10.6 Variational Problems of Functionals with Parallel-Type Inner …
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τ · N = X (on S σ )
(6)
It should be pointed out that in this example the deformation gradient F does not
involve in variation.
Example 10.6.2 Let V be the region of the super-elastic body, A is the boundary
surface that surrounds V, and A = A P , N is the exterior normal unit vector on the
boundary surface A, the surface force per unit area is given on the stress boundary A P :
P N | A P = P · N| A P = P 1N , u is the displacement vector, ¯
ρ is the mass density, b
is the volumetric force per unit mass, F is the deformation gradient, F = I + u∇,
here I is the unit tensor of second order. The systematic total potential energy
functional of the function class in possible displacement field can be expressed as
Π [u] =
V
[Σ(F) − u · ρb]dV −
A P
u · P 1N d A
(1)
where, Σ(F) is the strain energy density of the body, with conjugate constitutive
relation P =
dΣ(F)
d F
, F =
dΣ
c2 ( P)
d P
, Σ
c2
( P) is the complementary strain energy
density. Find the Euler equation and the boundary condition of the functional.
Solution Let the integrand F = Σ(F) − u · ρb, where the variation of the function
Σ(F) is
δΣ(F) =
dΣ(F)
d F
: δ F =
dΣ(F)
du∇
: δ F = P : δu∇
(2)
so that
∂ F
∂ u
= − ¯
ρb,
∂ F
∂ u∇
= P
(3)
The Euler equation of the functional is
P · ∇ + ¯
ρb = 0 (in V )
(4)
On the stress boundary A P of Eq. (1) there is
P · N − P 1N = 0 (on A P )
(5)
Owing to P · N = P N , the stress boundary condition can also be written as
P N = P 1N (on A P )
(6)
Example 10.6.3 The functional of the function class in the possible displacement
field is