ZZZ
V
σ ij δε ij dV þ
ZZZ
V
m ij δx ij dV ¼
ZZ
S
σ
n
ð Þ
i δu i dS þ
ZZ
S
q i δθ i dS
ð5:129Þ
The concept of “reduced Cosserat” will be discussed later in the chapter.
5.6.3 Toupin-Mindlin Higher-Order Stress Theory
The Cosserat couple stress theory introduces curvatures as gradients of rotation.
Toupin (1962), Mindlin and Tiersten (1962) and Mindlin (1964, 1965) extended the
original Cosserat formulation by considering also gradients of the normal strain
components, therefore, including all the components of the second gradient of
displacement. The additional gradients of strain as variables give rise to new work
conjugates in the form of force couples per unit area or higher-order stresses. The
physical interpretation of these force couples and higher-order stresses is explained
in Fig. 5.12. In this type of solid, Newtonian mechanics equilibrium (neglecting
body forces) is given by
Fig. 5.10 Isothermal fatigue at strain rate 1.67 Â 10
–4
/s at 22
C with inelastic strain range 0.022
242
5 Unified Mechanics of Thermo-mechanical Analysis
V
σ ij δε ij dV þ
ZZZ
V
m ij δx ij dV ¼
ZZ
S
σ
n
ð Þ
i δu i dS þ
ZZ
S
q i δθ i dS
ð5:129Þ
The concept of “reduced Cosserat” will be discussed later in the chapter.
5.6.3 Toupin-Mindlin Higher-Order Stress Theory
The Cosserat couple stress theory introduces curvatures as gradients of rotation.
Toupin (1962), Mindlin and Tiersten (1962) and Mindlin (1964, 1965) extended the
original Cosserat formulation by considering also gradients of the normal strain
components, therefore, including all the components of the second gradient of
displacement. The additional gradients of strain as variables give rise to new work
conjugates in the form of force couples per unit area or higher-order stresses. The
physical interpretation of these force couples and higher-order stresses is explained
in Fig. 5.12. In this type of solid, Newtonian mechanics equilibrium (neglecting
body forces) is given by
Fig. 5.10 Isothermal fatigue at strain rate 1.67 Â 10
–4
/s at 22
C with inelastic strain range 0.022
242
5 Unified Mechanics of Thermo-mechanical Analysis