Page 310 - Introduction to Continuum Mechanics

294 The Elastic Solid

The matrix C is known as the stiffness matrix for the elastic solid. In the notation of
Eq, (5.223), the strain energy U is given by

We require that the strain energy U be a positive definite function of the strain components.
That is, it is zero if and only if all strain components are zero and otherwise it is positive. Thus,
the stiffness matrix is said to be a positive definite matrix which has among its properties : (1)
All diagonal elements are positive, i.e., C/, > 0 (no sum on i ) (2) the determinant of C is
positive, i.e. detC > 0, and (3) its inverse S = C~ exists and is also symmetric and positive
definite. (See Example 5.22.1). The matrix S (the inverse of C ) is known as the compliance
matrix.
As already mentioned in the beginning of this chapter, the assumption of the existence of
a strain energy function is motivated by the concept of elasticity which implies that all strain
states of an elastic body requires positive work to be done on it and the work is completely
used to increase the strain energy of the body.

Example 5.22.1

Show that (a) Q,- > 0 (no sum on i) (b) the determinant of C is positive (c) the inverse
of C is symmetric and (d) the inverse is positive definite, (e) the submatrices

etc. are positive definite.
Solution, (a) Consider the case where only E\ is nonzero, all other EI = 0, then the strain
1 2
energy is U = -€"11^11. Since t/>0 , therefore Cu >0. Similarly if we consider the case
£*
1 2 an > etc
s
where £"2 * nonzero, all other £/ = 0, then U = -^22^22 ^ ^22 0 -
^C

f An obvious consequence of these restrictions is that in uniaxial loading, a positive strain gives rise to a positive
stress and vice versa.