Special functions with bounded variation and with weakly differentiable traces on the jump set

1998 ◽  
Vol 5 (2) ◽  
pp. 219-243 ◽  
Author(s):  
Luigi Ambrosio ◽  
Andrea Braides ◽  
Adriana Garroni
Keyword(s):  
Bounded Variation ◽  
Special Functions ◽  
Jump Set

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 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.

ON THE VARIATIONAL APPROXIMATION OF FREE-DISCONTINUITY PROBLEMS IN THE VECTORIAL CASE

2001 ◽  
Vol 11 (04) ◽  
pp. 663-684 ◽  
Author(s):  
M. FOCARDI
Keyword(s):  
Bounded Variation ◽  
Special Functions ◽  
Variational Approximation ◽  
Free Discontinuity Problems ◽  
Vector Valued

We provide a variational approximation for quasiconvex energies defined on vector valued special functions with bounded variation. We extend the Ambrosio–Tortorelli's construction to the vectorial case.

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.

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.

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