Page 138 - Introduction to Continuum Mechanics
P. 138
Kinematics of a Continuum 123
(b)At the deformed state, the line OQ quadruple its original length and remains parallel to
the *2~ ®x*&\ stretch =^2 — 4-
(c)The line OB has an original length of 1.414. In the deformed state, it has a length of 5,
thus, the stretch is 5/1.414. Originally, the line OB makes an angle of 45° with the x\ -axis; in
1
the deformed state, it makes an angle of tan~ (4/3). In other words, the material line OB
changes its direction from OB to OB' (see Fig. 3.9).
Fig. 3.9
Example 3.20.2
For a material sphere with center at X and described by \dX\ = e, under a symmetric
deformation gradient U, what does the sphere become after the deformation?
Solution. Let e^, 62,63 be the principal directions for U, then with respect (e^ 62,63 ) a
material element dX can be written
In the deformed state, this material vector becomes
Since Fis diagonal, with diagonal element A x, A 2, A 3, therefore dx=FdX gives
thus, the sphere:
