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# Tsai-Wu - continuous analytic function?

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## Tsai-Wu - continuous analytic function?

(OP)
Hello,

I was reading an article recently about composite failure criteria, and it stated that the Tsai-Wu criterion is often preferred, since it is a “continuous analytic function that adequately represents all modes of ply failure while including stress interaction effects.”

I understand what continuous means, and I understand what analytic means mathematically (infinitely differentiable), but I’m not sure I understand why this makes the Tsai-Wu criterion preferable.
Can anyone help me out?

Thanks!

### RE: Tsai-Wu - continuous analytic function?

The Tsai-Wu criterion in an interactive failure criterion, as opposed to a non-interactive failure criterion. In theory, it is possible that an interactive ply level failure criterion is more accurate for ply level predictions (though even that is highly debatable). However, what is not debatable, is that when an interactive failure criterion is applied to a practical laminate, it becomes difficult to determine what the subcritical failure modes are (such as a matrix failure mechanisms). Because of this (and other reasons), non-interactive failure criterion tend to be preferred for laminates. This is because the fiber and matrix failure modes are decoupled, which is useful for practical laminate level predictions. More specifically, max fiber strain can be used for laminates that exhibit fiber dominated failure. Fiber dominated failure is usually the case for well designed laminates and works well for carbon fiber/epoxy material systems (glass fiber is a little different).

Moving beyond simple unnotched failure prediction, ANY ply level failure criterion, especially the interactive ones, lose physical meaning. For example, laminates with holes, fastened joints, strength after impact, are what structures are usually designed to. Because of the notch sensitively, hole filling effects, installation torque, etc., strength prediction using ply level criteria has little meaning. Instead, one can use analytical and test methods (semi-empirical) to account for the practical effects. The Tsai-Wu criterion is probably best used for academic study and comparison to other ply level failure criterion. Beyond that, it has limited value.

From a historical perspective, it was hoped that a ply level criterion could ultimately lead to accurate laminate level failure predictions. This would be a best case scenario because the required test data would be minimal. Eventually it became clear that this was insufficient for practical laminates and practical considerations. However, failure criteria such as Tsai-Wu remain popular in academia. This is because academic approaches do not go beyond the unnotched strength prediction level. Unfortunately, structures are almost never designed to the unnotched strength.

### RE: Tsai-Wu - continuous analytic function?

I agree with Brain's view that non-interactive criteria are more useful for composites for general analysis. Max stress and strain criteria would be a typical criteria I'd use for sizing laminates and layups; not something like Tsai-Wu for the reasons Brian has outlined.

However, and speaking as a (sort of) academic, I disagree with Brian's view that Tsai-Wu or Tsai-Wu like criteria are, or should be, used for academic work... . In the academic sphere, criteria like Tsai-Wu are basically used for teaching, not research. Students learning about composites will learn about Tsai-Wu along with max stress, strain ,etc..., so their eyes can then be opened to the world of damage based models, traction-separation laws, multi-scale approaches, etc...(at least this is what we do at Imperial College).

For example; The most well known, recent failure 'criteria' (they're more like methodologies than discrete criteria) to come out of Imperial College are probably the LaRC04 and LaRC05. Both of these have multiple, competing failure criteria to account for various failure/damage modes and some of them are 'interactive' (e.g. matrix compression failure) so factors like the relative proportion of hydrostatic and deviatoric stress components can be accounted for. This is very much in contrast to Tsai-Hill or -Wu type approaches and I'd be surprised if any researchers here would seriously suggest that criteria for composites that can't discriminate between various observed failure modes would be a worthwhile matter to research.

### RE: Tsai-Wu - continuous analytic function?

I think you may have misinterpreted what I said/meant about the Tsai-Wu criterion. Or perhaps I was not clear. I was indicating that Tsai-Wu tends to be a popular criterion in academia, at least in the US. This is one reason why people frequently ask about it. So in that sense, it should be addressed. It also has some value from a historical perspective and helps to facilitate an understanding of shortcomings of ply level criterion. I would not recommend it be used as a practical approach. That said, because of the confusion it creates, I would not be disappointed to see it removed from being taught in academia. People tend to get the impression that an improved ply level criterion is the answer, when it will never be the answer.

Moving beyond that to something like Puck, Larc04, Larc05 gives a more physically consistent result. But even these improved criterion tend to be academic unless you are strictly interested in unnotched strength. For most practical structures, you usually consider open hole strength, filled hole strength (function of installation torque and clearance, etc), the hole size effect, BVID, bearing-bypass interaction, in-situ vs ex-situ effects, scatter, processing effects, stacking sequence effects, etc. Once you consider the practical effects, the failure mechanisms (especially at the local and microscopic level) become too much for the criterion that attempt to physically capture all of these mechanisms. Instead, you think about an approach that works with OHT, OHC, FHT, FHC, CAI, etc. coupon data.

### RE: Tsai-Wu - continuous analytic function?

adfergusson, I reread what you said. I think the confusion is that we have had two different definitions of "academic". I considered academia to be what is taught in schools, since that is the experience of a large majority of practicing engineers. Alternatively, if you consider academia to be the research aspect, that is a different story. No one uses Tsai-Wu for that. Puck, Larc04, Larc05 make more sense from that perspective. There is also SIFT and MCT. But the real problem is when you move beyond the simple case of unnotched strength (as previously stated). None of them really work well for that.

### RE: Tsai-Wu - continuous analytic function?

In the past (about twenty years ago) we have had a method based on Tsai-Wu approved for use in a check stress. It had restrictions on the layup and the usual fudge factors to make it (probably) conservative. This superseded a truncated max strain method and was thought to have some advantage. However, that was then. More recent basis for MS/RF is laminate-based either using extensive compression after impact testing and/or using a laminate compressive strain allowable based on open hole compression tests. We might use Tsai-Wu with open hole data for an initial rough sizing situation where there's limited data and a combination of applied 1, 2 and 12-directions present, but we'd always use max stress/strain as well.

As to why it might be preferred in some situations by some people it may boil down to the fact that it's easy to implement without a huge amount of testing and gives a nice simple single number. However, that also makes it fairly inaccurate and uninformative—you can separate out the different terms that get added up for a clue as to what loading is contributing most to the final number, but that's not much different from checking with max stress/strain. Use at your peril (as adfergusson and Brian Esp have observed, generating any margin on composite is fairly perilous and keeping it simple with max stress/strain is often best for preliminary work).

### RE: Tsai-Wu - continuous analytic function?

(OP)
Thanks all for your detailed responses. Very helpful!

Andrew

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