Erik Kostson
Ansys Employee
An example would be predicting the ultimate load a structure can take before it fails (say concrete slab). If we went by the load it takes to first crack (or first plastify). When designing structure, one might also want to find out what the ultimate load is before it fails, so we want to push beyond first cracking in concrete (or initial plasticity). ALso for buckling of structure the linear buckling might give very conservative results, while in reality if we include all of the nonlinear (nl) effects (including material nl) we might find the buckling load is different than the linear buckling prediction. Another example is the designing of connections in buildings where we need to use nl material, e.g., see this example for details :

So these are just a few examples, and sure there are many more.
For the first mentioned above See an example of a nonlinear material analysis (concrete nonlinear material), where we predict the failure load of a slab (this analysis requires nonlinear material and large deflections).