the kinematic constraint in Eq. (5.119) and with the definitions of small strain and
curvatures expressed in Eqs. (5.120) and (5.121):
θ i ¼
1
2
e ijk u k,j
ð5:119Þ
ε ij ¼
1
2
u i,j þ u j,i
À
Á
ð5:120Þ
x ij ¼ θ i,j
ð5:121Þ
Second, there is a general couple stress theory in terms of a micro rotation ω i ,
which is regarded as an independent kinematic degree of freedom. The rotation and
micro rotation are related by a relative rotation tensor α ij = e ijk ω k 2 e ijk θ k . For the
particular choice of ω k = θ k , the general couple stress theory reduces to the reduced
couple stress theory. Equilibrium of the differential element shown in Fig. 5.12, after
neglecting body forces and body couples, yields, [in Newtonian Mechanics]
Fig. 5.7 Cyclic shear simulation vs. test data at 22
C, strain rate 1.67 Â 10
–3
/s, and ISR ¼ 0.012
5.6 Thermo-mechanical Analysis of Cosserat Continuum: Length-Scale Effects
239