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.

When stressed beyond a critical stress, ductile materials such as metals and alloys display a nonlinear plastic response. This is sketched in Fig. 26.1 for a uniaxial tensile test of a smooth specimen, where some relevant terms are defined. In general, plastic yielding is gradual when resolved at typical strain levels (e.g., ∼ 10−4). A critical stress, called the yield stress, is defined, which is the stress in uniaxial tension, or compression, required to cause a small, yet finite, permanent strain that is not recovered after unloading. It is common to take this onset yield strain as ey = 0.002 = 0.2%. Some common general features of plastic flow, with reference to stress vs. strain curves, are:

• The σ vs. e response is nonlinear and characterized by a decreasing intensity of strain hardening, measured by the slope dσ/de, as the strain increases. Generally, dσ/de ≥ 0.

• Unloading is nearly elastic.

• Plastic deformation of nonporous metals is essentially incompressible, i.e., volume preserving. A discussion of the physical basis for plastic deformation in subsequent chapters will explain why this is so.

As noted, a schematic stress-strain curve during uniaxial loading and unloading of an elastoplastic material is shown in Fig. 26.1. The initial yield stress is Y (later terms such as σy will be used to denote yield stress). Note that the yield stress is now ideally represented as a stress level at which an abrupt transition from linear, purely elastic, to nonlinear, elastic-plastic deformation occurs.

Fundamental concepts concerning the micromechanics of crystalline plasticity are reviewed in this chapter. An overview of deformation mechanisms is given for crystalline materials that possess grain sizes that are said to be “traditional,” i.e., larger than about 2 µm in diameter. Some brief comments are made about the trends in deformation mechanisms when the grain sizes are much below this range (nanograins).

Early Observations

In a series of articles published between 1898 and 1900 Ewing and Rosenhain summarized their metallographic studies of deformed polycrystalline metals. The conclusion they reached concerning the mechanisms of plastic deformation provided a remarkably accurate picture of crystalline plasticity. Figure 27.1 is a schematic diagram, including some surrounding text, taken from their 1900 overview article. Figure 27.2 is one of their many excellent optical micrographs of deformed polycrystalline metals; the particular micrograph in Fig. 27.2 is of polycrystalline lead. They identified the steps a-e in Fig. 27.1 as “slip-steps” caused by the emergence of “slip bands,” which formed along crystallographic planes, at the specimen surfaces (thereby coining these two well-known phrases).

Traces of the crystalline slip planes were indicated by the dashed lines. The line labeled C was indicated by them to be a grain boundary separating two grains; the grains, they concluded, were crystals with a more or less homogeneous crystallographic orientation. Slip steps corresponding to the diagram of Fig. 27.1 are clearly visible in the micrograph of Fig. 27.2.

Introduction

In a micropolar continuum the deformation is described by the displacement vector and an independent rotation vector. The rotation vector specifies the orientation of a triad of director vectors attached to each material particle. A particle (material element) can experience a microrotation without undergoing a macrodisplacement. An infinitesimal surface element transmits a force and a couple vector, which give rise to nonsymmetric stress and couple-stress tensors. The former is related to a nonsymmetric strain tensor and the latter to a nonsymmetric curvature tensor, defined as the gradient of the rotation vector. This type of the continuum mechanics was originally introduced by Voigt (1887) and the brothers Cosserat (1909). In a simplified micropolar theory, the so-called couple-stress theory, the rotation vector is not independent of the displacement vector, but related to it in the same way as in classical continuum mechanics.

The physical rationale for the extension of the classical to micropolar and couplestress theory was that the classical theory was not able to predict the size effect experimentally observed in problems which had a geometric length scale comparable to material's microstructural length, such as the grain size in a polycrystalline or granular aggregate. For example, the apparent strength of some materials with stress concentrators such as holes and notches is higher for smaller grain size; for a given volume fraction of dispersed hard particles, the strengthening of metals is greater for smaller particles; the bending and torsional strengths are higher for very thin beams and wires.