CONTINUUM MECHANICS for ENGINEERS

The deformation is volume preserving, having V = A oL o = AL where the
subscript o denotes the initial area and length. Since the stretch ratio in this
case is λ = L/L o, it is clear that dλ/dL = 1/L o. Using this result and differentiating
Eq 8.1-16 results in
which may be written as
Materials satisfying this equation are called neo-Hookean material.
8.2 A Strain Energy Theory for Nonlinear Elasticity
(8.1-17)
The theory developed in the previous section does not well represent experimental
data at large strains. A better approach to modeling the response of
rubbers comes from assuming the existence of a strain energy which is a
function of the deformation gradient in the form of the left deformation tensor
B ij = F i,AF j,A. This approach, first published by Mooney (1940) and furthered
by Rivlin (1948), actually predates the molecular approach discussed in
Section 8.1. The basis of Mooney’s and subsequent theories is the initially
isotropic material has to obey certain symmetries with regards to the functional
form of the strain energy function.
Assume the strain energy per unit volume to be an isotropic function of
the strain in the form of the left deformation tensor invariants I 1, I 2, and I 3
where
dW dW d
F =
L dL d dL
∂ψ
λ
= =
∂ λ
F VG ⎡ 1 ⎤ ⎡ 1 ⎤
= λ − = AG o λ −
2 2
L ⎣⎢ λ ⎦⎥ ⎣⎢ λ ⎦⎥
o
F
f
A G
⎡ ⎤
= = λ −
⎣⎢ λ ⎦⎥
W = W( I1, I2, I3)
I = B
1
ii
1
I2= B B −B
B
2
o
[ ii jj ij ij]
1 2
I ε B B B det
B
= = { }
3 ijk 1i 2j 3k
ij
(8.2-1)
(8.2-2)