Which special functions of bounded deformation have bounded variation?

Self Cite

We prove an integral representation result for functionals with growth conditions which give coercivity on the spaceThe space SBD p of functions whose distributional strain is the sum of an L p part and a bounded measure supported on a set of finite H 1 -dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by W 1,p functions. We also obtain a generalization of Korn's inequality in the SBD p setting.

“…For ρ <ρ we define ρ k := (2 k ρ) ∧ρ and we adopt the notation k in place of ρ k for the subscriptions. As above, using [21,Lemma 4.3] and the triangular inequality we infer…”

Section: Reflectionmentioning

confidence: 89%

Integral Representation for Functionals Defined on SBDp in Dimension Two

Conti

Focardi

Iurlano

2016

Self Cite

“…However, our primary purpose comes from the study of free-discontinuity problems defined on the space GSBD p , see [38], which has obtained steadily increasing attention over the last years, cf., e.g., [30,31,32,33,34,35,46,47,48,50]. We have indeed already mentioned before how the analysis of partition problems has proved to be a relevant tool in the study of freediscontinuity problems on SBV .…”

Section: Introductionmentioning

confidence: 94%

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

Friedrich

Solombrino

2020

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.

“…Therefore, following the argument of [27,Lemma 4.3] for the rectangles Q 0 ∩ Q i in place of B (notice that for a given i the shape of these rectangles is the same independently of r, that is the ratios between the sidelengths are independent of r) one has that for any affine function a :…”

Section: )mentioning

confidence: 99%

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 ).