Biomedical Engineering Reference
In-Depth Information
can be applied in such a way that:
=
2
∂ψ
∂C
ij
S
ij
(6.1)
where
I
1
,
I
2
,and
I
3
are the strain invariants of the symmetric right Cauchy - Green tensor
C
,and
λ
a
,
a
= 1,2,3 are the principal stretches. The Second Piola-Kirchoff stress tensor
S
, is related to the Cauchy stress tensor
σ
,by
σ
=
J
-1
FSF
, where
F
is the deformation
gradient of the motion, that is,
F
(
X
,
t
) = grad
x
. It can be shown that the relationship
between the Cauchy - Green tensor
C
and the deformation gradient
F
is:
C=F
T
F
.The
Jacobian of the motion,
J
,isdefinedas
J
=det[
F
ij
].
In this chapter, to denote scalar, vector, and tensor quantities we use uppercase let-
ters when they are evaluated in the reference configuration, and lowercase letters for
corresponding quantities in the current configurations. For example, symbols
X
and
x
represent the positions in the referential (original) and deformed (current) configurations,
respectively.
The general form of the Cauchy stress tensor
σ
for an isotropic and incompressible
material can be derived as [8]:
=−
pI
+
2
∂ψ
∂I
1
C
−
2
∂ψ
∂I
2
C
−
1
σ
(6.2)
where
I
1
,
I
2
(and
I
3
), the invariants of
C
ij
are:
⎨
⎩
I
1
=
λ
1
+
λ
2
+
λ
3
I
2
=
λ
1
λ
2
+
λ
2
λ
3
+
λ
3
λ
1
I
3
=
λ
1
λ
2
λ
3
=
J
2
(6.3)
The symbol
J
is also a measure of compressibility, that is, the ratio of the deformed
elastic volume to the undeformed (reference) volume of material. For incompressible
material, therefore, a kinematic constraint, namely
J
=
λ
1
λ
2
λ
3
= 1 can be considered.
Hence, as can be seen in Equation 6.2, the two principal invariants
I
1
and
I
2
are the only
independent deformation variables.
6.2.1 Hyperelastic Relationships in Uniaxial Loading
All strain energy density functions that are introduced for hyperelastic materials contain
some unknowns referred to as
material constants
and it is important these are accurately
computed for the material under the test conditions. The conventional way of deriving
these constants is by using the experimental stress - strain data. Technically, it is recom-
mended that these test data should be taken from several modes of deformation over
a wide range of strain values. The number of modes of deformation in experimental
tests should be at least as many deformation states as will be experienced during the
analysis [9].
For the present study, the test data of compression, which is the dominant mode in MIS
grasping, was used in order to find the material constants of the strain energy function.
For a uniaxial compression,
λ
1
=
λ
applied
=
λ
is the stretch in the direction being loaded.
