Corrugated Membrane Nonlinear Deformation Process Calculation

Abstract

Elastic elements are widely used in instrumentation. They are used to create a particular interference between the parts, for accumulating mechanical energy, as the motion transmission elements, elastic supports, and sensing elements of measuring devices. Device reliability and quality depend on the calculation accuracy of the elastic elements. A corrugated membrane is rather common embodiment of the elastic element.

The corrugated membrane properties depend largely on its profile i.e. a generatrix of the meridian surface.

Unlike other types of pressure elastic members (bellows, tube spring), the elastic characteristics of which are close to linear, an elastic characteristic of the corrugated membrane (typical movement versus external load) is nonlinear. Therefore, the corrugated membranes can be used to measure quantities, nonlinearly related to the pressure (e.g., aircraft air speed, its altitude, pipeline fluid or gas flow rate). Another feature of the corrugated membrane is that significant movements are possible within the elastic material state. However, a significant non-linearity of membrane characteristics leads to severe complicated calculation.

This article is aimed at calculating the corrugated membrane to obtain the elastic characteristics and the deformed shape of the membrane meridian, as well as at investigating the processes of buckling. As the calculation model, a thin-walled axisymmetric shell rotation is assumed. The material properties are linearly elastic. We consider a corrugated membrane of sinusoidal profile. The membrane load is a uniform pressure.

The algorithm for calculating the mathematical model of an axisymmetric corrugated membrane of constant thickness, based on the Reissner’s theory of elastic thin shells, was realized as the author's program in C language. To solve the nonlinear problem were used a method of changing the subspace of control parameters, developed by S.S., Gavriushin, and a parameter marching method developed by N.V. Valishvili. The principle of the method of changing the subspace of control parameters is piecewise smooth parameter marching process. In each smooth section a numerical analysis is reduced to the one-parameter problem.

The problem is solved by two-stage predictor-corrector scheme. The predictor stage uses extrapolation to predict initial values of unknown on the basis of historical data. At the corrector stage a modified method of Newton - Raphson is used to specify initial approximation solutions.

As a result of the programme, the following results were obtained: elastic characteristic of the corrugated membrane and deformed shapes of a corrugated shell meridian in appropriate points of the elastic characteristic.

The paper has considered a phenomenon of local buckling. It has shown the elastic characteristic obtained and a deformed shape of the corrugated membrane meridian. The method to have an isolated solution by changing a subspace of control parameters has been proposed. The proposed algorithm enables efficient investigation of membrane behaviour during nonlinear deformation.