The previous three chapters have laid out some of the basic phenomenological features of plastic deformation in crystals and have developed a mathematical constitutive framework for analyzing crystalline deformation. It is not the purpose herein to provide an exhaustive treatment of particular case studies, in particular through the review of various numerical studies that have been performed, as this is the subject of a rather different volume. We do, however, explore some of the phenomenological implications of the mechanisms and theory developed above vis-à-vis the nature of crystalline deformation. In particular, we will explore the natural tendency of plastic deformation to become highly nonuniform and in fact localized into patterns that can, inter alia, evolve into bands of intensely localized slip, kinking patterns, and the sort of heterogeneous patterns of slip on different systems that were referred to as “patchy slip” in Chapter 27. These examples of localized deformation are important because they often lead to material failure, as well as to the evolution of internal substructure that, in turn, directly influences evolving material response. On the other hand, the analysis of these deformation patterns serves to highlight some rather fundamental aspects of the process of crystalline deformation via the process of slip. This serves to reveal and, in part explain, some of the basic implications of the type of theory we have outlined herein.

The problem considered here has found application to a legion of physical applications including, inter alia, the theory of solid state phase transformations where the transformation (arising from second phase precipitation, allotropic transition, or uptake of solutes, or changes in chemical stoichiometry) causes a change in size and/or shape of the transformed, included, region; differences in thermal expansion of an included region and its surrounding matrix, which in turn causes incompatible thermal strains between the two; and, perhaps surprisingly, the concentrated stress and strain fields that develop around included regions that have different elastic modulus from those of their surrounding matrices. For the reason that the results of this analysis have application to such a wide variety of problem areas, and because the solution approach we adopt has heuristic value, we devote this chapter to the inclusion problem.

The Problem

In an infinitely extended elastic medium, a region – the “inclusion” – undergoes what would have been a stress free strain. Call this strain the “transformation strain,”eT. Due to the elastic constraint of the medium, i.e., the matrix, there are internal stresses and elastic strains. What is this resulting elastic field and what are its characteristics? In particular, can an exact solution be found for this involved elastic field? The region of interest is shown in Fig. 20.1 and is denoted as VI; the outward pointing unit normal to VI is n.

The breakdown of an initially flat, or smooth, surface into one characterized by surface roughness is an important type of phenomena occurring, inter alia, during the growth of thin films or at surfaces of solids subject to remotely applied stress in environments that induce mass removal or transport. In the case of thin films, stresses arise due to lattice mismatch and/or differences in coefficients of thermal expansion. The sources of stress are, indeed, legion but the effect can be to induce roughness, and surface restructuring, that may be either deleterious, or in some cases desirable, if the patterning can be controlled. The phenomena was first studied by Asaro and Tiller (1972) and has since been pursued by others. Our purpose is to develop some of the guiding principles, but we note that the topic is far from being thoroughly worked out. In particular, we make many simplifying assumptions, one being the assumption of surface isotropy. We also ignore some important physical attributes of surfaces, such as surface stress, which have recently been added to the description of surface patterning (Freund and Suresh, 2003).

Stressed Surface Problem

We consider here the phenomena of the breakdown of planar interfaces subject to stress into interfaces characterized by undulated topology. The phenomena is governed by those same driving forces that lead to crack growth and the growth of defects, such as inclusions, that cause internal stresses.

In this chapter we explore the transition from the plastic response of single crystals to that of polycrystalline aggregates. The treatment given here is not meant to be exhaustive but rather to reveal some of the more fundamental issues involved. Suggested reading provides the link to the rather large volume of research conducted during the past two decades on the subject. The basic issues to be explored include the link between the micromechanical mechanisms of deformation on the scale of individual grains and macroscopic elastic-plastic response. One particular aggregate model is developed in detail and used to examine several physical phenomena. Among these are the development of crystallographic texture and anisotropic macroscopic response. We use the model to perform “numerical experiments” to define yield surfaces as they might be measured experimentally. We note how such surfaces naturally develop structure that is described as corners and explore the significance of this vis-à-vis the plastic strain response to sudden changes in strain path. We study this path dependent behavior further by appealing to simple rate-independent flow and deformation theories thus completing the link between microscopic and macroscopic behavior. The development of anisotropic plastic behavior is shown to occur after only modest deformation of initially isotropic aggregates.

Perspectives on Polycrystalline Modeling and Texture Development

Polycrystals are continuous 3D collections of grains (crystallites), which, as assumed herein, can deform by cyrstallographic slip. As such, the actual solution to a problem of a deforming polycrystal is that of a highly complex elastic-plastic boundary value problem for a large collection of anisotropic, continuous, and fully contiguous crystals.

Introduction

A general constitutive theory of the stress-modulated growth of biomaterials is presented in this chapter with a particular accent given to pseudoelastic living tissues. The governing equations of the mechanics of solids with a growing mass are derived within the framework of finite deformation continuum thermodynamics. The analysis of stress-modulated growth of living soft tissues, bones, and other biomaterials has been an important research topic in biomechanics during past several decades. Early work includes a study of the relationship between the mechanical loads and uniform growth by Hsu (1968) and a study of the mass deposition and resorption processes in a living bone by Cowin and Hegedus (1976a, 1976b). The latter work provided a set of governing equations for the so-called adaptive elasticity theory, in which an elastic material adopts its structure to applied loading. In contrast to hard tissues which undergo only small deformations, soft tissues such as blood vessels, tendons, or ligaments can experience large deformations. Fundamental contributions were made by Fung and his co-workers (e.g., Fung 1993, 1995) in the analytical description of the volumetrically distributed mass growth and by Skalak et al. (1982) for the mass growth by deposition or resorption on a surface. Hard tissues, such as bones and teeth, grow by deposition on a surface (apposition). Changes in porosity, mineral content and mass density are because of internal remodeling. Soft tissues grow by volumetric, also referred to as interstitial, growth.

Crystalline materials deform by a process of crystalline slip, whereby material is transported via shear across distinct crystal planes and only in certain distinct crystallographic directions in those planes. This process imparts a strong directionality to the plastic flow process and specifies a clear kinematic definition to the plastic spin. In what follows the theory is developed around a model for a laminated material; this is done to demonstrate the generality of the approach to a broader range of materials where slip is kinematically mediated by fixed directions.

Laminate Model

We consider the fiber reinforced plastic (FRP) material to be composed of an essentially orthotropic laminate, which contains a sufficient number of plies so that homogenization is a reasonable way to describe the material behavior. The principal directions of the fibers are described by a set of mutually orthogonal unit base vectors, ai, as depicted in Fig. 31.1. The resulting orthotropic elastic response of the laminated composite will thus be fixed on and described by these vectors. The material can also deform via slipping in the plane of the laminate, i.e., via interlaminar shear, and this slipping is confined to the interlaminar plane. Slipping is possible in all directions in the plane, but not necessarily with equal ease. We thus introduce two slip systems, aligned with the slip directionss1 and s2. The normal to the laminate plane is m, so that s1 · m = 0 and s2 · m = 0.

This book is written for graduate students in solid mechanics and materials science and should also be useful to researchers in these fields. The book consists of eight parts. Part 1 covers the mathematical preliminaries used in later chapters. It includes an introduction to vectors and tensors, basic integral theorems, and Fourier series and integrals. The second part is an introduction to nonlinear continuum mechanics. This incorporates kinematics, kinetics, and thermodynamics of a continuum and an application to nonlinear elasticity. Part 3 is devoted to linear elasticity. The governing equations of the three-dimensional elasticity with appropriate specifications for the two-dimensional plane stress and plane strain problems are given. The applications include the analyses of bending of beams and plates, torsion of prismatic rods, contact problems, semi-infinite media, and three-dimensional isotropic and anisotropic elastic problems. Part 4 is concerned with micromechanics, which includes the analyses of dislocations and cracks in isotropic and anisotropic media, the well-known Eshelby elastic inclusion problem, energy analyses of imperfections and configurational forces, and micropolar elasticity. In Part 5 we analyze dislocations in bimaterials and thin films, with an application to the study of strain relaxation in thin films and stability of planar interfaces. Part 6 is devoted to mathematical and physical theories of plasticity and viscoplasticity. The phenomenological or continuum theory of plasticity, single crystal, polycrystalline, and laminate plasticity are presented. The micromechanics of crystallographic slip is addressed in detail, with an analysis of the nature of crystalline deformation, embedded in its tendency toward localized plastic deformation.