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Enriched $C^1$ finite elements for crack problems in simplified strain gradient elasticity

Yury Solyaev, Vasiliy Dobryanskiy

TL;DR

This paper addresses crack problems in plane-strain simplified strain gradient elasticity (SGE) by introducing enriched triangular $C^1$ finite elements. The enrichment embeds near-field SGE crack asymptotics into the standard Bell-triangle $C^1$ interpolation, preserving $C^1$ continuity and enabling direct computation of the asymptotic amplitudes $K_1\!\ldots\!K_4$ and the J-integral. Numerical tests on mode I and II cracks show improved convergence on coarse meshes, and reveal a linear dependence of the amplitudes on crack length for longer cracks, with gradient-strengthening reducing the energy release rate relative to the classical theory. The approach provides a practical tool for identifying the SGE length scale $\ell$ and validating size-effects in fracture of quasi-brittle materials, with implications for extended FEM formulations and 3D extensions in future work.

Abstract

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

Enriched $C^1$ finite elements for crack problems in simplified strain gradient elasticity

TL;DR

This paper addresses crack problems in plane-strain simplified strain gradient elasticity (SGE) by introducing enriched triangular finite elements. The enrichment embeds near-field SGE crack asymptotics into the standard Bell-triangle interpolation, preserving continuity and enabling direct computation of the asymptotic amplitudes and the J-integral. Numerical tests on mode I and II cracks show improved convergence on coarse meshes, and reveal a linear dependence of the amplitudes on crack length for longer cracks, with gradient-strengthening reducing the energy release rate relative to the classical theory. The approach provides a practical tool for identifying the SGE length scale and validating size-effects in fracture of quasi-brittle materials, with implications for extended FEM formulations and 3D extensions in future work.

Abstract

We present a new type of triangular finite elements developed for the plane strain crack problems within the simplified strain gradient elasticity (SGE). The finite element space contains a conventional fifth-degree polynomial interpolation that was originally developed for the plate bending problems and subsequently adopted for SGE. The enrichment is performed by adding the near-field analytic SGE solutions for crack problems preserving continuity of interpolation. This allows us an accurate representation of strain and stress fields near the crack tip and also results in the direct calculation of the amplitude factors of SGE asymptotic solution and related value of J-integral (energy release rate). The improved convergence of presented formulation is demonstrated within mode I and mode II problems. Size effects on amplitude factors and J-integral are also evaluated. It is found that amplitude factors of SGE asymptotic solution exhibit a linear dependence on crack size for relatively large cracks.
Paper Structure (13 sections, 45 equations, 6 figures)

This paper contains 13 sections, 45 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Simplified strain gradient elasticity theory
  3. Asymptotic solution for crack problems
  4. Finite element formulation
  5. Weak form
  6. Standard $C^1$ finite elements
  7. Enriched $C^1$ finite elements
  8. Integration scheme
  9. Examples
  10. Comparison between the conventional and enriched elements
  11. Convergence analysis
  12. Amplitude factors of asymptotic solution and J-integral
  13. Conclusions

Figures (6)

  • Figure 1: Integration Gauss points used for the conventional elements and enriched elements (a -- reduced 13-points scheme, b -- 25 points, c -- 30 points, b -- 37 points)
  • Figure 2: Boundary conditions and examples of finite element mesh used in the test problems. (a): mode I problem, (c): mode II problem, (d): Enriched elements (blue color) around the crack tip. Cracks are highlighted with red color.
  • Figure 3: Distribution of field variables along the lower boundary of the domain containing crack. (a) - normal displacements in mode I problem, (b) - Cauchy stresses in mode I problem, (c) - magnified view of figure (b) around the crack tip, (d) - Cauchy stresses in mode II problem. Classical asymptotic solution is shown by black dotted line. In plot (a), SGE asymptotic solution is show by blue dotted line
  • Figure 4: Dependence of maximum stress concentration in mode I (a) and mode II (b) on the relative size of elements placed around the tip of crack. Dashed orange lines correspond to the solutions obtained with the smallest mesh
  • Figure 7: Dependence of amplitude factors of SGE asymptotic solution (left) and J-integral on the relative length of crack under mode I (a) and mode II (b) loading conditions. Classical solutions for J-integral are shown by black dotted line
  • ...and 1 more figures