Biomedical Engineering Reference
In-Depth Information
Q
T
g
ð
Q
F
Þ¼
Q
g
ð
F
Þ
(11.52)
R
T
, where
R
is the rigid object rotation
and
U
the right stretch tensor which are related to
F
by
F
for all orthogonal tensors
Q
. If we take
Q
¼
¼
R
U
(Eq. (
11.8
)), it
follows from (
11.52
) that
g
(
R
T
R
T
F
)
¼
g
(
U
)
¼
g
(
F
)
R
; thus
T
¼
g
(
F
)
¼
R
g
(
U
)
R
T
,or
R
T;
T
¼
R
f
ð
C
Þ
(11.53)
U
2
(Eq. (
11.27
))). In terms of the first
Piola-Kirchhoff stress tensor
T
1PK
the constitutive equation for an elastic material is
f
(
U
2
) (since
C
where
g
(
U
)
¼
f
(
C
)
¼
¼
T
1
PK
¼
h
ð
F
Þ;
(11.54)
or, due to the invariance of constitutive equations under rigid object rotations,
h
(
Q
R
T
,
h
(
U
)
R
T
F
)
¼
Q
h
(
F
), and taking
Q
¼
¼
h
(
F
); thus
h
(
F
)
¼
R
h
(
U
), and
T
1
PK
¼
R
h
ð
U
Þ:
(11.55)
In terms of the second Piola-Kirchhoff stress tensor
T
2PK
, the constitutive
equation for an elastic material has the form
T
2
PK
¼
t
ð
C
Þ:
(11.56)
These constitutive equations are said to describe a material with Cauchy elastic-
ity; that is to say a material in which stress is a function of some measure of the
strain or deformation.
11.8 The Isotropic Finite Deformation Stress-Strain Relation
The assumption of isotropic symmetry of a material is an adequate model for many
materials. Recall from Chapter
5
that, in the case of a stress strain relation, isotropy
means that the response of stress to an applied strain is the same in any direction in
the material. The mathematical statement of this notion is that the stress tensor, say
T
2PK
in (
11.56
), is an isotropic function of the right Cauchy-Green tensor
C
.In
order for the tensor valued function
T
2PK
t
(
C
), given by (
11.56
) to be isotropic
an function, it must satisfy the relation for all orthogonal tensors
Q
:
¼
Q
T
Q
T
T
2
PK
Q
¼
t
ð
Q
C
Þ:
(11.57)
As one can see from the transformation law for a second order tensor (A83) that
the definition of an isotropic function (
11.57
) requires that, when the value of the
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