Variational Treatment and Stability Analysis of Coupled Electro-Mechanics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by a representative model problem. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of Micro- and Macro-Instability Phenomena in Computational Homogenization of Finite Electro-Statics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by representative model problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Multi-Scale Stability Analysis of Periodic Electroactive Polymer Composites at Finite Strains

This work is dedicated to multi-scale stability analysis, especially macroscopic and microscopic stability analysis of periodic electroactive polymer (EAP) composites with embedded fibers. Computational homogenization is considered to determine the response of materials at macro-scale depending on the selected representative volume element (RVE) at micro-scale [4, 5]. The quasi-incompressibility condition is considered by implementing a four-field variational formulation on the RVE, see [7]. Based on the works [1–3, 6, 8] the macroscopic instabilities are determined by the loss of strong ellipticity of homogenized moduli. On the other hand, the bifurcation type microscopic instabilities are treated exploiting the Bloch-Floquet wave analysis in context of finite element discretization, which allows to detect the changed critical size of periodicity of the microstructure and critical macroscopic loading points. Finally, representative numerical examples are given which demonstrate the onset of instability surfaces, the stable macroscopic loading ranges, and further a periodic buckling mode at a microscopic instability point is presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter–identification of macroscopic material models based on virtual testing of given material mesostructures

For many heterogeneous materials such as composites and polycrystals, the material modeling for the constituents on a representative mesoscale can be considered as known, including concrete values of their inherent material parameters. Typical examples are isotropic elastic–plastic models for the constituents of composites or anisotropic crystal–plasticity models for the grains of polycrystals. This knowledge can be exploited with regard to the modeling of the homogenized macroscopic response. In particular, parameters in macroscopic models may be identified by virtual experiments provided by a computational deformation–driving of representative mesostructures. This paper outlines the general concept for the parameter–identification of macroscopic materialmodels based on the virtual testing of given material mesostructures. The virtual test data are obtained in the form of multi–dimensional stress–strain paths by applying different deformation gradients to a given mesostructure. After specifying a corresponding macroscopic material model covering the observed effects on the macroscale, the material parameters are identified by a least–square–type optimization procedure that optimizes the macroscopic material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structural damage accumulation and stable postcritical deformation of composite materials

The macroscopic failure of composite materials is preceded by complex multilevel processes accompanied by accumulation and localization of damaged centers and formation of a failure cluster. Therefore, the study of these mechanisms is one of the basic problems for the mechanics of modern composite materials used in aerospace engineering. The formation of a theory of the stable postcritical deformation of the work-softening media is considered. The pseudo-plastic deformation affected by structural damage of granular composites is investigated within the framework of the considered two-level structurally phenomenological model of heterogeneous media. The stable evolution of the interconnected processes is accompanied by stress redistributions, partial or complete unloading, and strain or damage localization that are one of the main causes of implementation of the postcritical deformation stage. The numerical calculation results of inelastic deformation and failure of the periodic unidirectional fiber-reinforced composites are presented under conditions of the displacement-controlled transverse proportional loading mode. The main mechanisms of the work-softening behavior for the indicated type of materials are described in the macro-homogeneous stress-strain states. Macroscopically, the failure of heterogeneous media as a result of postcritical deformation and the loss of stability of damage accumulation depends on the stiffness of the loading system. When a deformable body is fixed on the closed surface with sufficiently but not infinitely large coefficients of stiffness, it is possible to observe the equilibrium development of the localized volumes of work-softening and damage. The constitutive equations for the work-softening isotropic, transverse isotropic, and orthotropic media are presented. The effect of the loading system on the stability of deformation, damage accumulation, and failure under monotone and nonmonotone triaxial loading was studied. The growth of failure strains with increase in stiffness of the loading system and unequal resistance of heterogeneous body are registered and investigated. A preventive unloading method is offered for the mathematical modeling of the damage accumulation during the testing of the materials on the servo-controlled systems. The displacement-controlled mode is simulated by a series of soft loading and unloading cycles. The detected phenomenon of failure where the unloading leads to stress-strain diagrams with a negative slope of the descending branch was not found either in the displacement or stress-controlled monotone loading mode.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998, Riga, Latvia.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 234–250, March–April, 1998.     

An exact deformation analysis for the magneto-electro-elastic fiber-reinforced thin plate

A closed form expressions for bending problem of magneto-electro-elastic (MEE) rectangular thin plates are derived, the exact solutions for the deformation behaviors of the fiber-reinforced BaTiO₃/CoFe₂O₄ composites subjected to certain types of surface loads are analytically obtained. Based on Kirchhoff thin-plate theory, structural characteristics such as elastic displacements, electric potential and magnetic induction for magneto-electro-elastic (MEE) rectangular plates are investigated, the governing equation in terms of the transverse displacement is presented in a rather compact form due to the omission of the transverse shear deformation and rotatory inertia. The material coefficients for the MEE plate can be uniquely expressed by the volume-fraction (v.f.) of piezoelectric constituent BaTiO₃ in the fiber-reinforced composite, and are tabulated with 25% offset of the volume-fraction. The deformation variations of the MEE thin plate with closed-circuit electric restriction are evaluated analytically according to their specified boundary conditions, and the effects of the volume-fractions on the deformations variations are discussed. It can be found that all the results obtained by using the proposed model have reached good agreements with the other available research works, whereas, the present study provides a much simpler way in seeking the analytic solutions for the interactively coupled quantities of a multiphase medium.     

Material stability analysis of particle methods

Material instabilities are precursors to phenomena such as shear bands and fracture. Therefore, numerical methods that are intended for failure simulation need to reproduce the onset of material instabilities with reasonable fidelity. Here the effectiveness of particle discretizations in reproducing of the onset of material instabilities is analyzed in two dimensions. For this purpose, a simplified hyperelastic law and a Blatz–Ko material are used. It is shown that the Eulerian kernels used in smooth particle hydrodynamics severely distort the domain of material stability, so that material instabilities can occur in stress states that should be stable. In particular, for the uniaxial case, material instabilities occur at much lower stresses, which is often called the tensile instability. On the other hand, for Lagrangian kernels, the domain of material stability is reproduced very well. We also show that particle methods without stress points exhibit instabilities due to rank deficiency of the discrete equations. AMS subject classification 74S30     

FEM-analysis of a multiferroic nanocomposite – comparison of experiment and numerical simulation

The combination of electric and magnetic materials opens new possibilities in the field of sensor technologies and data storage [1]. These magneto-electric (ME) materials have the property to change a physical ferroic quantity into another, i.e. a magnetic field can change the electric polarization and vice versa. The combination of multiple ferroic characteristics within materials is called multiferroic. Since magneto-electric single-phase materials are rare in nature and typically operate only at very low temperature, they are not favorable in technical applications. However, ME composites, consisting of ferroelectric and ferromagnetic phases, produce a strain-induced magneto-electric product property at room temperature [2]. In these composites, two different effects can be differentiated, the direct and the converse ME effect. The first one describes a polarization which is magnetically caused. In detail, a magnetic field is applied which produces a deformation of the magneto-active phase which is transferred to the electro-active phase and as a consequence this phase exhibits a polarization. Therefore, one can discover a strain-induced polarization. The second effect to observe is a magnetization caused by an electric field. In our contribution, we focus on a (1-3) composite, where cobalt ferrite nanopillars are embedded in a barium titanate matrix, see the experiments described in [3]. In the numerical simulations we compare the changes of the strain-induced inplane polarizations of the ferroelectric matrix with experimental measurements. Furthermore, we analyze the magneto-electric coupling coefficient. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Well-Posedness of Dynamical Equations of Magneto-electro-elasticity

A mathematical model of wave propagation in magneto-electro-elastic materials is obtained in the form of a symmetric hyperbolic system of the first-order partial differential equations. This model is a result of the qualitative analysis of the coupled time-dependent Maxwell’s equations and equations of elastodynamics which are considered together with constitutive relations in non-homogeneous anisotropic magneto-electro-elastic materials. Applying the theory and methods of symmetric hyperbolic systems, we have proved that the reported model of wave propagation in magneto-electro-elastic materials satisfies the Hadamards correctness requirements: solvability, uniqueness and stability with respect to perturbation of data.     

压电压磁复合材料中一对平行裂纹对弹性波的散射

利用Schmidt方法对压电压磁复合材料中一对平行对称裂纹对反平面简谐波的散射问题进行了分析,借助富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程．在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式,最终获得了应力强度因子、电位移强度因子、磁通量强度因子三者之间的关系．结果表明,压电压磁复合材料中平行裂纹动态反平面断裂问题的应力奇异性与一般弹性材料中的动态反平面断裂问题的应力奇异性相同,同时讨论了裂纹间的屏蔽效应．     

Analysis of Elastomeric Composites Based on Fiber Systems. 4. 3D Composites

The elastic properties of 3D elastomeric composite materials under large deformations are considered. The investigation is based on the structural macroscopic theory of stiff and soft composites. The results of micro- and macromechanical analyses of composite materials with compressible and poorly compressible matrices are presented. The character of interaction between the fibers of various reinforcing systems in these matrices is revealed. The deformation characteristics of the composites in tension and shear are presented as functions of their orientation and loading parameters. The evolution of the configuration of a composite material with a compressible matrix during loading is traced.     

Application of the finite-element method to improve the quasi-exact reinforcement theory of fibrous polymeric composites

In the paper, the WL quasi-exact reinforcement theory of fibrous polymeric composites is improved. An optimum compatibility condition related to the transverse shear problem for a unit cell, which brings solutions closest to reality, is derived. This condition is formulated in the form of a linear combination of maximum radial and circumferential displacements. Optimum coefficients of this combination are determined by comparing analytical and numerical solutions for a test specimen in the form of a rectangular thin plate, which is in a plane strain state and is subject to selected loading schemes. The analytic solutions are obtained for a homogenized material by using the WL reinforcement theory. The numerical solutions are found for an actual heterogeneous composite material by using the finite-element method, and they verify the WL reinforcement theory, in particular, the admissibility of Hills assumption. An analysis performed for two composite materials shows that the improved WL reinforcement theory gives adequate displacement fields.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 79–92, January–Febrauary, 2005.     

Homogenization methods for two-phase composites

Temperature, magnetic, and electric fields in composite materials are of interest in many engineering applications. In order to use them effectively in modern constructions, it is necessary to predict their effective homogenized properties. The aim of the present paper is to describe, both theoretically and numerically, the effective properties of two-phase composites and to compare the results obtained. The analysis is conducted for 2D and 3D cases.     

A 4D Composite Reinforced along Cube Diagonals with a Small Content of Yarns under Large Shear Deformations

The deformation behavior of a 4D composite reinforced along cube diagonals under large shear deformations is examined. The investigation is based on an applied theory which allows one to perform a macromechanical analysis of composite materials with small volume contents of reinforcing yarns to an accuracy sufficient in practice. Qualitative differences between the properties of such composites under large and small shear deformations are revealed. The evolution of the structural geometry of the deformed composite material is traced.     

Computational Homogenization of Piezoelectric Materials using FE2

Due to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, two-scale classical (first-order) homogenization of electro-mechanically coupled problems using a FE²-approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. The homogenization of the coupled problem is done using different representative volume elements (RVEs), which capture the microstructure of the inhomogeneous material, to represent the macro material response. Later this technique is used to determine the macroscopic and microscopic configurational forces on certain defects. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semi-analytic modelling of transversely isotropic magneto-electro-elastic materials under frictional sliding contact

Functionally graded magneto-electro-elastic (FGMEE) materials has been increasingly used in engineering applications, particularly in smart material or intelligent structure systems. This paper proposes a semi-analytical approach for sliding frictional contact problem between a rigid insulating sphere and a transversely isotropic FGMEE film and half-space based on frequency response functions (FRFs). Multilayered approximation is used to model the functionally graded material (FGM), and the FRFs for each MEE layer are derived explicitly. The unknown coefficients in FRFs are formulated by two matrix equations, and their efficient solution process is proposed. Based on the obtained FRFs, a highly efficient semi-analytical model (SAM) is developed which is able to solve the three-dimensional frictional contact of FGMEE materials with arbitrary layer designs. The model is validated with finite element method and the literature. Furthermore, the pressure/stress distribution and electric/magnetic potential are studied in different FGM designs to investigate the influence of material layout.     

压电压磁复合材料中界面裂纹对弹性波的散射

利用Schmidt方法分析了压电压磁复合材料中可导通界面裂纹对反平面简谐波的散射问题．经过富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程A·D2在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式．数值模拟分析了裂纹长度、波速和入射波频率对应力强度因子、电位移强度因子、磁通量强度因子的影响A·D2从结果中可以看出,压电压磁复合材料中可导通界面裂纹的反平面问题的应力奇异性形式与一般弹性材料中的反平面问题应力奇异性形式相同．     

A consistent second-order plate theory for monotropic material

Mathematical homogenization (or averaging) of composite materials, such as fibre laminates, often leads to non-isotropic homogenized (averaged) materials. Especially the upcoming importance of these materials increases the need for accurate mechanical models of non-isotropic shell-like structures. We present a second-order (or: Reissner-type) theory for the elastic deformation of a plate with constant thickness for a homogeneous monotropic material. It is equivalent to Kirchhoff's plate theory as a first-order theory for the special case of isotropy and, furthermore, shear-deformable and equivalent to R. Kienzler's theory as a second-order theory for isotropy, which implies further equivalences to established shear-deformable theories, especially the Reissner-Mindlin theory and Zhilin's plate theory. Details of the derivation of the theory will be published in a forthcoming paper [3]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)