This chapter discusses the main elements of continuum theory, which will be utilized further in later sections dealing with a variety of applications. First, a basic review of vector and tensor operations is presented, followed by descriptions of stress and strain tensors, and their relationships through thermodynamic functions. The special conditions of anisotropic crystals are also discussed. Green's tensor functions are derived for the displacement field of a material in response to a point force, for both isotropic and anisotropic crystals. Then, the methods of Eshelby for inclusions and inhomogeneities are presented, with analytical solutions to special problems of a second phase within a perfect crystals. This is followed by a presentation of the basic elements of crystal plasticity, and its connection with dislocation (or incompatibility) fields. The elastic fields of complex dislocation loop ensembles are finally given for both isotropic and anisotropic crystals.
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