Localized Effects in Periodic Elastoplastic Composites

Research Article

Ann J Materials Sci Eng. 2015;2(1): 1017.

Localized Effects in Periodic Elastoplastic Composites

Jacob Aboudi* and Michael Ryvkin

Department of Engineering, Tel Aviv University, Israel

*Corresponding author: Jacob Aboudi, Department of Engineering, School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel

Received: November 24, 2014; Accepted: February 14,2014; Published: February 18, 2015

Abstract

A method is applied for the study of the field distributions in metal matrix fiber reinforced composites with periodic microstructure in which localized damage exists in the form of complete or partial fiber loss and crack. In addition, the behavior of ceramic/metal periodically layered composites with a single broken ceramic layer is determined. The pro-posed analysis is based on continuum damage mechanics considerations, and the method of solution combines three distinct approaches. In the first one, referred to as the representative cell method, the periodic composite domain is reduced, in conjunction with the discrete Fourier transform to a finite domain problem of a single representative cell. This method has been previously applied on linear thermoelastic, smart and electrostrictive composites, but is presently extended and applied on elastoplastic composites (presently deformation and incremental plasticity). In the second approach, the appropriate far-field boundary conditions in the transform domain are applied in conjunction with the high-fidelity generalized method of cells micromechanical model for the prediction of the macroscopic behavior of the inelastic composite. The third approach consists of the application of the inelastic higher-order theory for the computation of the elastoplastic field in the transform domain. An inverse transform provides the actual field. The effect of damage is included in the analysis in the form of eigenstresses which are a priori unknown. Hence an iterative procedure is employed to obtain a convergent solution.

The proposed method is verified by a comparison with an analytical solution, and several applications illustrate the applicability of the method for metal matrix composites with localized damage in the form of a crack or fiber loss.

Keywords: Localized damage; Cracked fiber reinforced materials; Representative cell method; High-Fidelity generalized method of cells; Inelastic higher-order theory; Elastoplastic composites

Introduction

The micromechanical analysis of composites with periodic microstructure is usually carried out by identifying and analyzing a repeating unit cell. However, when localized effects such as one or several cracks occur in the composite, the periodicity is lost and its behavior cannot be determined directly by analyzing a repeating unit cell. If these effects are nevertheless included in the analysis of the repeating unit cell, the resulting behavior would correspond to that of a composite with periodic (i.e., not localized) effects which obviously is an unrealistic situation.

In a recent article, Aboudi and Ryvkin [1] proposed the analysis of linearly elastic composites with localized damage by representing the effect of the latter by eigenstresses. This analysis combines continuum damage mechanics considerations with three different approaches. In the first one the idea of using the eigenstresses to represent the nonlinear effects enables application of the representative cell method, Ryvkin and Nuller [2], based on the discrete Fourier transform which is applicable to linear problems. As a result the initial problem formulated for a domain comprising a large number of cells is reduced to a problem for a single representative cell. Appropriate far-field boundary conditions (which are not influenced by the localized effects) in the transform domain are applied in conjunction with the highfidelity generalized method of cells (HFGMC) [3]) micromechanical model which forms the second approach. The third approach consists of the application of the higher-order inelastic theory, Aboudi et al. [4], for the computation of the field in the transform domain. An inverse transform provides the actual field. The effect of damage is included in the analysis in the form of eigenstresses which are a priori unknown. In Ryvkin and Aboudi [5], this approach has been also proven to be successful and effective in the analysis of cracked layered elastic composites, where one or several combinations of a transverse and two longitudinal cracks (H-cracks) caused branching have been investigated. Furthermore, it has been successfully applied for the prediction of the field distributions in electro-magneto-thermoelastic composites with cracks, cavities and inclusions, Aboudi [6]. A brief review of various methods for the analysis of localized effects in thermoelastic composites has been recently presented by Aboudi and Ryvkin [3].

Thus far, the representative cell method has been employed in the analysis of linearly thermoelastic [1], electro-magneto-thermo-elastic [6] and electrostrictive (nonlinear) [7] composites. In the present investigation, this method is extended to elastoplastic materials (presently incremental and deformation elastoplasticity). In addition, the approach of Aboudi and Ryvkin [1] in analyzing localized damage in composites is presently further extended to enable the prediction of the behavior of elastoplastic periodic composites which include localized damage in the form of a cavity (a fiber loss) or a crack. In this approach, the plasticity effects are represented in the form of eigenstresses which extend over the entire considered region. In addition, these eigenstresses include the effects of localized damage which are operative over the damaged region only as has been proposed by Aboudi and Ryvkin [1] in the case of linear elasticity. Thus, according to our present proposed approach these eigenstresses include the combined contributions of the effects of plasticity as well as the localized damage. These eigenstresses are not known in advance and, therefore, an iterative procedure is employed to establish a convergent solution.

This article is organized as follows. In the next section the problem is formulated which followed by the method of solution section. The verification of the method is performed by a comparison with analytical solution for an infinite elastoplastic (deformation plasticity) medium with embedded cavity subjected to a remote biaxial loading, Ishikawa [8]. The application section presents the applicability of the proposed method for the prediction of the behavior of elastoplastic solids with embedded cavity, fiber reinforced metal matrix composites with a complete and partial fiber loss, and layered metal matrix composite with a single broken layer. The final section presents the conclusions and several possible future generalizations.

Problem Statement

The present investigation deals with elastoplastic composites that possess a periodic microstructure and include a localized damage such as a crack, fiber loss or cavity. This type of composites is illustrated in Figure 1(a) where the fibers, oriented in the x1-direction and arranged in a doubly periodic manner, are embedded in an elastoplastic material.