Browsing by Author "Toader, Rodica"
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Item An artificial viscosity approach to quasistatic crack growth(2006-07-26T10:13:39Z) Toader, Rodica; Zanini, Chiara; Mathematics; Functional Analysis and ApplicationsWe introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\epsilon$-gradient flow of the energy functional, as the "viscosity" parameter $\epsilon$ tends to zero.Item Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition(2015-11-18) Dal Maso, Gianni; Larsen, Christopher J.; Toader, Rodica; MathematicsThere are very few existence results for fracture evolution, outside of globally minimizing quasi-static evolutions. Dynamic evolutions are particularly problematic, due to the difficulty of showing energy balance, as well as of showing that solutions obey a maximal dissipation condition, or some similar condition that prevents stationary cracks from always being solutions. Here we introduce a new weak maximal dissipation condition and show that it is compatible with cracks constrained to grow smoothly on a smooth curve. In particular, we show existence of dynamic fracture evolutions satisfying this maximal dissipation condition, subject to the above smoothness constraints, and exhibit explicit examples to show that this maximal dissipation principle can indeed rule out stationary cracks as solutions.Item Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition(2018-03) Dal Maso, Gianni; Larsen, Christopher J.; Toader, Rodica; MathematicsWe consider a model of elastodynamics with fracture evolution, based on energy-dissipation balance and a maximal dissipation condition. We prove an existence result in the case of planar elasticity with a free crack path, where the maximal dissipation condition is satisfied among suitably regular competitor cracks.Item Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case(SISSA, 2015-04) Dal Maso, Gianni; Orlando, Gianluca; Toader, Rodica; MathematicsWe study the asymptotic behavior of a variational model for damaged elastoplastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter " , which forces damage concentration on regions of codimension one. We determine the -limit as " tends to zero and show that it contains an energy term involving the crack opening.Item Homogenisation problems for free discontinuity functionals with bounded cohesive surface terms(2023-07-11) Dal Maso, Gianni; Toader, Rodica; mathematicsWe study stochastic homogenisation problems for free discontinuity func- tionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimum problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.Item Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation(SISSA, 2015-09-24) Dal Maso, Gianni; Orlando, Gianluca; Toader, Rodica; MathematicsWe study the lower semicontinuity of some free discontinuity functionals, whose volume term depends on the Euclidean norm of the symmetrized gradient.Item A lower semicontinuity result for a free discontinuity functional with a boundary term(2015-12-15) Almi, Stefano; Dal Maso, Gianni; Toader, Rodica; MathematicsWe study the lower semicontinuity in $GSBV^{p}(\Om;\R^{m})$ of a free discontinuity functional~$\F(u)$ that can be written as the sum of a crack term, depending only on the jump set~$S_{u}$, and of a boundary term, depending on the trace of~$u$ on~$\partial\Om$. We give sufficient conditions on the integrands for the lower semicontinuity of~$\F$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of~$\F$ can be represented by the sum of two integrals on~$S_{u}$ and~$\partial\Om$, respectively.Item A new space of generalised functions with bounded variation motivated by fracture mechanics(2022-01-11) Dal Maso, Gianni; Toader, Rodica; mathematicsWe introduce a new space of generalised functions with bound ed variation to prove the existence of a solution to a minimum problem that arises in the variational approach to fracture mechanics in elasto plastic materials. We study the fine properties of the functions belonging to this space and prove a compactness result. In order to use the Direct Method of the Calculus of Variations we prove a lower semicontinuity result for the functional occurring in this minimum problem. Moreover, we adapt a nontrivial argument introduced by Friedrich to show that every minimizing sequence can be modified to obtain a new minimizing sequence that satisfies the hypotheses of our compactness result.Item On the Cauchy problem for the wave equation on time-dependent domains(SISSA, 2018-04) Dal Maso, Gianni; Toader, Rodica; MathematicsWe introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous de nition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.Item On the pure jump nature of crack growth for a class of pressure-sensitive elasto-plastic materials(2021-01-19) Dal Maso, Gianni; Toader, Rodica; mathematicsIn the framework of a model for the quasistatic crack growth in pressure sensitive elasto-plastic materials in the planar case, we study the properties of the length l (t) of the crack as a function of time. We prove that, under suitable technical assumptions on the crack path, the monotone function l is a pure jump function.Item Rate-independent damage in thermo-viscoelastic materials with inertia(SISSA, 2014-10-15) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, RodicaWe 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.Item Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics(SISSA, 2014-10) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, RodicaThis 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.Item VISCO-ENERGETIC SOLUTIONS FOR A MODEL OF CRACK GROWTH IN BRITTLE MATERIALS(2021) Dal Maso, Gianni; Rossi, Riccarda; Savarè, Giuseppe; Toader, Rodica; mathematicsVisco-energetic solutions have been recently advanced as a new solution concept for rate-indepen dent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introduced by Almgren, Taylor, and Wang in the study of mean curvature flows. With our main result we prove the existence of a visco-energetic solution with a given initial crack. We also show that, if the cracks have a finite number of tips evolving smoothly on a given time interval, visco-energetic solutions comply with Griffith’s criterion.Item Γ-CONVERGENCE AND INTEGRAL REPRESENTATION FOR A CLASS OF FREE DISCONTINUITY FUNCTIONALS(2023-05-05) Dal Maso, Gianni; Toader, Rodica; mathematicsWe study the Γ -limits of sequences of free discontinuity functionals with linear growth, assuming that the surface energy density is bounded. We determine the relevant properties of the Γ -limit, which lead to an integral representation result by means of integrands obtained by solving some auxiliary minimum problems on small cubes.