Mechanics of Composite Materials

Constitutive Relationships for Composite Materials . Material Behavior in Principal Material AxesIsotropic materialsuniaxial loading

2-D loadingWhere [ S ]: compliance matrixWhere [Q]: stiffness matrix

Isotropic MaterialsNote: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,PolymersRandomly oriented fiber-reinforced compositesParticulate-reinforced composites

Transversely isotropic materialsPrincipal material axesL: longitudinal directionT: transverse directionIn LT plane

Transversely isotropic materialsPrincipal material axesL: longitudinal directionT: transverse directionIn T1, T2 plane Same as those for isotropic materials:

Transversely isotropic materialsWhere EL: elastic modulus in longitudinal directionET: elastic modulus in transverse directionGLT: shear modulus in L T planeGTT: shear modulus in transverse planeLT: major Poissons ratio(strain in T direction caused by stress in L direction)TL : minor Poissons ratioAnd 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 materials1.2.3: principal material axesFor example in 1-2 plane

Orthotropic MaterialsNote: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

QuestionEx.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 , xyxy are tensorial shear strains

LetThen

Transformed stiffness matrixWhere = transformed stiffness matrix

Transformed compliance matrix: transformed compliance matrix

Off-axis loading - deformation1. 4 material constants in 1-2 plane.2. There is normal stress and shear strain coupling (for0, 90 ), or shear stress and normal strain coupling.

Transformation of engineering constantsFor uni-axial tensile testing in x-direction stresses in L T axesStrains in L T axes

And strains in x y axes

Recall for uni-axial tensile testing

Define cross-coefficient, mxSimilarly, 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 byvolume fraction of constituent materials (fiber, matrix, void, etc.)form of the reinforcement (fiber, particle, )orientation of fibers

Volume fraction & Weight fractionVi=volume, vi=volume fraction=Wi=weight, wi=weight fraction=Where subscripts i = c: compositef: fiberm: matrix

Conservation of mass:Assume composite is void-free:

Density of composite Generalized equations for n constituent composite

Void content determinationExperimental 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 gWeight of crucible +composite = 50.1817 gWeight of crucible +glass fibers = 49.4476 gFindSol:

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

Longitudinal Strength

Modes of Failurematrix-controlled failure:fiber-controlled failure:

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

Factors influencing EL and scu mis-orientation of fibersfibers of non-uniform strength due to variations in diameter, handling and surface treatment, fiber lengthstress concentration at fiber ends (discontinuous fibers)interfacial conditionsresidual stresses

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

Transverse StrengthDue to stress (strain) concentrationFactors influence scu:properties of fiber and matrixthe interface bond strengththe presence and distribution of voids (flaws)internal stress and strain distribution (shape of fiber, arrangement of fibers)

In-plane Shear ModulusFor linearly elastic fiber and matrix:

Major Poissons Ratio

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

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

Laminates of Special ConfigurationsSymmetric laminates Unidirectional (UD) laminatesspecially orthotropic off-axis Cross-ply laminatesAngle-ply laminatesQuasi-isotropic laminates

Strength of Laminates

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

Maximum Strain CriterionLamina 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 LoadingMaximum stress criterionTsai-Hill criterion

Strength of a LaminateFirst-ply failureLast-ply failure