Measure Theory and Fine Properties of Functions, Revised Edition

Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [4] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimilable to ductile and cohesive fracture. Here, we further expand this approach considering specific constitutive functions frequently used in phase-field models of brittle fracture. A numerical solution technique of the coupled elastodamage-plasticity problem, based on an alternate minimisation algorithm, is proposed and tested against semi-analytical results. Considering a one-dimensional traction test, we illustrate the properties of four different regimes obtained by a suitable tuning of few key constitutive parameters. Namely, depending on the relative yield stresses and softening behaviours of the plasticity and the damage criteria, we obtain macroscopic responses assimilable to (i) brittle fracturè a la Griffith, (ii) cohesive fractures of the Barenblatt or Dugdale type, and (iii) a sort of cohesive fracture including a depinning energy contribution. The comparisons between numerical and analytical results prove the accuracy of the proposed numerical approaches in the considered quasi-static time-discrete setting, but they also emphasise some subtle issues occurring during time-discontinuous evolutions.

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“…Clearly any convex function admits also a distributional Hessian D 2 D u. Recalling that a positive distribution is a measure, it is simple to show that D 2 D u is a matrix valued measure [45,Chapter 6]. Then one can show that the "pointwise" Hessian D 2 u defined in the Alexandrov theorem is actually the density of the absolutely continuous part of D 2 D u with respect to the Lebesgue measure, i.e.,…”

Section: Theorem 32 (Alexandrov)mentioning

confidence: 99%