Composite Strength and Failure Criteria

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.

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

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

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

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

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:

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:

Failure in longitudinal tension

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.

Failure in longitudinal compression Microbuckling Shear failure mode

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

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

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

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.

Typical composite strengths (MPa) UD CFRP UD GRP woven GRP SiC/Al s1T* 2280 1080 367 1462 s1C* 1440 620 549 2990 s2T* 57 39 367 86 s2C* 228 128 549 285 t12* 71 89 97 113

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.

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:

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...

Maximum stress failure envelope s1C* s2C*

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

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:

Orientation dependence of strength

Daniel & Ishai (1994)

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.

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

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:

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

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

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:

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

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

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

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: