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.

1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttps://www.math.sissa.it/publication/lower-semicontinuity-class-integral-functionals-space-functions-bounded-deformation01104nas a2200145 4500008004100000245009100041210006900132300001200201490000800213520063400221100001800855700002100873700001900894856004500913 2017 en d00aA lower semicontinuity result for a free discontinuity functional with a boundary term0 alower semicontinuity result for a free discontinuity functional a952-9900 v1083 aWe 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.

1 aAlmi, Stefano1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/20.500.11767/1597900965nas a2200169 4500008004100000022001400041245012900055210006900184260000800253300000700261490000700268520041200275100002100687700002200708700001900730856004600749 2016 eng d a1432-083500aFracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case0 aFracture models for elastoplastic materials as limits of gradien cApr a450 v553 aWe 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.

1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttps://doi.org/10.1007/s00526-016-0981-z01267nas a2200121 4500008004100000245009800041210006900139520082000208100002101028700002601049700001901075856005101094 2015 en d00aExistence for constrained dynamic Griffith fracture with a weak maximal dissipation condition0 aExistence for constrained dynamic Griffith fracture with a weak 3 aThere 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.1 aDal Maso, Gianni1 aLarsen, Cristopher J.1 aToader, Rodica uhttp://urania.sissa.it/xmlui/handle/1963/3504501118nas a2200145 4500008004100000245013100041210006900172260001000241520052100251653010200772100002100874700002200895700001900917856003600936 2014 en d00aLaplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length0 aLaplace equation in a domain with a rectilinear crack higher ord bSISSA3 aWe 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.

10acracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttp://hdl.handle.net/1963/727100712nas a2200157 4500008004100000245005200041210005100093260001300144300001200157490000800169520027400177100001800451700002100469700001900490856004500509 2014 en d00aQuasi-static crack growth in hydraulic fracture0 aQuasistatic crack growth in hydraulic fracture bElsevier a301-3180 v1093 aWe 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.

1 aAlmi, Stefano1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/20.500.11767/1735001353nas 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/1963/746300848nas a2200133 4500008004100000245009600041210006900137260003400206520037800240653002300618100001800641700001900659856003600678 2014 en d00aA variational model for the quasi-static growth of fractional dimensional brittle fractures0 avariational model for the quasistatic growth of fractional dimen bEuropean Mathematical Society3 aWe 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.

10aVariational models1 aRacca, Simone1 aToader, Rodica uhttp://hdl.handle.net/1963/698300835nas 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/1963/420600728nas 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-000919nas a2200121 4500008004300000245013400043210006900177260004600246520042800292100002200720700001900742856003600761 2011 en_Ud 00aQuasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach0 aQuasistatic crack evolution for a cohesive zone model with diffe bCambridge University Press / EDP Sciences3 aA 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.1 aCagnetti, Filippo1 aToader, Rodica uhttp://hdl.handle.net/1963/235500550nas a2200109 4500008004300000245008300043210006900126520016900195100002100364700001900385856003600404 2010 en_Ud 00aQuasistatic crack growth in elasto-plastic materials: the two-dimensional case0 aQuasistatic crack growth in elastoplastic materials the twodimen3 aWe study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/296400386nas a2200109 4500008004300000245006300043210006300106260003100169100002100200700001900221856003600240 2008 en_Ud 00aDecomposition results for functions with bounded variation0 aDecomposition results for functions with bounded variation bUnione Matematica Italiana1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/353500613nas a2200109 4500008004300000245006800043210006200111520025400173100002100427700001900448856003600467 2007 en_Ud 00aOn a notion of unilateral slope for the Mumford-Shah functional0 anotion of unilateral slope for the MumfordShah functional3 aIn 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.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/205900611nas a2200109 4500008004300000245006500043210006200108520025700170100001900427700001900446856003600465 2006 en_Ud 00aAn artificial viscosity approach to quasistatic crack growth0 aartificial viscosity approach to quasistatic crack growth3 aWe 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.1 aToader, Rodica1 aZanini, Chiara uhttp://hdl.handle.net/1963/185000706nas a2200121 4500008004300000245005300043210005300096520033400149100002100483700002500504700001900529856003600548 2005 en_Ud 00aQuasistatic Crack Growth in Nonlinear Elasticity0 aQuasistatic Crack Growth in Nonlinear Elasticity3 aIn 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.1 aDal Maso, Gianni1 aFrancfort, Gilles A.1 aToader, Rodica uhttp://hdl.handle.net/1963/229300759nas a2200121 4500008004300000245007800043210006900121520034600190100002100536700002500557700001900582856003600601 2004 en_Ud 00aQuasi-static evolution in brittle fracture: the case of bounded solutions0 aQuasistatic evolution in brittle fracture the case of bounded so3 aThe 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 $L^\\\\infty$.1 aDal Maso, Gianni1 aFrancfort, Gilles A.1 aToader, Rodica uhttp://hdl.handle.net/1963/222900630nas a2200121 4500008004300000245007700043210006900120260002900189520021200218100002300430700001900453856003600472 2003 en_Ud 00aA note on the integral representation of functionals in the space SBD(O)0 anote on the integral representation of functionals in the space bRendiconti di Matematica3 aIn 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.1 aEbobisse, Francois1 aToader, Rodica uhttp://hdl.handle.net/1963/306401609nas a2200121 4500008004100000245009900041210006900140260001800209520118400227100002101411700001901432856003601451 2002 en d00aA model for the quasi-static growth of a brittle fracture: existence and approximation results0 amodel for the quasistatic growth of a brittle fracture existence bSISSA Library3 aWe 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.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/157101599nas a2200121 4500008004100000245008900041210006900130260001800199520118400217100002101401700001901422856003601441 2002 en d00aA model for the quasi-static growth of brittle fractures based on local minimization0 amodel for the quasistatic growth of brittle fractures based on l bSISSA Library3 aWe 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.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/162101237nas a2200121 4500008004300000245009800043210006900141260001300210520081600223100002101039700001901060856003601079 2002 en_Ud 00aA Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results0 aModel for the QuasiStatic Growth of Brittle Fractures Existence bSpringer3 aWe 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.1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/305600409nas a2200109 4500008004100000245010100041210006900142260001000211653002300221100001900244856003600263 1997 en d00aSome Problems in the Asymptotic Analysis of Partial Differential Equations in Perforated Domains0 aSome Problems in the Asymptotic Analysis of Partial Differential bSISSA10aDirichlet problems1 aToader, Rodica uhttp://hdl.handle.net/1963/569800411nas a2200109 4500008004100000245009800041210006900139260001800208100002100226700001900247856003500266 1996 en d00aA capacity method for the study of Dirichlet problems for elliptic systems in varying domains0 acapacity method for the study of Dirichlet problems for elliptic bSISSA Library1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/98900448nas a2200109 4500008004300000245007400043210006900117260007600186100002100262700001900283856003600302 1994 en_Ud 00aLimits of Dirichlet problems in perforated domains: a new formulation0 aLimits of Dirichlet problems in perforated domains a new formula bUniversitÃ degli Studi di Trieste, Dipartimento di Scienze Matematiche1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/3649