σ ji,j þ τ ji,j ¼ 0
ð5:136Þ
τ jk þ
1
2
e ijk m pi,p ¼ 0
ð5:137Þ
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
ð5:138Þ
q i ¼ m ij n j
ð5:139Þ
where σ
(n)
i is the surface traction vector, q i is the surface couple tractions vector, and
n i is a surface normal vector. In order to establish a general framework for elastic
material behavior, Cosserat couple stress continuum can be obtained after postulating a strain energy density function dependent on strains and curvatures (rotation
gradients). Toupin (1962) and Mindlin (1965) extended this theory to include also
stretch gradients. In particular, they considered invariants of the strain and strain
gradients into the strain energy density function in terms of the following generalized
von Mises strain invariant which is given by (Fleck and Hutchinson 1997)
1
x
2
x
'
1
x
'
2
x
1,2
u
1,2
u
21
ˆ
γ
21
ψ
21
1,2
21
ˆ
u
γ
ψ
=
−
ˆ ij
γ
Relative
ij
ψ Micro deformation
Fig. 5.14 Relative deformation in the couple stress theory by Mindlin (1964)
246
5 Unified Mechanics of Thermo-mechanical Analysis