70
3
E
E
V
V
L
m
T f
T f
=
+
−
1 2
1
η
η
(3.21)
where
5 E L = Longitudinal modulus of short fibre-reinforced composite
5 E m = Modulus of matrix
5 E f = Modulus of fibre
5 l = Length of fibre
5 d = Diameter of fibre
5 η L = Longitudinal coefficient of fibre
5 η T = Transverse coefficient of fibre
5 V f = Volume fraction of fibre
The coefficients of fibre, η L and η T , are given as
η L
f
m
f
m
=
(
)−
(
)+ ( )
E E
E E
l d
/
/
/
1
2
(3.22)
η T
f
m
f
m
=
(
)−
(
)+
E E
E E
/
/
1
2
(3.23)
Nevertheless, it is important to appreciate under what service conditions advanced composites and structures operate. These conditions demand the design of composites must suit to a loading
condition. The principles and equations mentioned herein only
pave the way for such designs and provide a broader understanding
of the response of advanced composites.
3.2 Macromechanics of Polymeric Composites
Macromechanics is the study of stresses and strains at the macro
level in a material. In composites, macromechanics is the study of
the gross behaviour of a lamina or a laminate when forces are
applied on it. A single lamina is generally orthotropic, which has
three mutually perpendicular planes of symmetry and is often
transversely isotropic.
3.2.1 Analysis of Anisotropic Composite Laminae
In order to understand the macromechanics of anisotropic composite laminae, it is essential to understand three-dimensional
(3-D) distribution of stresses on a unit element like a cube. In a
cube, the edges are parallel to the coordinate axes x, y, and z. The
forces are acting on three perpendicular faces of the cube, and
the forces are resolved into nine stress components, as shown in
. Fig. 3.3. The stress at point in a body is defined by nine components of the stress tensor σ ij . A tensor is a mathematical or physical expression that transforms according to a specific law of
transformation with a change in the coordinate system. A stress
Chapter 3 · Micromechanics and Macromechanics of Polymeric Composites
3
E
E
V
V
L
m
T f
T f
=
+
−
1 2
1
η
η
(3.21)
where
5 E L = Longitudinal modulus of short fibre-reinforced composite
5 E m = Modulus of matrix
5 E f = Modulus of fibre
5 l = Length of fibre
5 d = Diameter of fibre
5 η L = Longitudinal coefficient of fibre
5 η T = Transverse coefficient of fibre
5 V f = Volume fraction of fibre
The coefficients of fibre, η L and η T , are given as
η L
f
m
f
m
=
(
)−
(
)+ ( )
E E
E E
l d
/
/
/
1
2
(3.22)
η T
f
m
f
m
=
(
)−
(
)+
E E
E E
/
/
1
2
(3.23)
Nevertheless, it is important to appreciate under what service conditions advanced composites and structures operate. These conditions demand the design of composites must suit to a loading
condition. The principles and equations mentioned herein only
pave the way for such designs and provide a broader understanding
of the response of advanced composites.
3.2 Macromechanics of Polymeric Composites
Macromechanics is the study of stresses and strains at the macro
level in a material. In composites, macromechanics is the study of
the gross behaviour of a lamina or a laminate when forces are
applied on it. A single lamina is generally orthotropic, which has
three mutually perpendicular planes of symmetry and is often
transversely isotropic.
3.2.1 Analysis of Anisotropic Composite Laminae
In order to understand the macromechanics of anisotropic composite laminae, it is essential to understand three-dimensional
(3-D) distribution of stresses on a unit element like a cube. In a
cube, the edges are parallel to the coordinate axes x, y, and z. The
forces are acting on three perpendicular faces of the cube, and
the forces are resolved into nine stress components, as shown in
. Fig. 3.3. The stress at point in a body is defined by nine components of the stress tensor σ ij . A tensor is a mathematical or physical expression that transforms according to a specific law of
transformation with a change in the coordinate system. A stress
Chapter 3 · Micromechanics and Macromechanics of Polymeric Composites