The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of $\Gamma$-convergence, in the framework of finite plasticity. Denoting by $\epsilon$ the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order $\epsilon^{2 \alpha -2}$, with $\alpha \geq 3$. According to the value of $\alpha$, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von K{\'a}rm{\'a}n plate theory and the linearized plate theory.

}, doi = {10.1051/cocv/2013081}, author = {Elisa Davoli} } @article {doi:10.1142/S021820251450016X, title = {Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {24}, number = {10}, year = {2014}, pages = {2085-2153}, abstract = {In this paper we deduce by $\Gamma$-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by $\epsilon$ the thickness of the plate, we study the case where the scaling factor of the elasto-plastic energy is of order $\epsilon^{2 \alpha -2}$, with $\alpha\geq 3$. These scalings of the energy lead, in the absence of plastic dissipation, to the Von K{\'a}rm{\'a}n and linearized Von K{\'a}rm{\'a}n functionals for thin plates. We show that solutions to the three-dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on $\alpha$.

}, doi = {10.1142/S021820251450016X}, url = {https://doi.org/10.1142/S021820251450016X}, author = {Elisa Davoli} } @article {DAVOLI2013615, title = {A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence}, journal = {Annales de l{\textquoteright}Institut Henri Poincare (C) Non Linear Analysis}, volume = {30}, number = {4}, year = {2013}, pages = {615 - 660}, abstract = {The subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic{\textendash}perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ-convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl{\textendash}Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff{\textendash}Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.

}, keywords = {-convergence, Perfect plasticity, Prandtl{\textendash}Reuss plasticity, Quasistatic evolution, Rate-independent processes, Thin plates}, issn = {0294-1449}, doi = {https://doi.org/10.1016/j.anihpc.2012.11.001}, url = {http://www.sciencedirect.com/science/article/pii/S0294144912001035}, author = {Elisa Davoli and Maria Giovanna Mora} } @article {1106.6245, title = {Thin-walled beams with a cross-section of arbitrary geometry: derivation of linear theories starting from 3D nonlinear elasticity}, year = {2011}, abstract = {The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and δ_h, respectively, the diameter and the thickness of the cross-section, we analyse the case where the scaling factor of the elastic energy is of order ε_h^2, with ε_h/δ_h^2 \rightarrow l \in [0, +\infty). Different linearized models are deduced according to the relative order of magnitude of δ_h with respect to h.

}, author = {Elisa Davoli} } @article {2010, title = {Convergence of equilibria of thin elastic rods under physical growth conditions for the energy density}, number = {SISSA;67/2010/M}, year = {2010}, abstract = {The subject of this paper is the study of the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod, as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads, that correspond at the limit to different rod models: the constrained linear theory, the analogous of von K{\'a}rm{\'a}n plate theory for rods, and the linear theory.}, url = {http://hdl.handle.net/1963/4086}, author = {Elisa Davoli and Maria Giovanna Mora} }