Biomedical Engineering Reference
In-Depth Information
The time rate computed by holding the places x fixed is called the local time rate.
In general, the material time rate is related to the local time rate by the following
operator expression that follows from ( 2.29 ),
x fixed þ
Dt ¼ @
D
v
r;
(2.30)
@
t
where D / Dt is the material time rate of change,
/
t is the local time rate of change
and v
determines the convective change of the quantity.
The second order tensor formed by taking the spatial gradient of the velocity
field v
¼
v ( x , t ) is called the tensor of velocity gradients and is denoted by L , thus
2
3
@
v 1
@
v 1
@
v 1
4
5
@
x 1
@
x 2
@
x 3
¼
@
v i
@
v 2
@
v 2
@
v 2
T
L
¼r
½
v
¼
:
(2.31)
@
x j
@
x 1
@
x 2
@
x 3
@
v 3
@
v 3
@
v 3
@
x 1
@
x 2
@
x 3
L is decomposed into a symmetric part D called the rate-of-deformation tensor , and
a skew symmetric part W called the spin tensor , thus
L T
L T
L
¼
D
þ
W
;
D
¼ð
1
=
2
Þð
L
þ
Þ;
W
¼ð
1
=
2
Þð
L
Þ:
(2.32)
The three nonzero components of W can be formed into an axial vector (1/2)
(
v ) which represents the local rotational motion and is called the angular
velocity or one-half the vorticity.
The rate-of-deformation tensor D defined by the second of ( 2.32 ) has the
component representation
2
4
3
5
2 @
v 1
@
v 1
x 2 þ @
v 2
@
v 1
x 3 þ @
v 3
@
x 1
@
@
x 1
@
@
x 1
1
2
@
v j
x i þ @
v i
1
2
@
v 1
x 2 þ @
v 2
2 @
v 2
@
v 2
x 3 þ @
v 3
D
¼
¼
:
(2.33)
@
@
x j
@
@
x 1
@
x 2
@
@
x 2
@
v 1
x 3 þ @
v 3
@
v 2
x 3 þ @
v 3
2 @
v 3
@
@
x 1
@
@
x 2
@
x 3
The components of D along the diagonal are called normal rates of deformation
and the components off the diagonal are called shear rates of deformation. The
normal rates of deformation, D 11 , D 22 , and D 33 , are measures of instantaneous time
rate of change of the material filament instantaneously coincident with the 1, 2, and
3 axes, respectively, and the shear rates of deformation, D 23 , D 13 , and D 12 , are equal
to one-half the time rate of decrease in an originally right angle between material
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