Nonlinear elastic constitutive modeling of α-Ge

We present a constitutive framework for modeling the nonlinear elastic behavior of alpha-Germanium (Ge). Starting with all possible phase changes Ge sustains under compression, we correspond to each space group of every phase the arithmetic symmetry group by viewing Ge as a multilattice. We then focus on the mother alpha phase of Ge. Confining ourselves to weak transformation neighborhoods and adopting the Cauchy-Born rule we work with the classical symmetries of continuum mechanics. Since there is no available representation theory for the symmetry group of alpha-Ge, we use integrity basis theory to find the most general expression for the arguments of the energy. Using energy's expression we evaluate the stress tensor in its most general form. We then study two problems of increasing difficulty: simple tension/compression and anti-plane shear. For the first class of problems we give closed form solutions, while the anti-plane shear problem complicates the equations dramatically; we give conditions for the solvability of the field equations ruling the shift vector.