Biomedical Engineering Reference
In-Depth Information
Fig. 7.3
An illustration of a
block of a homogeneous
material containing
cylindrical pores all aligned
in one direction
stress assumption in the
x
1
,
x
2
plane, that the effective elastic constants are given by
(Nemat-Nasser and Hori
1999
; Chap.
2
; (5.1.18a, b, c, d) and (5.1.27a, c)):
G
eff
13
G
m
¼
G
eff
23
G
m
¼
E
eff
1
E
m
¼
E
eff
2
E
m
¼
E
eff
3
E
m
¼
4
f
c
1
þ n
m
;
1
3
f
c
;
1
f
c
;
1
f
c
;
(7.26)
eff
12
eff
21
eff
31
eff
32
eff
13
eff
23
n
n
m
¼
n
1
n
m
n
n
m
¼
n
n
n
m
¼
n
n
m
¼
1
3
n
m
¼
1
;
n
m
¼
1
2
f
c
:
As noted, the cylindrical cavities aligned in the
x
3
direction change that material
symmetry but the isotropic character of the plane perpendicular to the
x
3
direction is
retained. The material in the plane perpendicular to the
x
3
direction is isotropic; all
the elastic constants associated with that plane will be isotropic, as shown in the
following exercise.
Example Exercise 7.4.2
Problem
: The effective elastic constants (
7.23
) for the composite composed of an
isotropic matrix material containing cylindrical cavities aligned in the
x
3
direction
are isotropic in the plane perpendicular to the
x
3
direction. Verify that this is the
case showing that if the matrix material satisfies the isotropy relationship 2
G
m
¼
E
m
/(1
eff
21
also satisfy the isotropy relationship. However, due to the notation, there is a
multitude of equivalent forms: 2
G
eff
þ n
m
), the effective elastic constants
G
eff
G
eff
23
,
E
eff
E
eff
2
eff
13
¼
¼
and
n
12
¼ n
1
2
G
eff
E
eff
1
eff
E
eff
2
eff
13
¼
23
¼
=ð
1
þ n
12
Þ¼
=ð
1
þ n
12
Þ¼
E
eff
1
=ð
þ n
eff
21
Þ¼
E
eff
2
=ð
þ n
eff
21
Þ
1
1
.
Solution
: The first formula of (
7.26
) is rewritten as
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