A relaxation result in the framework of structured deformations in a bounded variation setting

Abstract

We obtain an integral representation of an energy for structured deformations of continua in the space of functions of bounded variation, as a first step to the study of asymptotic models for thin defective crystalline structures, where phenomena as slips, vacancies and dislocations prevent the effectiveness of classical theories.

The general form of the relaxation of a purely interfacial energy for structured deformations

“…The bulk relaxed energy is shown to coincide with the subadditive envelope of the unrelaxed interfacial energy while the relaxed interfacial energy is the restriction of the envelope to rank-1 tensors. Moreover, it is shown that the minimizing sequence required to define the bulk energy in the relaxation scheme of Choksi and Fonseca (1997) can be realized in the more restrictive class required in the relaxation scheme of Baía, Matias and Santos (2012), thus establishing the equality of relaxed energies of the two approaches for general purely interfacial energies. The relaxations of the specific interfacial energies of Owen and Paroni (2015) and Barroso, Matias, Morandotti and Owen (2017) are simple consequences of our general results.…”

Second-Order Structured Deformations: Relaxation, Integral Representation and Applications

“…(2) The hypotheses listed above are similar to the ones in [12] and [7] where there is no explicit dependence on x , and with the hypotheses in [9] where the density functions depended explicitly on the variable x . (3) It is well known that the bulk energy may have potential wells and for this reason it is desirable to consider…”

“…This interpretation of I dis (g, G) is justified by considering a sequence {u n } in SBV 2 (Ω, R 3 ) with u n → g and ∇u n → G both in L 1 and by writing 9) showing that M L 3 := (∇g − G) L 3 is the absolutely continuous part of the limit of the singular measures [u n ] ⊗ ν un H 2 that capture the submacroscopic disarrangements associated with (g, G) . Moreover, the energy density (A, L) −→ W 2 (A, L) of [7] provides the remaining portion…”

“…Indeed, in a variety of settings [7,12,15,32], one can prove an approximation theorem to the effect that there exist a sequence of mappings u n : Ω → R d that converges to g and whose gradients ∇u n : Ω → R d×N converge to G. In addition, one obtains a formula that identifies the difference M := ∇g − G = ∇ lim n→∞ u n − lim n→∞ ∇u n as a limit of "disarrangements", i.e., of averages of directed jumps [u n ] ⊗ ν un in the approximating mappings (here, ν un denotes the normal to the jump-set of u n ). These disarrangements include the formation of voids, slips, and separations occuring at submacroscopic levels.…”

Structured Deformations of Continua: Theory and Applications

“…It is worth noticing that the second minimization problem in (10) is the one proposed in [3] in a different context for relaxation; the proof of the third inequality in (10) is achieved by an explicit construction and on the notion of isotropic vectors [9].…”