σ ji,j þ τ ji,j ¼ 0
ð5:122Þ
τ jk þ
1
2
e jik m pi,p ¼ 0
ð5:123Þ
where σ ij and τ ij are the symmetric and antisymmetric components of the Cauchy
stress tensor, respectively, and m ij is the couple stress tensor. Surface stress tractions
and surface couple stress tractions are given by
σ
n
ð Þ
i ¼ σ ij þ τ ij
À
Á
n j
ð5:124Þ
q i ¼ m ij n j
ð5:125Þ
where σ
n
ð Þ
i is the traction vector, q i is the couple tractions vector, and n i is a surface
outward normal vector. For a linear elastic solid, the strain energy density is given by
Eq. (5.126) where l is a length scale associated to elastic processes and may be
regarded to be of the order of interatomic distances (Koiter 1964):
Fig. 5.8 Cyclic shear simulation vs. test data at 22
C, strain rate 1.67 Â 10
–3
/s, and ISR ¼ 0.02
240
5 Unified Mechanics of Thermo-mechanical Analysis
ð5:122Þ
τ jk þ
1
2
e jik m pi,p ¼ 0
ð5:123Þ
where σ ij and τ ij are the symmetric and antisymmetric components of the Cauchy
stress tensor, respectively, and m ij is the couple stress tensor. Surface stress tractions
and surface couple stress tractions are given by
σ
n
ð Þ
i ¼ σ ij þ τ ij
À
Á
n j
ð5:124Þ
q i ¼ m ij n j
ð5:125Þ
where σ
n
ð Þ
i is the traction vector, q i is the couple tractions vector, and n i is a surface
outward normal vector. For a linear elastic solid, the strain energy density is given by
Eq. (5.126) where l is a length scale associated to elastic processes and may be
regarded to be of the order of interatomic distances (Koiter 1964):
Fig. 5.8 Cyclic shear simulation vs. test data at 22
C, strain rate 1.67 Â 10
–3
/s, and ISR ¼ 0.02
240
5 Unified Mechanics of Thermo-mechanical Analysis