Integral Representation for Functionals Defined on SBDp in Dimension Two

We analyse integral representation and Γ -convergence properties of functionals defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for Γ -convergence and a careful adaption of the global method for relaxation [17,18] to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.2010 Mathematics Subject Classification. 49J45, 49Q20, 70G75, 74R10.

Section: Introductionmentioning

confidence: 94%

Section: Functionalsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

Friedrich

Solombrino

2020

“…Hence in a second step, we shall further introduce "boundary good cubes". On these we can clean up the jump before mollification using a construction due to [16], which we only know to hold true in dimension 2. The details are found in Subsection 3.2.…”

Section: A General Proof In Dimensionmentioning

confidence: 99%

Approximation of a Brittle Fracture Energy with a Constraint of Non-interpenetration

Chambolle

Conti

Francfort

2017

Self Cite

Abstract. Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over a SBD type space. The corresponding functional can in turn be approximated in the sense of Γ-convergence by a sequence of functionals involving a phase field as well as the displacement field. We show that a similar approximation persists if additionally imposing a non-interpenetration constraint in the minimization, namely that only nonnegative normal jumps should be permissible.2010 Mathematics subject classification: 26A45

A Density Result in GSBDp with Applications to the Approximation of Brittle Fracture Energies

Chambolle

Crismale

2018

105

We prove that any function in GSBD p (Ω), with Ω a n-dimensional open bounded set with finite perimeter, is approximated by functions u k ∈ SBV (Ω; R n ) ∩ L ∞ (Ω; R n ) whose jump is a finite union of C 1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies´Ω W (e(u)) dx + H n−1 (Ju), e(u) and Ju being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1. The difference between u k and u is small in L p outside a sequence of sets E k ⊂ Ω whose measure tends to 0 and if |u| r ∈ L 1 (Ω) with r ∈ (0, p], then |u k −u| r → 0 in L 1 (Ω). Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce Γ-convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in L 1 (Ω; R n ).