Page 111 - Introduction to Continuum Mechanics

96 Kinematics of a Continuum

= U
Solution, (a) For the material line OA, Xi = 0, therefore, u\ = «2 3 ~ 0 • That is, the
line is not displaced. For the material CB, X^ = 1, «i = fc, the line is displaced by k units to
the right. For the material line OC and AB, «j = kX^, the lines become parabolic in shape.
Thus, the deformed shape is given by OAB 'C ' in Fig. 3.4.
(b) For the material point C, the matrix of the displacement gradient is

Therefore, from Eq. (3.7. la)

2 V2
(1
(2}
(c) From Eqs.(iii) and (iv), we have |dx >| =dX l, \d\ \ = dX 2(l+4k ) , thus,

and

If k is very small, we have the case of small deformations and by the binomial theorem, we
have, keeping only the first power of k,