(9 .1 .2 ) V(t)
and rrotn ( 9 . 1 . 8 ) , It follows that
f i f = S : A ‘ » (9 .1 .3 )
since the volume '/ can be selected arbitrarily at any Initial time. Using the fact that the energy E should remain a scalar Invariant of the tensors sj and d ; for an isotropic medium, Thomas / ^ ) _ / has obtained the equations which determine the function d in the case of linear elastic theory. In case of hypo
elasticity of grade zero, the basic equations of which are given in chapter 711 of this thesis, these equations reduce to
r + 4 — =
0,
(8.1.4)r f
f M + ( * & ) » £ + ! . » . ! £ . ( 9 1 s )
i f • * / * • » « r- ’ i » .i .s )
128
, we get
(9 .1 .6 )
where it is understood that we take double the shear modulus i . e . , t m as the unit of stress.
(
The expression for E in (9 .1 .6 ) is obviously the same as the one in the ordinary theory of elasticity- proving that hypo-elasticity reduces to elasticity under certain conditions. The expression (9 .1 .6 ) was obtained by Veraa / ^7 _ / for the first time introducing the idea of deformation energy in hypo-elasticity. Later on, Sinha £ S'2- _ / derived the corresponding expressions for grade one and two. Sricksen £ t>l _ / deduced the condi
tions for the existence of hypo-elastic potential.
9 . 2 . Existence of hypo-elastic potential.
Ericksen £ 5 1 _ / introduced the scalar potential analogous to the elastic potential by talcing
Sj
^C j ~ f t >
(9 .2 .1 )where
I f we assume that £ does not depend on |=
from (9 .1 .4 ) and ( 9 . 1 . K):
E = — ^ X 61 1 i (a m p )
where is a solution of the hypo-elastic equations
^<j ~ ^<jkl 3 (9*2*2)
j> and 'j' are scalar Invariants of stress and It being
assumed that ^BS an inver*® such that
1 C4 U s = s-CCjMb ^ k L (9 .2 .3 ) - ^ t A ^ S + ^ 1 5 '
The equation (9 .2 .1 ) can be put in its equivalent form
V i ‘4 =
f * £ *li ‘ t
E%
' Viu ’
{’ -M >o V
1
which, i f satisfied for arbitrary dcj * gives
( 9 ,2 -s) Hence
o
where L . . is a tensor invariant of the stress and as
• 130 -
such Is expressible in the form
(9 .2 .7 )
where A‘ ~ are scalar invariants of £{,• . The equations ( 9 .2 .6 ) have a solution i f , and only I f ,
r c*ki.
Vi)
ki~ 1V
Sj i -Vi VJ)C;
■is.- -bS j \ r °- (9 .2 .8 ) .VS
The hypo-elastic materials are characterized by the existence of hypo-elastic potential i f equation (9 .2 .3 ) is identically satisfied. To show that this condition is also sufficient, we must formally show that i f (9 .2 .8 ) is satisfied, the solutions <j> and ^ of ( 9 .2 .1 ) are
scalar invariants of S£; . That is to show that, for arbitrary stress
(9*2*9)
where > « • is related to by an orthogonal trana formation
^i(‘ ~ f 7 K ' h i ~ S'-- (9 .2 .1 0 )
(9.2*11)
and linearize \ $ f S* ) — ^ f ) \ r 1 '
to ,31
1
', obtainingC
with respect
* /
^ I S r s . i » *£• _ c . - \ „ c lj j “ f , jl£,} n. j , Q _ (9 .2.12).
"D- where
E . = ! X $, v • ’ _ 'b f ... O .• ;, . 'L fi:Ju *4 V , *
(9.2.13)
Using ( 9 .2 .7 ) , we see that E q = o when 'f is a solution of equations ( 9 .2 .6 ) . The equations =■ o constitute a system of three independent linear first
order partial differential equations for vp . A general solution is an arbitrary function of the three stress invariants and hence vp is a scalar invariant function of S, ■, . Therefore from equations ( 9 .2 .6 ) ,
V t
0
- 132
which shows that <f> is also a scalar invariant func
tion of . Hence a hypo-elastic potential vj/
0
exists i f equation (9 .2 .3 ) is satisfied.
9 . 3 . Thermal stresses in hypo-elasticity.
In this section, we propose to derive the hypo
elastic constitutive equations which also take tempera
ture into account. We give a general method to introduce temperature in hypo-elasticity.
We briefly recapitulate conservation laws in
classical mechanics. Conservation of mass, conservation of momentum, conservation of moment of a omentum and con
servation of energy give respectively
( 9 .3 .1 )
(9 .3 .2 )
* «
( 9 .3 .3 )
= W u _ i r u + f A ' • ( 9 -3-4 )
where ft Is the supply of enery, £ is the total less kinetic energy, p is the volume density, A is the rate of deformation tensor, S is the stress tensor,
vector, JLu is the unit of stress, comma denotes the covariant differentiation and dot means the material tine derivative.
A hypo-elastic material in a thermal state is characterized by two sets of constitutive equations, one for stress rate and ths other for heat . The stress rate tensor S la a polynomial function of S., <A and
f* t {.
f is the body force vector, V is the velocity
k , where Is the thermal gradient bivector. The heat flux bivector •&. is also taken as polynomial func
tion of d and |j . This assumption is vslid since b is antisymmetric and d and S are symmetric.
Now we write
( 9 .3 .6 )
( 9 .3 .7 )
(9 .3 .8 )
134
where 6 is the temperature, 6 ^ the usual -&
tensor and X and -f are hemitropic functions of A
their arguments. Since ^ and f are hemitropic func
tions of matrices mentioned above, we can obtain
S = I + ^ S + ^ ci f ^ s % / r ^-t- ^(^+<^5)+
+ ^g (ib-bol)+ ^ (s b - b + ^i.(.Sc^ +<^ -S) + *^n
+ W ) -f ^-Sbfc+ W4" + ^|i, (^s1'-S1'b_)4 *<15 i>)+
+ (li b J M ^ b <9 f ( b 5 ^ b ^ - W vt( U k - W * ) *
+
^ (b n ^ m )+ ^ C ^ b - b is )f^ (d 4 - b s c L )W 27(c i^ A >
+ (sud j> - b a sL) -f ^ (?t<L-ikS) +* U a^ - ^ ) +^ ^ < 9 *
f ^ (ft d + i 5 ^ * x31( W s t sd s V b 5 t y
f ^ (s v * - fc ^ ^ v b - b ^ ^ w > i f
+ ^ ( m - + A b V >
f ^ ( 2 j)d s~ sLd bs)+ ^ ( U sdl- d*'sUj+^/Asw" a
+ ^ ( b sdb*-- t M ^ b ) + (^b Asb1*-b^s JL
(9*3*9}
i ( s ) , 1 ( d ) , t L ( s 4 ) ,
% C s L) )
t i ( s b % k ( d s y , k ( d P ) / 1 * ( M ) Jk ( ‘l %
(^V); fct £ is i ol1) } t»». £ s Jlkf) , tx. (d lo S^) } tv (s bd.jv'k ^ ;
fc. (^ Y ), fc. (d ip'i1') ,tx. (ds b^/fc^dtbci*) ;k(cUsi^,
•bi (sl^ T ),k (W N") * t. (M 4-Ml) ? ti (s b d?-^,
fc O c i y ) ; * k ( A t
(9 .3.10) We also have
i = ,\ u + + ^ c ^ i+ a i y
lf P ' ' (9 .3 .1 1 )
.J
.
where f ■* are polynomials in the invariants (
k(lj, k<Xl>),&(£-),-k (it), k(^J,fe(d^k(<iVj.
(9 .3 .1 2 ) Now we have to see that no constitutive coefficient of
thermohypoelastic material shall carry a dimension
136
independent of the dimension of stress. Since 'S and d are of dimension f " 1 In ( 9 .3 .9 ) , we have to drop *11 tha second and higher powers of c[ to balance the dimen
sions of both sides. We also decide to neglect terms containing second and higher powers of \> snd third and higher powers of S in (9 .3 .9 ).
Then, we shall get
? = ^ 0 l + 3 A + 3 i t < U - U ) + s 3/ + s i H (si>-kii)+>'afI
^ t ^ to - bAsj (4sb~ bsci
+ iV " + f V t + S 9o( K L-4l V + f l'( J i s % f J . } f-6 ^ ^ - i> J- 4 ^ ^ d. - ^ ^ S 3 ^ ? i 7 (^ b - b o l ) i' m 3 « s V V +
whare M = j ! N = S1.- f j d ^ . . (8.3.13) k L <1 * t
Dropping d 1 terms in ( 9 .3 .1 1 ) , we get
^ B ft ^ (tdl 4- (9.3.14)
* Thus (9 .3 .1 3 ) and (9 .3 .1 4 ) constitute, the basic constitu
tive equations of thermo-hypoelasticity. The other equa
tions used are grouped in ( 9 .3 .1 ) to ( 9 .3 .4 ) .