1. Mechanics of Composite Materials

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

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

7. 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

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

9. 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

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

15. 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.

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
Vi=volume, vi=volume fraction=
Wi=weight, wi=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 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

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

35. 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)

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

37. Major Poisson’s Ratio

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

39. Resultant Forces and Moments
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