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.
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.
We obtain a cohesive fracture model as -limit, as ε → 0, of scalar damage models in which the elastic coefficient is computed from the damage variable v through a function f ε of the form f ε (v) = min{1, ε 1 2 f (v)}, with f diverging for v 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 s at small values of s and has a finite limit as s → ∞. If in addition the function f is allowed to depend on the parameter ε, 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.
Abstract.We provide an approximation result in the sense of Γ-convergence for energies of the form Ω Q 1 (e(u)) dx, where Ω ⊂ R n is a bounded open set with Lipschitz boundary, Q 0 and Q 1 are coercive quadratic forms on M n×n sym , a, b are positive constants, and u runs in the space of fields SBD 2 (Ω); i.e., it's a special field with bounded deformation such that its symmetric gradient e(u) is square integrable, and its jump set Ju has finite (n − 1)-Hausdorff measure in R n . The approximation is performed by means of Ambrosio-Tortorellitype elliptic regularizations, the prototype example being where (u, v) ∈ H 1 (Ω, R n )×H 1 (Ω), ε ≤ v ≤ 1, and γ > 0.
Building upon the recent results in [15] we provide a thorough description of the free boundary for solutions to the fractional obstacle problem in R n+1 with obstacle function ϕ (suitably smooth and decaying fast at infinity) up to sets of null H n−1 measure. In particular, if ϕ is analytic, the problem reduces to the zero obstacle case dealt with in [15] and therefore we retrieve the same results:(i) local finiteness of the (n − 1)-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) H n−1 -rectifiability of the free boundary, (iii) classification of the frequencies and of the blow-ups up to a set of Hausdorff dimension at most (n − 2) in the free boundary. Instead, if ϕ ∈ C k+1 (R n ), k ≥ 2, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function ϕ is less than k + 1.2010 Mathematics Subject Classification. Primary 35R35, 49Q20.
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