Browsing by Author "Almi, Stefano"

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    A lower semicontinuity result for a free discontinuity functional with a boundary term
    (2015-12-15) Almi, Stefano; Dal Maso, Gianni; Toader, Rodica; Mathematics
    We 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.
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    Quasi-static hydraulic crack growth driven by Darcy's law
    (2016) Almi, Stefano; Mathematics
    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.