Analysis of adhesive bonded joints: a unified approach

Composites Science and Technology 62 (2002) 1011–1031 www.elsevier.com/locate/compscitech

Analysis of adhesive bonded joints: a unified approach F. Mortensen, O.T. Thomsen* Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, DK-9220 Aalborg East, Denmark Received 2 October 2001; accepted 28 January 2002

Abstract This paper presents a newly developed unified approach for the analysis and design of adhesive bonded joints. The adherends are modelled as beams or wide plates in cylindrical bending, and are considered as generally orthotropic laminates using classical laminate theory. Consequently, adherends made as asymmetric and unbalanced composite laminates can be included in the analysis. The adhesive layer is modelled in two ways. The first approach assumes the adhesive layer to be a linear elastic material and the second approach takes into account the inelastic behaviour of many adhesives. The governing equations are formulated in terms of sets of first order ordinary differential equations. The multiple-point boundary value problem constituted by the differential equations together with the imposed boundary conditions is solved numerically by direct integration using the ‘multi-segment method’ of integration. The approach is validated by comparison with finite element models and a high-order theory approach. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Adhesive bonded joints; C. Laminates

1. Introduction The use of polymeric fibre reinforced composite materials has gained widespread acceptance as an excellent way to obtain stiff, strong and very lightweight structural elements. However, load introduction into composite structural elements through joints, inserts and mechanical fasteners is associated with considerable difficulties. The primary reason for this is the layered structure of composite laminates, which results in poor strength properties with respect to loading by interlaminar shear and transverse normal stresses. Thus, the interaction between composite elements and adjoining parts often proves to be among the most critical areas of a structural assembly. Joining of composite structures can be achieved through the use of bolted, riveted or adhesive bonded joints. The performances of the mentioned joint types are severely influenced by the characteristics of the layered composite materials, but adhesive bonded joints provide a much more efficient load transfer than mechanically fastened joint types. Accurate analysis of adhesive bonded joints, for instance using the finite ele* Corresponding author. Tel.: +45-9635-9319; Fax :+45-98151411. E-mail address: [email protected] or [email protected] (O.T. Thomsen).

ment method, is an elaborate and computational demanding task, see Crocome and Adams [1] and Harris and Adams [2], and there is a specific need for analysis and design tools that can provide accurate results with little computational efforts involved. Such tools would be very useful for preliminary design purposes, i.e. in the stages of design where fast estimates of stress and strain distributions as well as joint strengths and margin of failure are needed. The first attempt of analyzing adhesive bonded joints was carried out by Volkersen [3] who modelled the adhesive layer as continuously distributed shear springs. The modelling was later refined by adopting a twoparameter elastic foundation approach, in which the adhesive layer is described in terms of uniformly distributed transverse normal and shear springs. This type of model was originally suggested by Goland and Reissner [4] for the analysis of single lap joints, but has later formed the basis for many investigations of the structural response of various types of adhesive bonded joints such as Hart-Smith [5–7], Tong [8], Yuceoglu and Updike [9–12], Renton and Vinson [13,14], Pickett [15, 16] and Thomsen [17,18]. A different approach that overcomes the limitations of the elastic foundation formulation has been suggested by Frostig et al. [19] following a high-order theory approach.

0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00030-1

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Most of the references reviewed, except for HartSmith [6], Pickett [15] and Thomsen [17,18] assume the adhesive layer to behave as a linear elastic material. Analyses based on this assumption do provide useful information about the intensities of stress concentrations and their location, but do not include the fact that most polymeric structural adhesives behave non-linearly, and that many of them show pronounced viscous and plastic effects even at temperatures below their glass transition temperature. Thus, the results do not reflect the true stress distribution at appreciable levels of loading because of the non-linear behaviour of the polymeric structural adhesives. A full treatment of the visco-elastic–plastic behaviour of the polymeric structural adhesives, characterized by their sensitivity towards strain rate and environmental effects such as temperature and moisture, is a very difficult task, and not many investigations have been carried out on this topic. Instead, several investigations using an elastoplastic approach which neglects strain rate and temperature effects and includes plastic residual strains have been carried out, see Hart-Smith [5–7], Pickett [15,16], Adams et al. [20], Gali and Ishi [21] and Thomsen [17,18]. The main objective of the present paper is to consider a newly developed unified approach for the analysis of adhesive bonded joints. The approach can be used for the analysis of most types of adhesive bonded joints. The approach has been used for the analysis of the following types of joints (see Mortensen and Thomsen [22–24] and Mortensen [25]): single lap joints, scarfed single lap joints, bonded doubler lap joints, double lap joints, stepped lap joints (single and double sided) and scarfed lap joints (single and double sided). The different structural bonded joints that have been modelled using the proposed unified approach are shown schematically in Figs. 1 and 2. In the analysis the adherends are modelled as beams or wide plates in cylindrical bending and are considered as generally orthotropic laminates using classical laminate theory. Consequently, the effect of having adherends made as asymmetric and unbalanced composite laminates is included in the analysis. The adhesive layer is modelled as continuously distributed linear tension/ compression and shear springs. As non-linear effects in the form of adhesive plasticity play an important role in the load transfer, the analysis allows inclusion of nonlinear adhesive properties. The load and boundary conditions can be chosen arbitrarily. For simplicity, the description of the unified approach is restricted to the case of a single lap joint. For a detailed description of the approach used on all the joint types mentioned above see Mortensen [25]. Analysis procedures for all the joint types mentioned have been developed and implemented in the new composite analysis and design software package ESAComp 2.0. ESAComp, which is being developed for the European Space Agency, provides an easy-to-use environment

for preliminary evaluation and analysis of plies, laminates as well as structural composite components. The developed analysis tools for adhesive bonded joints are completely integrated in the ESAComp environment, which offers complete access to the ESAComp design system, see Saarela et al. [26]. Thus, the system provides access to for instance linear/non-linear adhesive load response analysis, adhesive/adherend failure analysis etc.

2. Structural modelling The structural modelling is carried out by adopting a set of basic restrictive assumptions for the behaviour of bonded joints. Based on those from which the constitutive and kinematic relations for the adherends are derived, and constitutive relations for the adhesive layers are adopted. Finally, the equilibrium equations for the joints are derived, and by combination of these equations and relations, the set of governing equations is obtained. This results in a set of first order ordinary differential equations, called the governing system equations, describing the system behaviour. The governing system equations are solved numerically using the ‘multi-segment method of integration’. As an example of one of the different bonded joint types, Fig. 3 shows a single lap composed of similar or dissimilar generally orthotropic laminates subjected to general loading conditions. 2.1. Basic assumptions for the structural modelling The basic restrictive assumptions adopted for the structural modelling are the following:
The adherends:  Beams or plates in cylindrical bending, which are described by use of ordinary ‘Kirchhoff’; plate theory (‘Love–Kirchhoff’; assumptions).  Generally orthotropic laminates using classical lamination theory (e.g. asymmetric and unbalanced composite laminates can be included in the analysis).  The laminates are assumed to obey linear elastic constitutive laws.  The strains are small, and the rotations are very small.
The adhesive layer:  Modelled as continuously distributed linear tension/compression and shear springs.  Inclusion of non-linear adhesive properties, by using a secant modulus approach for the nonlinear tensile stress–strain relationship in conjunction with a modified von Mises yield criterion.

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Load and boundary conditions:  Can be chosen arbitrarily. The system of governing equations is set up for two different cases, i.e. the adherends are modelled as plates in cylindrical bending or as wide beams. 2.2. Modelling of adherends as plates in cylindrical bending For the purposes of the present investigation, and with references to Fig. 3, cylindrical bending can be defined as a wide plate (in the y-direction), where the displacement field can be described as a function of the

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longitudinal coordinate only. As a consequence of this, the displacement field in the width directions will be uniform. Thus, the displacement field can be described as: ui0 ¼ ui0 ðxÞ;

vi0 ¼ vi0 ðxÞ;

wi ¼ wi ðxÞ

ð1Þ

where u0 is the midplane displacement in the longitudinal direction (x-direction), v0 is the midplane displacement in the width direction (y-direction), and w is the displacement in the transverse direction (z-direction). The displacement components u0, v0, w are all defined relative to the middle surfaces of the laminates, and i corresponds to the laminate/adherend number (see Fig. 3).

Fig. 1. Schematic illustration of the different standard bonded joints analysed using the developed unified approach.

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Fig. 2. Schematic illustration of the different advanced bonded joints analysed using the developed unified approach.

As a consequence of this, the following holds true: ui0;y ¼ vi0;y ¼ w;iy ¼ w;iyy ¼ 0

ð2Þ

The boundary conditions at the boundaries in the width direction are not well defined within the concept of ‘cylindrical bending’’. However, it is assumed that there are some restrictive constraints on the boundaries, such that they not are capable of moving and rotating freely. This is illustrated conceptually in Fig. 4. It should be noted that the concept of ‘cylindrical bending’; is not unique, and that other definitions than the one used in the present formulation can be adopted, see Whitney [27]. Substitution of the quantities in Eq. (2) into the constitutive relations for a laminated composite material

Fig. 3. Schematic illustration of adhesive single lap joint with straight adherends in the overlap zone subjected to general loading conditions.

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2.3. Modelling of adherends as beams Modelling of the adherends as wide beams can be considered as a special case of cylindrical bending. When the adherends are modelled as beams the width direction displacements are not considered, and only the longitudinal and vertical displacements are included. Thus, the displacement field in Eq. (1) is reduced to: u0 ¼ u0 ðxÞ;

w ¼ wðxÞ

ð5Þ

For this case the constitutive relations for a composite beam are reduced to: i i i i Nxx ¼ A11 u0;x  B11 w;ixx i i i i Mxx ¼ B11 u0;x  D11 w;ixx

Fig. 4. Schematic illustration of adhesive single lap joint ‘clamped’ between two vertical laminates, which prevent the adherends of the single lap joint from moving and rotating freely in the width direction. This represents the conceptual interpretation of cylindrical bending as defined in the present formulation.

(Whitney [27]) gives the constitutive relations for a laminate (i) in cylindrical bending: i i i i i i ¼ A11 u0;x þ A16 v0;x  B11 w;ixx Nxx i i i i i i Nyy ¼ A12 u0;x þ A26 v0;x  B12 w;ixx i i i i i i Nxy ¼ A16 u0;x þ A66 v0;x  B16 w;ixx i i i i i i Mxx ¼ B11 u0;x þ B16 v0;x  D12 w;ixx

ð3Þ

i i i i i i Myy ¼ B12 u0;x þ B26 v0;x  D12 w;ixx i i i i i i Mxy ¼ B16 u0;x þ B66 v0;x  D12 w;ixx

where Ajki , Bjki and Djki ðj; k ¼ 1; 2; 6Þ are the extensional, coupling and the flexural rigidities as defined by classii i i cal lamination theory, see Jones [28]. Nxx , Nyy and Nxy i i i are the in-plane stress resultants and Mxx , Myy and Mxy are the moment resultants. For the advanced joint types such as a scarfed or stepped lap the rigidities Ajki , Bjki and Djki ðj; k ¼ 1; 2; 6Þ are determined as functions of the longitudinal direction of the joint within the overlap zone, since the adherend thicknesses are variable within the overlap. From the ‘Love–Kirchhoff’; assumptions, the following kinematic relations for the laminates are derived: ui ¼ ui0 þ zix;

ix ¼ w;ix;

iy ¼ 0

ð4Þ

where ui is the longitudinal displacement, ui0 is the longitudinal displacement of the midplane, and wi is the vertical displacement of the ith laminate.

ð6Þ

The kinematic relations [(Eq. (4)] are the same as for the cylindrical bending case except that all variables associated with the width direction are nil. 2.4. Constitutive relations for the adhesive layer The coupling between the adherends is established through the constitutive relations for the adhesive layer, which as a first approximation is assumed to be homogeneous, isotropic and linear elastic. The constitutive relations for the adhesive layer are established by use of a two-parameter elastic foundation approach, where the adhesive layer is assumed to be composed of continuously distributed shear and tension/compression springs. The constitutive relations of the adhesive layer are suggested in accordance with Thomsen [17,18] and Tong [8]:   9  Ga i ti ðxÞ i Ga
i tj ðxÞ j > j j > > ax ¼   u0   u u ¼ u0  > 2 x 2 x > ta ta > > = 
 Ga
i G a j j i ay ¼ v v ¼ v0  v0 > ta ta > > > >  Ea
i > j > ; w w a ¼ ta ði; j ¼ 1; 2; 3Þ; ð7Þ ði 6¼ jÞ

where i and j are the numbers of the adherends, Ga is the shear modulus, and Ea is the elastic modulus of the adhesive layer. The consequence of using the simple spring model approach for the modelling of the adhesive layers is,

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9 > > > > > > > > > > > > =

that it is not possible to satisfy the equilibrium conditions at the edges (free) of the adhesive, where the shear stresses must equal zero, see Fig. 3. However, in real adhesive joints no free edges are present at the ends of the overlap, since a fillet of surplus adhesive, a so-called spew-fillet, is formed at the ends of the overlap zone. This spew fillet allows for the transfer of shear stresses at the overlap ends. A detailed discussion of the validity of the adhesive layer spring model approach is presented later in this paper.

1 Nxx;x ¼ ax;

2 Nxx;x ¼ ax

1 Nxy;x ¼ ay ;

2 Nxy;x ¼ ay

1 Qx;x ¼ a ;

2 Qx;x ¼ a

2.5. Equilibrium equations

where t1(x) and t2(x) are the adherend thicknesses and ta is the adhesive layer thickness. For the single lap joint and for the bonded doubler the adherend thicknesses will remain the same in the entire overlap zone. For the single sided stepped lap joint, the adherend thicknesses will change inside the overlap zone between each step.

The equilibrium equations are derived based on equilibrium elements in- and outside the overlap zone for each of the considered joint types. The equilibrium equations are derived for plates in cylindrical bending, since the equilibrium equations for the beam modelling can be considered as a reduced case of this. The general equilibrium equations outside the overlap zone for each of the adherends, i.e. in the regions  L1 4 x 4 0 and L 4 x 4 L þ L2 , are all the same (see Fig. 3) and are derived based on Fig. 5: i Nxx;x ¼0 i Nxy;x ¼0 i ¼0 Qx;x i Mxx;x ¼ Qxi i Mxy;x ¼ Qyi

9 > > > > > > = > > > > > > ;

 L1 4 x 4 0 and L 4 x 4 L þ L2 : ð8Þ

where i corresponds to the adherends i=1, 2, 3. In cylindrical bending the stress and moment resultants are only a function of the longitudinal coordinate x, and the derivatives with respect to the width direction y are all equal to zero. The equilibrium equations derived inside the overlap zones can be divided into the following two groups:
Joints with one adhesive layer inside the overlap zone.
Joints with two adhesive layers inside the overlap zone. These two groups are further divided into joints with straight or scarfed adherends within the overlap. However, in the following only the equilibrium equations for joints with two straight adherends within the over lap will be shown, i.e. single lap joint (see Fig. 3); bonded doubler and single sided stepped lap joint. For a full description of the derivation of the equilibrium equations for the rest of the joint types, see Mortensen [25]. The equilibrium equations for joints with two straight adherends within the over lap are derived based on Fig. 6:

t1 ðxÞ þ ta ; 2 t1 ðxÞ þ ta 1 ; ¼ Qy1  ay Mxy;x 2 0 4 x 4 L:

1 Mxx;x ¼ Qx1  ax

t2 ðxÞ þ ta 2 t2 ðxÞ þ ta 2 ¼ Qy  ay 2

2 Mxx;x ¼ Qx2  ax 2 Mxy;x

> > > > > > > > > > > > ; ð9Þ

2.6. The complete set of system equations From the equations derived, it is possible to form the complete set of system equations for each of the bonded joint configurations. Thus, combination of the constitutive and kinematic relations, i.e. Eqs. (3) and (4), together with the constitutive relations for the adhesive layers, i.e. Eq. (7), and the equilibrium equations lead to a set of 8 linear coupled first-order ordinary differential equations describing the system behaviour of each of the adherends. The total set of coupled first-order ordinary differential equations within the overlap zone is therefore 16 for joints with 2 adherends inside the overlap zone, and 24 for the joints with 3 adherends inside the overlap zone. The actual derivations of the governing equations for the bonded joint types included in this study are quite lengthy and tedious (although straightforward). Therefore the derivations are not included in this paper, but the resulting sets of governing equations for the single lap joint are shown in Appendix A. The governing equations can be expressed in the following general form within each region, ie insite and outside the overlap zone: 2 6 6 6



n i r

y ðx Þ ; x ¼ 6 6 6 4 

r ðxr Þ A11



...



r A1n ðxr Þ

3 7 7 7 7 7 7 5

 r r ð10Þ r An1 ðxr Þ . . . Ann ðx Þ







n yi ðxr Þ þ  n Bir ðxr Þ









n yi ðxr Þ ¼ y1 ðxr Þ ; y2 ðxr Þ ; . . . ; yn ðxr Þ









 n Bir ðxr Þ ¼ B1r ðxr Þ ; B2r ðxr Þ ; . . . ; Bnr ðxr Þ

where n is the number of adherends within the region considered. The values of n are between 1 and 3 (both

F. Mortensen, O.T. Thomsen / Composites Science and Technology 62 (2002) 1011–1031

Fig. 5. Equilibrium elements of adherend outside the overlap zone;  L1 4 x 4 0 or L 4 x 4 L þ L2 .

Fig. 6. Equilibrium element of adherends inside the overlap zone for joints with one adhesive layer and straight adherends; 0 4 x 4 L.

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included) depending  on the type of joint and the region considered. Aijr ðxr Þ ði; j ¼ 1; . . . nÞ is a ðm; mÞ sub-coeffimatrix for the system of governing equations, and cient

Bir ðxr Þ is a (m,1) sub-matrix of non-homogeneous load terms, where m is the number of equations for each adherend, i.e. 8 for the cylindrical bending case and 6 for the beam

case. For the cylindrical bending case the vector yi is the vector containing the fundamental variables, which are those quantities that appear in the natural boundary conditions at an edge x=constant defined by: o i n i i i i i i i Nxy ; Mxx ; Qxi y ¼ u0 ; w ; x ; v0 ; Nxx; ð11Þ i ¼ 1; 2; 3: These variables will be determined through the analysis. In addition, the quantities o i n i i i yres ¼ Nyy ; Myy ; Mxy ; Qyi

i ¼ 1; 2; 3

ð12Þ

can be determined from the equilibrium equations and the constitutive relations. These quantities can be considered as the stress and moment resultants necessary to keep the structure in a state of cylindrical bending. For the beam case the problem is reduced to a set of 6 coupled first-order ordinary differential equations for each adherend, and the solution vector containing the fundamental variables for each adherend for this problem is defined by: i i i i

i i y ¼ u0 ; w ; x ; Nxx ; Mxx ; Qxi i ¼ 1; 2; 3: ð13Þ In this case, no additional quantities need to be determined. 2.7. The boundary conditions To solve the adhesive bonded joint problems (see Fig. 1) the boundary conditions and continuity conditions have to be stated. In the following the boundary conditions and continuity conditions are stated for a single lap joint, see Fig. 3: ) i x ¼ L1 ; L þ L2 : prescribed : ui0 or Nxx ; wi or Qxi ; i i ix or Mxx ; vi0 or Nxy

The boundary conditions for adherend 1 at x=L and for adherend 2 at x=0 are derived from the assumption that the adherend edge is free, see Fig. 3. 2.8. Multi-segment method of integration The governing equations, together with the boundary conditions constitutes a multiple-point boundary value problem to which no general closed-form solution is obtainable. The multiple-point boundary value problem is therefore solved using the ‘multi-segment method of integration’. The method is based on a transformation of the original ‘multiple-point’ boundary value problem into a series of initial value problems. The principle behind the method is to divide the original problem into a finite number of segments, where the solution within each segment can be accomplished by means of direct integration. Fulfilment of the boundary conditions, as well as fulfilment of continuity requirements across the segment junctions, is assured by formulating and solving a set of linear algebraic equations. For a detailed description of the method, see [29] and Mortensen [25].

3. Examples and discussion To show the applicability of the developed linear solution procedures an example of a single lap joint is presented. The adherends used in the examples are chosen to be laminates, and the lay-ups are made such that asymmetric laminates, and thereby coupling effects appear in the example presented. This is chosen to show the capabilities of the approach, i.e. the modelling of the adherends as plates in cylindrical bending including coupling effects. The basic adherend and adhesive properties assumed in the examples are shown in Table 1. The adhesive used, AY103 from Ciba-Geigy, is a twocomponent plasticized epoxy, which is considered to be a general structural adhesive forming strong semi flexible bonds. The lay-up of the adherends, the dimensions as well as the boundary conditions used in the example, are shown in Table 2.

i ¼ 1; 2 Table 1 Specification of ply and adhesive material properties used for the single lap joint example

x ¼ 0 : adherend 1 : continuity across junction 2 2 2 adherend 2 : Nxx ¼ Nxy ¼ Mxx ¼ Qx2 ¼ 0

Plies

Graphite/epoxy E1=164.0 GPa, E2=E3=8.3 GPa, G12=G31=G23=2.1 GPa, v12=v13=v23=0.34, t=0.125 mm

Adhesive

Epoxy AY103 (Ciba Geigy), Ea=2800 MPa, va=0.4

1 1 1 x ¼ L : adherend 1 : Nxx ¼ Nxy ¼ Mxx ¼ Qx1 ¼ 0

adherend 2 : Continuity across junction

ð14Þ

F. Mortensen, O.T. Thomsen / Composites Science and Technology 62 (2002) 1011–1031 Table 2 Laminate lay-ups, thicknesses, lengths and boundary conditions used for the single lap joint example (see Fig. 3) Laminate 1 Laminate 2 Lengths Adhesive Modelling Load and B.C.’’s

Graphite/epoxy [0 , 30 , 60 ]4, t1=1.5 mm Graphite/epoxy [60 , 30 , 0 ]4, t2=1.5 mm L1=L2=30.0 mm, L=20.0 mm ta=0.05 mm Wide plates in cylindrical bending 1 ¼ 0; x=L1: u10 ¼ w1 ¼ v10 ¼ Mxx 2 2 x=L+L2: w2 ¼ v20 ¼ Mxx ¼ 0; Nxx ¼ 100 N=mm

The overlap length is L=20 mm, see Fig. 3 for reference, and the adherends are modelled as plates in cylindrical bending. The lay-up of the adherend laminates is made such that the joint is symmetric seen from the adhesive layer midplane, and such that the plies facing the adhesive layers are 0 plies. The joint is assumed to be simply supported at both ends, and, additionally, prevented from moving in the width direction, i.e. vi0 ¼ 0 for (i=1,2) at x=-L1 and x=L+L2. In the examples only in-plane normal loading has been applied, and the adhesive layer stresses are therefore normalized with respect to the prescribed load on the loaded edge, which is given by: Ni ¼ Ni =ti ;

ði ¼ 1; 2Þ

ð15Þ

where Ni is in-plane normal stress resultant applied. In the example the results for a selected number of the fundamental variables for the single lap joint problem are presented to show the output from the analysis, where the adherends are asymmetric and unbalanced laminates modelled as plates in cylindrical bending. The single lap joint configuration assumed in the example is shown in Fig. 7.

Fig. 7. Single lap joint simply supported at both ends (clamped in the width direction), t1=t2=1.5 mm, ta=0.05 mm, L1=L2=30 mm, 2 L=20 mm, Nxx =100 N/mm $ N ¼ 67 MPa.

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The vertical and the width direction displacements of the adherends are shown in Figs. 8 and 9, respectively. From Fig. 8 it is seen that relatively large deflections occur due to the eccentricity in the load path, and that the joint rotates around the center of the overlap zone, i.e. at x=10 mm. As described previously the joint configuration is symmetric seen from the adhesive layer midplane, and this is the reason for the perfect skewsymmetric deflection pattern displayed. The width direction displacements v10 and v20 , shown in Fig. 9, occur because of the coupling effects in the laminates. Comparison of the lateral deflections in Fig. 8 and the width direction displacements in Fig. 9 shows that the displacements in the width direction are much smaller. The in-plane normal stress resultants in the longitudinal direction (x-direction) for the adherends are shown in Fig. 10. From Fig. 10 it is observed that the in-plane normal stress resultants are constant outside the overlap zone, and that the load is transferred between the adherends through the adhesive layer in the overlap zone. The in-plane shear stress resultants for the adherends are shown in Fig. 11. Fig. 11 shows a similar pattern of the in-plane shear stress resultants as for the in-plane normal stress resultants shown in Fig. 10. The shear stress resultants occur since the adherends are asymmetric and unbalanced laminates, and since displacements and rotations are prevented in the width direction at the boundaries at both ends (see Fig. 4). The peak values of the shear stress resultants are about 24% of the in-plane normal stress resultants applied to the bonded joint. The bending moment resultants in the adherends in the longitudinal direction are shown in Fig. 12. From Fig. 12 it is observed that the maximum bending moment resultants occur at the ends of the overlap. The large bending moment resultants occur due to the eccentricity of the load path. Fig. 13 shows the out-of-plane shear stress resultants. It is observed from Fig. 13 that highly localized shear stress resultants are present at the ends of the overlap zone, and this demonstrates the presence of strong local bending effects. The peak values of the out-of-plane shear stress resultants are about 17% of the in-plane normal stress resultants applied to the structure. The results shown in Figs. 8–13 are the fundamental variables determined through the analysis according to Eq. (11). In Fig. 14 the distributions of the normalized adhesive layer stresses are shown. From Fig. 14 it is seen that the peek stresses are located at the ends of the overlap. As a consequence of the coupling effects in the laminates, it is seen that width direction shear stresses (ay ) are non-zero (although of minor magnitude). Considering Fig. 8 it is difficult to distinguish any noticeable differences between the

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Fig. 8. Vertical displacements w1, w2 of the adherends.

Fig. 9. Width direction displacements v10 , v20 of the adherends.

deflections of the adherends, but from the adhesive layer stress distribution displayed in Fig. 14, it is seen that significant transverse normal stresses do occur. These stresses are induced from differences in the lateral deflections of the adherends. If the adherends of a bonded joint fulfil the assumptions for cylindrical bending, i.e. if the width of the bonded joint is considerable or if the movement of the sides of the bonded joint is subjected to restraints, the analysis should be carried out following this approach. The ‘width direction’; variables under such conditions can be of considerable magnitude.

4. Non-linear adhesive formulation The structural modelling described is based on the assumption that the adhesive layer behaves as a linear

elastic material. This is a good approximation for most brittle adhesives, especially at low load levels, and the approach is useful to predict the stress distribution and the location of the peak stress values. However, most polymeric structural adhesives exhibit inelastic behavior, in the sense that plastic residual strains are induced even at low levels of external loading. 4.1. Non-linear formulation and solution procedure The concept of effective stress/strain is one way of approaching this problem, and it assumes, for a ductile material, that the plastic residual strains are large compared with the creep strains at normal loading rates. Therefore, a plastic yield hypothesis can be applied, and the multidirectional state of stress can be related to a simple unidirectional stress state through a function similar to that of von Mises.

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1 2 Fig. 10. In-plane normal stress resultants Nxx , Nxx in the adherends in the longitudinal direction.

1 2 Fig. 11. In-plane shear stress resultants Nxy , Nxy in the adherends.

1 2 Fig. 12. Bending moment stress resultants Mxx , Mxx in the adherends.

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Fig. 13. Out-of-plane shear stress resultants Qx1 , Qx2 in the adherends.

Fig. 14. Normalized adhesive layer stresses, ax =N ; ay =N ; a =N .

However, it is widely accepted that the yield behaviour of polymeric structural adhesives is dependent on both deviatoric and hydrostatic stress components. A consequence of this phenomenon is a difference between the yield stresses in uniaxial tension and compression, see Adams et al. [20], Gali [30], Adams [2] and Thomsen [17,18]. This behaviour has been incorporated into the analysis by the application of a modified von Mises criterion suggested by Gali et al. [30]:

s ¼ CS ðJ2D Þ

1=2

þCV J1 ;

l1 ; CV ¼ 2l

c l¼ t

pffiffiffi 3ð 1 þ lÞ ; CS ¼ 2l

ð16Þ

where s is the effective stress, J2D is the second invariant of the deviatoric stress tensor, J1 is the first invariant of the general stress tensor and l is the ratio between the compressive and tensile yield stresses. For l ¼ 1, Eq. (16) is reduced to the ordinary von Mises criterion. At the failure load level, the first of Eq. (16) is transformed into the expression: sult ¼ CS;ult ðJ2D Þ1=2 ult þCV;ult ðJ1 Þult

ð17Þ

where the subscript ‘ult’; denotes ‘ultimate’’. Eq. (17) describes the failure envelope for the general case of a ductile material, and in three-dimensional stress space Eq. (17) represents a paraboloid with its axis coincident with the line 1 ¼ 2 ¼ 3 .

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The effective strain e is given by Gali [30]: e ¼ CS

1 1 ðI2D Þ1=2 þCV ð I1 Þ 1þv 1  2v

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been used for the analysis of non-linear adhesive behaviour in tubular lap joints by Thomsen [17,18]. ð18Þ

where v is Poisson’s ratio, I2D is the second invariant of the deviatoric strain tensor and I1 is the first invariant of the general strain tensor. The non-linear adhesive properties are included by implementing an effective stress–strain relationship derived experimentally from tests on adhesive bulk specimens, see Thomsen [17,18] and Tong [8]. Thus it is assumed that the bulk and ‘in situ’; mechanical properties of the structural adhesive are closely correlated as discussed by Gali et al. [30] and shown experimentally by Lilleheden [32]. Based on a secant modulus approach for the non-linear effective stress–strain relationship for the adhesive, the solution procedure for determining the stress distribution in the adhesive layer can be described by the following steps: 1. Calculate the effective strains e1 and stresses s 1 [Eqs. (16) and 18)] for each point of the adhesive layer, using the linear elastic solution procedure, assuming a uniform elastic modulus E1 for the adhesive. 2. If the calculated effective stresses s 1 are above the proportional limit denoted by sprop, determine the effective stresses s1 for each point of the adhesive layer according to the corresponding effective strains e1 calculated in step (1). 3. Calculate the difference %s1 ¼ s 1  s1 between the ‘calculated’; and the ‘experimental’; effective stresses, and determine the specific secant-modulus E2t defined by:

E2t ¼ 1  ð%s1 =s1 Þ E1  is a weight-factor, which determines the change of the modulus in each iteration. 4. Rerun the procedure [steps (1) and (2)] with the elastic modulus E1 for each adhesive point modified as per step (3). 5. Compare the ‘calculated’; effective stresses s for each adhesive point with the ‘experimental’; values s obtained from the effective stress–strain curve. 6. Repeat steps (4) and (5) until the difference between the ‘calculated’; and ‘experimental’; stresses (%s) drops below a specified fraction (2%) of the ‘experimental’ stress value. Convergence is usually achieved within a few iterations. The procedure described above has previously

4.2. Examples and discussion The effects of non-linear adhesive behaviour is illustrated by the following example. The basic differences between the linear and non-linear analysis is that the adhesive stresses, at higher levels of loading, are reduced and smoothed out in the regions adjacent to the ends of the overlap zone. To illustrate the effects of non-linear adhesive behaviour the single lap joint example shown in Section 3 will be used. Therefore, the results obtained can be compared with the results shown in Section 3. The dimensions, laminates, loads etc. are all shown in Table 2 in Section 3. The adhesive used in the example is AY103 from CibaGeigy, which is a two-component plasticized epoxy adhesive as described in Section 3. The tensile stress– strain curve for this adhesive has been obtained from bulk specimens (see Adams [20,31] and Harris and Adams [2]), which have been subjected to the cure cycle usually used for joints bonded with this adhesive. The stress–strain relation for the used adhesive material is shown in Fig. 15 and the material properties are given in Table 3. Analysis of the single lap joint example shown in Section 3 by use of the non-linear solution procedure, based on the adhesive properties given in Table 1 and the non-linear adhesive stress–strain relation shown in Fig. 15, results in the adhesive layer stress distribution shown in Fig. 16. Comparison of the adhesive layer stress distribution obtained for the linear case, shown in Fig. 14, and for the non-linear case, shown in Fig. 16, shows that inclusion of the non-linear effects reduce the maximum predicted stresses with about 25%. It is also seen that the non-linear effects are only influential very close to the ends of the overlap zone. The differences between the linear and non-linear solutions are strongly dependent on the load level. Failure has been predicted to occur at a load level corresponding to N ¼ 80 MPa by use of the ultimate stress criterion. Fig. 17 displays the differences between the linear and non-linear solutions, by showing the maximum adhesive layer stresses as a function of the load level obtained using the same single lap joint configuration. It is observed from Fig. 17, that the adhesive non-linearity starts to affect the adhesive layer stresses at very low load levels, and the non-linear effects become increasingly important as the external loading is increased. Thus, prediction of the adhesive stresses using the linear solution procedure will underestimate the load bearing capability of the adhesive joint, except for very brittle adhesives where the non-linear effects are of minor importance. This observation is in close

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F. Mortensen, O.T. Thomsen / Composites Science and Technology 62 (2002) 1011–1031

Fig. 15. Tensile stress/strain curve for the adhesive AY103 obtained from bulk specimen, load controlled: 20.0 [MPa/min].

Table 3 Specification adhesive material properties Adhesive

Epoxy AY103
(Ciba Geigy), Ea=2800 MPa, va=0.4, l ¼ 1:3 l ¼ ct sprop ¼ 27:0 MPa; sult ¼ 71:5 MPa, eult ¼ 0:049 MPa

agreement with experimental results, see Hart-Smith [5–7], Adams [20,31] and Harris and Adams [2]. 4.3. Summary From the comparison between the linear elastic solution, shown in Section 3, and the non-linear adhesive

solution, shown in Section 4.1, it can be concluded that the non-linear behaviour shown by many polymeric adhesives exert a strong influence on the adhesive layer stress distribution. The severe stress concentrations, predicted by the linear solution procedure, tend to smooth out when a non-linear solution procedure is applied. The non-linear effects become influential even at low load levels, and become very influential at higher load levels. Thus, in most cases non-linear effects are unavoidable, and a certain degree of plasticity in the adhesive layer close to the ends of the overlap cannot be prevented. Linear elastic solution procedures will therefore underestimate the strength of adhesive bonded joints unless very brittle adhesives with approximately linear elastic properties are considered.

Fig. 16. Normalized adhesive layer stresses, ax =N ; ay =N ; a =N for a single lap joint obtained by use of the non-linear solution procedure.

F. Mortensen, O.T. Thomsen / Composites Science and Technology 62 (2002) 1011–1031

5. Validation of the adhesive layer model To investigate the validity of the adhesive layer model a high-order theory approach developed for the analysis of sandwich structures has been used, see Frostig et al. [33] and Frostig [34], together with FE-analyses. The investigation is partially presented in the paper by Frostig et al. [19]. In this paper an extended validation of the adhesive layer model is presented. As a consequence of modelling the adhesive layer as continuously distributed springs (tension/compression and shear springs) it is not possible to fulfil the condition of zero shear stresses (ax ¼ 0) at the free edges of the adhesive layers (see Fig. 3) as assumed in the boundary conditions for all the joints. However, in real adhesive bonded joints no free edges at the ends of the overlap zones are present, since a fillet of surplus adhesive, a so-called spew fillet, is formed at the ends of the overlap. The differences between the model joint and a real joint is illustrated in Fig. 18. Consequently, the shear stresses (ax ) in a real joint will not be zero at the ends of the overlap zone. It can therefore be stipulated that using the spring model

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approach in some sense is equivalent to assuming the existence of a spew-fillet at the ends of the overlap zone. The actual spew-fillet size, however, is not addressed by adopting the spring model approach. The validity of the adhesive layer spring model approach has been investigated by comparison with:
A high-order theory approach including the presence of a spew-fillet;
FE-analyses including a spew-fillet; In the following, the method described in this paper is called the ‘spring model approach’ for simplicity. 5.1. Structural modelling using a high-order theory approach The high-order theory approach used has been developed by Frostig et al. [33] and Frostig [34] for the study of localized effects in sandwich beams and plates. The theory includes the transverse flexibility of the core material. Thus, the core thickness is allowed to change during the deformation of the sandwich panel, and the

Fig. 17. Maximum adhesive layer stresses, tax ; ay ; a for a single lap joint using the linear and the non-linear solution procedure as a function of the applied in-plane nominal stresses.

Fig. 18. Illustration of a model joint and a real joint.

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face sheets are allowed to deflect differently. In the present study the high-order theory approach has been used for the analysis of a single lap joint, since the bonded joint in the overlap zone are comparable with a sandwich plate, i.e. the adhesive layer can be seen as the core material and the adherends can be seen at the face sheets. The structural modelling using the high-order theory approach can be described in the following way (for more details see Frostig et al. [19]:
The adherends: - Modelled as beams or plates in cylindrical bending.
The adhesive: - Assumed to be a 2-D or 3-D (in case of cylindrical bending) elastic continuum only possessing stiffness in the thickness direction, and therefore only capable of transferring transverse normal and shear stresses. The constitutive relations for the adhesive layer can be written as: xz ðx; za Þ ¼ Ga xz zz ðx; za Þ ¼ Ea "zz

ð20Þ

where za is the local z-coordinate in the adhesive layer. za=0 is at the upper interface of the adhesive layer, and za=ta is at the lower interface of the adhesive layer. The stress field in the adhesive layer is derived using point equilibrium conditions together with the constitutive relations in Eq. (20), and are given by (for more detail see Frostig et al. [19] and Frostig [32]): xz ðx; za Þ ¼ xz ðxÞ ¼ ax ax;x ð2za  ta Þ Ea ðw2  w1 Þ þ zz ðx; za Þ ¼  2 ta ð21Þ Compared with the stress field for the adhesive layer obtained using the spring model approach [Eq. (7)], it is observed, that the relation for the shear stresses are the same. In the high-order theory approach, however, the transverse normal stresses are not constant across the thickness of the adhesive layer, but are predicted to vary linearly across the adhesive layer thickness, and are a function of the derivative of the shear stresses with respect to x. Thus, comparing the equation for the transverse normal adhesive layer stresses from the spring model approach, i.e. Eq. (7), with the same for the high-order theory approach, i.e. Eq. (21), it is seen that the term:

a;x ð2za  ta Þ 2

ð22Þ

has been added. Since the adhesive layer is modelled as a continuum it is possible to prescribe the value of the shear stresses at the ends of the overlap.
The spew-fillet:  First approach: ax;x ¼ 0 in the adhesive layer at the ends of the overlap. Usually this requirement leads to prediction of non-zero shear stresses at the adhesive edges, thus resembling the presences of a spew-fillet. The size of the spew-fillet is not addressed by this approach, however. By this approach the predicted stress field for the adhesive layer is nearly exactly the same as predicted using the spring model approach.  Second approach: the spew-fillet is modelled as an equivalent elastic bar extending between the two adherends. Thus, the capability of the spew-fillet to transfer loads directly between the two adherends is included in the model. Using the high-order theory approach for the analysis of adhesive bonded joints it is possible to prescribe the shear stresses at the ends of the overlap ax to be equal to zero, thus assuming the boundaries of the adhesive layer to be free. Doing this the peeling stresses will be unrealistically large at the ends of the overlap zone, see Mortensen [25]. Instead the two approaches described above for the modelling of the spew-fillet can be used. In the second approach, the spew-fillet is modelled as an equivalent elastic bar extending between the two adherends. This concept is based on a practical assumption, indicated by finite element results, see Adams et al. [20,35,36], Harris and Adams [2], that the stress state in the spew-fillet consists mainly of unidirectional stresses that are parallel to the free edge of the spew-fillet, see Fig. 18. However, using this approach it is necessary to specify the spew-fillet size, which can be very difficult to determine in practice, and is usually unknown in the design phase. Comparison of the two approaches shows, however, that the first approach provides reasonable results for small spew-fillet sizes, and that the second approach provides more meaningful results for larger spew-fillets. 5.2. FE-models used for validation The aim of this comparison is to investigate the validity of the spring model approach, and to establish to which extent the approach provides reasonably accurate results in predicting the adhesive layer stresses.

F. Mortensen, O.T. Thomsen / Composites Science and Technology 62 (2002) 1011–1031

Therefore, two finite element analyses have been carried out with different spew-fillet sizes. In the first analysis the size of the spew-fillet is hsp=2ta and in the second analysis the size is hsp=1.2ta, where the size of the spewfillet is defined according to Fig. 18. The finite element analysis has been performed using the finite element code ODESSY developed at Aalborg University, see Rasmussen et al. [37] and Rasmussen and Lund [38]. A zoom of the finite element models at the left end of a single lap joint including a spew-fillet with the size hsp=1.2ta hsp=2ta is shown in Fig. 19. The structure has been modelled using a mixture of 6 and 8 node isoparametric 2D solid elements. The adhesive layer has been divided into six 8 node isoparametric elements through the thickness. In the near vicinity of the overlap edges the adhesive layer elements are quadratic, whereas irregular elements are used in the middle of the overlap regions where low stress gradients are present. The finite element model for the single lap joint, including a spew-fillet with the size hsp=2ta, contains 8525 elements with 26208 nodes. The analysis has been carried out as a plane strain model, since the proposed simple approach is based on the assumption that the laminates behave as plates or wide beams. It should be noted that the prediction of the stresses in the adhesive layer is affected by the singularities present at the corners of the adherends facing the adhesive layer at the ends of the overlap zone. To support the claim that the presence of a spew-fillet plays a very important structural role as a ‘stress reliever’ at the ends of the overlap zones, a finite element analysis of the considered single lap joint configuration without a spew-fillet has also been performed. The accuracy of the finite element results obtained near the ends of the overlap zone is of course again affected by the singularity present at the corners of the adherends facing the adhesive layer at the ends of the overlap zone.

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5.3. Comparison of results The comparison has been carried out analysing a single lap joint composed of two identical aluminium adherends with the properties shown in Table 4. Analysing the single lap joint problem described in Table 4 and using the three different methods: (1) spring model approach, (2) high-order theory approach and (3) the finite element method give the adhesive layer stress distributions shown in Figs. 20 and 21. From Figs. 20 and 21 it is seen that the stresses calculated by the three methods compare very well and display only small differences. The results calculated by the high-order theory approach, and displayed in the figures, are determined by the assumption that ax;x ¼ 0 at the ends of the overlap zone, i.e. the actual size of the spew-fillet is not addressed. In Fig. 20 the transverse normal stresses in the adhesive layer are displayed in the top and bottom interfaces of the adhesive layer and are marked by HOTAT and HOTAB. Only small differencies between the two curves HOTAT and HOTAB are observed, however. The stresses calculated by the finite element method and displayed in the figures are determined from the single lap joint configuration with the spew-fillet size hsp=2ta. It is seen that the stresses determined by the finite element method are a bit lower than the stresses determined by the two other approaches, but if the stresses are determined using the configuration with the spew-fillet size hsp=1.2ta, the stresses will be a bit higher than by the two other approaches. Thus, it is concluded that the adhesive layer stresses predicted by the spring model approach essentially equal the stresses predicted by the high-order theory approach including a spew-fillet (ax;x ¼ 0 condition at x=0,L), which again equal the stresses predicted by the finite element method including a spew-fillet with a size hsp between 1 and 2 times the adhesive layer thickness ta. In the manufacturing process of real adhesive bonded joints, it is unavoidable to have a spew-fillet of at least

Fig. 19. Zoom of finite element models at the left end of a single lap joint including a spew-fillet with the sizes hsp=1.2ta and hsp=2ta.

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1–2 times the adhesive layer thicknesses. However, it is possible to have larger spew-fillet sizes, and in this case, the stresses predicted by the spring model approach will overestimate the stresses. Since the real spew-fillet size is Table 4 Adherend and adhesive properties, thicknesses, lengths and boundary conditions assumed for the single lap joint to investigate the validity of the spring model approach Adherend 1 and 2

Aluminium E1=E2=70 GPa, t1=1.62 mm, b1=b2=25.4 mm

Adhesive

Epoxy Ea=4.82 GPa, v=0.4, ta=0.25 mm

Lengths

L1=L2=50.8 mm, L=12.7 mm

Load and B.C.

Simply supported at both ends, PN=1.0 kN(N ¼ PN =b)

usually not known, and, moreover, can be difficult to determine, it is concluded that the stresses predicted by the spring model approach represent reasonable and conservative estimates of the ‘real’ adhesive layer stresses. Finally, to emphasize the importance of the presence of a spew-fillet at the ends of the overlap zone, the adhesive layer stresses obtained for the single lap joint configuration without a spew-fillet are displayed in Figs. 22 and 23. The stress distributions displayed have been determined by use of the spring model approach, the high-order theory approach and the finite element method. From Fig. 23 it is seen that the adhesive layer shear stresses display the same pattern and almost same peak values as obtained with the presence of spew-fillets. The only exception being that the shear stresses are zero at the adhesive edges when there is no presence of a spew-fillet.

Fig. 20. Normalized adhesive layer transverse normal stresses, a =N using the spring model approach (SMA), the high-order theory approach (HOTA, ax;x ¼ 0 at x=0, L) and the finite element method (FEM) for a single lap joint with a spew-fillet of size hsp=2ta.

Fig. 21. Normalized adhesive layer shear stresses, ax =N using the spring model approach (SMA), the high-order theory approach (HOTA, ax;x ¼ 0 at x=0,L) and the finite element method (FEM) for single lap joint with a spew-fillet of size hsp=2ta.

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Fig. 22. Normalized adhesive layer transverse normal stresses, a =N using the spring model approach (SMA), the high-order theory approach (HOTA, ax ¼ 0 at x ¼ 0; L) and the finite element method (FEM) for a single lap joint without a spew-fillet.

Fig. 23. Normalized adhesive layer shear stresses, a =N using the spring model approach (SMA), the high-order theory approach (HOTA, ax ¼ 0 at x=0, L) and the finite element method (FEM) for single lap joint without a spew-fillet.

Fig. 22 shows, however, that the transverse normal stresses reach extreme values at the adhesive layer free edges. Furthermore it is seen that the transverse normal stresses at the upper (HOTAT) and lower (HOTAB) adhesive interface have different signs, i.e. tremendous gradients across the adhesive layer thickness are induced. Thus, from this and by comparison with the stresses determined for a single lap joint with a spewfillet it can be concluded that adhesive bonded joints without spew-fillets will be structurally very weak and useless for any practical applications. This conclusion is well known from engineering pratice.

6. Conclusions A general method for the analysis of adhesive bonded joints between composite laminates has been presented. The analysis accounts for coupling effects induced by adherends made as asymmetric and unbalanced laminates. The analysis allows specification of any combination of boundary conditions and external loading. The analysis can be carried out with the adherends modelled as wide beams or as plates in cylindrical bending. The adhesive layers are as a first approximation assumed to behave as a linear elastic material. The

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thickness of the adhesive layer is assumed to be small compared with the thickness of the adherends, and the adhesive layers are modelled as continuously distributed linear tension/compression and shear springs. The results obtained using this approach have been compared with finite element results and results obtained using a high-order theory approach (both including spew-fillets), and the comparison shows that the results are in very good agreement. The linear solution procedure has been used together with a modified von Mises criterion and a secant modulus approach for the effective stress–strain relationship of the adhesive, to perform a non-linear solution for the adhesive bonded joint problems. Comparison of the results obtained using the linear and non-linear solution procedures has shown that non-linear adhesive behaviour influences the adhesive layer stresses even at low load levels, and that the non-linear adhesive behaviour tends to smooth out the severe stress concentrations induced at the ends of the overlap zones.

i  cli ¼ A16

i i i i B16 B11 B16 B16 i ; c ¼ A  ; 2i 66 i i D11 D11

c3i ¼

i B16 Bi B i cli ; hli ¼ 11  16 ; i i i c D11 D11 D11 2i

h3i ¼

i B16 A i cli i i ; mli ¼ A11  B11 hli  16 ; i c2i D11 c2i

m2i ¼ 

i A16 c3i i  B11 h2i; c2i

m3i ¼

h2i ¼ 

i B16 c3i 1  i ; i c D11 D11 2i

i A16 i  B11 h3i c2i

ð25Þ

Within the overlap zone, i.e. for 0 4 x 4 L (see Fig. 3), combination of Eqs. (3), (4), (7) and (9) yields for laminates 1 and 2: 1 1 1 u10;x ¼ k11 Nxx þ k21 Nxy þ k31 Mxx

w;1x ¼ 1x 1 1 1 1x;x ¼ k41 Nxx  k51 Nxy  k61 Mxx 1 1 1 v10;x ¼ k71 Nxx þ k81 Nxy þ k91 Mxx

Ga 1 Ga t1 1 Ga 2 Ga t2 2 u þ   u þ  ta 0 2ta x ta 0 2ta x Ga 1 Ga 2 1 Nxy;x ¼ v  v ta 0 ta 0 Ga ðt1 þ ta Þ 1 Ga t1 ðt1 þ ta Þ 1 1 Mxx;x ¼ Qx1 þ u0 þ x 2ta 4ta Ga ðt1 þ ta Þ 2 Ga t2 ðt1 þ ta Þ 2  u0 þ x 2ta 4ta Ea 1 Ea 2 1 Qx;x ¼ w  w ta ta 2 2 2 u20;x ¼ k12 Nxx þ k22 Nxy þ k32 Mxx 1 Nxx;x ¼

Appendix A. Governing equations The governing equations shown here are for a single lap joint. From the equations derived, it is possible to form the complete set of system equations for the problems. Thus a combination of Eqs. (3), (4) and (8) yields for laminates 1 and 2 in the areas  L1 4 x 4 0 and L 4 x 4 L þ L2 (outside of overlap; see Fig. 3): i i i ui0;x ¼ kli Nxx þ k2i Nxy þ k3i Mxx w;ix ¼ ix i i i ix;x ¼ k4i Nxx  k5i Nxy  k6i Mxx i i i i v0;x ¼ k7i Nxx þ k8i Nxy þ k9i Mxx i Nxx;x ¼0 i Nxy;x ¼0 i Mxx;x ¼ Qxi i Qx;x ¼0

9 > > > > > > > > > > > = > > > > > > > > > > > ;

w;2x ¼ 2x 2 2 2 2x;x ¼ k42 Nxx  k52 Nxy  k62 Mxx

i ¼ 1; 2:

ð23Þ

Eq. (23) constitutes a set of eight linear coupled firstorder ordinary differential equations. The coefficients kli  k9i ði ¼ 1; 2Þ contain laminate stiffness parameters and are a result of isolating ui0;x ; vi0;x and w;ixx from i i i Nxx ; Nxy and Mxx in Eq. (3): 1 m3i m2i ; k2i ¼  ; k3i ¼  ml i mli mli k4i ¼ hli kli ; k5i ¼ hli k2i þ h3i ; k6i ¼ hli k3i þ h2i cli 1 cli cli c3i k7i ¼  kli ; k8i ¼  k2i ; k9i ¼  k3i  c2i c2i c2i c2i c2i ð24Þ kli ¼

where the coefficients cji, hji and mji (j=1,2,3) are:

2 2 2 v20;x ¼ k72 Nxx þ k82 Nxy þ k92 Mxx

Ga 1 Ga t1 1 Ga 2 Ga t2 2 u   þ u   ta 0 2ta x ta 0 2ta x Ga Ga 2 2 Nxy;x ¼  v10 þ v ta ta 0 Ga ðt2 þ ta Þ 1 Ga t1 ðt2 þ ta Þ 1 2 Mxx;x ¼ Qx2 þ u0 þ x 2ta 4ta Ga ðt2 þ ta Þ 2 Ga t2 ðt2 þ ta Þ 2  u0 þ x 2ta 4ta Ea Ea 2 Qx;x ¼  w1 þ w2 ta ta 2 Nxx;x ¼

ð26Þ

Eq. (26) constitutes a set of 16 linear coupled firstorder ordinary differential equations. References [1] Crocombe AD, Adams RD. An effective stress/strain concept in mechanical characterization of structural adhesive bonding. Journal of Adhesion 1981;13(2):141–55.

F. Mortensen, O.T. Thomsen / Composites Science and Technology 62 (2002) 1011–1031 [2] Harris JA, Adams RD. Strength prediction of bonded single lap joints by non-linear finite element methods. International Journal of Adhesion and Adhesives 1984;4:65–78. [3] Volkersen O. Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten. Luftfahrtforschung 1938;15:41–7. [4] Goland M, Reissner E. The stresses in cemented joints. Journal of Applied Mechanics 1944;1:A17–A27. [5] Hart-Smith LJ. Adhesive bonded single lap joints. Technical report NASA CR 112236, Douglas Aircraft Company, USA: McDonnell Douglas Corporation; 1973. [6] Hart-Smith LJ. Adhesive bonded double lap joints. Technical report NASA CR 112237, Douglas Aircraft Company, USA: McDonnell Douglas Corporation; 1973. [7] Hart-Smith LJ. Adhesive bonded scarf and stepped-lap joints. Technical report NASA CR 112235, Douglas Aircraft Company, USA: McDonnell Douglas Corporation; 1973. [8] Tong L. Bond strength for adhesive-bonded single-lap joints. Acta Mechanic 1996;117:101–13 [Springer-Verlag]. [9] Yuceoglu U, Updike DP. Bending and shear deformation effects in lap joints. Journal of Engineering Mechanics 1981:55–76. [10] Yuceoglu U, Updike DP. The effect of bending on the stresses in adhesive joints. Lehigh University, Bethlehem, Pensylvania, NASA NGR-39–007–011, 1975. [11] Yuceoglu U, Updike DP. Stress analysis of adhesive bonded stiffener plates and double joints. Lehigh University, Bethlehem, Pensylvania, NASA NGR-39–007–011, 1975. [12] Yuceoglu U, Updike DP. Comparison of continuum and spring models of adhesive layers in bonded joints. ASME, Advances in Aerospace Structures and Materials 1981;AD-01:75–83. [13] Renton WJ, Vinson JR. On the behavior of bonded joints in composite material structures. Engineering Fracture Mechanics 1975;7:41–60. [14] Renton WJ, Vinson JR. The efficient design of adhesive bonded joints. Journal of Adhesion 1975;7:175–93. [15] Pickett AK. Stress analysis of adhesive bonded lap joints. PhD thesis. University of Surrey, 1983. [16] Pickett AK, Hollaway L. The analysis of elasto-plastic adhesive stress in adhesive bonded lap joints in F.R.P.-structures. Composite Structures 1985;4:135–60. [17] Thomsen OT. Analysis of adhesive bonded generally orthotropic circular cylindrical shells. PhD thesis. Institute of Mechanical Engineering. Aalborg University, Denmark, Special Report, 4, 1989. [18] Thomsen OT. Elasto-static and elasto-plastic stress analysis of adhesive bonded tubular lap joints. Composite Structures 1992; 21:249–59. [19] Frostig Y, Thomsen OT, Mortensen F. Analysis of adhesive bonded joints, square-end and spew-fillet: higher-order theory approach. Journal of Engineering Mechanics, ASCE Engineering Mechanics Division 1997;125(11):1298–307. [20] Adams RD, Coppendale J, Peppiatt NA. Failure analysis of aluminium–aluminium bonded joints. Adhesion 1978;2:105–19. [21] Gali S, Ishai O. Interlaminar stress distribution within an adhesive layer in the nonlinear range. Journal of Adhesion 1978;9: 253–66.

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[22] Mortensen F, Thomsen OT. A simple unified approach for the analysis and design of adhesive bonded composite laminates. Proc of Eleventh International Conference on Composite Materials 1997;VI(18):129–39. [23] Mortensen F, Thomsen OT. Simplified linear and non-linear analysis of stepped and scarfed lap adhesive joints between composite laminates. Composite Structures 1997;38(1–4):281–94. [24] Mortensen F, Thomsen OT. Facilities in {ESAComp} for analysis and design of adhesive joints between composite laminates. Proc. Composite and Sandwich Structures, (NESCOII), ISBN 0 94781794 8. Chameleon Press Ltd., London, UK; 1997. p. 247– 266. [25] Mortensen F. Development of tools for engineering analysis and design of high-performance FRP-composite structural elements, PhD thesis. Institute of Mechanical Engineering, Aalborg Universitet, Special Report No. 37, ISSN 0905–2305, 1998. [26] Saarela O, Palantera M, Haberle J, Klein M. ESAComp: a powerful tool for analysis and design of composite materials. Proc. of Int. Symp. on Advanced Materials for Lightweight Structures, ESA-WPP-070, ESTEC. Noordwijk, The Netherlands; 1994. p. 161–9. [27] Whitney JM. Structural analysis of laminated anisotropic plates. Lancaster: Technomic Publishing Company, 1987. [28] Jones RM. Mechanics of composite materials. 2nd ed. Taylor and Francis; 1999. [29] Kalnins A. Analysis of shell of revolutions subjected to symmetrical and non-symmetrical loads. Journal of Applied Mechanics 1964;31:1355–65. [30] Gali S, Dolev G, Ishai O. An effective stress/strain concept in mechanical characterization of structural adhesive bonding. International Journal of Adhesion and Adhesives 1981;1:135– 40. [31] Adams RD. Stress analysis: a finite element analysis approach. In: Developments in Adhesives. 2nd ed. London: Applied Science Publishers; 1981. [32] Lilleheden L. Properties of adhesive in situ and in bulk. International Journal of Adhesion and Adhesive 1994;14(1):31–7. [33] Frostig Y, Baruch M, Vilnay O, Sheinman I. Bending of nonsymmetric sandwich beams with transversely flexible core. ASCE Journal of Engineering Mechanics 1991;117(9):1931–52. [34] Frostig Y. Behaviour of delaminated sandwich beam with transversely flexible core-high order theory. Composite Structures 1992;20:1–16. [35] Adams RD, Peppiatt NA. Stress analysis of adhesive-bonded lap joints. Journal of Strain Analysis 1974;9:185–96. [36] Adams RD, Wake WC. Structural adhesive joints in engineering. 1st ed. Elsevier Applied Science Publishers, ISBN 0–85334–263–6, 1984. [37] Rasmussen J, Lund E, Olhoff N. Integration of parametric modeling and structural analysis for optimum design. In: Gilmore, et al., editors. Proceedings of advances in design automation. The American Society of Mechanical Engineers. Albequerque, New Mexico, USA, 1993. [38] Rasmussen J, Lund E. The issue of generality in design optimization systems. Engineering Optimization 1997;29:23–37.