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Mechanics of Composite Materials

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Constitutive Relationships for Composite Materials. 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 Principal material axes L: longitudinal direction T: transverse direction In L–T plane

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Transversely isotropic materials Principal material axes L: longitudinal direction T: transverse direction In T 1, T 2 plane Same as those for isotropic materials:

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Transversely isotropic materials Where E L : elastic modulus in longitudinal direction E T : elastic modulus in transverse direction G LT : shear modulus in L – T plane G TT : shear modulus in transverse plane LT : major Poissons ratio (strain in T – direction caused by stress in L – direction) TL : minor Poissons ratio And Note:1. 4 independent material constants (E L, E T, G LT, LT ) in L – T plane while 5 (E L, E T, G LT, LT, G TT ) 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 1.2.3: principal material axes For example in 1-2 plane

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Orthotropic Materials Note: 1. 4 independent constants in 2-D state (e.g. 1-2 plane, E 1, E 2, G 12, 12 )while 9 in 3-D state ( E 1, E 2, E 3, G 12, G 13, G 23, 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, xy xy 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 : 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 (forθ0, 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 V i =volume, v i =volume fraction= W i =weight, w i =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 E L and cu 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, E T Assume all constituents are in linear elastic range: Generalized equation for n – constituent composite:

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Transverse Strength Due to stress (strain) concentration Factors influence cu : 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 Poissons Ratio

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Analysis of Laminated Composites Classical Laminate Theory (CLT) Displacement field:

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Resultant Forces and Moments Resultant forces: 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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