Skip to main content

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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17


  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.


  1. 1.

    Alnæs, M. S., Hake, J., Kirby, R. C., Langtangen, H. P., Logg, A., Wells, G. N.: The FEniCS manual. FEniCS Project, version October 31st (2011)

  2. 2.

    Barroso, A., Cañas, J.: Uniones en estructuras aeronáuticas, uniones adhesivas y remachadas. Universidad de Sevilla, Seville (2019)

    Google Scholar 

  3. 3.

    De Sterck, H., Manteuffel, T., McCormick, S., Nolting, J., Ruge, J., Tang, L.: Efficiency-based h-and hp-refinement strategies for finite element methods. Numer. Linear Algebra Appl. 15(2–3), 89–114 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Jiménez-Alfaro, S., Villalba, V., Mantič, V.: Singular elastic solutions in corners with spring boundary conditions under anti-plane shear. Int. J. Fract. 223(1–2), 197–220 (2020)

    Google Scholar 

  5. 5.

    Johnson, W.S.: Adhesively Bonded Joints: Testing, Analysis, and Design, vol. 981. ASTM International, West Conshohocken (1988)

    Book  Google Scholar 

  6. 6.

    Mantič, V., Távara, L., Blázquez, A., Graciani, E., París, F.: A linear elastic—brittle interface model: application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads. Int. J. Fract. 195, 15–38 (2015)

    Article  Google Scholar 

  7. 7.

    Massabò, R.: The bridged-crack model. Nonlinear Crack Models for Nonmetallic Materials, pp. 141–208. Springer, Dordrecht (1999)

    Chapter  Google Scholar 

  8. 8.

    Pantelakis, S., Tserpes, K.: Revolutionizing Aircraft Materials and Processes. Springer, New York (2020)

    Book  Google Scholar 

  9. 9.

    Perelmuter, M.: Structures with bridged cracks and weak interfaces. Proc. Struct. Integr. 13, 793–798 (2018)

    Article  Google Scholar 

  10. 10.

    Távara, L., Mantic, V., Graciani, E., Cañas, J., París, F.: Analysis of a crack in a thin adhesive layer between orthotropic materials: an application to composite interlaminar fracture toughness test. Comput. Model. Eng. Sci. (CMES) 58(3), 247–270 (2010)

    MATH  Google Scholar 

  11. 11.

    Valoroso, N., Champaney, L.: A damage-mechanics-based approach for modelling decohesion in adhesively bonded assemblies. Eng. Fract. Mech. 73(18), 2774–2801 (2006)

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to S. Jiménez-Alfaro.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



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

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

whereas for the D–R

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


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