CONTINUUM MECHANICS for ENGINEERS

By inserting Eq 6.6-2 into Eq 6.6-1 and using Hooke’s law for the mechanical
strains in that equation, we arrive at the thermoelastic constitutive equation
1+v
v
εij = σij − δijσkk + α( θ −θ0)
δij
E E
(6.6-3)
This equation may be easily inverted to express the stresses in terms of the
strains as
E
σij = vδijεkk + ( 1−2v ) εij − ( 1+v) α( θ −θ0)
δij
( 1+v) 1−2v ( )
[ ]
(6.6-4)
Also, in terms of the deviatoric and spherical components of stress and strain,
the thermoelastic constitutive relations appear as the pair of equations
S
E
η
1+v
= ( )
ij ij
E
σii = εii + α θ −θ0
1−2v ( )
[ ( ) ]
(6.6-5a)
(6.6-5b)
If the heat conduction in an elastic solid is governed by the Fourier law,
Eq 5.7-10, which we write here as
q i = –κθ, i
(6.6-6)
where κ is the thermal conductivity of the body (a positive constant), and if
we introduce the specific heat constant c through the equation
-qi,i = ρcθ (6.6-7)
˙
the heat conduction equation for the uncoupled theory becomes
κθ, ii = (6.6-8)
ρcθ˙ This equation, along with the thermoelastic stress-strain equations Eq 6.6-3
or Eq 6.6-4, the equilibrium equations Eq 6.4-1, and the strain-displacement
relations Eq 6.4-2, constitute the basic set of field equations for uncoupled,
quasi-static, thermoelastic problems. Of course, boundary conditions and
the strain compatibility equations must also be satisfied.