Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

The homogenization results obtained by Bacca et al. (2013a), to identify the effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i) orthotropic nonlocal effects follow from homogenization of a dilute distribution of aligned elliptical holes and, in the limit case, of cracks; (ii) even under the dilute assumption and isotropic local effective behaviour, homogenization may lead to effective nonlocal orthotropic properties.     

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i) is positive definite only when the discrepancy tensor is negative defined; (ii) the non-local material symmetries are the same of the discrepancy tensor, and (iii) the non-local effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthotropic matrix.     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Constitutive equations for porous plane-strain gradient elasticity obtained by homogenization

This paper addresses the problem of plane-strain gradient elasticity models derived by higher-order homogenization. A microstructure that consists of cylindrical voids surrounded by a linear elastic matrix material is considered. Both plane-stress and plane-strain conditions are assumed and the homogenization is performed by means of a cylindrical representative volume element (RVE) subjected to quadratic boundary displacements. The constitutive equations for the equivalent medium at the macroscale are obtained analytically by means of the Airy’s stress function in conjunction with Fourier series. Furthermore, a failure criterion based on the maximum hoop stress on the void surface is formulated. A mixed finite-element formulation has been implemented into the commercial finite-element program Abaqus. Using the constitutive relations derived, numerical simulations were performed in order to compute the stress concentration at a hole with varying parameters of the constitutive equations. The results predicted by the model are discussed in comparison with the results of the theory of simple materials.     

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

基于迭代法的非线性弹性均质化研究

微观结构对复合材料的宏观力学性能具有至关重要的影响, 通过合理设计复合材料微观结构可以得到期望的宏观性能. 均质化方法作为一种有效的设计方法, 它从微观结构的角度出发, 利用均匀化的概念, 实现了对复合材料宏观力学性能的预测和设计. 而当考虑非线性因素, 均质化的实现就非常困难. 本文利用双渐近展开方法, 将位移按照宏观位移和微观位移展开, 推导了非线性弹性均质化方程. 通过直接迭代法, 对非线性弹性均质化方程进行了求解, 并给出了具体的迭代方法和实现步骤. 本文基于迭代步骤和非线性弹性均质化方程编写MATLAB 程序, 对3种典型本构关系的周期性多孔材料平面问题进行了计算, 对比细致模型的应变能、最大位移和等效泊松比, 对程序及迭代方法的准确性进行了验证. 之后对一种三元橡胶基复合材料进行多尺度均质化, 将其分为芯丝尺度和层间尺度. 用线弹性的均质化方法得到了芯丝尺度的等效弹性参数, 并将其作为层间尺度的材料参数. 在层间尺度应用非线性弹性均质化方法对结构进行计算, 得到材料的宏观等效性能, 并以实验结果为基准进行评价.      

Three-dimensional stochastic analysis using a perturbation-based homogenization method for elastic properties of composite material considering microscopic uncertainty

This paper discusses evaluation of influence of microscopic uncertainty on a homogenized macroscopic elastic property of an inhomogeneous material. In order to analyze the influence, the perturbation-based homogenization method is used. A higher order perturbation-based analysis method for investigating stochastic characteristics of a homogenized elastic tensor and an equivalent elastic property of a composite material is formulated.As a numerical example, macroscopic stochastic characteristics such as an expected value or variance, which is caused by microscopic uncertainty in material properties, of a homogenized elastic tensor and homogenized equivalent elastic property of unidirectional fiber reinforced plastic are investigated. The macroscopic stochastic variation caused by microscopic uncertainty in component materials such as Young’s modulus or Poisson’s ratio variation is evaluated using the perturbation-based homogenization method. The numerical results are compared with the results of the Monte-Carlo simulation, validity, effectiveness and a limitation of the perturbation-based homogenization method is investigated. With comparing the results using the first-order perturbation-based method, effectiveness of a higher order perturbation is also investigated.     

考虑电场梯度的挠曲电纳米板弯曲性能分析

具有尺度依赖的挠曲电效应在器件的设计中扮演着越来越关键的角色, 研究人员在微纳米尺度多物理场分析中进行了大量工作. 基于考虑挠曲电和电场梯度效应的弹性介电材料非经典理论, 以二维纳米板为例, 通过理论建模, 分析纳米板在弯曲问题中的力?电耦合行为. 根据Mindlin假设给出板的位移场和电势场的一阶截断, 选取板的材料为立方晶体(m3m点群), 将广义三维本构方程代入到高阶应力、高阶偶应力、高阶电位移和高阶电四极矩的表达式中得到相应的二维本构方程, 利用弹性电介质变分原理得到板的控制方程和边界上的线积分等式, 分别将二维本构方程和边界上外法线的方向余弦代入, 得到板的高阶弯曲方程、高阶电势方程以及对应的四边简支边界条件. 利用四边简支矩形板的高阶弯曲方程、高阶电势方程和相应的边界条件, 根据Navier解理论, 求解纳米板的电势场, 重点分析电场梯度对板内一阶电势的影响. 数值计算结果表明: 电场梯度对纳米板中由挠曲电效应产生的一阶电势有削弱作用, 且材料参数g₁₁越大, 一阶电势受到的削弱越大; 同时电场梯度的存在消除了纳米板在受横向集中载荷作用时一阶电势的奇异性. 本文是对具有挠曲电效应和电场梯度效应的纳米板结构分析理论的一个扩展, 为微纳米尺度器件的结构设计提供参考.      

Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models

Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell.     

Distribution-enhanced homogenization framework and model for heterogeneous elasto-plastic problems

Multi-scale computational models offer tractable means to simulate sufficiently large spatial domains comprised of heterogeneous materials by resolving material behavior at different scales and communicating across these scales. Within the framework of computational multi-scale analyses, hierarchical models enable unidirectional transfer of information from lower to higher scales, usually in the form of effective material properties. Determining explicit forms for the macroscale constitutive relations for complex microstructures and nonlinear processes generally requires numerical homogenization of the microscopic response. Conventional low-order homogenization uses results of simulations of representative microstructural domains to construct appropriate expressions for effective macroscale constitutive parameters written as a function of the microstructural characterization. This paper proposes an alternative novel approach, introduced as the distribution-enhanced homogenization framework or DEHF, in which the macroscale constitutive relations are formulated in a series expansion based on the microscale constitutive relations and moments of arbitrary order of the microscale field variables. The framework does not make any a priori assumption on the macroscale constitutive behavior being represented by a homogeneous effective medium theory. Instead, the evolution of macroscale variables is governed by the moments of microscale distributions of evolving field variables. This approach demonstrates excellent accuracy in representing the microscale fields through their distributions. An approximate characterization of the microscale heterogeneity is accounted for explicitly in the macroscale constitutive behavior. Increasing the order of this approximation results in increased fidelity of the macroscale approximation of the microscale constitutive behavior. By including higher-order moments of the microscale fields in the macroscale problem, micromechanical analyses do not require boundary conditions to ensure satisfaction of the original form of Hill's lemma. A few examples are presented in this paper, in which the macroscale DEHF model is shown to capture the microscale response of the material without re-parametrization of the microscale constitutive relations.     

A two-dimensional,higher-order,elasticity-based micromechanics model

A new, two-dimensional (2D) homogenization theory is proposed. The theory utilizes a higher-order, elasticity-based cell model (ECM) analysis. The material microstructure is modeled as a 2D periodic array of unit cells where each unit cell is discretized into four subregions (or subcells). The analysis utilizes a (truncated) eigenfunction expansion of up to fifth order for the displacement field in each subcell. The governing equations for the theory are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through an order consistent with the order of the subcell displacement field. The formulation is carried out independently of any specified constitutive models for the behavior of the individual phases (in the sense that the general governing equations hold for any constitutive model). The fifth order theory is subsequently specialized to a third order theory. Additionally, the higher order analyzes reduce to a theory equivalent to the original 2D method of cells (MOC) theory when all higher order terms are eliminated. The proposed 2D theory is the companion theory to an equivalent 3D theory [T.O. Williams, A three-dimensional, higher-order, elasticity-based micromechanics model, Int. J. Solids Struc., in press].Comparison of the predicted bulk and local responses with published results indicates that the theory accurately predicts both types of responses. The high degree of agreement between the current theory results and published results is due to the correct incorporation of the coupling effects between the local fields.The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, two-dimensional microstructures.     

Fissuration multiple par corrosion sous contrainte : Modélisation du comportement mécanique par une méthode d'homogénéisation

Stress Corrosion Cracking (SCC) phenomenon is characterised by initiation and propagation of surface cracks frequently multiple. In order to model the mechanical behaviour of such materials we propose, as for composites, the use of homogenization techniques. Two materials are considered, first one corresponding to the cracked external volume and the second to the internal safe material. The cracked volume is considered as a two phase material, i.e., elastic matrix containing elliptical voids. The overall behaviour of the equivalent material is obtained applying the usual homogenization rules. Comparison between simulations and experimental results is done.     

A micromechanical model of elastoplastic and damage behavior of a cohesive geomaterial

The present study is devoted to the development and validation of a nonlinear homogenization approach of the mechanical behavior of Callovo-Oxfordian argillites. The material is modeled as an heterogeneous composite composed of an elastoplastic clay matrix and of linear elastic or elastic damage inclusions. The macroscopic constitutive law is obtained by adapting the incremental method proposed by Hill [Hill, R., 1965. Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89–101]. The approach consists in formulating the macroscopic tangent operator of the material by considering the nonlinear local behavior of each phase. Due to the matrix/inclusion morphology of the microstructure of the argillite, a Mori–Tanaka scheme is considered for the localization step. The developed model is first compared to Finite Element calculations and then validated and applied for the prediction of the macroscopic stress–strain responses of argillites.     

Numerical simulation of fiber reinforced composite materials––two procedures

In this work, two methodologies for the analysis of unidirectional fiber reinforced composite materials are presented.The first methodology used is a generalized anisotropic large strains elasto-plastic constitutive model for the analysis of multiphase materials. It is based on the mixing theory of basic substance. It is the manager of the several constitutive laws of the different compounds and it allows to consider the interaction between the compounds of the composite materials. In fiber reinforced composite materials, the constitutive behavior of the matrix is isotropic, whereas the fiber is considered orthotropic. So, one of the constitutive model used in the mixing theory needs to consider this characteristic. The non-linear anisotropic theory showed in this work is a generalization of the classic isotropic plasticity theory (A Continuum Constitutive Model to Simulate the Mechanical Behavior of Composite Materials, PhD Thesis, Universidad Politécnica de Cataluña, 2000). It is based in a one-to-one transformation of the stress and strain spaces by means of a four rank tensor.The second methodology used is based on the homogenization theory. This theory divided the composite material problem into two scales: macroscopic and microscopic scale. In macroscopic level the composite material is assuming as a homogeneous material, whereas in microscopic level a unit volume called cell represents the composite (Tratamiento Numérico de Materiales Compuestos Mediante la teorı́ de Homogeneización, PhD Thesis, Universidad Politécnica, de Cataluña 2001). This formulation presents a new viewpoint of the homogenization theory in which can be found the equations that relate both scales. The solution is obtained using a coupled parallel code based on the finite elements method for each scale problem.     

Stress energy minimization as a tool in the material layout design of shallow shells

The present research deals with the compliance minimization problem of an elastic thin shallow shell subjected to simultaneous in-plane and bending loads. In this context, our goal is to lay out a given amount of material in the volume of a shell assuming that the distribution in the direction transversal to its middle surface S is homogeneous. The discussion hence reduces to the question of finding the optimal material arrangement on S. Similar problems were solved in the framework of two dimensional elasticity or Kirchhoff plate theory and the present research attempts to generalize these results. Following the pattern emerging from the above mentioned considerations, our research starts from the minimum compliance problem of a structure made of two elastic materials whose volumetric fractions are fixed. The existence of a solution to thus posed optimization task is guaranteed if the fine-scale microstructural composites are admitted in the analysis. Their constitutive tensors can be obtained by certain averaging ensuing from the theory of homogenization for periodic media. Additionally, by the Castigliano Theorem, the compliance minimization problem is equivalent to the one for structural stress energy. In turn, the lower estimation of the energy is achieved in two steps: (i) its modification by a certain energy-like functional, and (ii) utilizing the quasiconvexity property of thus obtained expression. As a result, formulae describing the effective stress energy of one-material shallow shell and the material distribution function are explicitly derived.     

Residual stresses in random elastic composites: nonlocal micromechanics-based models and first estimates of the representative volume element size

Random elastic composites with residual stresses are examined in this paper with the aim of understanding how the prestress may influence the overall mechanical properties of the composite. A fully non-local effective response is found in perfect analogy with the un-prestressed case examined in (Drugan and Willis, J. Mech. Phys. Solids 44(4):497–524, 1996). The second gradient approximation is considered and the impact of the residual stresses on the estimate of the RVE size is studied whenever the local response is used to describe the mechanical properties of the heterogeneous medium. To this aim, total and incremental formulations are worked out in this paper and the influence of both uniform and spatially varying prestresses are studied. Among other results, it is shown how rapid oscillations of relatively “small” residual stresses in most cases may result in the impossibility of describing the overall behavior of the composite with a local constitutive equation. On the other hand, prestresses with relatively high amplitudes and slow spatial oscillations may even reduce the RVE size required for approximating the mechanical properties of un-prestressed heterogeneous media with a local constitutive equation.