New variational principles in plasticity and their application to composite materials

I , a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte castañeda (J. Mech. Phys. Solids 39, 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly.     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

Hill–Mandel condition and bounds on lower symmetry elastic crystals

Despite advances in contemporary micromechanics, there is a void in the literature on a versatile method for estimating the effective properties of polycrystals comprising of highly anisotropic single crystals belonging to lower symmetry class. Basing on variational principles in elasticity and the Hill–Mandel homogenization condition, we propose a versatile methodology to fill this void. It is demonstrated that the bounds obtained using the Hill–Mandel condition are tighter than the Voigt and Reuss [1], [2] bounds, the Hashin–Shtrikman [3] bounds as well as a recently proposed self-consistent estimate by Kube and Arguelles [4] even for polycrystals with highly anisotropic single crystals.     

The boundaries of the effective elastic moduli for inhomogeneous solids

The calculation of the effective elastic moduli of inhomogeneous solids, which connect the stresses and strains averaged for the material, is accompanied by certain mathematical difficulties owing to correlation relationships of arbitrary orders. Neglect of correlation relationships leads to average elastic moduli, where averaging according to Voigt and Reuss establishes boundaries containing the effective elastic moduli [1]. Approximate values of the latter can be found by taking into account the correlation relationships of the second order in both calculation schemes [2, 3]. Another method of evaluating the true moduli consists of narrowing the boundaries of Voigt and Reuss on the basis of model representations [4-6]. The approximate effective elastic moduli for a series of polycrystals with various common-angle values are presented in [7]. An analysis of the effect of the correlation relationships between the grains of a mechanical mixture of isotropic components on the effective elastic moduli is carried out in [8], although in all the papers just mentioned the use of correlative corrections to narrow the range of elastic moduli is not investigated. Below it is shown that the calculation of the correlation corrections in the second approximation allows the range for the effective moduli to be narrowed.     

On the Generation of Discrete Isotropic Orientation Distributions for Linear Elastic Cubic Crystals

We consider a model for the elastic behavior of a polycrystalline material based on volume averages. In this case the effective elastic properties depend only on the distribution of the grain orientations. The aggregate is assumed to consist of a finite number of grains each of which behaves elastically like a cubic single crystal. The material parameters are fixed over the grains. An important problem is to find discrete orientation distributions (DODs) which are isotropic, i.e., whose Voigt and Reuss averages of the grain stiffness tensors are isotropic. We succeed in finding isotropic DODs for any even number of grains N≥4 and uniform volume fractions of the grains. Also, N=4 is shown to be the minimum number of grains for an isotropic DOD. This revised version was published online in August 2006 with corrections to the Cover Date.     

A variational approach to the theory of the elastic behaviour of polycrystals

Variational principles for anisotropic and nonhomogeneous elasticity, established by the authors in a previous paper, have been applied to the derivation of lower and upper bounds for the elastic moduli of polycrystals in terms of the moduli of the constituting crystals. The results hold for arbitrary crystal shapes. Explicit results tor cubic polycrystals showed that the present bounds are a considerable improvement of the well-known Voigt and Reuss bounds. Good agreement with experimental results has been obtained.     

Bounds and estimates for the effect of strain gradients upon the effective plastic properties of an isotropic two-phase composite

Predictions are made for the size effect on strength of a random, isotropic two-phase composite. Each phase is treated as an isotropic, elastic-plastic solid, with a response described by a modified deformation theory version of the Fleck-Hutchinson strain gradient plasticity formulation (Fleck and Hutchinson, J. Mech. Phys. Solids 49 (2001) 2245). The essential feature of the new theory is that the plastic strain tensor is treated as a primary unknown on the same footing as the displacement. Minimum principles for the energy and for the complementary energy are stated for a composite, and these lead directly to elementary bounds analogous to those of Reuss and Voigt. For the case of a linear hardening solid, Hashin-Shtrikman bounds and self-consistent estimates are derived. A non-linear variational principle is constructed by generalising that of Ponte Castañeda (J. Mech. Phys. Solids 40 (1992) 1757). The minimum principle is used to derive an upper bound, a lower estimate and a self-consistent estimate for the overall plastic response of a statistically homogeneous and isotropic strain gradient composite. Sample numerical calculations are performed to explore the dependence of the macroscopic uniaxial response upon the size scale of the microstructure, and upon the relative volume fraction of the two phases.     

Stiffness and strength of hierarchical polycrystalline materials with imperfect interfaces

In this study we investigate the effect of imperfect (not perfectly bonded) interfaces on the stiffness and strength of hierarchical polycrystalline materials. As a case study we consider a honeycomb cellular polycrystal used for drilling and cutting tools. The conclusions of the analysis are, however, general and applicable to any material with structural hierarchy. Regarding the stiffness, generalized expressions for the Voigt and Reuss estimates of the bounds to the effective elastic modulus of heterogeneous materials are derived. The generalizations regard two aspects that are not included in the standard Reuss and Voigt estimates. The first novelty consists in considering finite thickness interfaces between the constituents undergoing damage up to final debonding. The second generalization consists of interfaces not perpendicular or parallel to the loading direction, i.e., when isostress or isostrain conditions are not satisfied. In this case, approximate expressions for the effective elastic modulus are obtained by performing a computational homogenization approach. In the second part of the paper, the homogenized response of a representative volume element (RVE) of the honeycomb cellular polycrystalline material with one or two levels of hierarchy is numerically investigated. This is put forward by using the cohesive zone model (CZM) for finite thickness interfaces recently proposed by the authors and implemented in the finite element program FEAP. From tensile tests we find that the interface nonlinearity significantly contributes to the deformability of the material. Increasing the number of hierarchical levels, the deformability increases. The RVE is tested in two different directions and, due to different orientations of the interfaces and Mixed Mode deformation, anisotropy in stiffness and strength is observed. Stiffness anisotropy is amplified by increasing the number of hierarchical levels. Finally, the interaction between interfaces at different hierarchical levels is numerically characterized. A condition for scale separation, which corresponds to the independence of the material tensile strength from the properties of the interfaces in the second level, is established. When this condition is fulfilled, the material microstructure at the second level can be efficiently replaced by an effective homogeneous continuum with a homogenized stress–strain response. From the engineering point of view, the proposed criterion of scale separation suggests how to design the optimal microstructure of a hierarchical level to maximize the material tensile strength. An interpretation of this phenomenon according to the concept of flaw tolerance is finally presented.     

Evaluation of the elastic moduli of a transversely isotropic aggregate of cubic crystals

The crystals and the aggregate have the same bulk modulus. The other three overall elastic moduli in the simplified stress-strain relations of Walpole (1985) are placed between upper and lower, Voigt and Reuss, bounds and some exact calculations are given for particular fibre textures.     

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Stationary heat conduction in a macro- and microperiodically layered solid

Summary This paper deals with the stationary heat conduction in a solid consisting of planar, isotropic or transversely isotropic layers. The following cases are considered: (1) arbitrarily layered solid, (2) macroperiodically layered solid, consisting of equal pairs of different basic layers with finite thicknesses, (3) microperiodically layered solid, consisting of equal layer groups of two or more different basic layers of infinitesimal thicknesses. Assuming a perfect contact between the layers, exact solutions for the plane, the axisymmetric and the general three-dimensional problem of macro- and microperiodically layered semispaces, respectively, are derived using integral transforms and the transfer-matrix method. It is proved that the microperiodically layered solid with isotropic or transversely isotropic basic layers is equivalent to a homogeneous transversely isotropic solid. Received 31 March 1999; accepted for publication 21 May 1999     

Elastic moduli of cubic polycrystals

The averaged elastic constants of polycrystals can be found by averaging the stresses (Voigt method [1]) or the strains (Reuss method [2]). Comparison of the elastic moduli, averaged according to Voigt and Reuss, with the experimental values shows that in the first case averaging gives values that are too high, and in the second values that are too low [3]. The reason for this is that direct averaging of the moduli with respect to arbitrary orientations of the crystallites does not take account of correlation effects. There are two ways of allowing for such correlations between polycrystal grains.     

The analysis of thermoplastic characteristics of special polymer sulfur composite

Specific chemical environments step out in the industry objects. Portland cement composites (concrete and mortar) were impregnated by using the special polymerized sulfur and technical soot as a filler (polymer sulfur composite). Sulfur and technical soot was applied as the industrial waste. Portland cement composites were made of the same aggregate, cement and water. The process of special polymer sulfur composite applied as the industrial waste is a thermal treatment process in the temperature of about 150–155 \(^{\circ }\hbox {C}\). The result of such treatment is special polymer sulfur composite in a liquid state. This paper presents the plastic constants and coefficients of thermal expansion of special polymer sulfur composites, with isotropic porous matrix, reinforced by disoriented ellipsoidal inclusions with orthotropic symmetry of the thermoplastic properties. The investigations are based on the stochastic differential equations of solid mechanics. A model and algorithm for calculating the effective characteristics of special polymer sulfur composites are suggested. The effective thermoplastic characteristics of special polymer sulfur composites, with disoriented ellipsoidal inclusions, are calculated in two stages: First, the properties of materials with oriented inclusions are determined, and then effective constants of a composite with disoriented inclusions are determined on the basis of the Voigt or Rice scheme. A brief summary of new products related to special polymer sulfur composites is given as follows: Impregnation, repair, overlays and precast polymer concrete will be presented. Special polymer sulfur as polymer coating impregnation, which has received little attention in recent years, currently has some very interesting applications.     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.     

Multiphase laminates of extremal effective conductivity in two dimensions

This paper deals with two-dimensional composites made of several isotropic linearly conducting phases in prescribed volume fractions. The primary focus is on the three-phase case; the generalization to a larger number of phases is straightforward.A class of high- but finite-rank laminates is introduced. The laminates saturate the known inequality bounds—due to the work of Hashin and Shtrikman, Lurie and Cherkaev, Tartar, and Murat and Tartar—on the effective conductivity tensor of any composite. These bounds depend only on the constituent material properties and volume fractions and not on the placement of these materials in the composite. The bounds are known not to be optimal for all admissible choices of the conductivities and volume fractions. However, they are now known to be realizable in a much larger range of these parameters than was previously known.The range of effective properties of our multiphase laminates strictly includes those corresponding to the composites found earlier by Milton and Kohn, Lurie and Cherkaev, and Gibiansky and Sigmund. The new optimal laminates are found in a systematic fashion by satisfying sufficient conditions on the fields in each layer. This leads to a simple algorithm for generating optimal laminates.In addition a new supplementary bound for multiphase structures is also proven which must be satisfied by composites with smooth interfaces.     

Wave propagation in magneto-electro-elastic multilayered plates

An analytical treatment is presented for the propagation of harmonic waves in magneto-electro-elastic multilayered plates, where the general anisotropic and three-phase coupled constitutive equations are used. The state-vector approach is employed to derive the propagator matrix which connects the field variables at the upper interface to those at the lower interface of each layer. The global propagator matrix is obtained by propagating the solution in each layer from the bottom of the layered plate to the top using the continuity conditions of the field variables across the interfaces. From the global propagator matrix, we finally obtain the dispersion relation by imposing the traction-free boundary condition on the top and bottom surfaces of the layered plate. Dispersion curves, modal shapes, and natural frequencies are presented for layered plates made of orthotropic elastic (graphite–epoxy), transversely isotropic PZT-5A, piezoelectric BaTiO₃ and magnetostrictive CoFe₂O₄ materials. While the numerical results show clearly the influence of different stacking sequences as well as material properties on the field response, the general methodology presented in the paper could be useful to the analysis and design of layered composites made of smart piezoelectric and piezomagnetic materials.     

Homogenization of layered elastoplastic composites: Theoretical results

Exact closed-form solutions are derived that completely characterize the effective behavior of a composite material made of elastic-perfectly plastic parallel plane layers perfectly bonded together. The derivation is framed within a rigorous theory of homogenization for elastoplastic composites, and based on the fundamental fact that the in-plane part of the strain tensor and the out-of-plane part of the stress tensor are uniform throughout the composite provided no free-edge effects occur. The obtained expressions are coordinate-free and valid in the general anisotropic case. As an example, a layered composite material with isotropic constituents is examined in detail.     

Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions

In the present work, unified formulae for the overall elastic bounds for multiphase transversely isotropic composites with different geometrical types of inclusions embedded in a matrix are calculated, including the spherical and long or short continuous cylindrical fiber cases. The influence of the different geometrical configurations of the inclusions on the composites is studied. The transversely isotropic effective bounds are obtained by applying the variational formulation for anisotropic composites developed by Willis, which relies on expressions for the static transversely isotropic Green’s function. Some numerical calculations and comparisons with the effective coefficients derived from the self-consistent approach, asymptotic homogenization method, and finite element method (FEM) are shown for different aspect ratio values, exhibiting good agreement.     

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.