Background

Failure criteria and laminate strength prediction

This section presents information covering

Failure criteria for unidirectional materials

Maximum stress criterion

Maximum strain criterion

Tsai-Hill criterion

Tsai-Wu criterion

Summary

Laminate strength prediction

Introduction

The strengths of materials used will typically be established initially (by prediction or test) in the principal material directions. Loadings in service may introduce multi-axial stress states hence it is important that the failure loads under arbitrary stress states can be predicted. To this end, a wide variety of failure criteria have been developed for composite materials. Many of these criteria focus on prediction of the failure loads for a single unidirectional material under any in-plane stress state. Prediction of failure for a single unidirectional layer is seen as a starting point for predicting the strength of a laminate overall.

Failure criteria for unidirectional materials

A large number of criteria have been proposed but they can be categorised according to how they consider interaction between the effects of different stress components. The simplest approaches are those where no interaction is considered - e.g. the strength in one direction is not influenced by the magnitude of stresses in the other directions. In these criteria there is typically one equation for each of the three in-plane stress (or strain) components.

Interaction between stress components is considered by a number of criteria. These typically suggest that the failure load in one direction is affected by the application of loads in the other directions. In these criteria there is just a single equation used to define the failure envelope.

Some criteria have a combination of interactive and non-interactive conditions, e.g. the failure stress in the fibre direction might be taken as independent of the transverse and in-plane stresses while the failure stress in each of the latter two directions might be based on interactive considerations.

Some of the more commonly applied criteria are presented below.

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Maximum stress criterion

The maximum stress criterion does not consider any interaction and judges failure to occur when the stress in any direction exceeds the strength in that direction. Hence the conditions for failure can be stated as

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Maximum strain criterion

The maximum strain criterion does not consider any interaction and judges failure to occur when the strain in any direction exceeds the strain to failure in that direction. Hence the conditions for failure can be stated as

This criterion defines a failure envelope similar to that of the maximum stress criterion but it takes into account Poisson's deformations.

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Tsai-Hill failure criterion

This interactive criterion is a development of theories derived for metals considering distortional energy. A single equation is used to define the failure envelope, namely

X and Y are taken as tensile or compressive strengths depending on the signs of s₁ and s₂ , e.g. if s₁>0 then X=X_t .

Note that within each quadrant (as defined by the s₁ - s₂ axes) this failure criterion is represented by a smooth and continuous curve. The non-interactive criteria are not smooth and continuous within each quadrant.

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Tsai-Wu failure criterion

This is a phenomenological criterion with its derivation being based on linking experimental constants rather than on a physical interpretation of material behaviour. This criterion states that failure occurs when

The coefficients are determined according to the material strengths as follows

The term A₁₂ is obtained from bi-axial tests made on the lamina material.

This criterion is also represented by smooth continuous curves in each quadrant.

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Summary

There is not widespread agreement at present on what represents the best failure criterion to use. There are, for example, some fairly readily identifiable limitations in each of the criteria presented above. A clear limitation of the maximum stress and maximum strain criteria is that they do not take interactions into account - this effect is has been found to be marked under stress states where s₂ and t₁₂ dominate. On the other hand, an argument against the interactive criteria is that they do not take into account the failure mode in any meaningful way. In order to represent the underlying physical phenomena correctly one would expect that there would be at least one equation for each failure mode whereas the interactive criteria often use just a single equation to represent the entire failure envelope. Several other anomalies have been identified - for example, in the compression-compression quadrant, a rise in failure load is predicted on decreasing the material's transverse compressive strength.

Work in this area is continuing and major programmes are underway to improve understanding and arrive at a more consistent approach. While present understanding in this area is not entirely satisfactory, the implications for the designer are not, in the majority of cases, so severe. There are several reasons for this including;

• In many design situations rules have been developed that do not require implicit use of failure criteria, e.g. strain limited design rules.
• The designer will often specify laminates in which failure is dominated by the fibre direction stresses, i.e. the stress state is such that most of the failure criteria available would predict similar results.
• Failure criteria typically consider single layers only. When embedded in a laminate a single layer can typically tolerate higher transverse and shear stresses than when considered in isolation. This means that layers for which transverse and in-plane shear is likely to affect failure, and consequently for which the differences between failure criteria are expected to be most significant, will usually be stronger than predicted.

It is worth noting that most commercial finite element software packages incorporate a number of lamina based failure criteria. Given that each will have some limitations it is important that the user have a good understanding of these and the situations in which they will be important.

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Laminate strength prediction

The failure loads for laminates are usually predicted by consideration of the stresses in each of the individual layers making up the laminate. The stress state within each layer is compared to that allowed by the failure criterion being applied and the load at which the first layer fails is calculated. This is usually referred to as the first ply failure load. Failure of a single layer in a laminate will not always mean that the laminate as a whole cannot continue to sustain load. Analysis can therefore proceed by recalculation of stresses with the elastic properties of the failed layer reduced according to its type of failure. The load is then increased (if possible) until the next layer failure is predicted and this procedure is continued until no further load can be sustained, i.e. the ultimate strength is reached.

The behaviour up until ultimate load will be determined to a large extent by the type of material. For most practical glass fibre reinforced plastic laminates there will usually be some ply failure (transverse failure in 90 degree plies or shear failure in off axis plies) prior to ultimate failure. This is a consequence of the relatively high strain to failure of the glass fibres (typically > 2%) which means that some matrix failures will occur prior to fibre failure. The designer in this case should consider what is considered as laminate failure, e.g. is it when some matrix cracking occurs (this could be detrimental where the matrix is relied upon to provide resistance to a chemically aggressive environment) or is it when the ultimate strength is reached.

For practical carbon fibre laminates it is unlikely that any transverse or shear failures will occur prior to the ultimate strength being attained. This applies to laminates that have been constructed with a reasonable proportion of material in each of the 0, 90 and ± 45 degree orientations and where individual layer thicknesses are small. Strength prediction for these laminate types depends on predicting the longitudinal failure load for the layers most closely aligned to the principal stress or strain directions and this is not heavily influenced by the failure criterion used.