26 Eigen Stresses and Dislocations
287
In order to achieve a better understanding in the mathematical structure of these
equations, we consider an isotropic continuum. Hooke’s tensor C is then already
defined by the two Lamé parameters μ and λ according to (344), and for the relationship between stress and strain (345) is valid,
σ ik = 2 μ ε ik + λ δ ik ε .
(318)
We intend on completely ignore external forces,
f i = 0 .
For (384a), we thus receive
ρ
∂
∂t
v i − μ ε ri , r −λ ε rr , i = 0 .
(385)
By differentiation, we receive (386) instead of equations (343a)
∂
∂t
1
2
(v i , k + v k , i ) − μ (ε ir , kr + ε kr , ir ) − λ ε rr , ik = 0 .
(386)
Equation (384b) we transform to
ε ik , rr + ε rr , ik − (ε ir , rk + ε kr , ri ) =
1
2
( mni α nk , m + mnk α ni , m )
+
1
2
( mni α mn , k + mnk α mn , i ) .
(387)
The actual equivalence of (387) to (384b) can be verified by inserting all values for
i, k = 1, 2, 3, whereby one should still take α ri , r = 0 into consideration. Equation
(384c) can be differentiated with respect to time, so that
∂
2
∂t 2 ε ik −
∂
∂t
1
2
(v i , k + v k , i ) =
∂
∂t
1
2
(J ik + J ki ) .
(388)
We insert Eqs. (387) and (388) into (386) and find after some simple calculations,
ε ik +
μ + λ
μ
ε rr , ik −
ρ
μ
∂
2
∂t 2 ε ik = −
1
2
( mni α nk , m + mnk α ni , m )−
−
1
2
( mni α mn , k + mnk α mn , i )−
−
ρ
μ
∂
∂t
1
2
(J ik + J ki ) .
(389)
We also find by differentiating (386) with respect to time and including (384c) the
field equation for material velocities