How interface size, density, and viscosity affect creep and relaxation functions of matrix-interface composites: a micromechanical study

Mehran Shahidi,Bernhard Pichler,Christian Hellmich

doi:10.1007/s00707-015-1429-9

Abstract

Matrix-inclusion composites are known to exhibit interaction among the inclusions. When it comes to the special case of inclusions in form of flat interfaces, interaction among interfaces would be clearly expected, but the two-dimensional nature of interfaces implies quite surprising interaction properties. This is the motivation to analyze how interaction among two different classes of microscopic interfaces manifests itself in macroscopic creep and relaxation functions of matrix-interface composites. To this end, we analyze composites made of a linear elastic solid matrix hosting parallel interfaces, and we consider that creep and relaxation of such composites result from micro-sliding within adsorbed fluid layers filling the interfaces. The latter idea was recently elaborated in the framework of continuum micromechanics, exploiting eigenstress homogenization schemes, see Shahidi et al. (Eur J Mech A Solids 45:41–58, 2014). After a rather simple mathematical exercise, it becomes obvious that creep functions do not reflect any interface interaction. Mathematical derivation of relaxation functions, however, turns out to be much more challenging because of pronounced interface interaction. Based on a careful selection of solution methods, including Laplace transforms and the method of non-dimensionalization, we analytically derive a closed-form expression of the relaxation functions, which provides the sought insight into interface interaction. The seeming paradox that no interface interaction can be identified from creep functions, while interface interaction manifests itself very clearly in the relaxation functions of matrix-interface materials, is finally resolved based on stress and strain average rules for interfaced composites. They clarify that uniform stress boundary conditions lead to a direct external control of average stress and strain states in the solid matrix, and this prevents interaction among interfaces. Under uniform strain boundary conditions, in turn, interfacial dislocations do influence the average stress and strain states in the solid matrix, and this results in pronounced interface interaction.

Highlights

Matrix-inclusion composites are known to exhibit interaction among the inclusions [1,2,3,4,5,6,7]. When it comes to the special case of inclusions in form of flat interfaces, i.e., to that of matrix-interface composites, interaction among interfaces would be clearly expected as well; the two-dimensional nature of interfaces is responsible for surprising interaction properties [8,9], reminiscent of the situation encountered with micro-cracked materials [10,11]
We analyze matrix-interface composites consisting of a linear elastic solid matrix and parallel viscous interfaces, and we consider that the creep and relaxation behavior of such composites results from micro-sliding within adsorbed fluid layers filling the interfaces
There, we discuss the seemingly paradox situation that no interface interaction can be identified from the mathematical structure of the creep functions, while interface interaction is clearly manifested in the relaxation functions

Summary

Introduction

Matrix-inclusion composites are known to exhibit interaction among the inclusions [1,2,3,4,5,6,7]. 2, where we briefly recall fundamental state equations governing the time-dependent behavior of matrix-interface composites comprising two-dimensional interfaces filled with viscous fluids [15] This review comprises both types of so-called Hashin boundary conditions [17], i.e., uniform strain boundary conditions and uniform stress boundary conditions, as means for the study of relaxation and creep properties, respectively. 3 of the present paper is fully devoted to: In order to (i) derive a compact closed-form solution of the sought relaxation functions and to (ii) provide detailed insight into interface interaction, we carefully select solution methods for coupled systems of linear differential equations They are: Laplace transformation, a decoupling strategy in time domain based on an elimination scheme, and the method of non-dimensionalization. Thereby, we consider that the matrix-interface composite of Fig. 1 is subjected to a specific type of so-called Hashin boundary conditions [17], i.e., either to uniform strain boundary conditions or to uniform stress boundary conditions, respectively

State equations for uniform strain boundary conditions

State equations for uniform stress boundary conditions

Review of creep functions

Determination of dislocation histories based on Laplace transformation

Relaxation functions

Study of interface interaction in a relaxation test