to symmetry conditions, one quarter
of each plate was modelled and discretised into 10 x 10 elements. The identical undeformed
configuration of both plates is illustrated overleaf in Figure 5.8.
Figure 5.6. Simply supported plates - Case I (Sansour and Wagner, 2003)
Figure 5.7. Simply supported plates - Case II (Sansour and Wagner, 2003)
Table 5.3. Simply supported plates - elastic and plastic material parameters
Parameter Value (kN/mm2
)
µ 88.0
K 238.3
σy 0.450
σ∞ 0.715
H 0.129
η 0
37
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GHopkins_BSc_2014

  • 1. Department of Civil Engineering Highly Non-linear Post-buckling Analysis of Shell Structures A thesis in partial fulfilment of the requirements for a Bachelor of Sciences degree in the field of Civil Engineering Prepared for: University of Cape Town Author: Mr. Gary Hopkins Supervisor: Dr. Sebastian Skatulla November, 2014
  • 2. Plagiarism Declaration ˆ I know that plagiarism is wrong. Plagiarism is to use anothers work and to pretend that it is ones own. ˆ I have used the Harvard Convention for citation and referencing. Each significant contri- bution to and quotation in this report from the work or works of other people has been attributed and has been cited and referenced. ˆ This thesis is my own work. ˆ I have not allowed and will not allow anyone to copy my work with the intention of passing it as his or her own work. Name Student No Date Signed Gary Hopkins HPKGAR001 10 November 2014 i
  • 3. Abstract Shell structures have been widely used in many engineering applications such as pipelines, liquid-retaining structures and aerospace structures. As shell structures are commonly con- structed with thin walls, buckling is generally the primary failure mechanism of concern. Many non-linear studies concerned with buckling have been conducted with regards to shell struc- tures. However, limited studies have been conducted in which the post-buckling behaviour of such structures has been investigated, and for this reason the post-buckling behaviour of shell structures remains misunderstood. In the post-buckling regime of ductile metallic structures, large deformations are often ex- perienced with corresponding plastic deformation. Therefore, it is required that analyses are conducted in a non-linear manner with respect to deformation and stress-strain relationships. The Finite Element Method is commonly employed as a tool to model such behaviour, how- ever, problems such as shear locking result in computational difficulties. Such difficulties are often avoided with the use of specialised shell elements making simplifications in terms of geometry and behaviour. This study aims to implement an elasto-plastic constitutive law based on the logarithmic right Cauchy-Green deformation tensor, in order to make possible computational analysis of shell structures in the post-buckling regime using three-dimensional continuum mechanics and finite strain theory, avoiding as far as possible the simplification of shell geometry and behaviour. The constitutive law is implemented within the existing framework of SESKA, a C++ code developed by Doctor Sebastian Skatulla with the purpose of solving equation systems as required within this study. Implementation within the current framework allows for the use of highly smooth meshfree approximations of the Element-free Galerkin Method, avoiding the usual limitations of the Finite Element Method. Following implementation, simple shell structures are modelled and analysed in order to gain a preliminary understanding of the behaviour of such structures in the post-buckling regime. Additionally, analysis is conducted in order to ascertain whether use of the methods employed within this study are feasible and suitable for further use in non-linear post-buckling analyses of shell structures. This study is unique in that it appears to be the first known attempt to implement an elasto- plastic post-buckling analysis tool for shell structures making use of three-dimensional con- tinuum mechanics and meshfree methods, as opposed to specialised shell elements. Therefore, to conclude this study, results are benchmarked against verified analyses of similar structures which have made use of specialised shell elements and a visco-plastic material law. ii
  • 4. Acknowledgements Firstly, I would like to thank my supervisor, Dr Sebastian Skatulla, for his guidance and sup- port throughout this study, and for providing me with such an interesting topic to undertake. While I have been overwhelmed with an abundance of new information and skills required, Dr Skatulla has continuously put in great effort to guide me through this study. Additionally, his patience and enthusiasm while explaining countless topics is highly appreciated and I would have not been able to undertake this study without his expert guidance. Secondly, my parents, for funding me through my schooling and most of my university career; providing me with the opportunity to excel academically and as a person, by setting a flawless example for me to aspire to. I extend many thanks to my brother Ryan, girlfriend Michaela and all other friends who have supported and guided me. The funding of Stefanutti Stocks Marine for the past two years is acknowledged and highly appreciated, relieving my parents of financial pressure. This thesis project has been supported by the Centre for High Performance Computing, Mowbray. iii
  • 5. Table of Contents Plagiarism Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Notation and list of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background to study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research project motivation, aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Motivation for research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Aim of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.3 Objectives of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Scope and limitations of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Expected results of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.1 Benchmarking of constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.2 Geometric results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.3 Load carrying capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Layout of this document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 iv
  • 6. 2.2 Shell structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Methods of investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Geometry of deformed structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Post-buckling strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Continuum mechanics theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.1 Elastic and plastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.2 Balance law of continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Stress measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 The constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5.2 Elastic constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5.3 Plastic constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Computational analysis principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 General analysis considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.1 Buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.2 Linear and non-linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.3 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Modelling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.2 Gauss quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3.3 Element-free Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 v
  • 7. 4.4 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4.1 Time integration and local iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4.2 Stress and algorithmic tangent operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.6 Assembly of equation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2.2 Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2.3 Dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Simply supported plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4 Pinched cylinder with rigid diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.4.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.4.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.5 Axially compressed cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.5.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.5.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6 Conclusions and recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Implementation of constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3 Benchmarking of constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.4 Pinched cylinder with rigid diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.5 Axially compressed cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 vi
  • 8. 6.6 Further use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Appendix A Source code of implemented constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vii
  • 9. List of Figures 2.1 Simple geometry of a shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 A buckled cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Typical computational buckling dimples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 Stress versus strain of a ductile material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Deformation of a solid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.1 Cantilever beam - undeformed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Cantilever beam - static load deformation graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 Cantilever beam - deformed configuration displacement contour plot . . . . . . . . . . 35 5.4 Cantilever beam - deformed configuration plastic strain contour plot . . . . . . . . . . 35 5.5 Cantilever beam - time deformation graph during dynamic analysis. . . . . . . . . . . 36 5.6 Simply supported plates - Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.7 Simply supported plates - Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.8 Simply supported plates - undeformed configuration . . . . . . . . . . . . . . . . . . . . . . . . 38 5.9 Simply supported plate - Case I load deformation graph . . . . . . . . . . . . . . . . . . . . 38 5.10 Simply supported plate - Case II load deformation graph. . . . . . . . . . . . . . . . . . . . 39 5.11 Simply supported plate - case I deformed configuration . . . . . . . . . . . . . . . . . . . . . 39 5.12 Simply supported plate - case I plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.13 Simply supported plate - case II deformed configuration . . . . . . . . . . . . . . . . . . . . 40 5.14 Simply supported plate - case II plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.15 Pinched cylinder - problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.16 Pinched cylinder - undeformed configuration with coarse mesh . . . . . . . . . . . . . . . 42 5.17 Pinched cylinder - undeformed configuration with fine mesh . . . . . . . . . . . . . . . . . 42 5.18 Pinched cylinder (coarse mesh) - load deformation graph . . . . . . . . . . . . . . . . . . . . 43 viii
  • 10. 5.19 Pinched cylinder - maximum deformation and plastic strain at end of first loading cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.20 Pinched cylinder - final deformation and plastic strain . . . . . . . . . . . . . . . . . . . . . . 45 5.21 Pinched cylinder (fine mesh) - load deformation graph . . . . . . . . . . . . . . . . . . . . . . 46 5.22 Pinched cylinder - deformed configuration at 5mm displacement . . . . . . . . . . . . . 46 5.23 Pinched cylinder - deformed configuration at 15mm displacement . . . . . . . . . . . . 47 5.24 Pinched cylinder - effective plastic strain at 15mm displacement . . . . . . . . . . . . . 47 5.25 Pinched cylinder - deformed configuration at 20mm displacement . . . . . . . . . . . . 47 5.26 Compressed cylinder - undeformed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.27 Compressed cylinder - load deformation graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.28 Compressed cylinder - deformed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.29 Compressed cylinder - final plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.30 Compressed cylinder - experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ix
  • 11. List of Tables 5.1 Cantilever beam elastic and plastic material parameters. . . . . . . . . . . . . . . . . . . . . 33 5.2 Cantilever beam dynamic material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Simply supported plates - elastic and plastic material parameters . . . . . . . . . . . . 37 x
  • 12. Notation and list of symbols Symbols used are defined as they appear throughout the text. In the following, the general scheme of notation and list of symbols is assembled: a roman lower-case letters denote scalars A roman upper-case bold-face letters denote tensors a roman lower-case bold-face letters denote vectors A−1 inverse of a tensor AT transpose of a tensor A calligraphic upper-case letters denote sets := definition of equivalence Subscripts e elastic p plastic ref in the reference configuration xi
  • 13. Chapter 1 Introduction This thesis serves the purpose to document the process followed in order to implement a non- linear elasto-plastic material law to conduct qualitative computational investigations into the post-buckling behaviour of shell structures. 1.1 Background to study Shell structures are commonly used in engineering practice due to their many advantages over other engineering structures. When compared to other engineering structures, such advantages include high stiffness, efficiency and strength to weight ratio (Fung and Sechler, 1974). Due to such properties, shell structures are able to serve the purpose of other engineering structures while requiring less material. Contemporary examples of shell structures include arch dams, domed roofs, liquid-retaining structures, aircraft and carbon nanotubes. Shell structures have been extensively studied over the past century. With increasing com- puting power, many experimental and theoretical investigations have been conducted into the strength and stability of these structures. According to Mandal and Callandine (2000), when shell structures are loaded in compression, buckling constitutes the most common form of failure. Furthermore, as elastic buckling is often catastrophic, many studies concerned with failure of shell structures have been aimed at predicting initial buckling loads. For this reason, few studies have been undertaken to explore the post-buckling behaviour of shell structures deep into the non-linear deformation and plastic regimes. (Vaziri, 2009) Additionally, as mentioned by Vaziri (2009), recent advancements in nanotechnology have increased the potential use of shell structures. However, understanding of the post-buckling behaviour of shells remains ambiguous and requires further investigation. Chapelle and Bathe (1998) refer to shells as the prima donnas of structures, and raise the problem that analysis of such structures is difficult due to their unpredictable behaviour with small changes in geometry. According to Vaziri (2009), “Our understanding of shell and membrane structures is still in its infancy”, and many further studies are required to fully understand the mechanics of these highly significant structures. 1
  • 14. 1.2 Research project motivation, aims and objectives In order to successfully further the understanding of the post-buckling behaviour of shell structures, the following research motivation, aims and objectives are specified. 1.2.1 Motivation for research While shell structures have been extensively studied, few studies have been concerned with plastic post-buckling behaviour. This is partly due to failure concerns related to initial buck- ling, and partly due to the difficulty of conducting such experiments caused by shear locking and excessive mesh distortion when making use of the Finite Element Method. Additionally, studies which have investigated the post-buckling regime have predominantly investigated elastic buckling. However, during buckling of metallic structures, plastic deformation is expe- rienced and it is required that plasticity is accounted for. As structures require larger and more efficient spans, are required to contain more water as population increases, or as nanotechnology advances, a further understanding of shell struc- tures will be required. Therefore, it is critical that tools be developed in order to obtain a better knowledge of the post-buckling behaviour of shell structures in terms of stability, buck- ling mechanisms and load carrying capacity to allow for more efficient and safe use of such structures. 1.2.2 Aim of research This study aims to implement and utilise an elasto-plastic constitutive law based on the loga- rithmic right Cauchy-Green deformation tensor, in order to make possible the computational analysis of shell structures in the post-buckling regime using three-dimensional continuum mechanics, avoiding as far as possible the simplification of shell geometry and behaviour. It is aimed that the constitutive law be programmed within the existing framework of the in-house structural analysis software SESKA. Following implementation, this study aims to ascertain the feasibility of using the material law in further studies in order to gain an improved understanding of the post-buckling behaviour of shell structures. Therefore, it is aimed that the implemented material law is benchmarked against verified results of analysis conducted with the use of specialised shell elements. Subsequently, it is aimed that preliminary qualitative analyses concerning the post-buckling behaviour of shell structures be conducted taking into consideration stability, geometric changes during deformation, and load carrying capacity. 2
  • 15. 1.2.3 Objectives of study The objectives of this research project are: 1. Familiarisation with the in-house structural analysis software SESKA; 2. Presentation of background knowledge required for the analysis of such structures; 3. Implementation of an elasto-plastic constitutive law within the existing framework of SESKA; 4. Creation of simple models of shell structures using the commercial pre-processing software GiD; 5. Computational analysis of models created using the implemented material law; 6. Benchmarking against verified results of similar shell structures in order to determine the feasibility of further use of the implemented law; and 7. Post-processing of results using GiD in order to gain a preliminary qualitative understand- ing into the post-buckling regime of shell structures with respect to stability, deformation behaviour and load carrying capacity. 1.3 Scope and limitations of investigation Prior to the implementation of the constitutive law used in this study, a von Mises elasto- plastic constitutive law based on the logarithmic unsymmetric stretch tensor had been imple- mented. However, it was required that a constitutive law making use of the right Cauchy-Green deformation tensor be implemented due to the unsymmetric nature of the previously imple- mented law; causing increased computational time. Due to the many similarities between the two laws, the existing law was modified in order to implement the law used in this study. The law used in this study makes use of the existing computational continuum mechanics framework contained within SESKA, and therefore the documentation of implementation is primarily concerned with detailing the additional methods followed during implementation. Subsequently, simple models of thin-walled and specifically shell structures are created and analysed. This study is aimed at qualitative analysis of such structures in the post-buckling and plastic regimes, and therefore aspects such as initial material imperfections are not ac- counted for as initial buckling loads are not of high importance within this study. However, plasticity of the materials analysed is taken into account with the inclusion of plastic history variables, and strain hardening with the inclusion of linear and non-linear hardening terms. This study does not attempt to conduct a full analysis of such structures, but rather to gain a preliminary understanding of behaviour of such structures when analysed with the use of elasto-plasticity and three dimensional continuum mechanics. Additionally, included within the scope of this study is the use of the implemented constitutive law in order to 3
  • 16. determine feasibility for use in further, more detailed investigations. This investigation and the constitutive law implemented are limited to the use of metallic structures only. 1.4 Expected results of investigation The following results in terms implementation, benchmarking and analysis are expected: 1.4.1 Benchmarking of constitutive law It is expected that in comparison to the visco-plastic shell element analysis against which the implemented law is to be benchmarked, similar behaviour within the elastic regime is to be experienced. However, it is not known how the two methods of analysis will compare in the plastic regime; elasto-plasticity is rate-independent, whereas visco-plasticity is rate- dependant. Despite this difference, similar qualitative behaviour of the two laws is expected. 1.4.2 Geometric results During post-buckling analysis of cylindrical shells, the formation of inward ellipsoidal de- flections in the metal known as buckling “dimples” is expected. However, the locations and number of such dimples is not known. It is expected that the greatest deformation is to be expected at such dimples, and dimple behaviour is to govern overall behaviour of the struc- tures in terms of structural integrity. Additionally, it is expected that buckling is to occur in the locations of these so-called dimples. 1.4.3 Load carrying capacity Following initial buckling and movement into the plastic regime, a decrease in load resisted is expected. However, the amount by which the load is expected to decrease is not known and will most likely be influenced by initial material parameters assigned to the materials tested, and the behaviour of buckling dimples. It is expected that in the post-buckling and plastic regimes, a portion of the load carrying capacity will be retained. It is expected that the formation of the previously mentioned dimples is related to the loss of load carrying capacity, however, it is not known whether these dimples form as a result of decreased load carrying capacity or vice-versa. The presence of a post-buckling plateau, during which large deformations are experienced with small increases in stress; similar to the onset of plastic deformation, is expected. It is expected that this plateau will occur once structures have reached a stable mode of deformation. 4
  • 17. 1.5 Layout of this document The following breakdown contains the primary information which will be addressed within each chapter: Chapter one provides an introduction to the topic of this document by providing the reader with relevant background information. The motivation, aims and objectives of this research project are subsequently addressed. Furthermore, this chapter contains the scope and limita- tions of this document. Chapter two contains a review of the available literature concerning buckling and post- buckling studies related to shell structures. The chapter begins with an introduction into shell structures gleaned from the review of various studies and texts. Subsequently, methods of analysis and results of previously conducted studies are presented. Within this chapter, conclusions are made concerning the need for further study in the field. Chapter three is concerned with the continuum mechanics theory used to implement the mate- rial law. General considerations when dealing with elasto-plasticity are illustrated, following which the kinematics of an elastic-plastic body is described. Subsequently, the continuum mechanics of the constitutive law is presented. Chapter four deals with computational aspects of implementation, such as types of analy- sis and modelling methods. Additionally, the computational methods followed in order to implement the constitutive law are described. Chapter five describes all preliminary models created and analysed, and provides the problem definitions for each model respectively in terms of geometry, loading conditions and type of analysis conducted. Initial configurations of all models are presented, along with material parameters used. Additionally, this chapter illustrates the results of analysis of all preliminary models and comments on relevant aspects of these results. Chapter six makes conclusions and recommendations with regards to the investigation con- ducted and whether the implemented law is feasible for further use in the post-buckling analysis of shell structures. Additionally, conclusions are made with respect to analysis of models created. Appendix A contains the source code of the implemented material law as compiled. 5
  • 18. Chapter 2 Literature Review 2.1 Introduction This chapter contains a review of available literature concerned with shell structures and previously conducted investigations into the buckling and post-buckling behaviour of such structures. Initially, an introduction to shell structures is presented with the intention to fa- miliarise the reader with such structures. Subsequently, previously conducted investigations are reviewed with attention being given to methods of investigation, geometry and load bear- ing capacity in the post-buckling regime with the aim of illustrating that further research in the field is required. 2.2 Shell structures As explained by Lowe (1970), shells may be defined as a body bound by two curved surfaces, where the distance between the two surfaces is small in comparison with the other dimensions. The line which is equidistant from both curved surfaces is defined as the middle surface of a shell, as illustrated overleaf in Figure 2.1. Shell structures are commonly used in engineering applications due to the many advantages they possess over alternative types of structures. Some advantages, as listed by Venstel and Krauthammer (2001), are efficiency, high strength to weight ratio and a very high stiffness. Contemporary examples of shell structures are liquid-retaining structures, arch dams, aircraft and nano structures. Shells are particularly similar to plates; however, they differ in that shells have a curved surface (Venstel and Krauthammer, 2001). Various classifications of shells exist, based upon surface curvature, and shells are generally classified in such a manner as the behaviour of a shell is governed primarily by curvature. Additionally, according to Lowe (1970), the middle surface, thickness and edges define the geometry of a shell. As shells gain their unique properties largely from geometric configuration (Venstel and Krauthammer, 2001), multiple theories have been developed in order to predict the be- haviour of shell structures of varying geometry. Shells may be divided into varying forms of geometry according to surface curvature; such as cylindrical, conical, spherical, ellipsoidal or paraboloidal. Furthermore, shells may be classified as either thick or thin. 6
  • 19. Figure 2.1. Simple geometry of a shell (Venstel and Krauthammer, 2001) According to Lowe (1970), in engineering applications, a shell is regarded as thin if max h R < 1 20 (2.1) where h is the thickness of the shell, and R is the radius of curvature of the middle surface. Otherwise the shell is defined as thick. As a result of the wide use of shells, many experimental and theoretical studies have been conducted in order to gain insight into the behaviour of such structures, taking into account varying geometry, initial material imperfections and loading conditions. Mandal and Callan- dine (2000) state that when shells are loaded in compression, buckling constitutes the most common form of failure. Therefore, many of the studies mentioned have been concerned with investigating buckling of shells. Zhu, Mandal and Callandine (2002) mention that often failure of shell structures is catas- trophic. As shell structures are commonly used in industries where risks of failure are high, many studies concerned with buckling of shells have focused on predicting the load at which certain shells fail due to buckling. Consequently, there has been limited investigation into the post-buckling behaviour of shell structures. 2.3 Methods of investigation During investigation into the post-buckling behaviour of shell structures, various aspects which influence this behaviour are of interest. Such aspects are material used, loading and boundary conditions, methods of analysis, geometry of the structure and presence of initial imperfections. The post-buckling behaviour of steel cylindrical shells was investigated by Aghajari, Abedi and Showkati (2006), and the behaviour analysed with respect to pressure versus displacement and the number of inward indentations, referred to buckling waves or dimples as seen in Figure 2.2 overleaf. Analysis of the post-buckling regime is generally investigated with reference to 7
  • 20. these criteria, and has been made use of by various other authors. While investigating metallic cylindrical shells, Mandal and Callandine (2000) also used the presence of such dimples in an attempt to explain the loss of strength of cylindrical shells following initial buckling. Fakhim, Showkati and Abedi (2009) did not record the presence of such dimples, but made use of pressure versus displacement as an analysis tool. The presence of buckling dimples has not been limited to the analysis of metallic shells, however. Figure 2.2. A buckled cylindrical shell (Fakhim, Showkati and Abedi, 2009) Zhu, Mandal and Callandine (2002) investigated the post-buckling behaviour of cylindrical silicone rubber shells under axial compression, in an attempt to explain observations not pre- dicted by classical shell theory. Similar to this, Mandal and Callandine (2000) also investigated the behaviour of silicone rubber shells. Both authors observed the post-buckling behaviour making notes of the deformed geometry with reference to buckling waves. While the stud- ies conducted on silicone-rubber shells were experimental and lab-based, computational and theoretical analyses may also be conducted, employing various forms of numerical analysis. In order to conduct analyses on post-buckling behaviour, many authors have conducted non- linear computational analyses, as buckling of shells is a non-linear problem due to large changes in geometry and loading conditions. Aghajari, Abedi and Showkati (2006) conducted a non-linear analysis of cylindrical shells, and made use of the Arc-Length-Type method, as according to Memon and Su (2004), ordinary solution techniques such as the Newton-Raphson technique are not able to limit instability near limit points and fail to accurately predict the load versus displacement response. According to Chatzi (2014), the Newton-Raphson tech- nique performs poorly in buckling analysis as the slope at limit points is equal to zero. As 8
  • 21. described by Memon and Su (2004), the Arc-Length-Type method is conducted by modifying the load factor at each iteration step. While conducting a non-linear analysis of shells, Zhu, Mandal and Callandine (2002) made use of the commercial analysis software ABAQUS, solv- ing non-linear equations with the Arc-Length-Type method. In addition to various methods of analysis, different types of mechanical behaviour may be analysed. The previously mentioned investigations were concerned with elastic buckling. However, Cortivo, et al (2009) investigated the plastic buckling and collapse of thin shell structures. In order to carry out the non-linear investigation, the Assumed Natural Deviatoric Strain (ANDES) finite element was utilised. While this appears to have been successful, during this current investigation, it will not be required that a shell element be used to model shell struc- tures, as the analysis will be conducted making use of three dimensional continuum mechanics, thus reducing simplification of shell behaviour. In order to better predict the plastic behaviour of shells the ANDES shell element was modified to be elasto-plastic with the introduction of the von Mises yield criterion, while taking into consideration linear and isotropic hardening in a rate independent analysis. The introduction of the yield criterion allowed for the identi- fication of the point at which plastic deformation occurs. Furthermore, the Arc-Length-Type method was utilised. Additionally, this study made use of the stress resultant approach to predict non-linear behaviour of the structures. The use of the stress resultant approach incor- porates the change in stress in the material due to plastic deformation. Skallerud and Haugen (1999) conducted a similar study in which the same method of analysis was used; however, the effectiveness of such an analysis was investigated. It was concluded that the use of a modified ANDES element yielded results with low error. While some of the previously mentioned investigations conducted laboratory experiments, all made use of the Finite Element Method in order to test theoretical ideas against observed data. Two interesting studies were conducted by Mandal and Callandine (2000) and Zhu, Mandal and Callandine (2002) in which silicone-rubber was used to investigate elastic buck- ling of cylindrical and domed shells, as previously discussed. The use of silicone-rubber as a material was interesting as it allowed one to easily observe typical post-buckling behaviour. Cylindrical shells were tested under self weight, and domed shells under a point load. Mandal and Callandine (2000) concluded that buckling heights under self weight were proportional to the thickness of the shell, and noted the presence of dimples during buckling. Under a point load, it was observed that domed shells develop various buckling patterns with varying thick- ness. It was noted that as thickness is varied, the number of vertices formed during buckling varies. While conducting similar experiments, Zhu, Mandal and Callandine (2002) concluded that the method conducted illustrated well-characterised post-buckling behaviour of shell structures. While the materials and methods outlined will not be utilised in this research project, the outcomes are of interest as it is suspected that a plastic analysis of metallic shells will yield similar qualitative results characteristic of the post-buckling regime of shell structures. During these studies, initial material imperfection was taken into account. Both studies incorporated such imperfections during finite element analysis. It was not investi- gated whether these initial imperfections are influential in the post-buckling regime. However, it was concluded by studies such as Fakhim, Showkati and Abedi (2009), and various other 9
  • 22. authors that initial geometric imperfections in shell structures are instrumental in decreasing the initial buckling load. During their investigations, Mandal and Callandine (2000) and Zhu, Mandal and Callandine (2002) did take initial imperfections into account, but concluded that there was no correlation between initial geometric imperfections and buckling load. Mandal and Callandine (2000) suggested that during their study, the presence of initial stress was minimised reducing scatter of results and increasing initial buckling load. Buckling loads observed to be lower than that of predicted theoretical loads were explained in terms of static determinacy. Mandal and Callandine (2000) suggested that because their experimental structures were statically determinate, residual stresses were lower; the cause of scatter in other studies. It is the view of the author that the material choice may have impacted these results, as there may have been lower residual stresses in the silicone-rubber as opposed to metallic structures investigated, increasing initial buckling loads. It was noted by Fung and Sechler (1974) that initial geometric imperfections are thought to be the underlying cause of scatter in experimental results. In order to account for initial imperfections, specimen thickness at strategic points was mea- sured and included in models to be computationally analysed. Alternatively, initial imperfec- tions were included making use of trigonometric functions with dampening effects applied. In the latter case, imperfections introduced by Mandal and Callandine (2000) took a circular form and were deflected inwards, in order to ascertain whether the initial imperfection affected buckling behaviour. It was noted that a dimple did form near the initial imperfection, but subsequently migrated upwards after forming. However, other theories have been put forward concerning the differences between experimental and theoretical results. Shamass, Alfano and Guarracino (2014) conducted a non-linear plastic investigation into the post-buckling behaviour of cylindrical shells loaded under axial compression, making use of the deformation and flow theories of plasticity. In this study, it was concluded that the dis- crepancy between theoretical and experimental results may be addressed through careful and appropriate use of plastic theory. They did, however, take initial imperfections into account. Additionally, it was hypothesised that plastic buckling of shell structures generally occurs in thicker shells. The study tested cylinders made of an aluminium alloy. Various boundary conditions were tested, such as fully fixed, clamped and hinged. Cylindrical specimens were modelled using ABAQUS and a general purpose 4-noded shell element was used, based upon thick shell theory. Additionally, a structured mesh was used. In order to trace the non-linear response, the modified Arc-Length-Type approach was used. While this study is similar to the proposed study of this document, the use of shell elements differs to the proposed use of a three dimensional continuum approach. 2.4 Geometry of deformed structures During the analysis of shell structures, it has been noted that such structures primarily fail due to buckling (Mandal and Callandine, 2000). Therefore, loading which places these structures in compression is often investigated. While investigating buckling of shell structures, axial and pressure loading are common. During various studies making use of such loading, it appears that there are common post-buckling deformation patterns which occur. 10
  • 23. Indentations which form on shell structures in the post-buckling regime, often referred to as dimples or buckles, have been used in some studies to analyse the nature of buckling. While dimples of various quantities and size were noted in many studies mentioned in the previous section, all studies did note the presence of such dimples. An example of buckling dimples is illustrated in Figure 2.3 as red and yellow indentations on the wall of the cylindrical shell under axial compression. Figure 2.3. Typical computational buckling dimples (Skatulla, Sansour and Hijiaj, 2014) While conducting experimental tests on cylindrical shells, thickness was varied by Fakhim, Showkati and Abedi (2009). Each shell tested had a constant thickness throughout the length. However, it was concluded that the height of cylindrical shells influenced the location and size of the dimples, and thickness had no effect. In comparison, Aghajari, Abedi and Showkati (2006) investigated cylindrical shells which varied in thickness throughout the length of the cylinder. It was concluded that the variation in thickness had a direct influence on final buckling waves, and subsequently recommended that cylindrical shells with less thickness variation be used in construction making use of shells. Due to the presence of dimples, shells often lose the property of symmetry in the post- buckling regime. When loaded axially, eccentricities may form causing the structure to bend perpendicular to the loading axis. 11
  • 24. 2.5 Post-buckling strength As discussed by Mandal and Callandine (2000), experimental studies conducted in the 1930’s with connection to the aircraft industry revealed that buckling loads were lower than that of loads predicted by classical theory. Furthermore, there was a wide scatter in results and failure of shell structures was catastrophic. This, therefore, led to the buckling load of shell structures being studied in great detail. However, a significant amount of studies were concerned with predicting buckling loads of shell structures, and not with post-buckling strength. Studies which have investigated the post-buckling strength of shell structures, such as Mandal and Callandine (2000) Zhu, Mandal and Callandine (2002) and Fakhim, Showkati and Abedi (2009) have noted an interesting phenomenon. Whether loaded axially or by pressure, it has been observed that at a certain point during loading, shell structures exhibit large deflections with small corresponding changes in load - referred to in these studies as the post-buckling plateau. Mandal and Callandine (2000) noted that during the formation of dimples, in the region of the dimple formed, the structure is no longer able to support axial loading due to an inward tensile force experienced at the centre of the dimple. The formation of such dimples may therefore be an explanation for the post-buckling plateau, along with plastic deformation. While testing cylindrical shells, Fakhim, Showkati and Abedi (2009) noted that all specimens exhibited load resistance in the post-buckling regime. Therefore, during this current study, it is expected that a post-buckling plateau will be encountered, and that the structures will still exhibit some structural integrity following buckling. 2.6 Concluding remarks Some of the methods of analysis available for shell structures have been illustrated in this chapter. Through review of available literature, it has been concluded that some studies have been conducted investigating the post-buckling behaviour of shell structures while most stud- ies have been concerned with events leading up to buckling. It appears as though the causes of the observed post-buckling behaviour are not entirely known; but may be accurately pre- dicted taking imperfections into account and a careful use of constitutive laws. Additionally, all computational studies have been conducted with the use of a shell element in one form or another, thus simplifying analysis of the structures. It appears that a three dimensional continuum post-buckling plastic buckling analysis has not been conducted to date. 12
  • 25. Chapter 3 Continuum mechanics theory 3.1 Introduction During engineering analysis, it is common to analyse bodies which are far larger than the matter from which the body is composed. Therefore, it is valid to assume that the body is continuous, meaning that elements contained within the body share equal properties with the entire body. This implies that if the body is analysed as a whole, or an element within the body is analysed under identical loading and boundary conditions, both analyses will yield identical results; based on the assumption that the underlying molecular structure of the material analysed is generalised by the overlying material properties. This assumption that a body is made up of elements of matter which behave identically to the entire body gives way to the definition of a continuum. Following this, continuum mechanics is concerned with the analysis of such bodies, making use of discretisation techniques and the tools of algebra and calculus of vectors and tensors. This chapter begins with a description of general theory to be taken into account when mod- elling such bodies. Subsequently, relevant non-linear equations of continuum mechanics are presented, and their relation to stress and strain illustrated to finally present the methodol- ogy with respect to continuum mechanics followed in order to implement the elasto-plastic constitutive law. 3.2 General considerations 3.2.1 Elastic and plastic behaviour Consider a ductile, metallic material loaded under simple tension by a force F as illustrated in Figure 3.1 with a length l and area A. Stress and strain within the material may be defined respectively as σtrue = F A and (3.1) true = l l0 dl l (3.2) for small changes in l. Initially, as σ is increased, small changes in l occur resulting in a small value of . During this initial stage, it is assumed that if unloading were to occur, the material 13
  • 26. will deform back to the original length l0. Such a situation describes elastic behaviour and gives way to the principle of elasticity, stating that deformations are reversible and occur at small strains. When in the elastic regime of deformation, the relation between σ and is of a linear manner and related by the scalar Young’s Modulus E. However, if further loading were to occur resulting in larger strains, the relation between σ and becomes non-linear and the elastic relation assumed is no longer valid. Figure 3.1. Stress versus strain of a ductile material (Knutsen, 2012) Assume the material under consideration is further deformed. As l increases, deformation is resisted by internal forces within the crystal lattice constituting the underlying material structure of the metal. However, a point is reached where some metallic bonds within the lattice are no longer able to resist further loading. At this point, these bonds are broken causing movement within the lattice. If unloading were to occur at this point, the structure of the crystal lattice would not return to the original configuration. Such movement within the lattice is termed dislocation, occurring along a so-called slip plane within the lattice. Following dislocation, the material is said to have yielded and the relation between stress and strain has changed, and further increase in l is termed plastic deformation. (Hibbeler, 2011) Disque (1971) explains that there is a small increase in strength during plastic deformation due to strain hardening, as illustrated in Figure 3.1. Following yielding, a process of partly irreversible deformation takes place as a result of further formation and movement of dislo- cations. Due to dislocations being formed, and slip of the crystal structure along dislocation planes caused by shear stress placed on the crystal lattice, deformation of this nature is partly irreversible. When modelling plastic deformation due to large strains, it is no longer permissible to make use of the theory of elasticity due to non-linear behaviour, nor is it acceptable to make use of 14
  • 27. theory derived for the use of small strains as detailed for computation of σ and . Therefore, the use of finite strain theory is required. 3.2.2 Balance law of continuum mechanics Preceding the descriptions of methods used to model finite strain behaviour, a general descrip- tion of the fundamental laws underlying such methods is required. When subjected to any form of loading (such as force or temperature), the behaviour of the body may be described by the following laws fundamental to continuum mechanics (Mase, 1999): ˆ The law of Conservation of Mass ensures that there is no mass loss or gain when the body deforms; ˆ The law of Linear and Angular Momentum Conservation which is related to New- ton’s Second Law which defines the rate of change of linear and angular momentum over time, resulting in force and momentum; and ˆ The law of Energy Conservation which is related to the Principle law of Energy which states that energy can neither be created nor destroyed but can only change forms. 3.3 Kinematics Consider a body B, which is to be analysed during deformation. During analysis, two states or configurations are commonly referred to: the reference (undeformed) configuration and the current (deformed) configuration. The state of the material in these two configurations is illustrated overleaf in Figure 3.2. In order to characterise the deformation of B, a second order tensor referred to as the de- formation gradient F is utilised. F characterises the local deformation of point x contained within B, and thus the change in position of a particle with respect to time and may be expressed as F = F i j ei ⊗ Ij = ∂xi ∂Xj ⊗ Ij (3.3) where dX and dx represent material line elements in the reference and current configurations respectively, and ⊗ represents the dyadic product of terms. Defining the change in position of a particle is important, as it has been assumed that the body being analysed is constituted of many infinitesimally small particles. Therefore, by characterising the deformation of all discrete particles within B, it is possible to characterise the deformation of B itself (Mase, 1999). As discussed in Section 3.2, deformation may be of an elastic or plastic nature. During defor- mation of B, before the yield stress of the material is reached, deformation is purely elastic, and is described by F e. However, following yielding of the material, deformation consists of elastic and plastic components, with the plastic component being denoted F p (Haupt, 2000). 15
  • 28. Figure 3.2. Deformation of a solid body (Sanpaz, 2011) In such a situation, the final deformation gradient is expressed with the use of multiplicative decomposition as by Sansour and Kollmann (1997) as F = F eF p. (3.4) Next, the change in length of a line during deformation is to be accounted for. Two Cauchy- Green deformation tensors are available, namely the left and right, and are defined so by their relation to the left and right stretch tensors. The deformation tensors may be thought of as the squared length of a deformed fibre dx in a solid body (Bower, 2009). The total right Cauchy-Green deformation tensor C, and its inverse C−1 are expressed respectively as (Capaldi, 2012) C = F T F and (3.5) C−1 = F −1 F −T . (3.6) Similar to F , C is decomposed into its elastic and plastic components with Ce = F T e F e and (3.7) Cp = F T p F p. (3.8) as detailed by Sansour and Kollmann (1997). 16
  • 29. 3.4 Stress measures In Section 3.2, stress was defined with the use of Eq. (3.1). However, as mentioned, an alter- nate means of describing the stress within the material is required during this investigation. The stress tensor to be used is the second Piola-Kirchoff stress tensor, S. Capaldi (2012) explains S as a measure of the transformed force per unit area of an element in the reference configuration. S is expressed as S = C−1 Ξ (3.9) and is symmetric (Chandrasekharaiah & Debnath, 1994). Ξ describes the elastic material stress as is addressed later in Section 3.5.2. 3.5 The constitutive model With the requirement to make use of an alternate method to linear elasticity, and the kine- matics of B defined, the following sections detail the theory and implementation of the chosen constitutive model, following the methods described by Sansour and Kollmann (1997). 3.5.1 Background and motivation Capaldi (2012) explains that the model developed and used for linear elasticity reasonably describes the behaviour of metals at small values of strain. However, this model is no longer able to accurately predict the response of the material to prescribed boundary and loading conditions at larger values of strain due to non-linear behaviour. Ramos (1992) states that constitutive laws may be defined as a set of equations applied to a specific material which mathematically represent experimental and empirical observations concerning responses of particular materials to applied loads and boundary conditions, and may be applied during analysis of finite strain. Elasto-plasticity is a theory which attempts to predict the effect of plastic deformation on a material. During plastic behaviour, the metal is subject to a stress which causes a deformation which permanently influences the geometry of the metal, as a portion of the deformation is not recovered elastically. Therefore, it is useful to discern between the elastic (recoverable) and plastic (permanent) components of strain with the use of elasto-plastic strain decomposition. History of previous deformation is accounted for in order to account for unloading behaviour and strain hardening. Elasto-plastic constitutive laws are rate independent; the rate at which the material is loaded or unloaded is inconsequential. (Bower, 2009) Classical elasto-plasticity is based upon the assumption that when a material is loaded such that the stress experienced is significantly lower than the yield stress of the material, elastic behaviour will be experienced. However, when loaded such that stress within the material reaches a critical value (generally the yield stress denoted σy), plastic flow occurs leaving the material permanently deformed. (Haupt, 2000) Haupt (2000) explains that elasto-plastic models generally describe material behaviour realis- tically, and apply particularly to metals; characterised by a crystalline structure. As described 17
  • 30. in Section 3.2.1, plastic flow may be described by the formation and destruction of dislocations within the crystal lattice, for which complicated dynamic processes are responsible. However, such models do not deal with these processes, and rather make use of a phenomenological representation of the material under study. For this reason, only the relevant macroscopic effects are described with the use of an elasto-plastic model, even though the effects are fundamentally caused by dislocation behaviour. In the phenomenological representation, a critical quantitative load is defined - above which dislocations are predicted to occur and below which the effect of dislocations is ignored. To this end, elasto-plastic models are not able to model plastic deformation such as creep whereby low loads over sustained time periods are able to cause plastic deformation. The critical load used to discern the onset of dislocation, and hence plastic deformation, may be represented by means of the yield surface in the space of the relevant strain tensors of the constitutive law. (Haupt, 2000) Macroscopic effects of dislocations depend on the previous history of loading and unloading, and are dealt with by a variable Y which accounts for an increase in the yield stress following plastic deformation. Additionally, kinematic or isotropic hardening is taken into account and may be interpreted as the process-dependant obstacle impeding the motion of further dislo- cation. Following initial plastic deformation, further dislocation movement and formation is hindered due to an increased energy required for further dislocation formation and movement within the crystal lattice structure of the metal. (Haupt, 2000) Another fundamental aspect of the phenomenological representation is the concept of a yield function, which is a scalar-valued function depending on stresses, strains and internal vari- ables. The yield function determines the yield surface and consequently the nature of defor- mation of the material. The yield function defines whether deformation is of an elastic or plastic nature. Furthermore, an associated plastic flow rule is determined with the use of the yield function. (Haupt, 2000) With the use of the above assumptions and the yield surface, any state of stress which lies outside the yield surface is excluded, with the state of stress moving exclusively within the yield surface in the elastic state. However, during plastic deformation, it is required that the stress tensor used stay on the yield surface. In order to achieve this, a plastic multiplier is introduced. Therefore, hardening can be easily interpreted as the change in location, size or shape of the yield surface. (Haupt, 2000) An elasto-plastic constitutive model generally contains the following components listed below (de Souza Neto, Peric and Owen, 2008): ˆ elasto-plastic strain decomposition; ˆ an elastic law; ˆ a yield criterion, with the use of a yield function; ˆ a plastic flow rule, defining the evolution of plastic strain; and ˆ a hardening law, characterising the change of the yield limit. 18
  • 31. 3.5.2 Elastic constitutive model Due to the linear nature of elastic deformation and the non-linear nature of plastic deforma- tion, different methods were required to compute stress and strain of the material depending on the nature of deformation occurring at a particular point in time. The multiplicative decomposition of the deformation gradient tensor and the right Cauchy-Green deformation tensor allowed two models to be developed; each to capture the elastic and plastic effects of deformation respectively. The methodology presented in this section follows that of Sansour and Kollmann (1997), and was not developed by the author of this paper. Firstly, consider the elastic material stress tensor Ξ used in this study expressed as Ξ = 2ρref F T p Ce ∂ψ (Ce, Z) ∂Ce F −T p . (3.10) In the above expression, for a unified inelastic constitutive model, a phenomenological internal variable Z is taken into account (corresponding to the strength increase due to plastic hard- ening) and the existence of a stored free energy function ψ is assumed such that ψ (Ce, Z). ρref is the density of the material before deformation has taken place i.e. in the reference configuration. When concerned with the elastic constitutive model, ψ is decomposed into two parts via additive decomposition; the first depending on Ce and the second on Z: ψ = ψe (Ce) + ψZ (Z) (3.11) Following the definition of Ce in Eq. (3.7), the logarithmic measure of strain α may be introduced: α = ln Ce (3.12) Making use of the assumption that an elastically isotropic material is being dealt with, the following holds true: Ce ∂ψe (Ce, Z) ∂Ce = ∂ψe (α) ∂α (3.13) Thus, making use of Eq. (3.11) and Eq. (3.13), Eq. (3.10) is expressed alternatively as Ξ = 2ρref F T p ∂ψe (α) ∂α F −1 p . (3.14) At this point, it is required that α be modified in order to simplify the computation of the material stress Ξ. Therefore, the modified logarithmic measure of strain α is introduced as α = F T p αF −1 p . (3.15) 19
  • 32. During computation, however, it is preferable to make us of an alternative definition of α in order to increase computational efficiency. Therefore, α is alternatively expressed as α = ln C−1 p C or (3.16) αT = ln CC−1 p . (3.17) In application, it is useful to make use of the logarithmic strain measure, as further strain is taken into account via an additive process, as opposed to a multiplicative process. The additive process is illustrated as follows: αt+1 = αt + ∆α (3.18) Following the introduction of α in Eq. (3.15), Eq. (3.14) is further simplified and expressed as Ξ = 2ρref ∂ψe (α) ∂α . (3.19) A linear constitutive model is chosen such that Ξ is finally expressed as Ξ = KtrαT 1 + µ devαT (3.20) where K and µ are elastic material parameters denoting the bulk modulus and shear modulus respectively. 3.5.3 Plastic constitutive model The inelastic constitutive model was used in order to capture the effects of plastic deformation. As this investigation was concerned with the post-buckling behaviour of shell structures, it was expected that plastic deformation of the metal tested would occur at larger, finite strains. Therefore, it was required that a method be employed allowing plastic deformation to be accounted for, making it possible to compute stress and strain due to plastic deformation. The methodology presented in this section follows that of Sansour and Kollmann (1997), and was not developed by the author of this paper. As a distinction was made between the elastic and plastic deformation measures, it was required that the type of deformation at each time step of the calculation be discerned. To this end, a second internal variable is required; namely Lp, referred to as the plastic rate. In order to derive the evolution equations for the internal variables Lp and Z an elastic region is considered which is confined by the yield function f and is assumed to depend on the thermodynamical quantities Ξ and Y such that the following relation holds: E := {(Ξ, Y ) : f (Ξ, Y ) 0} (3.21) 20
  • 33. The often used von Mises yield function is defined by Sansour et al. (2006) as f (λ, Lp) = ||devΞ|| − 2 3 (σy − Y ) (3.22) where ||devΞ|| denotes the Frobenius norm defined as tr (devΞ)2 , and Y is expressed as Y (λ) = −H Zn + 2 3 ∆tλ − (σ∞ − σy) 1 − exp −η Zn + 2 3 ∆tλ . (3.23) In Eq. (3.23) above, H is a plastic material parameter controlling linear plastic hardening which is a special case of a work hardening material as the yield surface expands uniformly, and may be thought of as the slope of the stress-strain curve in the plastic regime, analogous to E in the elastic regime. η is a second plastic material parameter which serves a similar purpose to H, with the addition of a non-linear hardening component. For a work hardening material the plastic multiplier λ is inversely related to the hardening parameter η. It should be noted in Equation (3.22) that Y is subtracted from the initial yield stress σy. This is due to Y (λ) being computed as a negative value. Initially, the term was expressed as (σy + Y ) in the C++ code implemented by the author as detailed an Appendix A and was incorrect causing computational issues during analysis. The expression as originally imple- mented is incorrect due to the implication that yield stress decreases during strain hardening. Therefore, Eq. (3.22), and subsequently Eqs. (4.11) and (4.12) were additionally corrected ac- cordingly in order to allow for an increase in yield stress during plastic deformation attributed to the processes within the crystal lattice of the metal impeding further dislocation. The yield function f may be viewed as a surface in the three-dimensional stress space which encompasses all possible stress states for which elastic deformation occurs. f is used to de- termine whether deformation is elastic or plastic in the following manner: for f < 0 elastic deformation occurs, for f = 0 and λ = 0 neutral loading occurs. Plastic deformation occurs in the case of f = 0 and λ > 0. For f > 0, the function is outside the yield surface; such a situation is not possible and the calculation step is re-estimated and the yield function recalculated. With the introduction of λ and f, the associated flow rule Lp and Z are expressed as Lp = λ ∂f ∂Ξ = λ devΞT ||devΞ|| = λνT and (3.24) Z = λ ∂f ∂Y = 2 3 λ (3.25) where ν is termed the yield-locus normal, and may be thought of as the normal vector to the yield surface. 21
  • 34. 3.6 Concluding remarks In this section, the kinematics required in order to describe the deformation of a body were presented, along with general considerations to be taken into account, such as elastic and plas- tic material behaviour. Furthermore, the continuum mechanics of the elasto-plastic material law was illustrated. However, it is apparent that making use of the theory presented in this chapter to analyse shell structures analytically would not be practically possible. Therefore, it was required that computational methods be employed for efficient and accurate analysis. 22
  • 35. Chapter 4 Computational analysis principles 4.1 Introduction In Chapter 3, the continuum mechanics principles required in order to implement the elasto- plastic material law were addressed. It was concluded that the use of computational techniques were required in order to apply the law to shell structures. Therefore, this chapter begins with a description of relevant considerations to be made when conducting computational analysis of such structures. Subsequently, computational modelling techniques which were employed are discussed, with methods of analysis being illustrated. Furthermore, detailed computational methodology with respect to the constitutive law is presented, which follows the work of Sansour and Kollmann (1997). Additionally, techniques used to set-up and solve for the global equation system are addressed. 4.2 General analysis considerations Various types of analysis are available, depending on the problem at hand. Therefore, distinc- tion is made between buckling and post-buckling analysis and linear and non-linear analysis, in order to motivate the choices of each technique for use is this study. Additionally, general theory of dynamic behaviour and analysis is addressed. 4.2.1 Buckling analysis Members which are considered long or slender often exhibit a form of failure referred to as buckling when loaded axially under compression. Global buckling failure is observed when slender members subject to axial compression deflect laterally or side-sway, changing from one configuration to another over a short period of time, referred to as bifurcation. This concept may be further applied to members such as cylinders, where loading causes a sudden increase in strain within a local area of the cylinder, while stress remains constant or decreases. While the cylinder has not deflected laterally or side-swayed, indentations within the cylinder walls are said to have buckled locally. In both conditions, the load at which this initial sudden deformation occurs is termed the critical buckling load, and is the maximum load which may be supported when the structure is on the verge of buckling. During a critical buckling load analysis, experiments are conducted and theory developed in order to predict the load at which a structure will begin to buckle. During such an analysis, the primary concerns are factors which influence initial buckling load. (Hibbeler, 2011) 23
  • 36. In contrast to initial buckling, a post-buckling analysis may be conducted. During such an analysis, aspects following initial buckling of a structure are investigated. Typical aspects which are investigated are structural integrity with increased loading and structural strength in the post-buckling regime. In the analysis of shell structures, other post-buckling aspects of interest are those as investigated by Mandal and Callandine (2000) Zhu, Mandal and Callandine (2002) and Fakhim, Showkati and Abedi (2009) such as the formation of the post- buckling plateau and dimples. In order to observe such post-buckling behaviour, structures are further loaded after the onset of initial buckling. During post-buckling deformation, large deformations are often experienced, and specific analysis is required in order to account for corresponding non-linear behaviour. 4.2.2 Linear and non-linear analysis As explained by Bower (2009), linear analysis of models is only suited for analyses in which specimens exhibit purely reversible deformation with small displacements; if a specimen de- formed during some loading condition, upon removal of the load, returns back to its original shape and state of stress. Furthermore, linear analysis is only appropriate to specimens with which strain does not depend on the rate of loading applied, and history of loading and de- formation. Additionally, in the linear regime of material behaviour, stress and strain of the material are related in a linear manner. In the case of metallic materials, this relationship only holds true if the strain is small. As is the case with buckling analyses as described in Section 4.2.1, a non-linear analysis is required (Cortivo et al, 2009). This is due to a non-linear stress-strain relationship, and large deformations of the material being investigated. As explained in Chapter 3, at strains within the strain hardening and necking regions or deformation of a material, the relationship between stress and strain is no longer of a linear relationship. As there is an increase in strain with a lower increase in stress, the stiffness of the material has decreased, decreasing the Young’s modulus of elasticity E. It is at this point that stress is now considered a function of strain and no longer linearly related to E. When there is a decrease in stiffness of a material altering the relationship between stress and strain, behaviour becomes non-linear and a non-linear analysis of the material is required. (Solidworks, 2008) Various types of non-linearity during analysis may occur. The first type of non-linearity is as previously discussed, where the material being deformed exhibits varying properties (stiffness) under altered loading. A second type of non-linearity which occurs at large deformations is geometrically non-linear behaviour. Global non-linear geometry may be observed in the case of a slender cylindrical shell under axial compression. As deformation and load increase, the shell begins to deform and exhibits bending about the horizontal axis. Initially, it was assumed that the axial load was only a cause of compressive forces. However, due to imperfections in the material and eccentricity of loading, the shell begins to deform under flexure causing bending of the element. As bending occurs, the loading is no longer purely axial, thus altering the loading conditions. At extreme deformations, the load may no longer be axial at all. 24
  • 37. 4.2.3 Dynamic analysis In order to conduct a post-buckling analysis, it is often required that dynamics of the structure be accounted for. This is the case as buckling is often sudden and results in an increased velocity of portions undergoing buckling behaviour. In order to account for dynamic effects, material density ρ is required to be specified, as inertia and thus acceleration are affected. Additionally, a damping parameter c is required to specify the rate at which the material returns to a point of stable equilibrium. Furthermore, the choice of time increment per loading step is required in spite of the use of a rate-independent material law, as large loading rates may nullify the effects of dynamics being accounted for, and small loading rates may cause computational difficulties and unrealistic material behaviour. 4.3 Modelling methods As stated by Rock, Zhang and Wilkinson (2008), numerical modelling is a tool that is often employed and extremely useful for engineering design and analysis. Numerical modelling may be broadly defined as solving physical problems making use of appropriate simplification of reality. In order to simplify problems in engineering analysis, it is required that key variables be identified or introduced and relations between such variables established. Once variables and relations have been established, it is common to construct a set of differential equations to be solved allowing engineers to predict the outcomes of complex models and scenarios without conducting physical testing. 4.3.1 Finite Element Method The continuous development of new structural materials leads to ever increasingly complex structural designs that require careful analysis. Although analytical techniques are important, the use of numerical methods to solve mathematical models of complex structures has become an essential tool in the design process. The Finite Element Method (FEM) has been the fundamental numerical procedure for the analysis of shells. (Bucalem and Bathe, 1997) During this study, shell structures are subjected to large deformations in the post-buckling and plastic regimes. According to Bower (2009), FEM is the most widely used and appropriate method available to account for large shape changes and non-linear material behaviour. FEM is a technique applied to computational analysis used to solve partial differential equations. An application of solving such equations is to predict the deformation and stress fields within solid bodies subjected to external forces such as concentrated and distributed loads. According to Roylance (2001), the finite element method is usually carried out making use of three steps: pre-processing, analysis and post-processing. During pre-processing, a model of the material to be analysed is created. The geometry of the model is discretised into so-called elements, which is commonly assisted by various computer software programs. In order to carry out a finite element analysis, a finite element mesh is required, as mentioned by Bower (2009). The finite element mesh is used to specify the geometry of the solid to be modelled, 25
  • 38. and is additionally used to describe the displacement field within the solid. With the use of later described meshfree methods, a mesh is also required in order to discretise the geometry of the structure to be analysed. During the pre-processing phase, a mesh is created and is defined by a set of nodes together with a set of finite elements, which as a whole represent the model to be analysed. A node is described by Bower (2009) as a discrete point within the solid body. The boundaries of elements within the body are represented by such nodes. Furthermore, support and loading conditions are applied to nodes. With the input geometry and boundary conditions, a dataset to be analysed is created. Following pre-processing, analysis of the model is conducted. During analysis, the dataset created during pre-processing is used as the input into analysis. With this input, a set of linear or non-linear algebraic equations is constructed and solved based on the fundamental equations of physics governing the problem. The general system of equations is defined as Ku = F (4.1) where u is the displacement matrix of the nodes, and F is the force matrix applied at the nodes or upon elements bound by nodes in the case of line, traction, pressure and volume loads. The K matrix is dependent on the type of problem to be analysed. During analysis, it is possible to either solve for a matrix of unknown forces, or a matrix of unknown displacements. (Roylance, 2001) Once the system of equations has been solved using numerical methods, the results are inter- preted during the post-processing phase. Generally, results are visualised using post-processing software, which creates, for example, a coloured contour image of the initial or deformed body illustrating stress, strain, displacement, rotation and temperature. (Roylance, 2001) 4.3.2 Gauss quadrature During computational analysis of continuum mechanics problems, if a function to be solved is not simple or given in a tabulated form, analytical analysis of its corresponding integral can be difficult or impossible. In such cases, the use of numerical methods is required, and subsequent formulas used for numerical integration are named quadratures. If functions used to approxi- mate such integrals are evaluated using unequally spaced base points, the Gauss quadrature formulas are obtained. In such formulas, integration points are strategically selected in order to obtain the best possible accuracy. (Shabana, 2012) Shabana (2012) states that in Gauss quadrature, an integral to be solved is evaluated by approximation of a function f (x) by a polynomial Pn (x). When using such an approximation, an error may be realised. Therefore, base points are selected in order to make the integral of the error function equal to zero. 26
  • 39. 4.3.3 Element-free Galerkin Method As previously mentioned in Section 4.3.1, FEM is a tool used to calculate approximate so- lutions to a set of differential equations in order to analyse the behaviour of materials. As described by Belytschko, Lu and Gu (1994), the Element-free Galerkin Method (EFGM) is a method used to solve such equations by converting the governing set of equations to the weak form, known as the principle of virtual work making use of moving least-squares interpolants. The EFGM was developed in order to reduce computational time required for mesh generation. When utilising the EFGM, Belytschko, Lu and Gu (1994) explain that models to be analysed are free of any mesh, and only an array of nodes in the domain of consideration is required. Advantages of the EFGM which aid in the post-buckling analysis of shell structures are that the EFGM does not seem to exhibit volumetric and shear locking and the rate of convergence is quicker than that of a conventional finite element (Belytschko, Lu and Gu, 1994). Therefore, the use of the EFGM is essential in this study in order to conduct analysis of three dimensional shells avoiding volumetric and shear locking. 4.4 Implementational aspects Certain aspects of the constitutive law presented in Chapter 3 require specific computational techniques and modification in order to be used in computational analysis. Therefore, the fol- lowing computational issues were dealt with following the methods described by Sansour and Kollmann (1997). The corresponding C++ code used in order to implement the constitutive law within the framework of SESKA may be viewed in Appendix A. 4.4.1 Time integration and local iteration During computation, the state of deformation was computed over an interval of time. The time interval was incremented with a number of calculation steps, such that the sum of time of all calculation steps was equal to the total calculation time. Therefore, it was required that various parameters be updated during each respective time step, taking into account the history of variables in the previous calculation step. The incorporation of history into calculation steps was crucial in order to track plastic history variables in the case of strain hardening. For discrete times tn and tn+1 over an increment ∆t, the following updating algorithms are applied to the plastic deformation gradient F p and the strain measure CC−1 p : F p|n+1 = F p|n exp (∆tLp) and (4.2) CC−1 p |n+1 = C|n+1 exp (−∆tLp) C−1 p |n exp −∆tLT p . (4.3) 27
  • 40. Furthermore, it is required that the increment of the plastic multiplier ∆λ be computed; which is determined with the use of the von Mises yield function - described by Eq. (3.22). It may be noted that Lp and λ depend on Ξ. However, with CC−1 p depending also on Lp, and Ξ depending on CC−1 p in the computation of α, an alternative method is required to compute the terms in consideration. Therefore, the stress and plastic deformation state variables are computed via the so-called corrector-predictor method making use of the Newton-Raphson method. With the use of the plastic corrector-predictor method, a separate sequence of plastic itera- tion steps j is applied within each iteration step i of the time step ∆t, during which global equilibrium of the system is sought. Therefore, within each iteration conducted to solve the global equation system, a second Newton-Raphson iteration is conducted. The plastic-corrector method is carried out over two steps during each Newton-Raphson iteration. During the first step, referred to as the trial step with j = 0, the elastic constitutive model is used in order to determine a trial stress Ξtrial , with Ce|i = Ce|i n+1C−1 p |n. During this step, C is kept constant, while λ is updated via λ|j+1 = λ|j + ∆λ. (4.4) Updating λ as above has the result of C−1 p being updated as C−1 p |n+1 = exp (−∆tLp) C−1 p |n exp −∆tLT p . (4.5) Additionally, the state variable Z is updated as Z|n+1 = Z|n + 2 3 ∆tλ|j+1. (4.6) In order to compute the increment of the plastic multiplier ∆λ, firstly an expression for Ξ is required. Due to the modified logarithmic strain measure being used, Eq. (4.3) is modified in order to compute the expression of the logarithmic strain as ln CC−1 p |n+1 = ln C|n+1 exp (−∆tLp) C−1 p |n exp −∆tLT p = ln C|n+1C−1 p |n exp −2∆tLT p or αT |n+1 = αtrial T − 2∆tLT p . (4.7) Therefore, with the substitution of Eq. (4.7) into Eq. (3.20), the elastic material stress Ξ is updated as Ξ|n+1 = Ξtrial − 2µ∆tλ devΞtrial ||devΞtrial || . (4.8) Secondly, Yn+1 is substituted into the yield function resulting in the expression used to com- pute f: f (λ, Lp) = ||devΞtrial || − 2µ∆tλ − 2 3 (σy − Y |n+1) = 0 (4.9) 28
  • 41. With the use of Eq. (4.9), a non-linear equation for λ is obtained. However, it is still required that the increment ∆λ be computed. In order to compute this increment, Eq. (4.9) is linearised for each plastic step j where: ∆λ = −f ∂λ ∂f (4.10) In order to implement Eq. (4.10), it is required that the yield function tangent ∂f ∂λ be computed as ∂f ∂λ = −2µ∆t + 2 3 ∂Y |n+1 ∂λ (4.11) where ∂Y |n+1 ∂λ is referred to as the Y tangent. Finally, the Y tangent is computed as ∂Y |n+1 ∂λ = − 2 3 ∆t H + η (σ∞ − σy) exp −η Zn + 2 3 ∆tλ . (4.12) 4.4.2 Stress and algorithmic tangent operator In order to compute the second Piola-Kirchoff stress tensor S over each iteration, the following expression is used: S = C−1 Ξtrial − 2µ∆tλν (4.13) Additionally, the algorithmic tangent operator H is obtained by linearising S with respect to C: H = ∂S ∂C = ∂C−1 ∂C Ξtrial − 2µ∆tλν + C−1 ∂Ξtrial ∂C − 2µ∆t ∂λ ∂Ξtrial ∂Ξtrial ∂C C−1 ν − 2µ∆tλC−1 ∂ν ∂Ξtrial ∂Ξtrial ∂C (4.14) However, it is further required that the derivatives in Eq. (4.14) be evaluated in order to make use of the expression. Therefore, the final equation conveniently expressed in index notation which is used in order to compute the algorithmic tangent operator is as follows: ∂Sij ∂Crs = −C−1 ir C−1 sk Ξtrial kj − 2µ∆tλνkj + C−1 ik K − 1 3 µ δkjC−1 rs + µδksC−1 rj − 2µ∆t νba 2µ∆t 2 3 ∂Y |n+1 ∂λ K − 1 3 µ δabC−1 rs + µδasC−1 rb C−1 ik νkj − 2µ∆tC−1 ik 1 ||devΞtrial|| δkaδjb − 1 3 δkjδab − νkjνba K − 1 3 µ δabC−1 rs + µδasC−1 rb where δ represents the Kronecker delta. 29
  • 42. 4.5 Variational formulation Following the computation of S and subsequently H, a set of equation systems may finally be set up and solved. In the analysis of non-linear initial value boundary problems, a coupled system of partial differential equations has to be considered. During the analysis of shell structures, such relations are governed by equilibrium equations, and the complex geometry to which they are applied requires computational approximation methods to be employed. The previously mentioned Element Free Galerkin Method (the so-called meshfree method) was therefore utilised and is based on a variational formulation of the governing equations, also known as the principle of virtual work or the weak form. (Sack, 2013) The variational formulation of continuum mechanics states that the external potential, Wext, is equal to the internal potential, Wint. The external potential corresponds to the work done by the external forces such as traction forces acting on B. Taking the above into consideration, the principle of virtual work is expressed in the following manner: δΨ = Wint − Wext = 0 (4.15) The relevance of the computation of S in apparent with the definition of Wint as Wint = B S : δEdV = B 1 2 S : δCdV (4.16) where E is the Green strain tensor and : signifies the multiplication of two tensors. The relevance in computation of F is apparent with the definition of E as E = 1 2 F T F − 1 = 1 2 (C − 1) . (4.17) The final variational formulation is defined as δΨ = B 1 2 S : δCdV − Wext = 0 (4.18) and is governed by the set of binding equilibrium equations. The linearisation of the variational formulation making use of a Taylor expansion is expressed as δΨ = v 1 2 S : δCdV + v 1 2 H∆C : δC + 1 2 S : ∆δCdV − Wext = 0. (4.19) 30
  • 43. 4.6 Assembly of equation system At the assembly stage, the objective is to obtain a discrete set of equations that can be subsequently solved for the unknown field, such as displacement in the case of this study. With Eq. (4.1) in mind, the equation system to be solved takes the form of K∆u − [fext − fint] = 0 (4.20) where K is the stiffness or tangent matrix, ∆u is the unknown displacement field increment, and fext and fint are the external and internal loading and reaction vectors respectively. However, for non-linear problems such as in this study, no equality is present in Eq. (4.20), and hence an iterative procedure is required to find the unknown displacement field. To this end, the Newton-Raphson method is employed to finally solve for the unknown displacement field. 4.7 Concluding remarks This chapter has introduced general considerations to be taken into account during the com- putational analysis of structures with specific relevance to shell structures. Methods of com- putational modelling were dealt with, and subsequently the specific methods employed in order to computationally make use of the constitutive law implemented such as the use of the plastic corrector-predictor method. Additionally, the variational formulation and assem- bly of the final equation system were addressed in order to illustrate the method used to finally set-up and solve for the global equation system of structures analysed. Following the implementational aspects of this study, attention is now paid to the analysis of simplified structures. 31
  • 44. Chapter 5 Numerical examples 5.1 Introduction With the computational analysis tools described in Chapters 3 and 4, analysis was possible of structures making use of the implemented constitutive law. Various types of simplified structures were modelled with increasing complexity in order to meet the modelling objec- tives of this study. This chapter contains the problem set-up for structures analysed, with corresponding results and discussions being presented. Initially, a simple cantilever beam was modelled for benchmarking purposes, in order to de- termine whether performance of the implemented material law was similar to that of the previously implemented stretch based material law. Subsequently, two plates were modelled in order to make comparisons against the verified results of Sansour and Wagner (2003) who analysed identical structures making use of a visco-plastic material law. Furthermore, two cylindrical shell structures were modelled with varying loading and boundary conditions. While results of these analyses were also benchmarked against verified results, these models served an additional purpose of providing a preliminary insight into the post-buckling regime of shell structures with the use of an elasto-plastic material law, three dimensional continuum mechanics and meshfree methods. 5.2 Cantilever beam 5.2.1 Problem set-up The simple cantilever beam modelled was defined geometrically by a length L = 10 m and a square cross-section of h = d = 1 m. One end surface of the beam was prescribed displacement boundary conditions such that no displacement in all three degrees of freedom occurred, emulating a built-in boundary constraint. The initial configuration of the cantilever beam is illustrated overleaf in Figure 5.1. The free end of the beam was assigned a traction load in the vertical direction, being controlled by Dirichlet loading conditions; such that the end of the beam was displaced by a user defined increment during each loading step. With the use of such loading, following the displacement 32
  • 45. Figure 5.1. Cantilever beam - undeformed configuration (Essack, 2014) Table 5.1. Cantilever beam elastic and plastic material parameters Parameter Value (kN/mm2 ) µ 88.0 K 238.3 σy 0.450 σ∞ 0.715 H 0.129 η 16.29 Table 5.2. Cantilever beam dynamic material parameters Parameter Value ρ 4.5 x 10−9 kg/mm3 c 10−6 N · s/mm increment being applied, the force required for displacement is computed. Material parameters used were those for titanium, as detailed in Tables 5.1 and 5.2. Two types of analyses were conducted on the cantilever beam; namely static and dynamic. The static analysis was conducted in order to make comparisons with the previously mentioned stretch based material law, and therefore two simulations on the beam were run, making use of the respective constitutive laws. Subsequently, a dynamic analysis was conducted in order to determine the initial expected behaviour with the use of an elasto-plastic material law taking dynamic effects into consideration. 5.2.2 Static analysis During analysis, a load deformation graph was generated for the tip of the beam undergoing displacement. It may be observed overleaf in Figure 5.2 that both material laws predict very similar behaviour of the beam in the elastic region and early stages of plastic deformation. Elastic deformation is apparent from 0 m to 0.15 m displacement preceding a change in gradient of the curve. Following a displacement of 0.15 m, yielding of the material is evident 33
  • 46. with a change in the slope of the load deformation graph. Following initial yielding, a slight increase in load during further deformation is required, due to to the effects of strain hardening which have been incorporated into both material laws. 0 1 2 3 4 5 6 7 8 9 deformation [m] 0 50 100 150 200 250 loadfactor[kN] Cauchy-Green Stretch Figure 5.2. Cantilever beam - static load deformation graph However, it may be noted that the stretch based law extends only to 2.8 m of displacement; due to analysis ceasing to converge. Additionally, analysis time of this material law was significantly slower than that of the material law implemented in this study. The discrepancy in analysis time may be attributed to the theory of the material laws; as the stretch based law takes into account six degrees of freedom with the inclusion of rotation about all three axes. Following 2.8 m of displacement, the implemented material law continues calculation as fur- ther strain hardening occurs over a significant period of displacement in the plastic region; indicated by the increase in the load factor during further displacement. Within this region, effective plastic strain of the material continues to increase as stress further increases. A steep increase in the load factor following 6 m of deflection is attributed to tension forces within the beam. A second period of yielding begins at approximately 8 m where the material begins to yield under tensile forces as opposed to flexure. The deformed configuration at 4 m of displacement is illustrated in Figure 5.3 with a contour fill depicting effective plastic strain in Figure 5.4. 34
  • 47. Figure 5.3. Cantilever beam - deformed configuration displacement contour plot Figure 5.4. Cantilever beam - deformed configuration plastic strain contour plot 5.2.3 Dynamic analysis Following static analysis, a dynamic analysis was conducted in order to determine feasible dynamic material parameters for further testing, and secondly to gain an understanding of the behaviour of the elasto-plastic material law under dynamic loading. The time deformation graph produced is illustrated by Figure 5.5 overleaf. Figure 5.5 depicts displacement up to -0.23 m. Following the prescribed displacement being reached, the load applied was maintained while calculation steps continued. The change in deformation occurring over subsequent time steps is attributed to oscillation of the beam, and is observed to decrease over the remaining time period of analysis indicating that material parameters chosen were feasible with no under damping or over damping observed. 35
  • 48. 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 time [seconds] −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 deformation[m] Dynamic Cauchy-Green Figure 5.5. Cantilever beam - time deformation graph during dynamic analysis 5.3 Simply supported plates 5.3.1 Problem set-up The first thin structures modelled were those of two simply supported plates; serving the purpose of further calibration against verified results. Sansour and Wagner (2003) previously modelled two plates with the use of a visco-plastic material law, and specialised shell elements. Therefore, modelling of the two plates served the purpose of benchmarking the implemented law against verified results produced with the use of an alternate material law and analysis techniques. Additionally, a preliminary insight was to be gained into the analysis of thin walled structures with the use of the newly implemented material law. Geometrically, the two square plates modelled were identical, with lengths and breadths of 500 mm and a thickness of 1 mm. The first plate modelled (case I) was fixed in all three directions of displacement along the edges, while edges of the second plate (case II) were fixed against displacement in the vertical direction only, allowing inward movement during deformation as detailed overleaf by Figures 5.6 and 5.7 respectively. A uniformly distributed dead load was applied over the plate surfaces, with the centre particle of the plates being incremented at 0.1 mm per loading step. Material parameters were chosen in accordance with Sansour and Wagner (2003) described in Table 5.3 overleaf, for that of titanium facilitating comparison of results, and were adjusted to compensate for differences in elastic material law equations. The non-linear hardening parameter η was not used in order to allow for less stiff behaviour of the plates. 36
  • 49. As buckling during deformation is unlikely to occur in such structures, dynamics of the struc- tures was not accounted for, resulting in relatively simple initial models in order to illustrate differences between the elastic and plastic regimes. Due to symmetry conditions, one quarter of each plate was modelled and discretised into 10 x 10 elements. The identical undeformed configuration of both plates is illustrated overleaf in Figure 5.8. Figure 5.6. Simply supported plates - Case I (Sansour and Wagner, 2003) Figure 5.7. Simply supported plates - Case II (Sansour and Wagner, 2003) Table 5.3. Simply supported plates - elastic and plastic material parameters Parameter Value (kN/mm2 ) µ 88.0 K 238.3 σy 0.450 σ∞ 0.715 H 0.129 η 0 37