144
3 Continuum Mechanics and Nonlinear Elasticity
3.7.1 Governing Equations in Tensor Form
Kinematic Relations
F = (∇r)
T
= I + (∇u)
T
E =
1
2
F
T
· F − I
=
1
2
∇u + (∇u)
T
+ (∇u) · (∇u)
T
(3.235)
Incompressibility
J = det F = 1
(3.236)
Stresses
σ = J
−1 F · P = J
−1 F · S · F
T
(3.237)
Equations of Motion
∇ · σ + f = ρa
∇ · P + f 0 = ρ 0 a
∇ · (S · F
T )+f 0 = ρ 0 a
(3.238)
Constitutive Relations
σ = J
−1 F ·
∂W
∂E
· F
T
− p I = J
−1 F ·
∂W
∂F T − p I
P =
∂W
∂E
· F
T
− Jp F
−1
=
∂W
∂F T − Jp F
−1
S =
∂W
∂E
− Jp F
−1
· F
−T
(3.239)