![Schematic of a single edge dislocation with slip plane normal vector n. The origin of the coordinates is centred in the considered domain and the grey area indicates the domain where the plastic distortion is provided by the gradient theory solution; to match peridynamics and Helmholtz type gradient elasticity, the horizon δp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{\text {p}}$$\end{document} must be taken proportional to the length scale parameter lH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{\text {H}}$$\end{document} of the gradient theory](https://www.researchgate.net/publication/377980317/figure/fig1/AS:11431281232019980@1711597018575/Schematic-of-a-single-edge-dislocation-with-slip-plane-normal-vector-n-The-origin-of-the_Q320.jpg)
![Stress field components of a singular dislocation in peridynamics, the Burgers vector points in x-direction and the glide plane trace corresponds to the x-axis; left: σxy(x,y=0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{xy}(x,y=0)$$\end{document}, right: σxx(x=0,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{xx}(x=0,y)$$\end{document}; the top graphs show the original results for different values of the horizon where δp=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{\text {p}} =$$\end{document} 3nm, 6nm, 9nm and 12nm (marked H3, 6, 9, 12 in the graph labels); the bottom graphs show the same data after rescaling σ→δpσ,r→r/δp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\sigma } \rightarrow \delta _{\text {p}} \varvec{\sigma }, \varvec{r} \rightarrow \varvec{r}/\delta _{\text {p}}$$\end{document}; in these graphs the locations of the collocation points at which virial stresses are evaluated are marked by open circles](https://www.researchgate.net/publication/377980317/figure/fig3/AS:11431281232112112@1711597019039/Stress-field-components-of-a-singular-dislocation-in-peridynamics-the-Burgers-vector_Q320.jpg)
![Angle dependence of the dislocation stress field components, δp=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{\text {p}} =$$\end{document} 12nm, other parameters as in Fig. 3; top: 3D representation, bottom: contour plot](https://www.researchgate.net/publication/377980317/figure/fig4/AS:11431281232070932@1711597019263/Angle-dependence-of-the-dislocation-stress-field-components_Q320.jpg)
![Effect of micro-modulus distribution on stress of a singular dislocation; all parameters as in Fig. 3; left: σxx(x=0,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{xx}(x=0, y)$$\end{document} for δp=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{\text {p}} =$$\end{document} 12nm, simulated using a constant micro-modulus and a ’conical’ micro-modulus distribution; right: scaling collapse obtained by plotting lσxx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\sigma _{xx}$$\end{document} vs. y/l where l=δp3/5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l = \delta _{\text {p}} \sqrt{3/5}$$\end{document} for data evaluated with constant and l=δp2/5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l= \delta _{\text {p}} \sqrt{2/5}$$\end{document} for data evaluated with ’conical’ micro-modulus](https://www.researchgate.net/publication/377980317/figure/fig5/AS:11431281232070934@1711597019583/Effect-of-micro-modulus-distribution-on-stress-of-a-singular-dislocation-all-parameters_Q320.jpg)
February 2024
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103 Reads
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2 Citations
Journal of Materials Science: Materials Theory
Modeling dislocations is an inherently multiscale problem as one needs to simultaneously describe the high stress fields near the dislocation cores, which depend on atomistic length scales, and a surface boundary value problem which depends on boundary conditions on the sample scale. We present a novel approach which is based on a peridynamic dislocation model to deal with the surface boundary value problem. In this model, the singularity of the stress field at the dislocation core is regularized owing to the non-local nature of peridynamics. The effective core radius is defined by the peridynamic horizon which, for reasons of computational cost, must be chosen much larger than the lattice constant. This implies that dislocation stresses in the near-core region are seriously underestimated. By exploiting relationships between peridynamics and Mindlin-type gradient elasticity, we then show that gradient elasticity can be used to construct short-range corrections to the peridynamic stress field that yield a correct description of dislocation stresses from the atomic to the sample scale.
