arXiv subjects

Antonio Flavio Donnarumma

Publications and source records attributed to Antonio Flavio Donnarumma.

3 recordsLinked to original sources

arXiv2026

Asymptotic analysis for heterogeneous elastic energies with material voids

We study the effective behavior of heterogeneous energies arising in the modeling of material voids in geometrically linear elastic materials. Specifically, we consider functionals featuring bulk terms depending on the symmetrized gradient of the displacement and terms comparable to the surface area of the material voids inside the material. Under suitable growth conditions for the bulk and surface densities we prove that, as the microscale $\varepsilon$ tends to zero, the $Γ$-limit admits an integral representation that contains an additional surface term expressed by jump discontinuities of the displacement outside of the void region. This term is related to the phenomenon of collapsing of voids in the effective limit. Under a continuity assumption of the surface density at the $\varepsilon$-scale, we show that the limiting density related to jumps is twice the energy density for voids.

math.AP↗

arXiv2024

$Γ$-convergence and stochastic homogenization of second order singular perturbation models for phase transitions

We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physical-chemical systems. Specifically, we study the asymptotic behavior, as $ε$ goes to zero, of random heterogeneous anisotropic functionals in which the second order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals $Γ$-converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed.

math.AP↗

arXiv2023

Stochastic homogenisation for functionals defined on asymptotically piecewise rigid functions

We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show that there exists a $Γ$-limit almost surely and that it can be represented by a surface integral. In addition, the effective density can be characterised via a suitable cell formula and is deterministic under an ergodicity assumption. We also show via $Γ$-convergence that the homogenised functional defined on piecewise rigid functions can be recovered from a Griffith-type model by passing to the limit of vanishing elastic deformations.

math.AP↗