The Tsai–Hill failure criterion, a rather clinical term for what amounts to predicting when something decides to cease being functional, is one of those phenomenological material failure theories that engineers apparently find useful. It's particularly fond of anisotropic composite materials, which, if you haven't gathered, are the kind of substances that possess different strengths depending on the direction you prod them. Apparently, they can be robust in one direction and disappointingly fragile in another. This criterion, in its infinite wisdom, predicts failure when a certain "failure index" within a laminate reaches the rather definitive number of 1. Anything less, and it presumably carries on. Anything more, and it's a done deal.

Tsai–Hill failure criterion in plane stress

The Tsai–Hill criterion is, at its core, an energy theory. It’s not a simple add-them-up kind of calculation; it involves interactions between stresses, which is to say, the forces in different directions don't operate in isolation. They conspire. Ply rupture, the point where a layer of the composite gives up the ghost, is predicted when the following rather imposing equation is met:

$$\left({\frac {\sigma _{11}}{X_{11}}}\right)^{2}-\left({\frac {\sigma _{11}\sigma _{22}}{X_{11}^{2}}}\right)+\left({\frac {\sigma _{22}}{X_{22}}}\right)^{2}+\left({\frac {\tau _{12}}{S_{12}}}\right)^{2}\geq 1$$

Let's break down this delightful arrangement of symbols, shall we?

• $X_{11}$: This represents the allowable strength of the ply in the longitudinal direction. Think of it as the material's polite refusal to break when you push it along its grain. The "0° direction," if you're being precise.
• $X_{22}$: This is the allowable strength of the ply in the transversal direction. The strength in the direction that cuts across the grain. It’s often, though not always, considerably less than $X_{11}$.
• $S_{12}$: This is the allowable in-plane shear strength. It deals with the strength when forces are trying to slide adjacent layers of the material past each other, a rather messy business. It’s measured between the longitudinal and transversal directions.

The Tsai–Hill criterion is what they call "interactive." This means the stresses in different directions don't just sit there independently; they actively influence each other's contribution to potential failure. It’s a bit like a poorly managed office where everyone’s bad mood affects everyone else’s productivity.

Furthermore, it’s a failure mode independent criterion. It tells you that failure will occur, but not how. It won't predict whether the material will snap, tear, delaminate, or simply disintegrate into a cloud of regret. This is in contrast to criteria like the Hashin criterion or the Puck failure criterion, which attempt to distinguish between different types of failure. Knowing the how can be rather important, mind you. Some ways of failing are considerably more catastrophic than others. It’s like knowing you’re going to be late for a meeting versus knowing you’re going to be late because your car exploded. The latter is generally considered more critical.