A plastic analysis of Griffith crack problem in 1D hexagonal piezoelectric quasicrystals

The elastic–plastic fracture mechanics of one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) Griffith crack under a small-scale yielding is studied. Due to the properties of this material, the atomic cohesive force zone of the phonon field is the smallest. Based on the theory of distributed dislocation, the mechanical and electrice coupling model for the elastic–plastic fracture mechanics of 1D hexagonal piezoelectric QCs Griffith crack under a small yield range is established for the first time. The crack opening is arrested by prescribing the cohesive loads of yield point phonon field, phason field and electric field over the phonon field atomic cohesive force zone, the phason field atomic cohesive force zone and saturation zone rims, respectively. Without loss of generality, two cases are considered. Using Dugdale method, the corresponding size of atomic cohesive force zone is obtained. By using Fourier transform and the integral equation method, the closed analytical expressions of phonon field crack opening displacement (COD), phason field COD, crack opening potential drop (COP) and J-integral are obtained. Numerical analysis results show that crack arrest is possible in 1D hexagonal piezoelectric QCs under small-scale yielding, which provides a theoretical basis for the application of QCs materials.