Biomedical Engineering Reference
In-Depth Information
A second common set of bounds for composite materials are the
Hashin-Shtrikman(H-S)bounds,whichhavebeenfoundtobemorephysically
realistic for many composites, particularly those in which the reinforcing phase is
particulate and not continuous aligned fibers. These bounds are formulated in terms
of the shear and bulk moduli
G
and
K
, remembering that, for isotropic materials,
E
2
(
1
+
ν)
E
3
(
1
−
2
ν)
G
=
and
K
=
(3.8)
The lower bounds are [69]
V
2
−
1
1
(
K
2
−
K
1
)
+
3
(
1
−
V
2
)
3
K
1
+
4
G
1
K
L
=
K
1
+
+
V
2
−
1
1
6
(
K
1
+
2
G
1
)(
1
−
V
2
)
G
L
=
G
1
)
+
(
G
2
−
G
1
5
G
1
(
3
K
1
+
4
G
1
)
9
K
L
G
L
3
K
L
E
L
=
(3.9)
+
G
L
and the upper bounds are
−
1
1
(
K
1
−
K
2
)
+
3
V
2
3
K
2
+
4
G
2
K
U
=
K
2
+
(
1
−
V
2
)
−
1
1
6
(
K
2
+
2
G
2
)
V
2
G
U
=
G
2
+
(
1
−
V
2
)
)
+
(
G
1
−
G
2
5
G
2
(
3
K
2
+
4
G
2
)
9
K
U
G
U
3
K
U
E
U
=
(3.10)
+
G
U
A comparison of the V-R and H-S bounds is shown in Figure 3.2 for a stiff phase
with
E
2
E
1
) of 10, 100, and 1000.
The difference between the upper and lower bounds (either V-R or H-S) increases
with increasing modulus mismatch, making it increasingly difficult to predict the
expected behavior of the composite material without
apriori
knowledge of the
mechanism of strain transfer between the two different phases. The two variations
(V-R or H-S) on the lower bound expression become indistinguishable (on linear
coordinates) at large modulus mismatch, but the upper bounds remain distinct.
The Halpin-Tsai relationships for composite materials are approximations to
exact elasticity solutions in easy-to-calculate simple analytical forms. There are
well-known Halpin-Tsai expressions for the elastic modulus of a composite with
oriented short fibers or whiskers, precisely the situation being modeled with the FE
analysis presented in this section. The analytical expressions for the longitudinal
and transverse elastic moduli (
E
L
,
E
T
) relative to the compliant matrix modulus
(
E
M
) are [70]
E
L
E
M
=
=
100GPaandamodulusmismatchfactor(
E
2
/
1
+
(
2
AR
) η
L
V
F
(3.11)
1
−
η
L
V
F
E
T
E
M
=
1
+
2
η
T
V
F
1
−
η
T
V
F
(3.12)
