Abstract

A finite deformation hyper-elastic membrane theory based on inter-atomic potentials for crystalline films composed of a single atomic layer is developed. For this purpose, an extension of the standard Born rule that exploits the differential geometry concept of the exponential map is proposed to deal with the curvature of surfaces. The exponential map is approximated locally and strain measures based on the stretch and the curvature of the membrane arise. The methodology is first particularized to atomic chains in two dimensions, and then to graphene sheets. A reduced model for the transverse mechanics of carbon nanotubes is developed in detail. This model is a hyper-elastic constrained membrane which fully exploits the symmetry of the transverse deformation. Additionally, a continuum version of the non-bonded interactions is provided. The continuum model is discretized using finite elements and very good agreement with molecular mechanics simulations is obtained. Finally, several simulations illustrate the strong effect of the van der Waals interactions in the transverse deformation of carbon nanotubes.

Abstract

We formulate a material model for micro-magneto-mechanics based on the generalized standard material approach. Our model includes exchange, elastic, anisotropy, demagnetizing and Zeeman energy. Furthermore we account for dissipative micro-magnetic behavior by means of a dissipation potential. For the constrained optimization problem w.r.t. magnetization we rely on the exponential map algorithm. We demonstrate our ideas with numerical examples. In particular we apply our model to a thin film composite. With this composite we represent the magneto-mechanical part of a magneto-electric composite sensor (resp. small sensor segment). Our numerical experiments focus on FeCoSiB as the magnetostrictive material. We discuss the coupling effects for the considered thin film composite in detail.