Non-Classical Mechanical Behavior of Elastic Membranes of Two Independent Bending Rigidities

Abstract

The non-classical mechanical behavior of an elastic membrane of two independent bending rigidities has been studied. The major interest focuses on the case when the ratio of the Gaussian bending rigidity to the common flexural rigidity falls within the non-classical ranges which cannot be covered by a classical elastic plate with an admissible positive Poisson ratio. In this work, variational method is applied to derive the governing equation and boundary conditions for a non-classical elastic membrane characterized by two independent bending rigidities. Mechanical responses of rectangular and circular non-classical elastic membranes with different boundary conditions have been analyzed systematically and compared with those of a classical elastic plate with an admissible positive Poisson ratio under otherwise identical conditions. For a rectangular non-classical membrane with two opposite free edges, it is shown that its deflection under a uniform transverse pressure could be considerably (even more than twice) larger than a classical elastic plate under otherwise identical conditions, while its lowest fundamental frequency and critical buckling force could be considerably (even more than 50%) lower than a classical elastic plate under otherwise identical conditions. These unexpected results suggest that, unlike classical elastic plates whose actual mechanical behavior are often not sensitive to the exact value of the admissible positive Poisson ratio, actual mechanical behavior of such a rectangular non-classical elastic membrane is very sensitive to the ratio of the Gaussian bending rigidity to the common flexural rigidity. In particular, the overall mechanical stiffness of such a rectangular non-classical membrane could be vanishingly low when the bending rigidity ratio approaches its upper limit 0. On the other hand, the mechanical behavior of a hinged circular non-classical elastic membrane monotonically depends on the bending rigidity ratio even in the two non-classical ranges. Actually, its overall mechanical stiffness becomes vanishingly low when the bending rigidity ratio approaches its lower limit -2, while its overall mechanical stiffness is higher than a classical elastic plate with an admissible positive Poisson ratio under otherwise identical conditions when the bending rigidity ratio goes to its upper limit 0. For both rectangular and circular non-classical elastic membranes studied in this work, the obtained results indicate that the exact value of the Gaussian bending rigidity could be crucial, and only knowing the values of the flexural rigidity and Poisson ratio is insufficient for accurate prediction of mechanical modeling of such non-classical elastic membranes (such as biomembranes and atom-thick graphene membranes).