Mechanics of Composite Materials

Constitutive Relationships for Composite Materials Ⅰ. Material Behavior in Principal Material Axes Isotropic materials uniaxial loading

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

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

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

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

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

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

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

Question Ex. Find the deformed shape of the following composite: Possible answers?

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

Let Then

Transformed stiffness matrix Where = transformed stiffness matrix

Transformed compliance matrix

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.

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

And strains in x – y axes

Recall for uni-axial tensile testing

Define cross-coefficient, mx Similarly, for uni-axial tensile testing in y-direction

For simple shear testing in x – y plane stresses in L – T axes Strains in L – T axes

Strains in x – y axes

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

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

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

Conservation of mass: Assume composite is void-free:

Density of composite Generalized equations for n – constituent composite

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

Burnout test of glass/epoxy composite Weight of empty crucible = 47.6504 g Weight of crucible +composite = 50.1817 g Weight of crucible +glass fibers = 49.4476 g Find Sol:

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

Longitudinal Strength

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

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

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

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

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)

In-plane Shear Modulus For linearly elastic fiber and matrix:

Major Poisson’s Ratio

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

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

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

Strength of Laminates

Maximum Stress Criterion Lamina fails if one of the following inequalities is satisfied:

Maximum Strain Criterion Lamina fails if one of the following inequalities is satisfied:

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

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

Strength of Off-Axis Lamina in Uni-axial Loading Maximum stress criterion Tsai-Hill criterion

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