On type-2 twins in crystals

Abstract

In this article I propose an extension of results of Ericksen and Gurtin on twinning in elastic crystals. Ericksen considered an example arising in cubic-tetragonal phase transitions. Gurtin produced a compatibility theorem under the assumption that the material symmetry group of the crystal be a subgroup of the orthogonal group. Roughly, the condition for twinning is QSF = FH, where F is the deformation gradient, S = 1 + a ⊗ n is a simple shear, Q is orthogonal and such that Q2 = 1, and H is an element of . When H is orthogonal, Gurtin showed that the condition above had two classes of solutions. Roughly, in the first class the axis of Q has the direction of n, whereas it has the direction of a in the second class. Here I show that, when H belongs to the group G proposed by Ericksen, and G contains nonorthogonal tensors, then the compatibility theorem holds if either H2 = 1 and Q is orthogonal, or Q satisfies in addition the condition Q2 = 1 and H has finite order. I show by an example that in the latter case the condition on the order of H is necessary. In the former case, Ericksen showed that the condition on H was also necessary. Lastly, I show that the two solutions for twinning under the aforementioned hypotheses are what crystallographers call conjugate twins.