Homogenization In Linear Elasticity Research Articles

A micromechanical approach for homogenization of elastic multiphase periodic composites

This work presents a micromechanical model for the linear elastic homogenization of composites with periodic microstructures. The model is constructed for composites with an arbitrary number of phases and geometric shapes of the inclusions, differently from the most homogenization approaches available in the literature. In addition, no restriction is made with respect to the mismatch between the properties of the constituent phases and volume fractions of the inclusions. The model formulation is based on the Eshelby equivalent inclusion method and corresponds to an extension of a procedure originally derived to evaluate the effective elastic moduli of periodic composites with only two constituent phases. The fluctuating elastic fields within of the repeating unit cell (RUC) are represented by Fourier series, resulting in Lippmann-Schwinger integral equations involving the unknown eigenstrain fields of the inclusions. These integral equations are solved by using a straightforward approach that employs a scheme of partition of the domain of each inclusion. Applications to composites with different arrays of coated fibers and constituent materials are presented to show the efficiency of the model.

An eigenstrain-based micromechanical model for homogenization of elastic multiphase/multilayer composites

This paper presents a novel micromechanical procedure for the linear elastic homogenization of composites with periodic microstructures. The procedure is developed for composites with arbitrary number of phases and geometric shapes of the inhomogeneities, in contrast with most existing homogenization approaches. Also, no restriction is made in relation to the mismatch between the properties of the phases and volume fractions of the inhomogeneities. The proposed procedure is based on the Eshelby equivalent inclusion approach and extends a model originally derived for evaluating the effective elastic moduli of periodic two-phase composites. The procedure represents the fluctuating elastic fields within each multiphase repeating unit cell (RUC) using Fourier series, resulting in Lippmann-Schwinger integral equations governing the unknown eigenstrain fields of the inclusions. Unlike traditional iterative algorithms used in Fast Fourier Transform (FFT)-based approaches, the procedure solves the integral equations straightforwardly from a scheme of partition of the domain of each inclusion. The efficiency of the proposal procedure is demonstrated through applications to composites with different arrays of coated fibers and constituent materials.

A computational multi-scale model for the stiffness degradation of short-fiber reinforced plastics subjected to fatigue loading

In this work, we investigate a model for the anisotropic and loading-direction dependent stiffness degradation of short-fiber reinforced thermoplastics subjected to high-cycle fatigue loading. Based upon the variational setting of generalized standard materials, we model the stiffness degradation of the matrix in cycle space by a simple isotropic fatigue-damage model with minimum number of model parameters, and consider the fibers to be linear elastic. The stiffness degradation upon cyclic loading is determined, for fixed damage state, by standard linear elastic homogenization. We thoroughly investigate the influence of the involved numerical and material parameters on the effective stiffness degradation of short-fiber reinforced volume elements, and also pay close attention on the dependence on geometric parameters of the composite like fiber-volume fraction and the fiber-orientation state.To enable component-scale simulations, we use a model-order reduction strategy for identifying a suitable macroscopic model. This reduction is possible because our free energy involves polynomial powers in the fields not exceeding two, and our dissipation potential is assumed quadratic. For resolving the balance of linear momentum on the microscopic scale, we rely upon a fast Fourier transform based non-local micromechanics solver.We investigate the clever identification of the modes and demonstrate the capabilities of our model on the microscopic and on the component scale by pertinent numerical examples.

Loss of Strong Ellipticity Through Homogenization in 2D Linear Elasticity: A Phase Diagram

Since the seminal contribution of Geymonat, M\"uller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Guti\'errez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.

A Review of Analytical Micromechanics Models on Composite Elastoplastic Behaviour

Elasto-plastic behavior of composite structures is a key problem in light-weight application, which has been investigated vastly for decades. Most approaches for composites elasto-plastic simulation can be split into three categories: 1) numerical methods also known as computational micromechanics methods, 2) variational approaches for upper and lower bounds, and 3) analytical homogenization models. In this study only focuses on the analytical models. Two steps are essential to establish an analytical model, i.e. choosing a suitable linear elastic homogenization models and a rational linearization theory so that linear elastic models can be extended to nonlinear elasto-plastic regime. Besides, considering the situation that isotropic materials may become anisotropic due to plastic deformation, three approaches are found in the literature to deal with the anisotropic properties and corresponding Eshelby's tensor. In summary, a comparative study is made among nine different models to evaluate their capabilities on elasto-plastic behavior simulation. Experiment data of a series of unidirectional composites under different loading cases are used for the purpose of comparison.

Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity

Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved by homogenization in linear elasticity. This phenomenon is typically related to microscopic buckling of the composite material. The present contribution concerns the interplay between isotropy and strong ellipticity in the framework of periodic homogenization in linear elasticity. Mixtures of two isotropic phases may indeed lead to loss of strong ellipticity when arranged in a laminate manner. We show that if a matrix/inclusion type mixture of isotropic phases produces macroscopic isotropy, then strong ellipticity cannot be lost.

Computational homogenization of elasticity on a staggered grid

SummaryIn this article, we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers. Our method shares some properties with the FFT‐based homogenization technique of Moulinec and Suquet, which has received widespread attention recently because of its robustness and computational speed. These similarities include the use of FFT and the resulting performing solvers. The staggered grid discretization, however, offers three crucial improvements. Firstly, solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by Moulinec–Suquet's discretization. Secondly, the iteration numbers of our solvers are bounded independently of the grid size and the contrast. In particular, our solvers converge for three‐dimensional porous structures, which cannot be handled by Moulinec–Suquet's method. Thirdly, the finite difference discretization allows for algorithmic variants with lower memory consumption. More precisely, it is possible to reduce the memory consumption of the Moulinec–Suquet algorithms by 50%. We underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale. Copyright © 2015 John Wiley & Sons, Ltd.

Loss of ellipticity through homogenization in linear elasticity

In 1993, it was shown by Geymonat, Müller and Triantafyllidis that, in the setting of linearized elasticity, a Γ-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant in ℝN is zero while the corresponding coercivity constant on the torus remains positive. We illustrate the range of applicability of that result by finding sufficient conditions for such a situation to occur. We thereby justify the degenerate laminate construction given by Gutiérrez in 1999. We also demonstrate that the predicted loss of strict strong ellipticity resulting from the construction by Gutiérrez is unique within a "laminate-like" class of microstructures. It will only occur for the specific micro-geometry investigated there. Our results thus confer both a rigorous, and a canonical character to those of Gutiérrez.

Micromechanical modeling of the elastic behavior of unidirectional CVI SiC/SiC composites

The elastic behavior of SiC/SiC composite is investigated at the scale of the tow through a micromechanical modeling taking into account the heterogeneous nature of the microstructure. The paper focuses on the sensitivity of transverse properties to the residual porosity resulting from the matrix infiltration process. The full analysis is presented stepwise, starting from the microstructural characterization to the study of the impact of pore shape and volume fraction. Various Volume Elements (VEs) of a virtual microstructure are randomly generated. Their microstructural properties are validated with respect to an experimental characterization based on high definition SEM observations of real materials, using various statistical descriptors. The linear elastic homogenization is performed using finite elements calculations for several VE sizes and boundary conditions. Important fluctuations of the apparent behavior, even for large VEs, reveal that scales are not separated. Nevertheless, a homogeneous equivalent behavior is estimated by averaging apparent behaviors of several VEs smaller than the Representative Volume Element (RVE). Therefore, the impact of the irregular shape of the pores on the overall properties is highlighted by comparison to a simpler cylindrical porous microstructure. Finally, different matrix infiltration qualities are simulated by several matrix thicknesses. A small increase in porosity volume fraction is shown to potentially lead to an important fall of transverse elastic moduli together with high stress concentrations.

The effective stiffness and stress concentrations of a multi-layer laminate

Using the jump conditions for the geometrically linearized strains and the Cauchy stresses and assuming homogeneous strain and stress fields inside the layers, we determine the effective stiffness tetrad of a laminate consisting of linearly elastic layers. From a two-layer reference solution, an explicit solution for the multi-layer case is derived. Unlike classical methods, the present approach applies to any loading case and an arbitrary number of layers with arbitrary stiffness tetrads, and can be considered as the explicit analytical solution of the multi-layer homogenization in linear elasticity.

A particular implementation of the Modified Secant Homogenization Method for particle reinforced metal matrix composites

The stress–strain relationship for Particle Reinforced Metal Matrix Composites (PMMCs) in the nonlinear regime has been frequently assessed by means of the Modified Secant Homogenization Method. This nonlinear Homogenization uses a Linear Elastic Homogenization scheme to calculate the Secant Compliance tensor of the composite in terms of the known Secant Compliance tensors of the composites’ constituent phases.The subject of the present work is the development of a particular implementation of the Modified Secant Homogenization Method for the case of PMMCs using, as the required underlying Linear Elastic Homogenization scheme, the Halpin–Tsai equation. The developed implementation, valid for PMMCs of geometrically isotropic microstructure, results in a relatively simple iterative procedure for the estimation of the nonlinear macroscopic stress–strain response. It has only two explicit parameters: the reinforcement volume fraction F and the ‘s’ parameter of the Halpin–Tsai equation, which carries implicitly information about particle aspect ratio and orientation.The proposed scheme is applied to the prediction of the uniaxial hardening curve and to the study of the influence of macroscopic hydrostatic stress on composite’s yield.

Compatible model for herringbone bond masonry: Linear elastic homogenization, failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.

Random homogenization analysis in linear elasticity based on analytical bounds and estimates

In this work, random homogenization analysis of heterogeneous materials is addressed in the context of elasticity, where the randomness and correlation of components’ properties are fully considered and random effective properties together with their correlation for the two-phase heterogeneous material are then sought. Based on the analytical results of homogenization in linear elasticity, when the randomness of bulk and shear moduli, the volume fraction of each constituent material and correlation among random variables are considered simultaneously, formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from Random Factor Method. Results from the Random Factor Method and the Monte-Carlo Method are compared with each other through numerical examples, and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods. Moreover, the correlation coefficients of random effective properties are obtained by the Monte-Carlo Method. The Random Factor Method is found to deliver rapid results with comparable accuracy to the Monte-Carlo approach.

An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials

We discuss the consequences of a different application of the principle the Modified Secant Method is [C. R. Acad. Sci. Paris Sér. IIb 320 (1995) 563] based on. In fact, we directly compute the second-order averages of the local fields available from the linear elastic homogenization procedure exploited in order to evaluate the effective elastic moduli. This method, which can be seen either as a simplification of the Modified Secant Method or as an extension of the Secant Method [J. Mech. Phys. Solids 26 (1979) 325], may be useful for any composite whose overall elastic constants need to be estimated by modeling the microstructure through Morphologically Representative Patterns [J. Mech. Phys. Solids 44 (1996) 307], which is for instance the case of syntactic foams [Int. J. Solids and Structures 38/40–41 (2001) 7235]. In order to show the accuracy of the proposed method, we apply it to several examples and compare its results with those obtainable by means of other analytical methods available in the literature, with numerical results of Finite Element simulations, and with experimental results. Closed-form solutions are derived for the effective yield stress of porous metals and incompressible composites reinforced with rigid spheres.

Asymptotic analysis and boundary homogenization in linear elasticity

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of ℝ3, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ϵ distributed on the boundary of this domain, when the parameter ϵ goes to 0. We use epi-convergence arguments in order to establish this asymptotic behaviour. We then specialize this general situation to the case of identical strips of size rϵ ϵ-periodically distributed on the lateral surface of an axisymmetric body. We exhibit a critical size of the strips through the limit of the non-negative quantity −1/(ϵ ln rϵ) and we identify the different limit problems according to the values of this limit. Copyright © 2000 John Wiley & Sons, Ltd.