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.

%B Nonlinear Analysis %I Elsevier %V 109 %P 301-318 %G en %U http://hdl.handle.net/20.500.11767/17350 %N Nov %9 Journal article %1 34741 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by salmi@sissa.it (salmi@sissa.it) on 2015-09-24T08:10:23Z No. of bitstreams: 1 A-DM-T-070714.pdf: 283645 bytes, checksum: 68056ef27e9dcfa246029148c0016c0f (MD5) %& 301 %R 10.1016/j.na.2014.07.009 %0 Journal Article %J Journal of Dynamics and Differential Equations %D 2014 %T Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes %A Gianni Dal Maso %A Riccardo Scala %XWe introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

%B Journal of Dynamics and Differential Equations %V 26 %P 915–954 %8 Dec %G eng %U https://doi.org/10.1007/s10884-014-9409-7 %R 10.1007/s10884-014-9409-7 %0 Journal Article %J Calculus of variations and partial differential equations 44 (2012) 495-541 %D 2012 %T Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution %A Gianni Dal Maso %A Antonio DeSimone %A Francesco Solombrino %XCam-Clay plasticity is a well established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s , has been proposed in [8] to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t . We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.

%B Calculus of variations and partial differential equations 44 (2012) 495-541 %I Springer %G en_US %U http://hdl.handle.net/1963/3900 %1 809 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-07-26T09:17:50Z\\r\\nNo. of bitstreams: 1\\r\\nSolombrino_46M.pdf: 382911 bytes, checksum: dd118cc7f80d4cb1902713eb18747ac6 (MD5) %R 10.1007/s00526-011-0443-6 %0 Journal Article %J Calculus of Variations and Partial Differential Equations 40 (2011) 125-181 %D 2011 %T Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling %A Gianni Dal Maso %A Antonio DeSimone %A Francesco Solombrino %K Cam-Clay plasticity %XCam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of generalized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in [4] and [6], the proposed solution may be discontinuous with respect to the original time. Our formulation allows to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.

%B Calculus of Variations and Partial Differential Equations 40 (2011) 125-181 %I Springer %G en_US %U http://hdl.handle.net/1963/3670 %1 635 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-20T08:26:03Z\\r\\nNo. of bitstreams: 1\\r\\nDM-DeS-Sol-36_2009_preprint.pdf: 421582 bytes, checksum: 011806364200378d6deec80b88978550 (MD5) %R 10.1007/s00526-010-0336-0 %0 Journal Article %J Arch. Ration. Mech. Anal. 196 (2010) 867-906 %D 2010 %T Quasistatic crack growth in elasto-plastic materials: the two-dimensional case %A Gianni Dal Maso %A Rodica Toader %X 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. %B Arch. Ration. Mech. Anal. 196 (2010) 867-906 %G en_US %U http://hdl.handle.net/1963/2964 %1 1736 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-23T17:17:34Z\\nNo. of bitstreams: 1\\nDM-Toa-07-preprint.pdf: 320979 bytes, checksum: e4f2a1856f9bd91d63fc45557cbd6a16 (MD5) %R 10.1007/s00205-009-0258-1 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290 %D 2010 %T Quasistatic crack growth in finite elasticity with non-interpenetration %A Gianni Dal Maso %A Giuliano Lazzaroni %XWe 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.

%B Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290 %G en_US %U http://hdl.handle.net/1963/3397 %1 935 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-15T09:16:46Z\\nNo. of bitstreams: 1\\nDM-Laz-08pre.pdf: 360778 bytes, checksum: 14a35b4647fde0ea0931df8ae6cbfb73 (MD5) %R 10.1016/j.anihpc.2009.09.006 %0 Journal Article %J Netw. Heterog. Media 5 (2010) 97-132 %D 2010 %T Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case %A Gianni Dal Maso %A Francesco Solombrino %K Cam-Clay plasticity %XWe study the spatially uniform case of the problem of quasistatic evolution in small strain nonassociative elastoplasticity (Cam-Clay model). Through the introdution of a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE\\\'s in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity, the slow dynamics, when the strain evolves smoothly on the yield surface and plastic flow is produced, and the fast dynamics, which may happen only in the softening regime, where\\nviscous solutions exhibit a jump across a heteroclinic orbit of an auxiliary system.

%B Netw. Heterog. Media 5 (2010) 97-132 %G en_US %U http://hdl.handle.net/1963/3671 %1 634 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-20T08:35:17Z\\nNo. of bitstreams: 1\\nDM-Sol2-39_2009_preprint.pdf: 345995 bytes, checksum: 073ce622e3320be1f56a1cc9f8904fae (MD5) %R 10.3934/nhm.2010.5.97 %0 Journal Article %J Math. Models Methods Appl. Sci. 19 (2009) 1643-1711 %D 2009 %T Quasistatic evolution for Cam-Clay plasticity: examples of spatially homogeneous solutions %A Gianni Dal Maso %A Antonio DeSimone %X We study a quasistatic evolution problem for Cam-Clay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities.\\nThe behavior of the solutions at the jump times is studied by a viscous approximation. %B Math. Models Methods Appl. Sci. 19 (2009) 1643-1711 %G en_US %U http://hdl.handle.net/1963/3395 %1 937 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-10T11:56:05Z\\nNo. of bitstreams: 1\\nDal_Maso-DeSimone.pdf: 1160572 bytes, checksum: fd49c80a369183f1941b0fcf3cd65341 (MD5) %R 10.1142/S0218202509003942 %0 Report %D 2007 %T Quasistatic crack growth for a cohesive zone model with prescribed crack path %A Gianni Dal Maso %A Chiara Zanini %X In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes. %B Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253-279 %G en_US %U http://hdl.handle.net/1963/1686 %1 2447 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2005-06-20T12:05:32Z\\nNo. of bitstreams: 1\\nDM-Zan-04.pdf: 312529 bytes, checksum: 2373a684663d7901c85daad77c7aa8b6 (MD5) %R 10.1017/S030821050500079X %0 Journal Article %J Milan J. Math. 75 (2007) 117-134 %D 2007 %T Quasistatic evolution problems for pressure-sensitive plastic materials %A Gianni Dal Maso %A Alexey Demyanov %A Antonio DeSimone %X We study quasistatic evolution problems for pressure-sensitive plastic materials in the context of small strain associative perfect plasticity. %B Milan J. Math. 75 (2007) 117-134 %G en_US %U http://hdl.handle.net/1963/1962 %1 2231 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-05-15T10:32:12Z\\nNo. of bitstreams: 1\\nSissa27_2007M.pdf: 223083 bytes, checksum: a4cdbb7a1018403355c560749680e47a (MD5) %R 10.1007/s00032-007-0071-y %0 Journal Article %J Arch. Ration. Mech. Anal. 180 (2006) 237-291 %D 2006 %T Quasistatic evolution problems for linearly elastic-perfectly plastic materials %A Gianni Dal Maso %A Antonio DeSimone %A Maria Giovanna Mora %X The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain. %B Arch. Ration. Mech. Anal. 180 (2006) 237-291 %G en_US %U http://hdl.handle.net/1963/2129 %1 2114 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-18T10:14:12Z\\nNo. of bitstreams: 1\\n0412212v1.pdf: 419970 bytes, checksum: fa4e3c2a5db22bfa28aa4f682cb55da4 (MD5) %R 10.1007/s00205-005-0407-0 %0 Journal Article %J Arch. Ration. Mech. Anal. 176 (2005) 165-225 %D 2005 %T Quasistatic Crack Growth in Nonlinear Elasticity %A Gianni Dal Maso %A Gilles A. Francfort %A Rodica Toader %X 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. %B Arch. Ration. Mech. Anal. 176 (2005) 165-225 %G en_US %U http://hdl.handle.net/1963/2293 %1 1723 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-24T09:12:09Z\\nNo. of bitstreams: 1\\n0401196v1.pdf: 664295 bytes, checksum: cb1000c44e6ae356984e24b55ee97117 (MD5) %R 10.1007/s00205-004-0351-4 %0 Journal Article %J Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266 %D 2004 %T Quasi-static evolution in brittle fracture: the case of bounded solutions %A Gianni Dal Maso %A Gilles A. Francfort %A Rodica Toader %X 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 $L^\\\\infty$. %B Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266 %G en_US %U http://hdl.handle.net/1963/2229 %1 2015 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-15T15:04:28Z\\r\\nNo. of bitstreams: 1\\r\\n0401198v1.pdf: 166634 bytes, checksum: c21fba2b1fbbaec4fe14c56595b0664e (MD5)