scholarly journals Which special functions of bounded deformation have bounded variation?

2017 ◽  
Vol 148 (1) ◽  
pp. 33-50 ◽  
Author(s):  
Sergio Conti ◽  
Matteo Focardi ◽  
Flaviana Iurlano
Keyword(s):  
Bounded Variation ◽  
Special Functions ◽  
Functions Of Bounded Variation ◽  
Finite Area ◽  
Piecewise Affine ◽  
Positive Side ◽  
Almost Everywhere ◽  
Functions Of Bounded Deformation ◽  
Additional Regularity ◽  
Jump Set

Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.

scholarly journals A Korn-type inequality in SBD for functions with small jump sets

2017 ◽  
Vol 27 (13) ◽  
pp. 2461-2484 ◽  
Author(s):  
Manuel Friedrich
Keyword(s):  
Bounded Variation ◽  
Elastic Strain ◽  
Special Functions ◽  
Type Inequality ◽  
Rigid Motion ◽  
Exceptional Set ◽  
Finite Perimeter ◽  
Functions Of Bounded Deformation ◽  
Jump Set

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in [Formula: see text] with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in [Formula: see text] has bounded variation away from an arbitrarily small part of the domain.

scholarly journals Approximation of fracture energies with p-growth via piecewise affine finite elements

2019 ◽  
Vol 25 ◽  
pp. 34 ◽  
Author(s):  
Sergio Conti ◽  
Matteo Focardi ◽  
Flaviana Iurlano
Keyword(s):  
Finite Union ◽  
Generalized Functions ◽  
Lipschitz Continuous ◽  
Piecewise Affine ◽  
Functions Of Bounded Deformation ◽  
Affine Functions ◽  
Space Of Generalized Functions ◽  
Piecewise Affine Functions ◽  
Jump Set ◽  
Density Results

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBDp functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C1 hypersurfaces. The strains of the approximating functions converge strongly in Lp to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudenthal partition of a cubic grid.

Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials

2008 ◽  
Vol 138 (5) ◽  
pp. 1019-1041 ◽  
Author(s):  
Alessandro Giacomini ◽  
Marcello Ponsiglione
Keyword(s):  
Fracture Mechanics ◽  
Weak Convergence ◽  
Bounded Variation ◽  
Special Functions ◽  
Brittle Materials ◽  
Functions Of Bounded Variation ◽  
Variational Models ◽  
Stability Results

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

scholarly journals Integral representation for energies in linear elasticity with surface discontinuities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Vito Crismale ◽  
Manuel Friedrich ◽  
Francesco Solombrino
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Type Inequality ◽  
Representation Formula ◽  
Functions Of Bounded Variation ◽  
General Strategy ◽  
Global Method ◽  
Functions Of Bounded Deformation ◽  
Fracture Damage ◽  
Surface Discontinuities

AbstractIn this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation ({\mathrm{GSBD}^{p}}) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51–98] and a recent Korn-type inequality in {\mathrm{GSBD}^{p}}, cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré–Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in {\mathrm{SBD}^{p}}, obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on \mathrm{SBD}^{p} in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337–1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation ({\mathrm{GSBV}^{p}}).

NON-LOCAL VARIATIONAL LIMITS OF DISCRETE SYSTEMS

2000 ◽  
Vol 02 (02) ◽  
pp. 285-297 ◽  
Author(s):  
ANDREA BRAIDES
Keyword(s):  
Bounded Variation ◽  
Special Functions ◽  
Discrete Systems ◽  
Functions Of Bounded Variation ◽  
Step Size ◽  
One Dimensional ◽  
Continuum Energy ◽  
Non Local ◽  
Local Term

We show that the limit of a class of discrete energies defined on one-dimensional lattices of step size ε when ε→0 define a continuum energy with a local and a non-local term with domain a subspace of the special functions of bounded variation.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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