In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

%B Mathematical Models and Methods in Applied Sciences %V 28 %P 1371-1412 %G eng %U https://doi.org/10.1142/S0218202518500379 %R 10.1142/S0218202518500379 %0 Journal Article %J Annali di Matematica Pura ed Applicata (1923 -) %D 2017 %T Globally stable quasistatic evolution for strain gradient plasticity coupled with damage %A Vito Crismale %XWe consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin–Anand formulation (J Mech Phys Solids 53:1624–1649, 2005). The aim of the present model is to account for different phenomena: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material's degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the so-called energetic formulation). Furthermore, we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic damage model studied in Crismale (ESAIM Control Optim Calc Var 22:883-912, 2016).

%B Annali di Matematica Pura ed Applicata (1923 -) %V 196 %P 641–685 %8 Apr %G eng %U https://doi.org/10.1007/s10231-016-0590-7 %R 10.1007/s10231-016-0590-7 %0 Journal Article %J Nonlinear Differential Equations and Applications NoDEA %D 2017 %T Quasistatic crack growth based on viscous approximation: a model with branching and kinking %A Vito Crismale %A Giuliano Lazzaroni %XEmploying the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking.

%B Nonlinear Differential Equations and Applications NoDEA %V 24 %P 7 %8 Jan %G eng %U https://doi.org/10.1007/s00030-016-0426-6 %R 10.1007/s00030-016-0426-6 %0 Journal Article %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %T Globally stable quasistatic evolution for a coupled elastoplastic–damage model %A Vito Crismale %XWe show the existence of globally stable quasistatic evolutions for a rate-independent material model with elastoplasticity and incomplete damage, in small strain assumptions. The main feature of our model is that the scalar internal variable which describes the damage affects both the elastic tensor and the plastic yield surface. It is also possible to require that the history of plastic strain up to the current state influences the future evolution of damage.

%B ESAIM: Control, Optimisation and Calculus of Variations %I EDP Sciences %V 22 %P 883–912 %G eng %U https://www.esaim-cocv.org/articles/cocv/abs/2016/03/cocv150037/cocv150037.html %R 10.1051/cocv/2015037 %0 Thesis %D 2016 %T Some results on quasistatic evolution problems for unidirectional processes %A Vito Crismale %I SISSA %G en %1 35522 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by vcrisma@sissa.it (vcrisma@sissa.it) on 2016-09-18T20:26:55Z No. of bitstreams: 1 TesiPHD.pdf: 1721272 bytes, checksum: 95ef69bafc860616d3cdb08901da7ffb (MD5) %0 Journal Article %J Calculus of Variations and Partial Differential Equations %D 2016 %T Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model %A Vito Crismale %A Giuliano Lazzaroni %XEmploying the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions for elastoplastic materials with incomplete damage affecting both the elastic tensor and the plastic yield surface, in a softening framework and in small strain assumptions.

%B Calculus of Variations and Partial Differential Equations %V 55 %P 17 %8 Jan %G eng %U https://doi.org/10.1007/s00526-015-0947-6 %R 10.1007/s00526-015-0947-6