Failure Criteria for anisotropic composites

You have to have a failure criterion to assess the importance of an applied stress. You can't do without one (even for metal).

Accordingly the simplest failure criterion is probably
failure_index = applied_tensile_stress / allowable_ultimate_tensile_strength
or perhaps
FI = σ_1T / Ftu

If the value of FI is greater than or equal to 1 failure is predicted. If it is less than 1 then no failure is predicted. It is that simple.

Note that for this simple failure criterion, in terms of reserve factor RF = 1 / FI (or margin of safety MS = 1/FI - 1). There's a little bit of debate because it is usualy taken that an FI of 1.0 indicates failure but an RF of 1.00 (an MS of 0.00) does not.

Tsai-Wu and Hashin are just more complicated variants of this which can be used with orthotropic materials and a complicated stress state.

Stephen Tsai and Ed Wu (deceased) worked quite hard to come up with one equation which applies to any stress state and any orthotropic material. Hashin uses several separate failure indices depending on the type of failure being checked.

For reference Tsai-Wu is
FI = σ₁ / σ_1AT - σ₁ / σ_1AC + σ₂ / σ_2AT - σ₂ / σ_2AC + σ₁² / (σ_1AT * σ_1AC) + σ₂² / (σ_2AT * σ_2AC) + τ₁₂² / τ_12A² - σ₁ * σ₂ / Sqrt(σ_1AT * σ_1AC * σ_2AT * σ_2AC)

A = allowable.

In general Tsai-Wu when used with an isotropic material and when the equation is simplified accordingly boils down to von Mises with ultimate stresses and not yield.