Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 121

This result suggests that, if work is required to traverse the loading cycle in one
direction, then work may be extracted by traversing the cycle in the reverse
direction. It is common knowledge however, that it is not possible to extract work
from an inert material by mechanical methods. If it were, the world would be a
different place. To prevent the possibility of extracting work from an inert material,
it is required that C and S are symmetric,

¼ S T ¼ C T

S C

; : (5.24H)

There are further restrictions on the tensors of material coefficients and some of
them will be discussed in the next section.
The definition of a linear elastic material includes not only the stress-strain
relation T ¼ C T ,( 5.7H ) and ¼ C E

C

ð x ; t Þ , but also the symmetry restriction

¼ C T is equivalent to the
requirement that the work done on an elastic material in a closed loading cycle is
zero, ( 5.14H ). The work done is therefore an exact differential (see Sect. A.15).
This restriction on the work done allows for the introduction of a potential, the
strain energy U . Since the work done on an elastic material in a closed loading cycle
is zero, this means that the work done on the elastic material depends only on initial
and final states of stress (strain) and not on the path followed from the initial to the
final state. From an initial state of zero stress or strain, the strain energy U is defined
as the work done ( 5.15H ):

C

( 5.24H ), respectively. The symmetry restriction

ð ð

T

d E

E

d T U ¼ ¼ : (5.25H)

The strain energy U may be considered as a function of either T or E ðT

; U Þ or

ðE

U . From ( 5.25H ) and the fundamental theorem of the integral calculus, namely

that the derivative of an integral with respect to its parameter of integration yields
the integrand,

Þ

¼ @

U

¼ @

U
@T

¼ @

U
@E and E

¼ @

U
@T

T

@E and E

or T : (5.26H)

The following expressions for U are obtained substituting Hooke's law ( 5.7H )
into ( 5.25H ) and then integrating both of the expressions for U in ( 5.25H ), thus

1
2 E

1
2 T

C S E and U T U ¼ ¼ : (5.27H)

It is easy to verify that the linear form of Hooke's law is recovered if the
representations ( 5.27H )for U are differentiated with respect to T and E ,respectively
as indicated by ( 5.27H ). It then follows that ( 5.26H ) or ( 5.27H ) constitutes an
equivalent definition of a linear elastic material . The definition of the most