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FEM Benchmark Problems for Cracks with Spring Boundary Conditions Under Antiplane Shear Loadings

Abstract

A new analytical solution in the form of asymptotic series is proposed and studied for Mode III crack problems with spring boundary conditions, which are, in the mathematically-oriented literature, referred to as Robin boundary conditions. Under the assumption of antiplane shear loading, the corresponding elastic problem reduces to the Laplace equation for the out-of-plane displacement. Numerical solutions for benchmark problems are obtained, applying the Finite Element Method, to verify this asymptotic approximation. In particular, two problems are studied, Neumann–Robin and Dirichlet–Robin. Both are used to define a partially damaged adhesive interface in which the Linear Elastic Interface Model is applied. The numerical solution is obtained using the software FEniCS, for which the variational formulation of the problem is developed. Then, it is compared to the analytical expressions proposed for the problem, computing a normalized error. Finally, a convergence analysis is presented. Several parameters, such as the stress singularity or another error measure, are used to analyse two different ways to refine the mesh.

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Notes

  1. 1.

    It was created in 2003 by several universities around the world. What makes it different from other FEM software is the mathematical formulation, necessary to execute the software correctly. This formulation is implemented using Python language.

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Acknowledgements

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). The authors also thank to the NewFrac Innovative Training Network (H2020-MSCA-ITN-2019, Ref.: 861061) for the support received.

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Correspondence to S. Jiménez-Alfaro.

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Appendix

Appendix

The recursive problems for obtaining N–R and D–R solutions are the following, for N–R

$$\begin{aligned} \varDelta u_{j}^{(k)}&= 0 \quad \text {in } \quad \varOmega , \end{aligned}$$
(36)
$$\begin{aligned} \quad \frac{\partial u_{j}^{(k)}}{\partial \theta }&=0 \quad \text {on } \quad \varGamma _{{\text {N}}}, \end{aligned}$$
(37)
$$\begin{aligned} \frac{1}{r}\frac{\partial u_{j}^{(k)}}{\partial \theta }&=-\gamma u_{j}^{(k-1)} \quad \text {on } \quad \varGamma _{{\text {R}}}. \end{aligned}$$
(38)

whereas for the D–R

$$\begin{aligned} \varDelta u_{j}^{(k)}&= 0 \quad \text {in } \quad \varOmega , \end{aligned}$$
(39)
$$\begin{aligned} \quad u_{j}^{(k)}&=0 \quad \text {on } \quad \varGamma _{{\text {D}}}, \end{aligned}$$
(40)
$$\begin{aligned} \frac{1}{r}\frac{\partial u_{j}^{(k)}}{\partial \theta }&=-\gamma u_{j}^{(k-1)} \quad \text {on } \quad \varGamma _{{\text {R}}}. \end{aligned}$$
(41)

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Jiménez-Alfaro, S., Mantič, V. FEM Benchmark Problems for Cracks with Spring Boundary Conditions Under Antiplane Shear Loadings. Aerotec. Missili Spaz. 99, 309–319 (2020). https://doi.org/10.1007/s42496-020-00068-w

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Keywords

  • Adhesive joint
  • Bridged crack
  • Singular solution
  • Mode III crack
  • FEM
  • FEniCS
  • Robin boundary condition