FEM Benchmark Problems for Cracks with Spring Boundary Conditions Under Antiplane Shear Loadings
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ORIGINAL ARTICLE
FEM Benchmark Problems for Cracks with Spring Boundary Conditions Under Antiplane Shear Loadings S. Jiménez‑Alfaro1,2 · V. Mantič1 Received: 25 September 2020 / Revised: 25 October 2020 / Accepted: 30 October 2020 / Published online: 18 November 2020 © AIDAA Associazione Italiana di Aeronautica e Astronautica 2020
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 outof-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. Keywords Adhesive joint · Bridged crack · Singular solution · Mode III crack · FEM · FEniCS · Robin boundary condition
1 Introduction The main advantages of adhesive joints are their longer service life under fatigue loads and their lower weight in comparison with other kind of joints, for example, bolted joints. Nevertheless, they have also drawbacks, such as the high reduction of the joint’s mechanical properties when they are applied, at high or low temperatures. In spite of their disadvantages, as a consequence of the outcomes, they are widely used in aeronautics. Furthermore, their use is in constant growth, since composites are getting more and more important in the aircraft. In general, every engineering structure made of composite materials needs adhesives to join its plies. Moreover, some adhesives are used to join structural components. As * S. Jiménez‑Alfaro [email protected]; sara.jimenez‑[email protected] 1
Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain
Institut Jean Le Rond d’Alembert, Sorbonne Universités, CNRS, UMR 7190, 75005 Paris, France
2
an example, in the aircraft F18, the wing, made of carbon fiber, is bonded to the titanium fuselage, according to [5]. However, not only composite components need adhesives. For instance, in the A400M the flap track is partially bonded, see [8]. In this paper, we focus on the mechanical behaviour (displacements and stresses) in the neighbourhood of the tip of a crac
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