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 variation functions, and we qualitatively discuss their form by investigating the corresponding Euler-Lagrange equations. We show that, for sufficiently large values of bar elongation, minimizers of the energy are discontinuous and, most of all, the non-local term [u](x i )ρ(x-x i ) influences the relative position among the jump points, a finding that is of crucial importance to reproduce the experimental evidence.