Contact problem for magneto-electro-elastic half-plane materials indented by a moving punch. Part I: Closed-form solutions

A theoretical model is developed for the exact contact analysis of magneto-electro-elastic half-plane materials indented by a moving rigid punch in this paper, which is Part I of a series of papers. A numerical analysis based on this theoretical model will be presented in Part II. The Galilean transformation and the Fourier sine and cosine transforms are applied to make the transient problem tractable. Detailed analyses of the eigenvalue distributions of the double-biquadrate order characteristic equation related to the magneto-electro-elastic governing equations are performed. Real fundamental solutions are derived for each eigenvalue distribution. The punch may have a flat or cylindrical profile and may be electrically and magnetically conducting, electrically conducting and magnetically insulating, electrically insulating and magnetically conducting, or electrically and magnetically insulating. For each type of punch, the singular integral equations are derived with the surface contact stress, surface electric charge, and/or surface magnetic induction inside the contact region as the unknown functions. Exact solutions to the system of integral equations are obtained. In particular, closed-form expressions for the stresses, electric displacements, and magnetic inductions in terms of fundamental functions are derived, which provide a scientific basis for the interpretation of the contact behaviors of multiferroic materials as will be shown in Part II of this series of papers.