Page 109 - Introduction to Continuum Mechanics

94 Kinematics of a Continuum

Equation (3.6.3) states that the motion is described by a translation c(f), of an arbitrary
chosen material base point X=b plus a rotation R(t).

Example 3.6.2

From Eq. (3.6.3) derive the relation between the velocity of a general material point in the
rigid body with the angular velocity of the body and the velocity of the arbitrary chosen material
point.
Solution. Taking the material derivative of Eq. (3.6.3), we obtain

Now, from Eq. (3.6.3), we have

Thus

Since RR = I, RR +RR = 0, so that RR is antisymmetric which is equivalent to a dual
(or axial) vector <o [see Sect. 2B16], thus,

If we measure the position vector r for the general material point from the position at time
t of the chosen material base point, i.e., r = (x-c), then

3.7 Infinitesimal Deformations

There are many important engineering problems which involves structural members or
machine parts, for which the displacement of every material point is very small (mathemati-
cally infinitesimal) under design loadings. In this section, we derive the tensor which
characterizes the deformation of such bodies.
Consider a body, having a particular configuration at some reference time t a, changes to
another configuration at time t. Referring to Fig. 3.3, a typical material point P undergoes a
displacement u, so that it arrives at the position

A neighboring point Q at X+dX arrives at x+dx which is related to X+dX by:

Subtracting Eq. (i) from Eq. (ii), we obtain