Two-Dimensional Inclusion Problems

Abstract

This chapter will briefly review previous 2D work in the literature for elastic materials and then present a formulation for 2D viscous deformations using the Green function and integral formalism as Chap. 11 for 3D problems. In Chaps. 10 and 11 , we have developed formal and, where possible, explicit solutions for a 3D elastic (or viscous) ellipsoid in an infinite elastic (or viscous) medium. Two-dimensional (2D) inclusion problems may be regarded as elliptic cylinders in plane-strain or plane-stress deformations. One may wonder why we need to consider 2D inclusion problems separately. Historically, work on 2D inclusion problems was motivated by seeking more explicit solutions than the formal ones that involve “formidable” integrals [Yang and Chou (J Appl Mech-Trans Asme 43(3):424–430, 1976)]. There is a large amount of literature on 2D elastic inclusion problems since Jaswon and Bhargava (Paper presented at the Mathematical Proceedings of the Cambridge Philosophical Society, 1961), most of which can be found in the books by Mura (Micromechanics of defects in solids. Martinus Nijhoff, 1987) and Ting (Anisotropic elasticity: theory and applications. Oxford University Press, 1996). It is shown in Chap. 11 that there are no more formidable integrals for isotropic systems. The solutions for interior-point elastic problems can be expressed analytically or in terms of elliptic functions. The solutions for the exterior-point isotropic elastic problems can be evaluated quasi-analytically by combining the methods of Ju and Sun (J Appl Mech-Trans Asme 66(2):570–574, 1999) and Jin et al. (J Appl Mech-Trans Asme 78(3), 2011). The solutions for exterior-point viscous (incompressible) problems are also quasi-analytical from Jiang (Tectonophysics 693:116–142, 2016) and Chap. 11 . However, “formidable integrals” remain for anisotropic inclusion problems. The solution to these problems requires integrating the related Green functions and their derivatives, which are expressed in integral forms themselves. The numerical evaluation of these integrals is generally straightforward using appropriate quadrature but typically requires intensive computation resources, especially for the exterior field. Work on 2D inclusion problems is helpful in at least the following three aspects. First, 2D problems have less intensive numerical calculations or even analytical expressions which allow inclusion behaviors to be investigated more thoroughly. Second, some practical problems, including the deformation of thin elastic or viscous plates, may be approximated by 2D plane-strain or plane-stress problems. Third, 2D solutions, analytic or numerical, can serve as valuable verifications and benchmarks for more intensive and complex 3D inclusion systems.