Introduction of Composite Material:
A composite is a structural material that consists of two or more combined constituents that are combined at a macroscopic level and are not soluble in each other.
Reinforcing phase: fibres, particles, or flakes.
Matrix phase: polymers, metals, ceramics.
Composites are becoming an essential part of today’s materials because they offer advantages such as low weight, corrosion resistance, high fatigue strength, faster assembly, etc. Composites are used as materials in making aircraft structures to golf clubs, electronic packaging to medical equipment, and space vehicles to home building. Composites are generating curiosity and interest in students all over the world. They are seeing everyday applications of composite materials in the commercial market, and job opportunities are increasing in this field. The technology transfer initiative of the US federal government is opening new and large scale opportunities for use of advanced composite materials.

Reinforcement and Matrices:
Fibres are particularly attractive as reinforcement because
Spinning imparts strength and stiffness to them
· Their small diameter (usually 7 to about 20 µm) implies a low probability of defects and allows a higher fraction of ideal strength to be attained (strength decreases with increasing diameter); this having a positive influence on mechanical properties of fibres
· They are flexible and thus can undergo bending with a radius of curvature, r_{min} = 0.1 to 1 mm, without breaking (proceed from the simple theory of bending
where fibres are necessarily handled and should not break, and also for flexible composites like tyres.
A single fibre is often referred to as a filament. Several hundreds or thousands of filaments (count) can be put together to form so called strands, rovings, yarns or tows. So called heavy tows containing several tens of thousands (for example 100 k) of filaments have been introduced more recently.
Towards their high performance in composites, fibres are coated with sizing which
Generalized Hook’s Law
StressStrain Relation:
The generalized Hooke’s Law can be used to predict the deformations caused in a given material by an arbitrary combination of stresses.
The linear relationship between stress and strain applies for
The generalized Hooke’s Law also reveals that strain can exist without stress. For example, if the member is experiencing a load in the ydirection (which in turn causes a stress in the ydirection), the Hooke’s Law shows that strain in the xdirection does not equal to zero. This is because as material is being pulled outward by the yplane, the material in the xplane moves inward to fill in the space once occupied, just like an elastic band becomes thinner as you try to pull it apart. In this situation, the xplane does not have any external force acting on them but they experience a change in length.
Therefore, it is valid to say that strain exist without stress in the xplane.
We need to connect all six components of stress to six components of strain.
– Restrict to linearly elasticsmall strains.
– An isotropic materials whose properties are independent of orientation.
Elastic constants for anisotropic orthotropic and isotropic materials:
Some engineering materials, including certain piezoelectric materials (e.g. Rochelle salt) and 2ply fiberreinforced composites, are orthotropic.
By definition, an orthotropic material has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane. Such materials require 9 independent variables (i.e. elastic constants) in their constitutive matrices.
In contrast, a material without any planes of symmetry is fully anisotropic and requires 21 elastic constants, whereas a material with an infinite number of symmetry planes (i.e. every plane is a plane of symmetry) is isotropic, and requires only 2 elastic constants.
THREEDIMENSIONAL MATERIAL ANISOTROPY:
For a threedimensional elastic anisotropic body (Fig. 6.1), the generalized Hook’s law is expressed as
where and are the stress and strain tensors, respectively, and are the elastic constants. Here the indices i, j, k and l can assume values of 1, 2 and 3. This implies that there may exist 3^{4} = 81 independent elastic constants. However, it is known from the theory of elasticity, that both stress tensor and strain tensor are symmetric. As = , =
This results in reduction of possible independent elastic constants to thirtysix.
Further, if there exists a strain energy U such that
with the property that , then = (6.5)
Equation 6.5 in conjunction with Eq. 6.3 finally reduce the total number of independent elastic constants from thirtysix to twentyone only. Such an anisotropic material with twentyone independent elastic constants is termed as triclinic. Now, using the following contracted single index notations
the constitutive relations for the general case of material anisotropy are expressed as
Here, [ ] is the elastic constant matrix.
Conversely, { } = [S_{ij} ] { } ; i, j =1, 2,6 (6.9)
where [S_{ij}] is the compliance matrix.
Note that
[S_{ij}] = [C_{ij }]^{1} (6.10)
Also, [ ] =[ ] and [S_{ij}] = [S_{ji}] due to symmetry.
There may exist several situations when the distribution and orientation of reinforcements may give rise to special cases of material property symmetry. When there is one plane of material property symmetry (say, the plane of symmetry is x_{3 = }0, i.e., the rotation of 180 degree around the x3 axis yields an equivalent material), the elastic constant matrix [ ] is modified as
Thus there are thirteen independent elastic constants, and the material is monoclinic. The compliance matrix [S_{ij}] for a monoclinic material may accordingly be written from Eq. 6.11 by replacing ‘C ‘ with ‘S ‘.
If there are three mutually orthogonal planes of symmetry, the material behaviour is orthotropic. The elastic constant matrix is then expressed as
Thus there are nine independent elastic constants. Correspondingly there exist nine independent compliances.
Two special cases of symmetry, square symmetry and hexagonal symmetry, may arise due to packing of fibres in some regular fashion. This results in further reduction of independent elastic constant. For instance, if the fibres are packed in a square array (Fig. 6.2) in the X_{2,} X_{3} plane. Then [ ]
There exist now six independent elastic constants. Similarly, when the fibres are packed in hexagonal array (Fig. 6.3),
In the case of hexagonal symmetry, the number of independent elastic constants is reduced to
five only. The material symmetry equivalent to the hexagonal symmetry, is also achieved, if the fibres are packed in a random fashion (Fig. 6.4) in the X_{2}X_{3} plane. This form of symmetry is usually termed as transverse isotropy. The [ ] matrix due to the transverse isotropy is the same as that given in Eq. 6.14. The compliance matrices corresponding to Eqs. 6.12 through 6.14 can be accordingly written down. However, it may be noted that in the case of rectangular array (Fig. 6.5), C_{12 }≠ C_{13}, C_{22 }≠ C_{33} and C_{55 }≠ C_{66} (Eq. 6.13).
Material Isotropy
The material properties remain independent of directional change for an isotropic material. The elastic constant matrix [ ] for a three dimensional isotropic material are expressed as
The compliance matrix [S_{ij}] for an isotropic material can be accordingly derived.