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.

VL - 28 UR - https://doi.org/10.1142/S0218202518500379 ER - TY - RPRT T1 - On the 1D wave equation in time-dependent domains and the problem of debond initiation Y1 - 2017 A1 - Giuliano Lazzaroni A1 - Lorenzo Nardini AB -Motivated 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.

PB - SISSA UR - http://preprints.sissa.it/handle/1963/35302 U1 - 35608 U2 - Mathematics ER - TY - RPRT T1 - Analysis of a dynamic peeling test with speed-dependent toughness Y1 - 2017 A1 - Giuliano Lazzaroni A1 - Lorenzo Nardini AB -We 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.

UR - http://preprints.sissa.it/handle/1963/35292 U1 - 35598 U2 - Mathematics ER - TY - RPRT T1 - Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours Y1 - 2017 A1 - Roberto Alicandro A1 - Giuliano Lazzaroni A1 - Mariapia Palombaro AB - We 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. UR - http://urania.sissa.it/xmlui/handle/1963/35269 U1 - 35575 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - On the effect of interactions beyond nearest neighbours on non-convex lattice systems Y1 - 2017 A1 - Roberto Alicandro A1 - Giuliano Lazzaroni A1 - Mariapia Palombaro AB - We 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. UR - http://urania.sissa.it/xmlui/handle/1963/35268 U1 - 35574 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Linearisation of multiwell energies Y1 - 2017 A1 - Roberto Alicandro A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni A1 - Mariapia Palombaro AB - Linear 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. UR - http://preprints.sissa.it/handle/1963/35288 U1 - 35594 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Quasistatic crack growth based on viscous approximation: a model with branching and kinking JF - Nonlinear Differential Equations and Applications NoDEA Y1 - 2017 A1 - Vito Crismale A1 - Giuliano Lazzaroni AB -Employing 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.

VL - 24 UR - https://doi.org/10.1007/s00030-016-0426-6 ER - TY - RPRT T1 - Existence and uniqueness of dynamic evolutions for a peeling test in dimension one Y1 - 2016 A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni A1 - Lorenzo Nardini AB -In 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.

PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/35168 U1 - 35449 U2 - Mathematics ER - TY - JOUR T1 - On the quasistatic limit of dynamic evolutions for a peeling test in dimension one Y1 - 2016 A1 - Giuliano Lazzaroni A1 - Lorenzo Nardini AB -The 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.

UR - http://urania.sissa.it/xmlui/handle/1963/35260 U1 - 35564 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model JF - Calculus of Variations and Partial Differential Equations Y1 - 2016 A1 - Vito Crismale A1 - Giuliano Lazzaroni AB -Employing 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.

VL - 55 UR - https://doi.org/10.1007/s00526-015-0947-6 ER - TY - RPRT T1 - A bridging mechanism in the homogenisation of brittle composites with soft inclusions Y1 - 2015 A1 - Marco Barchiesi A1 - Giuliano Lazzaroni A1 - Caterina Ida Zeppieri AB - We 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7492 U1 - 7621 ER - TY - RPRT T1 - Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires Y1 - 2015 A1 - Giuliano Lazzaroni A1 - Mariapia Palombaro A1 - Anja Schlomerkemper AB - In 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 large PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7494 U1 - 7623 ER - TY - RPRT T1 - Rate-independent damage in thermo-viscoelastic materials with inertia Y1 - 2014 A1 - Giuliano Lazzaroni A1 - Riccarda Rossi A1 - Marita Thomas A1 - Rodica Toader AB - We 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7444 U1 - 7542 ER - TY - RPRT T1 - Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics Y1 - 2014 A1 - Giuliano Lazzaroni A1 - Riccarda Rossi A1 - Marita Thomas A1 - Rodica Toader AB - This 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7463 U1 - 7566 ER - TY - JOUR T1 - Some remarks on the viscous approximation of crack growth JF - Discrete Contin. Dyn. Syst. Ser. S Y1 - 2013 A1 - Giuliano Lazzaroni A1 - Rodica Toader KW - Variational models AB -We 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.

PB - SISSA VL - 6 UR - http://hdl.handle.net/1963/4206 U1 - 3945 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Crack growth with non-interpenetration : a simplified proof for the pure Neumann problem JF - Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1219-1231 Y1 - 2011 A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni AB - We 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. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3801 U1 - 526 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Energy release rate and stress intensity factor in antiplane elasticity JF - Journal de Mathematiques Pures et Appliquees 95 (2011) 565-584 Y1 - 2011 A1 - Giuliano Lazzaroni A1 - Rodica Toader AB - In 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. PB - Elsevier UR - http://hdl.handle.net/1963/3780 U1 - 546 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION JF - {MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES} Y1 - 2011 A1 - Giuliano Lazzaroni A1 - Rodica Toader KW - Brittle fracture KW - Crack propagation KW - energy derivative KW - energy release rate KW - free-discontinuity problems KW - Griffith's criterion KW - local minimizers KW - stress intensity factor} KW - vanishing viscosity KW - {Variational models AB -{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.}

PB - {WORLD SCIENTIFIC PUBL CO PTE LTD} CY - {5 TOH TUCK LINK, SINGAPORE 596224, SINGAPORE} VL - {21} ER - TY - JOUR T1 - Quasistatic crack growth in finite elasticity with Lipschitz data JF - {ANNALI DI MATEMATICA PURA ED APPLICATA} Y1 - 2011 A1 - Giuliano Lazzaroni KW - Brittle fracture KW - Crack propagation KW - Energy minimization KW - Finite elasticity KW - free-discontinuity problems KW - Griffith's criterion KW - Non-interpenetration} KW - Polyconvexity KW - Quasistatic evolution KW - Rate-independent processes KW - {Variational models AB -{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.}

PB - {SPRINGER HEIDELBERG} CY - {TIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY} VL - {190} ER - TY - JOUR T1 - Quasistatic crack growth in finite elasticity with non-interpenetration JF - Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290 Y1 - 2010 A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni AB -We 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.

UR - http://hdl.handle.net/1963/3397 U1 - 935 U2 - Mathematics U3 - Functional Analysis and Applications ER -