Singularities of homogeneous deformations in finite elasticity

Is classification of the singularities of the potential, concerning the homogeneous deformations in Finite Elasticity, an important material property? The present study demonstrates that the answer to the question is positive. Since the type of singularity prescribes Maxwell's sets in the neighborhood of a singularity, the emergence of multiphase strain states depends on the performed classification. Bifurcation analysis and singularity classification for homogeneous deformations of any hyperelastic, anisotropic, homogeneous material under any type of conservative quasi-static loading is performed. Critical conditions for branching of the equilibrium paths are defined and their post-critical behavior is studied. Singularities corresponding to simple and compound branching are classified according to Arnold's classification. Unconstrained materials are only considered. Special attention is given to the compound branching $(D_k)$ cases requiring elaborate mathematical techniques. The geometry of the umbilics is described with the equilibrium paths and the cusp lines. With the help of Maxwell's sets in the neighborhood of the umbilics, the emergence of discontinuous strain fields is demonstrated. The theory is applied to orthotropic and transversely isotropic materials as well. The proposed method may directly be applied to crystals stability problems.