Plate theory as the variational limit of the complementary energy functionals of inhomogeneous anisotropic linearly elastic bodies

We consider a sequence of linear hyper-elastic, inhomogeneous and fully anisotropic bodies in a reference configuration occupying a cylindrical region of height epsilon. We then study, by means of Gamma-convergence, the asymptotic behavior as epsilon goes to zero of the sequence of complementary energies. The limit functional is then identified as a dual problem for a two-dimensional plate. Our approach gives a direct characterization of the convergence of the equilibrating stress fields

Linear models for thin plates of polymer gels

Within the linearized three-dimensional theory of polymer gels, we consider a sequence of problems formulated on a family of cylindrical domains whose height tends to zero. We assume that the fluid pressure is controlled at the top and bottom faces of the cylinder, and we consider two different scaling regimes for the diffusivity tensor. Through asymptotic-analysis techniques we obtain two plate models where the transverse displacement is governed by a plate equation with an extra contribution from the fluid pressure. In the limit obtained within the first scaling regime the fluid pressure is affine across the thickness and hence it is determined by its instantaneous trace on the top and bottom faces. In the second model, instead, the value of the fluid pressure is governed by a three-dimensional diffusion equation

Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers

We first consider an elastic thin heterogeneous cylinder of radius of order epsilon: the interior of the cylinder is occupied by a stiff material (fiber) that is surrounded by a soft material (matrix). By assuming that the elasticity tensor of the fiber does not scale with epsilon and that of the matrix scales with epsilon square, we prove that the one dimensional model is a nonlocal system. We then consider a reference configuration domain filled out by periodically distributed rods similar to those described above. We prove that the homogenized model is a second order nonlocal problem. In particular, we show that the homogenization problem is directly connected to the 3D-1D dimensional reduction problem

The Gaussian stiffness of graphene deduced from a continuum model based on Molecular Dynamics potentials

We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations

A REBO-potential-based model for graphene bending by Γ\Gamma-convergence

An atomistic to continuum model for a graphene sheet undergoing bending is presented. Under the assumption that the atomic interactions are governed by a harmonic approximation of the 2nd-generation Brenner REBO (reactive empirical bond-order) potential, involving first, second and third nearest neighbors of any given atom, we determine the variational limit of the energy functionals. It turns out that the $\Gamma$-limit depends on the linearized mean and Gaussian curvatures. If some specific contributions in the atomic interaction are neglected, the variational limit is non-local

An atomistic-based F\"oppl-von K\'arm\'an model for graphene

We deduce a non-linear continuum model of graphene for the case of finite out-of-plane displacements and small in-plane deformations. On assuming that the lattice interactions are governed by the Brenner's REBO potential of 2nd generation and that self-stress is present, we introduce discrete strain measures accounting for up-to-the-third neighbor interactions. The continuum limit turns out to depend on an average (macroscopic) displacement field and a relative shift displacement of the two Bravais lattices that give rise to the hexagonal periodicity. On minimizing the energy with respect to the shift variable, we formally determine a continuum model of F\"oppl-von K\'arm\'a type, whose constitutive coefficients are given in terms of the atomistic interactions.Comment: arXiv admin note: text overlap with arXiv:1701.0746

A 2D metamaterial with auxetic out-of-plane behavior and non-auxetic in-plane behavior

Customarily, in-plane auxeticity and synclastic bending behavior (i.e. out-of-plane auxeticity) are not independent, being the latter a manifestation of the former. Basically, this is a feature of three-dimensional bodies. At variance, two-dimensional bodies have more freedom to deform than three-dimensional ones. Here, we exploit this peculiarity and propose a two-dimensional honeycomb structure with out-of-plane auxetic behavior opposite to the in-plane one. With a suitable choice of the lattice constitutive parameters, in its continuum description such a structure can achieve the whole range of values for the bending Poisson coefficient, while retaining a membranal Poisson coefficient equal to 1. In particular, this structure can reach the extreme values, $-1$ and $+1$, of the bending Poisson coefficient. Analytical calculations are supported by numerical simulations, showing the accuracy of the continuum formulas in predicting the response of the discrete structure

One-dimensional von K\'arm\'an models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von K\'arm\'an plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the "linearized" von K\'arm\'an energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von K\'arm\'an energy is a non-linear model that comprises stretching, bendings, and twisting. The "constrained" von K\'arm\'an energy, instead, leads to a new Sadowsky type of model

A variational model for anisotropic and naturally twisted ribbons

We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we derive a new variational one-dimensional model for naturally twisted ribbons by means of Gamma-convergence. Our result generalizes, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration