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Mechanics of Composite Materials. Constitutive Relationships for Composite Materials. Material Behavior in Principal Material Axes Isotropic materials.

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Presentation on theme: "Mechanics of Composite Materials. Constitutive Relationships for Composite Materials. Material Behavior in Principal Material Axes Isotropic materials."— Presentation transcript:

1 Mechanics of Composite Materials

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

3 –2-D loading Where [ S ]: compliance matrix Where [Q]: stiffness matrix

4 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

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

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

7 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

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

9 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

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

11 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

12 Let Then

13 Transformed stiffness matrix Where= transformed stiffness matrix

14 Transformed compliance matrix : transformed compliance matrix

15 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.

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

17 And strains in x – y axes

18 Recall for uni-axial tensile testing

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

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

21 Strains in x – y axes

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

23 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

24 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

25 Conservation of mass: Assume composite is void-free:

26 Density of composite Generalized equations for n – constituent composite

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

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

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

30 Longitudinal Strength

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

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

33 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

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

35 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)

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

37 Major Poissons Ratio

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

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

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

41 Strength of Laminates

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

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

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

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

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

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


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