Three-parameter, elastic foundation model for analysis of adhesively bonded joints Jialai Wang à , Chao Zhang Department of Civil, Construction, and Environmental Engineering, The University of Alabama, Tuscaloosa, AL 35487–0205, USA article info Article history: Accepted 25 October 2008 Available online 21 November 2008 Keywords: Interface stress Interface debonding Adhesively bonded joint Single-lap joint Stiffened joint Three-parameter elastic foundation model abstract A novel three-parameter, elastic foundation model is proposed in this study to analyze interface stresses of adhesively bonded joints. The classical two-parameter, elastic foundation model of adhesive joints models the adhesive layer as a layer of normal and a layer of shear springs. This model does not satisfy the zero-shear-stress boundary conditions at the free edges of the adhesive layer due to the inherent flaw of the two-parameter, elastic foundation model, which violates the equilibrium condition of the adhesive layer. To eliminate this flaw, this study models the adhesive layer as two normal spring layers interconnected by a shear layer. This new three-parameter, elastic foundation model allows the peel stresses along the two adherend/adhesive interfaces of the joint to be different, and therefore, satisfies the equilibrium condition of the adhesive layer. This model regains the missing ‘‘degree of freedom’’ in the two-parameter, elastic foundation model of the adhesive layer by introducing the transverse displacement of the adhesive layer as a new independent parameter. Explicit closed-form expressions of interface stresses and beam forces are obtained. The new model not only satisfies all boundary conditions, but also predicts correctly which interface has the strongest stress concentration. The new model is verified by continuum models existing in the literature and finite element analysis. The new three-parameter, elastic foundation model provides an effective and efficient tool for analysis and design of general adhesive joints. & 2008 Elsevier Ltd. All rights reserved. 1. Introduction Adhesively bonded joints are widely used in composite structures to connect components due to their many advantages compared with other joining methods. However, premature failure due to debonding and peeling of the joint is the major concern of this technique. To address this concern, numerous theoretical and experimental studies have been conducted to evaluate the strength of the adhesive joint. Goland and Reissner [1] modeled (G–R model) the adhesive layer as continuously distributed shear and vertical springs. In this model, no interac- tions are assumed between the shear and vertical springs, and therefore, the adhesive layer is modeled as a two-parameter, elastic foundation. Simple explicit closed-form expressions of interface stresses and beam forces can be obtained by this model as demonstrated by many researchers [2,3]. The inter- face stresses predicted by the two-parameter, elastic foundation model reach good agreements with those obtained through continuum analysis such as finite element analysis (FEA) [4] except in a small zone at the vicinity of the edge of the adhesive layer. To predict more accurate stress distribution of the adhesive joint, many refined models have been developed by modifying the two-parameter, elastic foundation model of the adhesive layer used in G–R model [5–17]. Martensen and Thomsen [18,19] considered the nonlinearity of the adhesive layer. Carpenter [20] used the solution based on finite element analysis as baseline to evaluate different lap-shear joint theories. The major drawback of the G–R model and its descendents mentioned above is that they do not satisfy the zero shear stress at the free edges of the adhesive layer [17]. As illustrated in [2,3], the governing differential equation of the two-parameter model is of the sixth order, which requires six boundary conditions; while there are eight boundary conditions available, including six forces and two shear stress boundary conditions. In the two-parameter, elastic foundation model, the zero shear stress boundary conditions are ignored. As a result, it predicts a maximum shear stress at the free edge of the adhesive layer. To overcome this drawback, some researchers modeled the adhesive layer as two-dimensional continuum medium [21–24]. However, these methods require complicated methods such as employing the variational principle of complementary energy or introducing higher order beam theory. This makes it difficult to use them in analysis and design [4]. In this study, we present a novel, three-parameter, elastic foundation model of adhesive joints. The model is a direct ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijadhadh International Journal of Adhesion & Adhesives 0143-7496/$- see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2008.10.002 à Corresponding author. Tel.: +1 205 348 6786; fax: +1 205 348 0783. E-mail address: jwang@eng.ua.edu (J. Wang). International Journal of Adhesion & Adhesives 29 (2009) 495–502