Christian Constanda - The Generalized Fourier Series Method: Bending of Elastic Plates

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The process of bending of elastic plates is of considerable importance in a variety of applications, from civil engineering projects to ventures in the aerospace industry. Since the inception of its study, many mathematical models have been constructed and used for the qualitative and quantitative investigation of its salient features. Leaving aside the particulars of individual cases, these models can be classified into two main categories: Kirchhoff type [24] and MindlinReissner type [27, 28]. The former, which reduces to the solution of one fourth-order partial differential equation, is easier to handle but produces only basic information. The latter consists generally of a system of three second-order equations and accompanying conditions, and is regarded as more refined in the sense that it trades the simplicity of the Kirchhoff model for additional details on the evolution of the physical phenomenon. However, the enhanced sophistication of the MindlinReissner models is achieved at the expense of a certain amount of lack of mathematical consistency. By contrast, the model that forms the object of our attention, described in [15] and other published material, has no such drawback, being based exclusively on the averaging (across the thickness of the plate) of the exact equations of three-dimensional linear elasticity, coupled with an assumption on the form of the displacement vector.

Our model has been studied extensively in the literature. The equilibrium problem is treated in [15] and [1] (see also [14]), where the questions of existence, uniqueness, and integral representation of the solution are answered in full; the comprehensive discussion of the time-dependent case can be found in [2]; the harmonic plate oscillations are analyzed in [29]; thermoelastic issues are examined in [313]; and a boundary element method is developed in [23] for the deformation of a plate on an elastic foundation.

As all scientists and engineers know only too well, it is very seldom that the solution of a mathematical model can be expressed in closed formthat is, by means of a specific, well-defined set of functions. In general, we have to be satisfied instead with a numerical approximation, accurate enough to be acceptable for practical purposes. Our book develops precisely this type of procedure, which makes use of generalized Fourier series and can be applied to any mathematical model governed by an elliptic equation or system with constant coefficients. We chose this particular one not only because of its utilitarian value but also because of its complexity, which showcases the power, sophistication, and adaptability of the proposed technique.

The idea behind such a method was put forward earlier in [26], in the context of three-dimensional elasticity. However, that construction does not work for our model, which is governed by a hybrid system of three equations with three unknown functions in only two independent variables, and with a very complicated far-field pattern that requires a novel way of handling.

In Chap. ), which is designed as a template that contains all the calculations and explanatory graphs supporting the mathematical and computational arguments. Some of these details, deemed more ancillary than essential, are not included in the other cases since the interested reader can develop them easily by following the patterns established for the interior Dirichlet problem.

Additional comments, together with a comprehensive guide to the use of the Mathematica software, the main tool for the numerical segments in the book, can be found in the Appendix. Since we wanted to help the reader by making independent, self-contained presentations of each problem, we saw it necessary to insert some identical portions of explicative text in all of them. But these are short passages that, we hope, will not detract from the fluency of the narrative.