2,194 research outputs found
Existence of minimizers for the d stationary Griffith fracture model
We consider the variational formulation of the Griffith fracture model in two
spatial dimensions and prove existence of strong minimizers, that is
deformation fields which are continuously differentiable outside a closed jump
set and which minimize the relevant energy. To this aim, we show that
minimizers of the weak formulation of the problem, set in the function space
and for which existence is well-known, are actually strong minimizers
following the approach developed by De Giorgi, Carriero, and Leaci in the
corresponding scalar setting of the Mumford-Shah problem
Phase field approximation of cohesive fracture models
We obtain a cohesive fracture model as a -limit of scalar damage
models in which the elastic coefficient is computed from the damage variable
through a function of the form , with diverging for close to the value describing undamaged
material. The resulting fracture energy can be determined by solving a
one-dimensional vectorial optimal profile problem. It is linear in the opening
at small values of and has a finite limit as . If the
function is allowed to depend on the index , for specific choices we
recover in the limit Dugdale's and Griffith's fracture models, and models with
surface energy density having a power-law growth at small openings
Approximation of functions with possibly infinite jump set
We prove an approximation result for functions
such that is -integrable, , and is
integrable over the jump set (whose measure is possibly
infinite), for some continuous, nondecreasing, subadditive function , with
. The approximating functions are piecewise affine
with piecewise affine jump set; the convergence is that of for and
the convergence in energy for and for
suitable functions . In particular, converges to -strictly,
area-strictly, and strongly in after composition with a bilipschitz map.
If in addition , we also have convergence of
to
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