# Singular elastic solutions in corners with spring boundary conditions under anti-plane shear

## Abstract

A new analytical procedure is developed for the deduction of the asymptotic series of the singular solutions in displacements and stresses near the vertex of the linear elastic isotropic corners with the Dirichlet–Robin (fixed-spring) and Neumann–Robin (free-spring) boundary conditions. Under the assumption of antiplane shear loading, the corresponding elastic problem reduces to the Laplace equation for the out-of-plane displacement. In the deduction of such singular solution, the complex variable is used to propose a harmonic function in the form of an asymptotic series including both power and logarithmic terms. This original procedure is suitable for its implementation in a computer algebra software which makes all the necessary symbolic computing, simplifications and rearrangements. This is a key issue due to the fact that the complexity of terms in these series may increase with increasing order of terms. These series are composed by the main terms (also called main singularities), solutions of the corresponding Dirichlet–Neumann or Neumann–Neumann problems, and the associated finite or infinite series of the so-called shadow terms (also called shadow singularities). These terms are determined by solving systems of recursive inhomogeneous Dirichlet–Neumann or Neumann–Neumann problems, respectively. A general classification of the behaviours of the asymptotic series covering all the considered corner problems is introduced. A few examples of the asymptotic series for corners with Dirichlet–Robin and Neumann–Robin boundary conditions are presented to illustrate the capabilities of this procedure.

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## Notes

1. 1.

The following subsets of real numbers will be used in this work: $${\mathbb {Z}}$$ the set of integer numbers, $${\mathbb {N}}$$ the set of natural numbers, $${\mathbb {Q}}$$ the set of rational numbers, and $${\mathbb {N}}_{0}={\mathbb {N}}\cup \{0\}$$ the set of non-negative integers.

2. 2.

The authors thank to Professor Serge Nicaise (Université Polytechnique Hauts-de-France, Valenciennes) for calling their attention to this non-uniqueness issue.

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## Acknowledgements

The authors thank to Professor Serge Nicaise (Université Polytechnique Hauts-de-France, Valenciennes) for calling their attention to the issue of non-uniqueness of shadow terms. The research was conducted with the support of the Spanish Ministry of Science, Innovation and Universities and European Regional Development Fund (Project PGC2018-099197-B-I00), and the Junta de Andalucía and European Regional Development Fund (Projects US-1266016 and P18-FR-1928).

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Jiménez-Alfaro, S., Villalba, V. & Mantič, V. Singular elastic solutions in corners with spring boundary conditions under anti-plane shear. Int J Fract 223, 197–220 (2020). https://doi.org/10.1007/s10704-020-00443-5

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