Pressure-driven wrinkling of soft inner-lined tubes

A simple equation modelling an inextensible elastic lining of an inner-lined tube subject to an imposed pressure difference is derived from a consideration of the idealised elastic properties of the lining and the pressure and soft-substrate forces. Two cases are considered in detail, one with prominent wrinkling and a second one in which wrinkling is absent and only buckling remains. Bifurcation diagrams are computed via numerical continuation for both cases. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. Our approach explains how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and shows how localised folds and mixed-mode states form in secondary bifurcations from wrinkled states. Our model aims to capture the wrinkling response of arterial endothelium to blood pressure changes but applies much more broadly.

STRENGTH OF SHIPS’ GRILLAGES UNDER LATERAL LOAD AND IN-PLANE COMPRESSION

The paper deals with strength of a grillage loaded by lateral load and in-plane compression load (in one direction). It consists of a system of prismatic girders crossing under 90°. The compression load is taken by the longitudinal girders that are elastically fixed on rigid supports. The system of aggregated differential equations is derived for solution of the problem using the Lagrange method. It allows for replacement of the system of aggregated differential equations by a system of independent differential equations. These equations for the case of simultaneous action of lateral and longitudinal compression load have the form of differential equations for bending of prismatic girders laying on elastic foundation and loaded with lateral and longitudinal compression forces. When only lateral load exists, the form of these equations coincides with the form of differential equations for bending of girders laying on elastic foundation and loaded with lateral load alone. When only longitudinal compression load exists, the form of these equations coincides with the form of differential equations for buckling of girders laying on elastic foundation. Solutions are given for bending of a grillage (the first two problems). Formulas are derived for calculation of the parameters of longitudinal girders’ bending when girders’ end sections are elastically fixed. Also, formulas are derived for calculation of the reaction forces at cross-points of transverse and longitudinal girders. When only longitudinal compression load exists (third problem), a solution is given for the connection between the coefficient of elastic foundation’s rigidity and the Euler force. Results obtained by using the proposed method are compared with FEA simulations.

AN APPROXIMATE METHOD TO PRESENT THE BUCKLING STRENGTH OF A GRILLAGE IN A PROBABILISTIC FORM

An approximate method for calculation in probabilistic terms of the buckling strength of a grillage under unidirectional in-plane compression is proposed. The geometric properties of longitudinals and transverses and the mechanical properties (yield stress and modulus of elasticity) of the material they are built from are treated as random parameters that may change over ship’s service life. The cumulative distribution function of the grillage’s critical buckling strength is calculated by using an analytical formula for multitude sets of input parameters while all of them having the same level of certainty. The assumption is that the critical buckling strength has the same (or very similar) level of certainty as that of the input parameters. The accuracy of the proposed approximate method is relatively high (the maximal error is around 2%). It is recommended for use when specialized computer programs for application of Monte Carlo simulation method are not available. The method does not require a complicated specialized computer program and can be run on EXCEL computer program.