A thorough review of the theoretical background of the invariant based approach to composite designs. The trace material property was explained, along with several topics from linear elasticity theory. The MATLAB implementation of a tool capable of determining the laminate stiffness properties from the lamina properties and stacking sequence was also presented. For access to the tool please contact the author: filipegiesteira@outlook.com
Assignment developed in the scope of the Composite Systems course, lecture at FEUP (Faculdade de Engenharia da Universidade do Porto).
1. Study of Invariant-based Method for
Accelerating Aerospace Composite Test Certification
Filipe Amorim Gonรงalves Giesteira
Supervisors:
Albertino Josรฉ Castanho Arteiro
Antรณnio Torres Marques
Composite Systems (SC) โ EM0115
Integrated Master in Mechanical Engineering
April, 2019
3. iii
Abstract
The present report was developed within the scope of the Composite Systems course, lectured at the
Faculty of Engineering of the University of Porto (FEUP).
In the first chapters a solid and concise overview of the theory of elasticity is made. Its correct
understanding is essential to grasp the theory and the rationale behind trace method. One of the main
goals of this report, is to fill some theoretical gap often observed in the technical literature. Thus, several
extensive and theoretical demonstration are detailed throughout the first chapters.
Based on the basic fundamental concepts of 2D elasticity, the lamina governing equations were
compiled and summarized in a single chapter, coherent and of simple notation. Which allows the
experienced reader to jump in the demonstrations and use this report also as search tool.
Classical Laminated Plate Theory (CLPT) is assumed perfectly assimilated in most of the research
material regarding Trace Invariant-approach. Keeping that in mind, it was dedicated a specific
introductory chapter detailing the laminate governing equations.
Carrying on the theoretical character of this report, it is first presented the complete background of Trace
and their mathematical and physical applications, from the Ply to the Laminate concept.
Finally, the Matlabยฎ code implementing the automatic computation process of the in-plane stiffness
properties of a given laminate is presented. The tool only requires the Ply longitudinal modulus as the
unique input.
Keywords
2D Elasticity, Generalized Hookโs Law, Plane Elasticity, Plane Stress, Plane Strain, Ply, Laminate,
Laminae, Classical Laminated Plate Theory, Master Ply Concept, Trace, Invariant-base Method,
Aerospace, Aeronautics, Composite Testing, Composite Design
5. v
Contents
Contents .............................................................................................................................................v
List of Acronyms ............................................................................................................................... vii
List of Figures .................................................................................................................................... ix
List of Tables ..................................................................................................................................... xi
1 Introduction..................................................................................................................................13
1.1 Context of the Report ....................................................................................................................13
1.2 Report Structure............................................................................................................................13
1.3 Basic Mathematical Nomenclature.................................................................................................14
2 Trace - Aerospace Industry ..........................................................................................................15
2.1 Composites in Aerospace Industry.................................................................................................15
2.2 Composite Certification and Testing in Aerospace Industry.............................................................16
3 Theory of Linear Elasticity for Continuum Medium ........................................................................17
3.1 Introduction...................................................................................................................................17
3.2 Stress Tensor ...............................................................................................................................17
3.3 Strain Tensor................................................................................................................................22
3.4 Generalized Hookeโs Law..............................................................................................................25
3.4.1 Theoretical Background...............................................................................................25
3.5 Transformation Matrix ...................................................................................................................29
3.5.1 General Definition........................................................................................................29
3.5.2 Modified Transformation Matrix....................................................................................30
3.6 2D Linear Elasticity .......................................................................................................................46
3.6.1 Types of Plane Linear Elastic Problems........................................................................46
3.6.2 Dynamic Equilibrium....................................................................................................47
3.6.3 2D Hookeโs Law โ Isotropic Material Behavior..............................................................48
3.6.4 Strain-Displacement Fields Relation.............................................................................52
3.6.5 Eliminating Stress and Strain in the z direction .............................................................52
3.7 Review of the Governing Equations for 2D Elasticity ......................................................................54
3.7.1 Displacement Field......................................................................................................54
3.7.2 Strain Field..................................................................................................................54
3.7.3 2D Hookeโs Law โ Isotropic Material Behavior..............................................................55
3.7.4 2D Hookeโs Law โ Orthotropic Material Behavior ..........................................................55
3.7.5 Total Stress-Strain Relation (Extra) ..............................................................................57
3.7.6 Dynamic Equilibrium....................................................................................................58
4 Orthotropic Lamina Constitutive Equations ...................................................................................59
4.1 Theory Background.......................................................................................................................59
4.2 Nomenclature Modification โ Composites Specifications ................................................................59
4.3 Equations Summary......................................................................................................................60
5 Classical Laminated Plate Theory ................................................................................................64
5.1 Plate Definition and Modeling ........................................................................................................64
5.2 Kirchhoff Assumptions...................................................................................................................65
5.3 Plate Kinematics and Governing Equations....................................................................................66
5.4 Kirchhoff vS Reissner Plate Theory................................................................................................69
5.5 Laminate Definition, Classification and Designation........................................................................71
5.6 Laminate Kinematics and Governing Equations .............................................................................75
5.6.1 Generalized Loads or Load Resultants.........................................................................75
5.6.2 Hygrothermal Behavior ................................................................................................79
5.6.3 Generalized Strains and Stress Field ...........................................................................79
6 Invariant-based Approach - The Master Ply Concept....................................................................81
6.1 Theoretical Background.................................................................................................................81
6. vi
6.2 Master Ply Concept.......................................................................................................................84
6.2.1 Engineering Elastic Parameters...................................................................................84
6.2.2 Plane Stress Stiffness Coefficients...............................................................................84
6.2.3 Trace-Normalized Factors ...........................................................................................86
6.2.4 Master Ply Stiffness Properties ....................................................................................92
6.3 Master Laminate Concept .............................................................................................................93
6.4 Trace Approach โ UD Coupons Testing and Ply Stiffness Matrix....................................................95
6.5 Trace Approach โ Laminate Testing and In-plane Stiffness Matrix................................................ 100
7 Conclusions and Future Work ....................................................................................................103
References.....................................................................................................................................105
7. vii
List of Acronyms
1D โ One Dimension/Dimensional
2D โ Two Dimension/Dimensional
3D โ Three Dimension/Dimensional
CAD โ Computer Aided Design
CAE โ Computer Aided Engineering
CFRP โ Carbon Fiber Reinforced Polymers
CLPT โ Classical Laminated Plate Theory
CSM โ Chopped Strand Mat
FEA โ Finite Element Analysis
FEM โ Finite Element Method
FEUP โ Faculty of Engineering of University of Porto
FRP โ Fiber Reinforced Polymers
GFRP โ Glass Fiber Reinforced Polymers
UD โ Unidirectional
List of Acronyms
9. ix
List of Figures
Figure 1- Illustration of the main goals aimed with this introductory chapter. ....................................15
Figure 2- a) Composite structure of A380 [9]. b) Composite materials used in Boeing 787 body [11].
.........................................................................................................................................................15
Figure 3- Definition of the nomenclature adopted for shear stresses acting on the differential volume
element [1]........................................................................................................................................18
Figure 4- a) Cauchy tetrahedron formed by slicing a parallelepiped along an arbitrary plane define by
the normal vector ๐ . b) Infinitesimal triangular portion of a generic 2D body...................................20
Figure 5- Nomenclature adopted for the shear stress definition, for the distortion of the differential
Cartesian element..............................................................................................................................24
Figure 6- a) Schematic representation of the algorithm used to codify Voigt notation in a second order
tensor. b) Illustration of the relation between Elastic and Hyperelastic materials for small strains
condition...........................................................................................................................................28
Figure 7- Illustration of the angles between the transformed ๐ฅโ-axis and the original cartesian coordinate
system. .............................................................................................................................................29
Figure 8- Illustration of the individual rotations of the Euler angles. Image adapted from [5]. ............29
Figure 9- Definition of the nomenclature used to define the coordinates transformation matrix. The ๐,
๐, and ๐, represent the rotation angle about the z, x and y axis respectively. .....................................30
Figure 10- a) Plane Stress schematic geometry. b) Plane Strain schematic geometry..........................46
Figure 11- Illustration of different fiber reinforcement architectures: a) Chopped Strand Mat (CSM) [4];
b) Woven fabric [10]; c) Knitted fabric [12]; and d) Ply stacking [13]................................................59
Figure 12- Schematic representation of the Orthotropic axes in a UD ply...........................................59
Figure 13- Schematic representation of the transformation of the Coordinate system. The blank square
indicates that the transformation relations are not bounded to a particular transformation direction โ, e.g.
off-axis โ principal axis or principal axis โ off-axis rotation. .............................................................60
Figure 14- Plate geometric definition along with the sign convention adopted for the displacements,
rotations, distributed and point loads and distributed and point momentums. Highlight of the middle
plane geometric reference [2]. ...........................................................................................................64
Figure 15- In-plane and out-of-plane displacement field in a thin plate [2].........................................66
Figure 16- Illustration of design capabilities using composite laminates, pointing out the two extreme
cases of mechanical behavior: a) Unidirectional laminate and b) Quasi-isotropic laminate [3]............71
Figure 17- Geometry and orientation of the fiber in a: a) Angle-ply and b) Cross-ply laminated panel
under transverse loading [8]. Two laminates with the exact same manufacturing orientation but with
different loading orientation. .............................................................................................................72
Figure 18- Representation of the two typical references used to define the beginning of the layup
direction, a) a callout line specifically introduced with this purpose; and b) the tool surface [6]. ........73
Figure 19- Schematic definition of the Clockwise warp direction, the sign convention adopted for the
(+) and (-) directions [6]....................................................................................................................73
Figure 20- Schematic illustration of the typical strain and stress fields shape, observed in composite
laminates when CLPT is applied. ......................................................................................................75
Figure 21- Schematic representation of the construction process of Table 6 and Table 7. ...................89
Figure 22- Schematic representation of the construction process of Table 1 and Table 2 from [7]. .....94
Figure 23- Schematic illustration of the experimental process use to determine the: a) longitudinal and
b) transverse elastic properties by Uniaxial Tensile Test of FRP coupons. .........................................95
10. x
Figure 24- Indirect measurement of Shear Modulus (๐บ12), elastic property of FRP, by Off-axis Tensile
Test. .................................................................................................................................................95
Figure 25- Schematic illustration of the +-45ยบ Tensile Shear Test method to determine the shear
modulus............................................................................................................................................96
11. xi
List of Tables
Table 1- Summary of the most important and distinct mathematical nomenclature used throughout the
report................................................................................................................................................14
Table 2- Summary of all simplifications made to the general stiffness matrix. ...................................27
Table 3- Examples of laminate stacking sequence notations and their description. .............................74
Table 4- Example of categories of laminates regarding the previous classifications............................74
Table 5- Summary of the mechanical quantities used to describe the mechanical behavior of thin plates.
.........................................................................................................................................................77
Table 6- Carbon Fiber/Epoxy composite coupons properties: engineering constants, plane stress
stiffness components and Trace [20]..................................................................................................87
Table 7- Carbon Fiber/Epoxy composite coupons trace and normalized properties: trace-normalized
engineering constants, and trace-normalized plane stress stiffness components [20]..........................88
Table 8- Additional statistics regarding the contribution of the plane stress stiffness coefficients for the
Trace. ...............................................................................................................................................90
Table 9- Basic statistics regarding the longitudinal stiffness coefficient for different orientations of
analysis.............................................................................................................................................91
Table 10- Master Ply mechanical and stiffness properties [20]...........................................................92
Table 11- Simple numerical exercise to illustrate the independence between the Trace-normalized
engineering and stiffness coefficients [20].........................................................................................92
13. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
13
1 Introduction
1.1 Context of the Report
This report was developed within the Composite System course, lectured in the Integrated
Master in Mechanical Engineering โ Specialty Structural Engineering and Machine Design, at Faculty
of Engineering of University of Porto (FEUP). The first (but not necessarily major) goal of this report
is divided in: (i) understand and produce a review type work, of the theory and applications of the master
ply concept; and (ii) develop an algorithm or script, based in MATLABยฎ programming language,
capable of computing the in-plane stiffness properties of a given laminate. However, the author was
slightly beyond this task and also sought to demonstrate the background behind some important concepts
of the Elasticity Theory and the Classical Laminated Plate Theory (CLPT).
1.2 Report Structure
The present report is divided in 8 main chapters, being the last two chapters dedicated to the
conclusions and future work, and literature references respectively. Chapter 2 presents itself as an
introductory chapter. Which aims to justifies the particularly important use and interest of the invariant-
based concept, in the aeronautics and aerospace industry. Even though the Master Ply concept had not
been detailed at this moment, its advantages and basic usage ideas are detailed.
Chapter 3 is essentially theoretical, and can be seen as an extra topic, which was exported and
condensed from several classic elasticity theory literature references. However, it was completely
reformulated in order to focus only on the demonstration and explanation of the concepts fundamental
to chapter 4 and necessary to really understand the Master Ply or Trace Concept. Thus, this chapter can
be omitted if a more practical reading is desired, without risks of misunderstanding the next chapters.
However, it is important to notice that some subchapters such as subchapter 3.4 and 3.5 have theoretical
derivations not found in a single textbook or article (at least to the best of the authorโs knowledge). They
resulted from research of several technical and theoretical literature, and from mathematical work from
the author. Thus, even if the reader only intends to use this report as a catalogue tool, it is very interesting
to explore subchapter 3.4 and 3.5 (even in a superficial way) and check that several typically used
formulae are indeed derived from considerations not completely explained and defined. For example,
the transformation matrices were derived for a generic transformation, from one axis to another. It
wasnโt assumed any kind of preference (as common practice in the majority of textbooks) related to the
transformation, e.g. from the off-axis to the principal axis of the orthotropic lamina, or vice-versa.
In chapter 4, a brief and concise summary of the formulae that govern the linear elastic behavior
of lamina is done. With the concepts fresh and clear in the readerโs mind from the previous chapter, the
lamina governing equations are detailed.
Chapter 5 exposes the hypothesis and assumptions necessary to formulate the Classical
Laminate Plate Theory. The governing equation of thin laminates will be derived and a brief comment
regarding thick laminates will also be done.
The Trace or Master Ply concept is finally explored in chapter 6, backed up in the basic and
more theoretical concepts explored in the previous chapters. Its theoretical background, definition and
applications are some of the topics approached in this chapter.
In chapter 7, the main conclusions and future works are drawn.
14. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
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1.3 Basic Mathematical Nomenclature
In order to ease the understanding of the (sometimes heavy) mathematical treatment, the author
slightly drifted away from the nomenclature usually seen in technical classic literature regarding,
Theory of elasticity, FEM and Composites Laminates [3], [14], [15], [16], [17], [18], [19], [20], [21].
The nomenclature used was similar to the one adopted in the Kinematics and Dynamics course, lectured
at FEUP, and considered by the author as more intuitive. Thus, in order to avoid misunderstandings,
Table 1 details the most relevant nomenclature adopted. This only concerns generic nomenclature; each
variable and symbol will be defined whenever necessary and convenient.
Table 1- Summary of the most important and distinct mathematical nomenclature used throughout the report.
| | Column Vector
| | ๐ Row Vector
[ ]
Matrix of any general dimension, with the exception of a column
vector
[ ๐พ ]
Stiffness matrix whose terms are structure/element properties
(depending on the geometry and material)
[ ๐ถ ] Stiffness tensor (or matrix) whose terms are material properties
[ ๐ ] Compliance tensor (or matrix) whose terms are material properties
[ ๐ ] Plane Stress reduced Stiffness Matrix
15. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
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2 Trace - Aerospace Industry
2.1 Composites in Aerospace Industry
The aim of this introductory chapter is to highlight the relation between the aerospace industry
and Trace - the invariant-based method applied to Fiber Reinforced Polymers (FRP), and presented
throughout chapter 6. Three important questions, illustrated in Figure 1, will be โansweredโ in this
chapter.
As first detailed by [20], trace is a novel invariant-based approach to describe the stiffness and
strength of Carbon Fiber Reinforced Polymers (CFRP). Several authors [20], [7], [22], [23], [24], [25]
already proposed and detailed strength-related applications of trace, and sizing and scaling methods
using this invariant-based approach. However, this report will only deal with the characterization of the
in-plane stiffness for plane stress by the Trace material property concept. Which forces the author to
leave the Trace-characterization of composite strength issues for future works.
FRP offer significant advantages over current conventional engineering materials in the
aerospace and aeronautic industry. Among other properties, at least their high fatigue and corrosion
resistance, and the capability of properties tailoring and material design optimization should be
highlighted. These superior properties promote relevant improvements such as reduced inspection and
maintenance costs, and increased passenger comfort level. In the particular case of CFRP, attractive
specific mechanical properties such as: high strength-to-weight ratio, high modulus-to-weight ratio
leading to a lower weight (and consequently higher fuel efficiency and lower emissions), have propelled
their increasing used in the aircraft industry [20], [26]. Hence, CFRP have been widely used to
manufacture different structural components such as aileron, flaps, landing-gear doors and other
structural parts. As illustrated in Figure 1 a)-b), these high-performance composite materials clearly rule
the aerospace composite material application spectrum.
As it will be detailed further in chapter 6, the Master Ply concept is only applicable with high
accuracy for composite systems based on carbon and aramid fibers. Satisfactory results were achieved
for GFRP but with lower accuracy when compared to CFRP [20], [7]. This could be considered as a
particular high limitation or drawback from this method. However, as already mentioned, the bigger
โpieceโ of FRP used in aerospace industry corresponds precisely to carbon fiber based composite
systems, ensuring that Trace perfectly โfitsโ for the aerospace industry.
Figure 2- a) Composite structure of A380 [9]. b) Composite materials used in Boeing 787 body [11].
Trace Applicability to
CFRP types?
FRP in the Aerospace
Indutry?
Trace &
Aerospace
What is Trace?
Composite testing in the
Aerospace Industry?
Revolution of CFRP
Testing by Trace
Figure 1- Illustration of the main goals aimed with this introductory chapter.
16. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
16
2.2 Composite Certification and Testing in Aerospace Industry
The procedure followed so far in this chapter was a top-down approach, starting with the higher-
level concerns. It was already proved the adequacy of Trace to aerospace industry. However, what are
the real advantages of Trace, and why is it used for? In order to clarify these questions, first is necessary
to establish some considerations regarding composite testing.
It is mandatory to test FRP in order to support design, quality management, and certification
programs. Characterizing the specific properties of composites through experimental testing is critical
to ensure their compliance with the client, industry, national, and international standard requirements
and specifications. When compared to polymeric materials (at some degree) and metals, it is quite
difficult and complex to measure material properties (stiffness and strength) of composites due to fiber
orientation. In other words, FRP systems do not exhibit isotropic behavior, demonstrating diverse
material properties and failure modes in different directions.
The complexity of composite systems testing is illustrated by the availability of the wide range
of standards and test procedures. There are over 150 standards available that outline and detail the
experimental testing of FRP. In addition to national and international standards from institutions such
as DIN, EN, ISO or ASTM, there are aircraft industry-specific standards designed by major companies
such as Boeing, Airbus and even NASA [26], [27].
Composite materials testing complexity is not only based on the stringency of the testing
procedures, but also in the number of necessary tests to fully characterize the material system. In one
hand, flexural and compressive properties must be tested independently since it is not possible to predict
them based on tensile properties (off-plane behavior). On the other hand, in order to be able to
completely characterize the shear properties in different directions, there are many different techniques
available for measuring shear properties (e.g. lap shear test, V-notch shear test, ยฑ45ยฐ in-plane shear test,
short beam shear test, etc.) [26], [27].
Due to the critical safety concerns and demanding service conditions of the aeronautics industry,
material testing goes beyond the โbasicโ mechanical properties (i.e. the three normal stresses which are
characterized in the nine-component stress tensor). Non-ambient conditions such as extreme
temperature and humidity/moisture (hygroscopic behavior) need to be considered along with fatigue
tests, which are critical for aerospace structural applications. Shifting from the mechanical design of
composite materials to the development of tough durable systems, the study of โeffects of defectsโ
emerged. Compression-after-impact (CAI) testing evaluates the tolerance of a composite material to
damage (e.g. structural damage caused by a bird strike or due to contact with other foreign objects during
flight or maintenance). In other words, evaluates the structural integrity after impact or damage tolerance
behavior. More specific tests such as open-hole compression (e.g. open-hole and filled-hole tensile), and
end-loading compression and shear tests ought also to be performed.
The relation between FRP testing and the aerospace industry was already described in the
previous two paragraphs. Finally, it must be highlighted where and how Trace influences these complex
FRP tests. At this point of the report, it can be said and unanimously accepted that the simplification of
CFRP unit testing implies cost and time reduction of the overall product testing in the aerospace
industry. This is precisely the main advantage of Trace, to simplify composite testing by reduction of
the number of necessary tests to fully characterize FRP. In chapter 6, how Trace reduces the number of
necessary tests is explained in detail.
17. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
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3 Theory of Linear Elasticity for Continuum Medium
3.1 Introduction
The linear elastic theory tries to model the mechanical behavior of continuum linear elastic
solids. And until the current century as proven its potential in a variety of engineering problems.
However, its usability lies on the capacity of assuming proper simplifications [28]
In this chapter, the basic constitutive equations for 2D linear elasticity will be derived. The
equations here demonstrated, are fundamental and will be directly used for the derivation of the ply
constitutive equations. Thus, this chapter can be considered as a literature review section. And, if the
reader already masters the basic concepts of linear elasticity, it can skip directly to chapter 4.
The approach followed is similar to the one typically carried in solid mechanics or strength of
materials classic literature. Basic concepts (valid for generic 3D anisotropic behavior) are progressively
simplified and particularized aiming the physical or engineering application in hands, in this case, the
2D orthotropic problem constitutive equations. The major difference might be the depth of study. The
starting point was the formulation of the stress and strain tensor in their generic form (considering
already the linear elastic assumptions). After deriving the two second order tensors, and underlining
their assumptions, the relation between the two was considered. Videlicet, the generalized Hookeโs Law
was stated and explored. Supported in concepts previously discussed, and some referred within the last
subchapter, the Generalized Hookeโs Law will be continuously simplified until reaching the most often
used and refined formula in 2D linear elasticity. The equations of motion will first be presented within
the stress tensor definition subchapter. However, later it will be dedicated a specific section for 2D
dynamic equilibrium. Due to the importance of the referential transformation matrix in the study of ply
mechanical behavior, a specific subchapter will also be dedicated to it.
3.2 Stress Tensor
In terms of continuum mechanics, anisotropic materials are materials that have different
mechanical properties depending on the direction of measurement. Concerning the mechanical behavior,
only the stiffness moduli and limit elastic stress parameters will be relevant. Concerning others fields of
interest, the anisotropy concept can be generalized, and we end up with anisotropy throughout the solid
relating to: thermal conductivity, magnetic permeability, refraction index, etc. [29].
From the solid mechanics of homogeneous materials1
[29], the tension matrix is a second order
tensor with 3x3 dimension. This second order Cartesian tensor is also called the Cauchy Stress Tensor
and has the form [30]:
[ ๐ ] = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
] ( 3.1 )
The nomenclature adopted in the definition of the stresses, is illustrated in Figure 3. In index
notation, the stress ( )๐๐ corresponds to the stress component acting in the j-direction, on a surface or
plane normal to i-direction. In other words, the first subscript refers to the plane in which the stress acts;
and the second subscript the direction about which the stress acts. Regarding the algebraic value, the
positive sign will be left for tension stresses and the negative for compression stresses.
1
Homogeneous materials are materials in which the mechanical properties of any given point are
equal to the specific properties of the solid. In other words, macroscopically, the specific properties are
independent of the point of analysis [29].
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The previous tensor shown in equation ( 3.1 ) has 9 terms; however, it can be shown that only
6 of them are independent. The stress matrix is symmetric to its main diagonal, and the symmetry
conditions or relations are also called the reciprocity property of the stress tensor. The symmetry
relations can be derived by the following principles or Cauchy Equations of Motion [1]:
โข According to the principle of conservation of linear momentum, if the continuum body is in
static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in
every material point in the body satisfies the linear equilibrium equation (equation of motion
for null acceleration)2
.
[ ๐ ] โ + | ๐ | = | ๐ | = | 0 | โ ( 3.2 )
[
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
]
[
๐
๐๐ฅ
๐
๐๐ฆ
๐
๐๐ฅ ]
๐๐๐ + |
๐๐ฅ
๐๐ฆ
๐๐ง
| ๐๐๐ = |
๐ ๐ฅ
๐ ๐ฆ
๐ ๐ง
| ๐๐๐ = |
0
0
0
| ( 3.3 )
Or making explicit each component of the vector equation comes:
(
๐๐๐ฅ๐ฅ
๐๐ฅ
+
๐๐ ๐ฅ๐ฆ
๐๐ฆ
+
๐๐ ๐ฅ๐ง
๐๐ง
) + ๐๐ฅ = ๐ โ ๐ ๐ฅ = ๐ โ
๐2
๐๐ก2
๐ข(๐ฅ, ๐ฆ, ๐ง) = 0 ( 3.4 )
(
๐๐ ๐ฆ๐ฅ
๐๐ฅ
+
๐๐ ๐ฆ๐ฆ
๐๐ฆ
+
๐๐ ๐ฆ๐ง
๐๐ง
) + ๐๐ฆ = ๐ โ ๐ ๐ฆ = ๐ โ
๐2
๐๐ก2
๐ฃ(๐ฅ, ๐ฆ, ๐ง) = 0 ( 3.5 )
(
๐๐ ๐ง๐ฅ
๐๐ฅ
+
๐๐ ๐ง๐ฆ
๐๐ฆ
+
๐๐๐ง๐ง
๐๐ง
) + ๐๐ง = ๐ โ ๐ ๐ง = ๐ โ
๐2
๐๐ก2
๐ค(๐ฅ, ๐ฆ, ๐ง) = 0 ( 3.6 )
๐ Total acceleration = local acceleration + convective acceleration
๐๐ฅ, ๐๐ฆ, ๐๐ง Volume forces acting on the x, y, and z direction respectively
๐๐ Differential of Volume, ๐๐ = ๐๐ฅ๐๐ฆ๐๐ง
2
The Cauchy Equation for the Conservation of Linear Momentum will be important in the formulation of the finite
element.
๐ ๐ฆ๐ฅ
๐ ๐ฅ๐ฆ
๐ฆ
๐ฅ๐ ๐ฆ๐ฅ
๐ ๐ฅ๐ฆ
๐ ๐ง๐ฅ
๐ ๐ฅ๐ง
๐ง
๐ฅ๐ ๐ง๐ฅ
๐ ๐ฅ๐ง
๐ ๐ง๐ฆ
๐ ๐ฆ๐ง
๐ง
๐ฆ๐ ๐ง๐ฆ
๐ ๐ฆ๐ง
Figure 3- Definition of the nomenclature adopted for shear stresses acting on the differential volume element [1].
19. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
19
โข According to the analogous principle regarding the conservation of angular momentum, the
angular equilibrium requires that the summation of moments with respect to an arbitrary axis is
null. Analytically it can be written:
[(๐ ๐ฆ๐ง +
๐๐ ๐ฆ๐ง
๐๐ฆ
๐๐ฆ
2
) + (๐ ๐ฆ๐ง โ
๐๐ ๐ฆ๐ง
๐๐ฆ
๐๐ฆ
2
) โ (๐ ๐ง๐ฆ +
๐๐ ๐ง๐ฆ
๐๐ง
๐๐ง
2
)
โ (๐ ๐ง๐ฆ โ
๐๐ ๐ง๐ฆ
๐๐ง
๐๐ง
2
)]
๐๐ฅ๐๐ฆ๐๐ง
2
= 0
( 3.7 )
[โ (๐ ๐ฅ๐ง +
๐๐ ๐ฅ๐ง
๐๐ฅ
๐๐ฅ
2
) โ (๐ ๐ฅ๐ง โ
๐๐ ๐ฅ๐ง
๐๐ฅ
๐๐ฅ
2
) + (๐ ๐ง๐ฅ +
๐๐ ๐ง๐ฅ
๐๐ง
๐๐ง
2
)
+ (๐ ๐ง๐ฅ โ
๐๐ ๐ง๐ฅ
๐๐ง
๐๐ง
2
)]
๐๐ฅ๐๐ฆ๐๐ง
2
= 0
( 3.8 )
[(๐ ๐ฅ๐ฆ +
๐๐ ๐ฅ๐ฆ
๐๐ฅ
๐๐ฅ
2
) + (๐ ๐ฅ๐ฆ โ
๐๐ ๐ฅ๐ฆ
๐๐ฅ
๐๐ฅ
2
) โ (๐ ๐ฆ๐ฅ +
๐๐ ๐ฆ๐ฅ
๐๐ฆ
๐๐ฆ
2
)
โ (๐ ๐ฆ๐ฅ โ
๐๐ ๐ฆ๐ฅ
๐๐ฆ
๐๐ฆ
2
)]
๐๐ฅ๐๐ฆ๐๐ง
2
= 0
( 3.9 )
The vector equilibrium equation will degenerate in the symmetry relations. They can now be
easily obtained by just solving the three angular momentum equilibrium equations. The final relations
are:
๐ ๐ฆ๐ง = ๐ ๐ง๐ฆ
๐ ๐ฅ๐ง = ๐ ๐ง๐ฅ
๐ ๐ฅ๐ฆ = ๐ ๐ฆ๐ฅ
( 3.10 )
From equation ( 3.1 ) and ( 3.10 ) we can finally write:
[ ๐ ] = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
] = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ฅ๐ง ๐ ๐ฆ๐ง ๐๐ง๐ง
] = [
๐ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
โฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
โฆ โฆ ๐๐ง๐ง
] ( 3.11 )
As indexed in the definition of second order tensor, equation ( 3.1 ) encloses the cartesian
components for a surface perpendicular to each one of the cartesian coordinate axis, as detailed in the
following equation:
[ ๐ ] = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
] = [
| ๐ ๐ ๐ฅ | ๐
| ๐ ๐ ๐ฆ | ๐
| ๐ ๐ ๐ง | ๐
] ( 3.12 )
Where:
| ๐ ๐ ๐ฅ | Stress vector acting on plane normal to x-direction
| ๐ ๐ ๐ฆ | Stress vector acting on plane normal to y-direction
| ๐ ๐ ๐ง | Stress vector acting on plane normal to z-direction
In a similar manner, the Cauchy Equation [1], allows to compute the resulting stress vector,
perpendicular to any arbitrary plane, acting on a generic point of coordinates (x,y,z). The Cauchy relation
20. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
20
can be given in two matrix forms, a condensed and a more explicit form. The two are respectively given
by:
| ๐ | = [ ๐ ] ๐ | ๐ | โ | ๐ | = [
| ๐ ๐ ๐ฅ | ๐
| ๐ ๐ ๐ฆ | ๐
| ๐ ๐ ๐ง | ๐
]
๐
| ๐ | ( 3.13 )
|
๐๐ฅ
๐๐ฆ
๐๐ง
| = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
]
๐
|
๐ ๐ฅ
๐ ๐ฆ
๐ ๐ง
| = [
๐๐ฅ๐ฅ ๐ ๐ฆ๐ฅ ๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ ๐ ๐ฆ๐ฆ ๐ ๐ง๐ฆ
๐ ๐ฅ๐ง ๐ ๐ฆ๐ง ๐๐ง๐ง
] |
๐ ๐ฅ
๐ ๐ฆ
๐ ๐ง
| ( 3.14 )
Where:
| ๐ | Vector of the direction cosines perpendicular to an arbitrary plane
[ ๐ ] Stress tensor matrix
| ๐ | Stress vector acting on a plane with normal unit vector | ๐|
Or considering the symmetry stated in the final equation ( 3.11 ), by the properties of the transposition
operation of a matrix it results:
[
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
]
๐
= [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
โฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
โฆ โฆ ๐๐ง๐ง
]
๐
= [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
โฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
โฆ โฆ ๐๐ง๐ง
] ( 3.15 )
|
๐๐ฅ
๐๐ฆ
๐๐ง
| = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐๐ง๐ง
]
๐
|
๐ ๐ฅ
๐ ๐ฆ
๐ ๐ง
| = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
โฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
โฆ โฆ ๐๐ง๐ง
] |
๐ ๐ฅ
๐ ๐ฆ
๐ ๐ง
|
( 3.16 )
The Cauchy equation can be usually demonstrated by writing the static equilibrium equations:
(i) for an infinitesimal interior tetrahedron element of a linear elastic body - in the case of 3D general
case; (ii) or for an infinitesimal triangle - in the case of 2D particular case. Figure 4 illustrates both cases.
z
x
y๐๐ด ๐ฆ
๐๐ด ๐ฅ
๐๐ด ๐ง
โ๐ ๐ ๐ง
โ๐ ๐ ๐ฅ
โ๐ ๐ ๐ฆ
๐
๐๐ = ๐๐๐
a) b)
Figure 4- a) Cauchy tetrahedron formed by slicing a parallelepiped along an arbitrary plane define by the
normal vector | ๐ |. b) Infinitesimal triangular portion of a generic 2D body.
๐๐ค
๐๐ฅ
๐๐ฆ
๐ ๐ฅ๐ฆ
๐ ๐ฆ๐ฅ
y
x
๐๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐ ๐ฆ
๐ ๐ฅ
๐โ ๐
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โข Verifying the Static Equilibrium condition comes:
{
โ ๐น๐ฅ = 0
โ ๐น๐ฆ = 0
โ {
โ๐๐ฅ๐ฅ ๐๐ฆ โ ๐ ๐ฆ๐ฅ ๐๐ฅ + ๐๐ฅ ๐๐ค = 0
โ๐ ๐ฆ๐ฆ ๐๐ฅ โ ๐ ๐ฅ๐ฆ ๐๐ฆ + ๐๐ฆ ๐๐ค = 0
( 3.23 )
โข Dividing both members of each equation by the length of the arbitrarily inclined surface ( dฮ ):
{
โ๐๐ฅ๐ฅ
๐๐ฆ
๐๐ค
โ ๐ ๐ฆ๐ฅ
๐๐ฅ
๐๐ค
+ ๐๐ฅ = 0
โ๐ ๐ฆ๐ฆ
๐๐ฅ
๐๐ค
โ ๐ ๐ฅ๐ฆ
๐๐ฆ
๐๐ค
+ ๐๐ฆ = 0
( 3.24 )
โข By the relations between the infinitesimals, equation ( 3.22 ), and manipulating the terms comes:
{
โ๐๐ฅ๐ฅ ๐ ๐ฅ โ ๐ ๐ฆ๐ฅ ๐ ๐ฆ + ๐๐ฅ = 0
โ๐ ๐ฆ๐ฆ ๐ ๐ฆ โ ๐ ๐ฅ๐ฆ ๐ ๐ฅ + ๐๐ฆ = 0
โ {
๐๐ฅ = ๐ ๐ฅ๐ฅ ๐ ๐ฅ + ๐ ๐ฆ๐ฅ ๐ ๐ฆ
๐๐ฆ = ๐ ๐ฅ๐ฆ ๐ ๐ฅ + ๐ ๐ฆ๐ฆ ๐ ๐ฅ
( 3.25 )
โข In the matrix form comes:
|
๐๐ฅ
๐๐ฆ
| = [
๐๐ฅ๐ฅ ๐ ๐ฆ๐ฅ
๐ ๐ฅ๐ฆ ๐ ๐ฆ๐ฆ
] |
๐ ๐ฅ
๐ ๐ฆ
| = [
๐๐ฅ๐ฅ ๐ ๐ฅ๐ฆ
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ
]
๐
|
๐ ๐ฅ
๐ ๐ฆ
| ( 3.26 )
3.3 Strain Tensor
The magnitude of the strains and displacements (linear displacements or rotations) can influence
the mathematical definition of strain. The main theories applied to the continuum mechanics are [32],
[33]:
โข Small Strains and small Displacements/rotations theory or infinitesimal strain theoryโ used to
solve most practical engineering problems that deal with common materials like wood, steel and
other alloys;
โข Small Strains and large Displacements theory โ essential to model materials and structures that
can withstand large displacements without entering the plastic domain, i.e. remaining elastic;
โข Finite Strains and Displacements theory โ necessary to model structures and materials where
the deformed and undeformed configuration is significantly different. These arbitrarily large
strains and displacements (linear or angular) can occur in materials with the mechanical
behavior of elastomers, fluids, biological (or not) soft tissues.
For small strains and small displacements (both linear and angular) the change in the geometry
and constitutive properties of the structure, due to deformation, doesnโt need to be considered after the
force is applied. In other words, physical and mechanical properties of the material e.g. density, stiffness,
etc. at each point of the infinitesimally deformed solid, can be assumed constant [29]. This definition of
strain is also designed by Cauchy strains, and it will be the strain concept used throughout the report.
The strain tensor or Cauchy strain tensor is also a second order tensor, and its 3x3 matrix is given by:
[ ๐ ] = [
๐ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐ ๐ง๐ง
] ( 3.27 )
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23
The nomenclature adopted in the definition of the strains is rather different from the stress
nomenclature. In index notation, the strain term ( )๐๐ means: when ๐ = ๐ , the term corresponds to
the extension along the ๐-direction; when ๐ โ ๐ , the term of the strain matrix corresponds to the rotation
about the ij plane. Regarding its algebraic value, as schematized in Figure 5, the positive sign will be
ascribed when the angle between the two faces of the conceptual parallelogram is reduced, and the
negative sign when the angle increases.
The geometric definition of strains is demonstrated and detailed in [30], [1]. The linear strain
(also designated by longitudinal strain, linear deformation, extension, etc.) is quantified by the on-
diagonal matrix components ๐ ๐ฅ๐ฅ , ๐ ๐ฆ๐ฆ , ๐ ๐ง๐ง . The remaining non-diagonal terms correspond to the
angular strain (also designated by shear strain, angular deformation, distortion, etc.). The relation of
each term of the strain tensor, with the displacement field is given by [1]:
๐ ๐ฅ๐ฅ =
๐๐ข
๐๐ฅ
; ๐ ๐ฆ๐ฆ =
๐๐ฃ
๐๐ฆ
; ๐ ๐ง๐ง =
๐๐ค
๐๐ง
( 3.28 )
๐ ๐ฅ๐ฆ = ๐ ๐ฆ๐ฅ =
1
2
(
๐๐ข
๐๐ฆ
+
๐๐ฃ
๐๐ฅ
) ; ๐ ๐ฆ๐ง = ๐ ๐ง๐ฆ =
1
2
(
๐๐ฃ
๐๐ง
+
๐๐ค
๐๐ฆ
) ; ๐ ๐ฅ๐ง = ๐ ๐ง๐ฅ =
1
2
(
๐๐ข
๐๐ง
+
๐๐ค
๐๐ฅ
) ( 3.29 )
The geometric relation between strain and displacements can also be written in matrix form as:
| ๐ | =
|
|
|
๐ ๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐ ๐ง๐ง
๐ ๐ฅ๐ฆ
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
|
|
|
=
[
๐
๐๐ฅ
0 0
0
๐
๐๐ฆ
0
0 0
๐
๐๐ง
1
2
๐
๐๐ฆ
1
2
๐
๐๐ฅ
0
0
1
2
๐
๐๐ง
1
2
๐
๐๐ฆ
1
2
๐
๐๐ง
0
1
2
๐
๐๐ฅ ]
|
๐ข(๐ฅ, ๐ฆ, ๐ง)
๐ฃ( ๐ฅ, ๐ฆ, ๐ง)
๐ค( ๐ฅ, ๐ฆ, ๐ง)
| ( 3.30 )
Other main contrast regarding the stress tensor and strain tensor, is the difference between
tensorial and engineering strain for angular distortion. The angular distortion can be quantified in terms
of engineering shear strain (also called global strain), or tensorial shear strain. The engineering shear
strain can be considered as the total rotation of the 2D cartesian element subjected to shear stresses or
the total change of the original angle formed by the undeformed element; whereas the tensorial shear
strain can be understood as the average of the two displacements or the amount that each edge rotates
in average. This difference is illustrated in Figure 5, and the two are related by the following vector
equation [30]:
|
|
|
๐พ๐ฅ๐ฆ
๐พ๐ฆ๐ฅ
๐พ๐ฆ๐ง
๐พ๐ง๐ฆ
๐พ๐ฅ๐ง
๐พ๐ง๐ฅ
|
|
|
=
|
|
|
2๐ ๐ฅ๐ฆ
2๐ ๐ฆ๐ฅ
2๐ ๐ฆ๐ง
2๐ ๐ง๐ฆ
2๐ ๐ฅ๐ง
2๐ ๐ง๐ฅ
|
|
|
( 3.31 )
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24
Their importance arises from the convenience of replacing the general symmetry of the stiffness
matrix (after continuous simplifications), as it will be explored in the next subchapter (see Page 26). In
the matrix form, the previous relations can be written as:
[ ๐ ] = [
๐ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐ ๐ง๐ง
] =
1
2
[
2๐ ๐ฅ๐ฅ ๐พ๐ฅ๐ฆ ๐พ๐ฅ๐ง
๐พ๐ฆ๐ฅ 2๐ ๐ฆ๐ฆ ๐พ๐ฆ๐ง
๐พ๐ง๐ฅ ๐พ๐ง๐ฆ 2๐ ๐ง๐ง
] ( 3.32 )
The properties of a tensor wonโt be remembered in this report [29]; however, it is always worth
notice that the following matrix is not a tensor!!
[
๐ ๐ฅ๐ฅ ๐พ๐ฅ๐ฆ ๐พ๐ฅ๐ง
๐พ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐พ๐ฆ๐ง
๐พ๐ง๐ฅ ๐พ๐ง๐ฆ ๐ ๐ง๐ง
] ( 3.33 )
The symmetry property for the strain tensor matrix is derived meticulously in [30]. By the
displacement field geometric definition and neglecting the second order terms (for small strains and
displacements, both linear and angular), it is possible to verify the strain tensor symmetry. The symmetry
relations relating the shear distortion come:
๐พ๐ฆ๐ง = ๐พ๐ง๐ฆ
๐พ๐ฅ๐ง = ๐พ๐ง๐ฅ
๐พ๐ฅ๐ฆ = ๐พ๐ฆ๐ฅ
( 3.34 )
From equation ( 3.27 ), ( 3.32 ) and ( 3.34 ) it is possible to finally write the strain tensor in
tensorial strains or engineering strains as:
[ ๐ ] = [
๐ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฆ๐ฅ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ง๐ฅ ๐ ๐ง๐ฆ ๐ ๐ง๐ง
] = [
๐ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
๐ ๐ฅ๐ง ๐ ๐ฆ๐ง ๐ ๐ง๐ง
] = [
๐ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ ๐ฅ๐ง
โฆ ๐ ๐ฆ๐ฆ ๐ ๐ฆ๐ง
โฆ โฆ ๐ ๐ง๐ง
] ( 3.35 )
[ ๐ ] =
1
2
[
2๐ ๐ฅ๐ฅ ๐พ๐ฅ๐ฆ ๐พ๐ฅ๐ง
๐พ๐ฆ๐ฅ 2๐ ๐ฆ๐ฆ ๐พ๐ฆ๐ง
๐พ๐ง๐ฅ ๐พ๐ง๐ฆ 2๐ ๐ง๐ง
] =
1
2
[
2๐ ๐ฅ๐ฅ ๐พ๐ฅ๐ฆ ๐พ๐ฅ๐ง
๐พ๐ฅ๐ฆ 2๐ ๐ฆ๐ฆ ๐พ๐ฆ๐ง
๐พ๐ฅ๐ง ๐พ๐ฆ๐ง 2๐ ๐ง๐ง
] =
1
2
[
2๐ ๐ฅ๐ฅ ๐พ๐ฅ๐ฆ ๐พ๐ฅ๐ง
โฆ 2๐ ๐ฆ๐ฆ ๐พ๐ฆ๐ง
โฆ โฆ 2๐ ๐ง๐ง
] ( 3.36 )
Figure 5- Nomenclature adopted for the shear stress definition, for the distortion of the differential Cartesian
element.
๐ฆ ๐ ๐ฅ๐ฆ
๐ ๐ฅ๐ฆ
๐๐ข
๐๐ฆ
๐ฅ
๐๐ฃ
๐๐ฅ
๐ฆ
๐ฅ
๐พ =
๐๐ข
๐๐ฆ
+
๐๐ฃ
๐๐ฅ
๐พ/2
๐ฆ
๐ฅ
๐พ =
๐๐ข
๐๐ฆ
+
๐๐ฃ
๐๐ฅ๐พ/2
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25
3.4 Generalized Hookeโs Law
3.4.1 Theoretical Background
Cauchy Elastic Materials or Simple Elastic Materials are materials for which the stress at a given
point is just function of the instantaneous strain. In other words, the stresses donโt depend of the strain
path, strain history, strain rate, and the time taken to achieve a given deformation field [34]. Cauchy
materials theoretical definition also implies:
โข Homogeneous materials โ the constitutive properties are independent of the point of analysis,
i.e. the terms of the stiffness matrix are not point functions;
โข Temperature effect is ignored โ even if there are thermal strains and/or residual stresses, the
effect of the temperature change in the properties of the material is neglected.
Assuming the previous hypothesis, the stress second order tensor is related by a second order-
valued function with the strain second order tensor as follows:
[ ๐ ] = ๐ ( [ ๐ ] ) ( 3.37 )
Considering that the stresses are a linear and homogeneous combination or function of the
strains, the contribution factors are in fact the elastic coefficients that characterize the mechanical
behavior of the material, i.e. are a property of the material. Historically the British engineer Robert
Hooke was the first to study this linear relation between the stress and strain [1]. Thatโs why the
generalize relationship of anisotropic materials - for spatial or triaxial stresses and strains - is called
Generalize Hookeโs Law. Itโs a constitutive model for infinitesimal deformation of a linear elastic
material, in which the relation between stress and strains is model by a 4th
order tensor that linearly maps
between second-order tensors [33].
The elasticity tensor will result in a 9x9 elastic coefficient matrix. Hookeโs law can be presented:
in terms of a stiffness tensor or matrix ([ ๐ถ ]), putting in evidence the stress; or in terms of compliance
tensor or matrix ([ ๐ ]), in which the response function linking strain to the deforming stress is the
compliance tensor of the material. The matrix form of Hookeโs Law can be written as:
| ๐ | = [ ๐ถ ] | ๐ | ( 3.38 )
| ๐ | = [ ๐ ] | ๐ | ( 3.39 )
Or explicitly as:
|
|
|
๐๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐๐ง๐ง
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ
๐ ๐ง๐ฆ
๐ ๐ง๐ฅ
๐ ๐ฆ๐ฅ
|
|
|
=
[
๐ถ ๐ฅ๐ฅ๐ฅ๐ฅ ๐ถ ๐ฅ๐ฅ ๐ฆ๐ฆ ๐ถ ๐ฅ๐ฅ ๐ง๐ง ๐ถ ๐ฅ๐ฅ ๐ฆ๐ง ๐ถ ๐ฅ๐ฅ ๐ฅ๐ง ๐ถ ๐ฅ๐ฅ ๐ฅ๐ฆ ๐ถ ๐ฅ๐ฅ ๐ง๐ฆ ๐ถ ๐ฅ๐ฅ ๐ง๐ฅ ๐ถ ๐ฅ๐ฅ ๐ฆ๐ฅ
๐ถ ๐ฆ๐ฆ ๐ฅ๐ฅ โฑ โฎ
๐ถ๐ง๐ง ๐ฅ๐ฅ โฑ โฎ
๐ถ ๐ฆ๐ง ๐ฅ๐ฅ โฑ โฎ
๐ถ ๐ฅ๐ง ๐ฅ๐ฅ โฑ โฎ
๐ถ ๐ฅ๐ฆ ๐ฅ๐ฅ โฑ โฎ
๐ถ๐ง๐ฆ ๐ฅ๐ฅ โฑ โฎ
๐ถ๐ง๐ฅ ๐ฅ๐ฅ โฑ โฎ
๐ถ ๐ฆ๐ฅ ๐ฅ๐ฅ โฆ โฆ โฆ โฆ โฆ โฆ โฆ ๐ถ ๐ฆ๐ฅ ๐ฆ๐ฅ ]
|
|
|
๐ ๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐ ๐ง๐ง
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ
๐ ๐ง๐ฆ
๐ ๐ง๐ฅ
๐ ๐ฆ๐ฅ
|
|
|
( 3.40 )
|
|
|
๐ ๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐ ๐ง๐ง
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ
๐ ๐ง๐ฆ
๐ ๐ง๐ฅ
๐ ๐ฆ๐ฅ
|
|
|
=
[
๐ ๐ฅ๐ฅ๐ฅ๐ฅ ๐ ๐ฅ๐ฅ ๐ฆ๐ฆ ๐ ๐ฅ๐ฅ ๐ง๐ง ๐ ๐ฅ๐ฅ ๐ฆ๐ง ๐ ๐ฅ๐ฅ ๐ฅ๐ง ๐ ๐ฅ๐ฅ ๐ฅ๐ฆ ๐ ๐ฅ๐ฅ ๐ง๐ฆ ๐ ๐ฅ๐ฅ ๐ง๐ฅ ๐ ๐ฅ๐ฅ ๐ฆ๐ฅ
๐ ๐ฆ๐ฆ ๐ฅ๐ฅ โฑ โฎ
๐๐ง๐ง ๐ฅ๐ฅ โฑ โฎ
๐ ๐ฆ๐ง ๐ฅ๐ฅ โฑ โฎ
๐ ๐ฅ๐ง ๐ฅ๐ฅ โฑ โฎ
๐ ๐ฅ๐ฆ ๐ฅ๐ฅ โฑ โฎ
๐๐ง๐ฆ ๐ฅ๐ฅ โฑ โฎ
๐๐ง๐ฅ ๐ฅ๐ฅ โฑ โฎ
๐ ๐ฆ๐ฅ ๐ฅ๐ฅ โฆ โฆ โฆ โฆ โฆ โฆ โฆ ๐ ๐ฆ๐ฅ ๐ฆ๐ฅ ]
|
|
|
๐๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐๐ง๐ง
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ
๐ ๐ง๐ฆ
๐ ๐ง๐ฅ
๐ ๐ฆ๐ฅ
|
|
|
( 3.41 )
26. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
26
The main root of the indexical notation is very similar for the coefficients of both stiffness and
compliance matrixes. However, its meaning is exactly the opposite:
โข The generic stiffness coefficient ๐ถ ๐๐, corresponds to the stress component acting on the i-
direction due to a strain imposed in j-direction, while constraining to zero the strains in the
remaining directions;
โข Whereas the generic compliance coefficient ๐ ๐๐, corresponds to the strain component about the
๐-direction due to a stress applied in the ๐-direction, while keeping null the remaining stresses.
Without making any further assumption, to apply the Generalized Hookeโs Law it would be
necessary to define 81 elastic terms to compute the coefficient matrix (whether in its Stiffness or
Compliance form). From the stress symmetry and strain symmetry relations (reciprocity relations),
detailed in subchapter 3.2 and 3.3 respectively, it is possible to further simplify this matrix to a more
treatable form, as schematized in the following schematic equation:
|
|
|
( ) ๐ฅ๐ฅ
( ) ๐ฆ๐ฆ
( ) ๐ง๐ง
( ) ๐ฆ๐ง
( ) ๐ฅ๐ง
( ) ๐ฅ๐ฆ
โ
โ
โ
|
|
|
=
[
๐ ๐ฅ๐ฅ ๐ฅ๐ฅ ๐ ๐ฅ๐ฅ ๐ฆ๐ฆ ๐ ๐ฅ๐ฅ ๐ง๐ง ๐ ๐ฅ๐ฅ ๐ฆ๐ง ๐ ๐ฅ๐ฅ ๐ฅ๐ง ๐ ๐ฅ๐ฅ ๐ฅ๐ฆ โ โ โ
๐ ๐ฆ๐ฆ ๐ฅ๐ฅ ๐ ๐ฆ๐ฆ ๐ฆ๐ฆ ๐ ๐ฆ๐ฆ ๐ง๐ง ๐ ๐ฆ๐ฆ ๐ฆ๐ง ๐ ๐ฆ๐ฆ ๐ฅ๐ง ๐ ๐ฆ๐ฆ ๐ฅ๐ฆ โ โ โ
๐๐ง๐ง ๐ฅ๐ฅ ๐๐ง๐ง ๐ฆ๐ฆ ๐๐ง๐ง ๐ง๐ง ๐๐ง๐ง ๐ฆ๐ง ๐๐ง๐ง ๐ฅ๐ง ๐๐ง๐ง ๐ฅ๐ฆ โ โ โ
๐ ๐ฆ๐ง ๐ฅ๐ฅ ๐ ๐ฆ๐ง ๐ฆ๐ฆ ๐ ๐ฆ๐ง ๐ง๐ง ๐ ๐ฆ๐ง ๐ฆ๐ง ๐ ๐ฆ๐ง ๐ฅ๐ง ๐ ๐ฆ๐ง ๐ฅ๐ฆ โ โ โ
๐ ๐ฅ๐ง ๐ฅ๐ฅ ๐ ๐ฅ๐ง ๐ฆ๐ฆ ๐ ๐ฅ๐ง ๐ง๐ง ๐ ๐ฅ๐ง ๐ฆ๐ง ๐ ๐ฅ๐ง ๐ฅ๐ง ๐ ๐ฅ๐ง ๐ฅ๐ฆ โ โ โ
๐ ๐ฅ๐ฆ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ฆ๐ฆ ๐ ๐ฅ๐ฆ ๐ง๐ง ๐ ๐ฅ๐ฆ ๐ฆ๐ง ๐ ๐ฅ๐ฆ ๐ฅ๐ง ๐ ๐ฅ๐ฆ ๐ฅ๐ฆ โ โ โ
โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ ]
|
|
|
( ) ๐ฅ๐ฅ
( ) ๐ฆ๐ฆ
( ) ๐ง๐ง
( ) ๐ฆ๐ง
( ) ๐ฅ๐ง
( ) ๐ฅ๐ฆ
โ
โ
โ
|
|
|
( 3.42 )
In order to simplify equation ( 3.42 ), it is not possible to directly eliminate all unnecessary
terms. Thus, in order that equation ( 3.42 ) preserves its meaning, the reciprocity property from both
stresses and strains implies the addition of the term 2 (due to the equal in value missing terms that were
eliminated).
|
|
( ) ๐ฅ๐ฅ
( ) ๐ฆ๐ฆ
( ) ๐ง๐ง
( ) ๐ฆ๐ง
( ) ๐ฅ๐ง
( ) ๐ฅ๐ฆ
|
|
=
[
๐ ๐ฅ๐ฅ ๐ฅ๐ฅ ๐ ๐ฅ๐ฅ ๐ฆ๐ฆ ๐ ๐ฅ๐ฅ ๐ง๐ง ๐๐ ๐ฅ๐ฅ ๐ฆ๐ง ๐๐ ๐ฅ๐ฅ ๐ฅ๐ง ๐๐ ๐ฅ๐ฅ ๐ฅ๐ฆ
๐ ๐ฆ๐ฆ ๐ฅ๐ฅ ๐ ๐ฆ๐ฆ ๐ฆ๐ฆ ๐ ๐ฆ๐ฆ ๐ง๐ง ๐๐ ๐ฆ๐ฆ ๐ฆ๐ง ๐๐ ๐ฆ๐ฆ ๐ฅ๐ง ๐๐ ๐ฆ๐ฆ ๐ฅ๐ฆ
๐๐ง๐ง ๐ฅ๐ฅ ๐๐ง๐ง ๐ฆ๐ฆ ๐๐ง๐ง ๐ง๐ง ๐๐๐ง๐ง ๐ฆ๐ง ๐๐๐ง๐ง ๐ฅ๐ง ๐๐๐ง๐ง ๐ฅ๐ฆ
๐ ๐ฆ๐ง ๐ฅ๐ฅ ๐ ๐ฆ๐ง ๐ฆ๐ฆ ๐ ๐ฆ๐ง ๐ง๐ง ๐๐ ๐ฆ๐ง ๐ฆ๐ง ๐๐ ๐ฆ๐ง ๐ฅ๐ง ๐๐ ๐ฆ๐ง ๐ฅ๐ฆ
๐ ๐ฅ๐ง ๐ฅ๐ฅ ๐ ๐ฅ๐ง ๐ฆ๐ฆ ๐ ๐ฅ๐ง ๐ง๐ง ๐๐ ๐ฅ๐ง ๐ฆ๐ง ๐๐ ๐ฅ๐ง ๐ฅ๐ง ๐๐ ๐ฅ๐ง ๐ฅ๐ฆ
๐ ๐ฅ๐ฆ ๐ฅ๐ฅ ๐ ๐ฅ๐ฆ ๐ฆ๐ฆ ๐ ๐ฅ๐ฆ ๐ง๐ง ๐๐ ๐ฅ๐ฆ ๐ฆ๐ง ๐๐ ๐ฅ๐ฆ ๐ฅ๐ง ๐๐ ๐ฅ๐ฆ ๐ฅ๐ฆ ]
|
|
( ) ๐ฅ๐ฅ
( ) ๐ฆ๐ฆ
( ) ๐ง๐ง
( ) ๐ฆ๐ง
( ) ๐ฅ๐ง
( ) ๐ฅ๐ฆ
|
|
( 3.43 )
After simplification of the 4th
order coefficientsโ tensor, the matrix lost its symmetry. The
importance of the engineering strains can now be fully understood. Instead of using the tensorial strains,
if the engineering strains were used, the symmetry of the matrix is restored, as detailed in [35].
Applying any energetic theorem e.g. Virtual Work Theorem, Minimum Potential Energy,
Maxwell-Betti Theorem, etc. [36], it is possible to prove that the matrix from the 4th
order tensor that
relates stress and strains in an elastic and loaded rigid body is symmetric. However, a different approach
was taken. In order to prove the symmetry of the elastic coefficient matrix, the concept of strain energy
density function is introduced. Conservative materials or Green Materials or Hyper-elastic materials are
a special case of Cauchy elastic materials (or simple elastic material), Figure 6 b). For this type of
materials, the stress-strain relation derives from a strain energy density function [37]:
โข Conservative materials possess a strain energy density function or energy potential, and this
energy potential is given by,
๐๐๐ =
๐๐๐๐
๐๐ ๐๐
( 3.44 )
27. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
27
โข Assuming linear stresses and strains,
| ๐ | = [ ๐ถ ] | ๐ | โ ๐๐๐ = ๐ถ๐๐ ๐๐ โ ๐ ๐๐ ( 3.45 )
โข The elastic energy is finally given by,
๐ถ๐๐ ๐๐ โ ๐ ๐๐ =
๐๐๐๐
๐๐ ๐๐
; ๐๐ = ๐๐ ( 3.46 )
โข Differentiating the previous equation to respect to ๐ ๐๐ or ๐๐๐ we get,
๐ถ๐๐ ๐๐ =
๐2
๐๐๐
๐๐๐๐ ๐๐ ๐๐
๐ถ๐๐ ๐๐ =
๐2
๐๐๐
๐๐ ๐๐ ๐๐๐๐
( 3.47 )
โข Which finally ends up in the symmetry relation:
๐ถ๐๐ ๐๐ =
๐2
๐๐๐
๐๐๐๐ ๐๐ ๐๐
=
๐2
๐๐๐
๐๐ ๐๐ ๐๐๐๐
= ๐ถ๐๐ ๐๐ โ ๐ถ๐๐ ๐๐ = ๐ถ๐๐ ๐๐ ( 3.48 )
The vast majority of engineering materials are conservative, as a result, the symmetry of the
stiffness and compliance matrices is verified for most of common engineering problems. After all
previous simplifications summarized in Table 2, the Generalized Hookeโs Law for a conservative
anisotropic material is a 6x6 elastic matrix, and now only involves the knowledge of 21 unknown elastic
terms or parameters (only 21 stiffness components are actually independent in Hooke's law), and it can
be written in the form bellow3
:
|
|
|
๐๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐๐ง๐ง
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ
|
|
|
=
[
๐ถ11 ๐ถ12 ๐ถ13 ๐ถ14 ๐ถ15 ๐ถ16
โฆ ๐ถ22 ๐ถ23 ๐ถ24 ๐ถ25 ๐ถ26
โฆ โฆ ๐ถ33 ๐ถ34 ๐ถ35 ๐ถ36
โฆ โฆ โฆ ๐ถ44 ๐ถ45 ๐ถ46
โฆ โฆ โฆ โฆ ๐ถ55 ๐ถ56
โฆ โฆ โฆ โฆ โฆ ๐ถ66 ]
|
|
|
๐ ๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐ ๐ง๐ง
๐พ๐ฆ๐ง
๐พ๐ฅ๐ง
๐พ๐ฅ๐ฆ
|
|
|
( 3.49 )
Table 2- Summary of all simplifications made to the general stiffness matrix.
Property No. Dependent terms Original Number of terms 81 = 9 x 9
Stress Reciprocity 18 + 9*
After Reciprocity Reduction 36 = 6 x 6
Strain Reciprocity 18 + 9*
Symmetry of the
Stiffness matrix
15
After Matrix Symmetry
Reduction 21 =
6 โ (6 + 1)
2
* 9 terms are automatically and simultaneously eliminated by the reciprocity property of both stresses and
strains
3
Stiffness Matrix written in Voigt notation, after eliminating the need for the stress and strain tensor matrix. See
next page to further clarifications
28. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
28
Voigt notation or Voigt form is a way to represent a symmetric tensor by reducing its order [38].
Voigt notation is commonly used in the study of composites, since it allows to: (i) reduce the second
order tensors (from the Stresses and Strains) to 6x1 vectors, and (ii) the 9x9 material stiffness matrix to
a 6x6 matrix. A fairly simple mnemonic to remember the codification of the indices in Voigt form is
illustrated in Figure 6. The previously mentioned mechanical quantities are then given by:
| ๐ | =
|
|
|
๐๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐๐ง๐ง
๐ ๐ฆ๐ง
๐ ๐ฅ๐ง
๐ ๐ฅ๐ฆ
|
|
|
=
|
|
|
๐11
๐22
๐33
๐23
๐13
๐12
|
|
|
=
|
|
|
๐1
๐2
๐3
๐4
๐5
๐6
|
|
|
( 3.50 )
| ๐ | =
|
|
|
๐ ๐ฅ๐ฅ
๐ ๐ฆ๐ฆ
๐ ๐ง๐ง
๐พ๐ฆ๐ง
๐พ๐ฅ๐ง
๐พ๐ฅ๐ฆ
|
|
|
=
|
|
|
๐11
๐22
๐33
๐พ23
๐พ13
๐พ12
|
|
|
=
|
|
|
๐1
๐2
๐3
๐4
๐5
๐6
|
|
|
( 3.51 )
[ ๐ถ ] =
[
๐ถ11 ๐ถ12 ๐ถ13 ๐ถ14 ๐ถ15 ๐ถ16
๐ถ21 ๐ถ22 ๐ถ23 ๐ถ24 ๐ถ25 ๐ถ26
๐ถ31 ๐ถ32 ๐ถ33 ๐ถ34 ๐ถ35 ๐ถ36
๐ถ41 ๐ถ42 ๐ถ43 ๐ถ44 ๐ถ45 ๐ถ46
๐ถ51 ๐ถ52 ๐ถ53 ๐ถ53 ๐ถ55 ๐ถ56
๐ถ61 ๐ถ62 ๐ถ63 ๐ถ64 ๐ถ65 ๐ถ66 ]
=
[
๐ถ11 ๐ถ12 ๐ถ13 ๐ถ14 ๐ถ15 ๐ถ16
โฆ ๐ถ22 ๐ถ23 ๐ถ24 ๐ถ25 ๐ถ26
โฆ โฆ ๐ถ33 ๐ถ34 ๐ถ35 ๐ถ36
โฆ โฆ โฆ ๐ถ44 ๐ถ45 ๐ถ46
โฆ โฆ โฆ โฆ ๐ถ55 ๐ถ56
โฆ โฆ โฆ โฆ โฆ ๐ถ66 ]
( 3.52 )
[ ๐ ] =
[
๐11 ๐12 ๐13 ๐14 ๐15 ๐16
๐21 ๐22 ๐23 ๐24 ๐25 ๐26
๐31 ๐32 ๐33 ๐34 ๐35 ๐36
๐41 ๐42 ๐43 ๐44 ๐45 ๐46
๐51 ๐52 ๐53 ๐53 ๐55 ๐56
๐61 ๐62 ๐63 ๐64 ๐65 ๐66 ]
=
[
๐11 ๐12 ๐13 ๐14 ๐15 ๐16
โฆ ๐22 ๐23 ๐24 ๐25 ๐26
โฆ โฆ ๐33 ๐34 ๐35 ๐36
โฆ โฆ โฆ ๐44 ๐45 ๐46
โฆ โฆ โฆ โฆ ๐55 ๐56
โฆ โฆ โฆ โฆ โฆ ๐66 ]
( 3.53 )
[ ] = [
( ) ๐ฅ๐ฅ ( ) ๐ฅ๐ฆ ( ) ๐ฅ๐ง
( ) ๐ฆ๐ฅ ( ) ๐ฆ๐ฆ ( ) ๐ฆ๐ง
( ) ๐ง๐ฅ ( ) ๐ง๐ฆ ( ) ๐ง๐ง
]
Figure 6- a) Schematic representation of the algorithm used to
codify Voigt notation in a second order tensor. b) Illustration
of the relation between Elastic and Hyperelastic materials for
small strains condition.
๐
๐a) b)
29. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
29
3.5 Transformation Matrix
3.5.1 General Definition
The transformation matrix allows to change the stress, the strain, or even the
stiffness/compliance tensor from one coordinate system to another generically transformed coordinate
system. In its completely general form, the transformation matrix is given by:
[ ๐ ] =
[
cos( ๐ผ ๐ฅโฒ ๐ฅ) cos(๐ผ ๐ฅโฒ ๐ฆ) cos( ๐ผ ๐ฅโฒ ๐ง)
cos(๐ผ ๐ฆโฒ ๐ฅ) cos(๐ผ ๐ฆโฒ ๐ฆ) cos(๐ผ ๐ฆโฒ ๐ง)
cos( ๐ผ ๐งโฒ ๐ฅ) cos(๐ผ ๐งโฒ ๐ฆ) cos( ๐ผ ๐งโฒ ๐ง) ]
( 3.54 )
The mathematical meaning of the angles of
the transformation matrix, equation ( 3.54 ), is
illustrated in Figure 7 for the particular case of the
transformed ๐ฅโ-axis. Using a similar principle, the
remaining rotation angles could also be drawn.
Usually, the direction cosines from equation ( 3.54 ), are hard to compute individually. So, the
transformation matrix can also be determined by the combination of three simpler transformations, each
one in respect to only one axis. The most used combination is designated as Euler Angles (or x-
convention) and is determined by [5]:
[๐] = [ โ
cos( ๐) sin( ๐) 0
sin( ๐) cos( ๐) 0
0 0 1
]
๐ง
[
1 0 0
0 cos( ๐) sin( ๐)
0 โsin( ๐) cos( ๐)
]
๐ฅ
[ โ
cos( ๐) sin( ๐) 0
sin( ๐) cos( ๐) 0
0 0 1
]
๐ง
( 3.55 )
The physical meaning of the three rotation angles is given in Figure 8 and Figure 9. Since the
matrix multiplication operation isnโt commutative, the order of rotation matters, Counter-clockwise
rotation was considered as a positive rotation for all angles.
๐ง
๐ฅ
๐ฆ
๐ฅโฒ
๐ผ ๐ฅโฒ ๐ฅ
๐ผ ๐ฅโฒ ๐ฆ
๐ผ ๐ฅโฒ ๐ง
Figure 7- Illustration of the angles between the
transformed ๐ฅโ-axis and the original cartesian coordinate
system.
Figure 8- Illustration of the individual rotations of the Euler angles. Image adapted from [5].
30. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
30
Since this report is essentially dedicated to plane elasticity, for the 2D case, the only possible
transformation consists in a rotation around the z-axis, and the transformation matrix s given by:
[ ๐ ] = [ โ
cos( ๐) sin( ๐) 0
sin( ๐) cos( ๐) 0
0 0 1
]
๐ง
( 3.56 )
Orthotropic material behavior will be considered further in the report, see chapter 4, Thus, it is
necessary to apply the transformation matrix to the stress and strain tensor as follows [35]:
[ ๐โฒ ] = [ ๐ ] [ ๐ ] [ ๐ ] ๐
( 3.57 )
[ ๐โฒ ] = [ ๐ ] [ ๐ ] [ ๐ ] ๐ ( 3.58 )
3.5.2 Modified Transformation Matrix
The previous transformation matrix [ ๐ ] changes the coordinate system of second order tensors.
However, it was shown in the previous subchapter that due to the several symmetries the stress and
strain can be related using the simplified Voigt notation. Matrix [ ๐ ] can be adapted precisely to be
applied directly to the coordinate transformation of both the stress and strain in Voigt notation. However,
in the case of the stress-strain value function matrix, i.e. for the stiffness or compliance tensor, the
approach is different. The transformation matrix is computed from the knowledge of the transformation
matrices for the stresses and strains, and the stress-strain relation.
The several simplifications detailed in [29], consist of a procedure merely algebraic that relies
on matrix manipulation. After computing the matrix product from equations ( 3.57 ) and ( 3.58 ), the
resulting terms are organized in a vector form following Voigt notation. It is important to notice that for
the shear strains, rather than Tensorial strains, Engineering strains were used. Thus, the ยฝ term must
affect the strain second order tensor in order to maintain the tensor properties. The final output consists
of three transformation matrices namely: [ ๐ โ ], [ ๐ โโ ], [ ๐ โโโ ], and [ ๐ โโโโ ] for the stress vector, strain
๐
๐ฅโฒ
๐ฆโฒ
๐ฆ
๐ฅ
๐
๐
๐ฆโฒโฒ
๐งโฒ
๐ง
๐ฆ
๐
Figure 9- Definition of the nomenclature
used to define the coordinates
transformation matrix. The ๐, ๐, and ๐,
represent the rotation angle about the z, x
and y axis respectively.
๐
๐ฅโฒโฒ
๐ฆโฒโฒโฒ
๐ฆโฒโฒ
๐ฅโฒโฒ
๐
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34
โข Considering the stress symmetry (eliminating the symmetric indexes not used and only working
the terms used) and organizing the trigonometric terms in a matrix compatible with the strain
vector in Voigt notation comes:
[
๐๐บ ๐๐
โฒ
๐ธ ๐๐
โฒ
๐ธ ๐๐
โฒ
๐พ21
โฒ
๐๐บ ๐๐
โฒ
๐ธ ๐๐
โฒ
๐พ31
โฒ
๐พ32
โฒ
๐๐บ ๐๐
โฒ
] = [
2 cos2( ๐) ๐บ ๐๐ + 2 sin( ๐) cos( ๐) ๐ธ ๐๐ + 2 sin2( ๐) ๐บ ๐๐ โฏ
โฏ
๐ ๐ฆ๐๐ โฏ
โฏ โ2 sin( ๐) cos( ๐) ๐บ ๐๐ + (cos2( ๐) โ sin2( ๐)) ๐ธ ๐๐ + +2 cos( ๐) sin( ๐) ๐บ ๐๐ โฏ
โฏ 2 sin2( ๐) ๐บ ๐๐ โ 2 sin( ๐) cos( ๐) ๐ธ ๐๐ + 2 cos2( ๐) ๐บ ๐๐ โฏ
โฏ โฏ
โฏ cos( ๐) ๐ธ ๐๐ + sin( ๐) ๐ธ ๐๐
โฏ โ sin( ๐) ๐ธ ๐๐ + cos( ๐) ๐ธ ๐๐
โฏ 2 ๐บ ๐๐
]
( 3.73 )
โข On the contrary to what was observed for the stress tensor4
, the strain tensor (left side of equation
( 3.69 )), was changed during manipulation of the expression. Thus, it is necessary to affect the
linear strains with the coefficient ยฝ in order to get the strain vector in Voigt notation, coming:
|
|
|
๐๐บโฒ ๐๐/๐
๐๐บโฒ ๐๐/๐
๐๐บโฒ ๐๐/๐
๐ธโฒ ๐๐
๐ธโฒ ๐๐
๐ธโฒ ๐๐
|
|
|
=
|
|
|
๐บโฒ ๐
๐บโฒ ๐
๐บโฒ ๐
๐บโฒ ๐
๐บโฒ ๐
๐บโฒ ๐
|
|
|
= โฏ
|
|
|
๐1
๐2
๐3
๐4
๐5
๐6
|
|
|
[
cos2( ๐) sin2( ๐) 0 0 0 sin( ๐) cos( ๐)
sin2( ๐) cos2( ๐) 0 0 0 โ sin( ๐) cos( ๐)
0 0 1 0 0 0
0 0 0 cos( ๐) โ sin( ๐) 0
0 0 0 sin( ๐) cos( ๐) 0
โ2 sin( ๐) cos( ๐) 2 sin( ๐) cos( ๐) 0 0 0 cos2( ๐) โ sin2( ๐) ]
( 3.74 )
Other approach, suggested and detailed in [39], could be followed to compute the transformation
matrix of the Strain vector. Compared with the previous approach, this one is more mathematically
elegant and convenient in terms of calculation effort, while the previous can be considered as the โbrute
forceโ version of it.
โข As already stated in the previous footnote, if Tensorial Strains were used, the transformation
matrix of the strain vector would be exactly the same as for the stress vector. First letโs
remember the definition of the Engineering Strain vector | ๐ | from equation ( 3.51 ), and let the
Tensorial Strain vector | ๐ ๐๐๐๐ ๐๐๐๐๐ | be defined as:
4
It should be noticed that Engineering Strains were used in the Strain Tensor. If Tensorial Strains were used, which
would be equivalent to the Stress Tensor, the transformation matrix for the strains vector would be equal to the
transformation matrix for the stress vector.
35. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
35
| ๐ | =
|
|
|
๐11
๐22
๐33
๐พ23
๐พ13
๐พ12
|
|
|
=
|
|
|
๐1
๐2
๐3
๐4
๐5
๐6
|
|
|
( 3.75 )
| ๐ ๐๐๐๐ ๐๐๐๐๐ | =
|
|
|
๐11
๐22
๐33
๐23
๐13
๐12
|
|
|
( 3.76 )
โข In a general different coordinate system, the two vectors would be redefined, and from the
premise stated in the footnote, itโs possible to use directly the stress transformation matrix, made
explicit in equation ( 3.67 ), as follows.
| ๐ | =
|
|
|
๐โฒ11
๐โฒ22
๐โฒ33
๐พโฒ23
๐พโฒ13
๐พโฒ12
|
|
|
=
|
|
|
๐โฒ1
๐โฒ2
๐โฒ3
๐โฒ4
๐โฒ5
๐โฒ6
|
|
|
( 3.77 )
| ๐ ๐๐๐๐ ๐๐๐๐๐ | =
|
|
|
๐โฒ11
๐โฒ22
๐โฒ33
๐โฒ23
๐โฒ13
๐โฒ12
|
|
|
( 3.78 )
| ๐โฒ ๐๐๐๐ ๐๐๐๐๐ | = [ ๐ โ ]| ๐ ๐๐๐๐ ๐๐๐๐๐ | ( 3.79 )
โข The two strain vectors can be related by the following two expressions as:
| ๐ | = [ ๐ ] | ๐ ๐๐๐๐ ๐๐๐๐๐ | ( 3.80 )
| ๐ ๐๐๐๐ ๐๐๐๐๐ | = [ ๐ ]โ1 | ๐ | ( 3.81 )
โข And the connection matrix is given by:
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For the Transformation Matrix of the Stiffness Matrix [ ๐ป โโโ ]
โข In order to compute the transformation matrix for the Stiffness Matrix, it will be used the
previous results already achieved for the stress and strain coordinates transformation, and
summarized in equations ( 3.67 ) and ( 3.100 ) respectively. From the simple stress-strain
relation in the original coordinate system comes:
| ๐ | = [ ๐ถ ] | ๐ | ( 3.101 )
Manipulating both sides in order to get the transformation of the stress vector in the left side,
but without changing the mathematical equality, comes
[ ๐โ] | ๐ | = [ ๐โ ] [ ๐ถ ] | ๐ | ( 3.102 )
Introducing a mathematical artifice operation โ equivalent to the Identity matrix that do not
change the mathematical equality โ in order to virtually get the transformed strain vector in the
right side, comes
[ ๐โ ] | ๐ | = [ ๐โ ] [ ๐ถ ] [ ๐โโ ]โ1 [ ๐โโ ] | ๐ | ( 3.103 )
Introducing equation ( 3.66 ) and equation ( 3.99 ) in the previous equation ( 3.103 ), comes:
| ๐โฒ | = [ ๐โ ] [ ๐ถ ] [ ๐โโ ]โ1 | ๐โฒ | ( 3.104 )
Introducing equation ( 3.98 ) in the previous equation ( 3.104 ), comes:
| ๐โฒ | = [ ๐โ ] [ ๐ถ ] ( [ ๐โ ]โ๐ )โ1 | ๐โฒ | ( 3.105 )
| ๐โฒ | = [ ๐โ ] [ ๐ถ ] [ ๐โ ] ๐ | ๐โฒ | ( 3.106 )
From the previous equation ( 3.106 ), it is possible to observe that the Transformed Stiffness
Matrix must be equal to:
[ ๐ถโฒ ] = [ ๐โ ] [ ๐ถ ] [ ๐โโ ]โ1 ( 3.107 )
[ ๐ถโฒ ] = [ ๐โ ] [ ๐ถ ] [ ๐โ ] ๐ ( 3.108 )
โข A similar theoretical exercise could have been done for the compliance matrix as summarized
below:
| ๐ | = [ ๐ ] | ๐ | ( 3.109 )
[ ๐ โโ] | ๐ | = [ ๐ โโ ] [ ๐ถ ] | ๐ | ( 3.110 )
[ ๐ โโ] | ๐ | = [ ๐ โโ ] [ ๐ ] [ ๐ โ ]โ1 [ ๐ โ ] | ๐ | ( 3.111 )
| ๐โฒ | = [ ๐ โโ ] [ ๐ ] [ ๐ โ ]โ1 | ๐โฒ | ( 3.112 )
[ ๐โฒ] = [ ๐ โโ ] [ ๐ ] [ ๐ โ ]โ1 ( 3.113 )
[ ๐โฒ] = [ ๐ โ ]โ๐ [ ๐ ] [ ๐ โ ]โ1 ( 3.114 )
Manipulating the previous equation ( 3.114 ), comes:
40. Study of Invariant-based Method for Accelerating Aerospace Certification Testing of Composite Systems
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[ ๐โฒ] = [ ๐ โ ]โ๐ [ ๐ ] [ ๐ โ ]โ1
( 3.115 )
([ ๐โฒ])โ1
= ([ ๐ โ ]โ๐ [ ๐ ] [ ๐ โ ]โ1)โ1 ( 3.116 )
[ ๐ถโฒ ] = ([ ๐ โ ]โ1)โ1 [ ๐ ]โ1 ([ ๐ โ ]โ๐)โ1 ( 3.117 )
[ ๐ถโฒ ] = [ ๐ โ ] [ ๐ถ ] [ ๐ โ ] ๐ ( 3.118 )
Comparing equations ( 3.118 ) and ( 3.108 ), it is possible to observe that:
[ ๐โโโ ] = [๐โโโโ]โ1 ( 3.119 )
โข For the sake of brevity and notation simplification, the following abbreviations will be made:
๐ = cos( ๐) ; ๐2
= cos2( ๐) ( 3.120 )
๐ = sin( ๐) ; ๐ 2
= sin2( ๐) ( 3.121 )
๐ ๐ = sin( ๐) cos( ๐) ( 3.122 )
The demonstration of the transformation matrix of the Stiffness matrix for the generic 3D case
is was done in the next pages throughout equation ( 3.125 ), ( 3.126 ), and ( 3.127 ). The final
transformation matrix is given in equation ( 3.128). The constitutive equation for the stress-strain
relation in the global coordinates is finally given by:
| ๐โฒ | = [ ๐ถโฒ] | ๐โฒ | ( 3.123 )
A final comment should be done, regarding the misguiding meaning of the transformation
matrix in technical literature of different fields (e.g. Solid Mechanics, Computer Engineering, Web
Design, etc.).
โข In solid mechanics and in the context of this report, the transformation matrix assumes that the
mathematical entities are static, while the coordinate system is changed, i.e. passive
transformation. In other words, the transformation matrix always refers to a matrix acting upon
a coordinate system, hence the designation of passive transformation;
โข Whereas this designation is also misused (without proper specification or a callout note) to refer
to the geometric transformation matrices of vectors and matrices (e.g. rotation, stretching,
squeezing, shearing, reflection, etc.). The geometric transformation matrices, usually designated
also as transformation matrices, change the entities while the coordinate system remains the
same, hence the designation of active transformation.
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