A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.     

Triply periodical particulate matrix compositesinvarying external stress fields

We consider a linear elastic composite medium, which consists of ahomogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in aperiodic arrayand subjected to inhomogeneous boundary conditions. The hypothesis of effectivefieldhomogeneity near the inclusions is used. The general integral equation obtained reducestheanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusionsinsome representative volume element (RVE) . The integral equation is solved by theFouriertransform method as well as by the iteration method of the Neumann series ( first-orderapproximation) . The nonlocal macroscopic constitutive equation relating the unit cellaverages ofstress and strain is derived in explicit closed forms either of a differential equation ofasecond-order or of an integral equation. The employed of explicit relations fornumericalestimations of tensors describing the local and nonlocal effective elastic properties aswell asaverage stresses in the composites containing simple cubic lattices of rigid inclusions andvoids areconsidered.     

Dielectric elastomer composites: A general closed-form solution in the small-deformation limit

A solution for the overall electromechanical response of two-phase dielectric elastomer composites with (random or periodic) particulate microstructures is derived in the classical limit of small deformations and moderate electric fields. In this limit, the overall electromechanical response is characterized by three effective tensors: a fourth-order tensor describing the elasticity of the material, a second-order tensor describing its permittivity, and a fourth-order tensor describing its electrostrictive response. Closed-form formulas are derived for these effective tensors directly in terms of the corresponding tensors describing the electromechanical response of the underlying matrix and the particles, and the one- and two-point correlation functions describing the microstructure. This is accomplished by specializing a new iterative homogenization theory in finite electroelastostatics (Lopez-Pamies, 2014) to the case of elastic dielectrics with even coupling between the mechanical and electric fields and, subsequently, carrying out the pertinent asymptotic analysis.Additionally, with the aim of gaining physical insight into the proposed solution and shedding light on recently reported experiments, specific results are examined and compared with an available analytical solution and with new full-field simulations for the special case of dielectric elastomers filled with isotropic distributions of spherical particles with various elastic dielectric properties, including stiff high-permittivity particles, liquid-like high-permittivity particles, and vacuous pores.     

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites. Part I: Closed form expression for the effective higher-order constitutive tensor

It is shown that second-order homogenization of a Cauchy-elastic dilute suspension of randomly distributed inclusions yields an equivalent second gradient (Mindlin) elastic material. This result is valid for both plane and three-dimensional problems and extends earlier findings by Bigoni and Drugan [Bigoni, D., Drugan, W.J., 2007. Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753] from several points of view: (i) the result holds for anisotropic phases with spherical or circular ellipsoid of inertia; (ii) the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material; (iii) a perfect energy match is found between heterogeneous and equivalent materials (instead of an optimal bound). The constitutive higher-order tensor defining the equivalent Mindlin solid is given in a surprisingly simple formula. Applications, treatment of material symmetries and positive definiteness of the effective higher-order constitutive tensor are deferred to Part II of the present article.     

正交各向异性Kagome蜂窝材料宏观等效力学性能

由于微结构的布局和尺寸的方向性,人造和天然的蜂窝材料都会不同程度呈现各向异性,其中正交各向异性的蜂窝材料较为常见.该文采用桁架模型推导了正交各向异性Kagome单胞蜂窝材料等效刚度和强度的解析表达式,给出了初始屈服函数和近似弹性屈曲强度,讨论了等效刚度与各向异性率和相对密度的关系.等效刚度的解析结果与单胞壁杆采用梁单元建模的刚架模型均匀化结果进行比较,结果令人满意.需要说明的是这类"组合蜂窝"材料具有多功能性和潜在的可设计性,正在受到人们关注.     

Thermoelastic characteristics of a composite solid with initial stresses

Thermoelastic problem for a composite solid with initial stresses is considered on the basis of the asymptotic homogenization method. The homogenized model is constructed by means of the two-scale asymptotic homogenization techniques. The major result of a present paper is that the effective (homogenized) thermoelastic characteristics of the composite material depend not only on local distributions of all types of material characteristics: local elastic properties, local thermoelastic properties, but also on local initial stresses. Therefore it is shown that for the inhomogeneous (composite) material local initial stresses contribute towards values of the effective characteristics of the material. This kind of interaction is not possible for the homogeneous materials. From the mathematical viewpoint, the asymptotic homogenization procedure is equivalent to the computation of G-limit of the corresponding operator. And the above noted phenomenon is based on the fact that in the considering case the G-limit of a sum is not equal to the sum of G-limits. The developed general homogenized model is illustrated in the particular case of the small initial stresses, which is common for the practical mechanical problems. The explicit formulas for the effective thermoelastic characteristics and numerical results are obtained for a laminated composite solid with the initial stresses.     

Nonlinear vibration analysis of laminated composite Mindlin micro/nano-plates resting on orthotropic Pasternak medium using DQM

The nonlocal nonlinear vibration analysis of embedded laminated microplates resting on an elastic matrix as an orthotropic Pasternak medium is investigated. The small size effects of micro/nano-plate are considered based on the Eringen nonlocal theory. Based on the orthotropic Mindlin plate theory along with the von Kármán geometric nonlinearity and Hamilton's principle, the governing equations are derived. The differential quadrature method (DQM) is applied for obtaining the nonlinear frequency of system. The effects of different parameters such as nonlocal parameters, elastic media, aspect ratios, and boundary conditions are considered on the nonlinear vibration of the micro-plate. Results show that considering elastic medium increases the nonlinear frequency of system. Furthermore, the effect of boundary conditions becomes lower at higher nonlocal parameters.     

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

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.     

Strain gradient elastic homogenization of bidimensional cellular media

The present paper aims at introducing an homogenization scheme for the determination of strain gradient elastic coefficients. This scheme is based on a quadratic extension of homogeneous boundary condition (HBC). It allows computing strain elastic effective tensors. This easy-to-handle computational procedure will then be used to construct overall behaviors and to verify some theoretical predictions on strain gradient elasticity.     

Spectral decomposition of the compliance fourth-rank tensor for orthotropic materials

Summary The compliance tensor related to orthotropic media is spectrally decomposed and its characteristic values are determined. Further, its idempotent tensors are estimated, giving rise to energy orthogonal states of stress and strain, thus decomposing the elastic potential in discrete elements. It is proven that the essential parameters, required for a complete characterisation of the elastic properties of an orthotropic medium, are the six eigenvalues of the compliance tensor, together with a set of three dimensionless parameters, the eigenangles θ, ϕ and ω. In addition, the intervals of variation of these eigenangles with respect to different values of the elastic constants are presented. Furthermore, bounds on Poisson's ratios are obtained by imposing the thermodynamical constraint on the eigenvalues to be strictly positive, as specified from the positive-definite character of the elastic potential. Finally, the conditions are investigated under which a family of orthotropic media behaves like a transversely isotropic or an isotropic one. Received 5 January 1999; accepted for publication 22 June 1999     

Effective thermoelastic properties of graded doublyperiodic particulate matrix composites in varying externalstress fields

We consider a linear elastic composite medium, which consists of a homogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in a doubly periodicarray and subjected to inhomogeneous boundary conditions. The hypothesis of effective fieldhomogeneity near the inclusions is used. The general integral equation obtained reduces theanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusions insome representative volume element (RVE) . The integral equation is solved by a modifiedversion of the Neumann series; the fast convergence of this method is demonstrated for concreteexamples. The nonlocal macroscopic constitutive equation relating the cell averages of stress andstrain is derived in explicit iterative form of an integral equation. A doubly periodic inclusion fieldin a finite ply subjected to a stress gradient along the functionally graded direction is considered.The stresses averaged over the cell are explicitly represented as functions of the boundaryconditions. Finally, the employed of proposed explicit relations for numerical simulations oftensors describing the local and nonlocal effective elastic properties of finite inclusion pliescontaining a simple cubic lattice of rigid inclusions and voids are considered. The local andnonlocal parts of average strains are estimated for inclusion plies of different thickness. Theboundary layers and scale effects for effective local and nonlocal effective properties as well as foraverage stresses will be revealed.     

An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate

Following a framework of elastic degradation and damage previously proposed by the authors, an ‘extended’ formulation of orthotropic damage in initially isotropic materials, based on volumetric/deviatoric decomposition, is presented. The formulation is founded on the concept of energy equivalence and makes use of second-order symmetric tensor damage variables. It is characterized by fourth-order damage-effect tensors (relating nominal to effective stresses and strains) built from the underlying second-order damage tensors and decomposed in product-form in isotropic and anisotropic parts. The formulation is developed in two steps. First, secant relations are established. In the isotropic case, the model embeds a path parameter allowing to range between pure volumetric to pure deviatoric damage. With the two undamaged material constants this makes a total of three constant parameters plus an evolving scalar damage variable, giving rise to a four-parameter model with two varying isotropic material coefficients. In the anisotropic case, the model is still characterized by the same three material constants plus three evolving variables which are the principal values of a second-order damage tensor. This leads to a six-parameter restricted form of orthotropic damage. In the second step, damage evolution rules are formulated in terms of a pseudo-logarithmic rate of damage. This allows to define meaningful conjugate forces that constitute a feasible space in which loading functions and damage evolution rules can be defined. The present ‘extended’ formulation is closed by the derivation of the tangent stiffness.     

Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases. The project supported by the National Natural Science Foundation of China (10372083, 90405016), 973 Program (2006CB601205) and the Aeronautical Science Foundation (04B53080). The English text was polished by Keren Wang.     

Electromechanical response of (3–0, 3–1) particulate,fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.     

Local Symmetry Group in the General Theory of Elastic Shells

We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.