Biomedical Engineering Reference
In-Depth Information
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#
2
þ
2
þ
2
e 33 ¼ @
u 3
1
2
@
u 1
@
u 2
@
u 3
x 3
;
@
@
x 3
@
x 3
@
x 3
1
2
@
u 1
x 2 þ @
u 2
x 1 @
u 1
x 1 @
u 2
x 1 @
u 2
x 1 @
u 2
x 2 @
u 3
x 1 @
u 3
e 12 ¼
;
(11.21)
@
@
@
@
@
@
@
@
x 2
1
2
@
u 1
x 3 þ @
u 3
x 1 @
u 1
x 1 @
u 3
x 1 @
u 2
x 1 @
u 2
x 3 @
u 3
x 1 @
u 3
e 13 ¼
;
@
@
@
@
@
@
@
@
x 3
1
2
@
u 2
x 3 þ @
u 3
x 2 @
u 1
x 2 @
u 2
x 3 @
u 2
x 3 @
u 2
x 3 @
u 3
x 2 @
u 3
e 23 ¼
;
@
@
@
@
@
@
@
@
x 3
respectively. The products of the displacement gradients appear in Eqs. ( 11.20 ) and
( 11.21 ) for the strain tensor E and e . This is called geometrical nonlinearity to
distinguish it from the physical or constitutive nonlinearity (e.g. the relation
between stress and strain) that will be considered later in this chapter. If the
deformation of the object is so small that the square of the displacement gradients
can be neglected, and thus the difference between the material and spatial
coordinates, then both the Lagrangian strain tensor E and the Eulerian strain tensor
e coincide with the infinitesimal strain tensor (2.44).
The geometrical interpretation of the Lagrangian strain tensor E and the Eulerian
strain tensor e are algebraically straightforward, but not very simple geometrically.
If
d I represents the change in length per unit length in the X I direction, then the
deformation gradient F and the Lagrangian strain tensor E are given by
2
4
3
5 ; E ¼
100
000
000
1
2 fF T
F ¼ 1 þ d I
F 1g
2
4
3
5 ;
100
000
000
1
2 d
2
I
¼ d I þ
(11.22)
thus all the components of E are zero except for,
1
2 d
2
I
E II ¼ d I þ
:
(11.23)
If
d II represents the change in length per unit length in the X II direction, and sin
f
indicates a shear, then the deformation gradient is given by
2
3
1
þ d I
ð
1
þ d I Þ
sin
f
0
4
5 ;
F
¼
0
1
þ d II
0
(11.24)
0
0
1
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