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Geometrically non-linear analysisof adhesively bonded doublecontainment cantilever j ointsM. Kemal Apalak a & Aysel Engin ba Department of Mechanical Engineering, University ofErciyes, Kayseri, 38039, Turkeyb Institute of Applied Science and Engineering,University of Erciyes, Kayseri, 38039, TurkeyPublished online: 02 Apr 2012.

To cite this article: M. Kemal Apalak & Aysel Engin (1997) Geometrically non-linearanalysis of adhesively bonded double containment cantilever j oints, Journal of AdhesionScience and Technology, 11:9, 1153-1195, DOI: 10.1163/156856197X00570

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Geometrically non-linear analysis of adhesively bonded

double containment cantilever j oints

M. KEMAL APALAK1,* and AYSEL ENGIN2

1Department of Mechanical Engineering, University of Erciyes, Kayseri, 38039, Turkey 2 Institute of Applied Science and Engineering, University of Erciyes, Kayseri, 38039, Turkey

Received in final form 5 May 1997

Abstract-Under an increasing load, the adhesively bonded joints may undergo large rotations and dis-

placements while strains are still small and even all joint members are elastic. In this case, the linear elasticity theory cannot predict correctly the nature of stress and deformation in the adhesive joints. In this study, an attempt was made to develop an analysis method considering the large displacements and rotations in the adhesive joints, assuming all joint members to be still elastic. An incremental finite ele- ment method was used in the application of the small strain-large displacement theory to the adhesively bonded joints. An adhesively bonded double containment cantilever (DCC) joint was analysed using this incremental finite element method under two different loadings: a tensile loading at the horizontal plate free end, Px. and one normal to the horizontal plate plane, Py. The adhesive and plates were assumed to have elastic properties, and some amount of adhesive, called spew fillet, that accumulated at the adhesive free ends was also taken into account. The analysis showed that the geometrical non-linear behaviour of adhesively bonded joints was strictly dependent on the loading and boundary conditions. Thus, a DCC joint exhibits a high non-linearity in the displacements, stresses, and strains in the critical sections of the adhesive and horizontal plate under a tensile loading at the free end of the horizontal plate, Px, while a similar behaviour in these regions was not observed for a loading normal to the horizontal plate plane, Py. However, an increasing non-linear variation in the stresses and deformations of the horizontal plate appeared from the free ends of the adhesive-horizontal plate interfaces to the free end of the hori- zontal plate for both loading conditions. Consequently, joint regions with a low stiffness always undergo high rotations and displacements, and if these regions include any adhesive layer, the non-linear effects will play an important role in predicting correctly the stresses and deformations in the joint members, especially at the adhesive free ends at which high stress concentrations occurred. In addition, the DCC joint exhibited a higher stiffness and lower stress and strain levels in the joint region in which the support and horizontal plate are bonded than those in the horizontal plate.

Keywords: Epoxy adhesives; steel adherends; containment joints; geometrical non-linearity; stress analysis; finite element method; adhesive fillet.

*To whom correspondence should be addressed.

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1. INTRODUCTION

A considerable world-wide expansion has taken place in the chemical industry during the last few decades. Intensive research has been carried out in synthetic organic

chemistry and as a result, many new materials have been developed. Among these

materials, plastics have an important place. Advances in synthetic adhesives have

provided engineers with polymeric and composite materials to be used in the design of structures. Thus, the adhesive bonding technique has been employed in the bonding of structural components with similar/dissimilar material properties. A high joint

strength can be achieved reliably in bonded metal assemblies with the correct design and the proper use of a suitable adhesive [1].

Adhesively bonded joints have attracted many researchers, since the adhesive bond-

ing technique is used widely as an alternative to more traditional joining methods.

Many theoretical and experimental studies are now available [2, 3]. The main objective in all the studies was to develop a full analysis including all properties of adhesively bonded joints. However, the studies so far have been limited to the elasto-plastic, viscoelastic properties of joints, and they are able to explain only the mechanical behaviour of materials. It is obvious that a full analysis should be able to explain the effects of all factors on the joint behaviour, i.e. geometrical and material non-

linearity, viscoelasticity, thermal and moisture effects, fatigue, etc. Therefore, the adhesion mechanism and the mechanics of stress and deformation in the adhesively bonded joints should be mastered. Stress analysis allows us to predict the strength of

the adhesive joints and to explain the mechanics of stress and deformations. However, the adhesion mechanism related to the physical and chemical properties of the adhe-

sive will play an important role in predicting the joint strength and failure mechanism.

Consideration of all the factors expressed above in the analysis of the adhesive joints

requires a realistic theoretical method supported by the adhesive properties determined

experimentally. The finite element method has been successfully applied to determine

the stress and deformations occurring in many structures with a complicated geometry and material properties [4]. The present study is an attempt to predict the geometrical non-linear behaviour of adhesively bonded joints under different loading and boundary conditions. The theoretical basis is given for a finite element model developed for

the stress analysis of adhesively bonded joints. The analysis method is also used for predicting the elastic stresses and deformations in an adhesively bonded double

containment cantilever joint.

2. BACKGROUND

The effects of the load eccentricity on the stresses of an adhesively bonded single lap

joint were taken into consideration by Goland and Reissner [5]. They assumed that the adhesive layer and adherends behaved as a linear elastic solid and considered two cases using a bending moment factor K: (i) the adhesive layer was extremely thin and of similar elastic stiffness to the adherends so that its deformations were of little

importance; (ii) the adhesive layer was thin but its deformation made a significant

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contribution to the stress distribution in the joint. The peel stress was found to be

very high at the adhesive free ends whilst the shear stress was zero for the first case. The maximum values of the normal and shear stresses occurred at the free ends in the second case. In this study, it was assumed that the adherends and adhesive were linear elastic and that the stress distribution across the adhesive layer was constant.

The effect of the internal bending moments resulting in normal stresses across the adhesive layer was taken into account by Volkersen [6]. The following studies are related to setting up the differential equations considering the effects of bending and transverse normal and shear deformations on an adhesively bonded joint with different

geometries and mechanical properties. Cornell [7] and Benson [8] are among others who carried out this type of study. Since the early studies showed that the shear and normal stresses were uniform along a long middle joint region but were maximum only at the joint ends, the following studies concentrated on the stresses and deformations at the adhesive free ends, called the end effects.

Based on Muki and Stenberg's study [9], Erdogan and Ratwani [10] analysed the stress distribution in the plates of a stepped joint under the assumption of a generalized plane stress for specific plate geometry and material combinations. They showed that the bending moments in the adherends were maximum at the free edge zones of the

adhesive layer. Chang and Muki [11] also dealt with the elastostatic load transfer of a tensile load in a model adhesive lap joint and showed that the effects of the adherend

geometry at the initial ends of the bond line were very important. Taking into account the transverse shear and normal strain effects, Renton and

Vinson [12-15] presented a comprehensive method of analysis for the linear elastic

response of two laminated composite material plates joined through a single bonded

lap joint and subjected to in-plane loads. Their analysis included the effects of the

anisotropic properties and they also determined accurately the state of stress in both the joint adhesive material and at any point in the adherends under a fatigue-producing dynamic load. They showed that both adhesive and adherend stiffnesses played an

important role in the peak stresses occurring at the adhesive free ends.

Yuceoglu and Updike [16-18] carried out a series of studies taking into account

bending deformations and shear deformations of the adherends. They developed a more general analytical model and formulation of the problem of adhesive joints and presented solutions to the transverse normal stress and longitudinal shear stress distributions in the adhesive layers of bonded stiffener plates, double lap joints, and

strap joints, all of dissimilar orthotropic adherends. Due to the complicated geometry of adhesively bonded joints, the previous studies

considered bending and shear deformations of the adherends but neglected the vari- ation of the stresses in the thickness direction of the adhesive layer. Almann [19] assumed a linear variation of the peeling stress across the adhesive thickness, but the adhesive shear stress was constant through the thickness. He assumed that two

orthotropic adherends were bonded together through the adhesive layer and deter- mined the average elastic stress through the thickness of the adhesive by minimizing the strain energy which was calculated from the equilibrium stress distribution in the

joint. Later, Ojalvo and Eidinoff [20, 21] presented an analytical investigation on the influence of bond thickness on the stress distribution in single lap adhesive joints

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which was an extension of the basic approach for bonded joints originally introduced

by Goland and Reissner [5]. They obtained completely new terms in the differential

equations and boundary conditions for peel stress, besides modifying some coefficients in the shear stress equations. Kline [22] also studied the effect of adhesive thickness on the stress distribution in an adhesively bonded joint, assuming a linear variation in adhesive stresses through the adhesive thickness. However, the assumptions were the same as those in the previous studies.

The studies referred to above showed that the peak stresses occurring at a very close distance to the adhesive free ends, called end effects, could be relieved by in-

creasing the overlap length. As an alternative, Thamm [23] considered lap joints with

partially thinned adherends and showed that sharpening the adherends of overlapped adhesive joints to a knife edge yielded a substantial increase in strength only when the sharpening was complete. Sainsbury-Carter [24], Wah [25-27], Ramamurthy and Rao [28], and Webber [29] are among those who have analysed adhesive scarf joints with sharpened adherends.

Delale et al. [30, 31] have continued Almann's method [19] and have formulated the problem of adhesively bonded structures consisting of two different orthotropic adherends by using a non-linear material for the adhesive under a generalized plane strain assumption. They neglected the effect of adherend thickness variation on the

stresses in the adhesive layer but took into account the transverse shear effects in the

adherends and in-plane normal strains in the adhesive. In addition, Hart-Smith [32-37] has presented a series of studies in which the stress

distribution in adhesives which can sustain a large plastic strain is investigated using the continuum mechanics approach. He tried to explain the elasto-plastic behaviour of different types of adhesively bonded joints. He believed that the adhesive layer was not a weak link and that the peeling stresses could be relieved by tapering the adherend edges. Harth-Smith's approach neglected the presence of normal stresses

across the adhesive layer. Later he recognized this and considered elastic peeling stresses and plastic shear stresses and showed that the peeling stresses did not have a

considerable effect if the adherends were sufficiently thin [38].

Considering the time-dependent properties of an adhesive, Delale and Erdogan [31] ]

analysed an adhesively bonded lap joint. They accounted for the transverse shear effects in the adherends and assumed that the adhesive was a linearly viscoelastic solid. Later Weitsmann [39, 40] investigated the effects of the non-linear viscoelastic behaviour of the adhesive material on the response of a symmetric lap joint and showed that the highly stressed regions were most substantially influenced by viscoelastic

creep, which tended to reduce the stress levels near the edges of the adhesive joint. Brinson [41] compared a linear version based on a modified Bingham model with

the measured response of neat bulk tensile specimens and symmetrical single-lap shear

specimens. He indicated that the most significant differences in the magnitudes of the

creep strains were encountered in the two geometries for the linear Bingham model and obtained good agreement at low stress levels but not at higher stress levels as a result of a non-linear analysis.

Due to the complex geometry of adhesively bonded joints and the non-linear prop- erties of the adhesive and adherend materials, the previous closed-form analytical so-

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lutions were limited in predicting the complete behaviour of adhesively bonded joints. The finite element method (FEM), which is a very powerful method, described by Zienkiewicz and Taylor [4], has found much use in the analysis of adhesively bonded

joints. Wooley and Carver [42] were among the first to apply this method to a single

lap joint, considering the plates and adhesive to be isotropic. Although they used a

linear quadrilateral element to model the adhesive layer and adherends by assuming the adhesive free ends to have square ends and by using an unrefined mesh around these regions, their analysis showed a good correlation with the closed-form solu- tions. Harrison and Harrison [43] carried out similar studies in which the effects of some parameters such the adhesive modulus, overlap length, and adhesive thickness were investigated using the FEM. However, the effects of adhesive fillet, bending, and

differential straining were ignored. Using the FEM, Barker and Hatt [44] analysed the behaviour of an adhesive layer

between a composite substrate and a metallic substrate, treating the adhesive layer as a separate elastic medium of finite thickness. They analysed a single lap joint and a smoothly tapered joint having similar and dissimilar materials and showed that maximum stresses occurred at the adhesive free ends and that increasing the adhesive thickness and tapering the adherend free ends relieved these peak stresses.

Adams and Peppiatt [45, 46] were the first researchers to take into account the

presence of high stress gradients across the adhesive layer close to the adherend

edges. They analysed single- and double-lap joints assuming the problem to be plane strain and using more than one triangular constant stress across the adhesive layer. In

addition, they considered the existence of adhesive spew fillets around the adhesive free ends. They showed that when the adhesive spew fillets were considered, the maximum adhesive shear stress was much smaller than that predicted in joints with

square adhesive free ends and the maximum principal stresses at the adhesive free ends occurred at right angles to the direction of cracks formed in the adhesive spew fillet of failed joints. Later, Adams and Peppiatt [47] analysed tubular lap joints subjected to tensile and torsional loadings whose adhesive free ends included adhesive spew fillets. They also predicted lower stress concentrations around the adhesive free ends for these joints. Similar results were obtained by Adams et al. [48] as a result of the stress analysis of axisymmetric circular annular and solid butt joints loaded in torsion.

Crocombe and Adams [49, 50] also studied the effect of a spew fillet on the ad- hesive stress distribution of a single lap joint over a range of material and geometric properties. They used a more advanced element mesh around the region subjected to

high stress gradients. All their results were in close agreement with the closed-form

analytical results, and the stress gradients across the adhesive layer were taken into consideration.

In previous finite element analyses of adhesively bonded joints, the adherend and in

particular the adhesive properties have been assumed to be linear elastic. Non-linear

properties of the adherends and adhesive material have often been ignored under high loads. Sawyer and Cooper [51] investigated the load transfer of a single lap joint in which the adherends were pre-formed so that the angle between the line of action of the applied in-plane force and the bond line was reduced. Since the bending moment is due to the eccentricity of the loading and to the deformation of the adherends as

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the load is applied, they considered that the dependence of the moment on the applied load made the problem geometrically non-linear. They found that pre-forming the adherends reduced the resultant moment in the adherend at the edge of the overlap region, which caused a reduction in both the peel and the shear stresses and gave a more uniform shear stress distribution in the adhesive layer.

In order to predict the failure modes and loads of single lap joints having adherends and adhesives with different mechanical properties, Adams [52] and Harris and Adams

[53] used a non-linear finite element technique that was able to account for the large displacements and rotations that may occur in a single lap joint under tensile loading and that allowed the effects of non-linear material behaviour of both the adhesive and

the adherends. They showed that the mechanical properties of both the adhesive and the adherends had a considerable effect on the failure mode and loads. Adams et al.

[54] also carried out a detailed analysis of the shear and transverse tensile stresses in carbon fibre-reinforced plastic/steel double-lap joints using an elastic-plastic model

for the rubber-modified epoxy adhesive in order to determine the strength of the joints.

They found that modifying the geometry of the double lap joint in the critical regions at the edge of the overlap caused significant increases in the strength of the joint.

Adams and Harris [55] investigated the influence of local geometry changes at the

edges of the overlap in single lap joints using a non-linear FEM. They modelled the effects on the stress distribution of geometrical changes which are small in relation to the dimensions of the local geometry in order to provide an improved model for failure

prediction. They found that the finite element analysis was capable of predicting the

significant strength increases that might be achieved in single lap joints by filleting the adhesive at the edges of the overlap and the ends of the adherends.

Reddy and Roy [56] presented a geometrically non-linear finite element approach for the analysis of adhesively bonded joints. Their analysis method was in good agreement with previous studies for single lap joints under different loading and

boundary conditions. Czamocki and Pierkaski [57] also studied the effect of the joint width on the adhesive stresses, treating the adhesive as a non-linear material.

More recently, Edlund and Klarbring [58, 59] have presented a general analysis method for determining the adhesive and adherend stresses and deformations in ad-

hesively bonded joints. They evaluated the joints as three-dimensional structures and assumed the adherends and adhesives to have non-linear material properties and also

took into account the geometrical non-linearity. The geometrical non-linear behaviour of engineering structures under increasing

load plays a very important role in predicting the load-bearing capability of the struc- tures. The analysis method, including this behaviour, using the finite element approach

generally requires iterative solutions of a large number of non-linear equilibrium equa- tions. However, many parameters that the analytical approach had to ignore can be

taken into account and a more realistic solution of the problem can be achieved. Wood et al. [60, 61], Stricklin et al. [62-66], Carey [67], and Bathe and Cimento

[68] are among those who have contributed to the development of the FEM including

geometrical non-linear effects. More details can be found in [4, 69, 70]. It is clear from the review of the literature related to the analysis of adhesively

bonded joints that the analysis methods for adhesively bonded joints are not complete.

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A full analysis would be possible only when all properties affecting the adhesion mechanics and the nature of stresses and deformations occurring in the adhesive joints are included. This study is an attempt to develop an analysis method considering the

large displacement effects occurring due to an increasing loading in adhesively bonded

joints. The method has been applied to an adhesively bonded double containment cantilever joint.

3. THE LARGE DISPLACEMENT THEORY

3.1. Introduction

In elasticity, the current configuration is compared with the initial state and the strain definition is based on this initial state. The detailed method by which the material has moved from the initial state to the current configuration is assumed not to affect the final state.

If the displacements and displacement gradients are not small, the characterization of the state of the strain is not simple, even with this assumption. In fact, only when the

displacements and their gradients are infinitesimal is this small strain theory strictly correct. In the case of the small displacement-small strain approach, in order to

develop expressions for the small strains and rotations in three dimensions, an arbitrary infinitesimal line vector dX in the initial position is considered and an expression for the components of the unit relative displacement vector du/dS is sought. In indexical

notation,

or in xl, x2, x3 notation, '

which may be written briefly as

where du/dS and n are the column matrix representations of the vector du/dS and the unit vector n in the direction of dX. If we transpose the square matrix Ju, the

- matrix of V u and the alternative form of equation ( 1 c)

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are obtained. Ju is the 3 x 3 Jacobian matrix or displacement gradient matrix. Each

row of the displacement gradient matrix contains the vector components of the gradient of one scalar component of the displacement vector. The Jacobian matrix is an

operator which is associated with each vector dX between two neighbouring particles, such as K and M - the unit relative displacement du/dS of M relative to K;

therefore, it is also called the unit relative displacement matrix.

The matrix Ju can be written as the sum of a symmetric matrix E and a skew-

symmetric matrix St. Generally, the notation E:ij is used to denote the components of

the small-strain tensor E. Thus, the small-strain tensor

and the rotation tensor

As can be seen, all components of the displacement gradient tensor should be small

compared to unity. Thus, when the components are small compared to one radian,

equation (3) exhibits a rigid-body rotation and the strain vector 8 exhibits that part of the unit relative displacement vector not attributable to local rigid body rotation of

the element initially at K.

These small strain-small displacement definitions appear when the squares and

products of the displacement-gradient components 8u; /8 X j are neglected in compar- ison with linear terms. Therefore, an approximate representation can be very good in

the usual elasticity theory of metals, but less good, and possibly not good at all, in

polymer materials where the elastic strains may be large. When the displacement-gradient components are not small compared to unity, the

problem of characterizing the strain from the initial state is more difficult than in the

small-strain case. The finite strain can be defined based on either the undeformed

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configuration or the deformed configuration. The first approach uses material co-

ordinates in the undeformed configuration, while the second one uses spatial co-

ordinates in the deformed configuration. The formulation in terms of the undeformed

configuration is usually called the Lagrangian formulation, while the one in terms of

the deformed configuration is called the Eulerian formulation. The simplest way of defining the deformation-gradient tensor is to use the defor-

mation equations. Since the deformation-gradient tensor includes the rotation as well as the deformation, constitutive equations employing it will have to be constructed so that they will not predict a stress due to rigid body rotation. A simple linear

homogeneous stress-strain equation will satisfy the requirement that no stress will be predicted to arise from rigid body rotation.

The Lagrangian formulation seems to be the more suitable one in elasticity, since in elasticity it is usually assumed that there is a natural undeformed state to which the body will return when it is unloaded. But the stress equations of motion or

equilibrium must be satisfied in the deformed or contemporary configuration, and stress is therefore defined in the deformed configuration. If a stress-strain equation is to be written, then either the stresses must be referred back to the undeformed

configuration or the strains must be referred to the deformed configuration in order to use the same reference for all the tensors appearing in the equation. We consider

only the second of the two alternatives. Let a particle displace from its initial position X to the current position x defined

by the deformation equations

The deformation gradients are the gradients of the functions on the right-hand side of these equations. The rectangular Cartesian components of the gradients are deriva- tives with respect to material co-ordinates. The deformation gradient referred to the

undeformed configuration is denoted by F or by x © ; its transpose is FT = Vx. The deformation gradient tensor F has rectangular Cartesian components being the partial derivatives 8xk /8 Xk and operating on an arbitrary infinitesimal material vector dX at X to associate with it a vector dx at x as follows:

since

The rectangular Cartesian component matrix forms are

both representing the same indexical form

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Although the components of the deformation gradient are finite in equation (5), we are

considering the deformation of an infinitesimal line element between two neighbouring particles. When the material vector dX is small but finite, the associated dx does not coincide exactly with the deformed position of the material dX. If we consider the

change in the squared length of the material vector dX, the strain tensor E, whose

component forms are rectangular Cartesians, is defined as follows:

The Green deformation tensor C is referred to the undeformed configuration and

gives the new squared length (ds)2 of the element into which the given element dX is deformed instead of giving the change in the squared length; thus,

Comparing equations (6) and (7), we obtain the following relationships between the deformation tensor and strain tensor:

In order to determine the expressions for the strain and deformation tensors in terms of the deformation gradient, using equation (5a), we can express (ds)2 as follows:

If we compare equation (9) with the defining equations for the deformation tensors,

equation (7) then shows that

The expressions for the strain tensors can be obtained from equation (8):

The strain and deformation tensors are symmetric and they each have three real

principal values. In order to compare them with the small-strain components, let us

write out E11 and E12. Using the same reference axes for both xi and X,, and using lower-case subscripts for both, we have

with displacement components

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Thus, we obtain

In terms of the displacements, the general expression for Eij in equation (11) takes the form - -

We see that if the partial derivatives of the displacements UI, U2, u3 with respect to the material co-ordinates X1, X2, X3 are all small compared to unity, the squares and

products of these derivatives may be neglected in comparison with the linear terms. The remaining terms are the small-strain components. It is obvious that the large displacement theory is an extension of the small displacement theory, including the

squares of the displacement gradients which introduce the non-linear effects. When the displacement gradients are not small compared to unity, these terms including the non-linear effects in the structure must be taken into account, since the strain is no

longer proportional to the displacement [71, 72].

3.2. Application of the large displacement theory to the FEM

For a two-dimensional problem, displacement gradient vectors and matrices can be defined as functions of x as

and

where typically

and the matrix including the displacement gradients

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The non-linear strain-displacement relation, Green's strain in equation (14), in terms

of linear Eo and non-linear EL displacement gradients can be written in matrix form

as follows:

where _

If a virtual displacement du is

then the variation in Green's strain dE that is necessary for developing finite element

formulations becomes

where

Let us consider equation (22b).

Substituting equation (23) into equation (22a), we obtain

In the finite element formulation, the displacement within an element is given as a

function of n discrete nodal displacements p as

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where

and N is a shape function array whose coefficients are functions of the initial position x within the element, i.e. for u 1,

From equation (27) the virtual displacements du are expressed in terms of the virtual nodal displacements dp as

that is, for u 1,

Similar expressions can be written for the other displacements and their derivatives. The displacement gradients 0 of equation (16) can be written in terms of the nodal

displacements p by the linear relation

where the coefficients of gradient matrix G are Cartesian derivatives of the shape functions contained in N; thus,

or in the matrix form

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and the nodal displacements

where n is the number of nodes in the element. The linear Green's strain vector Eo of equation (19) can be written in terms of p as

or in the matrix form

where the constant strain matrix Bo consists of Cartesian derivatives of N; thus,

dEo = Bo dp. (35)

Finally, the components of Green's strain vector E in terms of the nodal displacements

p may now be written as

where the non-linear strain matrix

Equation (36) can also be reduced in the form of

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Similarly, the variation dE of E from equation (24) can be written as

where the strain matrix

Bo is a function of the shape functions only, while BL is a function of the shape functions and displacements; therefore, E is a non-linear function of p because A(6) is a function of 0.

A further useful simplification of the strain-displacement relations can be made by

writing Bo as

where the Boolean matrix .........

In this way, B and B* defined in equations (41) and (39) can be written as

or

Similarly,

or

where

3.3. Non-linear equilibrium equations

The equilibrium equation for the deformed body is established from the virtual work

equation given in terms of the Lagrangian co-ordinate system, by equating virtual

internal and external work as [4, 60, 69, 70]

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where the external work is due to du acting on the surface tractions, extending over

the initial undeformed surface A and, given by

and the body forces per unit mass, acting within the undeformed volume V, given by

p being the density of the undeformed body. The total Lagrangian virtual work expression, equation (48), can be approximated

by the finite element idealization using equations (28) and (40), giving

Since the virtual nodal displacements dp are arbitrary, equation (51) can be written

HS all

If we represent the right-hand side of equation (52) by R for convenience, then the

non-linear equilibrium equations become

where # (p) is termed the residual, and the solution is achieved when # (p) is reduced

to zero or a given convergence criterion is satisfied. The solution is approached in

steps; i.e. a value of displacement is calculated and then the accuracy of the solution

is assessed by calculating the residual at this subsequent position.

3.4. Solution method to non-linear equilibrium equations

In order to determine a new displacement vector at each step, it is necessary to perform the solution of the following equation:

where Ks, the secant stiffness matrix, is non-symmetric as

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and D is a symmetric matrix of constitutive coefficients.

In the solution algorithms for the assembled non-linear equations (54), the Newton-

Raphson method is used, involving a series of solutions to linear incremental equilibri- um equations. In order to find a complete equilibrium path, R is applied as a series of

incremental loads. Iterations continue within a load increment until *(p) = 0 or sat- isfies a given convergence criterion. A new approximation to the total displacements is obtained as - _ ,

If we consider the variation dot of equation (53), provided that the loads do not vary with displacement, we obtain

or _

This expression can be reduced by rewriting

or I-

Let us consider the d[A(e)]TQ term on the right-hand side expression of equation (58):

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or

From the derivatives of the displacements with respect to x, and x2,

Substituting this last expression into equation (58), we obtain

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where the Piola-Kirchhoff stress matrix, S, is

We can write equation (59) as

where

is called the tangential stiffness matrix. Substituting equation (62) into equation (56), the Newton-Raphson formula becomes

Since the tangential stiffness matrix KT is symmetric, Ap?+1 can be determined

by any usual solution technique. It is necessary to correct displacements in order to reduce the residual *i to zero. Using equation (44) for B, KT can be written as

where

The matrix equation (54) can be solved by repeating the following procedures and

assessing the residual until satisfactory convergence is achieved:

(i) The displacement increment is determined:

(ii) The new displacement vector is calculated:

(iii) The residual is found at the new position:

(iv) Repeat loop from (i) if the convergence criterion is not satisfied.

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When the full Newton-Raphson approach is used, the tangential stiffness matrix

KT will be evaluated at every stage of the above procedure. As an alternative, it may be kept constant for a number of stages in which the non-linearity does not appear. This method is known as the modified Newton-Raphson approach. In the case of the

full Newton-Raphson approach, since recalculating the tangential stiffness matrix at

each new step requires more computing effort, convergence is reached very quickly.

4. JOINT CONFIGURATION

In this study, the geometrically non-linear finite element approach was applied to

a bonded double containment cantilever (DCC) joint shown in Fig. 1 to determine

its behaviour under an increasing load. The joint consists of a double containment

support, a horizontal plate, and an adhesive layer. Previous studies have shown that

peak peeling stresses arising around the adhesive free ends due to the bending moment

have a considerable effect on the strength of the adhesively bonded joints [2, 3]. In

order to reduce these peeling stresses, generally the overlap region, along which the

adherends are bonded, is increased. An alternative solution is to design the overlap

region so that the adhesive layer can be under compressive stresses. For this purpose, a double containment support including a slot into which the plate is bonded can be

used as shown in Fig. 1.

A bonded double containment cantilever joint was considered with a containment

support length a of 30 mm, slot depth b of 25 mm, adhesive thickness 8 of 0.5 mm,

plate thickness t and support thickness c of 5 mm each, joint length L of 160 mm, and joint width W of 500 mm. These dimensions of the joint were kept constant

throughout the study. Since the geometry along the width of the double containment

cantilever joint is uniform and the applied loads do not change in that direction, the

problem can be reduced to one of plane strain.

The DCC joint was fixed by giving zero displacements in the x- and y-directions of

the nodes along the back of its double containment support as shown in Fig. 2, and

was analysed for two loading conditions: a load Py of 5000 N applied at the end of

the horizontal plate in the normal direction to the horizontal plate plane, and a tensile

Figure 1. The double containment cantilever joint.

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Figure 2. Boundary and loading conditions of the double containment cantilever joint. (a) Loading normal to the horizontal plate plane; (b) tensile loading at the horizontal plate free end.

load Px of 5000 N/mm distributed among the nodes along the right free end of the horizontal plate.

An epoxy-based adhesive having a modulus of elasticity Ea = 3.33 GPa and Pois-

son's ratio va = 0.34 was used to bond the double containment support and the horizontal plate made of steel material having a modulus of elasticity E = 210 GPa

and Poisson's ratio v = 0.29. The materials of all joint members, i.e. the plate,

support, and adhesive, were assumed to have linear elastic properties. Their plastic

properties (material non-linearity) were not taken into account and only the behaviour

of the joint members, as the load was increased, was observed.

Analytical and photo-elastic studies of adhesively bonded joints have shown that the adhesive accumulating around the adhesive free ends, called the spew fillet, has

a considerable effect on the peak adhesive stresses and strains, and increasing the

adhesive fillet size reduces the peak stresses and strains [2, 73-75]. Therefore, the

presence of adhesive fillets was taken into account in the analysis and their shapes were idealized to a triangle of height and width twice the adhesive thickness due to

the ease of meshing them.

In order to apply the FEM to any continuous medium with a given boundary and

loading conditions, the continuum has to be divided into elements including nodes at its comers and edges. Therefore, eight-noded isoparametric quadratic quadrilateral

plane elements with four integration points were used to model the horizontal plate,

support, and adhesive layer. In addition, six-noded isoparametric quadratic quadrilat- eral plane elements with three integration points were used to model the joint regions in which the use of the other element type is not possible, i.e. transition regions be-

tween mesh areas and in the adhesive fillets. A series of analyses showed that the mesh refinement, particularly around adhesive free ends, in which high stress and strain gradients occurred, was necessary. Therefore, these regions were refined until

reasonable results were achieved, i.e. a negligible difference between the results of two different meshes. The adhesive free ends were divided into four elements across the

adhesive thickness while the other adhesive regions were divided into three elements. The final mesh details are shown in Fig. 3.

The pre- and post-processing modules of the general-purpose finite element software

ANSYSO (current version 5.1 ) were used to generate the mesh of the DCC joint and to process the analysis results [76]. All stress and strain contours were plotted using

averaged nodal values.

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Figure 3. Mesh details of the double containment cantilever joint. Support length a = 30 mm, slot depth b = 25 mm.

5. ANALYSIS AND RESULTS

Previous studies have generally concentrated on the analysis of unbalanced single- and

balanced double-lap joints due to their simple geometry and ease of testing [2, 3].

Although different types of adhesively bonded joints are used in practice, their di-

mensions, such as overlap length, plate thickness, etc., are determined considering the

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behaviour of adhesive joints with a simpler geometry, such as single- and double-lap

joints under similar loading conditions.

In this study, the geometrical non-linear analysis of an adhesively bonded DCC

joint, which has been used widely in practice and was presented first by Davies and Khalil [77], was carried out by using the incremental FEM. Davies and Khalil's

analysis considered only the displacements and general stress nature using a linear elastic FEM and ignored the presence of adhesive spew fillets around the adhesive free ends. Apalak and Engin [78] analysed this joint for different loading conditions

using a rather improved finite element mesh under the small strain -small displacement

assumption and showed that the adhesive fillet, overlap length, and slot depth were the main parameters having an effect of reducing the stress and strains at the adhesive free ends in which stress and deformations are very high.

The deformation and stresses in adhesively bonded joints are dependent on the

type of boundary condition. Therefore, the DCC joint was analysed for two types of loading condition shown in Fig. 2. The main dimensions of the DCC joint were used as determined in a previous study [78]. In the geometrical non-linear analysis of

the DCC joint, convergence values of 0.01 % and 0.004% were used for the loading conditions Px and Py, respectively. High stress and strain distributions occur around the free ends of the upper and lower horizontal adhesive layers for both loading conditions. In order to show the stress and strain distributions in detail, the joint region is enlarged and the normal and shear stress distributions are plotted in Figs 4 and 5 for the loading conditions Py and Px, respectively. The adhesive free ends are

subjected to stress concentrations for the loading condition Py and the stresses in a

large joint region remain at lower levels. In the case of the loading condition Px, the stresses concentrate around the adhesive free ends and propagate by reducing along the adhesive layer to the slot back end. In addition, this loading condition causes

lower stress levels. In order to determine the locations in which the maximum adhesive stresses and

strains occurred, the adhesive spew fillets at the upper and lower adhesive free ends

were magnified and the Von Mises stress and strain distributions were plotted for the loading conditions Py and Px, respectively (Figs 6 and 7). The adhesive stresses

and strains concentrate around the containment comer of the support for both loading conditions and these stress and strain values in both the upper and the lower adhesive

spew fillets are similar, due to the symmetry of the joint geometry. It is possible that the first crack, which may result in the joint failure, initiates from this comer and

propagates along the adhesive layer and to the inclined out-surface of the adhesive

spew fillet. In addition, higher stresses and strains occur in the adhesive spew fillet in the case of the loading condition P,,.

The general evaluation of the geometrical non-linear analysis of an adhesively bond- ed DCC joint shows that the behaviour of stresses and strains in the adhesive layer and other joint members is dependent on the loading and boundary conditions applied to the joint. However, the peak stresses and strains always take place around the

support comers inside the adhesive spew fillets at the adhesive free ends. In addition, the locations near the upper and lower adhesive spew fillets of the horizontal plate are subjected to high stress and strains for both loading conditions.

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Figure 4. Stress contours in a double containment cantilever joint subjected to a load normal to the horizontal plate plane, Py. (a) Normal stress axx; (b) normal stress O"yy; (c) shear stress txy.

In order to find out the non-linear effects in the stresses and strains in the critical

regions in the adhesive layer and the horizontal plate, linear elastic and geometrical non-linear analyses of the DCC joint were carried out for both loading conditions and then the variations of the normal and shear stresses and strains in the regions in

question were evaluated.

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Figure 5. Stress contours in a double containment cantilever joint subjected to a tensile load at the free end of the horizontal plate, Pr. (a) Normal stress or,,,,; (b) normal stress (c) shear stress rv.

In the case of the loading condition Pa , as the load is increased, the small strain-

large displacement (SSLD) analysis predicts lower displacements in the x- and

y-directions at the comer of the containment inside the adhesive spew fillet than

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the small strain-small displacement (SSSD) analysis, as shown in Fig. 8. Similarly, both the SSLD and the SSSD analyses present different variations of the adhesive peak normal and shear stresses and strains. Thus, they generally increase continuously as the load is increased, as shown in Figs 9 and 10; however, the SSLD analysis can achieve the solution for a given convergence value at higher stress and strain values than the SSSD analysis presents. This is an evident result of an internal bending moment resulting in large rotations and displacements at the adhesive free ends and at the horizontal plate. The differences between the displacement and stress and strain values in the critical region of the horizontal plate as a result of both analyses are very obvious, as shown in Figs 11-13. Thus, the SSLD analysis includes a large displace- ment effect and predicts higher displacement, normal stress and strain components than the SSSD analysis does. However, when the stress and strains are examined at different locations along the upper and lower horizontal edges of the horizontal

plate, an increasing non-linear effect is observed in the variations of the stresses and strains. These variations are not shown here to avoid unnecessary repetition. As the horizontal plate free end is approached, the stiffness of the horizontal plate decreases and therefore it undergoes higher displacements and rotations. In order to predict correctly the stresses and strains in all the critical members of the adhesively bonded

joint, it is necessary to take into account the non-linear effects of the large displace- ments arising from the internal bending moment in the analysis for the tensile loading condition.

In the case of the loading condition Py, as the load is increased both the SSSD and the SSLD analyses present very close results for the adhesive displacements and normal and shear stresses and strains around the adhesive free ends, as shown in

Figs 14-16. However, the stress and strain variations in the critical regions of the horizontal plate exhibit small differences for the SSSD and SSLD analyses, while all the displacement components are very close, as shown in Figs 17-19. The results of both analyses show that the nature of the stress and strain in all critical members around the adhesive free ends of the adhesively bonded DCC joint is not so sensitive to the non-linear effects. Thus, this type of joint does not represent large displacements and rotations in the support regions with double containment for the loading normal to the horizontal plate plane Py. However, the variations of the normal and shear stresses and strains at different nodal points along the upper horizontal edge of the horizontal

plate experience an increasing non-linear effect as the free end of the horizontal plate is

approached. These variations are also not shown here to avoid unnecessary repetition. In this loading condition, due to a lower stiffness of the horizontal plate, the DCC

joint undergoes higher displacements and rotations, especially along the horizontal

plate. It is obvious that the stiffness of the joint members can play an important role in the non-linear effects that the joint and its members may experience.

6. CONCLUSIONS

In this study, an attempt was made to develop an analysis method for adhesively bonded joints using the SSLD theory. The incremental FEM based on the SSLD

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Figure 8. The displacements at the lower comer of the upper containment inside the upper adhesive fillet of a DCC joint for the loading Pa predicted based on both the small strain-small displacement (SSSD) and the small strain-large displacement (SSLD) theories. (a) Displacement in the x-direction, Ux ; (b) displacement in the y-direction, Uy.

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Figure 9. The normal and shear stresses at the lower comer of the upper containment inside the upper adhesive fillet of a DCC joint for the loading Px predicted based on both the SSSD and the SSLD theories. (a) Normal stress or,, (b) normal stress oryy; (c) shear stress txy.

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Figure 10. The normal and shear strains at the lower comer of the upper containment inside the upper adhesive fillet of a DCC joint for the loading P, predicted based on both the SSSD and the SSLD theories. (a) Normal strain cxx; (b) normal strain Eyy; (c) shear strain Cxy.

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Figure 11. The displacements at the critical point in the horizontal plate of a DCC joint for the loading PX predicted based on both the SSSD and the SSLD theories. (a) Displacement in the x-direction, Ux; (b) displacement in the y-direction, Uy.

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Figure 12. The normal and shear stresses at the critical point in the horizontal plate of a DCC joint for the loading P, predicted based on both the SSSD and the SSLD theories. (a) Normal stress axx ; (b) normal stress any ; (c) shear stress rv.

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Figure 13. The normal and shear strains at the critical point in the horizontal plate of a DCC joint for the loading Px predicted based on both the SSSD and the SSLD theories. (a) Normal strain exx; (b) normal strain Eyy; (c) shear strain exy.

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Figure 14. The displacements at the lower comer of the upper containment inside the upper adhesive fillet of a DCC joint for the loading Py. predicted based on both the SSSD and the SSLD theories. (a) Displacement in the x-direction, UX; (b) displacement in the y-direction, Uv.

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Figure 15. The normal and shear stresses at the lower comer of the upper containment inside the upper adhesive fillet of a DCC joint for the loading Py predicted based on both the SSSD and the SSLD theories. (a) Normal stress O"xx; (b) normal stress ayy ; (c) shear stress rxy .

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Figure 16. The normal and shear strains at the lower comer of the upper containment inside the upper adhesive fillet of a DCC joint for the loading Pv predicted based on both the SSSD and the SSLD theories. (a) Normal strain cxx; (b) normal strain Cyy; (c) shear strain Cxy.

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Figure 17. The displacements at the critical point in the horizontal plate of a DCC joint for the loading Py predicted based on both the SSSD and the SSLD theories. (a) Displacement in the x-direction, Ux; (b) displacement in the y-direction, Uy.

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Figure 18. The normal and shear stresses at the critical point in the horizontal plate of a DCC joint for the loading Py predicted based on both the SSSD and the SSLD theories. (a) Normal stress axx ; (b) normal stress 0" yy; (c) shear stress ixy.

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Figure 19. The normal and shear strains at the critical point in the horizontal plate of a DCC joint for the

loading Py predicted based on both the SSSD and the SSLD theories. (a) Normal strain Exx; (b) normal strain Eyy; (c) shear strain Exy.

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theory was applied to an adhesively bonded DCC joint. The joint was analysed under two types of loading condition. Both the SSLD theory and the SSSD theory predicted the adhesive stresses and strains to concentrate around the adhesive free ends inside

the adhesive spew fillets, and the peak stresses and strains in the horizontal plate to occur around regions near the adhesive fillets. However, the SSLD theory predicts adhesive stresses and strains higher than those predicted by the SSSD theory for the tensile loading Px at the free end of the horizontal plate, and this non-linear variation becomes more apparent in the stresses and strains of the horizontal plate. However, the SSLD analysis found that the DCC joint did not show a similar behaviour for the loading normal to the horizontal plate plane Py. Thus, the displacement, stress, and strain components in the upper and lower adhesive fillets showed similar linear variations based on both analyses. However, in both loading cases, as the free end of

the horizontal plate is approached, an increasing non-linear effect is observed in the variations of the displacements, and the stresses and strains of the horizontal plate. This means that the adhesively bonded joints may undergo large displacements and rotations when they include members with low stiffness. In addition, the boundary conditions will affect considerably the behaviour of the adhesively bonded joints. In

order to predict correctly the stresses and deformations in adhesively bonded joints, the

geometrical non-linear effects arising from the large displacements and their gradients should be taken into account. This type of analysis may require much effort, but the results are always more realistic.

REFERENCES

1. S. Semerdjiev, Metal to Metal Adhesive Bonding. Business Books Ltd., London (1970). 2. R. D. Adams and W. C. Wake, Structural Adhesive Joints in Engineering. Elsevier Applied Science,

London (1984). 3. A. J. Kinloch, Adhesion and Adhesives. Chapman & Hall, London (1987). 4. O. C. Zienkiewicz and R. C. Taylor, The Finite Element Method (Solid and Fluid Mechanics,

Dynamics and Non-linearity), Vol. 2. McGraw-Hill, UK (1991). 5. M. Goland and E. Reissner, Trans. ASME, J. Appl. Mech. 11, A17-A27 (1944). 6. O. Volkersen, Constr Met. 4, 3-13 (1965). 7. R. W. Cornell, Trans. ASME, J. Appl. Mech. 75, 355 (1953). 8. N. K. Benson, in: Adhesion - Fundamentals and Practice. UK Ministry of Technology, London

(1969). 9. R. Muki and E. Stenberg, Int. J. Solids Structures 4, 75-94 (1968).

10. F. Erdogan and M. Ratwani, J. Composite Mater. 15, 378-393 (1971). 11. D. J. Chang and R. Muki, Int. J Solids Structures 10, 503-517 (1974). 12. W. J. Renton and J. R. Vinson, J. Adhesion 7, 175-193 (1975). 13. W. J. Renton and J. R. Vinson, Eng. Fract. Mech. 7, 41-60 (1975). 14. W. J. Renton and J. R. Vinson, J. Aircraft 12, 442-447 (1975). 15. W. J. Renton and J. R. Vinson, Trans. ASME, J. Appl. Mech. 44, 101-106 (1977). 16. U. Yuceoglu and D. P. Updike, in: Proc. 15th Annual Meeting Soc. Eng. Sci., pp. 69-74 (1978). 17. U. Yuceoglu and D. P. Updike, ASCE J. Eng. Mech. Div. 106 (EM1-15189) 37-56 (1980). 18. U. Yuceoglu and D. P. Updike, ASCE J. Eng. Mech. Div. 107 (EM1-16038) 55-76 (1981). 19. D. J. Almann, Q. J. Mech. Appl. Math. 30, 415-436 (1977). 20. I. U. Ojalvo and H. L. Eidinoff, AIAA J. 16, 204-211 (1978). 21. I. U. Ojalvo and H. L. Eidinoff, AIAA J. 18, 352 (1980).

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22. R. A. Kline, in: Adhesive Joints: Formation, Characteristics and Testing, K. L. Mittal (Ed.), pp. 587- 610. Plenum Press, New York (1984).

23. F. Thamm, J. Adhesion 7, 301-309 (1976). 24. J. B. Sainsbury-Carter, Trans. ASME, J. Eng. Ind. 95, 919-924 (1973). 25. T. Wah, J. Eng. Mater. Technol. 95, 174-181 (1973). 26. T. Wah, Int. J. Mech. Sci. 18, 223-228 (1976). 27. T. Wah, Int. J. Solids Structures 12, 491-500 (1976). 28. T. S. Ramamurthy and A. K. Rao, Int. J. Mech. Sci. 20, 721-728 (1978). 29. J. P. H. Webber, J. Adhesion 12, 257 (1981). 30. F. Delale, F. Erdogan and M. N. Aydinoglu, J. Composite Mater. 15, 249-271 (1980). 31. F. Delale and F. Erdogan, Trans. ASME, J. Appl. Mech. 48, 331-338 (1981). 32. L. J. Hart-Smith, McDonnell-Douglas Co. Report No: 6059A, Long Beach, CA (1972). 33. L. J. Hart-Smith, NASA CR-112235, Langley Research Center, Hampton, VA (1973). 34. L. J. Hart-Smith, NASA CR-112236, Langley Research Center, Hampton, VA (1973). 35. L. J. Hart-Smith, NASA CR-112237, Langley Research Center, Hampton, VA (1973). 36. L. J. Hart-Smith, NASA CR-112238, Langley Research Center, Hampton, VA (1973). 37. L. J. Hart-Smith, NASA CR-2218, Langley Research Center, Hampton, VA (1974). 38. L. J. Hart-Smith, in: Developments in Adhesives - 2, A. J. Kinloch (Ed.), pp. 1-44. Elsevier Applied

Science, London (1981). 39. Y. Weitsmann, J. Composite Mater. 11, 378-394 (1977). 40. Y. Weitsmann, J. Adhesion 11, 279-289 (1981). 41. H. F. Brinson, Composites 13, 377-382 (1982). 42. G. R. Wooley and D. R. Carver, J. Aircraft 8, 817-820 (1971). 43. N. C. Harrison and W. J. J. Harrison, J. Adhesion 3, 195 (1972). 44. R. M. Barker and F. Hatt, AIAA J. 11, 1650-1654 (1973). 45. R. D. Adams and N. A. Peppiatt, J. Strain Anal. 8, 134-139 (1973). 46. R. D. Adams and N. A. Peppiatt, J. Strain Anal. 9, 185-196 (1974). 47. R. D. Adams and N. A. Peppiatt, J. Adhesion 9, 1-18 (1977). 48. R. D. Adams, J. Coppendale and N.A. Peppiatt, J. Strain Anal. 13, 1-10 (1978). 49. A. D. Crocombe and R. D. Adams, J. Adhesion 12, 127-139 (1981). 50. A. D. Crocombe and R. D. Adams, J. Adhesion 13, 141-155 (1981). 51. J. W. Sawyer and P. A. Cooper, AIAA J. 19, 1443-1451 (1981). 52. R. D. Adams, in: Developments in Adhesives - 2, A. J. Kinloch (Ed.), pp. 45-81. Elsevier Applied

Science, London (1981). 53. J. A. Harris and R. D. Adams, Int. J. Adhesion Adhesives 4, 65-78 (1984). 54. R. D. Adams, R. W. Atkins, J. A. Harris and A. J. Kinloch, J. Adhesion 20, 29-53 (1986). 55. R. D. Adams and J. A. Harris, Int. J. Adhesion Adhesives 7, 69-80 (1987). 56. J. N. Reddy and S. Roy, Int. J. Non-linear Mech. 23, 97-112 (1988). 57. P. Czarnocki and K. Pierkaski, Int. J. Adhesion Adhesives 6, 157-160 (1986). 58. U. Edlund and A. Klarbring, Computer Methods Appl. Mech. Eng. 78, 19-47 (1990). 59. U. Edlund and A. Klarbring, Computer Methods Appl. Mech. Eng. 96, 329-350 (1992). 60. R. D. Wood and O. C. Zienkiewicz, Computers Structures 7, 725-735 (1977). 61. R. D. Wood and B. Schrefler, Int. J. Numerical Methods Eng. 12, 635-642 (1978). 62. J. A. Stricklin, W. E. Haisler and W. A. Von Riesemann, AIAA J. 9, 2066-2067 (1971). 63. D. A. Massett and J. A. Stricklin, AIAA J. 9, 2464-2466 (1971). 64. W. E. Haisler, J. A. Stricklin and F. J. Stebbins, AIAA J. 10, 264-272 (1972). 65. J. A. Stricklin, W. E. Haisler and W. A. Von Riesemann, AIAA J. 11, 292-299 (1973). 66. J. A. Stricklin and W. E. Haisler, Computers Structures 7, 125-136 (1977). 67. G. F. Carey, Computer Methods Appl. Mech. Eng. 4, 69-79 (1974). 68. K. J. Bathe and A. P. Cimento, Computer Methods Appl. Mech. Eng. 22, 59-85 (1980). 69. M. A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, Vol. I. John Wiley,

Chichester (1991).

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

13:

22 0

9 Se

ptem

ber

2013

1195

70. M. Kleiber, Incremental Finite Element Modelling in Non-linear Solid Mechanics. Ellis Horwood (1989).

71. L. E. Malwem, Introduction to the Mechanics of a Continuous Medium. Prentice-Hall (1969). 72. A. A. Saada, Elasticity - Theory and Applications. Pergamon Press, Oxford (1974). 73. C. Mylonas, Proc. Soc. Exp. Stress Anal. 12, 129-142 (1954). 74. I. Tuzi and H. Shimada, Bull. JSME 7, 263-267 (1964). 75. I. Tuzi and H. Shimada, Bull. JSME 8, 330-336 (1965). 76. ANSYS®, The General Purpose Finite Element Software (Version 5.1), User's Manuals 1-4. Swan-

son Analysis Systems, Inc., P.O. Box 65, Houston, TX. 77. R. Davies and A. A. Khalil, Int. J. Adhesion Adhesives 10, 25-30 (1990). 78. M. K. Apalak and A. Engin, in: Proceedings of the 7th International Machine Design and Production

Conference, pp. 517-527. METU, Ankara, Turkey (1996).

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

13:

22 0

9 Se

ptem

ber

2013