We consider an anisotropic body bounded by a cylindrical surface. Suppose that the body is infinitely long in the axial direction and is loaded by boundary conditions which do not vary along the generator. We show that our previous correspondence relations between plane piezoelectricity and generalized plane strain in elasticity can be extended to a more general loading situation. This is accomplished by incorporating a constant axial strain and a uniform temperature change in the formulation. Specifically, we show that by setting a linkage of 21 elastic (electroelastic) constants and seven thermal quantities, the deformation of a general anisotropic elastic solid can be characterized by a certain deformation mode of a piezoelectric solid with identical geometry. Applied to inhomogeneous media, they imply that the correspondence also holds for effective tensors. The formulation suggests that, for a solid whose microstructure and fields are invariant with the x3-axis, the derivations can be much clarified if the constitutive equations are properly rearranged. Applied to two-phase fibrous composites, we demonstrate that several known exact relations can be reconstructed in a systematic manner based on the uniform field approach. These include Hill's universal relations, Levin's correspondence results between thermal and mechanical properties, and Rosen and Hashin's formula between specific heat and thermo-mechanical moduli.