Page 121 - Introduction to Continuum Mechanics

106 Kinematics of a Continuum

Thus, for small deformation

This unit volume change is known as dilatation. Note also that

In cylindrical coordinates,

In spherical coordinates,

3.11 The Infinitesimal Rotation Tensor

Equation (3.7.1), i.e., dx = dX + (Vu)dX, can be written

where Q, the antisymmetric part of Vu, is known as the infinitesimal rotation tensor. We
see that the change of direction for dX in general comes from two sources, the infinitesimal
deformation tensor E and the infinitesimal rotation tensor Q. However, for any dX which is
in the direction of an eigenvector of E, there is no change of direction due to E, only that due
to Q. Therefore, the tensor Q represents the infinitesimal rotation of the triad of the
4
eigenvectors of E. It can be described by a vector f in the sense that

where (see Section 2B.16)

are ne
Thus, Q32,Qi3,Q2i * infinitesimal angles of rotation about ej, 63, and C3-axes, of the
triad of material elements which are in the principal direction of E.

3.12 Time Rate of Change of a Material Element
Let us consider a material element dx emanating from a material point X located at x at
time t. We wish to compute (D/Dt)dx, the rate of change of length and direction of the material
element d\. From x = x(X,f), we have