@article {2018, title = {On the Cauchy problem for the wave equation on time-dependent domains}, number = {SISSA;19/2018/MATE}, year = {2018}, institution = {SISSA}, abstract = {We 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.}, url = {http://preprints.sissa.it/handle/1963/35314}, author = {Gianni Dal Maso and Rodica Toader} } @article {2018, title = {Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition}, number = {SISSA;04/2018/MATE}, year = {2018}, abstract = {We 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.}, url = {http://preprints.sissa.it/handle/1963/35308}, author = {Gianni Dal Maso and Cristopher J. Larsen and Rodica Toader} } @article {dal2017lower, title = {Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation}, journal = {Advances in Calculus of Variations}, volume = {10}, number = {2}, year = {2017}, pages = {183{\textendash}207}, publisher = {De Gruyter}, abstract = {

We 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.

}, doi = {10.1515/acv-2015-0036}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2015, title = {A lower semicontinuity result for a free discontinuity functional with a boundary term}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {108}, year = {2017}, pages = {952-990}, chapter = {952}, abstract = {

We study the lower semicontinuity in $GSBV^{p}(\Omega;\mathbb{R}^{m})$ of a free discontinuity functional $\mathcal{F}(u)$ 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 $\partial\Omega$. We give sufficient conditions on the integrands for the lower semicontinuity of $\mathcal{F}$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of $\mathcal{F}$ can be represented by the sum of two integrals on $S_{u}$ and $\partial\Omega$, respectively.

}, doi = {10.1016/j.matpur.2017.05.018}, url = {http://hdl.handle.net/20.500.11767/15979}, author = {Stefano Almi and Gianni Dal Maso and Rodica Toader} } @article {DalMaso2016, title = {Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {55}, number = {3}, year = {2016}, month = {Apr}, pages = {45}, abstract = {

We 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.

}, issn = {1432-0835}, doi = {10.1007/s00526-016-0981-z}, url = {https://doi.org/10.1007/s00526-016-0981-z}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2015, title = {Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition}, year = {2015}, abstract = {There 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/35045}, author = {Gianni Dal Maso and Cristopher J. Larsen and Rodica Toader} } @article {2014, title = {Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length}, number = {Nonlinear Differential Equations and Applications}, year = {2014}, publisher = {SISSA}, abstract = {

We 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.

}, keywords = {cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions}, doi = {10.1007/s00030-014-0291-0}, url = {http://hdl.handle.net/1963/7271}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2014, title = {Quasi-static crack growth in hydraulic fracture}, journal = {Nonlinear Analysis}, volume = {109}, number = {Nonlinear Analysis, Theory, Methods and Applications;109}, year = {2014}, pages = {301-318}, publisher = {Elsevier}, type = {Journal article}, chapter = {301}, abstract = {

We present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.

}, doi = {10.1016/j.na.2014.07.009}, url = {http://hdl.handle.net/20.500.11767/17350}, author = {Stefano Almi and Gianni Dal Maso and Rodica Toader} } @article {2014, title = {Rate-independent damage in thermo-viscoelastic materials with inertia}, number = {SISSA;52/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {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.}, url = {http://urania.sissa.it/xmlui/handle/1963/7444}, author = {Giuliano Lazzaroni and Riccarda Rossi and Marita Thomas and Rodica Toader} } @article {2014, title = {Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics}, number = {SISSA;53/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {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.}, url = {http://urania.sissa.it/xmlui/handle/1963/7463}, author = {Giuliano Lazzaroni and Riccarda Rossi and Marita Thomas and Rodica Toader} } @article {10829, title = {A variational model for the quasi-static growth of fractional dimensional brittle fractures}, number = {Interfaces and Free Boundaries}, year = {2014}, publisher = {European Mathematical Society}, abstract = {

We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as $\Gamma$-limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero.

}, keywords = {Variational models}, doi = {10.4171/IFB/328}, url = {http://hdl.handle.net/1963/6983}, author = {Simone Racca and Rodica Toader} } @article {2001, title = {Some remarks on the viscous approximation of crack growth}, journal = {Discrete Contin. Dyn. Syst. Ser. S}, volume = {6}, number = {SISSA;24/2011/M}, year = {2013}, publisher = {SISSA}, chapter = {131-146}, abstract = {

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.

}, keywords = {Variational models}, url = {http://hdl.handle.net/1963/4206}, author = {Giuliano Lazzaroni and Rodica Toader} } @article {2011, title = {Energy release rate and stress intensity factor in antiplane elasticity}, journal = {Journal de Mathematiques Pures et Appliquees 95 (2011) 565-584}, number = {SISSA;67/2009/M}, year = {2011}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1016/j.matpur.2011.01.001}, url = {http://hdl.handle.net/1963/3780}, author = {Giuliano Lazzaroni and Rodica Toader} } @article { ISI:000296627000002, title = {A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION}, journal = {{MATHEMATICAL MODELS \& METHODS IN APPLIED SCIENCES}}, volume = {{21}}, number = {{10}}, year = {2011}, month = {{OCT}}, pages = {{2019-2047}}, publisher = {{WORLD SCIENTIFIC PUBL CO PTE LTD}}, type = {{Article}}, address = {{5 TOH TUCK LINK, SINGAPORE 596224, SINGAPORE}}, abstract = {

{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.}

}, keywords = {Brittle fracture, Crack propagation, energy derivative, energy release rate, free-discontinuity problems, Griffith{\textquoteright}s criterion, local minimizers, stress intensity factor}, vanishing viscosity, {Variational models}, issn = {{0218-2025}}, doi = {{10.1142/S0218202511005647}}, author = {Giuliano Lazzaroni and Rodica Toader} } @article {2011, title = {Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach}, journal = {ESAIM: COCV 17 (2011) 1-27}, number = {SISSA;56/2007/M}, year = {2011}, publisher = {Cambridge University Press / EDP Sciences}, abstract = {A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [6] is recovered. In this case, the convergence of the discrete time approximations is improved.}, doi = {10.1051/cocv/2009037}, url = {http://hdl.handle.net/1963/2355}, author = {Filippo Cagnetti and Rodica Toader} } @article {2010, title = {Quasistatic crack growth in elasto-plastic materials: the two-dimensional case}, journal = {Arch. Ration. Mech. Anal. 196 (2010) 867-906}, number = {SISSA;62/2008/M}, year = {2010}, abstract = {We study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity.}, doi = {10.1007/s00205-009-0258-1}, url = {http://hdl.handle.net/1963/2964}, author = {Gianni Dal Maso and Rodica Toader} } @article {2008, title = {Decomposition results for functions with bounded variation}, journal = {Boll. Unione Mat. Ital. (9) 1 (2008) 497-505}, year = {2008}, publisher = {Unione Matematica Italiana}, url = {http://hdl.handle.net/1963/3535}, author = {Gianni Dal Maso and Rodica Toader} } @article {2007, title = {On a notion of unilateral slope for the Mumford-Shah functional}, journal = {NoDEA 13 (2007) 713-734}, number = {SISSA;78/2004/M}, year = {2007}, abstract = {In this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional.}, doi = {10.1007/s00030-006-4054-4}, url = {http://hdl.handle.net/1963/2059}, author = {Gianni Dal Maso and Rodica Toader} } @article {2006, title = {An artificial viscosity approach to quasistatic crack growth}, number = {SISSA;43/2006/M}, year = {2006}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/1850}, author = {Rodica Toader and Chiara Zanini} } @article {2005, title = {Quasistatic Crack Growth in Nonlinear Elasticity}, journal = {Arch. Ration. Mech. Anal. 176 (2005) 165-225}, number = {SISSA;2/2004/M}, year = {2005}, abstract = {In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\\\\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.}, doi = {10.1007/s00205-004-0351-4}, url = {http://hdl.handle.net/1963/2293}, author = {Gianni Dal Maso and Gilles A. Francfort and Rodica Toader} } @article {2004, title = {Quasi-static evolution in brittle fracture: the case of bounded solutions}, journal = {Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266}, number = {SISSA;3/2004/M}, year = {2004}, abstract = {The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\\\\infty$.}, url = {http://hdl.handle.net/1963/2229}, author = {Gianni Dal Maso and Gilles A. Francfort and Rodica Toader} } @article {2003, title = {A note on the integral representation of functionals in the space SBD(O)}, journal = {Rend. Mat. Appl. 23 (2003) 189-201}, number = {SISSA;14/2001/M}, year = {2003}, publisher = {Rendiconti di Matematica}, abstract = {In this paper we study the integral representation in the space SBD(O) of special functions with bounded deformation of some L^1-norm lower semicontinuous functionals invariant with respect to rigid motions.}, url = {http://hdl.handle.net/1963/3064}, author = {Francois Ebobisse and Rodica Toader} } @article {2002, title = {A model for the quasi-static growth of a brittle fracture: existence and approximation results}, journal = {Math. Models Methods Appl. Sci., 12 (2002), no. 12, 1773}, number = {SISSA;115/00/M}, year = {2002}, publisher = {SISSA Library}, abstract = {We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.}, doi = {10.1142/S0218202502002331}, url = {http://hdl.handle.net/1963/1571}, author = {Gianni Dal Maso and Rodica Toader} } @article {2002, title = {A model for the quasi-static growth of brittle fractures based on local minimization}, journal = {Math.Models Methods Appl. Sci., 12 (2002) , p.1773-1800.}, number = {SISSA;51/2002/M}, year = {2002}, publisher = {SISSA Library}, abstract = {We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.}, doi = {10.1142/S0218202502002331}, url = {http://hdl.handle.net/1963/1621}, author = {Gianni Dal Maso and Rodica Toader} } @article {2002, title = {A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results}, journal = {Arch. Ration. Mech. Anal. 162 (2002) 101-135}, number = {SISSA;30/2001/M}, year = {2002}, publisher = {Springer}, abstract = {We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith\\\'s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.}, doi = {10.1007/s002050100187}, url = {http://hdl.handle.net/1963/3056}, author = {Gianni Dal Maso and Rodica Toader} } @mastersthesis {1997, title = {Some Problems in the Asymptotic Analysis of Partial Differential Equations in Perforated Domains}, year = {1997}, school = {SISSA}, keywords = {Dirichlet problems}, url = {http://hdl.handle.net/1963/5698}, author = {Rodica Toader} } @article {1996, title = {A capacity method for the study of Dirichlet problems for elliptic systems in varying domains}, journal = {Rend. Sem. Mat. Univ. Padova 96 (1996), 257--277}, number = {SISSA;87/95/M}, year = {1996}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/989}, author = {Gianni Dal Maso and Rodica Toader} } @article {1994, title = {Limits of Dirichlet problems in perforated domains: a new formulation}, journal = {Rend. Istit. Mat. Univ. Trieste 26 (1994) 339-360}, number = {SISSA;167/1994/M}, year = {1994}, publisher = {Universit{\`a} degli Studi di Trieste, Dipartimento di Scienze Matematiche}, url = {http://hdl.handle.net/1963/3649}, author = {Gianni Dal Maso and Rodica Toader} }