Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

Abstract

A boundary-only solution is presented for dynamic analysis of elastic membranes under large deflections. The solution procedure is based on the analog equation method (AEM). According to this method, the three coupled nonlinear second order hyperbolic partial differential equations in terms of displacements, which govern the response of the membrane, are replaced with three Poisson's quasi-static equations under fictitious time dependent sources. The fictitious sources are established using a BEM-based procedure and the displacements as well as the stress resultants at any point are evaluated from their integrals representations. Numerical examples are presented which illustrate the method.

On the dynamic electromechanical loading of dielectric elastomer membranes

“…The dynamic response of passive elastic membranes has been investigated by several researchers. Akkas (1978) and Verron et al (1999) studied the dynamic inflation of spherical elastic membranes, and Verron et al (2001) and Katsikadelis (2002) considered the dynamic inflation of planar membranes. More recently, Mockensturm and Goulbourne (2006) proposed a dynamic model of DEA spherical membranes.…”

Development and implementation of some BEM variants—A critical review

“…This method had been demonstrated in solving boundary value problems for many research areas, i.e. thermal conductivity in non-homogenous or nonlinear bodies [125], nonlinear flexural vibrations of plates [135], integration of nonlinear equations of motion [121], linear and nonlinear plate bending problems [132,160], plane elastostatic problems [128,129], finite deformation analysis of elastic cables [123,127], plate buckling problems [161], inverse problems [162], soap bubble problem [131], nonlinear analysis of shells [273], finite equationless problems in nonlinear bodies using only boundary data [133], large deflection analysis of beams [137], nonlinear static and dynamic analysis of membranes [122,130,136,138,139], ponding problem on membranes [134,163] and meshless approach on 2D elastostatic problem [124]. Since it is considered as boundary-only method, this method only deals with discretization and integration on the boundary only.…”

The analog equation method: A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies

“…Let u = u(x, y, t) be the sought solution to the problem (4)- (6). This function is two times continuously differentiable in Ω.…”

“…The method has been already successfully employed to solve a variety of engineering problems described by partial differential equations, among them potential flow problems in bodies whose material constants depend on the field function (e.g. temperature dependent conductivity) [3], determination of surface with prescribed mean or total curvature [3], the soap bubble problem [4], nonlinear static and dynamic analysis of homogeneous isotropic and heterogeneous orthotropic membranes [5,6,7,8], finite elasticity problems, inverse problems [9], equationless problems in nonlinear bodies using only boundary data [10], nonlinear analysis of shells [11]. The method has been also applied to problems described by coupled nonlinear ordinary differential equations, e.g.…”