Problems and Solutions in Thermoelasticity and Magneto-thermoelasticity

112 Pages·2017·3.008 MB·English

by  B. Das (auth.)

B. Das
Problems and
Solutions in
Thermoelasticity and
Magnetothermoelasticity
Problems and Solutions in Thermoelasticity
and Magnetothermoelasticity
B. Das
Problems and Solutions
in Thermoelasticity
and Magnetothermoelasticity
123
B. Das
Ramakrishna MissionVidyamandira
Howrah, West Bengal
India
ISBN978-3-319-48807-3 ISBN978-3-319-48808-0 (eBook)
DOI 10.1007/978-3-319-48808-0
LibraryofCongressControlNumber:2016955799
©SpringerInternationalPublishingAG2017
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Dedicated to my father late Nripendra
Nath Das and mother Smt. Gita Das
Preface
The theory of thermoelasticity began in the first half of the nineteenth century and
acquiredtheimportanceinthemiddleofthiscentury.Itiswellrecognizedthatthis
theory is of great practical utility in several new areas of engineering and tech-
nology such as acoustics, aeronautics, chemical, and nuclear engineering. During
thelastfourdecades,thetheoryhasbeeninrapidprogress,andnow,itisoneofthe
major disciplines of continuum mechanics wherein there is a vast scope for theo-
retical and practical research.
Rememberingtheutilityofthesetheories,thisbookdealswithsomeproblemsin
the mathematical theories of thermoelasticity (Coupled and Generalized) and
magnetothermoelasticity, and all these problems have been solved by eigenvalue
methodologyand presented infour chapters.The techniques ofintegraltransforms
such as Laplace and Fourier transforms as also Laplace–Fourier double integral
transformsandnormalmodeanalysishavebeenusedinthebasicequationstoform
a vector–matrix differential equation (with unknown variables in the transformed
domain) which is then solved by eigenvalue methodology.
ThisbookcontainsfivechaptersincludingoneintroductorychapternamedBasic
Ideas.
The first chapter is the introductory chapter containing basic ideas of stresses,
strains, thermoelasticity, magnetothermoelasticity, and constitutive relations in
between them.
In the second chapter, the rudiment discussions of vector-matrix differential
equationandsolutionmethodologyinthetheoriesandinversionofLaplacetransform.
The third chapter contains one problem on classical coupled thermoelasticity
which deals with the theory of coupled thermoelasticity in an isotropic elastic
medium with cylindrical cavity under the dependence of modulus of elasticity on
the reference temperature.
The fourth chapter contains four problems of generalized thermoelasticity.
The first problem is concerned with the thermoelastic interactions in an
unboundedisotropicbodywithsphericalcavity.Exactexpressionsfortemperature
distribution,stress,anddisplacementcomponentsareobtainedinLaplacetransform
domain for three different cases:
vii
viii Preface
(i) A known functional thermal load is given to the surface of the spherical
cavity which is also stress free.
(ii) The surface is stress free and exerted a ramp-type heat punch and
(iii) A ramp-type thermal load is given to the boundary surface which is main-
taining a constant reference temperature.
Anumerical approach isimplemented for the inversion of Laplace transform in
order to obtain the solution in physical domain. Finally, numerical computations
of the stress, temperature, and displacement have been made and presented
graphically.
The second problem of this chapter is concerned with the thermoelastic inter-
actions in an unbounded body with a cylindrical hole in the context of
Green-Naghdi [G-N Model II] theory. The fundamental equations have been
writtenintheformofavector–matrixdifferentialequationintheLaplacetransform
domain and solved by the eigenvalue approach methodology. Two different cases
arisinginthestudyofwavepropagationinaninfinitemediumarestudiedindetail
bytheexaminationofthenatureofthesolutioninspace–timedomainfordifferent
conditions of the boundary surface.
Thethirdproblemofthischapterdealswiththethermoelasticinteractionsdueto
instantaneous heat source in a homogeneous isotropic and unbounded rotating
elasticmediuminthecontextofgeneralizedthermoelasticity[L-SModel].Integral
transform techniques are adopted, namely the Laplace transform for the time
variableandtheexponentialFouriertransformfortwoofthespacevariablesinthe
basic equations of the generalized thermoelasticity, and finally, the resulting
equations are written in the form of a vector-matrix differential equation which is
then solved by the eigenvalue approach. Exact expressions for the temperature
distribution, thermal stresses, and displacement components are obtained in the
Laplace–double Fourier transform domain. A numerical approach is implemented
for the inversion of Laplace transform and double Fourier transforms in order to
obtain the solution in physical domain.
The fourth problem of this chapter deals with two-dimensional problem for a
half-space under the action of body force. A double integral transform (Laplace
transform for time variable and Fourier transform for space variable) is used to
obtain the expressions of stresses and temperature when surface is stress-free with
assigned thermal shock.
Fifth chapter contains three problems. These problems deal with generalized
magnetothermoelasticity.
The first problem is concerned with one-dimensional problem of generalized
magnetothermoelasticity for a half-space. Laplace transform for time variable is
used, and the resulting equations are written in the form of a vector–matrix dif-
ferential equation. To get the solution in the transformed domain, we apply the
method of eigenvalue approach. The inversion of Laplace transform is carried out
numerically by Bellman method. Finally, numerical computations have been done
for the expressions of displacement, temperature, stresses, and induced magnetic
Preface ix
field. Several figures characterizing these field variables in the form of graphs are
presented.
The second problem of this chapter deals with the thermoelastic interactions of
two-dimensional problem of generalized magnetothermoelasticity for a half-space
in a rotating medium under constant magnetic and electric intensities. The normal
mode analysis is used to obtain the expressions for displacements, temperature,
stresses, and induced magnetic field.
The third or last problem of this chapter deals with a two-dimensional problem
of a homogeneous isotropic perfectly conducting thick plate in the context of
generalized magnetothermoelasticity [L-S model]. A double integral transform
(Laplace transform for time variable and Fourier transform for space variable) is
used, and the resulting equations are written in the form of a vector–matrix dif-
ferential equation which is solved by the eigenvalue approach subjected to
time-dependent compression under constant magnetic and electric intensities. The
inversion of Laplace transform is carried out numerically by Zakian method.
Finally, numerical computations have been done for the expressions of displace-
ment,temperature,stresses,andinducedelectricandmagneticfield.Severalfigures
characterizing these field variables in the form of graphs are presented.
The book has been written for those whose interest is primarily in the applica-
tions of the problems of thermoelasticity and magnetothermoelasticity, which are
the branches of continuum mechanics related to the properties of elasticity and
plasticity. Further, the book is written for the research scholars who have a great
interest in thermoelastic problems (generalized or coupled). The chapters of this
bookshould,therefore,beaccessibletoastudentwellgroundedinthermoelasticity.
If the reader wishes to use the results of the theory, they may rapidly pick out the
results ofneeds. Thechaptersarequiteindependent andmay bereadinanyorder.
This book is very much applicable for the students from the different branches of
engineering, especially in the fields of material science and nuclear engineering.
Belur, Howrah B. Das
August 2016
Acknowledgements
Alotofpeoplehavehelpedmealottowritethisbook,andwithouttheirhelp,this
is very hard to complete my work. So I like to convey my thanks and regards to
them. First of all, I express my thanks and gratitude to my teacher, Dr. Abhijit
Lahiri, Professor, Department of Mathematics, Jadavpur University, Kolkata.
Withouthiscontinuouseffortsandinspirations,itwouldnothavebeenpossiblefor
me to complete this book. I would like to express my thanks to my friends and
fellow researcher for their uninterrupted moral support.
I am also grateful to all the teachers of Deparment of Mathematics of Jadvpur
University, Ramakrishna Mission Vidyamandira, Belur, and Prasanta Chandra
Mahalanobis Mahavidyalaya, Kolkata.
I amgrateful to myparents for theircontinuous encouragementsand supportin
all respect. I would like to express my thanks to my wife Sharmistha for her
inspiration. My love and blessings for my two little sons.
xi
Contents
1 Basic Ideas. .... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Definitions, Relations, and Theories.. .... .... .... ..... .... 3
1.1.1 Elastic Solids..... .... .... .... .... .... ..... .... 3
1.1.2 Thermal Stresses.. .... .... .... .... .... ..... .... 4
1.1.3 Equations of Motion ... .... .... .... .... ..... .... 6
1.1.4 Thermomechanical Coupling. .... .... .... ..... .... 7
1.1.5 Classical Coupled Thermoelasticity.... .... ..... .... 7
1.1.6 Lord–Shulman Model of Linear (1967)
Thermoelasticity (L-S Model) or Extended
Thermoelasticity (ETE). .... .... .... .... ..... .... 9
1.1.7 Green–Lindsay Model of Linear Thermoelasticity
(G-L Model (1972)) ... .... .... .... .... ..... .... 9
1.1.8 Green–Naghdi Model of Thermoelasticity... ..... .... 10
1.1.9 Basic Relations and Equations in Magnetoelasticity .... 11
2 Vector-matrix Differential Equation and Numerical Inversion
of Laplace Transform.... ..... .... .... .... .... .... ..... .... 13
2.1 Vector-matrix Differential Equation .. .... .... .... ..... .... 13
2.2 Solution of Vector-matrix Differential Equation. .... ..... .... 14
2.3 Applications... .... ..... .... .... .... .... .... ..... .... 17
2.4 Numerical Inversion of Laplace Transform. .... .... ..... .... 21
3 Coupled Thermoelasticity. ..... .... .... .... .... .... ..... .... 25
3.1 Problem (i) ... .... ..... .... .... .... .... .... ..... .... 25
3.2 Basic Equations and Formulation of the Problem.... ..... .... 26
3.3 Solution Procedure.. ..... .... .... .... .... .... ..... .... 28
3.4 Boundary Conditions ..... .... .... .... .... .... ..... .... 30
4 Generalized Thermoelasticity... .... .... .... .... .... ..... .... 33
4.1 Problem (i).... .... ..... .... .... .... .... .... ..... .... 33
4.2 Basic Equations and Formulation of the Problem.... ..... .... 34
4.3 Solution Procedure.. ..... .... .... .... .... .... ..... .... 35
xiii