Rules of Mixtures for Elastic Properties

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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…...

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