The Cauchy-Born hypothesis, nonlinear elasticity and mechanical twinning in crystals

The bridge between the molecular descriptions of crystalline configurations and the continuum theories of crystal mechanics such as linear and nonlinear elasticity is given by a natural hypothesis that goes back to Cauchy and Born, referred to as the 'Born rule'. This paper reports on an extensive investigation on the validity of the Born rule and the possibility of applying nonlinear elasticity to describe the behavior of crystalline solids. This is done by studying the phenomenon of mechanical twinning and its implications for the invariance group of the energy density of the crystal. The analysis leads to the conclusion that, in the 'generic' case, the Born rule does not hold and that nonlinear elasticity theory cannot provide an adequate model for crystal mechanics because an unphysical energy invariance is derived. However, the Born rule works and elasticity theory can be used for crystals whose twinning shears satisfy certain quite restrictive 'nongeneric' conditions. Relevant experimental data confirm these theoretical negative conclusions. It is remarkable that two very important classes of 'nongeneric' materials to which an elastic model safely applies do emerge experimentally: shape-memory alloys and materials whose crystalline structure is given by a simple Bravais lattice.