Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains

Highlights

• A Cosserat and a Cauchy medium are coupled for a two-scale homogenization.

• A higher-order kinematic map is formulated depending on the Cosserat macro-strains.

• Unit cells made of heterogeneous periodic materials are analyzed.

• The displacement perturbation distribution in the heterogeneous medium is studied.

• Boundary conditions on the UC more complex than the classical periodic ones emerge.

Abstract

In the framework of the computational homogenization procedures, the problem of coupling a Cosserat continuum at the macroscopic level and a Cauchy medium at the microscopic level, where a heterogeneous periodic material is considered, is addressed. In particular, non-homogeneous higher-order boundary conditions are defined on the basis of a kinematic map, properly formulated for taking into account all the Cosserat deformation components and for satisfying all the governing equations at the micro-level in the case of a homogenized elastic material. Furthermore, the distribution of the perturbation fields, arising when the actual heterogeneous nature of the material is taken into account, is investigated. Contrary to the case of the first-order homogenization where periodic fluctuations arise, in the analyzed problem more complex distributions emerge.

Introduction

Composite materials, either natural or manufactured, are widely used in many fields of engineering and for different types of structures. Although they may have very distinct features, they are all characterized by a heterogeneous micro-structure. The study of the constitutive response of such materials is essential both to reproduce the behavior of existing structures and to design innovative ones with optimized properties.

Various approaches, characterized by different formulations, have been proposed in literature to deal with this issue; among others the homogenization techniques have been widely used. In particular, these techniques analyze the actual heterogeneous material at two different scales: the macro-scale, where an effective homogenized medium is considered, characterized by overall mechanical properties, which are estimated from detailed information available at a lower scale, the micro-scale, where the texture, the geometry and the constitutive laws of the constituents are accurately described.

Several authors, in recent years, have adopted the so-called computational homogenization, also known as global-local analysis. The basic idea of such procedure is that the constitutive response at each macroscopic point is evaluated step-by-step, by the numerical analysis of the corresponding representative volume element (RVE), properly selected at the micro-level in order to account for all the information about the texture and the constitutive properties of the actual heterogeneous material. Depending on the texture of the constituents, the heterogeneous materials may be classified as periodic, i.e. made by a repetitive unit cell (UC) directly defined by observing the material geometry, or as random.

In particular, the finite element (FE) method is generally adopted for performing the numerical computations and both nonlinear material and geometric behavior can be considered. A fundamental contribution in literature dates back at Suquet (1985). Then, also the works by Guedes and Kikuchi (1990), Ghosh et al. (1996) and Miehe et al. (1999) are noteworthy. They all have in common the assumption that both continua adopted at macro- and micro-levels are classical Cauchy media. Although such a choice may be satisfactorily adopted when the intrinsic lengths at the micro-scale are very small compared to the macro-scale structural lengths, it is subjected to serious limitations in presence of strong strain and stress gradients at the macroscopic level or when the microscopic lengths of the constituents are comparable to the wavelength of variation of the mean fields, strain and stress, at the macro-level. The inner limit of this assumption is that no length scales can be accounted for, thus the information that the micro-level is able to pass to the macro-level is insensitive to the dimensions of the constituents.

Different approaches can be found in literature, in order to overcome such drawback, proposing the adoption of ‘generalized’ continua at the macro-level. A comprehensive discussion concerning the derivation of various extended ‘generalized’ continua is presented, among the others, by Forest et al., 2002, Forest and Trinh, 2011, where both the problems of defining properly non-homogeneous boundary conditions on the RVE and of identifying the homogenized properties of the generalized equivalent effective medium are addressed. Although it is possible to consider generalized continuum models at both the macro- and micro-scale, as for example proposed by Onck (2002), who adopts two Cosserat continua, herein the attention is focused on the homogenization procedures using the classical Cauchy model for the constituents at the micro-scale. The use of the classical Cauchy continuum at the micro-scale is motivated by the reason that the non-linear constitutive laws, modeling the constituents mechanical response, are well established in this framework, while for extended generalized continua the identification of the mechanical properties and the definition of the non-linear evolution laws is usually a hard task.

Different formulations for the ‘generalized’ continuum model adopted for the equivalent effective medium at the macro-scale are proposed in the literature, such as second gradient (Kouznetsova, 2002, Kaczmarczyk et al., 2008, Bacigalupo and Gambarotta, 2011), Cosserat (Forest and Sab, 1998, Feyel, 2001, De Bellis and Addessi, 2011, Addessi and Sacco, 2012) and micromorphic continua (Forest et al., 2002, Forest and Trinh, 2011). It has to be noted that various questions concerning the formulation of the homogenization procedures, when extended generalized continuum models are adopted at the macro-level, are still open, as remarked in Forest and Trinh (2011).

Aim of the present paper is to investigate on some relevant aspects of the homogenization procedure for periodic heterogeneous materials. A Cosserat continuum is adopted for modeling the effective macroscopic medium and polynomial expansions are used for defining the non-homogenous boundary conditions, as proposed in Forest and Sab (1998). The displacement field solution of the Boundary Value Problem (BVP) on the RVE is represented as the superposition of an assigned field, the polynomial kinematic map, whose coefficients depend on the macro-level strain components, and an unknown perturbation field due to the heterogeneity. To this end, it has to be noted that, contrary to the case of the classical first-order homogenization, where homogenous boundary conditions are applied to the RVE, the perturbation displacement field cannot be assumed a-priori as periodic, as remarked in Bouyge et al. (2001) and in Forest and Trinh (2011).

In particular, in this paper attention is focused on:

• the definition of the kinematic map adopted for enforcing the non-homogenous boundary conditions on the RVE;

• the characterization of the fluctuation field, in the case of higher order polynomial boundary conditions on the RVE.

The paper is organized as follows: in Section 2 the problem of deriving a suitable kinematic map linking the macro- and micro-level is addressed; in Section 3 the characterization of the perturbation fields is carried on; finally, in Section 4 some concluding remarks are reported.