A fundamental solution for the harmonic vibration of laminated composite plates with coupled dynamic bending and quasist
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O R I G I NA L
C. H. Daros
A fundamental solution for the harmonic vibration of laminated composite plates with coupled dynamic bending and quasistatic extension
Received: 9 January 2020 / Accepted: 27 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We obtain here a new fundamental solution for the harmonic vibration of asymmetric, laminated, anisotropic plates. The fundamental solution is derived via the Fourier transform and its final form is given in terms of definite integrals, which are evaluated numerically. Moreover, we present some of the higher order derivatives of the solution and their explicit spatial singularities, which are necessary for a boundary element method implementation. Keywords Fundamental solution · Fourier transform · Composite plates · Harmonic vibration · Bending extension coupling · Anisotropic plates 1 Introduction The aeronautical industry has witnessed an increasing use of laminated composites in the last few decades. These composites have allowed, e.g. the reduction in weight and the increase of strength in modern airplanes. One key structural component within the realm of laminated composites is the plate. Plates allow the reduction of certain 3-D problems into 2-D problems, hence reducing the computational costs with excellent results. There are several plate theories with different degrees of complexity. The most widespread plate theory is Kirchoff’s plate theory, which allows a simple and successful description of thin plates. Similarly, lamination theory is the classical laminated plate theory, which is an extension of the Kirchhoff plate theory to thin laminated composite plates. However, multilayer asymmetric laminated plates imply the existence of bending-stretching coupling, which considerably increases the mathematical complexity of the problem. Hence, numerical methods are unavoidable to assess plates with complicated contours or complex boundary conditions. There are several papers which focus on the use of the finite element method (FEM) and other numerical methods to model multilayer asymmetric laminated plates (see, e.g. [3,14–16]). Although FEM is undeniably the most general and successful numerical method in structural mechanics, the boundary element method (BEM) is a powerful alternative. Besides the well-known reduction in the dimensionality of the problem, the BEM is specially useful in assessing the results near the problem’s boundary. Conventional FEM formulations present much poorer results near boundaries when compared with the BEM. Hence, for special cases when high precision is required the BEM can outperform the conventional FEM. However, the BEM requires the existence of a fundamental solution for the considered differential equation, which can sometimes be difficult or even impossible to obtain. For the special case of static, laminated anisotropic plates we can cite some early fundamental solutions obtained by Becker and Zakharov [1,2,19,21]. An alternative to previous derivations is the use of complex C. H. Daros (B)
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