Page 238 - Introduction to Continuum Mechanics

Linear Elastic Solid 223

where -r~- are infinitesimal. Thus,
oXi

'That is,

(b) In Section 4.12, we derived the formula for computing the rate of work done by the stress
components (the stress power) as

Using Eq. (v), we have

Now if 7» = ~r[Eq. (5.2.7)], then,
1 d
&ij

That is, with the assumption given by Eq. (5.2.7), the rate at which the strain energy increases
is completely determined by the rate at which the stress components are doing work and if
jPy is zero then the strain energy remains a constant (i.e., stored). This result provides the
motivation for assuming the existence of a positive definite energy function through Eq. (5.2.7)
in association with the concept of "elasticity".

Example 5.2.2

t We are dealing here with a purely mechanical theory where temperature and entropy play no part in the model.
However, within the frame work of thermoelastic model, it can be proved that a stored energy function exists
if the deformation process is either isothermal or isentropic.