Continuum mechanics for engineers

380 Pages·1999·4.602 MB·English

by  G. Thomas Mase, George E. Mase

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
x , x , x or x or x Rectangular Cartesian coordinates
1 2 3 i
x *, x *, x * Principal stress axes
1 2 3
eˆ ,eˆ ,eˆ Unit vectors along coordinate axes
1 2 3
δ Kronecker delta
ij
ε Permutation symbol
ijk
∂ Partial derivative with respect to time
t
∂ Spatial gradient operator
x
(cid:2)φ = grad φ = φ Scalar gradient
,j
(cid:2)v = ∂v = v Vector gradient
j i i,j
∂v = v Divergence of vector v
j j j,j
ε v Curl of vector v
ijk k,j
b or b Body force (force per unit mass)
i
p or p Body force (force per unit volume)
i
f or f Surface force (force per unit area)
i
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
n or nˆ Unit normal in the current configuration
i
ˆ
N or N Unit normal in the reference configuration
A
t(nˆ) or t(nˆ) Traction vector
i
σ Normal component of traction vector
N
σ Shear component of traction vector
S
σ Cauchy stress tensor’s components
ij
σ* Cauchy stress components referred to principal axes
ij
( )
o Nˆ
p Piola-Kirchhoff stress vector referred to referential area
i
P First Piola-Kirchhoff stress components
iA
s Second Piola-Kirchhoff stress components
AB
σ ,σ ,σ Principal stress values
( ) ( ) ( )
1 2 3
or σ,σ ,σ
I II III
I II III First, second, and third stress invariants
σ, σ, σ
σ = σ/3 Mean normal stress
M ii
S Deviatoric stress tensor’s components
ij
I = 0, II , III Deviator stress invariants
S S S
σ Octahedral shear stress
oct
a Transformation matrix
ij
X or X Material, or referential coordinates
I
v or v Velocity vector
i
a or a Acceleration components, acceleration vector
i
u or u Displacement components, or displacement vector
i
d/dt = ∂/∂t + v ∂/∂x Material derivative operator
k k
F or F Deformation gradient tensor
iA
C or C Green’s deformation tensor
AB
E or E Lagrangian finite strain tensor
AB
c or c Cauchy deformation tensor
ij
e or e Eulerian finite strain tensor
ij
ε or ε Infinitesimal strain tensor
ij
ε( ),ε( ),ε( ) Principal strain values
1 2 3
or ε,ε ,ε
I II III
I , II , III Invariants of the infinitesimal strain tensor
ε ε ε
B = F F Components of left deformation tensor
ij iA jA
I , I , I Invariants of left deformation tensor
1 2 3