Hi,
FEniCS is a general purpose finite-element solver. What you are asking is really a constitutive material question. Implementing non-linear constitutive material laws is not straightforward from scratch in FEniCS.
For this purpose, we worked to link FEniCS with the constitutive law code generator MFront, the corresponding module is described here along with some documented demos.
What type of constitutive law are you investigating ?