Intrinsic Theory of Nonlinearly Elastic Plates and Shallow Shells
DescriptionWhen an elastic body is subjected to specific boundary conditions and applied forces, its “reference configuration”, i.e., the portion of space it occupies in the absence of forces, becomes a “deformed configuration”. One of the central themes of elasticity theory then consists in providing equations that allow to determine the displacement vector at each point of the reference configuration. The unknown is thus a vector field defined over the reference configuration, whose components are those of the unknown displacement field. These equations originally took the form of boundary value problems, i.e., partial differential equations and boundary conditions. However, it was subsequently recognized that minimizing an ad hoc “energy functional” over an appropriate “set of admissible displacements” was the most efficient way to obtain existence theorems.From the computational viewpoints, however, this “classical approach” is not fully satisfactory, since the unknowns of primary interest for a practicing engineer are not so much the components of the displacement field, but instead those of the stress tensor field inside the body, since large stresses, rather than large displacements, are more likely to provoke the collapse of an elastic structure. But computing the stresses from the displacement, by means of the so-called constitutive equation of the elastic material found in the structure, involves computing derivatives, a procedure well-known to be “unstable” numerically.By contrast, in an “intrinsic approach”, it is the components of the stress tensor, or more generally of any bona fide “measure of stress”, that are the only unknowns, instead of the components of the displacement vector field. Although the idea of this approach goes back to the forties, it is only recently that a group led by the present P.I has laid out the mathematically foundations of such methods, but only for linearly elastic bodies. The present proposal intends to go one significant step further, first by identifying the specific equations, then by laying down the mathematical foundations, of the intrinsic approach applied to specific nonlinearly elastic “thin” bodies of utmost importance in applications, namely nonlinearly elastic plates and shallow shells, modeled either by the well-known Kirchhoff-Love equations, or by the well-known von Kármán or Mar- guerre -von Kármán equations. In so doing, our main contributions will be to provide the existence of minimizers to each one of the “intrinsic” non-quadratic energy functionals associated with such nonlinearly elastic bodies.
|Effective start/end date||1/01/20 → …|