Indentation of a punch with chemical or heat distribution at its base into transversely isotropic half-space: Application to local thermal and electrochemical probes

The exact solution to the coupled problem of indentation of the punch, subjected to either heat or chemical substance distribution at its base, into three-dimensional semi-infinite transversely isotropic material is presented. The entire set of field components are derived in terms of integrals of elementary functions using methods of the potential theory and recently obtained, by the authors, results for the general solution of the field equations in terms of four harmonic potential functions. The exact solution for the stiffness relations that relate applied force, total chemical diffusion/heat flux in the domain of the contact, with indenter displacement, temperature, or chemical substance distribution of diffusing species at the base, and materials' chemo/thermo-elastic properties are obtained in closed form and in terms of elementary functions. These results can be used to understand the image formation mechanisms in techniques such as thermal scanning probe microscopy and electrochemical strain microscopy.