Fracture Surfaces and the Regularity of Inverses for BV Deformations

Abstract

Fracture surfaces and the regularity of inverses for BV deformations the date of receipt and acceptance should be inserted later Abstract Motivated by nonlinear elasticity theory, we study deformations that are approximately differentiable, orientation-preserving and one-to-one almost everywhere, and in addition have finite surface energy. This surface energy E was used by the authors in a previous paper, and has connections with the theory of currents. In the present paper we prove that E measures exactly the area of the surface created by the deformation. This is done through a proper definition of created surface, which is related to the set of discontinuity points of the inverse of the deformation. In doing that, we also obtain an SBV regularity result for the inverse.

Lusin's condition and the distributional determinant for deformations with finite energy

“…We thus proved existence of minimizers for the total energy In [18] we also showed that the condition E.u/ < 1 implies an SBV regularity property for the inverse of the deformation u, and, based on this result, we introduced a new notion of created surface. This paper is divided into two parts.…”

“…An important object in that study was the functional N E u (see Definition 3.1), which constitutes a generalization of the (singular part of the) distributional determinant, and provides information on the surface created by the deformation u. Just as Det Du det Du gives information on the volume and location of the cavities, we showed in [18] that N E u can be used to measure the area of the created surface in the deformed configuration, and to locate the singularities in the reference configuration that give rise to that surface. In contrast to the SBV situation, where the functional N E u provides a richer description of the fracture process than Det Du, Theorem 3.2 shows that in the Sobolev case, it is sufficient to know the structure of the distributional determinant so as to study the creation of surface.…”

“…This paper is divided into two parts. In the first part (Sections 3-5) we compare the approach of [17,18] with related work of Csörnyei, Hencl, Koskela, Malý, and Onninen (see [8,[19][20][21][22]29]) on the regularity of the inverse, and with the models for cavitation of Müller and Spector [28] and of Conti and De Lellis [7]. In particular, we show that:…”

Progress and puzzles in nonlinear elasticity

“…Thus a framework in which all these kinds of singularities can energetically compete is desirable. Recent progress in this direction, leading to a theory in which both cavitation and cracks are possible, has been made by 19].…”

A Calibration Method for Estimating Critical Cavitation Loads from Below in Three Dimensional Nonlinear Elasticity

“…This phenomenon was first analysed in the setting of hyperelasticity by Ball in [2]; since then, a large and sophisticated literature has developed, including but not limited to [15,12,9,13,14,7,8], part of which focuses on finding boundary conditions which, when obeyed by all competing deformations, ensure that cavitation does not occur. It is to the latter body of work that we contribute by considering the case of purely bulk energy…”