An analytical framework is developed for investigating the effect of viscoelasticity on irregular hexagonal lattices. At room temperature, many polymers are found to be near their glass temperature. Elastic moduli of honeycombs made of such materials are not constant, but changes in the time or frequency domain. Thus consideration of viscoelastic properties is essential for such honeycombs. Irregularity in lattice structures being inevitable from a practical point of view, analysis of the compound effect considering both irregularity and viscoelasticity is crucial for such structural forms. On the basis of a mechanics-based bottom-up approach, computationally efficient closed-form formulae are derived in the frequency domain. The spatially correlated structural and material attributes are obtained based on Karhunen-Lo\`{e}ve expansion, which is integrated with the developed analytical approach to quantify the viscoelastic effect for irregular lattices. Consideration of such spatially correlated behaviour can simulate the practical stochastic system more closely. Two Young's moduli and shear modulus are found to be dependent on the viscoelastic parameters, while the two in-plane Poisson's ratios are found to be independent of viscoelastic parameters. Results are presented in both deterministic and stochastic regime, wherein it is observed that the elastic moduli are significantly amplified in the frequency domain. The response bounds are quantified considering two different forms of irregularity, randomly inhomogeneous irregularity and randomly homogeneous irregularity. The computationally efficient analytical approach presented in this study can be quite attractive for practical purposes to analyse and design lattices with predominantly viscoelastic behaviour along with consideration of structural and material irregularity.
Homogeneous dynamic characteristics of damped 2 d elastic latticesUniversity of Glasgow
Lattice materials are characterised by their equivalent elastic moduli for analysing mechanical properties of the microstructures. The values of the elastic moduli under static forcing condition are primarily dependent on the geometric properties of the constituent unit cell and the mechanical properties of the intrinsic material. Under a static forcing condition, the equivalent elastic moduli (such as Young's modulus) are always positive. Poisson’s ratio can be positive or negative, depending only on the geometry of the lattice (e.g., re-entrant lattices have negative Poisson’s ratio). When dynamic forcing is considered, the equivalent elastic moduli become functions of the applied frequency and can become negative at certain frequency values. This paper, for the first time, explicitly demonstrates the occurrence of negative equivalent Young's modulus in lattice materials. In addition, we show the reversal of Poisson’s ratio. Above certain frequency values, a regular lattice can have negative Poisson’s ratio, while a re-entrant lattice can have a positive Poisson’s ratio. Using a titanium-alloy hexagonal lattice metastructure, it is shown that the real part of experimentally measured in-plane Young's modulus becomes negative under a dynamic environment as well as the reversal of the real part of the Poisson’s ratio. In fact, we show that the onset of negative Young's modulus and Poisson’s ratio reversal in lattice materials can be precisely determined by capturing the sub-wavelength scale dynamics of the system. Experimental confirmation of the negative Young's moduli and Poisson’s ratio reversal together with the onset of the same as a function of frequency provide the necessary physical insights and confidence for its potential exploitation in various multi-functional structural systems and devices across different length scales.
From the Egyptian, Roman to the Victorian era, important engineering structures were built "solid" for strength and durability. Although many such structures have passed the test of time and therefore validate the underlying design philosophy, they may not always represent the most efficient way of utilising materials. Due to the unprecedented pressure on sustainability and global focus on net-zero, using less material for future engineering structures is mandatory. The rise of 3D printing technology provides a timely solution towards creating "hollow" or "porous" structural members, which have the potential to utilise lesser materials compared to their equivalent solid counterparts. However, the mechanical behaviour of such hollow structures must be understood well and subsequently. They need to be designed carefully to withstand static and dynamic loads to be experienced during their lifetime. This talk will outline some recent results from my group towards understanding the mechanics of lattice structures. New results on equivalent elastic properties, buckling and wave propagation will be discussed.
Analysis and Design of One Dimensional Periodic Foundations for Seismic Base ...IJERA Editor
Periodic foundationis a new type of seismic base isolation system. It is inspired by the periodic material crystal
lattice in the solid state physics. This kind of material has a unique property, which is termed as frequency band
gap that is capable of blocking incoming waves having frequencies falling within the band gap. Consequently,
seismic waves having frequencies falling within the frequency band gap are blocked by the periodic foundation.
The ability to block the seismic waveshas put this kind of foundation as a prosperous next generation of seismic
base isolators. This paper provides analytical study on the one dimensional (1D) type periodic foundations to
investigate their seismic performance. The general idea of basic theory of one dimensional (1D) periodic
foundations is first presented.Then, the parametric studies considering infinite and finite boundary conditions are
discussed. The effect of superstructure on the frequency band gap is investigated as well. Based on the analytical
study, a set of equations is proposed for the design guidelines of 1D periodic foundations for seismic base
isolation of structures.
Determination of the equivalent elastic coefficients of the composite materia...eSAT Journals
Abstract We must first find a behaving law of a material, before using it in the structure. The establishment of this law in the case of a composite material requires the knowledge of his elastic equivalent coefficient. In this work, we will present the vibratory technic with which we can extract the equivalent elastic coefficients of some composite materials. The relationships we have obtained through this analysis permits to evaluate the equivalent elastic coefficients of composite materials as a function of their self throb. Those relationships are first validated with the determination of mechanical characteristics of conventional materials. After that, equivalent elastic coefficients of some composite materials are evaluated. Simulated results we have obtained are discussed in comparison of Voigt and Reuss boundaries.
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme.
It was found that: The convergence and accuracy of the DR solution is dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Homogeneous dynamic characteristics of damped 2 d elastic latticesUniversity of Glasgow
Lattice materials are characterised by their equivalent elastic moduli for analysing mechanical properties of the microstructures. The values of the elastic moduli under static forcing condition are primarily dependent on the geometric properties of the constituent unit cell and the mechanical properties of the intrinsic material. Under a static forcing condition, the equivalent elastic moduli (such as Young's modulus) are always positive. Poisson’s ratio can be positive or negative, depending only on the geometry of the lattice (e.g., re-entrant lattices have negative Poisson’s ratio). When dynamic forcing is considered, the equivalent elastic moduli become functions of the applied frequency and can become negative at certain frequency values. This paper, for the first time, explicitly demonstrates the occurrence of negative equivalent Young's modulus in lattice materials. In addition, we show the reversal of Poisson’s ratio. Above certain frequency values, a regular lattice can have negative Poisson’s ratio, while a re-entrant lattice can have a positive Poisson’s ratio. Using a titanium-alloy hexagonal lattice metastructure, it is shown that the real part of experimentally measured in-plane Young's modulus becomes negative under a dynamic environment as well as the reversal of the real part of the Poisson’s ratio. In fact, we show that the onset of negative Young's modulus and Poisson’s ratio reversal in lattice materials can be precisely determined by capturing the sub-wavelength scale dynamics of the system. Experimental confirmation of the negative Young's moduli and Poisson’s ratio reversal together with the onset of the same as a function of frequency provide the necessary physical insights and confidence for its potential exploitation in various multi-functional structural systems and devices across different length scales.
From the Egyptian, Roman to the Victorian era, important engineering structures were built "solid" for strength and durability. Although many such structures have passed the test of time and therefore validate the underlying design philosophy, they may not always represent the most efficient way of utilising materials. Due to the unprecedented pressure on sustainability and global focus on net-zero, using less material for future engineering structures is mandatory. The rise of 3D printing technology provides a timely solution towards creating "hollow" or "porous" structural members, which have the potential to utilise lesser materials compared to their equivalent solid counterparts. However, the mechanical behaviour of such hollow structures must be understood well and subsequently. They need to be designed carefully to withstand static and dynamic loads to be experienced during their lifetime. This talk will outline some recent results from my group towards understanding the mechanics of lattice structures. New results on equivalent elastic properties, buckling and wave propagation will be discussed.
Analysis and Design of One Dimensional Periodic Foundations for Seismic Base ...IJERA Editor
Periodic foundationis a new type of seismic base isolation system. It is inspired by the periodic material crystal
lattice in the solid state physics. This kind of material has a unique property, which is termed as frequency band
gap that is capable of blocking incoming waves having frequencies falling within the band gap. Consequently,
seismic waves having frequencies falling within the frequency band gap are blocked by the periodic foundation.
The ability to block the seismic waveshas put this kind of foundation as a prosperous next generation of seismic
base isolators. This paper provides analytical study on the one dimensional (1D) type periodic foundations to
investigate their seismic performance. The general idea of basic theory of one dimensional (1D) periodic
foundations is first presented.Then, the parametric studies considering infinite and finite boundary conditions are
discussed. The effect of superstructure on the frequency band gap is investigated as well. Based on the analytical
study, a set of equations is proposed for the design guidelines of 1D periodic foundations for seismic base
isolation of structures.
Determination of the equivalent elastic coefficients of the composite materia...eSAT Journals
Abstract We must first find a behaving law of a material, before using it in the structure. The establishment of this law in the case of a composite material requires the knowledge of his elastic equivalent coefficient. In this work, we will present the vibratory technic with which we can extract the equivalent elastic coefficients of some composite materials. The relationships we have obtained through this analysis permits to evaluate the equivalent elastic coefficients of composite materials as a function of their self throb. Those relationships are first validated with the determination of mechanical characteristics of conventional materials. After that, equivalent elastic coefficients of some composite materials are evaluated. Simulated results we have obtained are discussed in comparison of Voigt and Reuss boundaries.
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme.
It was found that: The convergence and accuracy of the DR solution is dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Electroelastic Response of a Piezoelectric Semi-infinite Body with D∞ Symmetr...IJERA Editor
In this paper, with the aim of developing a nondestructive evaluation technique using piezoelectric signals in
wooden materials, we theoretically study the electroelastic field in a semi-infinite body with D∞ symmetry
subjected to surface friction parallel to the ∞-fold rotation axis. By applying the analytical technique previously
proposed by us, we formulate expressions for electroelastic field quantities, including electric potential, electric
field, electric displacement, elastic displacement, strain, and stress by using two “elastic displacement potential
functions” and two “piezoelastic displacement potential functions.” These potential functions and, consequently,
the electroelastic field quantities are formulated using Fourier transforms in order to satisfy electroelastic
boundary conditions. We carried out numerical calculations to correctly evaluate field quantities inside the body
and at its surface. As a result, we were successful in quantitatively elucidating the surface electric displacement
in response to the elastic stimulus of surface friction and suggested the possibility of a nondestructive evaluation
technique using piezoelectric signals.
TRANSIENT ANALYSIS OF PIEZOLAMINATED COMPOSITE PLATES USING HSDTP singh
Piezoelectric materials have excellent sensing and actuating capabilities have made them the most practical smart materials to integrate with laminated structures. Integrated structure system can be called a smart structure because of its ability to perform self-diagnosis and quick adaption to environment changes. An analytical procedure has been developed in the work based on higher order shear deformation theory subjected to electromechanical loading for investigating transient characteristics of smart material plates. For analysis two displacement models are to be considered i.e., model-1 accounts for strain in thickness direction is zero whereas in model-2 in-plane displacements are expanded as cubic functions of the thickness coordinate. Navier’s technique has been adopted for obtaining solutions of anti-symmetric cross–ply and angle-ply laminates of both model-1 and model-2 with simply supported boundary conditions. For obtaining transient response of a laminated composite plate attached with piezoelectric layer Newmark’s method has been used. Effect of thickness coordinate of composite laminated plates attached with piezoelectric layer subjected to electromechanical loadings is studied.
Applications of layered theory for the analysis of flexible pavementseSAT Journals
Abstract
Unbound granular materials used in untreated base/sub-base layers, exhibit nonlinear behavior under repeated wheel loads. The
properties of the granular materials play a significant role in the performance of these pavements. Therefore, accurate modelling of
the granular layers is essential in the evaluation of critical pavement responses under the application of loads, these materials exhibit
stress dependent characteristics. Thus, consideration of non-linearity in these layers is necessary for accurate estimation of the
pavement responses of a flexible pavement structure. The pavement responses are computed using Kenlayer computer program
developed by Huang. Using the Kenlayer program, this paper examines the effect of nonlinearity in granular on critical pavement
responses by conducting parametric analysis. The results indicate that the consideration of nonlinearity yields 23.13% reduction in
tensile strains at the bottom of bituminous layers and 0.76% increase in compressive strains on the top of the sub grade layers and
same surface deflections compared to the value obtained using linear elastic analysis. This indicates that nonlinear analysis is more
realistic and accurate.
Keywords: Granular layers, Kenlayer, Nonlinearity, Pavement responses.
Applications of layered theory for the analysis of flexible pavementseSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme. It was found that: The convergence and accuracy of the DR solution were dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...IRJESJOURNAL
Abstract:- Buckling analysis of a carbon nanotube (CNT) embedded in Pasternak’s medium is investigated. Eringen’s nonlocal elasticity theory in conjunction with the first-order Donell’s shell theory is used. The governing equilibrium equations are obtained and solved for CNTs subjected to mechanical loads and embedded in Winkler-Pasternak’s medium. Effects of nonlocal parameter, radius and length of CNT, as well as the foundation parameters on buckling of CNT are investigated. Comparison with the available results is made.
Micromechanical Modeling of the Macroscopic Behavior of Soft Ferromagnetic Co...IJMERJOURNAL
ABSTRACT: Soft ferromagnetic composites consist of ferromagnetic particles embedded within nonmagnetic electrically insulating matrix. The aims of this article are twofold. In the first one, multiaxial nonlinear isothermal constitutive equations that govern the behavior of soft ferromagnetic materials are generalized to include the temperature and hysteretic effects. The second aim consists of developing a micromechanical analysis which takes into account the detailed interaction between the phases, and establishes the instantaneous concentration tensors which relate between the local magneto-thermo-elastic field and the externally applied loading. The ferromagnetic particles constituents employed in the micromechanical analysis are governed by the developed magneto-thermo-elastic coupled hysteretic constitutive relations. With the established concentration tensors, the macroscopic (global) re- sponse of the composite can be readily determined at any instant of loading. The offered micromechanical modeling is capable of predicting the response of soft ferromagnetic composites that are subjected to various types of magneto-thermo-elastic loadings, and it can be employed by the designerto easily determine the composite response for a particular desired application.
Numerical Analysis of Engineered Steel Fibers as Shear Reinforcement in RC BeamsP singh
Using suitable fibers and additives in concrete to enhance its performance is an important consideration in the concrete industry with regard to the structural aspects of concrete. The purpose of this project is to investigate numerically the effectiveness of the engineered steel fiber as shear reinforcement in RC beams. Here steel fibers completely replaces the shear reinforcement (stirrups & links). The dimension of beam taken was 1000*150*150 mm with aspect ratio 80. The beams were reinforced with 10 mm steel bars as secondary reinforcement and 12 mm bars as main reinforcement on the tension side. Numerical analysis using ANSYS R16.1 software package was carried out. The load-deflection curves for the beams with different dosage of fibers were drawn superimposing their numerical values. Initially, in all three cases the curve was linear elastic and about 80% of ultimate load they tend to be non-linear. It was observed that there was fair agreement between the results which indicates some favourable aspects concerning the use of steel fibres as shear reinforcement in concrete beams. It was investigated that the inclusion of steel fibres (Hook End Type) improves the shear strength of RC beams without stirrups by improving the matrix between concrete and steel fibers. Thus this project focuses in the design and analysis using the software ANSYS R16.1 for an alternative steel reinforcement with better or equivalent performance.
A composite material can be defined as a combination of two or more materials that
gives better properties than those of the individual components used alone. In contrast to
metallic alloys, each material retains its separate chemical, physical, and mechanical
properties. The two constituents are reinforcement and a matrix. The main advantages of
composite materials are their high strength and stiffness combined with low density when
compared to classical materials. Micromechanical approach is found to be more suitable for
the analysis of composite materials because it studies the volume proportions of the
constituents for the desired lamina stiffness and strength.
Linear And Nonlinear Analytical Modeling of Laminated Composite Beams In Thre...researchinventy
The large current development of aerospace and automotive technologies is based on the use of composite materials which provide significant weight savings compared to their mechanical characteristics. Correct dimensioning of composite structures requires a thorough knowledge of their behavior in small as in large deflection.This work aims to simulate linear and nonlinear behavior of laminates composites under threepoint bending test. The used modelization is based on first-order shear deformation theory (FSDT), classical plate theory (CPT) and Von-Karman’s equations for large deflection. A differential equation of Riccati, describing the variation of the deflection depending on the load, was obtained. Hence, the results deduced show a good correlation with experimental curves
Preliminary Research on Data Abnormality Diagnosis Methods of Spacecraft Prec...IJERA Editor
For precision measuring the satellite equipments, providing technical support for satellite assembly, combined
with satellite small size, complex structure, satellite equipment shapes vary, and other characteristics, presently,
indirect method that using electronic theodolite to measure cube mirror are commonly used to obtain the relative
attitude of the respective devices. But in the actual measurement process, there are measurement errors in the
measurement data. How to detect anomalies in the data is the focus of this study. This paper proposes two
methods to detect abnormal data, that is mathematical geometric method and outlier detection methods. This
paper analyzes their theoretical basis and verifies the feasibility of the two methods through part of the actual
measurement data to.
A comprehensive literature review on different theories of laminated plates have been
reviewed and discussed thoroughly. It has been found that there are two main theories of
laminated plates which are known as linear and nonlinear theories. The two theories are
depending on the magnitude of deformation resulting from loading the given plates. The
difference between the two theories is that the deformations are small in the linear theory,
whereas they are finite or large in the nonlinear theory.
In comparisons between FEM and different numerical methods it has been found that
FEM can be considered of acceptable accuracy, and can also be applied to different
complicated geometries and shapes.
A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Ho...drboon
In this work, a novel finite element model using the mechanical homogenization techniques of the human annulus fibrosus (AF) is proposed to accurately predict relevant moduli of the AF lamella for tissue engineering application. A general formulation for AF homogenization was laid out with appropriate boundary conditions. The geometry of the fibre and matrix were laid out in such a way as to properly mimic the native annulus fibrosus tissue’s various, location-dependent geometrical and histological states. The mechanical properties of the annulus fibrosus calculated with this model were then compared with the results obtained from the literature for native tissue. Circumferential, axial, radial, and shear moduli were all in agreement with the values found in literature. This study helps to better understand the anisotropic nature of the annulus fibrosus tissue, and possibly could be used to predict the structure-function relationship of a tissue-engineered AF.
A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Ho...drboon
In this work, a novel finite element model using the mechanical homogenization techniques of the human annulus fibrosus (AF) is proposed to accurately predict relevant moduli of the AF lamella for tissue engineering application. A general formulation for AF homogenization was laid out with appropriate boundary conditions. The geometry of the fibre and matrix were laid out in such a way as to properly mimic the native annulus fibrosus tissue’s various, location-dependent geometrical and histological states. The mechanical properties of the annulus fibrosus calculated with this model were then compared with the results obtained from the literature for native tissue. Circumferential, axial, radial, and shear moduli were all in agreement with the values found in literature. This study helps to better understand the anisotropic nature of the annulus fibrosus tissue, and possibly could be used to predict the structure-function relationship of a tissue-engineered AF.
Dynamic Relaxation (DR) method is presented for the analysis of
geometrically linear laterally loaded, rectangular laminated plates. The
analysis uses the Mindlin plate theory which accounts for transverse shear
deformations. A computer program has been compiled. The convergence
and accuracy of the DR solutions of isotropic, orthotropic, and laminated
plates for elastic small deflection response are established by comparison
with different exact and approximate solutions. The present Dynamic
Relaxation (DR) method shows a good agreement with other analytical and
numerical methods used in the verification scheme.
It was found that: The convergence and accuracy of the DR solution is
dependent on several factors which include boundary conditions, mesh size
and type, fictitious densities, damping coefficients, time increment and
applied load. Also, the DR small deflection program using uniform meshes
can be employed in the analysis of different thicknesses for isotropic,
orthotropic or laminated plates under uniform loads in a fairly good
accuracy.
Computational methods applied to materials modelingcippo1987Ita
Talk given to the 1st multidisciplinary conference of Italian researchers in Czechia.
This is a public engagement talk about computational tools to investigate materials properties.
Electroelastic Response of a Piezoelectric Semi-infinite Body with D∞ Symmetr...IJERA Editor
In this paper, with the aim of developing a nondestructive evaluation technique using piezoelectric signals in
wooden materials, we theoretically study the electroelastic field in a semi-infinite body with D∞ symmetry
subjected to surface friction parallel to the ∞-fold rotation axis. By applying the analytical technique previously
proposed by us, we formulate expressions for electroelastic field quantities, including electric potential, electric
field, electric displacement, elastic displacement, strain, and stress by using two “elastic displacement potential
functions” and two “piezoelastic displacement potential functions.” These potential functions and, consequently,
the electroelastic field quantities are formulated using Fourier transforms in order to satisfy electroelastic
boundary conditions. We carried out numerical calculations to correctly evaluate field quantities inside the body
and at its surface. As a result, we were successful in quantitatively elucidating the surface electric displacement
in response to the elastic stimulus of surface friction and suggested the possibility of a nondestructive evaluation
technique using piezoelectric signals.
TRANSIENT ANALYSIS OF PIEZOLAMINATED COMPOSITE PLATES USING HSDTP singh
Piezoelectric materials have excellent sensing and actuating capabilities have made them the most practical smart materials to integrate with laminated structures. Integrated structure system can be called a smart structure because of its ability to perform self-diagnosis and quick adaption to environment changes. An analytical procedure has been developed in the work based on higher order shear deformation theory subjected to electromechanical loading for investigating transient characteristics of smart material plates. For analysis two displacement models are to be considered i.e., model-1 accounts for strain in thickness direction is zero whereas in model-2 in-plane displacements are expanded as cubic functions of the thickness coordinate. Navier’s technique has been adopted for obtaining solutions of anti-symmetric cross–ply and angle-ply laminates of both model-1 and model-2 with simply supported boundary conditions. For obtaining transient response of a laminated composite plate attached with piezoelectric layer Newmark’s method has been used. Effect of thickness coordinate of composite laminated plates attached with piezoelectric layer subjected to electromechanical loadings is studied.
Applications of layered theory for the analysis of flexible pavementseSAT Journals
Abstract
Unbound granular materials used in untreated base/sub-base layers, exhibit nonlinear behavior under repeated wheel loads. The
properties of the granular materials play a significant role in the performance of these pavements. Therefore, accurate modelling of
the granular layers is essential in the evaluation of critical pavement responses under the application of loads, these materials exhibit
stress dependent characteristics. Thus, consideration of non-linearity in these layers is necessary for accurate estimation of the
pavement responses of a flexible pavement structure. The pavement responses are computed using Kenlayer computer program
developed by Huang. Using the Kenlayer program, this paper examines the effect of nonlinearity in granular on critical pavement
responses by conducting parametric analysis. The results indicate that the consideration of nonlinearity yields 23.13% reduction in
tensile strains at the bottom of bituminous layers and 0.76% increase in compressive strains on the top of the sub grade layers and
same surface deflections compared to the value obtained using linear elastic analysis. This indicates that nonlinear analysis is more
realistic and accurate.
Keywords: Granular layers, Kenlayer, Nonlinearity, Pavement responses.
Applications of layered theory for the analysis of flexible pavementseSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme. It was found that: The convergence and accuracy of the DR solution were dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...IRJESJOURNAL
Abstract:- Buckling analysis of a carbon nanotube (CNT) embedded in Pasternak’s medium is investigated. Eringen’s nonlocal elasticity theory in conjunction with the first-order Donell’s shell theory is used. The governing equilibrium equations are obtained and solved for CNTs subjected to mechanical loads and embedded in Winkler-Pasternak’s medium. Effects of nonlocal parameter, radius and length of CNT, as well as the foundation parameters on buckling of CNT are investigated. Comparison with the available results is made.
Micromechanical Modeling of the Macroscopic Behavior of Soft Ferromagnetic Co...IJMERJOURNAL
ABSTRACT: Soft ferromagnetic composites consist of ferromagnetic particles embedded within nonmagnetic electrically insulating matrix. The aims of this article are twofold. In the first one, multiaxial nonlinear isothermal constitutive equations that govern the behavior of soft ferromagnetic materials are generalized to include the temperature and hysteretic effects. The second aim consists of developing a micromechanical analysis which takes into account the detailed interaction between the phases, and establishes the instantaneous concentration tensors which relate between the local magneto-thermo-elastic field and the externally applied loading. The ferromagnetic particles constituents employed in the micromechanical analysis are governed by the developed magneto-thermo-elastic coupled hysteretic constitutive relations. With the established concentration tensors, the macroscopic (global) re- sponse of the composite can be readily determined at any instant of loading. The offered micromechanical modeling is capable of predicting the response of soft ferromagnetic composites that are subjected to various types of magneto-thermo-elastic loadings, and it can be employed by the designerto easily determine the composite response for a particular desired application.
Numerical Analysis of Engineered Steel Fibers as Shear Reinforcement in RC BeamsP singh
Using suitable fibers and additives in concrete to enhance its performance is an important consideration in the concrete industry with regard to the structural aspects of concrete. The purpose of this project is to investigate numerically the effectiveness of the engineered steel fiber as shear reinforcement in RC beams. Here steel fibers completely replaces the shear reinforcement (stirrups & links). The dimension of beam taken was 1000*150*150 mm with aspect ratio 80. The beams were reinforced with 10 mm steel bars as secondary reinforcement and 12 mm bars as main reinforcement on the tension side. Numerical analysis using ANSYS R16.1 software package was carried out. The load-deflection curves for the beams with different dosage of fibers were drawn superimposing their numerical values. Initially, in all three cases the curve was linear elastic and about 80% of ultimate load they tend to be non-linear. It was observed that there was fair agreement between the results which indicates some favourable aspects concerning the use of steel fibres as shear reinforcement in concrete beams. It was investigated that the inclusion of steel fibres (Hook End Type) improves the shear strength of RC beams without stirrups by improving the matrix between concrete and steel fibers. Thus this project focuses in the design and analysis using the software ANSYS R16.1 for an alternative steel reinforcement with better or equivalent performance.
A composite material can be defined as a combination of two or more materials that
gives better properties than those of the individual components used alone. In contrast to
metallic alloys, each material retains its separate chemical, physical, and mechanical
properties. The two constituents are reinforcement and a matrix. The main advantages of
composite materials are their high strength and stiffness combined with low density when
compared to classical materials. Micromechanical approach is found to be more suitable for
the analysis of composite materials because it studies the volume proportions of the
constituents for the desired lamina stiffness and strength.
Linear And Nonlinear Analytical Modeling of Laminated Composite Beams In Thre...researchinventy
The large current development of aerospace and automotive technologies is based on the use of composite materials which provide significant weight savings compared to their mechanical characteristics. Correct dimensioning of composite structures requires a thorough knowledge of their behavior in small as in large deflection.This work aims to simulate linear and nonlinear behavior of laminates composites under threepoint bending test. The used modelization is based on first-order shear deformation theory (FSDT), classical plate theory (CPT) and Von-Karman’s equations for large deflection. A differential equation of Riccati, describing the variation of the deflection depending on the load, was obtained. Hence, the results deduced show a good correlation with experimental curves
Preliminary Research on Data Abnormality Diagnosis Methods of Spacecraft Prec...IJERA Editor
For precision measuring the satellite equipments, providing technical support for satellite assembly, combined
with satellite small size, complex structure, satellite equipment shapes vary, and other characteristics, presently,
indirect method that using electronic theodolite to measure cube mirror are commonly used to obtain the relative
attitude of the respective devices. But in the actual measurement process, there are measurement errors in the
measurement data. How to detect anomalies in the data is the focus of this study. This paper proposes two
methods to detect abnormal data, that is mathematical geometric method and outlier detection methods. This
paper analyzes their theoretical basis and verifies the feasibility of the two methods through part of the actual
measurement data to.
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reviewed and discussed thoroughly. It has been found that there are two main theories of
laminated plates which are known as linear and nonlinear theories. The two theories are
depending on the magnitude of deformation resulting from loading the given plates. The
difference between the two theories is that the deformations are small in the linear theory,
whereas they are finite or large in the nonlinear theory.
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numerical methods used in the verification scheme.
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Dynamic Homogenisation of randomly irregular viscoelastic metamaterials
1. Dynamic homogenisation of randomly
irregular viscoelastic metamaterials
S. Adhikari1
, T. Mukhopadhyay2
, A. Batou3
1
Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea
University, Bay Campus, Swansea, Wales, UK, Email: S.Adhikari@swansea.ac.uk, Twitter:
@ProfAdhikari, Web: http://engweb.swan.ac.uk/~adhikaris
1
Department of Engineering Science, University of Oxford, Oxford, UK
3
Liverpool Institute for Risk and Uncertainty, University of Liverpool, Liverpool, UK
University of Texas at Austin: Swansea-Texas Strategic
Partnership
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 1
4. My research interests
• Development of fundamental computational methods for
structural dynamics and uncertainty quantification
A. Dynamics of complex systems
B. Inverse problems for linear and non-linear dynamics
C. Uncertainty quantification in computational mechanics
• Applications of computational mechanics to emerging
multidisciplinary research areas
D. Vibration energy harvesting / dynamics of wind turbines
E. Computational mechanics for mechanics and multi-scale
systems
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 4
5. Outline
1 Introduction
Regular lattices
Irregular lattices
2 Formulation for the viscoelastic analysis
3 Equivalent elastic properties of randomly irregular lattices
4 Effective properties of irregular lattices: uncorrelated
uncertainty
General results - closed-form expressions
Special case 1: Only spatial variation of the material properties
Special case 2: Only geometric irregularities
Special case 3: Regular hexagonal lattices
5 Effective properties of irregular lattices: correlated
uncertainty
6 Results and discussions
Spatially correlated irregular elastic lattices
Viscoelastic properties of regular lattices
Spatially correlated irregular viscoelastic lattices
7 Conclusions
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 5
6. Lattice based metamaterials
• Metamaterials are artificial materials designed to outperform
naturally occurring materials in various fronts. These include, but
are not limited to, electromagnetics, acoustics, optics, terahertz,
infrared, dynamics and mechanical properties.
• Lattice based metamaterials are abundant in man-made and
natural systems at various length scales
• Lattice based metamaterials are made of periodic
identical/near-identical geometric units
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 6
7. Hexagonal lattices in 2D
• Among various lattice geometries (triangle, square, rectangle,
pentagon, hexagon), hexagonal lattice is most common (note
that hexagon is the highest “space filling” pattern in 2D).
• This talk is about in-plane elastic properties of 2D hexagonal
lattice structures - commonly known as “honeycombs”
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 7
8. Lattice structures - nano scale
Illustrations of a single layer graphene sheet and a born nitride nano sheet
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 8
9. Lattice structures - nature
Top left: cork, top right: balsa, next down left: sponge, next down right: trabecular bone, next down left: coral, next down right: cuttlefish bone, bottom le
leaf tissue, bottom right: plant stem, third column - epidermal cells (from web.mit.edu)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 9
10. Some questions of general interest
• Shall we consider lattices as “structures” or “materials” from a
mechanics point of view?
• At what relative length-scale a lattice structure can be
considered as a material with equivalent elastic properties?
• In what ways structural irregularities “mess up” equivalent elastic
/ viscoelastic properties? Can we evaluate it in a quantitative as
well as in a qualitative manner?
• What is the consequence of random structural irregularities on
the homogenisation approach in general? Can we obtain
statistical measures?
• How can we efficiently compute equivalent elastic / viscoelastic
properties of random lattice structures?
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 10
11. Regular lattice structures
• Hexagonal lattice structures have been modelled as a continuous
solid with an equivalent elastic moduli throughout its domain.
• This approach eliminates the need of detail finite element
modelling of lattices in complex structural systems like sandwich
structures.
• Extensive amount of research has been carried out to predict the
equivalent elastic / viscoelastic properties of regular lattices
consisting of perfectly periodic hexagonal cells (El-Sayed et al.,
1979; Gibson and Ashby, 1999; Goswami, 2006; Masters and
Evans, 1996; Zhang and Ashby, 1992).
• Analysis of two dimensional hexagonal lattices dealing with
in-plane elastic properties are commonly based on an unit cell
approach, which is applicable only for perfectly periodic cellular
structures.
• For the dynamic analysis of perfectly periodic structures,
Floquet-Bloch theorem is normally employed to characterise
wave propagation.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 11
12. Equivalent elastic properties of regular hexagonal lattices
• Unit cell approach - Gibson and Ashby (1999)
(a) Regular hexagon (θ = 30◦) (b) Unit cell
• We are interested in homogenised equivalent in-plane elastic
properties
• This way, we can avoid a detailed structural analysis considering
all the beams and treat it as a material
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 12
13. Equivalent elastic properties of regular hexagonal lattices
• The cell walls are treated as beams of thickness t, depth b and
Young’s modulus Es. l and h are the lengths of inclined cell walls
having inclination angle θ and the vertical cell walls respectively.
• The equivalent elastic properties are:
E1 = Es
t
l
3
cos θ
(h
l + sin θ) sin2
θ
(1)
E2 = Es
t
l
3
(h
l + sin θ)
cos3 θ
(2)
ν12 =
cos2
θ
(h
l + sin θ) sin θ
(3)
ν21 =
(h
l + sin θ) sin θ
cos2 θ
(4)
G12 = Es
t
l
3 h
l + sin θ
h
l
2
(1 + 2h
l ) cos θ
(5)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 13
14. Finite element modelling and verification
• A finite element code has been developed to obtain the in-plane
elastic moduli numerically for hexagonal lattices.
• Each cell wall has been modelled as an Euler-Bernoulli beam
element having three degrees of freedom at each node.
• For E1 and ν12: two opposite edges parallel to direction-2 of the
entire hexagonal lattice structure are considered. Along one of
these two edges, uniform stress parallel to direction-1 is applied
while the opposite edge is restrained against translation in
direction-1. Remaining two edges (parallel to direction-1) are
kept free.
• For E2 and ν21: two opposite edges parallel to direction-1 of the
entire hexagonal lattice structure are considered. Along one of
these two edges, uniform stress parallel to direction-2 is applied
while the opposite edge is restrained against translation in
direction-2. Remaining two edges (parallel to direction-2) are
kept free.
• For G12: uniform shear stress is applied along one edge keeping
the opposite edge restrained against translation in direction-1
and 2, while the remaining two edges are kept free.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 14
15. Finite element modelling and verification
0 500 1000 1500 2000
0.9
0.95
1
1.05
1.1
1.15
1.2
Number of unit cells
Ratioofelasticmodulus E
1
E
2
ν
12
ν
21
G
12
θ = 30◦
, h/l = 1.5. FE results converge to analytical predictions after
1681 cells.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 15
16. Irregular lattice structures
(c) Cedar wood (d) Manufactured honeycomb core
(e) Graphene image (f) Fabricated CNT surface
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 16
17. Irregular lattice structures
• A significant limitation of the aforementioned unit cell approach is
that it cannot account for the spatial irregularity, which is
practically inevitable.
• Spatial irregularity may occur due to manufacturing uncertainty,
structural defects, variation in temperature, pre-stressing and
micro-structural variabilities.
• To include the effect of irregularity, voronoi honeycombs have
been considered in several studies.
• The effect of different forms of irregularity on elastic properties
and structural responses of hexagonal lattices are generally
based on direct finite element (FE) simulation.
• In the FE approach, a small change in geometry of a single cell
may require completely new geometry and meshing of the entire
structure. In general this makes the entire process time
consuming and tedious.
• The problem becomes worse for uncertainty quantification of the
responses, where the expensive finite element model is needed
to be simulated for a large number of samples while using a
Monte Carlo based approach.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 17
18. Examples of some viscoelastic materials
(g) Viscoelastic foam (h) Viscoelastic membrane
(i) Viscoelastic sheet (j) Viscoelastic sheet
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 18
19. Overview of the viscoelastic behaviour
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 19
20. Fundamental equation for the viscoelastic behaviour
• When a linear viscoelastic model is employed, the stress at
some point of a structure can be expressed as a convolution
integral over a kernel function as
σ(t) =
t
−∞
g(t − τ)
∂ (τ)
∂τ
τ (6)
• t ∈ R+
is the time, σ(t) is stress and (t) is strain.
• The kernel function g(t) also known as ‘hereditary function’,
‘relaxation function’ or ‘after-effect function’ in the context of
different subjects.
• In practice, the kernel function is often defined in the frequency
domain (or Laplace domain). Taking the Laplace transform of
Equation (6), we have
¯σ(s) = s ¯G(s)¯(s) (7)
Here ¯σ(s), ¯(s) and ¯G(s) are Laplace transforms of σ(t), (t) and
g(t) respectively and s ∈ C is the (complex) Laplace domain
parameter.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 20
21. Mathematical representation of the kernel function
• The kernel function in Equation (7) is a complex function in the
frequency domain. For notational convenience we denote
¯G(s) = ¯G(iω) = G(ω) (8)
where ω ∈ R+
is the frequency.
• The complex modulus G(ω) can be expressed in terms of its real
and imaginary parts or in terms of its amplitude and phase as
follows
G(ω) = G (ω) + iG (ω) = |G(ω)|eiφ(ω)
(9)
The real and imaginary parts of the complex modulus, that is,
G (ω) and G (ω) are also known as the storage and loss moduli
respectively.
• One of the main restriction on the form of the kernel function
comes from the fact that the response of the structure must not
start before the application of the forces.
• This causality condition imposes a mathematical relationship
between real and imaginary parts of the complex modulus,
known as Kramers-Kronig relations
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 21
22. Mathematical representation of the kernel function
• Kramers-Kronig relations specifies that the real and imaginary
parts should be related by a Hilbert transform pair, but can be
general otherwise. Mathematically this can be expressed as
G (ω) = G∞ +
2
π
∞
0
uG (u)
ω2 − u2
du
G (ω) =
2ω
π
∞
0
G (u)
u2 − ω2
du
(10)
where the unrelaxed modulus G∞ = G(ω → ∞) ∈ R.
• Equivalent relationships linking the modulus and the phase of
G(ω) can expressed as
ln |G (ω)| = ln |G∞| +
2
π
∞
0
uφ(u)
ω2 − u2
du
φ(ω) =
2ω
π
∞
0
ln |G(u)|
u2 − ω2
du
(11)
• Complex modulus derived using a physics based principle
automatically satisfy these conditions. However, there can be
many other function which would also satisfy these condition.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 22
23. Viscoelastic models
(k) Maxwell model (l) Voigt model
(m) Standard linear model (n) Generalised Maxwell model
Figure: Springs and dashpots based models viscoelastic materials.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 23
24. Viscoelastic models
The viscoelastic kernel function can be expressed for the four models
as
• Maxwell model:
g(t) = µe−(µ/η)t
U(t) (12)
• Voigt model:
g(t) = ηδ(t) + µU(t) (13)
• Standard linear model:
g(t) = ER 1 − (1 −
τσ
τ
)e−t/τ
U(t) (14)
• Generalised Maxwell model:
g(t) =
n
j=1
µj e−(µj /ηj )t
U(t) (15)
Models similar to this is also known as the Pony series model.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 24
25. Viscoelastic models
Viscoelastic
model
Complex modulus
Biot model G(ω) = G0 +
n
k=1
ak iω
iω+bk
Fractional deriva-
tive
G(ω) = G0+G∞(iωτ)β
1+(iωτ)β
GHM G(ω) = G0 1 + k αk
−ω2
+2iξk ωk ω
−ω2+2iξk ωk ω+ω2
k
ADF G(ω) = G0 1 +
n
k=1 ∆k
ω2
+iωΩk
ω2+Ω2
k
Step-function G(ω) = G0 1 + η 1−e−st0
st0
Half cosine model G(ω) = G0 1 + η 1+2(st0/π)2
−e−st0
1+2(st0/π)2
Gaussian model G(ω) = G0 1 + η eω2
/4µ
1 − erf iω
2
√
µ
Complex modulus for some viscoelastic models in the frequency
domain
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 25
26. The Biot Model
• We consider that each constitutive element of a hexagonal unit
within the lattice structure is modelled using viscoelastic
properties. For simplicity, we use Biot model with only one term.
Frequency dependent complex elastic modulus for an element is
expresses as
E(ω) = ES 1 +
iω
µ + iω
(16)
where µ and are the relaxation parameter and a constant
defining the ‘strength’ of viscosity, respectively. Es is the intrinsic
Young’s modulus.
• The amplitude of this complex elastic modulus is given by
|E(ω)| = ES
µ2 + ω2 (1 + )
2
µ2 + ω2
(17)
• The phase (φ) of this complex elastic modulus is given by
φ E(ω) = tan−1 µω
µ2 + ω2(1 + )
(18)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 26
27. Mathematical idealisation of irregularity in lattice structures
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 27
28. Irregular honeycombs
• Random spatial irregularity in cell angle is considered in this
study.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 28
29. Irregular lattice structures
• Direct numerical simulation to deal with irregularity in lattice
structures may not necessarily provide proper understanding of
the underlying physics of the system. An analytical approach
could be a simple, insightful, yet an efficient way to obtain
effective elastic properties of lattice structures.
• This work develops a structural mechanics based analytical
framework for predicting equivalent in-plane elastic properties of
irregular lattices having spatially random variations in cell angles.
• Closed-form analytical expressions will be derived for equivalent
in-plane elastic properties.
• An approach based on the frequency-domain representation of
the viscoelastic property of the constituent elements in the cells
is used.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 29
30. The philosophy of the analytical approach for irregular lattices
(a)
Typical representation of an irregular lattice (b) Representative unit
cell element (RUCE) (c) Illustration to define degree of irregularity (d)
Unit cell considered for regular hexagonal lattice by Gibson and
Ashby (1999).
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 30
31. The idealisation of RUCE and the bottom-up homogenisation approach
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 31
32. Unit cell geometry
(a) Classical unit cell for regular lattices (b) Representative unit cell
element (RUCE) geometry for irregular lattices
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 32
33. RUCE and free-body diagram for the derivation of E1
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 33
34. RUCE and free-body diagram for the derivation of E2
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 34
35. RUCE and free-body diagram for the derivation of G12
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 35
36. Equivalent E1, E2
Equivalent E1
E1v (ω) =
t3
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
l2
1ij l2
2ij l1ij + l2ij cos αij sin βij − sin αij cos βij
2
Esij 1 + ij
iω
µij + iω
( l1ij cos αij − l2ij cos βij
2
)
(19)
Equivalent Young’s moduli E2
E2v (ω) =
Lt3
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
Esij 1 + ij
iω
µij + iω
l2
3ij cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij (l1ij +l2ij )cos2 αij cos2 βij
(l1ij cos αij −l2ij cos βij )
2
−1
(20)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 36
37. Equivalent shear Modulus G12
Equivalent G12
G12v (ω) =
Lt3
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
Esij 1 + ij
iω
µij + iω
l2
3ij sin2
γij l3ij +
l1ij l2ij
l1ij +l2ij
−1
(21)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 37
38. Poisson’s ratios ν12, ν21
Equivalent ν12
ν12eq = −
1
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
cos αij sin βij − sin αij cos βij
cos αij cos βij
(22)
Equivalent ν21
ν21eq = −
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
l2
1ij l2
2ij l1ij + l2ij cos αij cos βij cos αij sin βij − sin αij cos βij
l1ij cos αij − l2ij cos βij
2
l2
3ij cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij (l1ij +l2ij )cos2 αij cos2 βij
(l1ij cos αij −l2ij cos βij )
2
(23)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 38
39. Only spatial variation of the material properties
• According to the notations used for a regular lattice by Gibson
and Ashby (1999), the notations for lattices without any structural
irregularity can be expressed as: L = n(h + l sin θ);
l1ij = l2ij = l3ij = l; αij = θ; βij = 180◦
− θ; γij = 90◦
, for all i and j.
• Using these transformations in case of the spatial variation of
only material properties, the closed-form formulae for compound
variation of material and geometric properties (equations 19–21)
can be reduced to:
E1v = κ1
t
l
3
cos θ
(h
l + sin θ) sin2
θ
(24)
E2v = κ2
t
l
3
(h
l + sin θ)
cos3 θ
(25)
and G12v = κ2
t
l
3 h
l + sin θ
h
l
2
(1 + 2h
l ) cos θ
(26)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 39
40. Only spatial variation of the material properties
• The multiplication factors κ1 and κ2 arising due to the
consideration of spatially random variation of intrinsic material
properties can be expressed as
κ1 =
m
n
n
j=1
1
m
i=1
1
Esij 1 + ij
iω
µij + iω
(27)
and κ2 =
n
m
1
n
j=1
1
m
i=1
Esij 1 + ij
iω
µij + iω
(28)
• In the special case when ω → 0 and there is no spatial variabilities in the
material properties of the lattice, all viscoelastic material properties
become identical (i.e. Esij = Es, µij = µ and ij = for i = 1, 2, 3, ..., m
and j = 1, 2, 3, ..., n) and subsequently the amplitude of κ1 and κ2
becomes exactly 1. This confirms that the expressions in 27 and 28 give
the necessary generalisations of the classical expressions of Gibson
and Ashby (1999) through 24–26.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 40
41. Only geometric irregularities
• In case of only spatially random variation of structural geometry
but constant viscoelastic material properties (i.e. Esij = Es,
µij = µ and ij = for i = 1, 2, 3, ..., m and j = 1, 2, 3, ..., n) the
19–21 lead to
E1v = ES 1 +
iω
µ + iω
ζ1 (29)
E2v = ES 1 +
iω
µ + iω
ζ2 (30)
G12v = ES 1 +
iω
µ + iω
ζ3 (31)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 41
42. Only geometric irregularities
• The random coefficients ζi (i = 1, 2, 3) are
ζ1 =
t3
L
n
j=1
m
i=1
(l1ij cos αij − l2ij cos βij )
m
i=1
l2
1ij l2
2ij (l1ij + l2ij ) (cos αij sin βij − sin αij cos βij )2
(l1ij cos αij − l2ij cos βij )2
(32)
ζ2 =
Lt3
n
j=1
m
i=1
(l1ij cos αij − l2ij cos βij )
m
i=1
l2
3ij cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij (l1ij +l2ij )cos2 αij cos2 βij
(l1ij cos αij −l2ij cos βij )2
−1
(33)
ζ3 =
Lt3
n
j=1
m
i=1
(l1ij cos αij − l2ij cos βij )
m
i=1
l2
3ij sin2
γij l3ij +
l1ij l2ij
l1ij +l2ij
−1
(34)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 42
43. Regular hexagonal lattices
• The geometric notations for regular lattices can be expressed as:
L = n(h + l sin θ); l1ij = l2ij = l3ij = l; αij = θ; βij = 180◦
− θ; γij = 90◦
, for
all i and j. Using these transformations, the expressions of in-plane
elastic moduli for regular hexagonal lattices (without the viscoelastic
effect) can be obtained.
• The in-plane Young’s moduli and shear modulus (viscosity dependent
in-plane elastic properties) can be expressed as
E1v = Es 1 +
iω
µ + iω
t
l
3
cos θ
(h
l
+ sin θ) sin2
θ
(35)
E2v = Es 1 +
iω
µ + iω
t
l
3
(h
l
+ sin θ)
cos3 θ
(36)
G12v = Es 1 +
iω
µ + iω
t
l
3 h
l
+ sin θ
h
l
2
(1 + 2h
l
) cos θ
(37)
• The amplitude of the elastic moduli obtained based on the above
expressions converge to the closed-form equation provided by Gibson
and Ashby (1999) in the limiting case of ω → 0.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 43
44. Regular uniform hexagonal lattices
• In the case of regular uniform lattices with θ = 30◦
, we have
E1v = E2v = 2.3ES 1 +
iω
µ + iω
t
l
3
(38)
• Similarly, in the case of shear modulus for regular uniform lattices
(θ = 30◦
)
G12v = 0.57ES 1 +
iω
µ + iω
t
l
3
(39)
• Regular viscoelastic lattices satisfy the reciprocal theorem
E2v ν12v = E1v ν21v = ES 1 +
iω
µ + iω
t
l
3
1
sin θ cos θ
(40)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 44
45. Random field model for material and geometric properties
• Correlated structural and material attributes can be modelled
random fields H (x, θ).
• The traditional way of dealing with random field is to discretise
the random field into finite number of random variables. The
available schemes for discretising random fields can be broadly
divided into three groups: (1) point discretisation (e.g., midpoint
method, shape function method, integration point method,
optimal linear estimate method); (2) average discretisation
method (e.g., spatial average, weighted integral method), and (3)
series expansion method (e.g., orthogonal series expansion).
• An advantageous alternative for discretising H (x, θ) is to
represent it in a generalised Fourier type of series as, often
termed as Karhunen-Lo`eve (KL) expansion.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 45
46. Karhunen-Lo`eve (KL) expansion
• Suppose, H (x, θ) is a random field with covariance function
ΓH(x1, x2) defined in the probability space (Θ, F, P). The KL
expansion for H (x, θ) takes the following form
H (x, θ) = ¯H (x) +
∞
i=1
λi ξi (θ) ψi (x) (41)
where {ξi (θ)} is a set of uncorrelated random variables.
• {λi } and {ψi (x)} are the eigenvalues and eigenfunctions of the
covariance kernel ΓH(x1, x2), satisfying the integral equation
N
ΓH(x1, x2)ψi (x1) dx1 = λi ψi (x2) (42)
• In practise, the infinite series of 41 must be truncated, yielding a
truncated KL approximation
˜H (x, θ) ∼= ¯H (x) +
M
i=1
λi ξi (θ) ψi (x) (43)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 46
47. Karhunen-Lo`eve (KL) expansion
• Gaussian and lognormal random fields have been considered.
The covariance function is represented as:
ΓαZ
= σ2
αZ
e(−|y1−y2|/by )+(−|z1−z2|/bz )
(44)
where by and bz are the correlation parameters at y and z
directions (that corresponds to direction - 1 and direction - 2
respectively). These quantities control the rate at which the
covariance decays.
• In a two dimensional physical space the eigensolutions of the
covariance function are obtained by solving the integral equation
analytically
λi ψi (y2, z2) =
a1
−a1
a2
−a2
Γ(y1, z1; y2, z2)ψi (y1, z1)dy1dz1 (45)
where −a1 y a1 and −a2 z a2.
• Assume the eigen-solutions are separable in y and z directions,
i.e.
ψi (y2, z2) = ψ
(y)
i (y2)ψ
(z)
i (z2) (46)
λi (y2, z2) = λ
(y)
i (y2)λ
(z)
i (z2) (47)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 47
48. Karhunen-Lo`eve (KL) expansion
• The solution of the integral equation reduces to the product of the
solutions of two equations of the form
λ
(y)
i ψ
(y)
i (y1) =
a1
−a1
e(−|y1−y2|/by )
ψ
(y)
i (y2)dy2 (48)
• The solution of this equation, which is the eigensolution (eigenvalues
and eigenfunctions) of an exponential covariance kernel for a
one-dimensional random field is obtained as
ψi (ζ) =
cos(ωi ζ)
a + sin(2ωi a)
2ωi
λi =
2σ2
αz
b
ω2
i + b2
for i odd
ψi (ζ) =
sin(ω∗
i ζ)
a −
sin(2ω∗
i
a)
2ωi ∗
λ∗
i =
2σ2
αz
b
ω∗2
i + b2
for i even
(49)
where b = 1/by or 1/bz and a = a1 or a2. ζ can be either y or z and ωi
presents the period of the random field.
• The final eigenfunctions are given by
ψk (y, z) = ψ
(y)
i (y)ψ
(z)
i (z) (50)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 48
49. Samples of the random fields
Spatial variability of the intrinsic elastic modulus (Es) with ∆m = 0.002
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 49
50. The degree of geometric irregularity
• To define the degree of irregularity, it is assumed that each
connecting node of the lattice moves randomly within a certain
radius (rd ) around the respective node corresponding to the
regular deterministic configuration. For physically realistic
variabilities, it is considered that a given node do not cross a
neighbouring node, that is
rd < min
h
2
,
l
2
, l cos θ (51)
• In each realization of the Monte Carlo simulation, all the nodes of
the lattice move simultaneously to new random locations within
the specified circular bounds. Thus, the degree of irregularity (r)
is defined as a non-dimensional ratio of the area of the circle and
the area of one regular hexagonal unit as
r =
πr2
d × 100
2l cos θ(h + l sin θ)
(52)
• The degree of irregularity (r) has been expressed as percentage
values for presenting the results.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 50
51. Samples of the random fields
Movement of the top vertices of a tessellating hexagonal unit cell with
respect to the corresponding deterministic locations (r = 6)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 51
52. Random geometric configurations
Structural configurations for a single random realisation of an irregular hexagonal lattice considering deterministic cell angle θ = 30◦ and
h/l = 1: (a) r = 0 (b) r = 2 (c) r = 4 (d) r = 6
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 52
53. Samples of random geometric configurations
Figure: Simulation bound of the structural configuration of an irregular
hexagonal lattice for multiple random realisations considering θ = 30◦
,
h/l = 1 and r = 6. The regular configuration is presented using red colour.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 53
54. Samples of random geometric configurations
• In randomly inhomogeneous correlated system, spatial variability
of the stochastic structural attributes are accounted, wherein
each sample of the Monte Carlo simulation includes the spatially
random distribution of structural and materials attributes with a
rule of correlation.
• The spatial variability in structural and material properties (Es, µ
and ) are physically attributed by degree of structural irregularity
(r) and degree of material property variation (∆m) respectively.
• As the two Young’s moduli and shear modulus for low density
lattices are proportional to Esρ3
(Zhu et al., 2001), the
non-dimensional results for in-plane elastic moduli E1, E2, and
G12, unless otherwise mentioned, are presented as:
¯E1 =
E1eq
Esρ3
, ¯E2 =
E2eq
Esρ3
¯G12 =
G12eq
Esρ3
• ρ is the relative density of the lattice (defined as a ratio of the
planar area of solid to the total planar area of the lattice).
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 54
55. Spatially correlated irregular elastic lattices: E1
(a) θ = 30◦; h
l
= 1 (b) θ = 30◦; h
l
= 1.5
(c) θ = 45◦; h
l
= 1 (d) θ = 45◦; h
l
= 1.5
Figure: Effective Young’s modulus (E1) of irregular lattices
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 55
56. Spatially correlated irregular elastic lattices: E2
(a) θ = 30◦; h
l
= 1 (b) θ = 30◦; h
l
= 1.5
(c) θ = 45◦; h
l
= 1 (d) θ = 45◦; h
l
= 1.5
Figure: Effective Young’s modulus (E2) of irregular lattices with different
structural configurations considering correlated attributes
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 56
57. Spatially correlated irregular elastic lattices: G12
(a) θ = 30◦; h
l
= 1 (b) θ = 30◦; h
l
= 1.5
(c) θ = 45◦; h
l
= 1 (d) θ = 45◦; h
l
= 1.5
Figure: Effective shear modulus (G12) of irregular lattices with different
structural configurations considering correlated attributes
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 57
58. Viscoelastic properties of regular lattices: E1, E2, G12
(a) Effect of viscoelasticity on the magnitude and phase angle of E1 for regular hexagonal lattices (b) Effect of viscoelasticity on the
magnitude and phase angle of E2 for regular hexagonal lattices (c) Effect of viscoelasticity on the magnitude and phase angle of G12 for
regular hexagonal lattices
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 58
59. Viscoelastic properties of regular lattices
(a) Effect of variation of µ on the viscoelastic modulus of regular hexagonal lattices (considering a constant value of = 2) (b) Effect of
variation of on the viscoelastic modulus of regular hexagonal lattices (considering a constant value of µ = ωmax /5). Here Z represents
the viscoelastic moduli (i.e. E1, E2 and G12) and Z0 is the corresponding elastic modulus value for ω = 0.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 59
60. Spatially correlated irregular viscoelastic lattices: E1
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 60
61. Spatially correlated irregular viscoelastic lattices: E2
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 61
62. Spatially correlated irregular elastic lattices: G12
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 62
63. Probability density function: random geometry
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 63
64. Probability density function: random material property
Probability density function plots for the amplitude of the elastic moduli considering randomly inhomogeneous form of stochasticity for
different values of ∆m (i.e. coefficient of variation for spatially random correlated material properties, such as Es, µ and ). Results are
presented as a ratio of the values corresponding to irregular configurations and respective deterministic values (for a frequency of 800 Hz).
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 64
65. Combined material and geometric uncertianty
Probabilistic descriptions for the amplitudes of three effective viscoelastic properties corresponding to a frequency of 800 Hz considering
individual and compound effect of stochasticity in material and structural attributes with ∆cov = 0.006
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 65
66. Conclusions
• The effect of viscoelasticity on irregular hexagonal lattices is
investigated in frequency domain considering two different forms of
irregularity in structural and material parameters (spatially uncorrelated
and correlated).
• Spatially correlated structural and material attributes are considered to
account for the effect of randomly inhomogeneous form of irregularity
based on Karhunen-Lo`eve expansion.
• The two Young’s moduli and shear modulus are dependent on the
viscoelastic parameters. Two in-plane Poisson’s ratios depend only on
structural geometry of the lattice structure.
• The classical closed-form expressions for equivalent in-plane and out of
plane elastic properties of regular hexagonal lattice structures have
been generalised to consider geometric and material irregularity and
viscoelasticity.
• Using the principle of basic structural mechanics on a newly defined unit
cell with a homogenisation technique, closed-form expressions have
been obtained for E1, E2, ν12, ν21 and G12.
• The new results reduce to classical formulae of Gibson and Ashby for
the special case of no irregularities and no viscoelastic effect.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 66
67. Future works and collaborations
• Explicit dynamic analysis (inertia effect).
• Optimally designed variability (perfectly imperfect system)
• Band-gap analysis of viscoelastic metamaterials
• More general metamaterials with complex geometry
• Investigation of possible unusual properties arising due to randomness
and viscoelasticity
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 67
68. Closed-form expressions: Elastic Moduli
E1v (ω) =
t3
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
l2
1ij
l2
2ij
l1ij + l2ij cos αij sin βij − sin αij cos βij
2
Esij
1 + ij
iω
µij + iω
( l1ij cos αij − l2ij cos βij
2
)
(53)
E2v (ω) =
Lt3
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
Esij
1 + ij
iω
µij + iω
l2
3ij
cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij
l1ij +l2ij cos2 αij cos2 βij
l1ij cos αij −l2ij cos βij
2
−1
(54)
G12v (ω) =
Lt3
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
Esij
1 + ij
iω
µij + iω
l2
3ij
sin2 γij l3ij +
l1ij l2ij
l1ij +l2ij
−1
(55)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 68
69. Closed-form expressions: Poisson’s ratios
ν12eq = −
1
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
cos αij sin βij − sin αij cos βij
cos αij cos βij
(56)
ν21eq = −
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
l2
1ij
l2
2ij
l1ij + l2ij cos αij cos βij cos αij sin βij − sin αij cos βij
l1ij cos αij − l2ij cos βij
2
l2
3ij
cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij
l1ij +l2ij cos2 αij cos2 βij
l1ij cos αij −l2ij cos βij
2
(57)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 69
70. Some of our papers on this topic
1 Mukhopadhyay, T. and Adhikari, S., “Effective in-plane elastic properties
of quasi-random spatially irregular hexagonal lattices”, International
Journal of Engineering Science, (revised version submitted).
2 Mukhopadhyay, T., Mahata, A., Asle Zaeem, M. and Adhikari, S.,
“Effective elastic properties of two dimensional multiplanar hexagonal
nano-structures”, 2D Materials, 4[2] (2017), pp. 025006:1-15.
3 Mukhopadhyay, T. and Adhikari, S., “Stochastic mechanics of
metamaterials”, Composite Structures, 162[2] (2017), pp. 85-97.
4 Mukhopadhyay, T. and Adhikari, S., “Free vibration of sandwich panels
with randomly irregular honeycomb core”, ASCE Journal of Engineering
Mechanics,141[6] (2016), pp. 06016008:1-5..
5 Mukhopadhyay, T. and Adhikari, S., “Equivalent in-plane elastic
properties of irregular honeycombs: An analytical approach”,
International Journal of Solids and Structures, 91[8] (2016), pp.
169-184.
6 Mukhopadhyay, T. and Adhikari, S., “Effective in-plane elastic properties
of auxetic honeycombs with spatial irregularity”, Mechanics of Materials,
95[2] (2016), pp. 204-222.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 70
71. Adhikari, S., May 1998. Energy dissipation in vibrating structures. Master’s thesis, Cambridge University Engineering Department,
Cambridge, UK, first Year Report.
Adhikari, S., Woodhouse, J., May 2001. Identification of damping: part 1, viscous damping. Journal of Sound and Vibration 243 (1), 43–61.
El-Sayed, F. K. A., Jones, R., Burgess, I. W., 1979. A theoretical approach to the deformation of honeycomb based composite materials.
Composites 10 (4), 209–214.
Gibson, L., Ashby, M. F., 1999. Cellular Solids Structure and Properties. Cambridge University Press, Cambridge, UK.
Goswami, S., 2006. On the prediction of effective material properties of cellular hexagonal honeycomb core. Journal of Reinforced Plastics
and Composites 25 (4), 393–405.
Li, K., Gao, X. L., Subhash, G., 2005. Effects of cell shape and cell wall thickness variations on the elastic properties of two-dimensional
cellular solids. International Journal of Solids and Structures 42 (5-6), 1777–1795.
Masters, I. G., Evans, K. E., 1996. Models for the elastic deformation of honeycombs. Composite Structures 35 (4), 403–422.
Zhang, J., Ashby, M. F., 1992. The out-of-plane properties of honeycombs. International Journal of Mechanical Sciences 34 (6), 475 – 489.
Zhu, H. X., Hobdell, J. R., Miller, W., Windle, A. H., 2001. Effects of cell irregularity on the elastic properties of 2d voronoi honeycombs.
Journal of the Mechanics and Physics of Solids 49 (4), 857–870.
Zhu, H. X., Thorpe, S. M., Windle, A. H., 2006. The effect of cell irregularity on the high strain compression of 2d voronoi honeycombs.
International Journal of Solids and Structures 43 (5), 1061 – 1078.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 70