Wave Propagation in Adhesively Bonded Single Lap Joints: Solution of Forward and Inverse Problems

Abstract

The use of adhesives for bonding composites as well as metals is rapidly increasing since it provides uniform load transfer and is free of stress concentration seen around the drilled holes in bolted or riveted joints. However, the strength of an adhesive bond depends on the surface preparation before bonding and curing of the adhesive, fatigue loading, and environmental factors. Therefore, we need a robust and fast technique for inspection of such joints, and one such technique is ultrasonic guided wave inspection technique. Ultrasonic inspection is a highly studied technique for various aspects of wave propagation and bond quality in different bonded joints, most of which is experimental in nature. The numerical studies on ultrasonic wave propagation are mainly carried out using Finite element method. However, it is essential to discretize the finite element model with a mesh size that is at least 8 - 20 times smaller than the smallest excitation wavelength, which leads to large and hardware intensive simulations. This severely restricts the use of high excitation frequencies and the dimensions of waveguides which can be analysed for bond strengths. To overcome this issue, we propose to use the frequency domain based spectral finite element method (SFEM) to study ultrasonic wave propagation in adhesively bonded single lap joints (ABSLJs). With the intention of studying the effects of the reduction of in-plane bond strength, we aim to develop a one-dimensional SFEM model for studying wave propagation in ABSLJs. We propose to develop a spectral elastically coupled double beam (ECDB) element, which is made up of two frames coupled using elastic springs distributed along the whole span of the joint. It is suggested that various levels of bonding of a bonded region can be represented by varying the spring stiffness. A general spectral ECDB element is developed and studied for four representative cases, namely, a symmetric metallic ECDB, a metallic ECDB with geometric asymmetry, a geometrically symmetric ECDB with symmetric laminates, and a geometrically symmetric ECDB with anti-symmetric laminates. Next, we develop the superconvergent elastically coupled double beam element (SECDB) using the exact solutions of the static part of the governing equations of motion to overcome the modelling issues associated with polynomial shape functions employed in the conventional finite element method. We then validate the superconvergent stiffness and mass matrices by solving static deflection, eigenvalue, and wave propagation problems. The spectral ECDB and SECDB elements are verified by FE simulations performed using commercial software. Then, we develop two semi-analytical models for ABSLJs. The first one is based on frequency domain SFEM using the spectral frame elements and spectral ECDB elements to represent the adherends and bonded region, respectively, of an ABSLJ. The second one is a time domain model in which adherends and bonded region are meshed using Lagrangian frame elements and SECDB elements, respectively. Both the computational models are verified using two-dimensional planar FE simulations. Experimental validation is also carried out for defect-free adherends and perfectly bonded aluminium and composite lap joints. In the last part, we investigate the capabilities of the spectral frame elements, spectral ECDB elements, and the SFEM model for ABSLJs in solving two inverse problems of unknown source force reconstruction and material properties determination, which are of high practical significance.