Browsing by Author "Lazzaroni, Giuliano"

Now showing 1 - 14 of 14
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    Analysis of a dynamic peeling test with speed-dependent toughness
    (2017) Lazzaroni, Giuliano; Nardini, Lorenzo; Mathematics
    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.
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    A bridging mechanism in the homogenisation of brittle composites with soft inclusions
    (SISSA, 2015) Barchiesi, Marco; Lazzaroni, Giuliano; Zeppieri, Caterina Ida
    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.
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    Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue
    (SISSA, 2016-07-19) Crismale, Vito; Lazzaroni, Giuliano; Orlando, Gianluca; Mathematics
    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 rst step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main di culty 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
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    Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
    (2017-01) Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    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.
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    Existence and uniqueness of dynamic evolutions for a peeling test in dimension one
    (SISSA, 2016) Dal Maso, Gianni; Lazzaroni, Giuliano; Nardini, Lorenzo; Mathematics
    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.
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    Interactions beyond nearest neighbours and rigidity of discrete energies: a compactness result and an application to dimension reduction
    (2016) Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    We analyse the rigidity of discrete energies where at least nearest and nextto- 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 surface-scaled energy and we give bounds on its possible Gamma-limit. In the second part of the paper we follow the approach developed in the first part to study a discrete model for (possibly heterogeneous) nanowires. In the heterogeneous case, by applying the compactness result shown in the first part of the paper, 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.
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    Linearisation of multiwell energies
    (2017-06) Alicandro, Roberto; Dal Maso, Gianni; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    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.
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    On the 1D wave equation in time-dependent domains and the problem of debond initiation
    (SISSA, 2017) Lazzaroni, Giuliano; Nardini, Lorenzo; Mathematics
    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.
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    On the effect of interactions beyond nearest neighbours on non-convex lattice systems
    (2017-01) Alicandro, Roberto; Lazzaroni, Giuliano; Palombaro, Mariapia; Mathematics
    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.
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    Quasistatic crack growth based on viscous approximation: a model with branching and kinking
    (2016) Crismale, Vito; Lazzaroni, Giuliano; Mathematics
    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.
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    Rate-independent damage in thermo-viscoelastic materials with inertia
    (SISSA, 2014-10-15) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, Rodica
    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.
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    Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires
    (SISSA, 2015-01) Lazzaroni, Giuliano; Palombaro, Mariapia; Schlomerkemper, Anja
    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
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    Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics
    (SISSA, 2014-10) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, Rodica
    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.
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    Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model
    (SISSA, 2015) Crismale, Vito; Lazzaroni, Giuliano
    Employing the technique of vanishing viscosity and time rescaling, we show the exis- tence 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.