ABSTRACT
FINITE STRAIN ELASTICITY
5.1. Green-Elasticity
Elastic deformation does not cause irreversible rearrangement of internal
structure, and the corresponding Helmholtz free energy is a function of stress
and temperature only. Restricting consideration to isothermal elastic defor-
mation (
_
= 0), Eqs. (4.3.7) and (4.3.9) give
_
=
@
@E
(n)
:
_
E
(n)
=
T
(n)
:
_
E
(n)
; (5.1.1)
i.e.,
T
(n)
=
@
@E
(n)
; =
E
(n)
: (5.1.2)
Alternatively, Eq. (5.1.2) can be deduced by adopting an experimentally
observed property that there is no net work left in a body upon any closed
cycle of elastic strain, i.e.,
I
T
(n)
: dE
(n)
= 0: (5.1.3)
This implies that
T
(n)
: dE
(n)
= d (5.1.4)
is a total dierential, which leads to Eq. (5.1.2). The function =
E
(n)
is the strain energy function per unit initial volume. It represents the work
done to isothermally deform a unit of initial volume to the state of strain
E
(n)
. The explicit representation of the function
E
(n)
depends on the
selected strain measure E
(n)
and the material properties.
