Acceleration Waves in Elastic Non-Conductors of Heat
Now, I shall examine the behaviour of acceleration waves in elastic bodies which do not conduct heat. For such a material body the values internal energy e, the stress T and the absolute temperature θ (strictly positive) are determined by the present valu
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No. 72
PETER CHEN SANDIA LA BORA TORIES ALBUQUERQUE
THERMODYNAMIC EFFECTS IN WAVE PROPAGATION
COURSE HELD AT THE DEPARTMENT FOR MECHANICS OF RIGID BODIES JULY 1971
UDINE 1971
SPRINGER-VERLAG WIEN GMBH
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©
1972 by Springer-Verlag Wien
Originally published by Springer- Verlag Wien- New York in 1972
ISBN 978-3-211-81176-4 DOI 10.1007/978-3-7091-4311-7
ISBN 978-3-7091-4311-7 (eBook)
PREFACE
In these leatures~ I have attempted to introduae the theory of singular surfaaes and illustrated the appliaation of this theory to the examination of the behaviour of shoak ~aves and aaaeleration ~aves propagating in nonlinear elastia bodies. The effeats of heat aonduation are ignored; and~ for aonvenienae~ the entire disaussion is restriated to the one dimensional aontext. I sho~ed that definite and aonarete results aan be obtained ~ithout having to speaify expliait aonstitutive relations. I am deeply indebted to Professor Luigi Sobrero for making it possible for me to speak at the Centre and for his kind hospitality.
Udine~
July 19?1
1. Preliminaries
In these lectures, I shall restrict my discussions to the one dimensional motions of material bodies. This restriction is adopted mainly for the sake of convenience, in that I need not be bogged down with the vast expanse of notation so as to cloud the interesting features of the behaviour of waves in material bodies. The one dimensional motion of a body is described by a function X(',') giving the position
xCX,-c)
X
( 1.1)
at time T of each material point of the body whose position in the reference configuration is
X. I shall, as is customary, iden
tify each material point with its position in the reference configuration. A property of the mot ion is that the function X( ·, "t) is invertible; the inverse function is denoted by X- 1( ' ''r)' and ( 1. 2)
which gives the material point X whose position at time T is x . The derivatives of the motion ()
F(X ;r) = DX X(X, 't'), i(X ;r)
=
fr: X (X;t),
(1.3a)
Preliminaries
6
(1. 3b)
::'J,x.(X,T:)
i(X,'l') =
are, respectively, the deformation gradient, the velocity and the acceleration of The function
(1.4)
X at time
'l' • Let t denote the present time.
FtX, '), with values F t(X , s)
= F(X, t - s) , s
t. [
0 , oo) ,
is called the history of the deformation gradient of
X up
to time
t . The restriction Frt(X, ') of the history to the open interval (0 , oo) is called the past history of the deformation gradient. Clearly, the present value Ft(X,O) of the history is given by
(1.5)
=
F t( X , o)
F (X , t )
.
The function Et(X, · ) , defined by
( 1.6)
&t(X, s) =
sex, t -
= F(X,
s)
t - s)- 1,
is called the strain history of
X up
S €
[O,oo)
,
to time t . Its present val
ue et(X,O) is giv
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