Complex Variable Formalism

Abstract

In engineering applications if a thin plate is considered, most of the stress analyses will concentrate on the plane variation instead of the distribution in thickness direction, and hence the complex variable formalism which combines two variables into one becomes a useful and powerful tool. When a thin plate is subjected to inplane loads, uniform distribution across the plate thickness is usually assumed for the deformations and stresses, and hence the two-dimensional analysis can be employed. When a thin plate is under transverse loads or bending moments, linear distribution across the plate thickness is usually assumed and the analysis is generally called plate bending analysis. By taking the stress resultants and the deformations of the middle surface as the basic functions, the plate bending analysis can also be treated with two plane coordinate variables. When the thin plates are made by laying up various layers, the inplane and plate bending deformations may uncouple or couple depending on the symmetry or unsymmetry with respect to the middle surface of the plate. In this case, the coupled stretching-bending analysis should be employed. Since all these kinds of problems have been well treated by using the theory of complex variable, summarizing the formalisms introduced in the book (Hwu Anisotropic Elastic Plates, Springer, New York, 2010) several complex variable formalisms are presented in this chapter such as Lekhnitskii formalism and Stroh formalism for two-dimensional analysis and plate bending analysis, and Stroh-like formalism for the coupled stretching-bending analysis. In addition, the extended versions of Stroh formalism for thermoelastic problems and piezoelectric materials are also presented in the related sections of this chapter. Further extensions to magneto-electro-elastic and viscoelastic materials will be described in Chaps. 11 and 12. Moreover, through the use of Radon transform, the Stroh formalism can also be applied to the three-dimensional analysis, whose detailed description will be presented in Sect. 15.9.1.