*to *Composites

**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:

**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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**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.

**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**

**E**_{cl} = E_{m}*V_{m} + E_{f}*V_{f}

**1/E**_{ct} = V_{m}/E_{m} + V_{f}/E_{f}

**Short fibers**

**E**_{cl} = η_{0}η_{L}V_{f} E_{f} + V_{m}E_{m}

**η**_{L} = 1 - 2/βL*tanh(βL /2)

**β = [8 G**_{m}/(E_{f}D²ln(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;

**η**_{0} - fiber orientation distribution factor.

**η**_{0} = 0.0 align fibers in transverse direction

**η**_{0} = 1/5 random orientation in any direction (3D)

**η**_{0} = 3/8 random orientation in plane (2D)

**η**_{0} = 1/2 biaxial parallel to the fibers

**η**_{0} = 1.0 unidirectional parallel to the fibers

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**G**_{ct} = G_{f} G_{m}/(V_{f} G_{m} + V_{m}G_{f})

Where:

**G**_{f} – shear modulus of elasticity of fiber material;

**G**_{m} – shear modulus of elasticity of matrix material;

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**μ**_{12} = v_{f} μ_{f} + V_{m}μ_{m}

Where:

**μ**_{f} – Poisson’s ratio of fiber material;

**μ**_{m} – Poisson’s ratio of matrix material;

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**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.

(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

(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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