CONTINUUM MECHANICS for ENGINEERS Second Edition Second Edition

CONTINUUM MECHANICS for ENGINEERS

G. Thomas Mase George E. Mase

CRC Press Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data

Mase, George Thomas. Continuum mechanics for engineers / G. T. Mase and G. E. Mase. -- 2nd ed. p. cm. Includes bibliographical references (p. )and index. ISBN 0-8493-1855-6 (alk. paper) 1. Continuum mechanics. I. Mase, George E. QA808.2.M364 1999 531—dc21 99-14604 CIP

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Preface to Second Edition

It is fitting to start this, the preface to our second edition, by thanking all of those who used the text over the last six years. Thanks also to those of you who have inquired about this revised and expanded version. We hope that you find this edition as helpful as the first to introduce seniors or graduate students to continuum mechanics. The second edition, like its predecessor, is an outgrowth of teaching con- tinuum mechanics to first- or second-year graduate students. Since my father is now fully retired, the course is being taught to students whose final degree will most likely be a Masters at Kettering University. A substantial percent- age of these students are working in industry, or have worked in industry, when they take this class. Because of this, the course has to provide the stu- dents with the fundamentals of continuum mechanics and demonstrate its applications. Very often, students are interested in using sophisticated simulation pro- grams that use nonlinear kinematics and a variety of constitutive relation- ships. Additions to the second edition have been made with these needs in mind. A student who masters its contents should have the mechanics foun- dation necessary to be a skilled user of today’s advanced design tools such as nonlinear, explicit finite elements. Of course, students need to augment the mechanics foundation provided herein with rigorous finite element training. Major highlights of the second edition include two new chapters, as well as significant expansion of two other chapters. First, Chapter Five, Fundamental Laws and Equations, was expanded to add material regarding constitutive equation development. This includes material on the second law of thermodynamics and invariance with respect to restrictions on constitu- tive equations. The first edition applications chapter covering elasticity and fluids has been split into two separate chapters. Elasticity coverage has been expanded by adding sections on Airy stress functions, torsion of noncircular cross sections, and three-dimensional solutions. A chapter on nonlinear elasticity has been added to give students a molecular and phenomenological introduction to rubber-like materials. Finally, a chapter introducing students to linear viscoelasticity is given since many important modern polymer applications involve some sort of rate dependent material response. It is not easy singling out certain people in order to acknowledge their help while not citing others; however, a few individuals should be thanked. Ms. Sheri Burton was instrumental in preparation of the second edition manuscript. We wish to acknowledge the many useful suggestions by users of the previous edition, especially Prof. Morteza M. Mehrabadi, Tulane University, for his detailed comments. Thanks also go to Prof. Charles Davis, Kettering

University, for helpful comments on the molecular approach to rubber and thermoplastic elastomers. Finally, our families deserve sincerest thanks for their encouragement. It has been a great thrill to be able to work as a father-son team in publish- ing this text, so again we thank you, the reader, for your interest.

G. Thomas Mase Flint, Michigan George E. Mase East Lansing, Michigan

Preface to the First Edition

(Note: Some chapter reference information has changed in the Second Edition.)

Continuum mechanics is the fundamental basis upon which several graduate courses in engineering science such as elasticity, plasticity, viscoelasticity, and fluid mechanics are founded. With that in mind, this introductory treatment of the principles of continuum mechanics is written as a text suitable for a first course that provides the student with the necessary background in con- tinuum theory to pursue a formal course in any of the aforementioned sub- jects. We believe that first-year graduate students, or upper-level undergraduates, in engineering or applied mathematics with a working knowledge of calculus and vector analysis, and a reasonable competency in elementary mechanics will be attracted to such a course. This text evolved from the course notes of an introductory graduate contin- uum mechanics course at Michigan State University, which was taught on a quarter basis. We feel that this text is well suited for either a quarter or semes- ter course in continuum mechanics. Under a semester system, more time can be devoted to later chapters dealing with elasticity and fluid mechanics. For either a quarter or a semester system, the text is intended to be used in con- junction with a lecture course. The mathematics employed in developing the continuum concepts in the text is the algebra and calculus of Cartesian tensors; these are introduced and discussed in some detail in Chapter Two, along with a review of matrix meth- ods, which are useful for computational purposes in problem solving. Because of the introductory nature of the text, curvilinear coordinates are not introduced and so no effort has been made to involve general tensors in this work. There are several books listed in the Reference Section that a student may refer to for a discussion of continuum mechanics in terms of general ten- sors. Both indicial and symbolic notations are used in deriving the various equations and formulae of importance. Aside from the essential mathematics presented in Chapter Two, the book can be seen as divided into two parts. The first part develops the principles of stress, strain, and motion in Chapters Three and Four, followed by the der- ivation of the fundamental physical laws relating to continuity, energy, and momentum in Chapter Five. The second portion, Chapter Six, presents some elementary applications of continuum mechanics to linear elasticity and clas- sical fluids behavior. Since this text is meant to be a first text in continuum mechanics, these topics are presented as constitutive models without any dis- cussion as to the theory of how the specific constitutive equation was derived. Interested readers should pursue more advanced texts listed in the

Reference Section for constitutive equation development. At the end of each chapter (with the exception of Chapter One) there appears a collection of problems, with answers to most, by which the student may reinforce her/his understanding of the material presented in the text. In all, 186 such practice problems are provided, along with numerous worked examples in the text itself. Like most authors, we are indebted to many people who have assisted in the preparation of this book. Although we are unable to cite each of them individually, we are pleased to acknowledge the contributions of all. In addi- tion, sincere thanks must go to the students who have given feedback from the classroom notes which served as the forerunner to the book. Finally, and most sincerely of all, we express special thanks to our family for their encour- agement from beginning to end of this work.

G. Thomas Mase Flint, Michigan George E. Mase East Lansing, Michigan

Authors

G. Thomas Mase, Ph.D. is Associate Professor of Mechanical Engineering at Kettering University (formerly GMI Engineering & Management Institute), Flint, Michigan. Dr. Mase received his B.S. degree from Michigan State Uni- versity in 1980 from the Department of Metallurgy, Mechanics, and Materials Science. He obtained his M.S. and Ph.D. degrees in 1982 and 1985, respec- tively, from the Department of Mechanical Engineering at the University of California, Berkeley. Immediately after receiving his Ph.D., he worked for two years as a senior research engineer in the Engineering Mechanics Depart- ment at General Motors Research Laboratories. In 1987, he accepted an assis- tant professorship at the University of Wyoming and subsequently moved to Kettering University in 1990. Dr. Mase is a member of numerous professional societies including the American Society of Mechanical Engineers, Society of Automotive Engineers, American Society of Engineering Education, Society of Experimental Mechanics, Pi Tau Sigma, Sigma Xi, and others. He received an ASEE/NASA Summer Faculty Fellowship in 1990 and 1991 to work at NASA Lewis Research Center. While at the University of California, he twice received a distinguished teaching assistant award in the Department of Mechanical Engineering. His research interests include design with explicit finite element simulation. Specific areas include golf equipment design and vehicle crashworthiness. George E. Mase, Ph.D., is Emeritus Professor, Department of Metallurgy, Mechanics, and Materials Science (MMM), College of Engineering, at Michi- gan State University. Dr. Mase received a B.M.E. in Mechanical Engineering (1948) from the Ohio State University, Columbus. He completed his Ph.D. in Mechanics at Virginia Polytechnic Institute and State University (VPI), Blacksburg, Virginia (1958). Previous to his initial appointment as Assistant Professor in the Department of Applied Mechanics at Michigan State Univer- sity in 1955, Dr. Mase taught at Pennsylvania State University (instructor), 1950 to 1951, and at Washington University, St. Louis, Missouri (assistant pro- fessor), 1951 to 1954. He was appointed associate professor at Michigan State University in 1959 and professor in 1965, and served as acting chairperson of the MMM Department, 1965 to 1966 and again in 1978 to 1979. He taught as visiting assistant professor at VPI during the summer terms, 1953 through 1956. Dr. Mase holds membership in Tau Beta Pi and Sigma Xi. His research interests and publications are in the areas of continuum mechanics, viscoelas- ticity, and biomechanics.

Nomenclature

x1, x2, x3 or xi or x Rectangular Cartesian coordinates * * * x1 , x2 , x3 Principal stress axes ˆ ˆ ˆ eee123, , Unit vectors along coordinate axes δ Kronecker delta ij ε Permutation symbol ijk ∂ t Partial derivative with respect to time ∂ x Spatial gradient operator φ φ φ = grad = ,j Scalar gradient ∂ v = jvi = vi,j Vector gradient ∂ jvj = vj,j Divergence of vector v ε ijkvk,j Curl of vector v bi or b Body force (force per unit mass) pi or p Body force (force per unit volume) fi or f Surface force (force per unit area) V Total volume V.o Referential total volume ∆V Small element of volume dV Infinitesimal element of volume S Total surface S o Referential total surface ∆S Small element of surface dS Infinitesimal element of surface ρ Density ni or nˆ Unit normal in the current configuration ˆ NA or N Unit normal in the reference configuration

()nnˆˆ () ti or t Traction vector

σ N Normal component of traction vector σ S Shear component of traction vector σ ij Cauchy stress tensor’s components σ * Cauchy stress components referred to principal axes ij

o ()Nˆ pi Piola-Kirchhoff stress vector referred to referential area

PiA First Piola-Kirchhoff stress components sAB Second Piola-Kirchhoff stress components σσσ Principal stress values ()123,, () () σσ σ or I,, II III

Iσ, IIσ, IIIσ First, second, and third stress invariants σ σ M = ii/3 Mean normal stress

Sij Deviatoric stress tensor’s components

IS = 0, IIS, IIIS Deviator stress invariants σ oct Octahedral shear stress aij Transformation matrix

XI or X Material, or referential coordinates vi or v Velocity vector ai or a Acceleration components, acceleration vector ui or u Displacement components, or displacement vector ∂/∂ ∂/∂ d/dt = t + vk xk Material derivative operator

FiA or F Deformation gradient tensor

CAB or C Green’s deformation tensor

EAB or E Lagrangian finite strain tensor cij or c Cauchy deformation tensor eij or e Eulerian finite strain tensor ε ε ij or Infinitesimal strain tensor εεε ()123,, () () Principal strain values εε ε or I,, II III Iε , IIε , IIIε Invariants of the infinitesimal strain tensor

Bij = FiAFjA Components of left deformation tensor

I1, I2, I3 Invariants of left deformation tensor

W Strain energy per unit volume, or strain energy density e Normal strain in the Nˆ direction ()Nˆ γ ij Engineering shear strain ∆ ε e = V/V = ii = Iε Cubical dilatation η ij or Deviator strain tensor ω ij or Infinitesimal rotation tensor ω j or Rotation vector Λ = dx dX Stretch ratio, or stretch in the direction on Nˆ ()Nˆ λ = dX dx Stretch ratio in the direction on nˆ ()nˆ

Rij or R Rotation tensor

UAB or U Right stretch tensor

VAB or V Left stretch tensor ∂ ∂ Lij = vi/ xj Spatial velocity gradient

Dij Rate of deformation tensor

Wij Vorticity, or spin tensor J = det F Jacobian

Pi Linear momentum vector K(t) Kinetic energy P(t) Mechanical power, or rate of work done by forces S(t) Stress work Q Heat input rate r Heat supply per unit mass qi Heat flux vector θ Temperature = θ gi ,i Temperature gradient u Specific internal energy η Specific entropy ψ Gibbs free energy ζ Free enthalpy χ Enthalpy γ Specific entropy production Contents

1 Continuum Theory 1.1 The Continuum Concept 1.2 Continuum Mechanics

2 Essential Mathematics 2.1 Scalars, Vectors, and Cartesian Tensors 2.2 Tensor Algebra in Symbolic Notation — Summation Convention 2.3 Indicial Notation 2.4 Matrices and Determinants 2.5 Transformations of Cartesian Tensors 2.6 Principal Values and Principal Directions of Symmetric Second-Order Tensors 2.7 Tensor Fields, Tensor Calculus 2.8 Integral Theorems of Gauss and Stokes Problems

3 Stress Principles 3.1 Body and Surface Forces, Mass Density 3.2 Cauchy Stress Principle 3.3 The Stress Tensor 3.4 Force and Moment Equilibrium, Stress Tensor Symmetry 3.5 Stress Transformation Laws 3.6 Principal Stresses, Principal Stress Directions 3.7 Maximum and Minimum Stress Values 3.8 Mohr’s Circles for Stress 3.9 Plane Stress 3.10 Deviator and Spherical Stress States 3.11 Octahedral Shear Stress Problems

4 Kinematics of Deformation and Motion 4.1 Particles, Configurations, Deformation, and Motion 4.2 Material and Spatial Coordinates 4.3 Lagrangian and Eulerian Descriptions 4.4 The Displacement Field 4.5 The Material Derivative 4.6 Deformation Gradients, Finite Strain Tensors 4.7 Infinitesimal Deformation Theory 4.8 Stretch Ratios 4.9 Rotation Tensor, Stretch Tensors 4.10 Velocity Gradient, Rate of Deformation, Vorticity 4.11 Material Derivative of Line Elements, Areas, Volumes Problems

5 Fundamental Laws and Equations 5.1 Balance Laws, Field Equations, Constitutive Equations 5.2 Material Derivatives of Line, Surface, and Volume Integrals 5.3 Conservation of Mass, Continuity Equation 5.4 Linear Momentum Principle, Equations of Motion 5.5 The Piola-Kirchhoff Stress Tensors, Lagrangian Equations of Motion 5.6 Moment of Momentum (Angular Momentum) Principle 5.7 Law of Conservation of Energy, The Energy Equation 5.8 Entropy and the Clausius-Duhem Equation 5.9 Restrictions on Elastic Materials by the Second Law of Thermodynamics 5.10 Invariance 5.11 Restrictions on Constitutive Equations from Invariance 5.12 Constitutive Equations References Problems

6 Linear Elasticity 6.1 Elasticity, Hooke’s Law, Strain Energy 6.2 Hooke’s Law for Isotropic Media, Elastic Constants 6.3 Elastic Symmetry; Hooke’s Law for Anisotropic Media 6.4 Isotropic Elastostatics and Elastodynamics, Superposition Principle 6.5 Plane Elasticity 6.6 Linear Thermoelasticity 6.7 Airy Stress Function 6.8 Torsion 6.9 Three-Dimensional Elasticity Problems 7 Classical Fluids 7.1 Viscous Stress Tensor, Stokesian, and Newtonian Fluids 7.2 Basic Equations of Viscous Flow, Navier-Stokes Equations 7.3 Specialized Fluids 7.4 Steady Flow, Irrotational Flow, Potential Flow 7.5 The Bernoulli Equation, Kelvin’s Theorem Problems

8 Nonlinear Elasticity 8.1 Molecular Approach to Rubber Elasticity 8.2 A Strain Energy Theory for Nonlinear Elasticty 8.3 Specific Forms of the Strain Energy 8.4 Exact Solution for an Incompressible, Neo-Hookean Material References Problems

9 Linear Viscoelasticity 9.1 Introduction 9.2 Viscoelastic Constitutive Equations in Linear Differential Operator Form 9.3 One-Dimensional Theory, Mechanical Models 9.4 Creep and Relaxation 9.5 Superposition Principle, Hereditary Integrals 9.6 Harmonic Loadings, Complex Modulus, and Complex Compliance 9.7 Three-Dimensional Problems, The Correspondence Principle References Problems