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.     

Influences of reinforcing particle and interface bonding strength on material properties of Mg/nano-particle composites

In the present study, an effective model is proposed to predict the effective elastic behavior of the three-phase composite containing spherical inclusions, each of which is surrounded by an interphase layer. The constitutive equations are derived for the stress and strain of each phase of the composite subjected to a far-field tension. Based on these constitutive laws, the effective bulk, shear and Young’s modulus are obtained. A statistical debonding criterion is adopted to characterize the varying probability of the evolution of interphase debonding. Influences of debonding damage, particle volume fraction, interphase properties and bonding strength on overall mechanical behavior of composites are also discussed. Numerical analyses are carried out on particle-reinforced composites and the predictions have a good agreement with the experimental results.     

The elastic moduli estimation of the solid-water mixture

Conceptually, the undrained elastic constants estimated by the poroelasticity theory should be identical to the effective moduli of the two-phase composite of a porous material saturated with pore water. Here we show numerically that the undrained elastic constants determined by an effective moduli estimate are almost identical with those calculated by poroelasticity theory, and if pore shapes are not exactly known and the porosity is around 50%, estimating the elastic constant as the average value of its Voigt and Reuss bounds is reasonably accurate. This is the situation in bone and dentin, the materials that are our primary intended application. This result will hold for situations in which the totally enclosed water phase is constrained to small deformations by virtue of its confinement. Importantly, in this work we assume that water is an isotropic elastic solid with a shear modulus that is 10^?4 times the bulk modulus of the water. Note that it is compressible, but almost incompressible with a Poisson’s ratio of 0.4999.     

Local and overall extremal properties of time-independent materials and non-linear elastic comparison materials

For time-independent materials which undergo non-linear deformations from some given reference configuration two (dual) hypotheses are considered. Firstly it is supposed that the work done to a given state of deformation is bounded below and that the bound is attainable on a physically possible path; secondly that the complementary work to a given state of stress is bounded above and that this bound too is attainable on a physically possible path. The consequences of these assumptions are analysed, and the results of Ponter and Martin [1] in the linear theory are generalized to account for non-linear deformations, due attention being paid to questions of stability.A non-linear elastic comparison material is defined whose strain energy is equal to the work done on a minimum path for the time-independent material. Extremum principles for non-linear elastic materials given in [2] are then applied to the comparison elastic material, and bounds are thereby placed on the work and complementary-work functional of the time-independent material. Corresponding overall properties of the time-independent and elastic materials are then compared by defining respective overall constitutive laws and overall stress and deformation variables.Following the definition of strengthening (weakening) of a non-linear elastic solid given by Ogden[2] a time-independent material is said to be strengthened (weakened) when its comparison elastic material is strengthened (weakened). Local and overall aspects of this definition are examined.     

An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

On correct account of finite rotations in finite plasticity theory

The non-uniqueness of the trantition from nonobjective constitutive relations to objective ones with the use of the principle of material frame-indifference (PMFI) is shown. To eliminate it, the concept of finite strain without rotations (FSWR) for a given material type and each strain component (elastic, plastic) is introduced. In FSWR the rotation is excluded with respect to the natural preferred configuration for a given material. Considered are a simple solid, a liquid, a monocrystal, a polycrystal and a composite. The proecedure is proposed for consistent generalization of known infinitesimal relations for finite strains and rotations. The structure of constitutive relations is derived for anisotropic elasto-plastic mono- and polycrystalline materials.     

Tailored heterogeneity increases overall stability regime of composites having a negative-stiffness inclusion

Recent theoretical and experimental results have shown the possibility of enormous increases in composite material overall elastic stiffness, damping, thermal expansion, piezoelectricity, etc., when the composite contains a tuned non-positive-definite (i.e., negative stiffness) constituent. For such composite materials to have practical utility, they must be stable. Recent research has shown they can be, for a limited range of constituent negative stiffness. This research has treated linear elastic composite materials with homogeneous phases, via the energy method and full dynamic stability analyses.In the present work, we first show how to analyze the composites previously treated by the comprehensive but simpler static stability approach, obtaining closed-form results. We then employ this approach to show that permitting heterogeneity of the positive-definite phase can substantially increase the range of constituent negative stiffness while maintaining overall composite stability. We first treat the positive-definite phase heterogeneity as piecewise homogeneous, and then treat it as continuously-varying. In the continuously-varying heterogeneity case, we seek the radially optimal distribution of the elastic moduli in the coatings, under constant coating average moduli constraint, to permit the most negative possible inclusion stiffness while maintaining overall composite stability. This is accomplished for three coating cases: constant bulk modulus but arbitrarily radially-varying shear modulus; constant shear modulus but arbitrarily radially-varying bulk modulus; and both moduli arbitrarily radially varying. We find the optimal coatings to be: a heterogeneous one with shear modulus being a specific continuously decreasing function of radius for the first case; a homogeneous one for the second case; and a heterogeneous one with both moduli being either Dirac-delta or Heaviside-step decreasing functions of radius for the last case (if the coating moduli are unrestricted in magnitude or have upper limits, respectively). The results show a substantial increase in the permissible inclusion negative stiffness range is provided by coating heterogeneity, while maintaining overall composite stability. Such an increased range of constituent negative stiffness provides an enlarged tuning parameter range for the development of novel, high-performance composite materials.     

含正交排列夹杂和缺陷材料的等效弹性模量和损伤

研究含正交排列夹杂和缺陷材料的等效弹性模量和损伤,推导了以Eshelby－Mori－Tanaka方法求解多相各向异性复合材料等效弹性模量的简便计算公式,针对含三相正交椭球状夹杂的正交各向异性材料,得到了由细观参量（夹杂的形状、方位和体积分数）表示的等效弹性模量的解析表达式．在此基础上,提出了一个宏细观结合的正交各向异性损伤模型,从而建立了以细观量为参量的含损伤材料的应力应变关系．最后,对影响材料损伤的细观结构参数进行了分析．     

Magnetostrictive composites in the dilute limit

We calculate the effective properties of a magnetostrictive composite in the dilute limit. The composite consists of well separated identical ellipsoidal particles of magnetostrictive material, surrounded by an elastic matrix. The free energy of the magnetostrictive particles is computed using the constrained theory of DeSimone and James [2002. A constrained theory of magnetoelasticity with applications to magnetic shape memory materials. J. Mech. Phys. Solids 50, 283-320], where application of an external field causes rearrangement of variants rather than rotation of the magnetization or elastic strain in a variant. The free energy of the composite has an elastic energy term associated with the deformation of the surrounding matrix and demagnetization terms. By using results from the constrained theory and from the Eshelby inclusion problem in linear elasticity, we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. The solution of the quadratic programming problem yields the effective properties of Ni₂MnGa and Terfenol-D composite systems. Numerical results show that the average strain of the composite depends strongly on the particle shape, the applied stress, and the elastic modulus of the matrix.     

Characterization of a class of polycrystals whose effective elastic bulk moduli can be exactly determinedCaractérisation d'une classe de polycristaux dont les modules de compression isotrope effectifs peuvent exactement être déterminés

Necessary and sufficient conditions are established for the stress response of a linearly elastic material to an isotropic stain to be hydrostatic. In the 3D case, these conditions are satisfied not only by the isotropic and cubic materials but also by all other anisotropic materials provided appropriate restrictions are imposed. In the 2D case, only the isotropic and square materials have an isotropic stress response to an isotropic strain. Using a uniform field argument, the elastic bulk modulus of a polycrystal consisting of monocrystals compatible with the established conditions is shown to equal that of any constituent monocrystal. Thus, Hill's relevant result about a polycrystal composed of cubic monocrystals is generalized. To cite this article: Q.-C. He, C. R. Mecanique 331 (2003).     

含界面相Voronoi单元的等效弹性常数的计算

采用含界面相Voronoi单元有限元法,根据广义胡克定律,计算了在给定边界条件下,颗粒增强复合材料的等效弹性常数。建立了含多个随机分布的椭圆形夹杂及界面相的VCFEM模型,分析了夹杂体分比,界面相厚度和界面相弹性模量等因素对颗粒增强复合材料等效弹性常数的影响,并利用普通有限元方法对比验证。结果表明,当界面相弹性模量小于基体与夹杂时,材料的等效弹性模量会随着界面相厚度的增大而减小,随着夹杂体分比的增大而减小,并且界面过薄时,材料的等效弹性模量会随着夹杂体分比的增大而增大;当界面相弹性模量大于基体或夹杂时,材料的等效弹性模量会随着夹杂体分比和界面相厚度的增大而增大。而界面相的厚度和弹性模量对材料的等效泊松比的影响较小,材料的等效泊松比主要受夹杂体分比的影响,与其呈反比关系。     

水泥砂浆界面相弹性常数的反演计算

为了计算水泥砂浆界面过渡区的弹性常数,采用广义自洽方法(GSCM),根据水泥砂浆内部的微细观结构,建立了由水泥浆基体、岩石离散夹杂(骨料)、界面过渡区(ITZ)和有效弹性材料组成的四相复合材料模型.推导了界面相的体积模量和剪切模量方程.利用已知水泥砂浆材料的实验数据,计算了界面相的弹性常数.发现界面相剪切模量约为水泥浆基体剪切模量的50%.     

An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure

In this paper a simplified three-dimensional constitutive equation for viscoelastic rubber-like solids is derived by employing a generalized strain measure and an asymptotic expansion similar to that used by Coleman and Noll (1961) in their derivation of finite linear viscoelasticity (FLV) theory. The first term of the expansion represents exactly the time and strain separability relaxation behavior exhibited by certain soft polymers in the rubbery state and in the transition zone between the glassy and rubbery states. The relaxation spectra of such polymers are said to be deformation independent. Retention of higher order terms of the asymptotic expansion is recommended for treating deformation dependent spectra.Certain assumptions for the solid theory are relaxed in order to obtain a constitutive equation for uncross-linked liquid materials which exhibit large elastic recovery properties.Apart from the strain energyW(I₁,I ₂), which alternatively characterizes the long-time elastic response of solids or the instantaneous elastic response of elastic liquids, only the linear viscoelastic relaxation modulus is required for the first-order theory. Both types of material functions can be obtained, in theory, from simple laboratory testing procedures. The constitutive equations for solids proposed by Chang, Bloch and Tschoegl (1976) and a special form of K-BKZ theory for elastic liquids are shown to be particular cases of the first-order theory.Previously published experimental data on a cross-linked styrene-butadiene rubber (SBR) and an uncross-linked polyisobutylene (PIB) rubber is used to corroborate the theory.     

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.     

Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite

A model is presented of a particulate composite containing spherical inclusions, each of which are surrounded by a localized region in which the elastic moduli vary smoothly with radius. This region may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. An exact solution is derived for the displacements and stresses around a single inclusion in an infinite matrix, subjected to a far-field hydrostatic compression, and is then used to derive an approximate expression for the effective bulk modulus of a material containing a random dispersion of these inclusions. The analogous conductivity (thermal, electrical, etc.) problem is then discussed, and it is shown that the expression for the normalized effective conductivity corresponds exactly to that for the normalized effective bulk modulus, if the Poisson ratios of both phases are set to zero.     

树脂基复合材料粘弹性损伤本构及其实验测定

粘弹性是复合材料最重要的力学性能之一。基于Ｂｏｌｔｓｍａｎ迭加原理和Ｓｃｈａｐｅｒｙ非线性粘弹性理论，本文提出了一变量分离式粘弹性本构关系，且由Ｋａｃｈａｎｏｖ“连续性”概念，将损伤变量引入本构关系式中，按不同的加载速度和加载方向对正交玻璃布－环氧材料进行了加、卸载实验研究。根据Ｌｅｍａｉｔｒｅ和Ｃｈａｂｏｃｈｅ由宏观弹性常数变化确定损伤量的方法，测得材料的损伤并用一二次函数表示。通过对实验数据的一系列处理方法，确定了材料常数，验证了本文理论的有效性。实验表明，当纤维做为主要承力者时，无粘性产生，材料的损伤和总应变相关而与应变率无关，并得到强度值随应变率增加而升高，极限应变不随应变率变化的结论。     

Thermomechanics of volumetric growth in uniform bodies

A theory of material growth (mass creation and resorption) is presented in which growth is viewed as a local rearrangement of material inhomogeneities described by means of first- and second-order uniformity “transplants”. An essential role is played by the balance of canonical (material) momentum where the flux is none other than the so-called Eshelby material stress tensor. The corresponding irreversible thermodynamics is expanded. If the constitutive theory of basically elastic materials is only first-order in gradients, diffusion of mass growth cannot be accommodated, and volumetric growth then is essentially governed by the inhomogeneity velocity “gradient” (first-order transplant theory) while the driving force of irreversible growth is the Eshelby stress or, more precisely, the “Mandel” stress, although the possible influence of “elastic” strain and its time rate is not ruled out. The application of various invariance requirements leads to a rather simple and reasonable evolution law for the transplant. In the second-order theory which allows for growth diffusion, a second-order inhomogeneity tensor needs to be introduced but a special theory can be devised where the time evolution of the second-order transplant can be entirely dictated by that of the first-order one, thus avoiding insuperable complications. Differential geometric aspects are developed where needed.     

Characterization of the Orthotropic Elastic Constants of a Micronic Woven Wire Mesh via Digital Image Correlation

Woven structures are steadily emerging as excellent reinforcing components in dual-phase composite materials subjected to multiaxial loads, thermal shock, and aggressive reactants in the environment. Metallic woven wire mesh materials in particular display good ductility and relatively high specific strength and specific resilience. While use of this class of materials is rapidly expanding, a significant gap in property characterization remains. This research classifies the homogenized, orthotropic material properties of a representative twill dutch woven wire mesh through the use of in-plane uniaxial tensile experiments incorporating a Digital Image Correlation (DIC) strain measurement technique. Values for elastic modulus and Poisson’s ratio are calculated from the experimental data, and shear modulus values are identified by means of constitutive modeling. This approach establishes a reproducible method for characterizing the in-plane elastic response of micronic metallic woven materials via macro-scale uniaxial tensile tests, and shows that a homogenous orthotropic constitutive model may be employed to describe the macro-scale elasticity of this class of materials with reasonable accuracy.     

Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.