On the localisation of damage under pure bending using a nonlocal approach

To reproduce the behaviour of quasibrittle materials mathematically, constitutive laws with softening are needed, which often leads to development of damage localisation bands. If a standard local formulation is used, this approach leads to a pathological mesh dependence, which can be eliminated by making use of alternative nonlocal formulations. The aim of this paper is to assess the localisation properties of damage models under pure bending using different nonlocal formulations; to permit a partially analytical treatment, the idealised case of pure bending is studied. Under these conditions, the localisation process starts at the tensile face of the beam, which belongs to the boundary of the domain on which the problem is solved. Consequently, localisation patterns are affected by the boundary treatment as well as other parameters, such as the characteristic length that defines the area contributing to the nonlocal averaging. This paper presents an analytical study of the onset of localisation of different nonlocal formulations for a beam under pure bending. In addition to it, the subsequent evolution of the localised solution is explored by numerical simulations, analysing the localisation bands spacing, the dissipated energy profile along the fracture plane and the Moment-ϕL diagrams, with ϕL being a parameter that represents the rotation that drives the loading process (ϕ stands for the relative rotation angle of the cross section and L for the beam length). An analysis of damage localisation on longer beams where damage localises in several areas is also carried out and, finally, the damage localisation due to shrinkage is studied as a more realistic example of the problem addressed here.