Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
CO U R S E SAN D L E C T U RES No. 119
IAN N. SNEDDON
UNIVERSITY OF GLASGOW
THE LINEAR THEORY OF
THERMOELASTICITY
COURSE HELD AT THE DEPARTMENT
OF MECHANICS OF SOLIDS
JULY 1972
UDINE 1974
SPRINGER-VERLAG WIEN GMBH
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© 1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81257-0 ISBN 978-3-7091-2648-6 (eBook)
DOI 10.1007/978-3-7091-2648-6
PRE PAC E
This monogpaph is made up of the notes fop
the lectupes I gave at the Intepnational Centpe fop
Mechanical Sciences in Udine in June 1972.
Chaptep I contains a bpief account of the
depivation of the basic equations of the lineap theopy
of thepmoelasticity. The ppopagation of thepmoelastic
waves is consideped in some detail in Chaptep 2, while
boundapy-value and initial value ppoblems ape considep
ed in the pemaining two chapteps. In Chaptep 3 an ac
count is given of coupled ppoblems of thepmoelasticity,
and in the final chaptep the methods tpaditionally used
in the solution of static ppoblems in thepmoelasticity
ape outlined. Since integpal tpansfopm methods ape used
whepe apppoppiate, the main ppopepties of the most fpe
quently used integpal tpansfopms ape collected togethep
in an Appendix.
It is my pleasant duty to pecopd hepe my
thanks to the officeps of C.I.S.M. and in papticulap to
ppofessop Luigi Sobpepo and ppofessop Woclaw OZszak fop
inviting me to give the lectupes and fop making my stay
in Udine so vepy enjoyable and stimuZating.
Udine, June 1972
Ian N. Sneddon
CHAPTER I
THE BASIC EQUATIONS OF THE LINEAR
THEORY OF THERMOELASTICITY
1.1 Introduction
The theory of thermo elasticity is concerned with
the effect which the temperature field in an elastic solid has
upon the stress field in that solid and with the associated ef
fect which a stress field has upon thermal conditions in the sol
id. In the "classical" theory attention is restricted to perfect
ly elastic solids undergoing infinitesimal strains and infinites
imal fluctuations in temperature. It turns out that these assump
tions are sufficiently strong to justify a purely thermodynamic
derivation of the form of the equation of state of an elastic sol
id. The resulting equations are linear only if the range of fluc
tuation of temperature is such that the variation with tempera
ture of the mechanical and thermal constants of the solid may be
neglected.
In these notes we shall also make the basic assump
tion that there exists a state in which all of the components of
strain, stress and temperature gradient vanish identically.
We shall assume, in addition, that the solid is
isotropic. This is not an essential assumption. It is simply one
which makes the equations much easier to handle. The correspond-
6 1.1 Introduction
ing equations for an isotropic body are still linear - but a
good deal more complicated than those derived here.
The study of thermal stresses was begun by
Duhamel (1838) who derived the additional terms which must be
added to the components of the strain tensor when a temperature
gradient has been set up in the body. Duhamel's results were re
discovered by Neumann (1885) who applied them in a study of the
double refracting property of unequally heated glass plates.
Neumann also showed that these additional terms in the strain
components can be interpreted as meaning that the effect of a
non-uniform distribution throughout an elastic solid is equiva
lent to that of a body force which is proportional to the tem
perature gradient.
The defect of the Duhamel-Neumann theory is that
although it predicts the effect of a non-uniform temperature
field on the strain field, it leads to the conclusion that the
process of the conduction of heat through an elastic solid is
not affected by the deformation of the solid. Both Duhamel and
Neumann were conscious of this defect in their theory and (again,
independently) suggested on purely empirical grounds that a term
proportional to the time rate of change of the dilatation should
be added to the heat-conduction equation. A number of authors no
tably Voigt (1910), Jeffreys (1930), Lessen and Duke (1953) and
Lessen (1956) adduced thermodynamic arguments to justify the
coupled equations postulated by Duhamel and Neumann. The failure
1.1 Introduction 7
of these authors to establish a satisfactory basis for thermoe
lasticity stems from their use of classical (i.e. reversible)
thermodynamics. For, although the deformation of a perfectly e
lastic solid is a reversible process, the diffusion of heat oc
curs irreversible so that the derivation of the field equations
has to be based on the theory of irreversible processes. The de
velopment of a satisfactory theory of irreversible thermodynamics
- an excellent account of which is given in de Groot (1952) -
provided the tools necessary to establish a proper theory of thermo
elasticity. This theory was established by Biot (1956) •.
In these lectures we shall be concerned more with
the solution of the equations of thermoelasticity than with a full
discussion of their derivation. In this first chapter, however,
a brief sketch is given of the derivation of the basic field
equations and of the variational principles which are their e
quivalent. Full accounts are given by Nowacki (1962) and Kovalenko
(1969) and a more brief outline by Chadwick (1960). The present
discussion leans heavily on the last two works quoted.
To facilitate the solution of problems in cylin
drical and spherical polar coordinates the equations are derived
in terms of these coordinates in § 8.
Throughout we have used the notation of Green and
Zerna (1954) for the stress tensor, denoting stress components
by dij (i,j = 1,2,3) or in particular problems by dxx ,dyy, dzz ,
0yz,dxz ,dxy • For the components of the strain tensor we use
8 1.2 The Basic Equations of Thermoelasticity
so that for an infinitesimal strain,
~;j
E. . = !(DUj + i)Ui)
IJ 2 Dx. ax.
I J
=
(i,j 1,2,3).
1.2 The Basic Equations of Thermoelasticity
We consider a perfectly elastic solid, initially
unstrained, unstressed and at a uniform temperature J~ • When
the solid is deformed by either mechanical or thermal means a
displacement field U and a non-uniform temperature fieldT are
v
set up. These in turn lead to a velocity field and a distribu
tion of strain and stress described respectively by the strain
tensor £ij and the stress tensor dij • Following such a disturb
ance of the equilibrium state, energy is transferred from one
part of the solid to another by the elastic deformation and by
the conduction of heat. The deformation of a perfectly elastic
solid is a reversible process but heat conduction takes place
irreversibly so the derivation of the partial differential equa
tions satisfied by the field quantities has to be based on the
thermodynamics of irreversible processes.
If d denotes the density of the solid in the ini
tial state and ("1 ,"2, "3) the components of the velocity vector
v
the conservation of mass is expressed by the equation
1.2 The Basic Equations of Thermoelasticity 9
(1.2.1)
where (x ,Xa,Xg) are the coordinates of a typical field point
1
and
is the operator of convective time differentiation.
Similarly, the conservation of "linear momen
tum is expressed by the three equations (1~)
=
(i 1,2,3) (1.2.2)
where (X1,X2,X3) are the components of the body force per unit
mass.
If we denote by Q the rate at which heat is gener
ated (perunitvolume) by internal sources and introduce the heat
flux vector "if = (q1 ,Q2,Q3) the equation expressing the conser
vation of energy is
i) ()q.
= oX· ". + -ax. (V. d.·) - _axI. + Q • (1.2.3)
1 1 1 IJ
J I
(1~) The summation convention is used consistently throughout
these notes
1.2 The Basic Equations of Thermoelasticity
10
The final equation of the set expresses the sec
ond law of thermodynamics. If U denotes the specific (~~) in
ternal energy and S the specific entropy of the system we have
the equation
(1.2.4)
If we form the scalar produce of both sides of
the vector equation (1.2.2) with the velocity vector vwe find
that
and substracting this equation from (1.2.3) that
DU = 6 .. _Ov_· aq· + Q •
I ___I
Dt IJ ax. Ox.
J 1
From the symmetry of the stress tensor we deduce that
(a".
()Vj)
a,,·
= 1
ff .. _ax_.I -6. . ---.!. + -ax.
2 OX·
IJ I)
J J 1
so that this last equation may be written in the form
( ~<) It should be observed that"specific" means "per unit
vo lume" throughout.
