A two-dimensional continuum model of pantographic sheets moving in a 3D space and accounting for the offset and relative rotations of the fibers

Recently growing attention has been paid to the particular class of metamaterials which has been called pantographic. Pantographic metamaterials have the following peculiar features: (i) their continuum model, at the macroscale, has to include a term of the deformation energy depending on the second gradient of placement, (ii) they can show an elastic behavior in large deformation regimes, and (iii) they are resilient and tough during rupture phenomena (dell'Isola et al. 2015). In order to predict pantographic metamaterials' mechanical behavior, it is possible to introduce a three-dimensional continuum micromodel, in which their internal geometrical microstructure is described in detail. However, the computational costs of this choice are presently prohibitive. In this paper, we introduce a reduced order model for pantographic sheets-which are an example of an elastic surface-whose kinematics include, for each of the two constituting families of fibers fully independent three-dimensional (i) placement and (ii) rotation fields. In this way it is possible to include, also in the reduced order model, (i) the initial and the actual offset between the fibers and (ii) the deformation energy of the interconnecting pivots. By postulating a simplified expression for the deformation energy we prove that also a reduced order model can describe some experimental observed buckling and postbuckling phenomena. The promising results which we present here motivate the quest of more general expressions for deformation energy capable of capturing the fully nonlinear behavior exhibited by pantographic sheets

Well-posedness of Hydrodynamics on the Moving Elastic Surface

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.Comment: The introduction is rewritte

Explicit asymptotic modelling of transient Love waves propagated along a thin coating

The official published version can be obtained from the link below.An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low-frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing the displacement at the interface between the coating and the substrate. Unusually, the model equation contains a term with a pseudo-differential operator. This result is confirmed and interpreted by analysing the exact solution obtained by integral transforms. The performance of the derived model is illustrated by numerical examples.This work is sponsored by the grant from Higher Education of Pakistan and by the Brunel University’s “BRIEF” research award