Implementation of С¹-continuous finite element method within strain gradient elasticity for crack problems
DOI:
https://doi.org/10.7242/1999-6691/2026.19.1.5Keywords:
Strain gradient elasticity, crack problems, asymptotic solutions, fracture mechanics, numerical calculations, UEL, enriched elementsAbstract
Strain gradient elasticity (SGE) offers the possibility of regularizing a wide class of problems in classical solid mechanics, including those involving cracks, sharp corners, concentrated loads, dislocations, etc. Within the framework of SGE, it is assumed that the strain energy density depends not only on the strains themselves but also on their spatial derivatives, which imposes additional smoothness requirements on the strain and stress fields realized in SGE solutions. This work presents an implementation of a С¹-continuous finite element method, based on a simplified strain gradient elasticity theory, in the Abaqus/CAE software suite. Standard and enriched three-node elements are presented; 5th-order polynomials are used as shape functions. The implementation was carried out using a user subroutine UEL and a custom user interface. Based on calculations performed for mode I, mode II, and mixed-mode crack opening problems, the mesh convergence of the obtained solutions is investigated, and estimates of the stress distribution near the crack tip are provided. The computational efficiency of the enriched elements compared to the standard ones is also demonstrated, due to their faster convergence rate. The use of enriched elements allows for obtaining the amplitude coefficients of the asymptotic SGE solutions, which, in turn, enable the calculation of the J-integral and the application of energy criteria from linear fracture mechanics while accounting for the size effect.
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