Biomedical Engineering Reference
In-Depth Information
For an incompressible isotropic material in an unconstrained compression test, the stretch
in the other two directions are equal,
λ
2
=
λ
3
and also
σ
2
=
σ
3
=0.
From the introduced relationships among the principal stretches and the compressibility
condition (
J
=
λ
1
λ
2
λ
3
= 1) we arrive at:
1
2
λ
2
=
λ
3
=
λ
−
(6.4)
For the uniaxial loading the motion
χ
can be expressed by the explicit equations:
⎨
⎩
x
1
=
λX
1
x
2
=
1
/
√
λ
X
2
(6.5)
1
/
√
λ
X
3
x
3
=
∂χ
a
Therefore, the deformation gradient,
F
aA
=
∂X
A
;
a, A
=
1
,
2
,
3 has the form:
⎡
⎣
⎤
⎦
λ
00
0
√
λ
[
F
]
=
0
(6.6)
00
√
λ
The Right Cauchy - Green tensor,
C
, can be obtained from Equation 6.6 as follows:
⎡
⎣
⎤
⎦
λ
2
00
0
λ
−
1
0
00
λ
−
1
[
F
]
T
[
C
]
=
[
F
]
=
(6.7)
The strain invariants, therefore, can be calculated directly from the principal stretches
given by Equation 6.3 or from the Cauchy - Green tensor:
⎧
⎨
I
1
=
tr(C)
=
λ
2
+
2
λ
−
1
2
[
tr(C)
]
2
−
tr(C
2
)
=
1
2
λ
+
λ
−
2
I
2
=
(6.8)
⎩
I
3
=
det
(C)
=
1
Considering relations given in Equation 6.7, the Cauchy stress given by Equation 6.2
will be reduced to the principal stresses:
σ
11
=−
p
+
2
∂ψ
−
2
∂ψ
∂I
1
λ
2
∂I
2
λ
−
2
(6.9)
σ
22
=−
p
+
2
∂ψ
−
2
∂ψ
∂I
1
λ
−
1
∂I
2
λ
=
0
(6.10)
Subtracting Equation 6.10 from Equation 6.9, we find that:
−
λ
−
1
∂ψ
∂I
2
λ
−
1
σ
11
=
2
λ
2
∂ψ
∂I
1
+
(6.11)
