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
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