748
10 Variational Problems of Functionals with Vector, Tensor …
J [ A] =
V
1
2
1
μ
(∇ × A)
2
+
1
μ
(∇ · A)
2
+ (σ + jεω)jω A
2
− 2 A · J f
dV
−
S 2
(H × n) S 2 · AdS −
S 2
1
μ
∇ · A
S 2
A · n dS
where, μH = ∇ × A.
10.46 The functional of the nonlinear generalized variational principle (potential
energy form) is
Π J 1
G [U, S, u] =
V
[Σ(U) + S : (R T · F − U) − u · F]dV −
S σ
u · PdS
−
S u
(u − u) · R · S · n dS
Write the Euler equations and boundary conditions of the functional.
10.47 The functional of the nonlinear generalized variational principle
(complementary energy form) is
Π
J 2
G [S, u] =
V
{Σ
c
(S) − R
T
: S + u · [(R · S) · ∇ + F]}dV
−
S σ
u · (R · S · N − P)dS −
S u
u · R · S · N dS
Given that the right polar decomposition of deformation gradient F is F =
R · U, write the Euler equations and boundary conditions of the functional.
10.48 The functional of the generalized variational principles (potential energy
form) in finite elasticity using the Atluri stresses T
and left stretch tensor V
as conjugate variables is
Π
p
g J =
V
[W (V ) + T
: (F · R
T − V ) − u · F]dV −
Sσ
u · PdS −
Su
(u − u) · T
· R · N dS
where, u is the displacement vector; F is the deformation gradient, F =
I + u∇ = V · R, here R is the orthogonal rotation tensor; F and P are
external forces per unit volume and area respectively; N is the unit vector
in the outward normal direction on surface boundary. Write out the Euler
equations and boundary conditions of the functional.
10.49 The functional of the generalized variational principles (complementary
energy form) in finite elasticity using the Atluri stresses T
and left stretch
tensor V as conjugate variables is
Π
c
g J [T
, V ] =
V
{W c (T
) − T
: R
T
+ u · [(T
· R) · ∇ + F]}dV