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.     

High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior

This work is concerned with the determination of the effective behavior of sequentially laminated composites with nonlinear behavior of the constituting phases. An exact expression for the effective stress energy potential of two-dimensional and incompressible composites is introduced. This allows to determine the stress energy potential of a rank-N sequentially laminated composite with arbitrary volume fractions and lamination directions of the core laminates in terms of an N-dimensional optimization problem.

Stress energy potentials for sequentially laminated composites with pure power-law behavior of the phases are determined. It is demonstrated that as the rank of the lamination becomes large the behaviors of certain families of sequentially laminated composite tend to be isotropic. Particulate composites with both, stiffer and softer inclusions are considered. The behaviors of these almost isotropic composites are, respectively, softer and stiffer than the corresponding second-order estimates recently introduced by Ponte Castañeda (1996).     

Optimal anisotropic three-phase conducting composites: Plane problem

The paper establishes tight lower bound for effective conductivity tensor K₁ of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin–Shtrikman and translation bounds to multiphase structures, it is derived using a combination of translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi, 1995, Cherkaev, 2009 for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites with effective conductivity tensor K₁. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of K₁, that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin et al., 2007a, Cherkaev, 2009. The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.     

Micro-stress prediction in composite laminates with high stress gradients

The objective of this research is to develop a macroscopic theory, which can provide the connection between macro-mechanics and micro-mechanics in characterizing the micro-stress of composite laminates in regions of high macroscopic stress gradients. The micro-polar theory, a class of higher-order elasticity theory, of composite laminate mechanics is implemented in a well-known Pipes–Pagano free edge boundary problem. The micro-polar homogenization method to determine the micro-polar anisotropic effective elastic moduli is presented. A displacement-based finite element method based on micro-polar theory in anisotropic solids is developed in analyzing composite laminates. The effects of fiber volume fraction and cell size on the normal stress along the artificial interface resulting from ply homogenization of the composite laminate are also investigated. The stress response based on micro-polar theory is compared with those deduced from the micro-mechanics and classical elasticity theory. Special attention of the investigation focuses on the stress fields near the free edge where the high macro-stress gradient occurs. The normal stresses along the artificial interface and especially, the micro-stress along the fiber/matrix interface on the critical cell near the free edge where the high macro-stress gradient detected are the focus of this investigation. These micro-stresses are expected to dominate the failure initiation process in composite laminate. A micro-stress recovery scheme based on micro-polar analysis for the prediction of interface micro-stresses in the critical cell near the free edge is found to be in very good agreement with “exact” micro-stress solutions. It is demonstrated that the micro-polar theory is able to capture the micro-stress accurately from the homogenized solutions.     

Growth of a penny-shaped crack with a nonsmall fracture process zone in a composite

The paper studies the stress rupture behavior of a reinforced viscoelastic composite through which a penny-shaped mode I crack propagates under a constant load. The composite has hexagonal symmetry and consists of elastic isotropic fibers and viscoelastic isotropic matrix. The material is modeled as a transversely isotropic homogeneous viscoelastic medium with effective characteristics. The crack is in the isotropy plane. The ring-shaped fracture process zone at the crack front is modeled by a modified Dugdale zone with time-dependent stresses. The viscoelastic properties of the matrix are characterized using a resolvent integral operator. Use is made of Volterra's principle, the method of operator continued fractions, and the theory of precritical crack growth in viscoelastic bodies. The problem is reduced to nonlinear integral equations. Numerical results are obtained for certain components of the composite, constant volume fractions, and different fracture strengths Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 45–51, August 2008.     

An exact solution for thermal stresses in a three-phase composite cylinder under uniform heat flow

An exact solution is given for the stress field in a three-phase composite cylinder induced by a uniform heat flow applied at infinity. Based on the method of analytical continuation in conjunction with the alternating technique, the general expressions of the temperature and stress functions are derived explicitly in each medium of a three-phase composite cylinder. It is discovered that the stress in the inclusion is always linearly proportional to the coordinate z. Comparison is made with the special case of a two-phase composite cylinder, which shows that our results presented here are exact and general.     

Extensions of the Prager-Shield theory of optimal plastic design

The Prager-Shield associated displacement field method for optimal plastic design is extended to multi-component specific cost functions and multiple load conditions, and a lower bound theorem based on kinematic requirements only is introduced. Since any statically admissible stress field results in an upper bound, the proposed theorem provides a simple method for establishing bounds on the optimal cost. By a simple substitution of parameters into the general equations presented, the optimality criteria can be obtained for particular design problems. Examples of optimal fibre-reinforced plates are given.     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

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.     

The effect of porosity distribution on ductile failure

The effect of a nonuniform distribution of porosity on flow localization and failure in a porous material is analyzed numerically. The void density distribution and properties used to characterize the material behavior were obtained from measurements on partially consolidated and sintered iron powder. The calculations were carried out using an elastic viscoplastic constitutive relation for porous plastic solids. Local material failure is incorporated into the model through the dependence of the flow potential on void volume fraction. The region modelled is a small portion of a larger body, subject to various triaxial stress conditions. Both plane strain and axisymmetric deformations are considered with imposed periodic boundary conditions. Interactions between regions with higher void fractions promote plastic flow localization into a band. Local failure occurs by void growth and coalescence within the band. The results suggest a failure criterion based on a critical void volume fraction that is only weakly dependent on stress history. The critical void fraction does. however, depend on the initial void distribution and material hardening characteristics.     

Homogenization of the viscoelastic heterogeneous materials with multi-coated reinforcements: an internal variables formulation

In this work, a new homogenization method to estimate the effective behavior of viscoelastic heterogeneous materials with multi-coated reinforcements is presented. Unlike classical methods that are based on the Laplace transform, the present internal variables formulation operates directly in the time domain. Using the Green’s function techniques, the micromechanical approach is based on establishing a new integral equation adapted to scale transition methods. Using this integral equation, we apply a generalized self-consistent scheme to determine the local stress concentration equations and the effective behavior of multi-coated inclusion-reinforced materials. To assess the reliability of our model, some applications to the isotropic viscoelastic heterogeneous materials with homothetic spherical inclusions are given. The model is applied to the case of two-phase and three-phase materials, and the results are compared to exact solutions. Results for three-phase materials are presented regarding the influence of soft and stiff viscoelastic interphase on the effective behavior of heterogeneous materials.     

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.     

Overall behavior of two-dimensional periodic composites

The overall properties of a binary elastic periodic fiber-reinforced composite are studied here for a cell periodicity of square type. Exact formulae are obtained for the effective stiffnesses, which give closed-form expressions for composites with isotropic components including ones for empty and rigid fibers. The new formulae are simple and relatively easy to compute. Examples show the dependences of the stiffnesses as a function of fiber volume fraction up to the percolation limit. The specific example of glass fibers in epoxy yields new curves, which correct those displayed before by Meguid and Kalamkarov. Comparison with experimental data is very good. Bruno, Hill and Hashin's bounds are compared with the exact solution. In most cases, the latter is very close to a bound in a given interval. A useful fact to know, where the easy formula afforded by the bound is advantageous. Plots of effective properties are also given for values of the shear moduli ratio of the two media. The overall parameters in the cases of empty and rigid fibers are also shown. The exact formulae explicitly display Avellaneda and Schwarts's microstructural parameters, which have a physical meaning, and provide formulae for them. The equations easily lead to Hill's universal relations.     

非对称铺层复合材料层合板质量优化设计

复合材料广泛应用于航空航天等领域,追求轻量化设计已经成为研究重点.对复合材料层合板质量优化设计,可以减少层合板的纤维用量,减小层合板的质量,降低成本.首先研究复合材料层合板在承受轴向载荷时,产生的形变量、应力示意图,分析容易发生失效部位;以层合板铺层厚度为设计变量,最大应变、铺层比例等为约束条件,最小化层合板质量为优化...     

Stress in elastic plates reinforced by fibres lying in concentric circles

We Consider fibre-reinforced elastic plates with the reinforcement continuously distributed in concentric circles ; such a material is locally transversely isotropic, with the circumferential direction as the preferred direction. For an annulus bounded by concentric circles, the exact solution of the traction boundary value problem is obtained. When the extension modulus in the fibre direction is large compared to other extension and shear moduli, the material is strongly anisotropic. For this case a simpler approximate solution is obtained by treating the material as inextensible in the fibre direction. It is shown that the exact solution reduces to the inextensible solution in the appropriate limit. The inextensible theory predicts the occurrence of stress concentration layers in which the direct stress is infinite ; for materials with small but finite extensibility these layers correspond to thin regions of high stress and high stress gradient. A boundary layer theory is developed for these regions. For a typical carbon fibre-resin composite, the combined boundary layer and inextensible solutions give an excellent approximation to the exact solution. The theory is applied to the problem of an isotropic plate, under uniform stress at infinity, containing a circular hole which is strengthened by the addition of an annulus of fibre-reinforced material.     

On the micromechanics of void growth by prismatic-dislocation loop emission

Experimental evidence and recent molecular dynamics simulations of void growth indicate that prismatic dislocation loop emission by externally applied stresses is a viable mechanism of void growth under shock loading conditions when diffusive processes are given no time to operate. In this paper, the process of growth by loop emission is studied in a model system comprised of a void in an infinite linearly elastic and isotropic solid loaded axisymmetrically by remote applied stresses. First, the interaction between applied stresses, the stress field of a single dislocation loop or a pile-up of loops next to the void, the surface energy expenditure on void surface change, and the lattice resistance to the motion of loops is reviewed. The necessary condition for interstitial loop emission is used to determine the equilibrium positions of the loops as well as the maximum number of loops in a pile-up under given applied stresses. For the parameters of the model-material with purely hydrostatic loading, the numerical results yield a volume change for the void, which when normalized by the initial undeformed volume, exhibits a strong dependence on the size of the void for radii less than ∼400 times the lattice Burgers vector. For larger voids, the normalized volume change was found to be independent of the void radius.     

Macroscopic strain potentials in nonlinear porous materials

By taking a hollow sphere as a representative volume element (RVE), the macroscopic strain potentials of porous materials with power-law incompressible matrix are studied in this paper. According to the principles of the minimum potential energy in nonlinear elasticity and the variational procedure, static admissible stress fields and kinematic admissible displacement fields are constructed, and hence the upper and the lower bounds of the macroscopic strain potential are obtained. The bounds given in the present paper differ so slightly that they both provide perfect approximations of the exact strain potential of the studied porous materials. It is also found that the upper bound proposed by previous authors is much higher than the present one, and the lower bounds given by Cocks is much lower. Moreover, the present calculation is also compared with the variational lower bound of Ponte Castañeda for statistically isotropic porous materials. Finally, the validity of the hollow spherical RVE for the studied nonlinear porous material is discussed by the difference between the present numerical results and the Cocks bound.     

Tension–torsion fracture experiments – Part II: Simulations with the extended Gurson model and a ductile fracture criterion based on plastic strain

An extension of the Gurson model that incorporates damage development in shear is used to simulate the tension–torsion test fracture data presented in Faleskog and Barsoum (2013) (Part I) for two steels, Weldox 420 and 960. Two parameters characterize damage in the constitutive model: the effective void volume fraction and a shear damage coefficient. For each of the steels, the initial effective void volume fraction is calibrated against data for fracture of notched round tensile bars and the shear damage coefficient is calibrated against fracture in shear. The calibrated constitutive model reproduces the full range of data in the tension–torsion tests thereby providing a convincing demonstration of the effectiveness of the extended Gurson model. The model reinforces the experiments by highlighting that for ductile alloys the effective plastic strain at fracture cannot be based solely on stress triaxiality. For nominally isotropic alloys, a ductile fracture criterion is proposed for engineering purposes that depends on stress triaxiality and a second stress invariant that discriminates between axisymmetric stressing and shear dominated stressing.     

Influence of corner stresses on the stability characteristics of composite skew plates

Stability characteristics of composite skew plates subjected to in-plane compressive load are investigated here using the shear deformable finite element approach. The influences of high prebuckling stresses at the corner regions of isotropic and composite skew plates on their stability characteristics are emphasized for different load direction, boundary condition and laminate stacking sequence. The non-linear governing equations based on von Kármán's assumptions are solved by Newton-Raphson technique to get the hitherto unreported postbuckling equilibrium paths of composite skew plates loaded between two rigid flat platens. The variation of out-of-plane deformation and end-shortening with compressive in-plane load are examined for simply supported and clamped skew plates made of isotropic, symmetric and unsymmetric laminates. Marguerre's shallow shell theory is employed to study the effect of sinusoidal imperfection on the non-linear behavior of composite skew plates.     

Bounds of the expansion coefficients of composites reinforced by spherically isotropic particles

I.IntroductionProblemsconcerningrigorousboundsofeffectivepropertiesofheterogeneousmediahaveattractedgreatattentionforalongtime.HashinandShtrikmannlllgavethebestpossiblelowerandupperboundsoftheeffectiveelasticmoduliofcompositesconsistingoftwoisotropicphase…