A comprehensive investigation of electromechanical brittle fracture in functionally graded piezoceramics using an adaptive phase-field approach

The assessment of brittle fracture behaviour of functionally graded piezoelectric materials (FGPMs) subjected to coupled electromechanical fields is crucial for ensuring their structural integrity and optimizing their performance. By diffusing sharp cracks into a continuous damage field, the phase-field approach offers a regularized representation of discrete cracks, enabling simulation of complex fracture mechanisms under coupled multiphysics loadings. A coupled phase-field and isogeometric-meshfree (IGA–MF) approach is developed herein to investigate fracture phenomena in FGPMs subjected to electromechanical loading. The IGA–MF formulation is built upon an equivalence between isogeometric and meshfree basis functions, through which the precise geometric description of IGA is effectively coupled with the adaptive refinement flexibility of meshfree methods. Computational efficiency is improved by embedding an adaptive mesh refinement scheme in the framework, where refinement is controlled through an error indicator evaluated from the phase-field variable and its spatial gradient. The developed framework is applied to examine the fracture behaviour of piezoelectric materials with horizontal and vertical gradation of material properties. Through a series of two-dimensional numerical studies, the influence of material gradation, electric field direction, and electromechanical operational modes on fracture behaviour is comprehensively explored. The findings highlight how gradation direction and profiles, and electric field polarity can be leveraged to tune fracture resistance, offering valuable insights for the design and optimization of smart graded piezoelectric structures.