Dr. Dmitri Kopeliovich

Composite materials may be either isotropic or anisotropic, which is determined by the Structure of composites.

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:

• Density
• Coefficient of Thermal Expansion
• Modulus of Elasticity
• Shear modulus
• Poisson's ratio
• Tensile strength

d_c = d_m*V_m + d_f*V_f

Where

d_c,d_m,d_f – densities of the composite, matrix and dispersed phase respectively;

V_m,V_f – volume fraction of the matrix and dispersed phase respectively.

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• Coefficient of Thermal Expansion (CTE) in longitudinal direction (along the fibers)

α_cl = (α_m*E_m*V_m + α_f*E_f*V_f)/(_m*V_m + E_f*V_f)

Where

α_cl, α_m, α_f – CTE of composite in longitudinal direction, matrix and dispersed phase (fiber) respectively;

E_m,E_f – modulus of elasticity of matrix and dispersed phase (fiber) respectively.

• Coefficient of Thermal Expansion (CTE) in transverse direction (perpendicular to the fibers)

α_ct = (1+μ_m) α_m *V_m + α_f* V_f

Where

μ_m – 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

• Modulus of Elasticity in longitudinal direction (E_cl)

E_cl = E_m*V_m + E_f*V_f

• Modulus of Elasticity in transverse direction (E_ct)

1/E_ct = V_m/E_m + V_f/E_f

Short fibers

E_cl = η₀η_LV_f E_f + V_mE_m

η_L = 1 - 2/βL*tanh(βL /2)
β = [8 G_m/(E_fD²ln(2R/D))]½

where:

E_f – modulus of elasticity of fiber material;
E_m – modulus of elasticity of matrix material;
G_m - shear modulus of matrix material;
η_L – 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

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G_ct = G_f G_m/(V_f G_m + V_mG_f)

Where:

G_f – shear modulus of elasticity of fiber material;
G_m – shear modulus of elasticity of matrix material;

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μ₁₂ = v_f μ_f + V_mμ_m

Where:

μ_f – Poisson’s ratio of fiber material;
μ_m – Poisson’s ratio of matrix material;

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• Tensile strength of long-fiber reinforced composite in longitudinal direction

σ_c = σ_m*V_m + σ_f*V_f

Where

σ_c, σ_m, σ_f – tensile strength of the composite, matrix and dispersed phase (fiber) respectively.

• Tensile strength of short-fiber composite in longitudinal direction

(fiber length is less than critical value L_c)

L_c = σ_f*d/τ_c

Where

d – diameter of the fiber;

τ_c –shear strength of the bond between the matrix and dispersed phase (fiber).

σ_c = σ_m*V_m + σ_f*V_f*(1 – L_c/2L)

Where

L – length of the fiber

• Tensile strength of short-fiber composite in longitudinal direction

(fiber length is greater than critical value L_c)

σ_c = σ_m*V_m + L* τ_c*V_f/d

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• Classification of composites
• Structure of composites
• Metal Matrix Composites
• Ceramic Matrix Composites
• Polymer Matrix Composites
• Tribological properties of alumina reinforced composites

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