Biomedical Engineering Reference
In-Depth Information
For transformation of the stress tensors (
3.96
) and (
3.98
) in spectral form, the
deformation gradient in spectral form is needed. Substituting (
3.182
)in(
3.59
)
leads to
!
F
¼
R
U
¼
R
X
¼
X
3
3
R
m
ð Þ
|{z}
n
i
m
i
;
i.e.
k
i
m
i
m
i
k
i
i
¼
1
i
¼
1
2
4
3
5
n
i
m
h i
k
1
00
0 k
2
0
00k
3
F
¼
X
3
k
i
n
i
m
i
and
½¼
ð
3
:
186
Þ
i
¼
1
with
n
i
¼
R
m
i
:
ð
3
:
187
Þ
In (
3.186
), n
i
¼
R
m
i
are the principal directions (eigen-vectors) rotated with
the versor R with respect to m
i
in the CCFG. Based on the ''mixed'' principal
direction dyads n
i
m
i
, the two-field tensor property of F becomes apparent. From
the simple form (
3.186
), the inverse of F and the J
ACOBI
-determinant J (third
invariant of F) derive to
2
3
k
1
1
00
0 k
1
2
0
00k
1
3
F
1
¼
X
3
k
1
i
4
5
m
i
n
h i
m
i
n
i
and
½¼
ð
3
:
188
Þ
i
¼
1
J
¼
det F
¼
k
1
k
2
k
3
:
ð
3
:
189
Þ
Split into Deviatoric (Isochoric) and Volumetric Parts. In the case of
compressible materials, a split of the constitutive equation into deviatoric and
volumetric parts is advantageous. This is done by performing a multiplicative
decomposition of the deformation gradient F into a volume-changing (dilational)
part F and a volume-preserving (distortional) part J
1/3
I so that (e.g. (Lee 1969))
F
J
1
=
3
F
F
¼
J
1
=
3
F
:
F
¼
J
1
=
3
I
and
ð
3
:
190
Þ
With (
3.190
) a relation between the modified principal stretches (deviatoric
principal stretches) k
i
and the eigen-values k
i
(principal stretches) of the right
stretch tensor U is obtained (the second term in (
3.191
) is obtained using (
3.189
)
and (
3.190
) and the relation det F
¼
det J
1
=
3
F
¼
J
1
=
3
3
det F
¼
J
1
det F
¼
J
1
J
¼
1 resulting in)
k
i
:
¼
J
1
=
3
k
i
i
¼
1
;
2
; ð Þ
and J :
¼
det F
¼
k
1
k
2
k
3
¼
1
: ð
3
:
191
Þ
Using (
3.190
) and (
3.66
)
1
the modified right C
AUCHY
strain tensor C (distor-
tional part) yields
