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.

1 aCrismale, Vito1 aLazzaroni, Giuliano1 aOrlando, Gianluca uhttps://doi.org/10.1142/S021820251850037901385nas a2200145 4500008004100000022001400041245009300055210006900148260000800217300001400225490000800239520092700247100001901174856004601193 2017 eng d a1618-189100aGlobally stable quasistatic evolution for strain gradient plasticity coupled with damage0 aGlobally stable quasistatic evolution for strain gradient plasti cApr a641–6850 v1963 aWe 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).

1 aCrismale, Vito uhttps://doi.org/10.1007/s10231-016-0590-700824nas a2200157 4500008004100000022001400041245009600055210006900151260000800220300000600228490000700234520033600241100001900577700002400596856004600620 2017 eng d a1420-900400aQuasistatic crack growth based on viscous approximation: a model with branching and kinking0 aQuasistatic crack growth based on viscous approximation a model cJan a70 v243 aEmploying 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.

1 aCrismale, Vito1 aLazzaroni, Giuliano uhttps://doi.org/10.1007/s00030-016-0426-600947nas a2200133 4500008004100000245008500041210006900126260001700195300001400212490000700226520047700233100001900710856008400729 2016 eng d00aGlobally stable quasistatic evolution for a coupled elastoplastic–damage model0 aGlobally stable quasistatic evolution for a coupled elastoplasti bEDP Sciences a883–9120 v223 aWe 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.

1 aCrismale, Vito uhttps://www.esaim-cocv.org/articles/cocv/abs/2016/03/cocv150037/cocv150037.html00428nas a2200097 4500008004100000245008000041210006900121260001000190100001900200856011100219 2016 en d00aSome results on quasistatic evolution problems for unidirectional processes0 aSome results on quasistatic evolution problems for unidirectiona bSISSA1 aCrismale, Vito uhttps://www.math.sissa.it/publication/some-results-quasistatic-evolution-problems-unidirectional-processes00786nas a2200157 4500008004100000022001400041245009300055210006900148260000800217300000700225490000700232520030000239100001900539700002400558856004600582 2016 eng d a1432-083500aViscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model0 aViscous approximation of quasistatic evolutions for a coupled el cJan a170 v553 aEmploying 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.

1 aCrismale, Vito1 aLazzaroni, Giuliano uhttps://doi.org/10.1007/s00526-015-0947-6