6,804 research outputs found
Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories
Long-wave models obtained in the process of asymptotic homogenisation of structures with a characteristic length scale are known to be non-unique. The term non-uniqueness is used here in the sense that various homogenisation strategies may lead to distinct governing equations that usually, for a given order of the governing equation, approximate the original problem with the same asymptotic accuracy. A constructive procedure presented in this paper generates a class of asymptotically equivalent long-wave models from an original homogenised theory. The described non-uniqueness manifests itself in the occurrence of additional parameters characterising the model. A simple problem of long-wave propagation in a regular one-dimensional lattice structure is used to illustrate important criteria for selecting
these parameters. The procedure is then applied to derive a class of continuum theories for a two-dimensional square array of particles. Applications to asymptotic structural theories are also discussed. In particular, we demonstrate how to improve the governing equation for the Rayleigh-Love rod and explain the reasons for the well-known numerical accuracy of the Mindlin plate theory
The limits of Hamiltonian structures in three-dimensional elasticity, shells, and rods
This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a three-dimensional elastic body converges, in a sense made precise, to that for a shell model described by a one-director Cosserat surface as the thickness goes to zero. We study limiting procedures that give rise to unconstrained as well as constrained Cosserat director models. The case of a rod is also considered and similar convergence results are established, with the limiting model being a geometrically exact director rod model (in the framework developed by Antman, Simo, and coworkers). The resulting model may or may not have constraints, depending on the nature of the constitutive relations and their behavior under the limiting procedure.
The closeness of Hamiltonian structures is measured by the closeness of Poisson brackets on certain classes of functions, as well as the Hamiltonians. This provides one way of justifying the dynamic one-director model for shells. Another way of stating the convergence result is that there is an almost-Poisson embedding from the phase space of the shell to the phase space of the 3D elastic body, which implies that, in the sense of Hamiltonian structures, the dynamics of the elastic body is close to that of the shell. The constitutive equations of the 3D model and their behavior as the thickness tends to zero dictates whether the limiting 2D model is a constrained or an unconstrained director model.
We apply our theory in the specific case of a 3D Saint Venant-Kirchhoff material andderive the corresponding limiting shell and rod theories. The limiting shell model is an interesting Kirchhoff-like shell model in which the stored energy function is explicitly derived in terms of the shell curvature. For rods, one gets (with an additional inextensibility constraint) a one-director Kirchhoff elastic rod model, which reduces to the well-known Euler elastica if one adds an additional single constraint that the director lines up with the Frenet frame
Straight rod with different order of thickness
In this paper, we consider rods whose thickness vary linearly between \eps
and \eps^2. Our aim is to study the asymptotic behavior of these rods in the
framework of the linear elasticity. We use a decomposition method of the
displacement fields of the form , where stands for the
translation-rotations of the cross-sections and is related to their
deformations. We establish a priori estimates. Passing to the limit in a fixed
domain gives the problems satisfied by the bending, the stretching and the
torsion limit fields which are ordinary differential equations depending on
weights.Comment: in Asymptotic Analysis, IOS Press, 201
Gaussian curvature as an identifier of shell rigidity
In the paper we deal with shells with non-zero Gaussian curvature. We derive
sharp Korn's first (linear geometric rigidity estimate) and second inequalities
on that kind of shells for zero or periodic Dirichlet, Neumann, and Robin type
boundary conditions. We prove that if the Gaussian curvature is positive, then
the optimal constant in the first Korn inequality scales like and if the
Gaussian curvature is negative, then the Korn constant scales like
where is the thickness of the shell. These results have classical flavour
in continuum mechanics, in particular shell theory. The Korn first inequalities
are the linear version of the famous geometric rigidity estimate by Friesecke,
James and M\"uller for plates [14] (where they show that the Korn constant in
the nonlinear Korn's first inequality scales like ), extended to shells
with nonzero curvature. We also recover the uniform Korn-Poincar\'e inequality
proven for "boundary-less" shells by Lewicka and M\"uller in [37] in the
setting of our problem. The new estimates can also be applied to find the
scaling law for the critical buckling load of the shell under in-plane loads as
well as to derive energy scaling laws in the pre-buckled regime. The exponents
and in the present work appear for the first time in any sharp
geometric rigidity estimate.Comment: 25 page
The time-dependent von Kármán plate equation as a limit of 3D nonlinear elasticity
The asymptotic behaviour of the solutions of three-dimensional
nonlinear elastodynamics in a thin plate is studied, as the thickness h of the
plate tends to zero. Under appropriate scalings of the applied force and of the
initial values in terms of h, it is shown that three-dimensional solutions of the
nonlinear elastodynamic equation converge to solutions of the time-dependent
von Kármán plate equation
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