Thermal Stresses – Advanced Theory and Applications

578 Pages·2009·7.97 MB·English

by  Richard B. Hetnarski, M. Reza Eslami (auth.)

The first paper ever published on thermal stresses and thermoelasticity: Duhamel,
J.-M.-C., Second mémoire sur les phénomènes thermo-mécaniques, J. de l’École
Polytechnique,tome15,cahier25,1837,pp.1–57.
Thermal Stresses – Advanced Theory
and Applications
SOLIDMECHANICSAND ITSAPPLICATIONS
Volume158
SeriesEditor: G.M.L.GLADWELL
DepartmentofCivilEngineering
UniversityofWaterloo
Waterloo,Ontario,CanadaN2L3GI
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Richard B. Hetnarski • M. Reza Eslami
Thermal Stresses – Advanced
Theory and Applications
123
RichardB.Hetnarski M.RezaEslami
DepartmentofMechanicalEngineering DepartmentofMechanicalEngineering
RochesterInstituteofTechnology AmirkabirUniversityofTechnology
Rochester,NewYork (TehranPolytechnic)
USA Tehran
[email protected] Iran
[email protected]
ISBN978-1-4020-9246-6 e-ISBN978-1-4020-9247-3
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The authors
dedicate this book
to the memory of their Parents
Jan Hetnarski Mohammad Sadegh Eslami
(1884 - 1966) (1900 - 1980)
and and
Izabela Hetnarska Zinat Shahrestani
(1893 - 1980) (1925 - 2006)
Love of their sons was overwhelming.
Non Omnis Moriar.
Preface
The authors are pleased to present Thermal Stresses – Advanced Theory and
Applications. This book will serve a wide range of readers, in particular, grad-
uate students, PhD candidates, professors, scientists, researchers in various
industrial and government institutes, and engineers. Thus, the book should be
considered not only as a graduate textbook, but also as a reference handbook
to those working or interested in areas of Applied Mathematics, Continuum
Mechanics, Stress Analysis, and Mechanical Design. In addition, the book pro-
vides extensive coverage of great many theoretical problems and numerous
references to the literature.
The field of Thermal Stresses lies at the crossroads of Stress Analysis, The-
ory of Elasticity, Thermodynamics, Heat Conduction Theory, and advanced
methods of Applied Mathematics. Each of these areas is covered to the extend
it is necessary. Therefore, the book is self-contained, so that the reader should
not need to consult other sources while studying the topic. The book starts
from basic concepts and principles, and these are developed to more advanced
levels as the text progresses. Nevertheless, some basic preparation on the part
of the reader in Classical Mechanics, Stress Analysis, and Mathematics, in-
cluding Vector and Cartesian Tensor Analysis is expected.
While selecting material for the book, the authors made every effort to
present both classical topics and methods, and modern, or more recent, devel-
opments in the field. The book comprises ten chapters.
Chapter 1 treats, among other topics, the basic laws of Thermoelastic-
ity,withdescriptionsandmathematicalformulationsofstresses,deformations,
constitutive laws, the equations of equilibrium and motion, the compatibility
conditions, and an introduction to two-dimensional thermoelasticity.
Chapter 2 is devoted to the necessary topics of thermodynamics. Detailed
attention is given to the first and second laws of thermodynamics, Fourier’s
law of heat conduction, and more advanced topics, namely, generalized ther-
moelasticity and second sound phenomenon, thermoelasticity without energy
dissipation, variational formulation of mechanics, the reciprocity theorem, and
the discussion of initial and boundary conditions. In the generalized theory of
thermoelasticity,theLord-Shulman,theGreen-Lindsay,andtheGreen-Naghdi
models are treated, and these models are then presented in a unified formula-
tion for heterogeneous/anisotropic materials.
vii
viii Preface
Basic problems of Thermoelasticity are discussed in Chapter 3, where the
analogy of thermal gradient and body forces is presented, and general solu-
tionsarederivedinrectangularCartesian,cylindrical,andsphericalcoordinate
systems.
Chapter 4 is devoted to problems of heat conduction, again treated in
various coordinate systems. Steady state one-, two-, and three-dimensional
problems are discussed, and necessary mathematical methods, like the use of
Fourier series and Bessel functions, are introduced.
Engineering applications are treated in Chapter 5. Various kinds of beams,
including rectangular, bimetallic, and curved beams are discussed in detail,
and more advanced or modern aspects, such as functionally graded beams, are
treated.
In Chapter 6, thermal stresses in disks, cylinders, and spheres are treated,
including functionally graded cylinders and spheres.
A short Chapter 7 presents an analysis of thermal expansion in piping
systems, a unique introduction to this frequently encountered engineering ap-
plication, a topic of importance, treated by advanced design codes.
In Chapter 8, the theories of coupled and generalized thermoelasticity are
presented. To the authors’ knowledge, such extensive treatment of these topics
has never before been given in a textbook.
Finite and boundary element methods are the topic of Chapter 9. The
Galerkin finite element is introduced, and the methods of generalized thermo-
elasticity are applied to disks and spheres. Also, problems of functionally
graded beams and layers are presented.
Thelastchapter,Chapter10,isdevotedtotheanalysisofcreep.First,gen-
eral definitions and the theory are presented, and then the problems related to
thermal effects are discussed. This chapter deals with useful and efficient nu-
mericaltechniquestohandlecreepproblemsofstructuressubjectedtothermal
stresses.
At the end of all chapters, except Chapters 9 and 10, there are a number
of problems for the students to solve. In total, there are 47 problems. Also, at
the end of each chapter, there is a list of literature.
The book was prepared over a number of years, with each author working
separately, except for a one-month period when both authors were Visiting
Scholars at the Institute of Fundamental Technological Research of the Polish
Academy of Sciences in Warsaw, where they worked jointly. The authors take
thisopportunitytothanktheauthoritiesoftheInstitutefortheinvitationand
for providing needed space and services.
The authors express their gratitude to Professor Theodore R. Tauchert of
the University of Kentucky for reading Chapters 1–3 and 7, to Professor Jo´zef
Ignaczak of the Polish Academy of Sciences in Warsaw for reading Chapters
3, 6, and 8, to Dr. Xiangmin Zhou of the University of Minnesota for reading
Preface ix
Chapter 9, and to Professor Jacek Skrzypek of Krako´w University of Technol-
ogy for reading Chapter 10. Their comments and corrections greatly improved
thetextandhelpedeliminateerrors.Also,thehelpofMr.A.Bagri,agraduate
student of the second author, in preparation and checking of the manuscript
is appreciated.
TheauthorsexpressthankstoProfessorG.M.L.Gladwell,theSeriesEditor,
for his support of the book’s publication and his advice regarding language
problems, and to Mrs. Nathalie Jacobs, Publishing Editor, Mechanical Engi-
neering, at Springer, for her kind offer to publish the book and her patience
with shifting deadlines for delivering of the manuscript.
Richard B. Hetnarski
M. Reza Eslami
September 2008
Contents
Preface vii
Notation – A Short List xv
Historical Note. Beginnings of Thermal Stresses Analysis xvii
1 Basic Laws of Thermoelasticity 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Stresses and Tractions . . . . . . . . . . . . . . . . . . . . . . . 2
3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Coordinate Transformation. Principal Axes . . . . . . . . . . . . 6
5 Principal Stresses and Stress Invariants . . . . . . . . . . . . . . 7
6 Displacement and Strain Tensor . . . . . . . . . . . . . . . . . . 10
7 Compatibility Equations. Simply Connected Region . . . . . . . 14
8 Compatibility Conditions. Multiply Connected Regions . . . . . 17
9 Constitutive Laws of Linear Thermoelasticity . . . . . . . . . . 21
10 Displacement Formulation of Thermoelasticity . . . . . . . . . . 23
11 Stress Formulation of Thermoelasticity . . . . . . . . . . . . . . 24
12 Two-Dimensional Thermoelasticity . . . . . . . . . . . . . . . . 27
13 Michell Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 35
14 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Thermodynamics of Elastic Continuum 43
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Thermodynamics Definitions . . . . . . . . . . . . . . . . . . . . 44
3 First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . 45
4 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . 46
5 Variational Formulation of Thermodynamics . . . . . . . . . . . 48
6 Thermodynamics of Elastic Continuum . . . . . . . . . . . . . . 50
7 General Theory of Thermoelasticity . . . . . . . . . . . . . . . . 54
8 Free Energy Function of Hookean Materials . . . . . . . . . . . 57
9 Fourier’s Law and Heat Conduction Equation . . . . . . . . . . 60
10 Generalized Thermoelasticity, Second Sound . . . . . . . . . . . 61
11 Thermoelasticity without Energy Dissipation . . . . . . . . . . . 70
xi