Enriched C 1 Finite Elements for Crack Problems in Simplified Strain Gradient Elasticity

We present a new type of triangular $C^1$ finite element developed for plane strain crack problems within the framework of simplified strain gradient elasticity (SGE). The finite element space incorporates a conventional fifth-degree polynomial interpolation originally developed for plate bending problems and later adopted for SGE. Enrichment is performed by adding near-field analytic SGE solutions for crack problems, preserving $C^1$ continuity of interpolation at the mesh nodes. This allows us an accurate representation of strain and stress fields near the crack tip and enables direct calculation of the amplitude factors of the SGE asymptotic solution, along with the corresponding value of the J-integral (energy release rate). The improved convergence of the proposed formulation is demonstrated for mode I and mode II problems. Size effects on the amplitude factors and the J-integral are also evaluated. It is found that the amplitude factors of the SGE asymptotic solution exhibit a linear dependence on crack size for relatively large cracks.