Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method

We propose a general formulation – which we believe to be new – for the mean-field homogenization of inclusion-reinforced elasto-viscoplastic composites assuming small strains. Our proposal is based on an interplay between constitutive equations and numerical algorithms, and the key ideas behind it are the following. The evolution equations for inelastic strain and internal variables at the beginning of each time interval are linearized around the ending time of the same interval. The linearized equations are then numerically integrated using a fully implicit backward Euler scheme. The obtained algebraic equations lead to an incrementally affine stress–strain relation which involves two important terms. The first one is the algorithmic tangent operator, obtained by consistent linearization of the time discretized constitutive equations. The second term is a new one and called an affine strain increment. The proposal leads to thermoelastic-like relations directly in the time domain, and not in the Laplace–Carson (L–C) one. There is no need for viscoelastic-type integral rewriting of the evolution equations, for L–C transforms, or for numerical inversion back from L–C to time domains. The proposed method can be readily applied to sophisticated elasto-viscoplastic models with an arbitrary set of scalar or tensor internal variables, and is valid for multi-axial, non-monotonic and non-proportional loading histories. The theory is applied in detail to a well-known constitutive model, and verified against finite element simulations of representative volume elements or unit cells, for a number of composite materials.     

Effect of crack position on stress intensity factor in particle-reinforced metal-matrix composites

In this study, a numerical model was developed to study the effects of mechanical properties of the particle and matrix materials, the crack position (in particle/in matrix) and loading conditions (mode 1 and mixed-mode) in particle-reinforced metal-matrix composites. The finite element technique was used to calculate the stress intensity factors for crack at and near-interface. The Displacement Correlation Method was used to calculate the stress intensity factors K₁ and K₂. In the present model, the particle and matrix materials were modeled in linear elastic conditions. The interface crack was considered between the particle and matrix, without the presence of the interface. For near-interface crack problem, two different crack positions (in particle/in matrix) were selected. The obtained results show the key role on the stress intensity factors played by the relative elastic properties of the particle and matrix. The results also show that loading condition has an important effect on the K₂ stress intensity factor and the crack deflection angle.     

A variational formulation for the incremental homogenization of elasto-plastic composites

This work addresses the micro–macro modeling of composites having elasto-plastic constituents. A new model is proposed to compute the effective stress–strain relation along arbitrary loading paths. The proposed model is based on an incremental variational principle (Ortiz, M., Stainier, L., 1999. The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171, 419–444) according to which the local stress–strain relation derives from a single incremental potential at each time step. The effective incremental potential of the composite is then estimated based on a linear comparison composite (LCC) with an effective behavior computed using available schemes in linear elasticity. Algorithmic elegance of the time-integration of J₂ elasto-plasticity is exploited in order to define the LCC. In particular, the elastic predictor strain is used explicitly. The method yields a homogenized yield criterion and radial return equation for each phase, as well as a homogenized plastic flow rule. The predictive capabilities of the proposed method are assessed against reference full-field finite element results for several particle-reinforced composites.     

A second-moment incremental formulation for the mean-field homogenization of elasto-plastic composites

In this paper, the incremental formulation for the mean-field homogenization (MFH) of elasto-plastic composites is enriched by including second statistical moments of per-phase strain increment fields, thus combining two advantages. The first one is to handle non-monotonic loading histories and the second is to better account for the heterogeneity of microscopic fields. The proposal is currently restricted to elasto-plasticity with J2 flow theory in each phase, under the small perturbation hypothesis. The formulation crucially exploits the return mapping algorithm for the J2 model, with its two steps: elastic predictor, and plastic corrections. It is shown that the second-moment measure of the average von Mises stress in each phase at the elastic predictor step plays a major role in the computation of both the average stress and the comparison tangent operator. The proposal is implemented for an extended Mori-Tanaka scheme. Predictions are compared to results provided by full-field, finite element computations of representative volume elements or unit cells, for various composite materials, with polymer or metal matrices. There are cases where the predictions of the proposed modeling improve significantly over those of a first-order incremental formulation.     

The homogenization method for the study of variation of Poisson's ratio in fiber composites

Summary Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This fact has been shown to be valid for certain mechanisms, composites with voids and frameworks and has recently been verified for microstructures optimally designed by the homogenization approach. For microstructures composed of beams, it has been postulated that nonconvex shapes (with reentrant corners) are responsible for this effect. In this paper, it is numerically shown that mainly the shape, but also the ratio of shear-to-bending rigidity of the beams do influence the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids, or for composites with irregular shapes of inclusions, even if the constituents are quite usual materials, provided that their porosity is strongly manifested. Elements of the numerical homogenization theory and first attempts towards an optimal design theory are presented in this paper and applied for a numerical investigation of such types of materials. Received 11 March 1997; accepted for publication 12 September 1997     

Numerical simulation of elasto-plastic deformation of composites: evolution of stress microfields and implications for homogenization models

The deformation of a composite made up of a random and homogeneous dispersion of elastic spheres in an elasto-plastic matrix was simulated by the finite element analysis of three-dimensional multiparticle cubic cells with periodic boundary conditions. “Exact” results (to a few percent) in tension and shear were determined by averaging 12 stress-strain curves obtained from cells containing 30 spheres, and they were compared with the predictions of secant homogenization models. In addition, the numerical simulations supplied detailed information of the stress microfields, which was used to ascertain the accuracy and the limitations of the homogenization models to include the nonlinear deformation of the matrix. It was found that secant approximations based on the volume-averaged second-order moment of the matrix stress tensor, combined with a highly accurate linear homogenization model, provided excellent predictions of the composite response when the matrix strain hardening rate was high. This was not the case, however, in composites which exhibited marked plastic strain localization in the matrix. The analysis of the evolution of the matrix stresses revealed that better predictions of the composite behavior can be obtained with new homogenization models which capture the essential differences in the stress carried by the elastic and plastic regions in the matrix at the onset of plastic deformation.     

Damage micromechanisms and notch sensitivity of glass-fiber non-woven felts: An experimental and numerical study

The deformation and damage micromechanisms of a glass-fiber non-woven felt were analyzed with a combination of experiments and simulations. Tensile tests were carried out on unnotched and notched rectangular panels to ascertain the physical phenomena which control the development of damage. It was found that fracture began by interbundle bond fracture followed by frictional sliding between bundles, leading to the localization of damage in a wide band. The mechanical behavior of the non-woven felt until final fracture was modeled by the finite element simulation of a 2D random network in which the geometric characteristics of the network and the constituent properties were obtained from experiments on the actual non-woven felt. The simulation results were in very good agreement with the experiments in terms of the macroscopic response and of the microscopic mechanisms. They contributed to explain different features of the experimental data, including the effect of specimen dimensions on the strength and the notch-insensitive behavior of the material.     

Macroscopic behavior and field fluctuations in viscoplastic composites: Second-order estimates versus full-field simulations

This work presents a combined numerical and theoretical study of the effective behavior and statistics of the local fields in random viscoplastic composites. The full-field numerical simulations are based on the fast Fourier transform (FFT) algorithm [Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linear and nonlinear properties of composites. C. R. Acad. Sci. Paris II 318, 1417-1423], while the theoretical estimates follow from the so-called “second-order” procedure [Ponte Castañeda, P., 2002a. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory. J. Mech. Phys. Solids 50, 737-757]. Two-phase fiber composites with power-law phases are considered in detail, for two different heterogeneity contrasts corresponding to fiber-reinforced and fiber-weakened composites. Both the FFT simulations and the corresponding “second-order” estimates show that the strain-rate fluctuations in these systems increase significantly, becoming progressively more anisotropic, with increasing nonlinearity. In fact, the strain-rate fluctuations tend to become unbounded in the limiting case of ideally plastic composites. This phenomenon is shown to correspond to the localization of the strain field into bands running through the composite along certain preferred orientations determined by the loading conditions. The bands tend to avoid the fibers when they are stronger than the matrix, and to pass through the fibers when they are weaker than the matrix. In general, the “second-order” estimates are found to be in good agreement with the FFT simulations, even for high nonlinearities, and they improve, often in qualitative terms, on earlier nonlinear homogenization estimates. Thus, it is demonstrated that the “second-order” method can be used to extract accurate information not only for the macroscopic behavior, but also for the anisotropic distribution of the local fields in nonlinear composites.     

Finite deformation of incompressible fiber-reinforced elastomers: A computational micromechanics approach

The in-plane finite deformation of incompressible fiber-reinforced elastomers was studied using computational micromechanics. Composite microstructure was made up of a random and homogeneous dispersion of aligned rigid fibers within a hyperelastic matrix. Different matrices (Neo-Hookean and Gent), fibers (monodisperse or polydisperse, circular or elliptical section) and reinforcement volume fractions (10–40%) were analyzed through the finite element simulation of a representative volume element of the microstructure. A successive remeshing strategy was employed when necessary to reach the large deformation regime in which the evolution of the microstructure influences the effective properties. The simulations provided for the first time “quasi-exact” results of the in-plane finite deformation for this class of composites, which were used to assess the accuracy of the available homogenization estimates for incompressible hyperelastic composites.     

Multi-scale modeling of the mechanical response of plain weave composites and cellular solids

Multi-component materials with customized mechanical properties, such as textile composites and sandwich materials (cellular core with metallic or composite skin), show a great prospective for use in aerostructures. Understanding of the mechanical response of these materials is still in progress. In the present paper, the tensile response of plain weave composites as well as the compressive response of cellular solids are investigated using a multi-scale damage model. The model, implemented by means of the FE method, is based on homogenized progressive damage modeling of a representative unit-cell. Four failure modes have been considered in the failure analysis of the tows, while material property degradation was performed using a damage mechanics approach which takes into account strain softening. For the cellular solids, two different types of FE models were considered namely, a beam model and a shell model. Failure analysis and material property degradation of the struts were integrated into a bilinear material model. Simulations show a non-linear tensile response of the plain weave mainly attributed to matrix cracking and shear failures occurring at warp tows and resin-rich areas. For the cellular solid, preliminary elastic analyses show a customizability of the normal stiffness with regard to strut’s dimensions.     

Nonlinear composites: a linearization procedure,exact to second-order in contrast and for which the strain-energy and affine formulations coincideComposites non linéaires : un schéma exact au second ordre pour lequel la formulation en énergie de déformation coı̈ncide avec la formulation affine

This Note presents a new approximate scheme for nonlinear composites. The approximation which is made preserves certain features of the original second-order scheme of Ponte Castañeda, exactness to second-order in the contrast and existence of an effective energy, but improves on one drawback, which is the gap between the strain-energy formulation and the affine formulation. A numerical example shows the accuracy of the present method. To cite this article: N. Lahellec, P. Suquet, C. R. Mecanique 332 (2004).     

Periodization of random media and representative volume element size for linear composites

Several existing numerical studies show that the effective linear properties of random composites can be accurately estimated using small volumes subjected to periodic boundary conditions – more suitable than homogeneous strain or stress boundary conditions – providing that a sufficient number of realizations are considered. Introducing the concept of periodization of random media, this Note gives a new definition of representative volume element which leads to estimates of its minimum size in agreement with existing theoretical results. A qualitative convergence criterion for the numerical simulations is proposed and illustrated with finite element computations. To cite this article: K. Sab, B. Nedjar, C. R. Mecanique 333 (2005).     

On the effect of small fluctuations in the volume fraction of constituents on the effective properties of composites

This study deals with three-scale composite materials comprised of nonlinear constituents. At the meso scale the composite can be considered as locally homogeneous with a macroscopic spatial variation of the constituents volume fraction. When these variations about a mean value are small, a Taylor expansion to second-order of the effective properties of the composite with respect to the fluctuations is given. This expansion can be used to discuss the beneficial or deleterious effects of clusters of inhomogeneities. It can also be used to derive new upper and lower bounds for the effective properties of nonlinear composites from dilute results. To cite this article: P. Suquet, C. R. Mecanique 333 (2005).     

Nonlinear shear behavior and interlaminar shear strength of unidirectional polymer matrix composites: A numerical study

Detailed finite element implementation is presented for a recently developed technique (He et al., 2012) to characterize nonlinear shear stress–strain response and interlaminar shear strength based on short-beam shear test of unidirectional polymeric composites. The material characterization couples iterative three-dimensional finite element modeling for stress calculation with digital image correlation for strain evaluation. Extensive numerical experiments were conducted to examine the dependence of the measured shear behavior on specimen and test configurations. The numerical results demonstrate that consistent results can be achieved for specimens with various span-to-thickness ratios, supporting the accurate material properties for the carbon/epoxy composite under study.     

Finite element simulations of fully non‐linear interaction between vertical cylinders and steep waves. Part 2: numerical results and validation

In Ma, Wu, Eatock Taylor [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure. International Journal for Numerical Methods in Fluids 2001], designated Part 1 hereafter, we have developed the methodology and solution procedure for simulating the three‐dimensional interaction between fixed bodies and steep waves based on a finite element method (FEM). This paper provides extensive numerical results and validation. The effectiveness of the radiation condition is investigated by comparing the results from short and long tanks; the accuracy of the computed data is confirmed through comparison with analytical solutions. The adopted mathematical model is also validated by comparing the obtained numerical results with experimental data. Various test cases, including non‐linear bichromatic and irregular waves and the interactions between waves and one or two cylinders, are analysed. Copyright © 2001 John Wiley & Sons, Ltd.