Crack analysis of size-dependent piezoelectric solids under a thermal load

General 2D boundary value problems of piezoelectric nano-sized structures with cracks under a thermal load are analyzed by the finite element method (FEM). The size-effect phenomenon observed in nano-sized structures is described by the strain-gradient effect. The strain gradients are considered in the constitutive equations for electric displacement and the high-order stress tensor. For this model, the governing equations and the corresponding boundary conditions are derived using the variational principle. Uncoupled thermoelasticity is considered; thus, the heat conduction problem is analyzed independently of the mechanical fields in the first step. The veracity of the derived formulations and their implementation into the finite element scheme is demonstrated by some numerical examples.