A superconvergent elastically coupled double beam element for analysis of adhesively bonded lap joints

Abstract

A novel elastically coupled double beam (ECDB) finite element with superconvergent properties is developed to primarily study wave propagation in the bonded region of an adhesively bonded single lap joint. An ECDB element consists of two axial-flexural-shear coupled beams coupled with continuously distributed linear elastic springs. A general formulation applicable to both metals and composites is proposed. An exact solution to the static part of six-coupled governing equations of motion is obtained, which is then used to formulate exact stiffness and mass matrices. Consequently, the ECDB element thus formulated has superconvergent properties and is inherently free of shear locking. It is shown that merely a single ECDB element is sufficient to accurately capture the tip deflections of a cantilever double beam type geometry subjected to static loading, following which eigenvalue analysis is performed, and comparisons are made with the commercial finite element software—Abaqus. Further, wave propagation studies are carried out to demonstrate the efficiency of the element in evaluating ultrasonic wave responses across various metallic and composite ECDBs, and comparisons are made with Abaqus and frequency-domain spectral finite element (SFEM) model described in Paunikar and Goapalakrishnan (“Wave propagation in adhesively bonded metallic and composite lap joints modelled through spectrally formulated elastically coupled double beam element,” Compos. Struct., under review). Following this, wave propagation in symmetric metallic, geometrically asymmetric metallic, and symmetric laminated composite lap joints with strong and weak bonding is studied by employing two-noded Lagrangian frame elements to mesh the adherends and superconvergent elastically coupled double beam (SECDB) elements to mesh the bonded region. The results obtained from the superconvergent finite element (SCFE) simulations are compared with those given by Abaqus and SFEM. Lastly, the SCFE models for the cases of perfectly bonded aluminium and symmetric ply laminated composite are experimentally validated. We have shown that the SECDB element developed in this work may be used to efficiently carry out static and dynamic analysis in any symmetric or asymmetric bonded joint comprising of only metals or composites or a combination of the two. It is also demonstrated that various levels of adhesion in a bonded joint can be simulated by varying the coupling spring stiffness value. Additionally, the SECDB element may also be used to study other double beam like coupled structures, for instance, space platforms or dynamic vibration absorbers.

Keywords:

• Superconvergent finite element method
• adhesively bonded single lap joint
• elastically coupled double beam
• wave propagation

Acknowledgements

The authors are grateful to their collaborators—Dr. David Barnoncel from Ariane Group for supplying the experimental samples, Prof. Krishnan Balasubramaniam and Prof. Prabhu Rajagopal from Centre of Non-destructive Evaluation, Indian Institute of Technology Madras, India, and Prof. Michel Castaings from University of Bordeaux, France for allowing the authors to use their experimental facilities. The authors would like to thank Mr. Sreedhar Puliyakote, University of Bordeaux, France and Mr. Dileep Koodalil, Indian Institute of Technology Madras, India for helping the authors in conducting the experiments.

Disclosure statement

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

Additional information

Funding

This work was supported by Centre Franco-Indien pour la Promotion de la Recherche Avancée—CEFIPRA Project No. 7119—EMAT.