Biomedical Engineering Reference
In-Depth Information
11.9.3. Derive the first result of (
11.81
), namely
ð@
I
C
=@
C
Þ¼
1
.
11.9.4. Derive the result (
11.83
),
T
2
PK
¼
2
½fð@
W
=@
I
C
Þþ
I
C
ð@
W
=@
II
C
Þþð@
W
=@
III
C
ÞC
2
III
C
Þ
II
C
g1 fð@
W
=@
II
C
Þþ
I
C
ð@
W
=@
III
C
ÞgCþð@
W
@
, from the
last of (
11.77
),
T
2
PK
¼
ð@
W
=@
C
Þ
, using (
11.80
),
W
¼
W
ð
I
C
;
II
C
;
III
C
Þ
2
,
III
C
C
1
},
(
11.81
){
ð@
I
C
=@
C
Þ¼
1
,
ð@
II
C
=@
C
Þ¼
I
C
1
C
,
ð@
III
C
=@
C
Þ¼
and the Cayley Hamilton theorem,
C
3
I
C
C
2
þ
II
C
C
III
C
¼
0.
11.9.5. Take the derivative of the Cayley Hamilton theorem,
C
3
I
C
C
2
þ
II
C
C
III
C
¼
0, with respect to
C
and employ the three formulas (
11.81
){
ð@
I
C
III
C
C
1
} to eliminate the
expressions that are the derivatives of the invariants with respect to
C
. Then
multiply through this result by
C
and simplify. What is the significance of
the final result? Is it a correct equation? Would it be a correct equation if the
Eq. (
11.81
) were not correct?
=@
C
Þ¼
1
,
ð@
II
C
=@
C
Þ¼
I
C
1
C
,
ð@
III
C
=@
C
Þ¼
11.10
Incompressible Elasticity
The assumption of incompressibility is an idealization that means that no agency
(stress, strain, electric field, temperature, etc.) can change the volume of the model
of the material. The Jacobian
J
Det
F
relates the element of volume d
V
in the
undeformed configuration to the volume d
v
in the deformed configuration
according to the rule (
11.36
), d
v
¼
¼
J
d
V
. The Jacobian
J
is related to the principal
stretches by
J
¼ l
I
l
II
l
III
. The requirement of incompressibility may then be
expressed in several different algebraic forms related to the deformation,
J
¼
1,
l
I
l
II
l
III
¼
1,
III
C
¼
III
B
¼
1, etc., and to algebraic forms related to the motion such
as tr
D
¼ ∇
v
¼
0 (Sect. 6.4). The assumption of incompressibility requires that
be a constant. The pressure field
p
in an incompressible material is a
Lagrange multiplier (see Example 6.4.1) that serves the function of maintaining the
incompressibility constraint, not a thermodynamic variable. Because the volume of
the model material cannot change,
p
does no work; it is a function of
x
and t,
p
(
x
, t),
to be determined by the solution of the system of differential equations and
boundary/initial conditions.
Recall from
Sect. 11.7
that the constitutive equation for an elastic material can
be written
T
r
the density
¼
g
(
F
) in terms of the Cauchy stress
T
and the deformation gradient
F
or as
T
1PK
h
(
F
)(
11.54
) in terms of the first Piola-Kirchhoff stress tensor
T
1PK
or
¼
as
T
2PK
t
(
C
)(
11.56
) in terms of the second Piola-Kirchhoff stress tensor
T
2PK
and the right Cauchy-Green tensor
C
. For incompressible elastic materials the
Cauchy stress tensor
T
must be replaced by
T
¼
p1
where
p
is the constitutively
indeterminate pressure described above and conveniently interpreted as a Lagrange
multiplier. The response functions, say
g
(
F
) above, are defined only for
deformations or motions that satisfy the condition
J
þ
¼
Det
F
¼
1. Thus
T
¼
g
(
F
) is replaced by
T
þ
p1
¼
g
(
F
) when the assumption of incompressibility is
made,
T
1PK
¼
h
(
F
), (
11.54
), becomes
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