Patrick Schneider

Comparison of various linear plate theories in the light of a consistent second-order approximation:

The uniform-approximation technique in combination with the pseudo-reduction technique is applied in order to derive consistent theories for isotropic and anisotropic plates. The approach has already been used to assess and validate theories established in the literature, e.g., the theories of Reissner and Zhilin. In this contribution, we also present a comparison with the theories of Vekua, Ambartsumyan, Steigmann and Reddy’s third-order theory. The current paper is a corrected and extended version of the one originally published in Kienzler and Schneider (Comparison of various linear plate theories in the light of a consistent second-order approximation. In Pietraszkiewicz, W and Górski, J (eds), Shell Structures: Theory and Applications. London: Taylor & Francis Group, 2014, pp. 109–112). In addition we put special emphasis on the derivation and validation of Ambartsumyan’s general and simplified theories.

An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-approximation Technique

One way to develop theories for the elastic deformation of two- or one-dimensional structures (like, e.g., shells and beams) under a given load is the uniform-approximation technique (see [2] for an introduction). This technique derives lower-dimensional theories from the general three-dimensional boundary value problem of linear elasticity by the use of series-expansions. It leads to a set of power series in one or two characteristic parameters, which are truncated after a given power, defining the order of the approximating theory. Finally, a so-called pseudo reduction of the resulting PDE system in the unknown displacement coefficients is performed, as the last step of the derivation of a consistent theory. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the variables of the main differential equation system, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. To find a valid pseudo reduction by inserting the PDEs of the original system into each other is a complicated and very time-consuming task for higher-order theories. Therefore, an structured algorithm seeking all possibilities of valid pseudo reductions (to a given number of PDEs in a given number of variables) is presented. The key idea is to reduce the problem to finding a solution of a linear equation system, by treating each product of different powers of characteristic parameters with the same variable as formally independent variables. To this end, all necessary equations, which can be build from the original PDE system, have to be identified and added to the system a-priori.

Direct Approach Versus Consistent Approximation

Relations between plate theories resulting from the direct approach and the consistent approximation are established and the resulting equations are compared. By introducing a scalar measure for the thickness strain, both theories can be reconciled within a consistent second-order approximation.

Teaching Nonlinear Mechanics: An Extensive Discussion of a Standard Example Feasible for Undergraduate Courses

Most courses on mechanics of materials use a linearized (second-order) buckling analysis of a simple elastic system as an introductory example. We propose to start with a third-order buckling analysis instead, to enable the students to understand the crucial load-response diagrams from the beginning of the course. We present an extensive mathematical discussion of an extended standard introductory example, leading to an easy-to-implement plotting routine for load-response diagrams. The resulting diagrams are interpreted in physical terms. An implementation of the plotting algorithm using Maplesoft MapleTM is attached.

Dimensioning of thick-walled spherical and cylindrical pressure vessels

In this contribution, we revisit the rather classical problem of Lamé and provide a novel and easy way to plot the stress distributions and the overall absolute maximum von Mises stress for arbitrary parameters in only two diagrams. We also provide a maximum hoop stress formula for combined loading and an extensive discussion covering the accuracy of dimensioning via the maximum hoop stress instead of the maximum von Mises stress, as well as the accuracy of the classical approximative hoop stress formulas.