1. Composite Strength and Failure Criteria

2. Micromechanics of failure in a unidirectional ply
In the fibre direction (‘1’), we assume equal strain in fibre and matrix. The applied stress is shared:
s1 = sf Vf + sm Vm
Failure of the composite depends on whether the fibre or the matrix reaches its failure strain first.

3. Brittle matrix - em < ef
composite strength
stress
fibres
sc = sf* Vf
sf*
sf1
sm*
em*
ef*
Vfx 1
matrix Vf
sc = sf1 Vf + sm* Vm

4. Brittle matrix - em < ef
composite strength
sf1 is the stress in the fibres when the matrix fails = Ef em*
For E-glass fibre/brittle polyester, Vfx ~ 3%
So for practical purposes, sc = sf* Vf
Vfx 1 Vf

5. Brittle fibres - em > ef
composite strength
stress
sc = sf* Vf + sm1 Vm
sf*
sm*
sm1
ef*
em*
Vfo 1
Vfcrit Vf
sc = sm* Vm

6. Brittle fibres - em > ef
composite strength
At Vfcrit, the strength is a minimum; at Vf > Vfo the composite is stronger than the matrix.
For carbon/epoxy,
Vfcrit ~ 2.5% Vfo ~ 2.6%
Vfo 1
Vfcrit
So for practical purposes, sc = sf* Vf + Em ef* (1-Vf)  sf* Vf

7. Micromechanics of failure in a unidirectional ply
ef* < em* (eg a polymer composite with brittle fibres):
At fibre failure: e
em*
ef*
sf*
If Ef >> Em:

8. Micromechanics of failure in a unidirectional ply
ef* > em* (eg a ceramic composite with brittle matrix):
At matrix failure:
sf* e
em*
ef*
If fibres continue to bear load, then:

9. Failure in longitudinal tension

10. Failure in longitudinal compression
Failure is difficult to model, as it may be associated with different modes of failure, including fibre buckling and matrix shear.
Composite strength depends not only on fibre properties, but also on the ability of the matrix to support the fibres.
Measurement of compressive strength is particularly difficult - results depend heavily on method and specimen geometry.

11. Failure in longitudinal compression
Microbuckling
Shear failure mode

12. Failure in transverse tension
High stress/strain concentrations occur around fibre, leading to interface failure. Individual microcracks eventually coalesce...

13. Failure in transverse compression
May be due to one or more of:
compressive failure/crushing of matrix
compressive failure/crushing of fibre
matrix shear
fibre/matrix debonding

14. Failure by in-plane shear
Due to stress concentration at fibre-matrix interface:

15. Five numbers are needed to characterise the strength of a composite lamina:
s1T* longitudinal tensile strength
s1C* longitudinal compressive strength
s2T* transverse tensile strength
s2C* transverse compressive strength
t12* in-plane shear strength
‘1’ and ‘2’ denote the principal material directions; * indicates a failure value of stress.

16. Typical composite strengths (MPa)
UD CFRP UD GRP woven GRP SiC/Al
s1T*
s1C*
s2T*
s2C*
t12*

17. The use of Failure Criteria
It is clear that the mode of failure and hence the apparent strength of a lamina depends on the direction of the applied load, as well as the properties of the material.
Failure criteria seek to predict the apparent strength of a composite and its failure mode in terms of the basic strength data for the lamina.
It is usually necessary to calculate the stresses in the material axes (1-2) before criteria can be applied.

18. Maximum stress failure criterion
Failure will occur when any one of the stress components in the principal material axes (s1, s2, t12) exceeds the corresponding strength in that direction.
Formally, failure occurs if:

19. Maximum stress failure criterion
All stresses are independent. If the lamina experiences biaxial stresses, the failure envelope is a rectangle - the existence of stresses in one direction doesn’t make the lamina weaker when stresses are added in the other...

20. Maximum stress failure envelope
s1C*
s2C*

21. Orientation dependence of strength
The maximum stress criterion can be used to show how apparent strength and failure mode depend on orientation: s2 s1 sx q
t12

22. Orientation dependence of strength
At failure, the applied stress (sx) must be large enough for one of the principal stresses (s1, s2 or t12) to have reached its failure value.
Observed failure will occur when the minimum such stress is applied:

23. Orientation dependence of strength

24. Daniel & Ishai (1994)

25. Maximum stress failure criterion
Indicates likely failure mode.
Requires separate comparison of resolved stresses with failure stresses.
Allows for no interaction in situations of non-uniaxial stresses.

26. Maximum strain failure criterion
Failure occurs when at least one of the strain components (in the principal material axes) exceeds the ultimate strain.

27. Maximum strain failure criterion
The criterion allows for interaction of stresses through Poisson’s effect.
For a lamina subjected to stresses s1, s2, t12, the failure criterion is:

28. Maximum strain failure envelope
For biaxial stresses (t12 = 0), the failure envelope is a parallelogram: s2 s1

29. Maximum strain failure envelope
In the positive quadrant, the maximum stress criterion is more conservative than maximum strain.
max strain s2
The longitudinal tensile stress s1 produces a compressive strain e2. This allows a higher value of s2 before the failure strain is reached.
max stress s1

30. Tsai-Hill Failure Criterion
This is one example of many criteria which attempt to take account of interactions in a multi-axial stress state.
Based on von Mises yield criterion, ‘failure’ occurs if:

31. Tsai-Hill Failure Criterion
A single calculation is required to determine failure.
The appropriate failure stress is used, depending on whether s is +ve or -ve.
The mode of failure is not given (although inspect the size of each term).
A stress reserve factor (R) can be calculated by setting

32. Orientation dependence of strength
The Tsai-Hill criterion can be used to show how apparent strength depends on orientation: s2 s1 sx q
t12

34. Tsai-Hill Failure Envelope
For all ‘quadratic’ failure criteria, the biaxial envelope is elliptical.
The size of the ellipse depends on the value of the shear stress: s2 s1
t12 = 0
t12 > 0

35. Comparison of failure theories
Different theories are reasonably close under positive stresses.
Big differences occur when compressive stresses are present.
A conservative approach is to consider all available theories: