On defect-preserving deformations in crystals

Equilibrium configurations of defective crystals

“…We follow the theory for slightly defective crystals proposed by DAVINI [9] and DAVINI & PARRY [10], [11]. We say that £ and Z* are elastically related if the Cauchy -Born hypothesis is satisfied,…”

“…over £>*, there is no invertible mapping u*: Cl* -»[R 3 which leads via an elastic deformation to a lattice matrix constant in u*(Q*). DAVTNI [9] and DAVINI & PARRY [10], [11] discussed subtleties related to this last idea, and in DAVINI & PARRY [11] they showed that any change of state can be factorized into an elastic deformation composed with specific types of non-elastic changes. It is natural to say that all non-elastic changes are plastic, however there are two major classes involved.…”

“…Essentially, a neutral change of state of a perfect crystal corresponds to a lattice matrix L(u(x)) = Vu(x){Vv(x)}-i, where u : Q. -»[R 3 is the elastic deformation, Cl is the reference configuration and v represents the slip or plastic deformation with det Vv = 1 a. e. in £1 ERICKSEN [13] and DAVINI & PARRY [10] have discussed the likelyhood that crystal equilibria correspond to some kind of variational principle, and offered the opinion that the relevant class of variations should encompass, at the most, the elastic changes and the rearrangements, so excluding any change of state which alters the invariants. In this paper we study the implications of including neutral changes of state in the class of admissible variations while taking the viewpoint that equilibria correspond to minimizers of an energy functional E(u, v) := Jw(Vu(x){ Vv(x)}' 1 ) dx (1.2)…”

“…Precisely, we want to judge whether or not allowing rearrangements leads to physically reasonable predictions in the context of plasticity. Thus, we consider the class of admissible pairs In DAVINI & PARRY [10] various properties of smooth minimizers were formally derived and convexity was assumed tacitly. Also, in DACOROGNA & FONSECA [7] existence and smoothness of minimizers for functionals of the type (1.2) were discussed.…”

“…In Section 2 we give a brief description of the model for defective crystals proposed by DAVINI [9] and DAVINI & PARRY [10], [11] and we recall the notion of neutral deformation.…”