Electromechanics of Dielectric and Piezoelectric Crystals With Point, Line, and Surface Defects

摘要

A mathematical continuum theory is developed to describe the electromechanical behavior of dielectric and piezoelectric solids containing imperfections. The macroscopic distortion field consists of recoverable elasticity, deviatoric plasticity arising from dislocation glide, and volumetric deformation from vacancies. A connection on the spatial manifold of deformed lattice vectors describes gradients of stretch and rotation at the microscale caused by various lattice defects. It is shown that parallel transport of a lattice director vector with respect to this connection about a closed loop yields a discontinuity with contributions from the torsion of the connection (physically, from dislocations) and curvature (physically, from domain walls and vacancy gradients). Classical balance laws of electrostatics and mass and momentum conservation are implemented. A free energy function dependent upon lattice distortion, polarization, temperature, and defect densities is suggested. Thermodynamically consistent kinetics relations for dislocation glide and vacancy diffusion are then derived, with the chemical potential for the latter depending upon defect density, electric potential, hydrostatic pressure, and elastic energy density. Vacancy migration and mass rearrangement at the free surface of the substance are also considered explicitly.