3.7 Boundary Value Problems
145
3.7.2 Governing Equations in Cartesian Coordinates
Kinematic Relations
F ij =
∂x i
∂X j
= δ ij +
∂u i
∂X j
E ij =
1
2 (F ki F kj − δ ij )
=
1
2
∂u i
∂X j
+
∂u j
∂X i
+
∂u k
∂X i
∂u k
∂X j
(3.240)
Incompressibility
J = det[F ij ] = 1
(3.241)
Stresses
σ ij = J
−1 F ik P kj = J
−1 F ik F jm S km
(3.242)
Equations of Motion
∂σ ji
∂x j
+ b i = ρa i
∂P ji
∂X j
+ b 0i = ρ 0 ¨
u i
∂
∂X j
(F ik S jk ) + b 0i = ρ 0 ¨
u i
(3.243)
Constitutive Relations
σ ij = J
−1 F ik F jl
∂W
∂E kl
− p δ ij
P ij = F jk
∂W
∂E ik
− Jp F
−1
ij =
∂W
∂F ji
− Jp F
−1
ij
S ij =
∂W
∂E ij
− Jp (F ki F kj )
−1
(3.244)
3.7.3 Governing Equations in Principal Coordinates
Some problems do not involve shear stress or shear strain in the chosen coordinate
system. In this case, the global coordinates also are principal coordinates at each