Page 137 - Introduction to Continuum Mechanics

122 Finite Deformation

Since U is real and symmetric, there exists three mutually perpendicular directions, with
respect to which, the matrix of U is diagonal. Thus, if e 1 ,e 2 ,e 3 are these principal directions,
(1) (1)
with eigenvalues A lf A 2, A 3, then, forrfX = dX&i, Eq. (3.20.1) gives dx = A 1dar le 1, i.e.,

We see that along each of these three directions, the deformed element is in the same direction
as the undeformed element. If the eigenvalues are distinct, these will be the only elements
which do not change their directions. The ratio of the deformed length to the original length
is called the stretch, i.e.,

Thus, the eigenvalues of U are the principal stretches; they include the maximum and the
minimum stretches.

Example 3.20.1
Given that at time t,

*s -~s
Referring to Fig. 3.9, find the stretches for the following material line (a)OP (b)OQ and (c)OB.
Solution. The matrix of the deformation gradient for this given motion is

which is a symmetric matrix and is independent of JSQ (i.e., the same for all material points).
Thus, the given deformation is a homogeneous pure stretch deformation. The eigenvectors
are obviously (see Sect. 2B.17, Example 2B17.2) e^^ with corresponding eigenvalues, 3,4
and 1. Thus:
(a)At the deformed state, the line OP triples its original length and remains parallel to the
xi -axis, i.e., stretch =Aj = 3.