In the setting of planar linearized elasticity, we study a fracture model depending on the crack opening. Assuming that the crack path is known a priori and sufficiently smooth, we prove that the energy release rate is well defined. Then, we consider the problem of quasi-static evolution for our model. Thanks to a vanishing viscosity approach, we show the existence of such an evolution satisfying a weak Griffith’s criterion.

PB - EDP Sciences VL - 23 UR - https://www.esaim-cocv.org/component/article?access=doi&doi=10.1051/cocv/2016014 ER - TY - JOUR T1 - A lower semicontinuity result for a free discontinuity functional with a boundary term JF - Journal de Mathématiques Pures et Appliquées Y1 - 2017 A1 - Stefano Almi A1 - Gianni Dal Maso A1 - Rodica Toader AB -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.

VL - 108 UR - http://hdl.handle.net/20.500.11767/15979 IS - 6 U1 - 34731 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Quasi-static hydraulic crack growth driven by Darcy's law Y1 - 2016 A1 - Stefano Almi AB -In the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain ε ∈ R and with a sufficiently regular crack Γ filled by a volume V of incompressible fluid. In particular, we are able to find the pressure p of the fluid inside the crack as a function of Γ, V , and ε. Then, we study the problem of quasi-static evolution for our model, imposing that the fluid volume V and the fluid pressure p are related by Darcy’s law. We show the existence of such an evolution, and we prove that it satisfies a weak notion of the so-called Griffith’s criterion.

UR - http://urania.sissa.it/xmlui/handle/1963/35198 U1 - 35492 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - THES T1 - Some results on the mathematical analysis of crack problems with forces applied on the fracture lips Y1 - 2016 A1 - Stefano Almi KW - Fracture mechanics AB - This thesis is devoted to the study of some models of fracture growth in elastic materials, characterized by the presence of forces acting on the crack lips. Working in the general framework of rate-independent processes, we first discuss a variational formulation of the problem of quasi-static crack evolution in hydraulic fracture. Then, we investigate the crack growth process in a cohesive fracture model, showing the existence of an evolution satisfying a weak Griffith's criterion. Finally, in the last chapter of this work we investigate, in the static case, the interaction between the energy spent in order to create a new fracture and the energy spent by the applied surface forces. This leads us to study the lower semicontinuity properties of a free discontinuity functional F(u) that can be written as the sum of a crack term, depending on the jump set of u, and of a boundary term, depending on the trace of u. PB - SISSA U1 - 35503 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Quasi-static crack growth in hydraulic fracture JF - Nonlinear Analysis Y1 - 2014 A1 - Stefano Almi A1 - Gianni Dal Maso A1 - Rodica Toader AB -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.

PB - Elsevier VL - 109 UR - http://hdl.handle.net/20.500.11767/17350 IS - Nov U1 - 34741 U2 - Mathematics U4 - 1 U5 - MAT/05 ER -