CONTINUUM MECHANICS for ENGINEERS

for an isotropic, incompressible material, show that
8.7 For biaxial loading of a thin vulcanized rubber sheet the strain energy
may be written as
(Rivlin and Saunders, 1951).
a. Use the definitions of invariants I1, I2, and I3 in terms of stretches
λ1, λ2, and λ3 to show
b. Substitute the results from (a) into the strain energy above to
obtain
where
3
3
∂ W ∂ W
λ1 = λ
2
2 2
∂λ ∂λ
∂λ ∂λ
8.8 Consider a material having reference configuration coordinates (R, Θ,
Z) and current configuration coordinates (r, θ, z). Assume a motion
defined by
Determine F iA and B ij in terms of g, g′, and f′.
1
2
W = C ( I −3)+
C ( I −3)+
C I −3
1 1 2 2 3 2
2
1
( )
2 1 1 1
I2−32I I
4 4 4 1 6 2 9
λ λ λ
( ) = + + + − +
1
( ) + −
2
2
−2− w( λ )= C + 2C λ C 6C
λ C λ
1 1 3 1
( 1 2 3)
− C + C −3C
3
( 2 3) 1 +
W = w( )+ w( )+ w
⎛ 1 ⎞
λ1 λ2
⎜ ⎟
⎝ λλ ⎠
r
1 2
R
, θ
= f( Θ)
, z= Z
g Θ
= ( )
2
4
3 1