Isotropic material is a material, properties of which do not depend on a direction of measuring.
Anisotropic material is a material, properties of which along a particular axis or parallel to a particular plane are different from the properties measured along other directions.
Rule of Mixtures is a method of approach to approximate estimation of composite material properties, based on an assumption that a composite property is the volume weighed average of the phases (matrix and dispersed phase) properties.
According to Rule of Mixtures properties of composite materials are estimated as follows:
dc = dm*Vm + df*Vf
dc,dm,df – densities of the composite, matrix and dispersed phase respectively;
Vm,Vf – volume fraction of the matrix and dispersed phase respectively.
αcl = (αm*Em*Vm + αf*Ef*Vf)/(m*Vm + Ef*Vf)
αcl, αm, αf – CTE of composite in longitudinal direction, matrix and dispersed phase (fiber) respectively;
Em,Ef – modulus of elasticity of matrix and dispersed phase (fiber) respectively.
αct = (1+μm) αm *Vm + αf* Vf
μm – Poisson’s ratio of matrix.
Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of applied force.
Long align fibers
Ecl = Em*Vm + Ef*Vf
1/Ect = Vm/Em + Vf/Ef
Ecl = η0ηLVf Ef + VmEm
ηL = 1 - 2/βL*tanh(βL /2)
β = [8 Gm/(EfD²ln(2R/D))]½
Ef – modulus of elasticity of fiber material;
Em – modulus of elasticity of matrix material;
Gm - shear modulus of matrix material;
ηL – length correction factor;
L – fibers length;
D – fibers diameter;
2R – distance between fibers;
η0 - fiber orientation distribution factor.
η0 = 0.0 align fibers in transverse direction
η0 = 1/5 random orientation in any direction (3D)
η0 = 3/8 random orientation in plane (2D)
η0 = 1/2 biaxial parallel to the fibers
η0 = 1.0 unidirectional parallel to the fibers
Gct = Gf Gm/(Vf Gm + VmGf)
Gf – shear modulus of elasticity of fiber material;
Gm – shear modulus of elasticity of matrix material;
μ12 = vf μf + Vmμm
μf – Poisson’s ratio of fiber material;
μm – Poisson’s ratio of matrix material;
σc = σm*Vm + σf*Vf
σc, σm, σf – tensile strength of the composite, matrix and dispersed phase (fiber) respectively.
(fiber length is less than critical value Lc)
Lc = σf*d/τc
d – diameter of the fiber;
τc –shear strength of the bond between the matrix and dispersed phase (fiber).
σc = σm*Vm + σf*Vf*(1 – Lc/2L)
L – length of the fiber
(fiber length is greater than critical value Lc)
σc = σm*Vm + L* τc*Vf/d