Interaction of fractures in tensile bars with non-local spatial dependence

Abstract

We propose to determine the displacement field u:ℐ⊂ℝ→ℝ of a 1-D bar extended in a hard device by minimizing a non-local energy functional of the type Π[u]:=∫ ℐ Uu ' (x)+1 K∑ x i ∈J u [u](x i )ρ(x-x i )dx+∑ x i ∈J u ϕ([u](x i )), where K is a material parameter, [u](x i ) denotes the jump of u at x i and J u ⊂ℐ is the set of all jump points. For appropriate choice of the bulk energy U(·), of the surface energy ϕ(·) and of the weight function ρ(·), we prove an existence theorem for minimizers in the space SBV(ℐ) of special bounded varia Show more