Differential Geometry And Kinematics Of Continua

John D Clayton

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814616058
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This book provides definitions and mathematical derivations of fundamental relationships of tensor analysis encountered in nonlinear continuum mechanics and continuum physics, with a focus on finite deformation kinematics and classical differential geometry. Of particular interest are anholonomic aspects arising from a multiplicative decomposition of the deformation gradient into two terms, neither of which in isolation necessarily obeys the integrability conditions satisfied by the gradient of a smooth vector field. The concise format emphasizes clarity and ease of reference, and detailed step-by-step derivations of most analytical results are provided.

Contents:

• Introduction
• Geometric Fundamentals
• Kinematics of Integrable Deformation
• Geometry of Anholonomic Deformation
• Kinematics of Anholonomic Deformation
• List of Symbols
• Bibliography
• Index

Readership: Researchers in mathematical physics and engineering mechanics.

Chapter 1

Introduction

In this chapter, an overview of motivation, objectives, and contents of each chapter of this book is given. Historical remarks provide context for the contents of this book. The general scheme of notation used in later chapters is described. Unique features of this book are summarized.

1.1Motivation, Objectives, and Scope

The main objective of this book is to provide a somewhat complete, yet still concise, collection of mathematical relationships associated with differential geometry in the context of deformable continua. Bodies whose points may undergo relative motion but do not physically separate or break, and thus can be described kinematically by smooth fields of continuum theory, are examples of such deformable continua. These mathematical relationships are of fundamental importance in historic and modern research activities in nonlinear continuum mechanics and continuum physics of solids, and analogous concepts from differential geometry and field theory exist among governing equations of various other branches of continuum physics such as fluid mechanics, biomechanics, electromagnetics, and gravitation.

As formally discussed in later chapters, the total deformation gradient F for an element of a material body is decomposed multiplicatively as

where

and

are generic two-point mappings from the reference to intermediate configuration and intermediate to current configuration, respectively. In general curvilinear coordinates,

is the tangent map from the reference to current configuration, where spatial coordinates x^a = φ^a (X, t) follow the motion φ that may depend on time t and material particle X.

The aforementioned multiplicative decomposition is encountered frequently in mechanics and physics literature. More specifically, some version of such a decomposition is often considered the most physically realistic and mathematically sound ingredient of geometrically nonlinear kinematic and constitutive models of many kinds of materials, including various kinds of solid crystals (metals, ceramics, minerals, molecular crystals, etc.), polymers, and biological materials. Thorough accounts of applications of multiplicative decompositions in the context of nonlinear elastic, plastic, and/or anelastic crystalline solids can be found in several monographs [3, 7, 23, 30]. Solutions to boundary value problems corresponding to discrete lattice defects—crystal dislocations, disclinations, and point defects—embedded in otherwise nonlinear elastic media have been derived [9, 10, 30, 33, 34]. Applications towards other classes of matter (e.g., biological materials, viscous solids and fluids) have likewise been described geometrically [13, 27]. Several other general theoretical and mathematically rigorous treatments of particular relevance to the present work are collected in [25]; see also [16, 29] for review articles of differential geometric fundamentals in the context of continuum mechanics and continuum physics.

In the terminology adopted throughout this book, the deformation gradient F is said to be integrable or “holonomic” since the differentiable one-to-one mapping φ(X, t) exists between referential and spatial positions of material particles at any given time t. Conversely,

and

need not be integrable functions of X^A and x^a, respectively; in such cases, these mappings are said to be “anholonomic” [28]. When

is not integrable at a given time instant, coordinates

that are differentiable one-to-one functions of X^A (x^a) do not exist, and the resulting intermediate configuration is labeled an anholonomic space [8] or anholonomic reference [21].

The first half of this book deals with geometry and kinematics of holonomic deformation. Chapter 2 introduces requisite notation and definitions associated with holonomic coordinate systems and various kinds of coordinate differentiation such as covariant differentiation. Identities from classical differential geometry associated with torsion and curvature, for example, are given. Chapter 3 introduces the deformation gradient and other fundamental derived quantities and identities, and it includes specific descriptions and examples in Cartesian, cylindrical, spherical, and convected coordinate systems.

Certain material in Chapters 2 and 3—which is necessary for comparison with or derivation of new results in later chapters—has been collected from the vast respected literature on continuum mechanics and nonlinear elasticity of solids [15, 22, 24, 31] and thus may already be familiar to many applied mathematicians and scholars of these disciplines. However, the organization, scope, stepwise derivations, and examples in these two chapters differ from those of any known earlier works and will thus presumably benefit the individual researcher (as a desktop reference), advanced student (as a practice guide for detailed tensor calculations), and instructor (as supplemental educational material). Furthermore, many of the example calculations expressed in indicial form in general curvilinear coordinates have not been presented elsewhere, to the author’s knowledge.

The second half of this book describes geometry and kinematics of anholonomic deformation. Chapter 4 develops rules for differentiation and derives numerous differential-geometric identities associated with a multiplicative decomposition of the deformation gradient into two generally anholonomic mappings. Extensive treatments of integrability conditions and choices of possibly curvilinear anholonomic coordinate systems are included. Chapter 5 contains important results pertaining to derived kinematic quantities and integral theorems in the context of anholonomic deformation. The treatment in this book for mathematical operations in anholonomic space associated with a multiplicative decomposition of the total deformation gradient is thought to be more comprehensive than any given elsewhere, at least in the context of literature known to the author. Although certain classical geometric identities [16, 28] are often used, much of Chapters 4 and 5 is new, with a number of identities, derivations, and example calculations not published elsewhere. The content of Chapters 4 and 5 of this book significantly extends a related study of anholonomic deformation, geometry, and differentiation published two years ago in a more brief research paper by the author [8].

A comprehensive list of symbols used throughout this book is provided in Appendix A, immediately after Chapter 5. The bibliography and subject index then follow.

1.2Historical Remarks

The history of anholonomic geometry is reviewed concisely here. In the context of classical differential geometry, analysis of anholonomic spaces began with work of G. Vrânceanu [32] and É. Cartan [5] in the 1920s. The notion of moving, possibly non-coordinate (i.e., non-holonomic), sets of director vectors or frames is also credited to Cartan [4]. With regard to construction of a unified field theory in physics, A. Einstein [12] considered geometric spaces with teleparallelism, meaning space-time described by a manifold equipped with a connection with vanishing curvature and non-vanishing torsion, the latter feature being related to anholonomicity. With regard to thermomechanical behavior of condensed matter, C. Eckart [11] formally conceived the notion of a locally relaxed configuration in anelasticity, corresponding geometrically to a non-Euclidean material manifold. Not long thereafter in the 1950s–1960s, fundamental linkages between anholonomicity associated with Cartan’s torsion tensor of a certain affine connection and the presence of dislocations in crystalline solids were established by B. Bilby [1, 2], E. Kröner [19, 20], and K. Kondo [17, 18]. These works established the geometric foundations for numerous more recent applications of the theory such as those cited in Section 1.1.

The history of the study of holonomic geometry and kinematics of continuous bodies is much older and much larger in scope. A comprehensive bibliography is contained in the 1960 treatise of C. Truesdell and R. Toupin [31], wherein proper credit is given to historical works of renowned European scholars dating back several centuries. In particular, correspondence between satisfaction of the local compatibility equations, for example those appearing in nonlinear elasticity theory, and vanishing of the curvature tensor in Riemannian geometry has been known since the middle of the 19th century [31].

1.3Notation

Notations of both ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. Introduction
• 2. Geometric Fundamentals
• 3. Kinematics of Integrable Deformation
• 4. Geometry of Anholonomic Deformation
• 5. Kinematics of Anholonomic Deformation
• Appendix A List of Symbols
• Bibliography
• Index