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Engineering LibreTexts

6.10: Failure of Laminates and the Tsai–Hill Criterion

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Similar to the discussions in the page on Strength of long-fibre composites, the failure of laminae can be understood by the same three failure modes: axial, transverse and shear. A number of failure criteria have been proposed for separate plies subjected to in-plane stress states , with the assumption that coupling stresses are not present. We will introduce two here.

Maximum Stress Criterion

This assumes no interaction between the modes of failure, i.e. the critical stress for one mode is unaffected by the stresses tending to cause the other modes. Failure then occurs when one of these critical values, σ1u , σ2u and τ12u , is reached. These values refer to the laminar principal axes and can be resolved from the applied stress system by using the equation

[σ1σ2τ12]=[T][σxσyτxy]

where

[T]=[cos2θsin2θ2cosθsinθsin2θcos2θ2cosθsinθcosθsinθcosθsinθcos2θsin2θ]

( See page on Off-axis loading )

It follows that under an applied uniaxial tension ( σy = τxy = 0) the critical values of σx for each failure mode are:

σxu=σ1ucos2θ,σxu=σ2usin2θ,σxu=τ12usinθcosθ

Tsai-Hill Criterion

Other treatments that take into account the interactions between failure modes are mostly based on modifications of yield criteria for metals (See TLP on Theory of Metal Forming - Stress States and Yielding Criteria ). The most important of these is the Tsai-Hill Criterion, which is an adaptation of the von Mises Criterion.

von Mises Criterion for Metals: ( σ1 - σ2 )2 + ( σ2 - σ3 )2 + ( σ3 - σ1 )2 = 2 σY2

where σY is the metal yield stress.

For in-plane stress states ( σ3 = 0) this reduces to

(σ1σY)2+(σ2σY)2σ1σ2σ2Y=1

This is then modified to take into account the anisotropy of composites and the different failure mechanisms to give the following expression.

(σ1σ1Y)2+(σ2σ2Y)2σ1σ2σ21Yσ1σ2σ22Y+σ1σ2σ23Y+(τ12τ12Y)2=1

The metal yield stresses can be regarded as composite failure stresses and since composites are transversely isotropic ( σ2u = σ3u ) we arrive at the Tsai-Hill Criterion for composites.

(σ1σ1u)2+(σ2σ2u)2σ1σ2σ21u+(τ12τ12u)2=1

Below, the Maximum Stress and the Tsai-Hill criteria are used to predict the dependence on the loading angle of the tensile stress required to cause failure of a single lamina.

Failure of Laminates

The above treatments only apply to single isolated plies. So, in order to extend this to laminates, we must obtain the in-plane stresses in each ply of a laminate subjected to an arbitrary in-plane stress state.

From Equation 13 the stress tensor for the kth ply is related to the strain tensor by:

[σ1kσ2kτ12k]=[C]k[ε1kε2kγ12k]

The strain tensor of the kth ply can be resolved from the strain tensor of the laminate by using Equation 14:

[ε1kε2kγ12k]=[T]k[εxεyγxy]

Now, from Equation 16, the laminate strain tensor is related to the laminate stress tensor by:

[εxεyγxy]=[ˉSL][σxσyτxy]

Combining these three equations gives:

[σ1kσ2kτ12k]=[C]k[T]k[ˉSL][σxσyτxy]

An appropriate failure criterion is then applied and the onset of laminate failure is taken to be the point at which one of the plies fail.

Note that the Maximum Stress Criterion suggests possible modes of failure whereas the Tsai-Hill criterion does not.


This page titled 6.10: Failure of Laminates and the Tsai–Hill Criterion is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS) via source content that was edited to the style and standards of the LibreTexts platform.

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