Page 110 - Introduction to Continuum Mechanics
P. 110
Infinitesimal Deformations 95
Fig. 33
Using the definition of gradient of a vector function [see Eq. (2C3.1)], Eq. (iii) becomes
where Vu is a second-order tensor known as the displacement gradient The matrk of Vu with
respect to rectangular Cartesian coordinates (with X = Xfa and u = «/e/) is
Example 3.7.1
Given the following displacement components
>•>
(a) Sketch the deformed shape of the unit square OABC in Fig. 3.4
1
(b) Find the deformed vector (i.e., die ' and dx (2) ) of the material elements dX^ = dX^i
and dlv ' - dX<£i which were at the point C.
(c) determine the ratio of the deformed to the undeformed lengths of the differential elements
(known as stretch) of part (b) and the change in angle between these elements.
