General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method

摘要

The three-dimensional equations of transversely isotropic magneto-electro-thermo-elasticity are simplified by the introduction of two displacement functions. A general solution is then rigorously derived by virtue of the operator theory, which is expressed in terms of two functions, satisfying a second-order and a tenth-order homogeneous partial differential equation, respectively. Utilizing the generalized Almansi's theorem, the general solution can be further simplified to the one expressed by six harmonic functions only. This allows us to extend the potential theory method to the mixed boundary value problems of magneto-electro-thermo-elastic materials. A flat crack in an infinite space subjected to symmetric mechanical, electric, magnetic as well as temperature loads at the crack surfaces is considered for instance. One integral equation and three integro-differential equations are derived, which are similar to those reported in the literature. For a penny-shaped crack subjected to a uniform loads, exact three-dimensional expressions for the full-space magneto-electro-thermo-elastic field are obtained in terms of elementary functions.