Biomedical Engineering Reference
In-Depth Information
Fig. 3.35
Simple shear deformation of a parallelepiped in the x
1
-x
2
-plane
Simple Shear Deformation: In the case of finite deformations the motion v and
the deformation gradient F of the homogenous plane-strain simple shear defor-
mation is given by (Silber and Steinwender 2005)
x
¼
v X
; ðÞ¼
X
1
þ
c
ðÞ
X
2
½
e
1
þ
X
2
e
2
þ
X
3
e
3
;
F
¼
x
r¼
I
þ
ce
1
e
2
ð
3
:
418
Þ
where c
ð
t
Þ
:
¼
K
ð
t
Þ=
H is the shear gradient with the constant shear velocity K
ð
t
Þ¼
v
0
t (see Fig.
3.35
).
The right C
AUCHY
-G
REEN
deformation tensor C is thus given by C :
¼
F
T
F
¼
I
þ
c
ð
e
1
e
2
þ
e
2
e
1
Þþ
c
2
e
2
e
2
:
Transformation of C in diagonal form leads to the
following eigenvalue problem
ð
C
j
i
I
Þ
m
i
¼
0
ð
i
¼
1
;
2
;
3
Þ;
where the j
i
are
the
eigenvalues
and
the
m
i
are
the
corresponding
eigenvectors
of
C
:
This
