A Cosserat multiparticle model for periodically layered materials

Abstract

In this paper, the Cosserat multiparticle model (CM2) for 3D periodically layered materials is proposed in order to reproduce both size and boundary effects in these materials. This model can handle $n-$ phase periodically layered materials with $4n+1$ kinematic variables at each 3D geometric point: two in-plane displacements and two rotations per phase plus one common out-of-plane displacement. The model gives excellent agreement with full finite element results for out-of-plane shearing.

Introduction

In this paper, a new continuum multiparticle model for 3D periodically layered materials is proposed in order to reproduce both size effects and boundary effects in these materials.

As example, a two-phase linear elastic periodically layered material with free body forces is submitted to out-of-plane shearing as following Fig. 1. The domain considered is infinite in the $x_2$ and $x_3$ directions, with $\mid x_1\mid ⩽\frac{L}{2}$, and the displacement vector, $\mathbf{u}=(u_1\text{,}{u}_2\text{,}{u}_3)$, is imposed at the boundary $x_1=\pm \frac{L}{2}\text{,}\mathbf{u}=(0\text{,}0\text{,}\pm w)$. The phase-1 material is situated between $x_3=0$ and $x_3=t_1$ and the phase-2 material is situated between $x_3=t_1$ and $x_3=t_1+t_2=t$. This pattern is reproduced by periodicity in the $x_3$ direction.

The full 3D solution of this $(x_1\text{,}{x}_3)$-plane strain problem is periodic in the $x_3$ direction. Hence, it can be obtained by restricting the analysis to the domain $(x_1\text{,}{x}_3)\in \mathcal{D}\equiv ]-\frac{L}{2}\text{,}\frac{L}{2}[\times ]0\text{,}t[$, and by applying suitable displacement and stress periodicity conditions to the boundaries $x_3=0$ and $x_3=t$.

One may also use the well-known standard homogenization procedure to solve the same problem: First, the effective overall elastic constants of the periodic material are determined by solving an auxiliary boundary value problem on the unit cell. Then, the overall elastic constants are used to solve the initial boundary value problem. Finally, an estimation of the real 3D stresses is obtained by a suitable localization of the overall stresses in the unit cell. When applying this procedure to periodically layered materials, the unit cell boundary value problem, as well as the stress localization problem, are 1D in the $x_3$ direction, and closed-form solution are obtained. Moreover, in the case of the above described out-of-plane shearing problem with isotropic constituents (Young’s modulus $E^{\gamma }$, shear modulus $G^{\gamma }$, Poisson’s ratio ${\nu }^{\gamma }\text{,}\gamma =1\text{,}2$), the homogenized solution is straightforward: the displacement is linear in $x_3\text{,}\left(0\text{,}0\text{,}\frac{2x_3}{L}w\right)$, and the predicted (1,3)-shear stress after localization is uniform in both layers, ${\sigma }_{13}^{\mathrm{hom}}=2G^{\mathrm{hom}}\frac{w}{L}$. Here, $G^{\mathrm{hom}}=(\overline{G^{-1}}{)}^{-1}$ is the homogenized (1,3)-shear modulus where the following notation is used:

$$\overline{X}=\frac{t_1}{t}{X}^1+\frac{t_2}{t}{X}^2$$

It is expected that the full 3D solution converges to the homogenized one as the slenderness ratio $\frac{L}{t}$ goes to infinity.

The finite element ABAQUS software (ABAQUS, 2007) has been used to numerically solve the above described plane strain problem on $\mathcal{D}$. Fig. 2 shows the normalized effective shear modulus¹, $G^{\mathit{eff}}/G^{\mathrm{hom}}$, versus $\frac{L}{t}$ for $\frac{t_1}{t_2}=4\text{,}\frac{E^1}{E^2}=10$ and ${\nu }^1={\nu }^2=0.3$ and Fig. 3 shows the normalized shear stress distribution with respect to its average value for $\frac{L}{t}=8$. Clearly, a size effect on the effective shear modulus is exhibited for small values of $\frac{L}{t}$. Moreover, even for large values of $\frac{L}{t}$, the stress distribution near the vertical boundaries does not fit the one predicted by the homogenization procedure because of its incompatibility with the displacement boundary conditions.

Many works have tried to capture these size and boundary effects by adopting higher-order homogenized models. See among others (Boutin, 1996, Adhikary and Dyskin, 1997, Forest and Sab, 1998, de Buhan et al., 2000, Kouznetsova et al., 2002, Bigoni et al., 2007, Yuan et al., 2008). In most of these papers, general procedures based on some modified unit cell boundary value problems are proposed to derive Cosserat-type or second-gradient-type macroscopic equivalent homogeneous descriptions of the periodic medium. Actually these procedures are not specific to periodically layered materials. Yet, the behavior of multilayered plate structures (non periodic with small thickness in the $x_3$ direction) has been studied for many years. See, for example, Carrera (2002) for a review. The most common approach is to substitute the heterogeneous plate with a homogeneous equivalent one, with or without taking into account shear effects. Another class of plate models initiated by Pagano (1978) is based on layerwise approach. These models called M4 (multiparticle model for multilayered materials) have been developed at Ecole des Ponts ParisTech by Ehrlacher, Caron, Foret, Sab and their co-workers (Diaz Diaz et al., 2001, Hadj-Ahmed et al., 2001, Caron et al., 2006, Diaz Diaz et al., 2007, Dallot and Sab, 2008). The main interest of these models is to make the study of local fields possible, especially at the interfaces between layers (stress concentration in adhesive joints, delamination in composite materials, limit analysis of reinforced plates…).

The idea of the present paper is to combine higher order homogenization procedures and layerwise plate models. It consists in using the M4 model for the periodically layered material, and then in homogenizing this model obtaining a Cosserat multiparticle model (CM2). The proposed CM2 is described in Section 2 and its application, in Section 3, to the shearing problem of Fig. 1 will demonstrate its ability to reproduce both size and boundary effects.