Second-order structured deformations

Structured deformations provide a multi-scale geometrical setting for resolving the fields occurring in continuum mechanics into two parts: the part arising from smooth changes at smaller lengths scales and the part due to slips and separations (disarrangements) at smaller length scales. The portion without disarrangements is known to be associated with limits of gradients of approximating deformations, and the portion due to disarrangements corresponds to the effects of jumps in the approximating deformations. This paper extends the principal concepts and results on structured deformations to include the effects of limits of second gradients and jumps in the first gradients of approximating deformations and provides examples illustrating these effects. The resulting second-order structured deformations provide a setting for a complete description of structured motions and for the systematic treatment of mesolevel phase transitions, including the effects of jumps in first gradients on the bulk density of the Helmholtz free energy.