Page 112 - Introduction to Continuum Mechanics
P. 112
Infinitesimal Deformations 97
and
if y denote the decrease in angle, then
That is, for small k,
We can write Eq. (3.7. la), i.e., dx = dX+(Vu)dX as
where
To find the relationship between ds, the length of d\ and dS, the length of rfX, we take the dot
product of Eq. (3.7.2) with itself:
i.e.,
T
If F is an orthogonal tensor, then F F = I, and
Thus, an orthogonal F corresponds to a rigid body motion (translation and/or rotation).
Now, from Eq. (3.7.3),
In this section, we shall consider only cases where the components of the displacement vector
as well as their partial derivatives are all very small (mathematically, infinitesimal) so that the
r
absolute value of every component of (Vu) Vu is a small quantity of higher order than those
of the components of Vu. For such a case, the above equation becomes:
