Second-order structured deformations: relaxation, integral   representation and applications
  Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications. 
Ana Cristina Barroso, Jos\'e Matias, Marco Morandotti, David R. Owen
math.OC