RULES OF MIXTURES FOR ELASTIC PROPERTIES
The paper is about the rules of mixtures which are used to express the dependencies of the physical properties and mechanical properties which depend on type, form, quality and arrangement of its constituents, but they are based on various assumptions so one should with caution, especially if they are used anything more than preliminary design. The paper mainly concentrates on expressions for elastic properties which are as follows:
Unidirectional Ply- longitudinal modulus: The Figure 1 clearly shows the orthogonal axes and fiber direction, the fiber directions. The first approximation made is E3= E4. And also for deriving the rules of mixtures the following assumptions are made:
Fibers are uniform, parallel and continuous.
Perfect bonding exists between fibers and matrix.
A longitudinal load produces equal strain in fiber and matrix. Using the above assumptions and approximations two rules of mixtures are derived which are E1 = EfVf + EmVm = EfVf + Em (1- Vf) V12= vfVf+vm+Vm
These two rules of mixtures are generally accepted as it goes well with experimental data.
Unidirectional Ply- Transverse modulus: In this the rules of mixtures are less reliable than those for longitudinal properties as they are based on assumptions of stress distribution. In this the poisson’s contraction is ignored and the stress is assumed to be the same.which leads to the result of: E2=EfEm/(VfEm+VmEf).
But the above rules of mixture has poor experimental agreement so an alternative Halpin-Tsai model for transverse modulus has been introduced. E2 = Em (1+ξηVf) (1 - ηVf)
Unidirectional Ply- Shear Modulus:
Shear modulus is defined as the ration of shear stress to the shear strain. The shear modulus is determined using Halpin-Tsai model and the rule of mixtures for shear modulus is based on same assumption as for transverse tensile modulus, and so same caution should be considered. It is obtained from:
G23 = E2 2(1+v23) Where v23 is the poisson’s ratio.
Unidirectional Ply- Poisson’s ratio:
If transverse isotropy is seen in the following equation changes in terms of bulk modulus, K. The three dimensional elastic constant is rarely used for preliminary use but is used for other numerical analysis. Kf=Ef/3(1-2vf)
Transverse isotropy is assumed here these values should be used for illustrative purpose only.
Multidirectional Ply- in-plane tensile modulus:
It is possible to modify Equations to give an estimate of tensile modulus for composites in which the fibres are neither continuous nor unidirectional. The correction factor is given as: The orientation correction factor is then included in Equation, giving the semi-empirical: The above equation applies strictly to in-plane reinforcement. In woven fabrics, the fibres exhibit ‘waviness’ in the through-thickness direction.
Short fiber reinforcement: In case of discontinuous fibers, the load transfer is at the fiber ends unlike the continuous fibers where uniform stress or strain is seen, therefore, shear stress and strain are maximum when relatively flexible matrix is embedded in short fiber and also tensile stress tends to zero at fiber end and increases towards center. Also short fibers are mostly well above their critical length, and elastic properties are dependent on orientation effects.
Thermal expansion: The coeffeicients of composites in this are considered as relevant as they various temperature changes