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**Mechanics of Composite Materials**

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**Constitutive Relationships for Composite Materials**

Ⅰ. Material Behavior in Principal Material Axes Isotropic materials uniaxial loading

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2-D loading Where [ S ]: compliance matrix Where [Q]: stiffness matrix

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**Isotropic Materials Note:**

1. Only two independent material constants in the constitutive equation. 2. No normal stress and shear strain coupling, or no shear stress and normal strain coupling. Examples: polycrystalline metals, Polymers Randomly oriented fiber-reinforced composites Particulate-reinforced composites

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**Transversely isotropic materials**

In L–T plane Principal material axes L: longitudinal direction T: transverse direction

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**Transversely isotropic materials**

In T1, T2 plane Same as those for isotropic materials: Principal material axes L: longitudinal direction T: transverse direction

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**Transversely isotropic materials**

Where EL: elastic modulus in longitudinal direction ET: elastic modulus in transverse direction GLT: shear modulus in L – T plane GTT: shear modulus in transverse plane LT: major Poisson’s ratio (strain in T – direction caused by stress in L – direction) TL : minor Poisson’s ratio And Note: 1. 4 independent material constants (EL, ET, GLT, LT ) in L – T plane while 5 (EL, ET, GLT, LT, GTT) for 3-D state. 2. No normal stress and shear strain coupling in L – T axes or no shear stress and normal strain coupling in L – T axes

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**Orthotropic materials**

For example in 1-2 plane 1.2.3: principal material axes

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**Orthotropic Materials**

Note: 1. 4 independent constants in 2-D state (e.g. 1-2 plane, E1, E2, G12, 12 )while 9 in 3-D state (E1, E2, E3, G12, G13, G23, 12 , 13 , 23 ) 2. No coupling between normal stress and shear strain or no coupling between shear stress and normal strain

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**Question Ex. Find the deformed shape of the following composite:**

Possible answers?

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**Off-axis loading of unidirectional composite**

For orthotropic material in principal material axes (1-2 axes) By coordinate transformation , xyxy are tensorial shear strains

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Let Then

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**Transformed stiffness matrix**

Where = transformed stiffness matrix

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**Transformed compliance matrix**

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**Off-axis loading - deformation**

1. 4 material constants in 1-2 plane. 2. There is normal stress and shear strain coupling (forθ≠0, 90˚ ), or shear stress and normal strain coupling.

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**Transformation of engineering constants**

For uni-axial tensile testing in x-direction ∴ stresses in L – T axes Strains in L – T axes

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And strains in x – y axes

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**Recall for uni-axial tensile testing**

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**Define cross-coefficient, mx**

Similarly, for uni-axial tensile testing in y-direction

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**For simple shear testing in x – y plane**

stresses in L – T axes Strains in L – T axes

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Strains in x – y axes

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**In summary, for a general planar loading, by principle of superposition**

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**Micromechanics of Unidirectional Composites**

Properties of unidirectional lamina is determined by volume fraction of constituent materials (fiber, matrix, void, etc.) form of the reinforcement (fiber, particle, …) orientation of fibers

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**Volume fraction & Weight fraction**

Vi=volume, vi=volume fraction= Wi=weight, wi=weight fraction= Where subscripts i = c: composite f: fiber m: matrix

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Conservation of mass: Assume composite is void-free:

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Density of composite Generalized equations for n – constituent composite

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**Void content determination**

Experimental result (with voids): Theoretical calculation (excluding voids): In general, void content < 1% Good composite > 5% Poor composite

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**Burnout test of glass/epoxy composite**

Weight of empty crucible = g Weight of crucible +composite = g Weight of crucible +glass fibers = g Find Sol:

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**Longitudinal Stiffness**

For linear fiber and matrix: Generalized equation for composites with n constituents: Rule-of-mixture

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**Longitudinal Strength**

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Modes of Failure matrix-controlled failure: fiber-controlled failure:

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**Critical fiber volume fraction**

For fiber-controlled failure to be valid: For matrix is to be reinforced:

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**Factors influencing EL and scu**

mis-orientation of fibers fibers of non-uniform strength due to variations in diameter, handling and surface treatment, fiber length stress concentration at fiber ends (discontinuous fibers) interfacial conditions residual stresses

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**Transverse Stiffness, ET**

Assume all constituents are in linear elastic range: Generalized equation for n – constituent composite:

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**Transverse Strength Factors influence scu:**

Due to stress (strain) concentration Factors influence scu: properties of fiber and matrix the interface bond strength the presence and distribution of voids (flaws) internal stress and strain distribution (shape of fiber, arrangement of fibers)

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**In-plane Shear Modulus**

For linearly elastic fiber and matrix:

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Major Poisson’s Ratio

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**Analysis of Laminated Composites**

Classical Laminate Theory (CLT) Displacement field:

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**Resultant Forces and Moments**

Resultant moments: [A]: extensional stiffness matrix [B]: coupling stiffness matrix [D]: bending stiffness matrix

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**Laminates of Special Configurations**

Symmetric laminates Unidirectional (UD) laminates specially orthotropic off-axis Cross-ply laminates Angle-ply laminates Quasi-isotropic laminates

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Strength of Laminates

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**Maximum Stress Criterion**

Lamina fails if one of the following inequalities is satisfied:

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**Maximum Strain Criterion**

Lamina fails if one of the following inequalities is satisfied:

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Tsai – Hill Criterion Lamina fails if the following inequality is satisfied: Where :

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**Comparison among Criteria**

Maximum stress and strain criteria can tell the mode of failure Tsai-Hill criterion includes the interaction among stress components

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**Strength of Off-Axis Lamina in Uni-axial Loading**

Maximum stress criterion Tsai-Hill criterion

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Strength of a Laminate First-ply failure Last-ply failure

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