Biomedical Engineering Reference
In-Depth Information
Fig. 11.14
An illustration
of a uniform dilation, a type
of pure homogeneous
deformation
Problems
11.4.1. For the six motions of the form (2.10) given in Problem 2.1.1, namely 2.1.1
(a)-2.1.1(f), compute the Lagrangian strain tensor
E
and the Eulerian strain
tensor
e
, the right and left Cauchy-Green tensors,
C
and
B
, respectively,
and the inverse of the right Cauchy-Green tensor. Discuss briefly the
significance of each of the tensors computed. In particular, explain the
form or value of the deformation strain tensors in terms of the motion.
11.4.2. Prove that, in the case of no deformation, the invariants of
C
and
c
satisfy
the following relationships:
I
c
¼
I
C
¼
II
c
¼
II
C
¼
3,
III
c
¼
III
C
¼
1.
11.4.3. Show that the Jacobian
J
is related to
III
C
by
J
2
III
C
.
11.4.4. A rectangular parallelepiped with a long dimension
a
o
and a square cross-
section of dimension
b
o
is deformed by an axial tensile force
P
into a
rectangular parallelepiped with a longer long dimension
a
and a smaller
square cross-section of dimension
b
.
(a) What are the stretch ratios in the long direction (
¼
l
L
) and the transverse
l
T
)?
(b) Express the volume of the deformed rectangular parallelepiped,
V
,asa
function of the volume of the undeformed rectangular parallelepiped,
V
o
, and the stretch ratios in the long direction (
direction (
l
L
) and the transverse
l
T
).
(c) Express the incompressibility condition for the rectangular parallelepi-
ped as a function of the stretch ratios in the long direction (
direction (
l
L
) and the
l
T
).
(d) Express the cross-sectional area of the deformed rectangular parallele-
piped,
A
, as a function of the cross-sectional area of the undeformed
rectangular parallelepiped,
A
o
, and the stretch ratio in the transverse
direction (
transverse direction (
l
T
).
11.5 Measures of Volume and Surface Change
in Large Deformations
In this section we will consider volume and area measures of deformation. Consider
volume deformation first. A material filament denoted by d
X
is mapped into its
present position d
x
by the deformation gradient
F
,d
x
¼
F
d
X
. By considering the
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