Fully-coupled nonlinear analysis of single lap adhesive joints

Transcription

1 International Journal of Solids and Structures (7) Fully-coupled nonlinear analysis of single lap adhesive joints Quantian Luo, Liyong Tong * School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 6, Australia Received 6 April 6; received in revised form 6 July 6 Available online July 6 Abstract This paper presents novel closed-form and accurate solutions for the edge moment factor and adhesive stresses for single lap adhesive bonded joints. In the present analysis of single lap joints, both large deflections of adherends and adhesive shear and peel strains are taken into account in the formulation of two sets of nonlinear governing equations for both longitudinal and transverse deflections of adherends. Closed-form solutions for the edge moment factor and the adhesive stresses are obtained by solving the two sets of fully-coupled nonlinear governing equations. Simplified and accurate formula for the edge moment factor is also derived via an approximation process. A comprehensive numerical validation was conducted by comparing the present solutions and those developed by Goland and Reissner, Hart-Smith and Oplinger with the results of nonlinear finite element analyses. Numerical results demonstrate that the present solutions for the edge moment factor (including the simplified formula) and the adhesive stresses appear to be the best as they agree extremely well with the finite element analysis results for all ranges of material and geometrical parameters. Ó 6 Elsevier Ltd. All rights reserved. Keywords: Single lap joint; Analytical solution; Nonlinear differential equation; Nonlinear finite element analysis; Adhesive joints. Introduction Single lap joint (SLJ) subjected to tensile loading as shown in Fig. (a) represents the simplest form of adhesive joints and is used as a standard test specimen for characterizing adhesive properties and strength in ASTM, BS, ES and ISO standards (Tong and Steven, 999). The main feature of a single lap joint in tension as shown in Fig. (a) is its eccentric loading path which results in large deflections, and thus the relationship between the bending moment at cross-section I in Fig. (a) and the applied tensile force is nonlinear. With an increase in the tensile loading, stress analysis of the single lap joint becomes high nonlinearity, and hence it represents a significant challenge (e.g., Dattaguru et al., 9; Tsai and Morton, 99; Tong and Steven, 999; Aydin et al., ; Magalhaes et al., ). Goland and Reissner (9) provided a well-known stress analysis of the single lap joint. They showed that adhesive peel stress is important in joint failure and also adhesive stresses are controlled by the normalized * Corresponding author. Tel.: ; fax: address: ltong@aeromech.usyd.edu.au (L. Tong). -76/$ - see front matter Ó 6 Elsevier Ltd. All rights reserved. doi:.6/j.ijsolstr.6.7.9

2 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 a F O c c l I x II F l O O b z P N F M F O x O x N z M z Q P Fig.. Coordinate systems and force definitions of the single lap joint. c Q edge moment or edge moment factor k at the overlap ends arisen from transverse deflection of adherends. In their paper, Goland and Reissner decoupled the determination of edge moment factor and the determination of adhesive stresses as shown in parts and of their paper. In part of their paper, the edge moment factor k was determined by combining the upper and lower adherends in the overlap as one classical homogeneous beam model and ignoring adhesive layer and its deformation. Hart-Smith (97) considered that decoupling of determination of the edge moment factor and adhesive stresses was a deficiency of the Goland and Reissner analysis. To correct this, he treated both adherends in the overlap as individual beams by adding the effects of adhesive shear strains only and then applied compatibility conditions to adherends independently at the overlap ends. In the course of this analysis, the effects of large transverse deflections of adherends and adhesive thickness deformation were not included, and hence there exists a remarkable discrepancy in the edge moment factor k between the Hart-Smith predictions and those of Goland and Reissner. Oplinger (99) provided an alternative analysis to the Hart-Smith modification for the single lap joint by considering large deflections of adherends and the effects of adhesive shear strains and by ignoring the effects of bond thickness deformation. In doing so, adhesive deflections are allowed to decouple the two halves of the joint in both bending deflection analysis and adhesive stress analysis. There exist a good correlation in the edge moment factor k between the Oplinger predictions and those of Goland and Reissner. The edge moment factor is an important parameter that controls adhesive stresses in single lap joints. It has been used in the development of two dimensional (D) analytic models in which assumed are linear variations of adhesive shear and/or peel stresses (e.g., Allman, 977; Chen and Cheng, 9; Ojalvo and Idinoff, 97; Carpenter, 9), and quadratic shear and cubic peel stress variations (Adams and Mallick, 99). Hence it is of significance to validate various formulas for the edge moment factor. Finite element method has been used to analyze single lap joints by many authors (e.g. Broughton and Hinopulos, 999; Dattaguru et al., 9; Li and Lee-Sullivan, ; Reddy and Roy, 9; Osnes and Andersen, ). Tsai and Morton (99) provided a comprehensive evaluation of the edge moment factors predicted using the formulas developed by Goland and Reissner (9), Hart-Smith (97) and Oplinger (99) as well as of the Goland and Reissner s adhesive stresses by comparing with the numerical results of geometrically nonlinear FEA. They found that the edge moment factor by Hart-Smith (97) is feasible and reasonable for the short single lap joints; whereas the edge moment factor by Oplinger (99) is reasonable for the long single lap joints; and the original Goland and Reissner shear and peel stresses are accurate enough to predict adhesive stress distributions for the short and long joints. An inspection of Figs 7 and in the paper by Tsai and Morton (99) reveals a remarkable discrepancy in the edge moment factor predicted by the nonlinear FEA and the three analytical predictions.

3 The edge moment factor obtained by Goland and Reissner (9) is widely used in the standard test and engineering designs because it is simple and easy to use. It is also a critical quantity to conduct stress analysis of SLJs. Therefore, an accurate equation derived by the analytical procedure would be very useful in engineering applications. To our best knowledge, there is no solution available for the edge moment factor and adhesive stresses in single lap joints that is obtained by simultaneously considering large deflections of the overlap and adhesive strains. In this paper, we present new accurate and closed-form solutions for the edge moment factor and adhesive stresses for the single lap joint. Large deflections of adherends and adhesive shear and peel strains are considered in the development of two sets of new fully-coupled nonlinear governing equations for the longitudinal and transverse deflections of adherends. Closed-form solutions are obtained for both edge moment factor and adhesive stresses. A simplified formula for the edge moment factor is derived with all material and geometric parameters included. A comprehensive numerical validation was performed to show that the present solutions for the edge moment factor and the adhesive stresses are more accurate than the analytical solutions given by Goland and Reissner (9), Hart-Smith (97) and Oplinger (99).. Review of previous solutions for outer adherend and adhesive stresses To analyze the single lap joint shown in Fig. (a), we divide it into four sections: O I, IIO, upper and lower adherends in the overlap. For the outer adherend O I section, considering the free body diagrams in Fig. (b), the axial force N and the bending moment M are: N ¼ F ; M ðx Þ¼Fðax w Þ where; a ¼ t þ t a ðl þ cþ ; ðþ where t a and t are the thickness of the adhesive and adherends, and the other geometric parameters and force components are shown in Fig.. The boundary conditions for this outer adherend section are: x ¼ : u ¼ u O and w ¼ ðor M ¼ Þ; x ¼ l : d w dx Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 ¼ M k D where; D ¼ E t ; where, u O is the axial displacement at O ; M k is the bending moment at the cross section I; E is Young s modulus of adherends. Displacements at O can be prescribed as zero due to symmetry, and thus u O ¼ u O, where u O is the axial displacement at point O. The displacements of the outer adherend are: u ¼ F rffiffiffiffiffi M k x þ u O ; w ¼ A D F sinh b k l sinh b F kx þ ax where; A D ¼ E t ; b k ¼ : ðþ D The displacements of outer adherend section IIO can be obtained similarly. In Eqs. () and (), plane stress state and unit width of the SLJ are assumed for brevity and these assumptions will also be employed in subsequent formulations. However, a practical SLJ is in a plane strain state and a non-unit width normally because its width is much larger than the thickness; in this case, A D and D should be modified accordingly. In all subsequent formulations, the following variables are introduced: u s ¼ u þ u ; w s ¼ w w ; u a ¼ u u ; w a ¼ w þ w ; N s ¼ N þ N ; Q s ¼ Q Q ; M s ¼ M M ; ðþ N a ¼ N N ; Q a ¼ Q þ Q ; M a ¼ M þ M ; where subscripts and refer to adherends and in the overlap, and the stress resultants are shown in Figs. and. The equilibrium equations for the free body diagrams in Fig. are: dn dx þ s ¼ ; dq dx þ r ¼ ; dm dx þ t s Q ¼ ; dn dx s ¼ ; dq dx r ¼ ; dm dx þ t ðþ s Q ¼ : ðþ

4 N N +dn Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 Q Q + dq N N + dn M M + dm Adherend Adhesive x Q Q + dq N N + dn M M + dm z Adherend dx Fig.. Free body diagrams of infinitesimal elements of the overlap part for linear analysis. M M +dm N x Q N +dn (dw /dx) Adherend τ Q +dq σ τ z Adhesive τ σ M τ M +dm Adherend Q Q +dq dx Fig.. Free body diagrams of the overlap for the geometrically nonlinear analysis. In Eq. (), s and r are shear and peel stresses in adhesive, whose definitions are (Goland and Reissner, 9): s ¼ G a ðu u Þþ t dw t a dx þ dw ¼ G a u a þ t dw a ; r ¼ E aðw w Þ ¼ E aw s : ð6þ dx t a dx t a t a In Eq. (6), E a and G a are Young s and shear moduli of the adhesive. Substantial investigations have shown that Eq. (6) is sufficiently accurate for the thin adhesive. In our previous work (Luo and Tong, ), we obtained analytical solutions of adhesive stresses with constant, linear and higher order variations through the thickness. The adhesive stresses based on the assumption of constant variation through bondline are: s ¼ b sðft þ 6M k Þ cosh b s x t sinh b s c þ ðft M k Þ ; ð7þ t c r ¼ E a ðb r sinh b t r x sin b r x þ B r cosh b r x cos b r xþ; ðþ a where, B r and B r are integration constants; the details can be found in our previous paper (Luo and Tong, ). The shear and peel stresses given in Eqs. (7) and () are the same as those of Goland and Reissner

5 (9) when the same boundary conditions of the overlap are prescribed. The eigenvalues b s and b r are defined as: sffiffiffiffiffiffiffiffiffiffi pffiffi sffiffiffiffiffiffiffiffiffiffiffi G a b s ¼ ; b A D t r ¼ a E a : ð9þ D t a Eigenvalues b k, b s and b r were used in the formulations of Goland and Reissner (9) and will be utilized in the present formulations for fully-coupled nonlinear analysis of SLJs.. Fully-coupled nonlinear governing equations and their solutions.. Governing equations Considering the free body diagram in Fig., we have the following equilibrium equations: dn dx þ s ¼ ; dq dx þ r þ s dw dx ¼ ; dm dx þ t s Q dw ¼ N dx ; dn dx s ¼ ; dq dx r s dw dx ¼ ; dm dx þ t s Q dw ¼ N dx : For a general geometrical nonlinearity of the Euler Bernoulli beam, constitutive equations are (Tsai and Morton, 99): " N ¼ A du dx þ du þ # dw B d w dx dx dx ; " M ¼ B du dx þ du þ # ðþ dw D d w dx dx dx where, A, B and D are extensional, extensional-bending and bending stiffness. For a beam made of an isotropic material, B =. Substituting Eq. () into (), we can obtain: dn s dx ¼ ; dq s dx r s dw a dx ¼ ; dm s dx Q dw s s ¼ N s dx N dw s a dx ; dn a dx s ¼ ; dq a dx s dw s dx ¼ ; dm a dx þ t ðþ s Q dw a a ¼ N s dx N dw s a dx : For the adhesive with the shear and peel stiffness, Eq. (6) is employed. Based on equilibrium, we have: N s ¼ N þ N ¼ F : ðþ To obtain closed-form analytical solutions, the nonlinear terms in Eqs. () and () are neglected. Noting Eq. (), we have the following equilibrium and constitutive equations: dn s dx ¼ ; dq s dx r ¼ ; dm s dx Q s ¼ F dw s dx ; dn a dx s ¼ ; dq a dx ¼ ; dm a dx þ t s Q a ¼ F N i ¼ A D du i dx ; Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 dw a dx ; M i ¼ D d w i dx ði ¼ s; aþ: ðþ A comparison with Eq. () reveals that the constant tensile load F is introduced in the present solvable nonlinear governing equations. Substituting Eqs. (6) and () into Eq. (), we derive the following governing equations: ðþ ðþ

6 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 d u s dx ¼ ; d w s D dx F d w s dx þ E a w s ¼ ; t a d u a A D dx G a u a þ t dw a ¼ ; t a dx d w a D dx þ G at du a t a dx þ t d w a þ F d w a dx dx ¼ : Closed-form solutions of Eqs. (6) and (7) can be readily obtained. ð6þ ð7þ.. Solutions of equation (6) The closed-form solutions of Eq. (6) can be expressed as: u s ¼ A s x þ A s ; w s ¼ðB s sinh b s x þ B s cosh b s xþ sin b s x þðb s sinh b s x þ B s cosh b s xþ cos b s x; where, A s, A s and B si (i =,,,) are the integration constants; the eigenvalues are: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ b s ¼ b r þ b k ; b s ¼ b r b k : ð9þ The solution procedure of the eigenvalues b s and b s is detailed in Luo and Tong (). The eigenvalue b s is a real number in general. When the tensile force F is so large that b s becomes a complex number, the SLJ will be failed because of the very high stress, which is easy to check out by substituting normal values of the geometrical sizes and materials properties into (9). Therefore, Eq. () is a correct solution form in engineering senses. The pure mathematical solutions will not be presented in this paper. The force components at the intersection I and II are (Goland and Reissner, 9): N I ¼ F ; V I ¼ V k ¼ M k b k coth b k l; M I ¼ M k ; ðþ N II ¼ F ; V II ¼ V k ¼ M k b k coth b k l; M II ¼ M k where V I, V II and V k are the shear forces at the deformed cross section and M k is the edge bending moment. By combining with Eqs. (), (), () and (), the boundary conditions for Eq. (6) are: du s x ¼ c : A D dx ¼ F ; D d w s dx ¼ V k ; D d w s dx ¼ M k ; du s x ¼ c : A D dx ¼ F ; D d w s dx ¼ V k ; D d w s dx ¼ M ðþ k : Substituting Eq. () into (), we can find the integration constants as follows (Luo and Tong, ): A s ¼ F ; A s ¼ ; B s ¼ B s ¼ ; A D B s ¼ a M k a V k ; B s ¼ a ðþ M k þ a V k ; D D where, a ¼ D b s b s sinh bs c sin b s c þ b s b s cosh b s c cos b s c ; a ¼ D b s b s cosh bs c cos b s c b s b s sinh b s c sin b s c ; a ¼ D b s b s b s cosh bs c sin b s c b s b s b s sinh bs c cos b s c ; a ¼ D b s b s b s sinh bs c cos b s c þ b s b s b s cosh bs c sin b s c ; D ¼ ða a a a Þ: ðþ

7 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 The closed-form solutions of Eq. (6) are: < u s ¼ F x; A : D w s ¼ B s sinh b s x sin b s x þ B s cosh b s x cos b s x: ðþ.. Solutions of equation (7) The closed-form solutions for Eq. (7) can be expressed as: u a ¼ A a sinh b a x þ A a cosh b a x þ A a sinh b a x þ A a cosh b a x þ A a ; ðþ w a ¼ B a sinh b a x þ B a cosh b a x þ B a sinh b a x þ B a cosh b a x þ B a x þ B a6 ; where, A ai and B aj (i =,,...,; j =,,...,6) are the integration constants, which are determined by the boundary conditions. The eigenvalues in Eq. () are: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b a b s þ b k þ b s þ b s b k þ b ka; b a b s þ b k b s þ b s b k þ b ka: ð6þ By referring to Eq. (), the boundary conditions for Eq. (7) can be obtained in a similar manner: du a x ¼ c : A D dx ¼ F ; D d w a dx þ t sð cþ ¼ V k ; D d w a dx ¼ M k ; du a x ¼ c : A D dx ¼ F ; D d w a dx þ t sðcþ ¼ V k ; D d w a dx ¼ M k : Substituting Eq. () into the st equation of (7), we have: A a ¼ K a B a ; A a ¼ K a B a ; A a ¼ K a B a ; A a ¼ K a B a ; A a ¼ K a B a ; ðþ where, K a ¼ b a K a ¼ K ab a b s b a ; K a ¼ b a K a ¼ K ab a b s b s b a ; K a ¼ t b : ð9þ s By substituting Eq. () into (7), the following equations can be obtained (Luo and Tong, ): ( K a b a cosh b ac Ba þ K a b a cosh b ac Ba ¼ ; b a cosh b ac Ba þ b a cosh b ðþ ac Ba ¼ ; K a b a sinh b ac Ba þ K a b a sinh b ac Ba ¼ F ; A D b a sinh b ac Ba þ b a sinh b ac Ba ¼ M ðþ k : D By solving Eqs. () and (), the integration constants are obtained: < B a ¼ B a ¼ A a ¼ A a ¼ ; K a M k t B a ¼ F E t b a ðk a K a Þ sinh b a c ; B K a M k t : a ¼ F E t b a ðk a K a Þ sinh b a c : ðþ The closed-form solutions of Eq. (7) are then obtained: u a ¼ A a cosh b a x þ A a cosh b a x þ A a ; w a ¼ B a sinh b a x þ B a sinh b a x þ B a x: In the solutions given by Eqs. (), () and (), there are three unknowns M k, u O and B a which can be solved by the continuity conditions at cross section I or II in Fig.. ð7þ ðþ

8 6 Q. Luo, L. Tong / International Journal of Solids and Structures (7) Solutions for the edge moment factor and adhesive stresses The continuity conditions at cross section I in Fig. are: < u ðlþ ¼u s ð cþ u a ð cþ; w ðlþ ¼w a ð cþ w s ð cþ; dw ðlþ ¼ dw að cþ dw sð cþ ðþ : ; dx dx dx Substituting Eqs. (), () and () into Eq. (), we have: Fl þ u ¼ Fc B a K a cosh b E t E t a c B a K a cosh b a c þ t B a; M k F al ¼ B a sinh b a c þ B a sinh b a c þ B a c þ B s sinh b s c sin b s c þ B s cosh b s c cos b s c; ðþ M kb k coth b k l þ a ¼ B a b F a cosh b a c þ B a b a cosh b a c þ B a þðb s b s þ B s b s Þsinh b s c cos b s c þ ðb s b s B s b s Þcosh b s c sin b s c: Solving Eq. (), we obtain the edge moment at cross section I: M k ¼ kf t þ t a þ d F ; k ¼ : ð6þ þ b k c coth b k l þ d M In Eq. (6), k is the edge moment factor; d F and d M are given by: D a F d F ¼ A D ðt þ t a Þ ; d M ¼ D a þ ða a b k coth b k lþd s ða a b k coth b k lþd s D D D where, D a ¼ ðsinh b ac b a c cosh b a cþ b a ðk a K a Þ sinh b a c þ ðsinh b ac b a c cosh b a cþ b a ðk a K a Þ sinh b a c ; D a ¼ K aðsinh b a c b a c cosh b a cþ b a ðk þ K aðsinh b a c b a c cosh b a cþ a K a Þ sinh b a c b a ðk ; ðþ a K a Þ sinh b a c D s ¼ sinh b s c sin b s c b s c cosh b s c sin b s c b s c sinh b s c cos b s c; D s ¼ cosh b s c cos b s c b s c sinh b s c cos b s c þ b s c cosh b s c sin b s c: The edge moment factor shown in Eq. (6) is quite complicated for practical design of single lap joints, and it needs to be simplified. Substituting the bending moment M k into Eq. (), we can solve the unknowns u O and B a, and thus all displacement components are found. Substituting the displacement components into Eq. (6), we can obtain the following expressions for adhesive shear and peel stresses: s ¼ G h a K a þ t b t a a B a cosh b a x þ K a þ t i b a B a cosh b a x ; ð9þ r ¼ E a ðb s sinh b t s x sin b s x þ B s cosh b s x cos b s xþ: ðþ a It is worth noting that the present shear and peel stresses are different from those of Goland and Reissner (9) even if the same boundary conditions of the overlap are prescribed... Simplification of the edge moment factor To simplify the edge moment factor k, the following approximations are used: sinh b s c cosh b s c eb sc ; sinh b sc cosh b s c ebsc ; sinh b ac cosh b a c ebac : ðþ F ; ð7þ

9 Q. Luo, L. Tong / International Journal of Solids and Structures (7) The approximations in Eq. () are sufficient accurate in most cases. By utilizing Eq. (), and cothb k l, which is accurate when l t for most cases. Eqs. () and () are simplified respectively as: a ¼ D b s b s sin bs c þ b s b s cos b s c e b ð s c =Þ; a ¼ D b s b s cos bs c b s b s sin b s c e b s ð c =Þ; a ¼ D b s b s b s sin bs c b s b s b s cos bs c e b ð s c =Þ; ðþ a ¼ D b s b s b s cos bs c þ b s b s b s sin bs c e b ð s c =Þ; D ¼ ða a a a Þ; ð b D a ¼ a cþ b a ðk a K a Þ þ ðsinh b ac b a c cosh b a cþ b a ðk a K a Þ sinh b a c ; D a ¼ K að b a cþ b a ðk a K a Þ þ K aðsinh b a c b a c cosh b a cþ b a ðk ; ðþ a K a Þ sinh b a c D s ¼½ð b s cþ sin b s c b s c cos b s cšðe bsc =Þ; D s ¼½ð b s cþ cos b s c þ b s c sin b s cšðe bsc =Þ: Substituting Eqs. () and () into (7), we have: d F ¼ a a ðb k cþ ; d M ¼ða a þ a s Þðb k cþ ; where, a s ¼ D ½ða a b k ÞD s ða a b k ÞD s Š t ; a c a ¼ D a D c ðt þ t a Þ ; a a ¼ D a c : Eqs. () and () give almost the same numerical results of the edge moment as those of Eqs. (7) and (). This approximate edge moment factor can be further simplified for engineering applications. In engineering applications, the following approximations are sufficiently accurate: b s b s b r ; b a b s b a ; b r b k : ð6þ Using Eq. (6), we have: K a ¼ t 6 ; K a ¼ t ; K a K a ¼ t : ð7þ The following inequalities hold for the long overlap: b r c ; b s c : ðþ Eq. () is used to simplify a s, a a and a a, whose influence on the bending factor is small for the short overlap (small b k c) by referring to Eqs. (6) and (). Therefore, these inequalities can be applied to both short and long overlaps. Applying Eqs. (6) () to (), we have: a s ¼ b r c ; a a ¼ b scf ðb a cþ b s cð þ t a =t Þ ; a a ¼ b scf ðb a cþþ ; ð9þ b s c where, f ðb a cþ¼ b ac coth b a c : ðþ ðb a cþ The simplified edge factor is: þðb k cþ b s cf ðb a cþ b k ¼ s cð þ t a =t Þ þðb k cþ coth b k l þðb k cþ b r c þ b : ðþ scf ðb a cþþ b s c ðþ ðþ

10 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 When l t,cothb k l ; the term cothb k l in Eq. () can be omitted. Eigenvalue b a is given in Eq. (6), which is dependent on b k and b s only. It can be seen that all variables in Eq. () are the eigenvalues used in stress analysis of SLJs. The simplified edge moment factor shown in Eq. () includes all geometric characteristics and material properties of the adhesive and adherends, which overcomes the deficiency of the formulations with no adhesive material properties and thickness presented by Goland and Reissner (9) and Hart-Smith (97). Tsai and Morton (99) indicated that both the material property and the thickness of the adhesive have significant influences on the edge moment factor for the long overlap. The precision of Eq. () will be numerically demonstrated. In some cases, the following equations may also be used to predict the edge moment factor: k ¼ þ b k c coth b k l or k ¼ when coth b þ b k c k l : ðþ Eq. () is much simpler but only applicable to short overlaps as it does not include the adhesive material property and thickness.. Numerical results, verification and comparison In this section, we will present numerical results, which will be verified with the nonlinear FEA results given by Tsai and Morton (99) and the present computation using MSC/NASTRAN. The numerical results are also compared with those predicted by the Goland and Reissner, Hart-Smith and Oplinger formulations. Table lists the relevant equations and abbreviations used in the figures of this paper. The relevant variable definitions in Table can be found in the corresponding literatures. The equation in the second row of the table was called the modified GR bending factor by Tsai and Morton (99). Symbol (S) in the bracket is referred to the simplified formula. Table Analytical and numerical solutions and abbreviate symbols Nomenclature Goland and Reissner formula (9) GR Goland and Reissner simplification GR (S) The Hart-Smith formula (97) HS The Hart-Smith simplification HS (S) Oplinger (99) OP The present formulation LT The present simplification LT (S) Tsai and Morton nonlinear finite element analysis (99) TM The present nonlinear finite element analysis NFEA Edge moment factor formula and numerical analysis p þ ffiffiffi p tanhðbk c= ffiffiffi Þ coth bk l p þ ffiffiffi p tanhðbk c= ffiffiffi Þ þ ðb kkþ ðk Þ þ k c k c tanh k c " #" # þ b k c þ ðb kcþ t þ t a ðb k kþ 6 t k b ðk Þ þ ðk cþ k c tanh k c þ b k c þðb k cþ =6 h i R þ ta þ R T h C t þ R T RC þ ta C h t h þ ta T h R þ R T RC p þ ffiffiffi i h ð þ R C Þ T h t þ d F ð þ b k c coth b k l þ d M Þ þðb k cþ b s cf ðb a cþ b s cð þ t a =t Þ þðb k cþ coth b k l þðb k cþ b r c þ b scf ðb a cþþ b s c The two dimensional nonlinear finite element analysis conducted by Tsai and Morton The two dimensional nonlinear finite element analysis using MSC/NASTRAN T h

11 In their D geometrically nonlinear FEA computations, Tsai and Morton (99) employed the following parameters: E = 7 (GPa), m = and t =.6 (mm); E a /E =.,. and. corresponding to inflexible, intermediate flexibility and flexible adhesive materials and m a =.; c/t =,, 6, and l/c = for different overlap and outer adherend lengths. These data are also used in this paper. The short and long overlaps are referred to as c/t =(l/c = ) and (l/c =. and ), respectively. Because Tsai and Morton (99) performed full D plane strain FEA computations, the numerical results of the analytical solutions will also be presented for the plane strain, which can be obtained by modifying material properties in the theoretical formulations... Edge moment factor Q. Luo, L. Tong / International Journal of Solids and Structures (7) The edge moment definition in Eq. (6) is the same as that of Hart-Smith (97), which is different from that of Goland and Reissner (9), Oplinger (99), and Tsai and Morton (99). When t a t, the bending factor of the HS definition approaches that of the GR definition. The GR equations in Table are also based on the HS report (97); the reasons can be seen from our numerical demonstrations. If the TM definitions for the edge moment factor and the bending moment update are employed, the GR numerical results presented in this paper for the edge moment and stresses should be divided by ( + t a /t ). Since the TM nonlinear FEA results are well accepted, the numerical results of the bending factor predicted by the present analytical solution will be verified using their data. The TM nonlinear FEA data are read from.... GR OP LT TM HS Dif..... Fig.. Edge moment factor comparisons of the single lap joint with parameters: E a /E =., t a /t =.7, c/t = and l/c = Difference (%) 9. GR OP LT TM HS Dif Fig.. Bending factor comparisons of the single lap joint with parameters: E a /E =., t a /t =.7, c/t = and l/c =. Difference (%)

12 6 Q. Luo, L. Tong / International Journal of Solids and Structures (7) GR OP LT 7 TM HS Dif Fig. 6. Edge moment factor comparisons of the single lap joint with parameters: E a /E =., t a /t =.7, c/t = 6 and l/c =. Difference (%) 9. GR OP LT 7 TM HS Dif Fig. 7. Edge moment factor comparisons of the single lap joint with parameters: E a /E =., t a /t =.7, c/t = and l/c =. Difference (%) 9. GR OP LT 7 TM HS Dif Fig.. Edge moment factor comparisons of the single lap joint with parameters: E a /E =., t a /t =.7, c/t = and l/c =. Difference (%)

13 Q. Luo, L. Tong / International Journal of Solids and Structures (7) GR OP LT 7 TM HS Dif Fig. 9. Edge moment factor comparisons of the single lap joint with parameters: E a /E =., t a /t =.7, c/t = and l/c =.. the TM figures using software Paint with pixels of 7, and thus the reading data are of high accuracy. The GR, HS and OP numerical results are calculated using the equations given in Table. Figs. 9 illustrate the edge moment factors given by TM and predicted by GR, HS, OP and the present formula of Eq. (6). In the figures, Dif. is referred to the difference defined by: difference ¼ jk LT k TM j ; ðþ k TM where k TM and k LT are edge moment factors given by TM and predicted by the present formula. The bending moment data taken from the TM paper is divided by ( + t a /t ) as the edge moment definitions are different. It is noted that the edge moment factor for the SLJ without considering large deflections of the outer adherends and the overlap is: k ¼ þ c=l ; ðþ when the applied force is close to zero, the bending factor for the SLJ with large deflection effects should approach Eq. (). In Figs. 9, the maximum values of b k c =,, and are used for cases of c/t =,, 6 and, respectively, which corresponds to the average stress of (MPa) in the outer adherends. When the edge moment factor is beyond these ranges, material nonlinearity has to be considered. The TM data in Fig. are taken from Figs. 9 and 6 given by Tsai and Morton. The difference of the edge moment factor k of TM and LT for the short overlap is less than %. The HS k values are also close to the TM results but the edge moment factor predicted by GR and OP are larger; Tsai and Morton (99) also indicated this observation. Figs. and 6 are the edge moment factors of the relatively longer overlap, whose TM data are read from the TM Fig.. The differences between TM and LT are also less than % for these two cases. However, the HS k values, or the GR and OP k values are evidently lower than or higher than the TM k values when the applied forces become large. Figs. 7 9 are the edge moment factors for the long overlaps. The TM data of Figs. and 7 are employed in our Figs. 7 and, which are referred to as the long overlap with intermediate flexibility and flexible adhesives. The TM data of Fig. are used in Fig. 9, in which the long overlap with the relatively short outer adherend (l/c =.) is considered. It can be seen that, for the long overlap, the edge moment factor predicted by HS is significantly lower than TM predictions, and the edge bending factors predicted by GR and OP are considerably higher than the TM results, which can also found in Figs. 7 and of Tsai and Morton (99). However, differences of TM and LT are less than %. Figs 9 show that the present formulation correlates remarkably well with the TM D nonlinear FEA for both short and long overlaps, and the maximum difference in all figures is less than %. When compared with the TM NFEA results, the maximum differences with TM of GR, OP and HS are 9.%, 7.% and 7.9%, respectively as shown in Fig. and are.%, % and 7.9%, respectively in Fig. 9. Difference (%)

14 6 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 Figs. 9 illustrate that the present formula for the edge moment factor has the best agreement with the TM results for all ranges of parameters considered, and thus the present formula represents a remarkable improvement over those abbreviated as GR, HS and OP... Deflections of the overlap Figs. and depict the deflection of adherend for the short and long overlap, in which, both TM and GR data are read from the TM Figs. and, respectively. The non-dimensional deflection is defined as: w n = w /t. Only the deflections in the overlap are plotted here, but the similar results can be obtained for the outer adherend. It is noted that non-dimensional axis of the overlap is used in the figures, which is defined by: n = x/c. Fig. indicates that the deflections predicted by the present formulation are close to those of GR and TM for the short overlap. It is noted that the present results are for the case of t a /t =.7. The SLJ with t a / t = cannot be modeled in the present formulation because of the stress definition in Eq. (6). Fig. shows that the present deflections correlates well with the TM numerical results whereas there exist significant differences between the GR and TM results for long overlap. -.E- Non-dimensional deflection w n -.E- -.E- -.E-.E+.E-.E-.E- TM GR LT.E Non-dimensional overlap axis ξ Fig.. Deflection of the short over lap (E a /E =.) predicted by the present formulation. (t a /t =.7), the TM and the GR formulation (t a /t = ). -.E- Non-dimensional deflection w n -.E- -.E- -.E-.E+.E-.E-.E- TM GR LT.E Non-dimensional overlap axis ξ Fig.. Deflection of the long overlap (E a /E =., t a /t =.7, c/t = and l/c = ) predicted by the present formulation, the TM NFEA and the GR formulation.

15 Q. Luo, L. Tong / International Journal of Solids and Structures (7) E+.E+ Shear stress (MPa ).E+ 6.E+.E+ LT - *Load NFEA - *Load GR - *Load LT - *Load NFEA - *Load GR - *Load.E+.E Non-dimensional overlap axis ξ Fig.. Shear stress of the long overlap predicted by the GR formulation, the present formulation and NFEA (E a /E =., t a /t =.7, c/t = and l/c =.)... Shear and peel stresses As the stress data in Fig. of Tsai and Morton (99) for the long overlap and the lower edge moment factor could not be read accurately, we conducted nonlinear FEA computations using MSC/NASTRAN. In the present computation, different mesh schemes have been used and the NFEA results in Figs. and are the convergent ones, which are adopted based on the mesh scheme: a four-node isoparametric element is employed for both the adhesive and the adherends; and elements are used through the adhesive and adherend thickness in the regions of. 6 jnj 6 ; and 6 elements are used in the other region of the overlap. The geometrically nonlinear FEA of the plane strain is also implemented in the present computation. The tensile force of 69.6 (N/mm) (b k c = ) is applied at point O by steps. The boundary conditions are: u O ¼ w O ¼ w O ¼, which are the same as those of the TM NFEA. The stresses in element centers of the central adhesive layer are used in Figs. and, which can represent the adhesive stresses based on the GR definitions. Figs. and illustrate the shear and peel stresses distributions for the long overlap with the intermediate flexible adhesive. in which, Load represents the maximum applied force. The load step of and are corresponding to b k c =. and, respectively. Fig. shows that the shear stress distributions predicted by the present formulation and NFEA are almost the same except for the edge area for b k c =. and. When Peel stress (MPa ).E+.E+.E+.E+ 6.E+.E+.E+.E+ -.E+ LT - *Load NFEA - *Load GR - *Load LT - *Load NFEA - *Load GR - *Load -.E Non-dimensional overlap axis ξ Fig.. Peel stress of the short overlap predicted by the GR formulation, the present formulation and NFEA (E a /E =., t a /t =.7, c/t = and l/c =.).

16 6 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 b k c =., the GR shear stress distribution is also close to that of NFEA, while the differences are observed between the GR and NFEA for b k c =. Fig. depicts the peel stresses predicted by GR, LT and NFEA, in which only the results in the region of 6 n 6. are given for the plotting quality. It is noted that the NFEA peel stress at the center line end is not equal to zero. The self-balance of the present peel stress can also be observed by referring the GR and NFEA results in this figure. It is evident that the present peel stress distribution correlates better to the NFEA results than that of Goland and Reissner (9). It is noted that, similar to the classical D model, the present theory could not model the edge stress-free conditions (Allman, 977), which can be satisfied in the D FEA when sufficiently fine meshes are used. In this sense, it might be reasonable to say that the FEA can capture better stress distribution than the present D model. Based on the mathematical elasticity for linear elastic body, stress singularity can be present at the intersectional points, which is not modeled by both the D analytical models and the D FEA without crack elements. Detail discussion on the issue of end stress singularity and distribution can be found in Tsai and Morton (99). Figs. and depict the peak edge shear and peel stresses, in which, GR-k represents the results calculated using the GR stress formulation and the present edge moment factor k. Whenb k c =, the GR peak shear stress is.9% larger than the present one, but the GR-k peak shear stress is.% lower than the present one. However, when the present edge moment factor is substituted into the GR peel stress formulation, the.e+ The maximum shear stress (MPa ).E+.E+ 6.E+.E+.E+ GR LT GR - k.e+ 6 7 Fig.. Shear stress of the long overlap predicted by the GR formulation, the present formulation and NFEA (E a /E =., t a /t =.7, c/t = and l/c =.)..E+ The maximum peel stress (MPa ).E+.E+.E+ 6.E+.E+.E+ GR LT GR - k.e+ 6 7 Fig.. Peel stress of the long overlap predicted by the GR formulation, the present formulation and NFEA (E a /E =., t a /t =.7, c/t = and l/c =.).

17 Q. Luo, L. Tong / International Journal of Solids and Structures (7) difference of the GR-k and LT peak peel stress is less than % in the range of < b k c 6. The GR stress similarity and difference with the NFEA can also be observed in Figs. and of TM paper (99), and the issues of adhesive edge stresses can be referred to Figs. and of the same paper.. Evaluation of the simplified edge moment factor The present analytical solutions appear accurate as benchmarked by the D NFEA results but quite complicated for practical applications, and thus we present a simplified form of the edge moment factor as given in Eq. (). In this section, we assess the simplified formula for the edge moment factor for SLJs with different material properties and geometric sizes. In the subsequent figures, the GR and HS (S) edge moment bending factors are also given as reference. It is noted that adhesive material properties and thickness are not included in the GR and HS (S) formulations, whilst all material properties and geometric sizes of SLJs are included in the present simplified edge moment factor... Length effects of the overlap and the outer adherend When b k l P, the approximation error of cothb k l is less than %. It is easy to show that the relation of b k l < is corresponding to the very low applied force when (l/t ) P. Therefore, the approximation of cothb k l can be used for the relatively long outer adherend. The edge moment factor for the short overlap predicted by the present full and simplified formulations is plotted in Fig. 6, which shows that the maximum error of the present simplification is less than %. Fig. 6 shows that the present simplified formula of the edge moment factor can be applied to the short overlap with different lengths of the outer adherends, even though we assumed the long overlap in the simplification. Figs. 7 9 show the numerical results of the edge moment factor predicted by the present full and simplified formulas, GR formula and HS simplified formula for different overlap lengths. The maximum error of the present simplification is less than % for these cases. Therefore, the present simplified edge moment factor gives accurate results for different overlap and adherend lengths... Effects of material properties and thicknesses of the adhesive and adherend Edge moment factors for the short overlap with flexible adhesive for different adhesive thickness predicted by TM and LT (S) are plotted in Fig., in which, R t = t a /t. The TM data are taken from Fig. 9 given by Tsai and Morton (99). When the adhesive is thick (t a /t = 6), the maximum difference between TM and LT (S) is less than %. Because the present formulation can not model SLJs with t a =,weuset a /t =.7 to.... LT - l/c= LT (S) - l/c= LT - l/c= LT (S) - l/c= Error - l/c = Error - l/c =.... Error (%) Fig. 6. Edge moment factor predicted by the present full and simplified formulas for the short overlap (c/t =, t a /t =.7, E a /E =. and l/c =, ).

18 66 Q. Luo, L. Tong / International Journal of Solids and Structures (7) GR LT LT (S) HS (S) Error Error (%) Fig. 7. Edge moment factor predicted by the present full and simplified formulas for the SLP when c/t =,t a /t =.7, E a /E =. and l/c = GR LT LT (S) HS (S) Error Error (%) Fig.. Edge moment factor predicted by the present full and simplified formulas for the SLJ when c/t = 6, t a /t =.7, E a /E =. and l/c = GR LT LT (S) HS (S) Error Error (%) Fig. 9. Edge moment factor predicted by the present full and simplified formulas for the long overlap (c/t =, t a /t =.7, E a /E =. and l/c = ).

19 Q. Luo, L. Tong / International Journal of Solids and Structures (7) TM - Rt = LT (S) - Rt =.7 TM - Rt = 6 LT (S) - Rt = 6 Dif. - Rt = Dif. - Rt = Fig.. Edge moment factor predicted by the present simplified formula and the TM NEA for the short overlap (E a /E =. c/t =, l/c =, R t = t a /t ) Differences (%) simulate the case of t a =. As compare to the TM results, the maximum difference is less than %. Errors of the present simplification are less than % for these cases and are not plotted in this figure. Fig. indicates that the present simplification can be applied to the short overlap with different adhesive thickness. Tsai and Morton (99) showed that the adhesive materials have minor effect on the k values for the short overlap. Fig. illustrates that the adhesive thickness has minor influence on the k values for the short overlap. Therefore, using the HS definition, both the material and the thickness of the adhesive have a little effect on the edge moment factor for the short overlap. This is another reason that we employ the HS edge moment factor definition in this paper.... Effects of material property and thickness of the adhesive on the long overlap Tsai and Morton (99) indicated that both the material property and the thickness of the adhesive have significant effects on the edge moment factor. In this subsection, we will verify the accuracy of the present simplification for the long overlap with different adhesive material and thicknesses. Fig. shows the comparisons of edge moment factors predicted by the present full and simplified formulations for the long overlap with the inflexible (E a /E =.) and flexible (E a /E =.) adhesive. Fig. is that for the long overlap (c/t =, l/c = ) with the thin (t a /t =.7 or t a =. mm) and thick adhesive (t a /t = 6) layers. Figs. and indicate that errors of the present simplification for the.... GR LT (S) - Inflex. LT (S) - Flex. Error - Inflex. LT - Inflex. LT - Flex. HS (S) Error - Flex 6 7 Error (%) Fig.. Edge moment factor predicted by the present full and simplified formulas for inflexible and flexible adhesives when c/t =, t a /t =.7 and l/c =.

20 6 Q. Luo, L. Tong / International Journal of Solids and Structures (7) GR LT (S) - Thick LT (S) - Thin Error - Thick LT - Thick LT - Thin HS (S) Error - Thin 6 7 Fig.. Edge moment factor predicted by the present full and simplified formulas for the thick and thin adhesive when c/t =, E a /E =. and l/c =. Error (%) demonstrated different adhesive material and thicknesses are less than %. It is also observed that the GR edge moment would be close to the present one when t a approaches to zero or E a becomes large.... Effects of material property and thickness of adherends The effects of the material property and thickness of the adherends on the edge moment factor are illustrated in Figs. and. The soft and hard materials in Fig. are referred to E = (GPa) and E = (GPa), respectively, Young s Modulus of the hard material is equivalent to that of steel. The Poisson s ratio of the adherends is, and the intermediate flexible adhesive (E a =. GPa) is used in this example. The thick and thin adherends in Fig. are t =.7 (mm) and t = (mm), respectively; the adhesive thickness is. (mm). Figs. and demonstrate that the errors of the present simplified formula k for different adherend materials and thicknesses are less than % compared to the present full formula k. On the basis of the above numerical results, we can see that the present simplified edge moment factor shown in Eq. () is of high precision. Effects of the material property and geometrical sizes on the edge moment factor can also be understood from the demonstrated numerical results. The accurate edge moment factor obtained in this paper is attributed to that the large deflection of the overlap is properly modeled. Eq. () is a general form of the nonlinear differential equilibrium equations and Edeg moment factor k.... GR LT (S) - Soft LT (S) - Steel Error - Soft LT - Soft LT - Steel HS (S) Error - Steel 6 7 Fig.. Edge moment factor predicted by the present full and simplified formulas for soft and hard adherend materials (E a =. GPa, c/t =, t a /t =.7 and l/c = ). Error (%)

21 Q. Luo, L. Tong / International Journal of Solids and Structures (7) GR LT (S) - Thin LT (S) - Thick Error - Thin LT - Thin LT - Thick HS (S) Error - Thick 6 7 Error (%) Fig.. Edge moment factor predicted by the present full and simplified formulas for thick and thin adherends (E a /E =., c = 6 mm and l/c =,t a =. mm). higher orders are neglected in Eq. (). The items on the right hand side of Eq. () reflect effects of the overlap large deflections. 6. Conclusions This paper presents two sets of new fully-coupled nonlinear governing equations for both longitudinal and transverse deflections of adherends for single lap joints by taking into account both large deflections of adherends and adhesive shear and peel strains. Novel closed-form and accurate solutions are developed for the edge moment factor (Eqs. (6) ()) and adhesive shear and peel stresses (Eqs. (9) and ()) for single lap adhesive bonded joints by solving the two sets of nonlinear equations. Simplified and accurate formula for the edge moment factor (Eq. ) is also derived via an approximation process. In the present formulas for the edge moment factor, all material and geometrical parameters are included. The comprehensive numerical evaluation demonstrates that the present solutions for the edge moment factor (including the simplified formula) and the adhesive stresses appear to be the best as they agree extremely well with the nonlinear finite element analysis results for all ranges of material and geometrical parameters considered in this paper. Acknowledgements The authors are grateful for the support of Australian Research Council via Discovery-Projects grants (DP69 and DP6666). References Adams, R.D., Mallick, V., 99. A method for the stress analysis of lap joints. Journal of Adhesion, Allman, D.J., 977. A theory for the elastic stresses in adhesive bonded lap joints. Quarterly Journal of Mechanics and Applied Mathematics, 6. Aydin, M.D., Ozel, A., Temiz, S.,. The effect of adherend thickness on the failure of adhesively-bonded single-lap joints. Journal of Adhesion Science and Technology 9, 7 7. Broughton, W.R., Hinopulos, G., 999. Evaluation of the single-lap joint using finite element analysis. NPL Report CMMT(A)6. Project PAJ Combined Cyclic Loading and Hostile Environments , Report No.. Carpenter, W., 9. Stresses in bonded connections using finite elements. International Journal for Numerical Methods in Engineering, Chen, D., Cheng, S., 9. An analysis of adhesive-bonded single lap joints. Journal of Applied Mechanics, 9. Dattaguru, B., Everett Jr., R.A., Whitcomb, J.D., Johnson, W.S., 9. Geometrically non-linear analysis of adhesively bonded joints. Transactions of ASME: Journal of Engineering Materials 6 (), 9 6. Goland, M., Reissner, E., 9. The stresses in cemented joints. Journal of Applied Mechanics, A7 A7. Hart-Smith, L.J., 97, Adhesive-bonded single-lap joints, CR-, NASA Langley Research Center.

22 7 Q. Luo, L. Tong / International Journal of Solids and Structures (7) 9 7 Li, G., Lee-Sullivan, P.,. Finite element and experimental studies on single-lap balanced joints in tension. International Journal of Adhesion and Adhesives,. Luo, Q., Tong, L.,. Linear and higher order displacement theories for adhesively bonded lap joints. International Journal of Solids and Structures, 6 6. Magalhaes, A.G., de Moura, M., Goncalves, J.,. Evaluation of stress concentration effects in single-lap bonded joints of laminate composite materials. International Journal of Adhesion and Adhesives, 9. Ojalvo, I.U., Idinoff, H.L., 97. Bond thickness upon stresses in single-lap adhesive Joints. AIAA Journal 6 (),. Oplinger, D.W., 99. Effects of adherend deflection on single lap joints. International Journal of Solids and Structures, 6 7. Osnes, H., Andersen, A.,. Computational analysis of geometric nonlinear effects in adhesively bonded single lap composite joints. Composite Part B, 7 7. Reddy, J.N., Roy, S., 9. Non-linear analysis of adhesively bonded joints. International Journal of Non-linear Mechanics, 97. Tong, L., Steven, G.P., 999. Analysis and Design of Structural Bonded Joints. Kluwer Academic, Boston. Tsai, M.Y., Morton, J., 99. An evaluation of analytical and numerical solutions to the single lap joints. International Journal of Solids and Structures, 7 6.