Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 28

x 2 , X II

(11, 9)

(8, 7)

t = 2

(9, 6)

5

(6, 5)

(4, 4)

(6, 4)

t = 1

(5, 3)

(3, 2)

1

t = 0 x 1 ,

X I

0

0 1 5 10

Fig. 2.3 The movement of a square at t ¼ 0 due to the motion ( 2.12 )

unit square whose corners are at the material points ( X I , X II ) ¼ (0, 0), ( X I , X II ) ¼

(1, 0), ( X I , X II ) ¼ (1, 1), ( X I , X II ) ¼ (0, 1). At t ¼ 0 the motion ( 2.12 ) specifies that

x 1 ¼ X I , x 2 ¼ X II , and x 3 ¼ X III so that t ¼ 0 has been taken as the reference

configuration. The square at t ¼ 0 is illustrated in Fig. 2.3 .At t ¼ 1 the motion

( 2.12 ) specifies the places x of the particles X as follows:

x 1 ¼ 2 X I þ X II þ 3 ; x 2 ¼ X I þ 2 X II þ 2 ; x 3 ¼ X III :

Thus the particles at the four corners of the unit square have the following new
places x at t ¼ 1:

ð 3 ; 2 Þ¼wð 0 ; 0 Þ; ð 5 ; 3 Þ¼wð 1 ; 0 Þ; ð 6 ; 5 Þ¼wð 1 ; 1 Þ; ð 4 ; 4 Þ¼wð 0 ; 1 Þ:

A sketch of the deformed and translated unit square at t ¼ 1 is shown in Fig. 2.3 .

At t

¼

2 the motion ( 2.12 ) specifies the places x of the particles X as follows:

x 1 ¼

3 X I þ 2 X II þ

; x 2 ¼

2 X I þ 3 X II þ

; x 3 ¼ X III :

6 4

Thus the particles at the four corners of the unit square have the following new
places at t ¼ 2:

ð 6 ; 4 Þ¼wð 0 ; 0 Þ; ð 9 ; 6 Þ¼wð 1 ; 0 Þ; ð 11 ; 9 Þ¼wð 1 ; 1 Þ; ð 8 ; 7 Þ¼wð 0 ; 1 Þ:

A sketch of the deformed and translated unit square at t ¼ 2 is shown in Fig. 2.3 .