Page 237 - Introduction to Continuum Mechanics

222 The Elastic Solid

Furthermore, we shall assume that the concept of "elasticity" is associated with the
existence of a stored energy function t/(E//)» also known as the strain energy function, which
4. '
is a positive definite function of the strain components such that

With such an assumption, (the motivation for Eq. (5.2.7) is given in Example 5.2.1), it can be
shown (see Example 5.2.2 below) that

Equations (5.2.8) reduces the number of elastic coefficients from 36 to 21.

Example 5.2.1
(a)In the infinitesimal theory of elasticity, both the displacement components and the com-
ponents of the displacement gradient are assumed to be very small. Show that under these
assumptions, the rate of deformation tensor D can be approximated by DE/Dt, where E is the
infinitesimal strain tensor.

(b) Show that if Tq is given by Ty = C,yjy EM [ Eq. (5.2.2b)], then the rate of work done P s by
the stress components in a body is given by

where U is the strain energy function defined by Eq. (5.2.7).

Since */ = Xf(Xi> X^, X$, t\ we can obtain

Now, from x m - X m + u m, where u m are the displacement components, we have

t By positive definite is meant that the function is zero if and only if all the strain components are zero, otherwise,
it is always positive.