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.

In this chapter we consider the behavior of essentially 2D membranes. The membranes may be linear or nonlinear and are generally considered to undergo arbitrarily large deformations. More specifically, the membranes considered here are modeled after biological membranes such as those that comprise cell walls or the layers that exist within biomineralized structures, e.g., shells or teeth. The discussion is preliminary and meant to provide a brief introduction to the basic concepts involved in the constitutive modeling of such structures.

Biological Membranes

Figure 33.1a illustrates an idealized view of the red blood cell that shows its hybrid structure consisting of an outer bilipid membrane and an attached cytoskeleton; Fig. 33.1b shows a micrograph of a section of the cytoskeleton that is illustrated schematically in Fig. 33.1a. The cell membrane is a hybrid, i.e., composite structure consisting of an outer bilipid layer that is supported (i.e., reinforced) by a network attached to it on the cytoplasmic side, which is on the inside of the cell. The cytoskeleton is built up from mostly tetramers, and higher order polypeptides, of the protein spectrin attached at actin nodes. We note that the spectrin network has close to a sixfold nodal coordination. It has been known that the nonlinear elastic properties of the membrane depend sensitively on the details of the topology that includes, inter alia, nodal coordination, spectrin segment length, and the statistical distribution of such topological parameters.