Higher-order electromechanical fracture analysis: Single- and mixed-mode nanocracks in flexoelectric solids

Fracture in flexoelectric and strain-gradient elastic solids is governed by higher-order electromechanical interactions that strongly alter the near tip behavior of cracks at the nanoscale. Yet most prior studies are restricted to the simplest crack modes or treat only selected aspects, yielding an incomplete picture. This work develops and verifies a higher-order fracture electromechanics framework that consistently incorporates strain gradient elasticity and flexoelectricity within an electric field based formulation. Analytical asymptotics are first established for Modes I-III, highlighting the role of true stress measures in capturing higher-order singularities. A mixed finite element scheme is then employed to resolve all strain-gradient components and to provide accurate post-processing of electrical and mechanical crack-tip fields. Numerical investigations address (i) a Mode I edge-cracked panel, and (ii) a mixed-mode cracked truncated pyramid. Results show that SGE introduces an r − 3 / 2 singularity near the crack tip and shifts the transition zone between higher-order and classical r − 1 / 2 regimes. Flexoelectric coefficients induce nonlinear variations of the crack-tip electric potential, while the material length scale governs the extent and shape of the crack-tip opening profile. In a truncated pyramid, surface cracks significantly bias electromechanical fields and may influence the experimental back calculation of flexoelectric parameters. The study provides a comprehensive framework for higher-order electromechanical fracture mechanics, with implications for the design and reliability of nano- and microelectromechanical systems. • A unified higher-order fracture framework combining strain-gradient elasticity and flexoelectricity. • Analytical asymptotics for Modes I-III showing r-3/2 true-stress singularities. • A mixed finite-element formulation resolving complete strain gradient tensors. • Nonlinear size-dependent crack-tip fields governed by material length scale l and flexoelectric coefficients are found. • Mixed-mode surface-crack effects on electromechanical fields in truncated pyramid specimens are demonstrated.