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/S021820251850037900988nas a2200157 4500008004100000245010900041210006900150260001500219300001400234490000700248520038700255100002100642700002200663700001900685856012600704 2017 eng d00aLower semicontinuity of a class of integral functionals on the space of functions of bounded deformation0 aLower semicontinuity of a class of integral functionals on the s bDe Gruyter a183–2070 v103 aWe study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to $L^1$ convergence.

1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttps://www.math.sissa.it/publication/lower-semicontinuity-class-integral-functionals-space-functions-bounded-deformation00965nas a2200169 4500008004100000022001400041245012900055210006900184260000800253300000700261490000700268520041200275100002100687700002200708700001900730856004600749 2016 eng d a1432-083500aFracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case0 aFracture models for elastoplastic materials as limits of gradien cApr a450 v553 aWe study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $\varepsilon$, which forces damage concentration on regions of codimension one. We determine the $\Gamma$-limit as $\varepsilon$ tends to zero and show that it contains an energy term involving the crack opening.

1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttps://doi.org/10.1007/s00526-016-0981-z01118nas a2200145 4500008004100000245013100041210006900172260001000241520052100251653010200772100002100874700002200895700001900917856003600936 2014 en d00aLaplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length0 aLaplace equation in a domain with a rectilinear crack higher ord bSISSA3 aWe consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.

10acracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttp://hdl.handle.net/1963/7271