Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 33

when ( 2.12 ) and ( 2.14 ) are employed. Often the base vectors of the coordinate
systems X and x are taken to coincide, in which case the position vector c of the
origin of the x system relative to the X system is zero. The selection of the
coordinate system is always the prerogative of the modeler and such selections
are usually made to simplify the analysis of the resulting problem.
Example 2.1.4
Compute the deformation gradient and the inverse deformation gradient for the
motion given by ( 2.12 ). Then compute the Jacobian of the motion and both the
spatial and material gradients of the displacement vector.

Solution : The deformation gradients and the inverse deformation gradients for this
motion are obtained from ( 2.16 )to( 2.12 ), thus

2 3 2 3

1 þ t t 0 1 þ t t 0

1

4 5 4 5 ;

and F 1

F ¼ t 1 þ t 0 ¼ t 1 þ t 0

1 þ 2 t

0 0 1 0 0 1 þ 2 t

a result that can be verified using FF 1 1 or F 1 F

¼ ¼ 1 . It is then easy to show

that J ¼ 1 þ 2 t . It also follows from ( 2.22 ) that

2
4

3
5

2
4

3
5:

110
110
000

110
110
000

t

T T

½r O u ¼ t and ½r u ¼

1 þ 2 t

Problems

2.1.1. Sketch the shape and position of the unit square with corners at (0, 0), (1, 0),
(1, 1), and (0, 1) subjected to the motion in ( 2.10 ) for the seven special cases,
(a) through (g) below. The shape and position are to be sketched for each of
the indicated values of t .

(a) Translation. A ( t ) ¼ 1, B ( t ) ¼ 1, C ( t ) ¼ 0, D ( t ) ¼ 0, E ( t ) ¼ 2 t , F ( t ) ¼ 2 t

0, 1, 2.
(b) Uniaxial extension. A ( t )

and values of t ¼

¼ 1 þ t , B ( t ) ¼ 1, C ( t ) ¼ 0, D ( t ) ¼ 0, E ( t ) ¼ 0,

F ( t ) ¼ 0 and values of t ¼ 0, 1, 2, 3.

(c) Biaxial extension. A ( t ) ¼ 1 þ t , B ( t ) ¼ 1 þ 2 t , C ( t ) ¼ 0, D ( t ) ¼ 0, E

( t ) ¼ 0, F ( t ) ¼ 0 and values of t ¼ 0, 1, 2.

(d) Simple shearing (R). A ( t ) ¼ 1, B ( t ) ¼ 1, C ( t ) ¼ t , D ( t ) ¼ 0, E ( t ) ¼ 0,

F ( t ) ¼ 0 and values of t ¼ 0, 1, 2.

(e) Simple shearing (U). A ( t ) ¼ 1, B ( t ) ¼ 1, C ( t ) ¼ 0, D ( t ) ¼ t , E ( t ) ¼ 0,

F ( t ) ¼ 0 and values of t ¼ 0, 1, 2.

(f) Rigid Rotation (cw). A ( t ) ¼ cos ( p t /2), B ( t ) ¼ cos ( p t /2), C ( t ) ¼ sin( p t /

2), D ( t ) ¼ sin( p t /2), E ( t ) ¼ 0, F ( t ) ¼ 0 and values of t ¼ 0, 1, 2, 3, 4.