Bistable elliptic equations with fractional diffusion

This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion. More precisely, the aim of this thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all space, at least up to dimension 8. This property on 1-D symmetry of monotone solutions for fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.

Fiber beam-columns models with flexure-shear interaction for nonlinear analysis of reinforced concrete structures

The aim of this study was to develop a model capable to capture the different contributions which characterize the nonlinear behaviour of reinforced concrete structures. In particular, especially for non slender structures, the contribution to the nonlinear deformation due to bending may be not sufficient to determine the structural response. Two different models characterized by a fibre beam-column element are here proposed. These models can reproduce the flexure-shear interaction in the nonlinear range, with the purpose to improve the analysis in shear-critical structures. The first element discussed is based on flexibility formulation which is associated with the Modified Compression Field Theory as material constitutive law. The other model described in this thesis is based on a three-field variational formulation which is associated with a 3D generalized plastic-damage model as constitutive relationship. The first model proposed in this thesis was developed trying to combine a fibre beamcolumn element based on the flexibility formulation with the MCFT theory as constitutive relationship. The flexibility formulation, in fact, seems to be particularly effective for analysis in the nonlinear field. Just the coupling between the fibre element to model the structure and the shear panel to model the individual fibres allows to describe the nonlinear response associated to flexure and shear, and especially their interaction in the nonlinear field. The model was implemented in an original matlab® computer code, for describing the response of generic structures. The simulations carried out allowed to verify the field of working of the model. Comparisons with available experimental results related to reinforced concrete shears wall were performed in order to validate the model. These results are characterized by the peculiarity of distinguishing the different contributions due to flexure and shear separately. The presented simulations were carried out, in particular, for monotonic loading. The model was tested also through numerical comparisons with other computer programs. Finally it was applied for performing a numerical study on the influence of the nonlinear shear response for non slender reinforced concrete (RC) members. Another approach to the problem has been studied during a period of research at the University of California Berkeley. The beam formulation follows the assumptions of the Timoshenko shear beam theory for the displacement field, and uses a three-field variational formulation in the derivation of the element response. A generalized plasticity model is implemented for structural steel and a 3D plastic-damage model is used for the simulation of concrete. The transverse normal stress is used to satisfy the transverse equilibrium equations of at each control section, this criterion is also used for the condensation of degrees of freedom from the 3D constitutive material to a beam element. In this thesis is presented the beam formulation and the constitutive relationships, different analysis and comparisons are still carrying out between the two model presented.