132
3 Continuum Mechanics and Nonlinear Elasticity
Under isothermal conditions, it is useful to introduce the Helmholtz free-energy
function (per unit mass)
ψ = u − T η,
(3.204)
which represents the part of the internal energy u that is not dissipated and,
therefore, is available to do work. For constant temperature, Eq. (3.203) can be
written as
S : ˙
E = ρ 0 ˙
ψ.
(3.205)
Setting
W (E) = ρ 0 ψ(E)
(3.206)
and noting that the initial density ρ 0 is independent of time, we have
S : ˙
E = ˙
W (E).
(3.207)
This relation states that the rate of work done by the stresses, i.e., the stress power,
is equal to the rate at which energy is stored in the material.
To compute ˙
W , we let φ = W (E) and T = E in Eq. (2.56), which is just the
chain rule. This gives
∂W
∂t
=
∂E
∂t
:
∂W
∂E
=
∂W
∂E
:
∂E
∂t
.
Substituting this expression into (3.207) yields
S −
∂W
∂E
:
∂E
∂t
= 0.
For an arbitrary strain field, this relation implies
S =
∂W
∂E
,
(3.208)
which is the 3D constitutive relation for a general hyperelastic material.
In 1D, Eq. (3.208) gives W =
S dE, where the reference geometry for S and
E is the undeformed configuration. Thus, W is given by the area under the S vs. E
curve (Fig. 3.23d) and represents the strain-energy density per unit undeformed
volume (with units of force/area). This is analogous to the strain energy in a
nonlinear spring (Fig. 3.23b), and Eq. (3.208) is analogous to (3.202).
Constitutive equations for the Cauchy and first Piola-Kirchhoff stress tensors
follow directly from Eqs. (3.119) and (3.120) 1 . Inserting (3.208) into these equations
gives