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/S021820251850037900999nas a2200121 4500008004100000245009100041210006900132260001000201520057300211100002400784700002100808856004800829 2017 en d00aOn the 1D wave equation in time-dependent domains and the problem of debond initiation0 a1D wave equation in timedependent domains and the problem of deb bSISSA3 aMotivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.

1 aLazzaroni, Giuliano1 aNardini, Lorenzo uhttp://preprints.sissa.it/handle/1963/3530201010nas a2200109 4500008004100000245007000041210006900111520062700180100002400807700002100831856004800852 2017 en d00aAnalysis of a dynamic peeling test with speed-dependent toughness0 aAnalysis of a dynamic peeling test with speeddependent toughness3 aWe analyse a one-dimensional model of dynamic debonding for a thin film, where the local toughness of the glue between the film and the substrate also depends on the debonding speed. The wave equation on the debonded region is strongly coupled with Griffth's criterion for the evolution of the debonding front. We provide an existence and uniqueness result and find explicitly the solution in some concrete examples. We study the limit of solutions as inertia tends to zero, observing phases of unstable propagation, as well as time discontinuities, even though the toughness diverges at a limiting debonding speed.

1 aLazzaroni, Giuliano1 aNardini, Lorenzo uhttp://preprints.sissa.it/handle/1963/3529200906nas a2200121 4500008004100000245009600041210006900137520045600206100002300662700002400685700002400709856005100733 2017 en d00aDerivation of a rod theory from lattice systems with interactions beyond nearest neighbours0 aDerivation of a rod theory from lattice systems with interaction3 aWe study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.1 aAlicandro, Roberto1 aLazzaroni, Giuliano1 aPalombaro, Mariapia uhttp://urania.sissa.it/xmlui/handle/1963/3526901090nas a2200121 4500008004100000245009000041210006900131520064600200100002300846700002400869700002400893856005100917 2017 en d00aOn the effect of interactions beyond nearest neighbours on non-convex lattice systems0 aeffect of interactions beyond nearest neighbours on nonconvex la3 aWe analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation.1 aAlicandro, Roberto1 aLazzaroni, Giuliano1 aPalombaro, Mariapia uhttp://urania.sissa.it/xmlui/handle/1963/3526801406nas a2200133 4500008004100000245004000041210004000081520101100121100002301132700002101155700002401176700002401200856004801224 2017 en d00aLinearisation of multiwell energies0 aLinearisation of multiwell energies3 aLinear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.1 aAlicandro, Roberto1 aDal Maso, Gianni1 aLazzaroni, Giuliano1 aPalombaro, Mariapia uhttp://preprints.sissa.it/handle/1963/3528800824nas 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/s00030-016-0426-600798nas a2200133 4500008004100000245008700041210006900128260001000197520034000207100002100547700002400568700002100592856005100613 2016 en d00aExistence and uniqueness of dynamic evolutions for a peeling test in dimension one0 aExistence and uniqueness of dynamic evolutions for a peeling tes bSISSA3 aIn this paper we present a one-dimensional model of a dynamic peeling test for a thin film, where the wave equation is coupled with a Griffith criterion for the propagation of the debonding front. Our main results provide existence and uniqueness for the solution to this coupled problem under different assumptions on the data.

1 aDal Maso, Gianni1 aLazzaroni, Giuliano1 aNardini, Lorenzo uhttp://urania.sissa.it/xmlui/handle/1963/3516800933nas a2200109 4500008004100000245008700041210006900128520053000197100002400727700002100751856005100772 2016 en d00aOn the quasistatic limit of dynamic evolutions for a peeling test in dimension one0 aquasistatic limit of dynamic evolutions for a peeling test in di3 aThe aim of this paper is to study the quasistatic limit of a one-dimensional model of dynamic debonding. We start from a dynamic problem that strongly couples the wave equation in a time-dependent domain with Griffith's criterion for the evolution of the domain. Passing to the limit as inertia tends to zero, we find that the limit evolution satisfies a stability condition; however, the activation rule in Griffith's (quasistatic) criterion does not hold in general, thus the limit evolution is not rate-independent.

1 aLazzaroni, Giuliano1 aNardini, Lorenzo uhttp://urania.sissa.it/xmlui/handle/1963/3526000786nas 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/s00526-015-0947-600903nas a2200133 4500008004100000245009000041210006900131260001000200520043700210100002100647700002400668700002700692856005000719 2015 en d00aA bridging mechanism in the homogenisation of brittle composites with soft inclusions0 abridging mechanism in the homogenisation of brittle composites w bSISSA3 aWe provide a homogenisation result for the energy-functional associated with a purely brittle composite whose microstructure is characterised by soft periodic inclusions embedded in a stiffer matrix. We show that the two constituents as above can be suitably arranged on a microscopic scale ε to obtain, in the limit as ε tends to zero, a homogeneous macroscopic energy-functional explicitly depending on the opening of the crack.1 aBarchiesi, Marco1 aLazzaroni, Giuliano1 aZeppieri, Caterina Ida uhttp://urania.sissa.it/xmlui/handle/1963/749201420nas a2200133 4500008004100000245009400041210006900135260001000204520094900214100002401163700002401187700002501211856005001236 2015 en d00aRigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires0 aRigidity of threedimensional lattices and dimension reduction in bSISSA3 aIn the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large1 aLazzaroni, Giuliano1 aPalombaro, Mariapia1 aSchlomerkemper, Anja uhttp://urania.sissa.it/xmlui/handle/1963/749401353nas a2200145 4500008004100000245007400041210006900115260001000184520088100194100002401075700002001099700001901119700001901138856005001157 2014 en d00aRate-independent damage in thermo-viscoelastic materials with inertia0 aRateindependent damage in thermoviscoelastic materials with iner bSISSA3 aWe present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is Independent of temperature.1 aLazzaroni, Giuliano1 aRossi, Riccarda1 aThomas, Marita1 aToader, Rodica uhttp://urania.sissa.it/xmlui/handle/1963/744400987nas a2200145 4500008004100000245008700041210006900128260001000197520050200207100002400709700002000733700001900753700001900772856005000791 2014 en d00aSome remarks on a model for rate-independent damage in thermo-visco-elastodynamics0 aSome remarks on a model for rateindependent damage in thermovisc bSISSA3 aThis note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubicek [1] with the methods from Lazzaroni/Rossi/Thomas/Toader [2] and extend the analysis to the setting of inhomogeneous time-dependent Dirichlet data.1 aLazzaroni, Giuliano1 aRossi, Riccarda1 aThomas, Marita1 aToader, Rodica uhttp://urania.sissa.it/xmlui/handle/1963/746300835nas a2200145 4500008004100000245006200041210006200103260001000165490000600175520040600181653002300587100002400610700001900634856003600653 2013 en d00aSome remarks on the viscous approximation of crack growth0 aSome remarks on the viscous approximation of crack growth bSISSA0 v63 aWe describe an existence result for quasistatic evolutions of cracks in antiplane elasticity obtained in [16] by a vanishing viscosity approach, with free (but regular enough) crack path. We underline in particular the motivations for the choice of the class of admissible cracks and of the dissipation potential. Moreover, we extend the result to a model with applied forces depending on time.

10aVariational models1 aLazzaroni, Giuliano1 aToader, Rodica uhttp://hdl.handle.net/1963/420600974nas a2200121 4500008004300000245009300043210006900136260004800205520051800253100002100771700002400792856003600816 2011 en_Ud 00aCrack growth with non-interpenetration : a simplified proof for the pure Neumann problem0 aCrack growth with noninterpenetration a simplified proof for the bAmerican Institute of Mathematical Sciences3 aWe present a recent existence result concerning the quasi-static evolution of cracks in hyperelastic brittle materials, in the frame-work of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in.1 aDal Maso, Gianni1 aLazzaroni, Giuliano uhttp://hdl.handle.net/1963/380100728nas a2200121 4500008004300000245007600043210006900119260001300188520032600201100002400527700001900551856003600570 2011 en_Ud 00aEnergy release rate and stress intensity factor in antiplane elasticity0 aEnergy release rate and stress intensity factor in antiplane ela bElsevier3 aIn the setting of antiplane linearized elasticity, we show the existence of the stress intensity factor and its relation with the energy release rate when the crack path is a C1,1 curve. Finally, we show that the energy release rate is continuous with respect to the Hausdorff convergence in a class of admissible cracks.1 aLazzaroni, Giuliano1 aToader, Rodica uhttp://hdl.handle.net/1963/378001532nas a2200277 4500008004100000022001600041245006500057210006300122260009400185300001600279490000900295520056700304653002100871653002200892653002200914653002400936653003200960653002500992653002101017653002901038653002401067653002401091100002401115700001901139856009601158 2011 eng d a{0218-2025}00aA MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION0 aMODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION a{5 TOH TUCK LINK, SINGAPORE 596224, SINGAPORE}b{WORLD SCIENTIFIC PUBL CO PTE LTD}c{OCT} a{2019-2047}0 v{21}3 a{In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.}

10aBrittle fracture10aCrack propagation10aenergy derivative10aenergy release rate10afree-discontinuity problems10aGriffith's criterion10alocal minimizers10astress intensity factor}10avanishing viscosity10a{Variational models1 aLazzaroni, Giuliano1 aToader, Rodica uhttps://www.math.sissa.it/publication/model-crack-propagation-based-viscous-approximation-001263nas a2200277 4500008004100000022001600041245007000057210006900127260008600196300001400282490001000296520028400306653002100590653002200611653002400633653002200657653003200679653002500711653002600736653001800762653002600780653003100806653002400837100002400861856010000885 2011 eng d a{0373-3114}00aQuasistatic crack growth in finite elasticity with Lipschitz data0 aQuasistatic crack growth in finite elasticity with Lipschitz dat a{TIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY}b{SPRINGER HEIDELBERG}c{JAN} a{165-194}0 v{190}3 a{We extend the recent existence result of Dal Maso and Lazzaroni (Ann Inst H Poincare Anal Non Lineaire 27:257-290, 2010) for quasistatic evolutions of cracks in finite elasticity, allowing for boundary conditions and external forces with discontinuous first derivatives.}

10aBrittle fracture10aCrack propagation10aEnergy minimization10aFinite elasticity10afree-discontinuity problems10aGriffith's criterion10aNon-interpenetration}10aPolyconvexity10aQuasistatic evolution10aRate-independent processes10a{Variational models1 aLazzaroni, Giuliano uhttps://www.math.sissa.it/publication/quasistatic-crack-growth-finite-elasticity-lipschitz-data00595nas a2200109 4500008004300000245007600043210006900119520021600188100002100404700002400425856003600449 2010 en_Ud 00aQuasistatic crack growth in finite elasticity with non-interpenetration0 aQuasistatic crack growth in finite elasticity with noninterpenet3 aWe present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking\\ninto account the non-interpenetration condition.

1 aDal Maso, Gianni1 aLazzaroni, Giuliano uhttp://hdl.handle.net/1963/3397