Biomedical Engineering Reference
In-Depth Information
AE
ε
+
AE
ε
11
2 2
=
ϕε
E
+
(1
−
ϕ ε
)
E
(5.19)
1
2
total
The combined modulus of elasticity is given by
(5.20)
EE
=+−
φ
(1
φ
)
E
1
2
Thus, by selecting two components with known properties, composites with
desired properties can be obtained by altering the volume of each component.
A similar analysis can be extended to composites with more components. In the
above derivation, the volume fraction of fibers is used, but while manufacturing the
weight fraction is easy to measure. Knowing the densities of each component, mass
fractions can be converted to volume fractions.
5.3.2.4 Young's Modulus of a Composite Material of Two Components in a Series
Confi guration
When a force
F
is applied, the stress in the fiber and the stress in the matrix are the
same as the cross sections of the two components are same (known as the isostress
conditition). If the force is in the transverse direction of the same composite, the
components are now in an isostrain situation. If the two components have varying
elastic properties, then each will experience a different strain. The strains can be
calculated knowing elastic modulus of each material.
For material 1, axial strain,
ε
1
=
σ
/
E
1
For material 2, axial strain,
ε
2
=
σ
/
E
2
The strain in the composite will be the volume average of the strain in each
material.
VL
VL
The volume fraction for material
as the cross-sectional
1
==
1
=
1
total
total
VL
VL
area is constant. Thus, the volume fraction equals
1
==
2
=
2
total
total
σ
σ
For the entire material, the axial deformation equals
LL
E
+
1
2
E
1
2
L
σ
L
σ
and the axial strain equals
1
+
2
=
φε
E
+
(1
−
φ ε
)
E
1
2
LE LE
total
1
total
2
σ
The combined modulus of elasticity equals
φσ
/
E
+−
(1) /
φ σ
E
1
2
EE
E
=
12
(5.21)
φ
E
+−
(1
φ
)
E
2
1
It is also important to determine the effective stiffness, which is an average
measure of the stiffness of the material, taking into account the properties of all
