# An analytical model for predicting the stress distributions within single-lap adhesively bonded beams.

1. Introduction

The use of adhesively bonded joint, which uses metallic, composite, and ceramic materials, is of great interest to many industrial sectors including aerospace, automotive, marine, machine tools, package, and appliance industries. This widespread use of adhesive joints is due to ease of application, time and cost savings, high corrosion and fatigue resistance, crack retardance, and good damping characteristics [1-3].

Any joint occurring in practice is designed to carry a given set of loads. Most of the adherends are loaded in tension. The subsequent loads on the adhesive are then a function of the geometry of the joint. Under most operating loads and environmental conditions, the adherends behave in a linearly elastic manner. However, the adhesive may exhibit viscoelastic or nonlinear behaviour. The exact analytical solution to the problem of stress distributions in the bonded area is complex. The existing analytical studies are, therefore, based on certain simplifying assumptions with regard to the modelling of the adhesive and adherends.

In Volkersen's shear-lag analysis, it was assumed that the adhesive deforms only in shear, while the adherend deformed only in tension . The consequences of the rotation of the adherends were first taken into account by Goland and Reissner . The authors derived equations to evaluate the shearing and normal stresses in the bond layer as well as those in the jointed plates, assuming that the peel and shear stresses were constants across the adhesive thickness. In Cornell's work , a variation and extension of Goland and Reissner's method were presented for determining the stresses in adhesive lap joints. The author assumed that the two lap joint plates act like simple beam and the more elastic adhesive layer is an infinite number of shear and tension springs. Differential equations were set up which describe the transfer of the load in one beam through the springs to the other beam. From the solution of these differential equations a fairly complete analysis of the stresses in the lap joint was obtained.

The objective of this work is to present an analytical model for predicting the stress distributions within a single-lap adhesively bonded beam under tension. This model is a development of the shear-lag analysis model of Volkersen . The derivations are similar to those of Goland and Reissner  but the geometry employed is different. By combining the governing equations of each adherend with the joint kinematics, the overall system of governing equations can be obtained. Both the adherends and the adhesive are assumed to be under plane strain condition, and the adhesive stresses are assumed to be uniform across the thickness. With suitable boundary conditions, the stress distribution of the adhesive in the longitudinal direction is determined.

2. Formulation of the Problem

Figure 1 shows the coordinate system used in this study; x and z are coordinates, while u and w represent displacements. A single-lap-jointed beam is shown in Figure 2. The joint length is 2c. The two adherends are considered to be of equal thickness t and equal length (l + 2c). Two points a and b are the points at the centre of two free edge of the single-lap-jointed beam. The adhesive thickness is [eta]. The joint width is assumed to be large compared with the adherend thickness. If the system is now loaded by tensile forces T per unit of adherend width at the points a and b, the line of action of the forces will be aob.

For studying the system deformation, two coordinate systems ([x.sub.1], [z.sub.1]) and ([x.sub.2], [z.sub.2]) are introduced in Figure 2. The first system is used to analyze the behavior of the left-hand adherend under load. The coordinate [x.sub.1] has its origin at the point a, extends along the longitudinal axis of the adherend, and is positive to the right. [w.sub.1] represents transverse deformation of the adherend from the unloaded condition and is positive upwards. A similar definition is taken for ([x.sub.2], [z.sub.2]) with reference to the joint.

If [M.sub.1] is equal to the bending moment in the adherend at station [x.sub.1] and [M.sub.2] is equal to the moment in the joint at station [x.sub.2], each per unit of width, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where

tan [[alpha].sub.n] = (t + [eta])/2/l + c. (2)

As [eta] [much less than] t and t [much less than] l, 1) then reduce to the simplified forms by using Taylor series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

As for deformations w1 and w2, we have

[d.sup.2][w.sub.1]/d[x.sup.2.sub.1] = - [[M.sub.1]/[D.sub.1]], [d.sup.2][w.sub.2]/d[x.sup.2.sub.2] = - [[M.sub.2]/[D.sub.2]], (4)

where [D.sub.1] and [D.sub.2] are the flexural rigidities of the adherend and joint, respectively.

From (3) and 4), we have

[d.sup.2][w.sub.1]/d[x.sup.2.sub.1] = - [T/[D.sub.1]] [[[alpha].sub.n][x.sub.1] - [w.sub.1]] (0 [less than or equal to] [x.sub.1] [less than or equal to] l), (5)

[d.sup.2][w.sub.2]/d[x.sup.2.sub.2] = - [T/[D.sub.2]] = [[[alpha].sub.n] (l + [x.sub.2]) - [w.sub.2] - [t/2]] (0 < [x.sub.2] [less than or equal to] c). (6)

From the four boundary conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The two preceding equations (5) and (6) have solutions of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where 

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

3. Stress Distributions in the Adhesive Layer

Figure 3 shows elements of the upper and lower adherends. M, V, and T are the bending moments, vertical shear, and axial tension in the adherends. The subscripts u and l designate quantities pertaining to the upper and lower adherend, respectively. [[sigma].sub.0] and [[tau].sub.0] are the transverse normal stress and the shear stress, respectively.

The conditions of moment equilibrium for the elements of the adherends are

d[M.sub.u]/dx - [V.sub.u] + [[tau].sub.0] [t/2] = 0, d[M.sub.l]/dx - [V.sub.l] + [[tau].sub.0] [t/2] = 0. (10)

The conditions of horizontal force equilibrium are

[d[T.sub.u]/dx] - [[tau].sub.0] = 0,

[d[T.sub.l]/dx] - [[tau].sub.0] = 0. (11)

The conditions of vertical force equilibrium for the elements of the adherends are

[d[V.sub.u]/dx] - [[sigma].sub.0] = 0,

[d[V.sub.l]/dx] - [[sigma].sub.0] = 0. (12)

The transverse deflections of the upper and lower adherends are denoted by [v.sub.u] and [v.sub.l], respectively, both measured positively upward. Then the thin plate theory gives

[d.sup.2][v.sub.u]/d[x.sup.2] = - [[M.sub.u]/[D.sub.1]],

[d.sup.2][v.sub.l]/d[x.sup.2] = - [[M.sub.l]/[D.sub.1]],

with [D.sub.1] = [Et.sup.3]/12(1 - [v.sup.2]), (13)

where E and v are Young's modulus and Poisson's ratio of adherends, respectively. [D.sub.1] refers to the flexural rigidity of the adherends.

The longitudinal displacements of the adherend at the adherend boundaries adjacent to the adhesive are denoted by [u.sub.u] and [u.sub.l]. Then from the stress-strain relations

d[u.sub.u]/dx = 1/E ([[T.sub.u]/t] - 6 [[M.sub.u]/[t.sup.2]]), d[u.sub.l]/dx = 1/E ([[T.sub.l]/t] - 6 [[M.sub.l]/[t.sup.2]]). (14)

[[sigma].sub.0]/[E.sub.ad] = [[v.sub.u] - [v.sub.l]]/[eta]. (15)

Combining equations from (10) to 14) and differentiating equation (15) we have

[[d.sup.3][[tau].sub.0]/d[x.sup.3]] - [8[G.sub.ad]/Et[eta]] [d[[tau].sub.0]/dx] = 0, (16) [[d.sup.4][[sigma].sub.0]/d[x.sup.4]] + [24 (1 - [v.sup.2])[E.sub.ad]/[Et.sup.3] [eta]] [[sigma].sub.0] = 0. (17)

The boundary conditions can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

From (8) and 9) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

On the basis of the differential equations (16) and 17) and he boundary conditions, the distributions of [[sigma].sub.0] and [[tau].sub.0] in the adhesive are obtained in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

If the materials and the thickness of the adherends are different, then (17) may become a 7th order differential equation. It is obvious that the analytical model is very complex and only the stress distributions of the adhesive in the longitudinal direction are determined. In addition, the adhesive stresses actually are not uniform across the thickness. In other words, the simplifications have restricted the results. Usually, when we consider different boundary conditions by a closed-form analysis, the limitation is how tractable a realistic mathematical model is within an algebraic solution. To overcome this limitation, the finite element analysis (FEA) technique can be employed.

4. Summary

An analytical model was presented for predicting the stress distributions within a single-lap adhesively bonded beam under tension. By combining the governing equations of each adherend with the joint kinematics, the overall system of governing equations can be obtained. Both the adherends and the adhesive are assumed to be under plane strain condition, and the adhesive stresses are assumed to be uniform across the thickness. With suitable boundary conditions, the stress distribution of the adhesive in the longitudinal direction is determined.

http://dx.doi.org/10.1155/2014/346379

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This study is partially supported by the Special Program of the Ministry of Science and Technology, China (Grant no. 2012ZX04012-031).

References

 X. He, "A review of finite element analysis of adhesively bonded joints," International Journal of Adhesion and Adhesives, vol. 31, no. 4, pp. 248-264, 2011.

 X. He, F. Gu, and A. Ball, "Fatigue behaviour of fastening joints of sheet materials and finite element analysis," Advances in Mechanical Engineering, vol. 2013, Article ID 658219, 9 pages, 2013.

 X. He, "Finite element analysis of torsional free vibration of adhesively bonded single-lap joints," International Journal of Adhesion and Adhesives, vol. 48, pp. 59-66, 2014.

 O. Volkersen, "Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten," Luftfarhtforschung, vol. 15, pp. 41-47, 1938.

 M. Goland and E. Reissner, "Stresses in cemented joints," Journal of Applied Mechanics, vol. 11, pp. A17-A27, 1944.

 R. W. Cornell, "Determination of stresses in cemented lap joints," Journal of Applied Mechanics, vol. 20, pp. 355-364, 1953.

 I. U. Ojalvo and H. L. Eidinoff, "Bond thickness effects upon stresses in single-lap adhesive joints," AIAA Journal, vol. 16, no. 3, pp. 204-211, 1978.

 F. Delale, F. Erdogan, and M. N. Aydinoglu, "Stress in adhesively bonded joints: a closed-form solution," Journal of Composite Materials, vol. 15, no. 3, pp. 249-271, 1981.

 J. N. Rossettos, P. Lin, and H. Nayeb-Hashemi, "Comparison of the effects of debonds and voids in adhesive joints," Journal of Engineering Materials and Technology, vol. 116, no. 4, pp. 533-538, 1994.

Xiaocong He and Yuqi Wang

Innovative Manufacturing Research Centre, Kunming University of Science and Technology, Kunming 650500, China

Correspondence should be addressed to Xiaocong He; hhxxcc@yahoo.co.uk

Received 13 November 2013; Accepted 23 December 2013; Published 11 February 2014

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