Adhesive 1

Chapter 4Complex Constitutive Adhesive Models

Erol Sancaktar

Abstract A complete approach to modeling adhesives and adhesive joints needs toinclude considerations for: deformation theories, viscoelasticity, singularity meth-ods, bulk adhesive as composite material, adhesively bonded joint as composite andthe concept of the “interphase”, damage models, and the effects of cure and pro-cessing conditions on the mechanical behavior. The adherend surfaces have distincttopographies, which result in a collection of miniature joints in micron, and evennano scale when bonded adhesively. The methods of continuum mechanics can beapplied to this collection of miniature joints by assuming continuous, or a combina-tion of continuous/discontinuous interphase zones.

4.1 Introduction

Adhesively bonded joints are complex composite structures with at least one of theconstituents, namely the adhesive, most often, being a composite material itself dueto the presence of secondary phases such as fillers, carriers, etc. The joint structurepossesses a complex state of stress with high stress concentrations, and often, sin-gularities due to the terminating adhesive layer where the substrates may possesssharp corners. Consequently, accurate analysis and modeling of adhesive materi-als and bonded joints require the use of the methods of composite materials andcomposite mechanics. The inclusion of the “interphase” region is necessary in thisanalysis as a distinct continuum. The presence of geometric discontinuities createsstress concentrations and, possibly, singularities adding additional complexity to thetopic of adhesively bonded joints. This problem, however, can be alleviated, at leastpartially, by making the proper changes in the geometry of the bonded joint. Further-more, since most adhesive materials are polymer-based, their natural viscoelastic

Erol SancaktarPolymer Engineering, Adjunct Professor, Mechanical Engineering, The University of Akron,Akron, OH 44325-0301, e-mail: [email protected]

L.F.M. da Silva, A. Ochsner (eds.), Modeling of Adhesively Bonded Joints, 95c© Springer-Verlag Berlin Heidelberg 2008

96 E. Sancaktar

behavior usually serves to reduce localized stress concentrations. In those caseswhere brittle material behavior prevails or, in general, when inherent material flawssuch as cracks, voids, disbonds exist, then the use of the methods of fracture me-chanics are called for. For continuum behavior, however, the use of damage modelsis considered appropriate in order to be able to model the progression of localizedand non-catastrophic failures.

Obviously, a technical person involved in adhesive development and/or appli-cations should keep the above mentioned issues in mind, along with insight intotypical joint stress distributions for adhesive joints as well. Stress distributions arerelevant from the mechanical adhesion point of view also, since they depend on sur-face topography, which can be considered a collection of many geometrical forms.Therefore, mechanical adhesion depends on the stress states of different adhesivejoint geometries on the scale of the surface topography, which may include manylap, butt and scarf joints in the interphase region.

A complete approach to modeling adhesives and adhesive joints, therefore,needs to include considerations for: deformation theories, viscoelasticity, singu-larity methods, bulk adhesive as composite material, adhesively bonded joint ascomposite – the concept of the “interphase”, damage models, and the effects of cureand processing conditions on the mechanical behavior.

4.2 Deformation Theories

The deformation theory was first introduced by Hencky (1924) as reported byHill (1956) and Kachanov (1971) in the form:

εi j = (σkk/9K)δi j +ψSi j (4.1)

where, Si j is the deviatoric stress tensor. Equation (4.1) reduces to the elastic stress-strain relations when ψ = 1/2G, where G is the elastic shear modulus. If the scalarfunction ψ is defined as

ψ = (1/2G)+Ω (4.2)

then Eq. (4.1) can be interpreted in the form

εi j = εi jE + εi j

V + εi jP (4.3)

whereεi j = (σkk/9K)δi j +(Si j/2G)+ΩSi j (4.4)

and,εi j

P = ΩSi j (4.5)

with Ω being a scalar function of the invariants of the stress tensor, and thesuperscripts E, V and P representing elastic, viscoelastic and plastic behaviors,respectively.

4 Complex Constitutive Adhesive Models 97

Consequently, the relation

ε = (σ/E)+Λ σ (4.6)

is obtained for uniaxial tension on the basis of Eqs. (4.4) and (4.5) with Λ = 2Ω/3.Ramberg and Osgood (1943) used a special form of Eq. (4.6) with Λ = Kσn−1

to result in:

ε = (σ/E)+Kσn (4.7)

where, K and r are material constants.They reported that Eq. (4.7) could be used successfully to describe uniaxial ten-

sion and compression behavior of various metal alloys. Equation (4.7) was latermodified by McLellan (1966, 1969) to accommodate strain rate effects. McLellaninterpreted the terms E, K and n of Eq. (4.7) as material functions with the func-tion E representing viscoelastic behavior and functions K and n representing work-hardening characteristics. The terms E, K and n were all described as functions ofthe strain rate (dε/dt) so that rigidity, stress and plastic flow respectively were allaffected by variations in the strain rate.

Renieri et al. (1976) used a bilinear form of rate dependent Ramberg-Osgoodequation to describe the stress-strain behavior of a thermosetting adhesive in thebulk tensile form. The bilinear behavior was obtained when log εp was plottedagainst log σ , where εp represents the second term on the right-hand side ofEq. (4.7). The model adhesive they used was an elastomer modified epoxy adhe-sive with and without carrier cloth. They made several modifications on the form ofthe equation previously used by McLellan. First, the plastic strain εp was assumedto be a function of the over-stress above the elastic limit stress (the developmentof over-stress approach will be presented subsequently) and second, the stress le-vels σ∗ defining the intersection point for the bilinear behavior were found to occurslightly below the stress whitening stress values. The equations they developed inthis fashion are given as:

ε = σ/E, 0 < σ < θ (4.8-1)

ε = (σ/E)+K1[σ −θ ]n1 θ < σ < σ∗ (4.8-2)

ε = (σ/E)+K2[σ −θ ]n2 σ∗ < σ < Y (4.8-3)

where, θ is the elastic limit stress (yield stress) in simple tension, Y is the maxi-mum stress, K1, K2, n1, and n2 are strain rate dependent material functions. Renieriet al. (1976) reported that the adhesive material properties were different beforeand after stress-whitening due to changes in material behavior and therefore bi-linear equations of the form (4.8) were needed to describe such behavior. Brinsonet al. (1975) determined that for the adhesive with the carrier cloth the parameter n,Eq. (4.7), varied much less with strain rate in comparison to the adhesive withoutthe carrier cloth. They concluded that the bilinear forms they proposed, Eq. (4.8),accurately predicted the rate dependent behavior of their model adhesive.

98 E. Sancaktar

Figures 4.1 and 4.2 show the application of the model described above to thestress-strain and stress whitening behaviors of a bulk thermosetting epoxy film ad-hesive containing a non-woven nylon mat, as obtained by Sancaktar and Beachtle(1993). In this application, two intersection points are identified by intersectingtwo pairs of bilinears, each of which containing a common slope line as partof the bilinear sets (Fig. 4.2). This procedure was called “dual bilinear fit” bySancaktar and Beachtle (1993). These intersection points define upper and lowerstress whitening stress limits shown in Fig. 4.1.

Ramberg-Osgood type equations were also used to describe shear behaviorof structural adhesives. Zabora et al. (1971) used a rate dependent form of theRamberg-Osgood equation to describe the shear behavior of structural adhesivesin the bonded form.

Simple power function type relations may be utilized in an empirical fashionto describe nonlinear elastic behavior of structural adhesives in tensile and shearmodes when their mechanical behavior is found to be relatively rate insensitive. Forexample Sancaktar and Schenck (1985-a) used a power function relation given by

τ = Kγ r (4.9)

where τ and γ are shear stress and strain respectively, to represent the nonlinearelastic shear behavior of a thermoplastic polyimidesulfone adhesive in the bondedform. Sancaktar and Dembosky (1986) showed that when the stress-strain curve hasa well defined initial elastic region and a yield point, τy, then a bimodal relationgiven by

Fig. 4.1 A stress-strain diagram identifying upper and lower stress whitening stress limits, σU andσLo

∗ respectively, as well as the visually observed stress whitening stress. The modulus of elastic-ity, E, and the elastic limit stress, θ for the bulk adhesive tested are 3.3× 105 and 3.5×103 psi,respectively (Sancaktar and Beachtle, 1993)


Fig. 4.2 A stress-strain diagram identifying upper and lower stress whitening stress limits, σU andσLo

∗ respectively, as well as the visually observed stress whitening stress. The modulus of elastic-ity, E, and the elastic limit stress, θ for the bulk adhesive tested are 3.3× 105 and 3.5× 103 psi,respectively (Sancaktar and Beachtle, 1993)

γ = τ/G, τ < τy (4.10-1)

γ = Bτm τ > τy (4.10-2)

can be utilized more appropriately.

4.3 Viscoelasticity Considerations

4.3.1 Linear Viscoelasticity

The differential forms of the fundamental constitutive relations are given as:

P{Skl} = 2Q{εkl} (4.11)

P′{σ} = Q′{ε}−Q′′{AT} (4.12)

where, A is the thermal expansion function, T is the temperature, Skl and εkl arestress and strain deviators respectively and the differential operators are defined as:

P =u

∑i=0

ai(x, t,T )∂ i/dti (4.13)

Q =v

∑i=0

bi(x, t,T )∂ i/∂ ti (4.14)

100 E. Sancaktar

P′ =u′

∑i=0

a′i(x, t,T )∂ i/∂ ti (4.15)

Q′ =v′

∑i=0

b′i(x, t,T )∂ i/∂ ti (4.16)

Q′′ =v′′

∑i=0

b′′i (x, t,T )∂ i/∂ ti (4.17)

Q′′{AT} =T (x,t)∫T0

Q′′′{α(T ′)}dT ′ (4.18)

Q′′′ =v′′′

∑i=0

b′′′i (x, t,T )∂ i/∂ ti. (4.19)

In Eqs. (4.13) through (4.19) the terms ai, ai′, bi, bi

′, bi′′ and bi

′′′ are materialproperties and the integers u, u′, v, v′, v′′ and v′′′ reflect the complexity of the ma-terial behavior, and t represents time. In Eq. (4.18) the quantity α represents thecoefficient of thermal expansion.

The same constitutive relations can also be represented in integral equation formgiven by:

σ(x, t) =t∫−∞

EV (x, t, t ′)[∂ε(x, t ′)/∂ t ′]dt ′ −t∫−∞

EV T (x, t, t ′)∂ [AT (x, t ′)]/∂ t ′dt ′

(4.20)and

Skl(x, t) = 2t∫−∞

E(x, t, t ′)∂εkl(x, t ′)/∂ t ′dt ′ (4.21)

where EV and EV T are the relaxation moduli as functions of time and time andtemperature, respectively, and E is the “relaxation memory function”. As shown byFlugge (1975), the assumptions of elastic dilatational behavior or no volume changeare frequently used in three dimensional analyses of polymeric materials. As fordistortion operations, however, any mechanical model (such as Maxwell, Kelvin,three-parameter solid, etc.) can be used.

4.3.2 Interrelationship Between the Tensile and Shear Properties

The constitutive equations proposed for characterization of solid polymer materialsin the tensile mode can be reformulated for the case of pure shear. One approach isto replace tensile stress and strain variables by their shear counterparts.

It is usually desirable to run a simple bulk tensile test program and subsequentlypredict (calculate) shear properties from their tensile counterparts. This approach re-quires a clearly defined relationship between shear and tensile elastic limit and yieldvariables and material properties. The elastic limit and yield stress values can be re-lated between tensile and shear conditions by using an appropriate failure criterion,


such as maximum normal stress, maximum shear stress, distortion energy criteria,etc. A material parameter that needs to be converted in addition to the usual elasticproperties is the viscosity coefficient. This can be done by using Tobolsky’s (1960)assumption of equivalent relaxation times in shear and tension.

Application of this assumption results in the relation

μS = μT /2(l +υ) (4.22)

where μS and μT refer to the viscosity coefficients in shear and tension respectively,and υ is Poisson’s ratio. An application of this procedure for a linear viscoelastic(over-stress) model was shown by Sancaktar and Brinson (1980).

4.3.3 Time-Temperature Superposition

The time-temperature superposition principle (TTSP) is used to extend the creep andrelaxation data obtained at higher temperatures to values at lower temperatures andlonger times in the same strain and stress ranges. Provided that no structural changesoccur in the material at high temperatures, TTSP can be used for amorphous andsemi-crystalline thermoplastics and thermosets for extrapolation purposes. In orderto apply this principle, one should first construct a “master curve,” where the com-plete compliance-time or relaxation modulus-time behavior is plotted at a constanttemperature. For the “master curve” a reference temperature To is chosen. As shownby Ahlonis et al. (1972), the relation for the compliance observed at any time t andtemperature To is given in terms of the experimentally observed compliance valuesat different temperatures T as:

D(To, t) = [ρ(T )T/ρ(To)To]D(T, t/aT ) (4.23)

whereD(t) = ε(t)/σ (4.24)

and ρ(T ) is the temperature dependent density. An illustration of this procedure isshown in Fig. 4.3, where adjustments for any change in the material density areignored.

A similar relation for the relaxation modulus is given as:

E(To, t) = [ρo(To)To/ρ(T )T ]E(T,aT t) (4.25)

where,E(t) = σ(t)/ε. (4.26)

The term aT is a time shift factor which can be obtained with the application of thefamous Williams-Landel-Ferry (WLF) equation, reported by Ferry (1961).

For most polymeric materials the polymer’s glass transition temperature (Tg) ischosen as the reference temperature for accurate results. Even though Ferry (1961)

102 E. Sancaktar

Fig. 4.3 An illustration of time-temperature (t-T ) superposition procedure for shifts in creep com-pliance, D(t). Adjustments for any change in the material density are ignored

reports that the above equations have not been proven to be valid below the Tg, datashowing its application for T < Tg is available in the literature. Examples of appli-cations to structural adhesives in the bulk form were given by Renieri et al. (1976)and Keuner et al. (1982).

The application of the TTSP can be extended into the nonlinear viscoelastic re-gion, as shown by Darlington and Turner (1978). Examples on the establishmentof rate-temperature superposition based on WLF equation as applied to peelingproblems have been given by Kaelble (1964, 1969), Hata et al. (1965) and Non-aker (1968). Nakao (1969) and Koizumi et al. (1970) report superposition based onArrhenius’ equation.

4.3.4 Nonlinear Viscoelasticity

Equations (4.11), (4.12) and (4.20), (4.21) can also be used to describe nonlinearmaterial behavior provided that ai, bi, etc. and E, EV are functions of stresses,strains, their derivatives and their invariants in addition to the functional variablesshown.

On the basis of their observations on polymers, composites and adhesives, Hielet al. (1983), Rochefert and Brinson (1983), Tuttle and Brinson (1984) and Zhanget al. (1985) report that the extent of nonlinearity is dependent on both the stresslevel and the time scale. Constitutive relations having the form of Eqs. (4.20)and (4.21) are in agreement with this observation. Schapery (1966, 1969a, b), re-ports that stress and strain dependent material properties of nonlinear constitutiveequations have a thermodynamic origin. For example the one-dimensional constitu-tive relation

ε = g0D(0)σ +g1

t∫o

ΔD(φ −φ ′)[d(g2σ)/dτ]dτ (4.27)

includes the variables g0, g1 and g2 which reflect stress dependence of the Gibbsfree energy. In Eq. (4.27) D(0) is the initial value of the linear viscoelastic creepcompliance, ΔD(φ) is the transient component and the quantities


φ = φ(t) =t∫o

dt ′/aσ (4.28)

and

φ ′ = φ(τ) =τ∫o

dt ′/aσ (4.29)

are reduced time parameters and depend on the shift function aσ . “aσ ” is a functionof stress and change with entropy production and free energy changes.

An equation similar to (4.27) can be written with strain as the independent statevariable. In this equation, the strain dependent properties h0, h1, h2 and aε reflectthe changes in the Helmholtz free energy.

Schapery’s Eq. (4.27) was applied extensively by Cartner et al. (1978), Peretzand Weitsman (1982) and by Tuttle and Brinson (1985) in characterization of resinmatrix composites.

In using nonlinear mechanical models, in addition to utilizing nonlinear elasticand shear thickening or thinning dashpot elements, one can also incorporate nonlin-earity by means of a perturbation technique. This is accomplished by adding smallperturbation terms which are functions of the current level of elastic strain and strainrate to the elastic and viscous coefficients respectively. This method was originallyproposed by Davis (1964) and later applied in material characterization of bulk ad-hesives by Renieri et al. (1976).

Further discussions on nonlinear viscoelastic theory are given by Green andAdkins (1960) and Hilton (1975).

4.3.5 Empirical Description of Creep Behavior

The constitutive equations obtained with the use of mechanical models can be solvedfor the creep condition given by

σ(t) = σoH(t) (4.30)

with σo and H(t) describing the level of constant stress and the unit step function re-spectively, to result in a creep equation. In some cases, however, empirical relationsdescribing time dependent strain functions are preferred over this method for prac-tical reasons. Actual creep data can be fitted with such functions to yield accurateanalysis in conjunction with equations of the form (4.20), (4.21) or (4.27).

A variety of mathematical forms have been proposed to describe the creep be-havior of adhesives and polymers in an empirical fashion. The simplest procedureis called the log-log method based on the assumption that the logarithm of the sec-ondary creep rate is linearly related to the logarithm of the stress level at which thecreep occurs. This assumption results in a creep relation which is a linear functionof time but nonlinear function of stress in the secondary range. An application ofthis method to polymeric materials was given by Sancaktar et al. (1987-a).

104 E. Sancaktar

The creep relation which is most commonly used in the literature is the “power-law” compliance given by

D(t) = D0 +D1 tn (4.31)

where, D0 is the instantaneous compliance, D1, and n are material constants. Notethat Eq. (4.31) allows time reduction by a stress-dependent shift-factor to describestress enhanced creep. Equations of this form were used by researchers such asWeitsman (1981) and Ravi-Chandar and Knauss (1984) for viscoelastic stress analy-sis and fracture mechanics considerations (respectively) of adhesives and polymericmaterials.

Various methods have been proposed to determine the parameters D0, D1 andn of Eq. (4.31). For example Findley and Peterson (1958) subtract the instanta-neous strain from the total strain and plot the transient strain on log-log paper.Other methods were described by researchers such as Boller (1957), Knauss andEnri (1981), Dillard et al. (1982), Dillard and Brinson (1983), Becher (1984) andHenriksen (1984). A comparison and discussion of these methods was given byDillard and Hiel (1985).

As shown by Sancaktar (1991), Figs. 4.4 and 4.5 illustrate the variation of ad-hesive shear stress at the overlap edges of double lap joints as predicted by linearand nonlinear viscoelastic analyses. In Fig. 4.4, the linear analysis was performedusing the Maxwell model with different relaxation times, T , and the nonlinear anal-ysis was performed using Eq. (4.31) with a stress-dependent shift factor. Figure 4.5illustrates the ability to obtain matching results between the two models by usingthe Maxwell model with variable relaxation time, T , as given by Sancaktar (1991).

Additional methods and discussions on constitutive equations for creep weregiven by Odquist (1954), Findley and Lai (1967), Gittus (1975) and Findley et al.(1976).

4.3.6 Increases in Joint Strength Due to Rate DependentViscoelastic Adhesive Behavior

Due to their viscoelastic nature, most adhesives exhibit rate dependent material be-havior, which can be modeled, for example, by using mechanical model characteri-zation.

In order to describe the rate dependence of limit stress and elastic limit strainfor polymeric and adhesive materials in the bulk form, Renieri et al. (1976) andSancaktar and Brinson (1980) utilized a semi-empirical approach proposed byLudwik (1909) in the form:

τult = τ ′ + τ ′′ log

[dγdt

/dγ ′

dt

](4.32)

where, τult is the ultimate shear stress, dγ/dt is the initial elastic strain rate, and τ ′,τ ′′, and dγ ′/dt are material constants.


Fig. 4.4 Variation of adhesive shear stress at the overlap edges of double lap joints as predicted bylinear and nonlinear viscoelastic analyses. The linear analysis was performed using the Maxwellmodel with different relaxation times, T , and the nonlinear analysis was performed using Eq. (4.31)with a stress-dependent shift factor. The joint dimensions used in analysis are also shown with δ ,L and d representing adhesive thickness, overlap length and the main plate thickness, respectively.Es and Ga represent the Young’s modulus for the substrate and the shear modulus for the adhesive,respectively (Sancaktar, 1991)

Renieri et al. (1976) and Sancaktar and Brinson (1980) used the same form ofEq. (4.32) to describe the variation of elastic limit shear stress (τel) and strains (γel)with initial elastic strain rates. These expressions may be written as:

τel = φ ′ +φ ′′ log

[dγdt

/dγ ′

dt

](4.33)

and,

γel = ζ ′ +ζ ′′ log

[dγdt

/dγ ′

dt

](4.34)

where, additional material constants are defined accordingly. Sancaktar et al. (1984,1985-b) later showed applicability of Eqs. (4.32) through (4.34) for adhesives inthe bonded form and also proposed superposition of temperature effects on theseequations.

As reported by Ward and Hadley (1993), the theoretical basis of Eq. (4.32) can befound in the Eyring Theorem, according to which the mechanical response of an ad-hesive is a process that has to overcome a potential barrier, (ΔE). The author thinks

106 E. Sancaktar

Fig. 4.5 Variation of maximum adhesive shear stress in double lap joints based on nonlinear vis-coelastic analysis, using Eq. (4.31) with a stress-dependent shift factor, and linear viscoelasticanalysis, using the Maxwell model with variable relaxation time, T (Sancaktar, 1991)

that this barrier decreases not only with increasing stress but also with increasingtemperature. Based on the Eyring Theorem, the relationship between a limit stress,say τel, and the strain rate can be written as:

dγ/dt = Aexp[(τelV −ΔE)/RT ] (4.35)

where, A is a pre-exponential factor, R is the gas constant, and V is the activationvolume.

Equation (4.35) can be expressed in logarithmic form as:

τel = {ΔE/V}+{

2.303RT log

[dγdt

/A

]}/V. (4.36)

Sharon et al. (1989) applied Eq. (4.36) to describe rate and temperature dependentvariation of four structural adhesives in the bulk form. They used the shift factor[log(1/A)]× 10−3 on the temperature (◦C) to describe the effect of rate. Anotherexample for bonded joint behavior was given by Chalkley and Chiu (1993).

If the energy barrier term ΔE/V of Eq. (4.36) is also affected by temperature thenone needs two shift factors in Eqs. (4.32) through (4.34) to describe rate-temperatureeffects on limit stress-strain equations:

τel = {aT 1}θ ′ +{{aT 2}θ ′′ log

[dγdt

/dγ ′

dt

]}(4.37)


and,

γel = {bT 1}ζ ′ +{{bT 2}ζ ′′ log

[dγdt

/dγ ′

dt

]}(4.38)

where aTi , bTi are shift factors as functions of temperature. Figure 4.6 illustrates anapplication of Eq. (4.37) by Sancaktar et al. (1984), showing the variation of elasticlimit shear stress with initial elastic strain rate and environmental temperature for anepoxy adhesive film with polyester knit fabric carrier cloth. The adhesive was testedin the bonded lap shear mode.

In order to analyze the effects of rate and temperature on the interfacial strengthof bonded joints, one can initially use a simple energy approach in the followingmanner: A critical energy level, Wc, is used to represent interfacial failure. In thepresence of adhesive, substrate and interphase (a distinct material layer betweenthe substrate and the adhesive layer which transmits the rigid substrate’s energy tothe adhesive layer), the combined elastic energy can be written as:

Va{(1/2)ε2adhesiveEadhesive}+Vip{(1/2)ε2

interphaseGinterphase}

+Vs{(1/2)ε2substrateEsubstrate} = Wc (4.39)

where, Va, Vip and Vs represent volume fractions of the adhesive layer, interphaseand the substrate, respectively. Now, if one considers Eq. (4.39) in conjunction withEq. (4.38) to include rate-temperature effects, then it can be easily deduced thathigher Wc levels are obtained at high rate and/or low temperature levels since theproportion of elastic strains increases with increasing elastic limit strains due tohigh rate and/or low temperature levels.

Fig. 4.6 Variation of elastic limit shear stress with initial elastic strain rate and environmental tem-perature, and comparison with Ludwik’s equation for an epoxy adhesive film with polyester knitfabric carrier cloth. The adhesive was tested in the bonded lap shear mode (Sancaktar et al. 1984)

108 E. Sancaktar

4.3.7 Reductions in Stress Concentration Due to ViscoelasticAdhesive Behavior

If the adhesive material exhibits viscoelastic behavior, then the high stress magni-tudes observed at the overlap end region of bonded joints are expected to diminishin time due to the creep process. For example Weitsman (1981) utilized the non-linear viscoelastic “power-law” response, which describes a stress enhanced creepprocess to illustrate time dependent reductions in shear stress peaks along the ad-hesive layers of symmetric double lap joints. Sancaktar (1991) later illustrated theapplicability of the correspondence principle to the same problem with the use ofMaxwell chain to approximate the continuous change in the relaxation time and tocoincide with the results calculated using nonlinear viscoelastic theory, as shown inFigs. 4.4 and 4.5.

4.4 Singularity Methods

Theoretically, geometrical singularities exist at sharp interface corners. The strengthof such singularities depends on the local geometry, including the actual corner ra-dius, and the material properties. Barsoum (1989) considered the cases of elasticadhesive at a rigid corner, and nonlinearly hardening adhesive at a square rigid cor-ner. He showed that for an elastic adhesive the obtuse angle notch gives a muchweaker singularity and, hence, a greater failure load.

Assuming linear elasticity, Groth (1988) proposed an exact solution for the ad-hesive stresses at an interface corner in the form of a power series:

σi j =∞

∑k=1

KkHki j(φ)ζ λk +Continuous part (4.40)

where σi j is the stress tensor, ζ = r/h is a non-dimensional radial distance from thesingularity, Hk

i j are functions depending on the geometry angle φ of the corner, Kk

constitutes generalized stress intensity factors, and λk defines the strength of sin-gularities governed by the eigenvalues of the λk in the open interval Re (λk) < 1.Due to restrictions on finite strain energy at the singularity, values of Re (λk) ≥ 1are not allowed. Obviously, based on Eq. (4.40), more than one singular term mayinfluence the stress field close to an interface corner. Dempsey (1995) showed thatin addition to the power type singularities, i.e. 0(r−λ ) as r → 0, and λ represent-ing the strength of singularity, power-logarithmic type singularities, i.e. 0(r−λ ln r)as r → 0, may occur for homogeneous boundary conditions of composite wedgeproblems.

Stress singularities at butt joint interface corners were discussed Reedy andGuess (1995) and Sawa et al. (1995).


4.5 Bulk Adhesive as Composite Material

Since most adhesives contain secondary phase materials such as fillers and carriercloth, they should be treated as composite materials in constitutive modeling. Theinteresting aspect in such treatment is the fact that such secondary phase constituentsform adhesive bonds with the adhesive matrix in small scale. Consequently, local-ized non-catastrophic failures within the bulk adhesive consisting of interfacial sepa-rations are expected to cause constitutive behavior changes for the adhesive bulk.Such behavior can be properly described by using “damage mechanics models”some of which will be described later in this chapter within the context of adhesivematerials.

It is known that particle size, shape and volume fraction all affect the mechanicalbehavior of filled polymers. For example, electrically and thermally conductive ad-hesives which usually contain dispersed metal particles of varying aspect ratios, butusually within the micron size range, require the methods of Eshelby (1957, 1959)for characterization. We note that these types of adhesives sometimes require direc-tional properties. Kerner (1956), Hashin and Shtrikman (1963), Budiansky (1965)and Hill (1965) provided methods for calculation of elastic moduli for isotropic ma-terials filled with spherical particles. Methods for elastic moduli of unidirectionalreinforced polymers with infinitely long fibers were given by Hill (1964) and Chowand Hermans (1969). Bulk and shear moduli for disk and needle shaped inclusionswere considered by Wu (1966) and Walpole (1969).

As reported by Ashton et al. (1969), Halpin and Tsai introduced approximateequations for square fiber reinforcement by reducing Hermans’ solution (1967) us-ing numerical solutions of elasticity theory. Equations for slender rigid inclusionsat low concentrations were developed by Russel and Acrivos (1972, 1973). As foreffective thermal expansion coefficient of filled polymers, the effects of filler geom-etry, and constituent material properties have been studied by Kerner (1956).

Using the generalized approach of Eshelby, Chow (1977, 1978-a) derived rela-tions for the bulk modulus, transverse and longitudinal Young’s moduli and, the twoeffective shear moduli, G12 and G13 of the filled polymer containing aligned ellip-soidal particles at finite concentration. Chow (1978-b) also derived relations for thelinear and volumetric thermal expansion coefficients for filled polymers containingaligned ellipsoidal inclusions at finite concentration. He concluded that in such filledsystems, the volumetric expansion varies only slightly with the aspect ratio of theellipsoid, while longitudinal and transverse linear expansions show strong depen-dence on the particle shape. All the modulus calculations developed by Chow arebased on the following assumptions: (i) The filler particles are well bonded to thematrix, and consequently, the isostrain condition at the particle/matrix interface canbe used to develop the method; (ii) The shapes of particles are uniform ellipsoidswith the aspect ratio Cρ , representing the ratio of the major to minor axes; (iii) Theparticles are distributed with their corresponding axes aligned.

Wei (1995) and Sancaktar and Wei (1998) applied the elastic moduli and thermalexpansion coefficient relations developed by Chow in describing the material

110 E. Sancaktar

behavior of electronically conductive (filled) adhesives in order to be able todetermine, accurately, the thermal mismatch stresses which occur between the adhe-sive and the substrates in the bonded form. Both finite element analysis and closedform solutions were used for stress analysis.

We note that the results given by Chow are consistent with the well known ruleof mixtures used in long fiber reinforced composites. When the aspect ratio, Cρ , ofthe particle is Cρ > 50, the moduli are not a function of Cρ . Therefore, for largerCρ the composite’s behavior is more like that of long fiber reinforced composite.For intermediate aspect ratio cases, i.e. 1 < Cρ < 50, Sancaktar and Wei (1998) pro-posed an approximation of Chow’s elastic moduli formulas by utilizing the methodused for randomly oriented short fibers in predicting the behavior of particle filledadhesives, so that the composite material can be treated as isotropic, as suggestedby Agarwal and Broutman (1990).

As for the thermal expansion coefficient, the isotropy assumption was accom-plished simply by using one-third of the volumetric thermal expansion coefficient,as suggested by Lipatov et al. (1991). We note, however, that in special cases ofconductive adhesive applications, directional conduction may be required, in whichcase the use of non-isotropic, directional material properties would be necessary.

The constitutive models cited above for predicting filled polymer behavior allassume perfect adhesive bond between the fillers and the polymer matrix. As suchfilled polymers are loaded, however, failure of adhesive bonds is expected in propor-tion to the applied force and the duration of its application. This, in turn, is expectedto result in changes in the constitutive properties of the filled polymer adhesive.Lipatov et al. (1991) reported a decrease in the modulus of elasticity of a filledelastomer at separation of filler particles from the binder and described it with theapproximate formula

Ex/E f = exp−4.3 (Vx) (4.41)

where E f is the initial elastic modulus for the filled polymer with unbroken bonds, Vx

is the volume fraction of unbonded filler, and Ex is the composite’s elastic moduluswhen the fraction, Vx, of the filler particles are no longer bonded to the matrix.

Garton et al. (1989) discussed stiff polar molecular additives which interactstrongly with the (thermoplastic or thermoset) matrix polymer adhesive, thus de-creasing the composite free volume, f , as given by the equation

f = V1 f1 +V2 f2 + k V1V2 (4.42)

where V1 and V2 are the volume fractions of the polymer matrix and additive, respec-tively, f1 and f2 are the corresponding fractional free volumes, and k is an interactionparameter. The composite properties: bulk elastic modulus, bulk strength (to a lesserdegree) as well as bonded joints properties of strength and stiffness are reported byGarton et al. (1989) to increase with reductions in composite free volume.

In another study, Sancaktar and Kumar (1999-a, 2000) obtained increases inthe lap joint strength by selectively rubber toughening the high stress concentra-tion regions of the adhesive overlap. This novel approach to increase the lap jointstrength by using functionally gradient adhesive introduced a method different from


the traditional methods of either increasing the lap joint area or altering the joint ge-ometry. For the different configurations of adhesive arrangements tried, the trendsobtained in finite element analysis matched the trends obtained by tensile shear test-ing. The use of such “functionally gradient” adhesives was later adapted by other re-searchers such as Pires et al. (2003, 2006), You et al. (2005), and Temiz (2006) underthe name “bi-adhesive”, and by Hu and Huang (2005), da Silva and Adams (2006)and Lei et al. (2008) under the names “mixed-adhesive” or “mixed-resin”.

4.6 Adhesively Bonded Joint Composite, the Conceptof the Interphase

There is strong interrelationship between the properties of the adhesive and the sub-strates in shaping the final joint behavior. A more accurate prediction of the jointbehavior can be obtained by including a distinct “interphase” region in modelingthe overall joint behavior. An early discussion of this concept was provided bySharpe (1972, 1974). This concept becomes especially relevant as efficient meth-ods for inducing mechanically distinct surface topography and chemically distinctinterlayers emerge. The use of excimer pulse lasers for surface treatment and to-pography inducement on metal adherends, shown by Sancaktar et al. (1995-a), cureand post-cure of interphase regions around carbon fibers embedded in epoxy matrixvia resistive electric heating by the fiber, shown by Sancaktar and Ma (1992-a), mi-crowave curing of carbon-epoxy composites, which produce “interfacial strengthshigher than that achievable in thermally cured system”, as shown by Agrawal andDrzal (1989), and the production of a transcrystalline region in crystalline polymersupon solidifying when in contact with a substrate, as shown by Hata et al. (1994),all provide examples of current methods used to create distinct interphases. Anothermain reason for the presence of an interphase region is the chemical primer (sizing)applied to the adherend surfaces to improve adhesion. The diffusion of this sizinginto the adhesive matrix can create a concentration gradient. There are other pos-sible mechanisms which could lead to an interphase region. Some examples are:possible polymeric diffusion into the substrate during the adhesion process, selec-tive adsorption of one or more of the components in the matrix before curing andthe presence of free volume differences between the bulk polymer and the polymernear the substrate-matrix boundary, possibly created by thermal stresses. Figure 4.7shows different “interphases” between an epoxy-based, approximately 80%wt silverflake filled electrically conductive adhesive and different substrates.

Adequate analysis and understanding of the interphase between an adherendand an adhesive is critical for design of efficient bonded structures and compos-ite materials. In adhesion science and technology circles it is a well acceptedfact that the mechanical properties of the (polymeric) adhesive material is al-tered in regions close to the adherend due to the adhesion process. For example,Knollman and Hartog (1985) reported approximately 10% reduction in the adhe-sive bulk shear modulus, in an interphase region comprising approximately 20% of

112 E. Sancaktar

Electrically Conductive Adhesive

Stainless Steel Substrate

Piezoelectric Device

Gold Layer

Fig. 4.7 Illustration of different “interphases” between an epoxy-based, approximately 80 %wtsilver flake filled electrically conductive adhesive and different substrates. The flake size isapproximately 7μm

the bulk adhesive thickness. Consequently, a satisfactory mechanical analysis on theadherend-adhesive interaction should include a finite thickness interphase with itsown material properties rather than assuming an interface with unknown propertiesor very high strength and negligible thickness. Modeling of the adhesive-adherendinterphase becomes especially relevant in understanding and design of compositematerials with “tailored” interphases which increase the toughness of the compositematerial.

Using the case of a carbon fiber embedded in epoxy matrix, Sancaktar andZhang (1990) and Sancaktar et al. (1992-b) reported that nonlinear viscoelastic be-havior of the matrix and the interphase may have a profound effect on the mecha-nism of stress transfer.

Sancaktar’s analytical model involves a cylindrical interphase surrounding anelastic fiber of finite length. The interphase region has nonlinear viscoelastic mate-rial properties. The region surrounding the interphase (which can be assumed to bethe bulk of the adhesive matrix) is also assigned nonlinear viscoelastic properties,which are different from those of the interphase. The nonlinear viscoelastic materialbehavior of the interphase zone and the matrix is represented using a stress enhancedcreep. Different interphase diameter values and different material properties for theinterphase and the matrix are used to determine their effects on the interfacial shearstress. Results of this analysis reveal that depending on the relative magnitudes ofinterphase and matrix viscoelastic material properties, it is possible to have stressreductions or increases along the fiber and, hence, the critical fiber length obtainedwill also vary accordingly, as illustrated in Fig. 4.8. This result reveals that not onlythe quality of adhesion but also the material properties of the bulk matrix and the in-terphase, all of which can be affected by cure conditions, can affect the critical fiberlength, a concept introduced by Kelly and Tyson (1965) to gage the fiber-matrixinterfacial strength.

The analytical and experimental results presented so far reveal the possibility formutual reduction between rate, temperature, and interphase thickness, as shown bySancaktar et al. (1992-b). If we assume that application of fiber sizing results inthicker interphases, and that the cross-head rate, v, can be related to the shear strainrate, dγ/dt, and the interphase thickness, δ , with the relation:


Fig. 4.8 The variation of the maximum shear stress with time and matrix transient creep compli-ance based on a nonlinear viscoelastic stress analysis of a cylindrical interphase zone for maximumtransferred load. The nonlinear analysis was performed using Eq. (4.31) with a stress-dependentshift factor. D0 and D1 are the interphase instantaneous and transient creep compliances, δ is theinterphase thickness, and the subscripts m and f indicate matrix and fiber, respectively (Sancaktarand Zhang, 1990)

v = f

(dγdt

)g(δ ) (4.43)

based on classic shear lag analysis (also based on definition of shear strain), wecan then argue that for a fixed strain rate (function) the effect of increased cross-head rate can be induced by increasing the interphase thickness (i.e., applicationof fiber sizing). Indeed, the experimental results reveal shorter fragment lengths forincreased cross-head rate and/or presence of fiber sizing. Based on this premise wecan write the following superposition relations for the interfacial strength:

τc(T,v,δ ) = τc(To,aT v,δ ) (4.44)

τc(T,v,δ ) = τc(To,v,δ/aT ) (4.45)

andτc(T,v,δ ) = τc(T,v/aδ ,δo) (4.46)

In order to represent the composite behavior of the adhesive/adherend interphase,where the polymeric adhesive is mechanically interlocked to the surface projections

114 E. Sancaktar

of a metal adherend, Sancaktar (1996) proposes a “smeared” modulus zone. Al-though deformation within the bulk adhesive is likely viscoelastic, simple elasticrelationships probably suffice, to describe deformation within the topographic sur-face region. Therefore, simple elastic relationships are used to relate strain gradientwithin those regions to topography.

The equivalent elastic modulus, GIP, for the mechanically interlocked interphaseis calculated by assuming an “isostress” condition for the interphase, illustrated inFig. 4.9. In other words, we assume that the adhesive and the adherend surface(projections) are subjected to the same level of shear stress, τIP, at the interphase,while the composite interfacial shear strain, γIP, is a volumetric weighted averageof the adhesive and adherend strains, i.e.

γIP = (Vf A)(γA)+(Vf S)(γS) (4.47)

In Eq. (4.47) γA and γS are the adhesive and substrate shear strains at the interphaseand Vf A and Vf S are the volume fractions of the adhesive and the substrate surface(projections) at the interphase. Assuming that the adhesive essentially fills the voids,the adhesive volume fraction can be equated to the sum of the void volume fractionsof each lamina.

With the assumption that the deformations in both the adhesive and the adherendare elastic, we can define the interphase shear modulus as:

GIP = τIP/γIP (4.48)

Fig. 4.9 The state of isostress shear at the interphase


Using Eqs. (4.47) and (4.48) and Hooke’s law to represent adhesive and adherend’smaterial behavior, we obtain the following expression for the interphase shearmodulus:

GIP = GS GA

Vf A GS +Vf S GA= GS GA

Vf S(GS−GA)+GA(4.49)

where GA and GS are the elastic moduli for the adhesive and the adherend, respec-tively.

We now assume that failure at the interphase is more likely with higher strain gra-dients, dγIP/dz. Of course, in the limit, a strain discontinuity defines failure basedon mechanics principles. Based on this premise, we can derive a mathematical ex-pression to gage the possibility of failure using the shear moduli of the adhesiveand the adherend along with the adhesive volume fraction, Vf A, at the interphase.For this purpose, we first note that differentiation of Eq. (4.48) along the interphasethickness direction (z-direction) results in

dGIP

dz= −τIP γ−2

IP

(dγIP

dz

)(4.50)

An equivalent expression can be obtained by differentiating Eq. (4.49) with respectto z:

dGIP

dz= − GS GA(GS−GA)

[Vf A(GS−GA)+GA ]2

(dV f A

dz

)(4.51)

Equating Eqs. (4.50) and (4.51) we can obtain an expression relating the shear straingradient, dγIP/dz at the interphase to the volume fraction, Vf A, and the differentialvolume fraction dVf A/dz, of the adherend voided during surface preparation. Bothparameters are experimentally determinable, as discussed by Sancaktar (1996).

Variation in mechanical properties of polymer transition layers was also consid-ered by Silberman et al. (1991). An approach involving macroscopic characteriza-tion of the interphase regions in polymer based composite systems by integratedmoduli over the thickness is given by Cardon et al. (1993). A three-phase, axisym-metric elasticity solution to predict the sensitivity of the stress state to the interphasematerial properties and temperature is presented by Sottos et al. (1994).

In order to overcome the difficulties arising from the geometrical complexityof bonded joints, a number of studies have been carried out by Adams and Peppiatt(1974), Cooper and Sawer (1979), Hashim et al. (1990), Groth and Nordlund (1991),Dorn and Weiping (1993) and Papini et al. (1994), using the finite element method.Lap and butt joints have been analyzed extensively due to their common usage inengineering applications.

Interfaces usually constitute a weak link in the chain of load transfer in bondedjoints. Also, the discontinuity of the material properties causes abrupt changes instress distribution, as well as causing stress singularities at the edges of the inter-faces. It is very desirable to optimize the substrate surface topography at the inter-faces to maximize the load bearing capacity of bonded joints, and to improve theirdeformational characteristics.

116 E. Sancaktar

During the second half of the twentieth century a considerable amount of researchwork has been published related to the interface stress distributions, including Bogy(1968, 1970, 1971-a) and Bogy and Wang (1971-b), Renton and Vinson (1977),Mittal and McKenzie (1980), Sih (1980), Okajima and Sinclair (1986), Suhir (1988),Sancaktar (1987-b, 1996), Groth (1988), Adams (1989), Bigwood and Crocombe(1989), Sawa et al. (1989, 1995), Ikegami et al. (1990), Temma et al. (1990,1991), Nakano et al. (1991), Carpenter (1991), Gao and Mura (1991), Keisler andLataillade (1995), Apalak et al. (1995), Maugis (1996) and Ganghoffer and Schultz(1996). These studies focused on flat surfaces between two different materials ex-cept when the effects of surface roughness were investigated. In general, stress dis-tributions, stress concentrations, singularities, and surface roughness effects werestudied. These studies, however, were not extended to the development of a generalmethodology to include surface topography effects for varying configurations whichcan be represented by different mathematical functions.

An investigation by Sancaktar and Narayan (1999-b) on adhesively bonded scarfjoints revealed that the angle between the bonded interface and the loading directionhad a significant influence on the stress distributions at the interface. Their analysisrevealed that the distributions of stresses became more uniform along the adhesivelayer when scarf angles varying from lap joint to butt joint in an increasing orderof scarf angle were considered. The transverse and shear stress distributions werein a decreasing order for scarf angle increments of 15◦ between 30◦ and 90◦, whilea reverse trend was observed for the normal stress distributions. The peak normalstress was minimum for the 30◦ scarf joint; on the other hand, the magnitudes oftransverse and shear stresses were minimum when a butt joint was used. Uniformlydistributed external shear loading on the joints led to higher stresses in lap joints,while 75◦ scarf joint had the lower values.

Sancaktar and Narayan (1999-b) stated that an attention to the type of stress(i.e., normal, transverse or shear) provided information relevant to the type of fail-ure (i.e., brittle or ductile type failures). The authors also introduced newly defineddesign parameters: stress times substrate volume, stress divided by substrate vol-ume, and stress (strain) gradients along and across the adhesive joint, as a generalmethodology to compare adhesive joints for design optimization.

In highly deformable adhesives, most of which are elastomeric in nature, suchas those used in pressure sensitive tape and tack applications, a high amount ofdeformation results in cavitation of the material under dilatation. High dilatation,which is a 3-D concept, may lead to cavitation in localized regions of the adhesivelayer starting at a microscopic scale.

It is also desirable to know the magnitudes of shear and normal stresses individu-ally since some adhesives are weak in shear loading (ductile adhesives) while othersare weak in normal stress loading (brittle adhesives).

High levels of stresses result in crack propagation in brittle adhesives and cavita-tion induced failures in deformable adhesives. Furthermore, not only the magnitudesbut also gradients of stresses and strains become an important issue in optimizingthe joint strength. Large strain gradients at adhesive-substrate interfaces may resultin bond failure.


The magnitude of strain gradients depends on the volume fractions of the adhe-sive and substrate materials within the effective bond region defined by the geometryof the surface topography. The effect of bonded substrate volume, corresponding tothe volume of substrate projections beyond a common base-line on the substrate, onthe strain gradients across the adhesive layer can be found by relating the volumefraction gradients to the strain gradients. For this purpose, the composite behaviorof the adhesive/adherend interphase, where the polymeric adhesive is mechanicallyinterlocked into the surface projections of an adherend can be utilized to determinean equivalent composite modulus, as suggested by Sancaktar (1996) and discussedabove.

We note that mechanical adhesion depends on surface topography, which can beconsidered a collection of many geometrical forms. Therefore, mechanical adhesiondepends on the stress states of different adhesive joint geometries on the scale of thesurface topography, which may include many lap, butt and scarf joints in the “in-terphase” region. To address this issue, Ma et al. (2001) compared the stress distri-butions in adhesive joints as functions of varying geometrical interfaces describedmathematically in polynomial or other functional forms, as well as the materialproperties of the adhesive and the adherend or two different substrates joined by aninfinitesimally thin adhesive layer. For this purpose, a mathematical procedure wasdeveloped to utilize the complementary energy method, by minimization, in order toobtain an approximate analytical solution to the 3-D stress distributions in bondedinterfaces of dissimilar materials. In order to incorporate the effects of surface to-pography, the interface was expressed as a general surface in Cartesian coordinates,i.e., F(x,y,z) = 0, and the results verified by comparison with Finite Element Anal-ysis (FEA). Their methodology involved the following steps: 1. Choose the appro-priate stress functions, Φmn, to satisfy stress boundary conditions and utilize equi-librium conditions to relate the stress functions to the stress distribution functions;2. Use stress interface boundary conditions to reduce the unknown variables in thestress distribution functions obtained; 3. Define a function Π as the sum of comple-

mentary energy I and the penalty function Ry= f (x,z)

∑i=1..3

(uiA −ui

B)2. R is chosen such

that it is the largest positive number which causes the total system complementaryenergy to change by only 0.1%; 4. Use the remaining unknown variables to mini-mize the function Π formed in step 3, using the Newton quadratic method, and solveall the unknown variables to obtain the complete stress distribution functions.

The continuity of displacements was enforced at the interface area, approxi-mately, using the fundamental theorem for surface theory of differential geometry,described by Struik (1950), instead of directly using displacements. This way, theintegrations of displacements were avoided, and the stress distributions could becalculated more accurately.

For example, for a flat interface (on x-z plane), such as a butt joint, they utilizedthe following stress functions:

ΦAxx = (z2 −1/4)2(y−1/2)2 f1(x,y,z) (4.52)

118 E. Sancaktar

ΦAyy = (z2 −1/4)2(x2 −1/4)2 f2(x,y,z) (4.53)

ΦAzz = (x2 −1/4)2(y−1/2)2 f3(x,y,z) (4.54)

ΦAxy = (x2 −1/4)(z2 −1/4)2(y−1/2) f4(x,y,z) (4.55)

ΦAxz = (x2 −1/4)(z2 −1/4)(y−1/2)2 f5(x,y,z) (4.56)

ΦAyz = (x2 −1/4)2(z2 −1/4)(y−1/2) f6(x,y,z) (4.57)

ΦBxx = (z2 −1/4)2(y+1/2)2 f7(x,y,z) (4.58)

ΦByy = (z2 −1/4)2(x2 −1/4)2 f8(x,y,z) (4.59)

ΦBzz = (x2 −1/4)2(y+1/2)2 f9(x,y,z) (4.60)

ΦBxy = (x2 −1/4)(z2 −1/4)2(y+1/2) f10(x,y,z) (4.61)

ΦBxz = (x2 −1/4)(z2 −1/4)(y+1/2)2 f11(x,y,z) (4.62)

ΦBzz = (x2 −1/4)2(z2 −1/4)(y+1/2) f12(x,y,z). (4.63)

Functions fn(x,y,z) are of the type: gn1 sinh(x)sinh(z) + gn2 sinh(x)cosh(z) +gn3 cosh(x)sinh(z)+gn4 cosh(x)cosh(z) where, gni are polynomials of hni1 +hni2x+hni3z, and, hni1, hni2 and hni3 are polynomials of ani j1 + . . .ani jmym−1. The selectionof sinhx, coshx, sinhz, coshz, and y facilitated numerical calculations.

A typical distribution for the normal stress, σyy, common to both substrates Aand B at the bonded interface of a three-dimensional flat interface between thesesubstrates is shown in Fig. 4.10. This stress is calculated by the novel mathematicalprocedure based on differential geometry and the penalty function as described byMa et al. (2001), and summarized above. Figure 4.11 shows the transverse stresses

Z

YA

B

Interface

Fig. 4.10 A three-dimensional flat interface between two substrates A and B, loaded as shown andanalyzed by FEA with the mesh pattern shown (left). A typical distribution for the normal stress,σyy, common to both substrates A and B at the bonded interface is shown on the right, as calcu-lated by a novel mathematical procedure based on differential geometry and the penalty function(Ma et al., 2001)


Fig. 4.11 The transverse stresses σxx, on substrates A (left) and B (right) at the bonded interface, il-lustrating the discontinuity of this stress across the interface, as sometimes called the “stress jump”,calculated by a novel mathematical procedure based on differential geometry and the penalty func-tion (Ma et al., 2001)

σxx, on substrates A and B at the bonded interface, illustrating the discontinuity ofthis stress across the interface, as sometimes called the “stress jump”. Again, thesestresses are calculated by the novel mathematical procedure based on differentialgeometry and the penalty function as described by Ma et al. (2001), and summarizedabove.

4.7 Damage Models

The incremental theory of plasticity assumes that the strains depend on the entirehistory of loading. The total increments of deviatoric strain Δεi j is composed of aviscoplastic component

Δεi jP = dλSi j (4.64)

where dλ is a variable depending on the loading history, and an elastic component

dεi jE = (dSi j/dt)/2G (4.65)

so thatdεi j/dt = dεi j

E/dt +dεi jP/dt (4.66)

Obviously, a yield criterion is needed to determine the actual magnitudes of plas-tic strain increments. This theory was first proposed by Prandtl (1925) and Reuss(1930).

Several constitutive forms were proposed to describe the viscoplastic term ofEq. (4.66). Among these Hohenemser et al. (1932), Sokolovsky (1948), Malvern(1951) and Perzyna (1963) proposed essentially the same idea. It involved the as-sumption that the viscoplastic component be a function of the over-stress above the

120 E. Sancaktar

yield point. On the basis of his experimental observations, Sokolovsky determinedthat the over-stress should be defined above the elastic limit, as a function F(σ −θ),so that

ε = σ/E, σ ≤ θ (4.67-1)

ε = (σ/E)+F(σ −θ)/E, σ > θ . (4.67-2)

The over-stress idea can be incorporated into the viscoelastic mechanical models bymeans of adding sliding elements to describe yield or termination points. With suchan application, equations containing viscosity terms are obtained in a form simi-lar to (4.67). For example Brinson et al. (1975), Renieri et al. (1976), Sancaktarand Brinson (1979, 1980), Sancaktar (1981), Sancaktar and Padgilwar (1982),Sancaktar et al. (1984) and Sancaktar and Schenck (1985-a) used the Modified Bing-ham model developed by Brinson (1974) to describe the shear and tensile materialbehavior of structural adhesives in the bulk and bonded forms:

σ = Eε σ ≤ θ (4.68-1)

dε/dt = (dσ/dt)+(σ −θ)/μ θ < σ ≤ Y (4.68-2)

σ = Y σ ≥ Y. (4.68-3)

The model describes linear elastic behavior below the elastic limit θ , linear vis-coelastic behavior between θ , and the maximum stress Y and perfectly plastic be-havior above Y . When the sliding element is activated at σ = θ , the model becomesessentially a Maxwell element, as illustrated in Fig. 4.12.

A similar application was also reported first by Chase and Goldsmith (1974) andlater by Sancaktar (1981), Sancaktar and Padgilwar (1982), Sancaktar et al. (1984),and Sancaktar and Schenck (1985-a), in modifying a three parameter solid modelwith a sliding element to describe the mechanical behavior of polymeric materialsand adhesives.

An isothermal theory of separation in (thermoplastic or pressure sensitive)polymer-solid adhering systems based on drawing of filaments is given by Goodand Gupta (1988).

Ganghoffer and Schultz (1996) stated that the influence of damage on the me-chanical behavior is specified through the concept of strain equivalence. They pro-pose that the constitutive law for the damaged material is given by that of the virginmaterial if the stress tensor σ is replaced by the effective stress tensor σ/(1−d) withthe term, d, representing the extent of damage between the virgin state, d = 0 andcomplete failure, d = 1. The damage parameter is further developed in Chapter 6.

Another example of empirical damage modeling in joints bonded using silverflake filled electrically conductive adhesives was given by Gomatam and Sancaktar(2006-a,b), who introduced a comprehensive fatigue life predictive model for singlelap joints subjected to variable loading. For the purpose of formulating a fatiguelife predictive model, a linear relationship is established first between the maxi-mum cyclic load Pmax (or maximum principal stress, σmax, as suggested by Go-matam, 2002, and Gomatam and Sancaktar, 2004), and the number of cycles N:


Modified Bingham Model

Strain,

V.E.

1

E

Plastic

Elastic

σ

σ = Y

σ

σ = θ

σ(t)

σ = Y

σ = θ

StressConstant Strain Rate (C.S.R.)

θ= Elastic Limit Stress

Maxwell behavior: when θ ≤ σ ≤ YPerfectly Plastic behavior: when σ ≥ Y

µ

E Y

( ) ( ))0t t Tt R 1 e

− −−⎡ ⎤= + ⋅ −⎢ ⎥⎣ ⎦

( )t E= ⋅

θ < (t) ≤ Y

(t) ≤

Strain,

Maxwell

1

E

Modified Bingham

Stress

Axis Shift forMaxwell Model

θ

εσ

σ

σ

σμθ

θ

Fig. 4.12 The Modified Bingham model and its comparison with the Maxwell model

Pmax = C′ +m′N (4.69)

where C′ and m′ are, respectively, the intercept and slope for this linear relation.For the purpose of the analytical fatigue life predictive model, two separate ana-

lytical equations were proposed depending on the intensity (severe or moderate) ofthe load-ratio first applied, at a maximum cyclic load level. The total fatigue life,NTotal, under severe-to-moderate condition was predicted by the relation,

NTotal = ns +ΔCmM

− (Cs −P)(

1ms

− 1mM

)(1− ns

Ns

)(4.70)

The total fatigue life under moderate-to-severe condition, on the other hand, waspredicted by,

NTotal = nM +Ns

(1− nM

NM

)(4.71)

In Eqs. (4.70) and (4.71) we have,

ΔC = CM −Cs

where CM and Cs are the intercepts, and mM and ms are the slopes for the lin-ear relationships between the maximum cyclic load Pmax (or maximum principalstress, σmax), and the number of cycles N under moderate and severe conditions,

122 E. Sancaktar

respectively. nM and ns are the number of cycles applied under moderate and severeconditions, respectively.

As illustrated in Fig. 4.13, in going from moderate to severe conditions, we sim-ply add to the number of cycles under moderate conditions (nM), the predicted re-maining life for the sample under the severe condition, Ns[1− (nM/NM)], based onexperimental predictions without making any slope corrections, Eq. (4.71). In go-ing from severe to moderate condition, however, we also make a slope correction,[Cs − P]{(1/ms)− (1/mM)}[1 − (ns/Ns)], to make the severe and moderate P-Ncurves parallel, and shift the intercepts to add the remaining life, ΔC/mM , to that al-ready used up under severe condition (i.e. ns). These particular methodologies werechosen based on experimental observations. Utilizing the above equations, the totalfatigue life of the joints subjected to varying loading conditions, namely severe-to-moderate, and moderate-to-severe were computed for a set of test cases, in additionto experimentally determined values, for the purpose of model validation. Com-parison between the experimental, and predicted values validated the accuracy, andefficiency of the proposed model.

In order to render the model usable with different stress components (σxx, σyy,σzz, τxy, τyz,τxz) instead of the maximum cyclic load values, Gomatam (2002) uti-lized the maximum and minimum principal and von Mises stresses obtained fromnon-linear elasto-plastic finite element analyses to calculate the ratios of peak stressvalues obtained at corresponding maximum load levels. These stresses were thencompared to the intercept ratios obtained from the P-N curves at given environmen-tal conditions. This comparison procedure revealed the maximum principal stress asthe appropriate parameter to be used in the form of S-N curves, i.e.,

σmax = C +mN (4.72)

S P

Cs

Cm

(M)

(S)

NsN

Δ (C)/m m

ns

(1 – ns/N s)

1) Severe to Moderate: NTotal = ns + (Δ(C)/mm) – (Cs–P) ((1/mm)) (1–(ns/Ns))

2) Moderate to Severe: NTotal = nm + Ns(1–(nm/Nm))

Δ(C) = Cm – Cs

mm, ms = slopes; nm, ns = number of cycles,under moderate, m, or severe, s, conditions.

Fig. 4.13 Illustration of the cumulative fatigue damage model (Gomatam and Sancaktar, 2006-a)


where C and m are, respectively, the intercept and slope values based on maximumprincipal stress. The mathematical basis for the maximum principal stress to replacethe maximum cyclic load in Eq. (4.69) was illustrated by performing fatigue testsunder three different maximum load, Pmax, conditions, and considering the follow-ing relations:

C′3

C′1

=C3

C1(4.73)

andC′

2

C′1

=C2

C1(4.74)

where C′1, C′

2, C′3 relate to Eq. (4.69), and C1, C2, C3 relate to Eq. (4.72), as illustrated

in Fig. 4.14.In order to account for environmental effects, such as elevated temperature and

humidity, a superposition method was applied for fatigue life prediction by shiftingthe slopes and the intercepts between different environmental conditions. For thispurpose, design charts were formulated based on experimental data. These designcharts comprise of two parts, one for shifting the slope, and the other for shiftingthe intercept between different environmental conditions. Once the slopes and theintercepts are determined, Eqs. (4.69) and (4.72) would yield the fatigue life of thejoint at the new environmental condition, as illustrated in Fig. 4.15 and shown byGomatam and Sancaktar (2006-b).

The proposed model was also extended to bonded cases with varying stress states.For this purpose, a non-linear elasto-plastic finite element analysis (FEA) for ambi-ent condition, and a non-linear thermo-elasto-plastic FEA for elevated temperatureconditions were performed using two different lap joint geometries and boundaryconditions to represent two distinct states of stress. Utilizing the FEA results, rela-tions between the stress components, slopes, and intercepts were established. Theintercepts were found to be inversely proportional to the maximum stresses, andthe slopes directly proportional to a shear stress component. Thus, by using theserelations along with a knowledge of the stress states for only two joint configura-tions, the slope, and intercept for any other lap joint with unknown P-N (σ −N)

C’3/C’3 = C3/C1

C’3

C’2

C’1Pmax(N)

Pmax = C’ + m’N

N

C3

C2

C1

σmax(MPa)

σmax = C + mN

N

Fig. 4.14 The similarity in P-N and σ −N relations (Gomatam and Sancaktar, 2006-a)

124 E. Sancaktar

StressRatio

Slope

RT50°C90°CHum

Stress Ratio

Intercept

RT50°C90°CHum

Fig. 4.15 The shifting mechanism implemented for the purpose of predicting fatigue life of jointssubject to a stress ratio (R), and maximum cyclic load of Pmax, from one test environment (28◦C,or 50◦C, or 90◦C) to another at the same stress ratio, and maximum cyclic load (Gomatam andSancaktar, 2006-a)

behavior could be computed, from which the fatigue life of the joint could be pre-dicted, as shown by Gomatam and Sancaktar (2006-b). Fatigue and environmentaldegradation are further discussed in Chaps. 7 and 8.

4.8 The Effects of Cure and Processing Conditionson the Mechanical Behavior

Cure and processing conditions have strong effects on the resulting mechanicalproperties of structural adhesives in the bulk and bonded forms.

Variation of various mechanical properties of an epoxy adhesive with or withouta carrier cloth as functions of cure temperature, time, and cool down conditions,and their effects on bulk tensile properties were studied by Sancaktar et al. (1983),Jozavi and Sancaktar (1989-a, b, c) and Sancaktar (1995-b), with the work by Jozaviand Sancaktar (1989-c) providing information on the effects of cure conditions onthe relaxation behavior of bulk thermosetting adhesives. A comparison of the degreeof cure with bulk strength was given by Jozavi and Sancaktar (1988). The effectsof the cure parameters on the stress whitening behavior were given by Jozavi andSancaktar (1989-a, b). The effects on the bonded joint behavior were discussed byShaw and Tod (1989), Sancaktar and Ma (1992-a) and Turgut and Sancaktar (1992).In general, mechanical properties are improved with increases in cure temperatureand time and with slow cool down. Such increases, however, have an upper limit,and further increases in cure parameter magnitudes lead to deterioration in adhe-sive mechanical properties. Consequently, in general, the variation of mechanicalproperties with the cure parameters are bell shaped functions.

Additional information on the effects of cure conditions were provided bySinclair (1992).


Matsui (1990) provides some empirical relations for the variation of shear mod-ulus and strength of joints bonded using various structural adhesives.

In the case of particle filled adhesive systems, such as electrically conductive ad-hesives, the viscosity, μ , during processing and cure is a function of filler volumefraction, Φ, shear rate, dγ/dt, and resin conversion level, α . The development ofviscosity from a chemorheological point of view is important as it relates to thewetting and diffusive characteristics of the adhesive during its cure. These charac-teristics ultimately determine the viability of the interphase and, therefore, the over-all adhesive bond. A comprehensive model may be represented by combining thefollowing models: the power law (shear rate effect), described by Wildemuth andWilliams (1985), Castro-Macosko model (conversion effect), described by Castroand Macosko (1980, 1982) and Castro et al. (1984), and Liu model (filler volumefraction effect), described by Liu (2000). Such a combined model takes the form:

μ (φ ,dγ/dt,α) = A · (φm −φ)−2 · (dγ/dt)C0+C1·α ·[αgel

/(αgel −α

)]D+E·α

(4.75)where, A, C0, C1, D and E are model constants that can be determined by multivari-able nonlinear regression analysis of isothermal data. The maximum filler volumefraction, Φm and conversion at gel point, αgel, values are usually known for typicalthermosets.

Zhou and Sancaktar (2008) incorporated the isothermal temperature effects toobtain a general-form comprehensive model in the form:

μ (T,φ ,dγ/dt,α) = A · exp(B/T ) · (φm −φ)−2 · (dγ/dt)C0+C1·α

·[αgel

/(αgel −α

)]D+E·α(4.76)

where, B is the activation temperature and other coefficients are same as in Eq. (4.75).In order to model realistic adhesive processing, usually with a broad tempera-

ture range, nonisothermal tests usually need to be undertaken. The comprehensiveisothermal chemoviscosity model, based on Eq. (4.76), can then be extended tononisothermal temperature cure cycle through nonlinear regression analysis of non-isothermal data.

In fact, nonisothermal temperature cure is a different process from isothermalcure. The reaction kinetics, total reaction order, and even the reaction energy forepoxy systems may not be constant and same, but process-dependent. Therefore,modifications need to be made to reflect the effects of such a difference.

An improvement in the model predictability is expected by allowing parame-ters D and E in Eq. (4.76) to change with temperature during nonisothermal cure.Furthermore, both of these parameters also show some variation with the volumefraction (Φ) and shear rate (dγ/dt). A modified comprehensive model was thus pro-posed as follows (Zhou and Sancaktar 2008):

μ (T,φ ,dγ/dt,α) = A · exp(B

/T

)· (φm −φ)−2 · (dγ/dt)C0+C1·α

·[αgel

/(αgel −α

)]D∗+E∗·α

126 E. Sancaktar

where:

D∗ = D0 +D1 · (dγ/dt)+D2 ·φ +D3 ·TE∗ = E0 +E1 · (dγ/dt)+E2 ·φ +E3 ·T

(4.77)

with Di and Ei values to be determined by data fitting.

4.9 Concluding Remarks

Interfaces usually constitute a weak link in the chain of load transfer in bondedjoints. Also, the discontinuity of the material properties causes abrupt changes instress distribution, as well as causing stress singularities at the edges of the inter-faces. It is very desirable to optimize the substrate surface topography at the inter-faces to maximize the load bearing capacity of bonded joints, and to improve theirdeformational characteristics. In most engineering joints the adherend surfaces havedistinct topographies, which result in a collection of miniature joints in micron, andeven nano scale when bonded adhesively. If the interphase is not considered as a dis-crete collection of individual chemical bonds, the methods of continuum mechanicscan still be applied to this collection of miniature joints by assuming continuous,or a combination of continuous/discontinuous interphase zones. Thus, the displace-ment at the interphase need not be continuous at every location. The analysis ofa miniature joint contributing to the overall adhesion in a macro joint can be per-formed in a fashion similar to that for the macro joint itself, which is usually studiedwith the employment of the methods of elasticity, viscoelasticity, plasticity, fracture,damage, and/or failure mechanics.

References

Adams RD, Peppiatt NA (1974) J. Strain Anal. 9 (3), p 185Adams RD (1989) J. Adhes. 30, p 219Agarwal BD, Broutman LJ (1990) Analysis and Performance of Fiber Composites, 2nd ed., John

Wiley & Sons, Inc., New York, p 131Agrawal R, Drzal LT (1989) J. Adhes. 29, p 63Ahlonis JJ, MacKnight WJ, Shen M (1972) Introduction to Polymer Viscoelasticity, Wiley-

Interscience, New YorkApalak MK, Davies R, Apalak ZG (1995) J. Adhes. Sci. Technol. 9, p 267Ashton JE, Halpin JC, Petit PH (1969) Primer on Composite Materials: Analysis Technomic Pub-

lication, Stamford, Connecticut, p 77Barsoum RS (1989) J. Adhes. 29, p 149Becher EB (1984) Viscoelastic Stress Analysis of Adhesively Bonded Joints Including Moisture

Diffusion, AFWAL TR-84-4057Bigwood DA, Crocombe AD (1989) Intl. J. of Adhes. Adhes. 9 (4), p 229Bogy DB (1968) J. Appl. Mech. 35, p 460Bogy DB (1970) Intl. J. Solids Struct. 6, p 1287


Bogy DB (1971-a) J. Appl. Mech. 38, p 377Bogy DB, Wang KC (1971-b) Intl. J. Solids Struct. 7, p 993Boller KH (1957) Tensile Stress Rupture and Creep Characteristics of Two Glass-Fabric-Base

Plastic Laminates, Forest Products Lab. Report, Madison, WisconsinBrinson HF (1974) Deformation and Fracture of High Polymers, Kausch H et al. (Eds.) Plenum

Press, New YorkBrinson HF, Renieri MP, Herakovich CT (1975) Fracture Mechanics of Composites, ASTM SPT

593, p 177Budiansky B (1965) J. Mech. Phys. Solids 13, p 223Cardon AH, De Welde WP, Van Hemelrijck D, Boulpaep F (1993) Proceedings of the 16th Annual

Meeting of the Adhesion Society, Boerio FJ (Ed.), The Adhesion Society, Blacksburg, Virginia,p 175

Carpenter W (1991) J. Adhes. 35, p 55Cartner JS, Griffith WI, Brinson HF (1978) The Viscoelastic Behavior of Composite Materials

for Automotive Applications, Virginia Polytechnic Institute and State University Report No.VPIE-78-15

Castro JM, Macosko CW (1980) SPE ANTEC Tech. Papers 26, p 434Castro JM, Macosko CW (1982) AIChE J. 28, p 250Castro JM, Macosko CW, Perry SJ (1984) Polym. Commun. 25, p 82Chalkley PD, Chiu WK (1993) Int. J. Adhes. Adhes. 13, p 237Chase KW, Goldsmith W (1974) Exp. Mech. 14, p 20Chow TS, Hermans JJ (1969) J. Compos. Mater. 3, p 382Chow TS (1977) J. Appl. Phys. 48, p 4072Chow TS (1978-a) J. Polymer Sci.: Polym. Phys. Ed. 16, p 959Chow TS (1978-b) J. Polymer Sci.: Polym. Phys. Ed. 16, p 967Cooper PA, Sawer JW (1979) A Critical Examination of Stresses in an Elastic Single Lap Joint,

NASA Tech. Paper 1507Darlington MW, Turner S (1978) Creep of Engineering Materials, Mechanical Engineering Publi-

cations, Londonda Silva LFM, Adams RD (2006) J. Adhes. Sci. Technol. 20, p 1705Davis JL (1964) J. Polym. Sci. Part A, V. 2, p 1311Dempsey JP (1995) J. Adhes. Sci. Technol. 9, p 253Dillard DA, Morris DH, Brinson HF (1982) Composite Materials: Testing and Design (Sixth Con-

ference), ASTM STP 787, Daniel IM (Ed.) ASTM, Philadelphia, p 357Dillard DA, Brinson HF (1983) Long-Term Behavior of Composites, ASTM STP 813, O’Brien

TK (Ed.) ASTM, Philadelphia, p 23Dillard DA, Hiel C (1985) Proceedings of the 1985 SEM Spring Conference on Experimental

Mechanics, p 142Dorn L, Weiping L (1993) Intl. J. Adhes. Adhes. 13 (1), p 21Eshelby JD (1957) Proc. R. Soc. A241, London, p 376Eshelby JD (1959) Proc. R. Soc. A252, London, p 561Ferry JD (1961) Viscoelastic Properties of Polymers, John Wiley and Sons, Inc., New YorkFindley WN, Peterson DB (1958) ASTM Proc. 58Findley WN, Lai JS (1967) Trans. Soc. Rheology 11, p 361Findley WN, Lai JS, Onaran K (1976) Creep and Relaxation of Nonlinear Viscoelastic Materials,

North Holland, New YorkFlugge W (1975) Viscoelasticity, 2nd ed., Springer-VerlagGanghoffer JF, Schultz J (1996) J. Adhes. Sci. Technol. 10 (9), p 775Gao Z, Mura T (1991) Intl. J. Eng. Sci. 29 (6), p 685Garton A, Haldankar GS, Mclean PD (1989) J. Adhes. 29, p 13Gittus J (1975) Creep, Viscoelasticity and Creep Fracture in Solids, John Wiley and Sons, Inc.,

New YorkGomatam RR (2002) Modeling Fatigue Behavior of Electronically Conductive Adhesives. Ph.D.

Dissertation, The University of Akron, Akron, Ohio

128 E. Sancaktar

Gomatam RR, Sancaktar E (2004) J. Adhes. Sci. Technol. 18, p 1833Gomatam RR, Sancaktar E (2006-a) J. Adhes. Sci. Technol. 20, p 69Gomatam RR, Sancaktar E (2006-b) J. Adhes. Sci. Technol. 20, p 87Good RJ, Gupta RK (1988) J. Adhes. 26, p 13Green AE, Adkins JE (1960) Large Elastic Deformations and Non-linear Continuum Mechanics,

Oxford University Press, New YorkGroth HL (1988) Int. J. Adhes. Adhes. 8, p 107Groth HL, Nordlund P (1991) Intl. J. Adhes. Adhes. 11 (4), p 204Hashim SA, Cowling MJ, Winkle IE (1990) Intl. J. Adhes. Adhes. 10 (3), p 139Hashin Z, Shtrikman S (1963) J. Mech. Phys. Solids 11, p 127Hata T, Gams M, Kojimer K (1965) Kobunshikagahn (Chem. High Polym., Japan) 22, p 160Hata T, Ohsaka K, Yamada T, Nakamue K, Shibata N, Matsumoto T (1994) J. Adhes. 45, p 125Hencky HZ (1924) Z. Angew. Math. Mech. 4, p 323Henriksen M (1984) Comput. Struct. 18, p 133Hermans JJ (1967) Proc. R. Acad. B70, Amsterdam, p 1Hiel C, Cardon AH, Brinson HF (1983) The Nonlinear Viscoelastic Response of Resin Ma-

trix Composite Laminates, Virginia Polytechnic Institute and State University Report No.VPI-E-83-6

Hill R (1956) The Mathematical Theory of Plasticity, Oxford University Press, LondonHill R (1964) J. Mech. Phys. Solids 12, p 199Hill R (1965) J. Mech. Phys. Solids 13, p 213Hilton HH (1975) Engineering Design for Plastics, Baer E (Ed.) Robert E. Krieger Publishing Co.

Huntington, New YorkHohenemser K, Prager W, Zeitschrift F (1932) Angew. Math. U. Mech. 12, p 216Hu XL, Huang PC (2005) Intl. J. Adhes. Adhes. 25 (4), p 296Ikegami K, Takeshita T, Matsuo K, Sugibayashi T (1990) Intl. J. Adhes. Adhes. 10 (3), p 199Jozavi H, Sancaktar E (1988) J. Adhes. 25, p 185Jozavi H, Sancaktar E (1989-a) J. Adhes. 27, p 143Jozavi H, Sancaktar E (1989-b) J. Adhes. 27, p 159Jozavi H, Sancaktar E (1989-c) J. Adhes. 29, p 233Kachanov LM (1971) Foundations of the Theory of Plasticity, North-Holland Publ. Co.,

AmsterdamKaelble DH (1964) J. Colloid Sci. 19, p 413Kaelble DH (1969) J. Adhes. l, p 102Keisler, C, Lataillade JL (1995) J. Adhes. Sci. Technol. 9, p 395Kelly A, Tyson WR (1965) Mech. Phys. Solids 13, p 329Kerner EH (1956) Proc. Phys. Soc. B69, p 808Keuner VH, Knauss WG, Chai H (1982) Exp. Mech. 22, p 75Knauss WC, Enri IJ (1981) Comput. Struct. 13, p 123Knollman GC, Hartog JJ (1985) J. Adhes. 17, p 251Koizumi S, Fuhue N, Matsunaga T (1970) Nihon J. Adhesion Soc. Japan 6, p 437Lei H, Pizzi A, Du GB (2008) J. Appl. Polym. Sci. 107, p 203Lipatov YS, Babich VF, Todosijchuk TT (1991) J. Adhes. 35, p 187Liu DM (2000) J. Mater. Sci. 35, p 5503Ludwik PG (1909) Elemente der Technologischen Mechanik, J. Springer, Berlin, p 9Ma W, Gomatam R, Fong R, Sancaktar E (2001) J. Adhes. Sci. Technol. 15, p 1533Malvern LE (1951) J. Appl. Mech. 18, p 203Matsui K (1990) Int. J. Adhes. Adhes. 10, p 277Maugis D (1996) J. Adhes. Sci. Technol. 10, p 161McLellan DL (1966) AIAA 5, No. 3McLellan DL (1969) Appl. Polym. Symp. 12, p 137Mittal RK, McKenzie HW (1980) J. Adhes. 11, p 91Nakano Y, Temma K, Sawa T (1991) J. Adhes. 34, p 137Nakao K (1969) Preprint, Symposium on Mechanisms of Adhesive Failure (Tokyo), p 47


Nonaker Y (1968) Nihon Setchahu Kyokaishi (J. Adhes. Soc. Japan) 4, p 207Odquist FKG (1954) Appl. Mech. Rev. 7, p 517Okajima M, Sinclair, GB (1986) SECTAM XIII Proceedings, The Southeastern Conference on

Theoretical and Applied Mechanics, Ranson WF, Biedenbech JM (Eds.) Vol. 2, EngineeringExtension Service, Auburn University, Alabama, p 367

Papini M, Fernlund G, Spelt JK (1994) Intl. J. Adhes. Adhes. 14 (1), p 5Peretz D, Weitsman Y (1982) J. Rheol. 26, p 245Perzyna P (1963) Q. Appl. Math. 20, p 321Pires I, Quintino L, Durodola JF, Beevers A (2003) Intl. J. Adhes. Adhes. 23 (3), p 215Pires I, Quintino L, Miranda RM (2006) J. Adhes. Sci. Technol. 20, p 19Prandtl L (1925) Proceedings of the lst International Congress on Applied Mechanics, Delft, Tech-

nisch Boekhandel en Dructerrg, p 43Ramberg W, Osgood WR (1943) Description of Stress-Strain Curves by Three Parameters, NACA

TN 902Ravi-Chandar K, Knauss WG (1984) Intl. J. Fract. 25, p 247Reedy Jr ED, Guess TR (1995) J. Adhes. Sci. Technol. 9, p 237Renieri MP, Herakovich CT, Brinson HF (1976) Rate and Time Dependent Behavior of Structural

Adhesives, Virginia Polytechnic Institute and State University, College of Engineering ReportNo. VPI-E-76-7

Renton WJ, Vinson JR (1977) J. Appl. Mech. 45 March, p 101Reuss E (1930) Z. Angew. Math. Mech. 10, p 266Rochefert MA, Brinson HF (1983) Nonlinear Viscoelastic Characterization of Structural Adhe-

sives, Virginia Polytechnic Institute and State University Report No. VPI-E-83-26Russel WB, Acrivos A (1972) Z. Angew, Math. Phys. 23, p 434Russel WB, Acrivos A (1973) Z. Angew, Math. Phys. 24, p 581Sancaktar E, Brinson HF (1979) Virginia Polytechnic Institute and State University, College of

Engineering Report No. VPI-E-79-33Sancaktar E, Brinson HF (1980) Adhesion and Adsorption of Polymers, Lee LH (Ed.) Vol. 12-A,

Polymer Science and Technology Series, Plenum Press, New York, p 279Sancaktar E (1981) Intl. J. Adhes. Adhes. 1, p 329Sancaktar E, Padgilwar S (1982) J. Mech. Des. 104, p 643Sancaktar E, Jozavi H, Klein RM (1983) J. Adhes. 15, p 241Sancaktar E, Schenck SC, Padgilwar S (1984) Ind. Eng. Chem. Prod. Res. Dev. 23, p 426Sancaktar E, Schenck SC (1985-a) Ind. Eng. Chem. Prod. Res. Dev. 24, p 257Sancaktar E (1985-b) Intl. J. Adhes. Adhes. 5, p 66Sancaktar E, Dembosky SK (1986) J. Adhes. 19, p 287Sancaktar E, Jozavi H, El-Mahallawy AH, Cenci NA (1987-a) Polym. Test. 7, p 39Sancaktar E (1987-b) Appl. Mech. Rev. 40 (10), p 1393Sancaktar E, Zhang P (1990) Transaction of ASME, J. Mech. Des. 112, p 605Sancaktar E (1991) J. Adhes. 34, p 211Sancaktar E, Ma W (1992-a) J. Adhes. 38, p 131Sancaktar E, Turgut A, Guo F (1992-b) J. Adhes. 38, p 91Sancaktar E, Beachtle, D (1993) J. Adhes. 42, p 65Sancaktar E, Babu SV, Zhang E, D’Couto GC, Lipshitz H (1995-a) J. Adhes. 50, p 103Sancaktar E (1995-b) J. Adhes. Sci. Technol. 9, p 119Sancaktar, E (1996) Appl. Mech. Rev. 49, p. S128Sancaktar E, Wei Y (1998) Mittal Festschrift on Adhesion Science and Technology, van Ooij WJ,

Anderson H (Eds.) VSP, Utrecht, The Netherlands, p 509Sancaktar E, Kumar S (1999-a) Reliability, Stress Analysis, and Failure Prevention Aspects

of Adhesive and Bolted Connections, Composite Materials and Components, ASME DE-Vol. 105, Sancaktar E (Ed.) p 113

Sancaktar E, Narayan K (1999-b) J. Adhes. 13, p 237Sancaktar E, Kumar S (2000) J. Adhes. Sci. Technol. 14, p 1265Sawa T, Temma K, Ishikama H (1989) J. Adhes. 31, p 33

130 E. Sancaktar

Sawa T, Temma K, Nishigaya T, Ishikawa H (1995) J. Adhes. Sci. Technol. 9, p 215Schapery RA (1966) Proceedings of the 5th U.S. National Congress of Applied Mechanics, ASME,

p 511Schapery RA (1969-a) Further Development of a Thermodynamic Constitutive Theory: Stress For-

mulation, Purdue University Report No. 69-2Schapery RA (1969-b) Poly. Eng. Sci. 9, p 295Sharon G, Dodiuk H, Kenig S (1989) J. Adhes. 31, p. 21Sharpe LH (1972) J. Adhes. 4, p 51Sharpe LH (1974) J. Adhes. 6, p 15Shaw SJ, Tod DA (1989) J. Adhes. 28, p 231Sih GC (1980) Polym. Eng. Sci. 20, p 977Silberman AB, Archireev VE, Vakula VL (1991) J. Adhes. 34, p 241Sinclair JW (1992) J. Adhes. 38, p 219Sokolovsky VV (1948) Prikladnaia Mathematiha I Mehhanika 12, p 261Sottos NR, Li L, Agrawal G (1994) J. Adhes. 45, p 105Struik DJ (1950) Classical Differential Geometry, Addison-Wesley, New YorkSuhir E (1988) J. Appl. Mech. 55, p 143Temiz S (2006) J. Adhes. Sci. Technol. 20, p 1547Temma K, Sawa T, Tsunoda Y (1990) J. Adhes. 10, p 294Temma K, Sawa T, Uchida H, Nakano, Y (1991) J. Adhes. 33, p 133Tobolsky AV (1960) Properties and Structure of Polymers, John Wiley and Sons, Inc., New YorkTurgut A, Sancaktar E (1992) J. Adhes. 38, p 111Tuttle ME, Brinson HF (1984) Accelerated Viscoelastic Characterization of T300/5208 Graphite-

Epoxy Laminates, Virginia Polytechnic Institute and State University Report No. VPI-E-9Tuttle ME, Brinson HF (1985) Proceedings of the 1985 SEM Spring Conference on Experimental

Mechanics, p 764Walpole LJ (1969) J. Mech. Phys. Solids 17, p 235Ward IM, Hadley, DW (1993) Mechanical Properties of Solid Polymers, John Wiley & Sons, Inc.,

New York, p 206Wei Y (1995) Electronically Conductive Adhesives: Conduction Mechanisms, Mechanical Behav-

ior and Durability, Ph.D. Dissertation, Clarkson University, Potsdam, New YorkWeitsman Y (1981) J. Adhes. 11, p 279Wildemuth CR, Williams MC (1985) Rheol. Acta. 24 (1), p 75Wu TT (1966) Int. J. Solids Struct. 2, p 1You M, Yan ZM, Zheng Y, Xiong WH, Zheng XL, Dai S (2005) Transactions of Nonferrous Metals

Soc. China 15 (Sp. Is. 3), p 344Zabora RF, Clinton WW, Bell JE (1971) Adhesive Property Phenomena and Test Techniques,

AFFDL-TR-71-68Zhang MJ, Straight MR, Brinson HF (1985) Proceedings of the 1985 SEM Spring Conference on

Experimental Mechanics, p 205Zhou J, Sancaktar, E (2008) Chemorheology of Epoxy/Nickel Conductive Adhesives During Pro-

cessing and Cure, J. Adhes. Sci. Technol. (accepted for publication)

Chapter 5Complex Joint Geometry

Andreas Ochsner, Lucas F.M. da Silva and Robert D. Adams

Abstract The finite element method is particularly suited to analyse complex jointgeometries. Adhesively bonded joints are increasingly being used in engineering ap-plications where the loading mode, the adherends shape and the material behaviourare extremely difficult to simulate with a closed form approach. A detailed descrip-tion of finite element studies concerning non-conventional adhesive joints is pre-sented in this chapter. Various types of joints, local geometrical features such as thespew fillet and adherend rounding, three dimensional analyses, hybrid joints and re-pair techniques are discussed. Special techniques to save computer power are alsotreated. It is shown that the finite element method offers unlimited possibilities forstress analysis but also presents some numerical problems at sharp edges.

5.1 Introduction

Stress analysis of bonded joints started 70 years ago with the well known Volkersenshear lag model (Volkersen, 1938). Since, various analytical models have been pro-posed that include the geometrical non-linearity (Goland and Reissner, 1944), theadhesive plasticity (Hart-Smith, 1973), fibre reinforced plastic adherends (Renton

Andreas OchsnerDepartment of Applied Mechanics, Faculty of Mechanical Engineering, Technical University ofMalaysia, 81310 UTM Skudai, Johor, Malaysia; University Centre for Mass and Thermal Trans-port in Engineering Materials, School of Engineering, The University of Newcastle, Callaghan,NSW, 2308, Australia, e-mail: [email protected]

Lucas F.M. da SilvaDepartamento de Engenharia Mecanica e Gestao Industrial, Faculdade de Engenharia, Universi-dade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal, e-mail: [email protected]

Robert D. AdamsDepartment of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK,e-mail: [email protected]

L.F.M. da Silva, A. Ochsner (eds.), Modeling of Adhesively Bonded Joints, 131c© Springer-Verlag Berlin Heidelberg 2008

132 A. Ochsner et al.

and Vinson, 1975), etc. However, most of the analytical models are for lap jointsof regular shape. Closed form analyses are very useful for an initial estimate ofthe stress distribution and are generally adequate for design purposes. For adhe-sive joints of complex shape, numerical techniques such as the finite element (FE)method are preferable. Simple lap joints have been modified because of the non-uniform stress distribution along the overlap. The load transfer and shear stress dis-tribution of various single lap joints (SLJ) are schematically represented in Fig. 5.1.It can be seen that there is a stress concentration at the ends of the overlap for thesingle lap joint with a square end. Modification of the joint end geometry with aspew fillet or a taper spreads the load transfer over a larger area and give a moreuniform shear stress distribution.

Adams and co-workers are among the first to have used the FE method foranalyzing the stresses in adhesive joints (Adams and Peppiatt, 1974; Crocombeand Adams, 1981; Harris and Adams, 1984; Adams et al., 1986; Adams andDavies, 2002). One of the first reasons for the use of the FE method was to assessthe influence of the spew fillet. The joint rotation and the adherends and adhesiveplasticity are other aspects that are easier to treat with a FE analysis. The study ofHarris and Adams (1984) is one of the first FE analyses taking into account thesethree aspects.

The increasing use of adhesive bonding with composite materials is another typ-ical example of the great advantage of using finite elements (Adams et al., 1986;Adams and Davies, 2002) since the anisotropic behaviour of these materials com-plicates drastically any analytical approach.

But it is when irregular geometries are involved that the FE method is an in-dispensable tool. Any type of geometry can be modelled, i.e., variations in theadherend shape, local adherend rounding, spew fillet, reinforcements, joints withrivets or bolts, etc. In addition, the simulation gives not only one, two or three stress

Fig. 5.1 Load transfer andshear stress distribution insingle lap joints

5 Complex Joint Geometry 133

components but all the stress components in all members of the joint. Three dimen-sional (3D) analyses for assessing width effects is another possibility offered by aFE analysis. As the model gets more complicated in terms of geometry and materialbehaviour, the computation time also increases. Also, if the user wants to changeone geometrical feature, a new model has to be created. For these reasons, paramet-ric studies are more difficult with a FE approach than with a closed form approach.However, special techniques are emerging for solving this problem. All these as-pects are discussed in this chapter with a mention to the difficult task of dealingwith the singular points present in any type of adhesive joint.

5.2 Types of Joints

Adhesive joint studies are generally related to lap joints with flat adherends. The firsttheoretical analysis of such joints was an analytical model proposed by Volkersen(1938). Since then, many analytical models have been proposed, being the over-whelming majority about lap joints with flat adherends. This is because joints of amore complex geometry are difficult to model using a closed form analysis. Numer-ical techniques, such as the FE method, can obviously be used for simple geometriessuch as single or double lap joints with regular flat adherends, but it is for complexgeometries that this method of analysis is truly advantageous. Within joints with flatadherends, one can have ‘wavy’ lap joints, ‘reverse bent’ joints, ‘tongue and groove’joints and scarf joints (see Fig. 5.2). Other types of joints include cylindrical joints,corner joints, T joints and peel joints (see Fig. 5.2). As for analytical models, themajority of the FE analyses are about lap joints with flat adherends. This type ofjoint is treated in detail in the following sections. However, numerous studies canbe found in the literature about other types of joints.

Avila and Bueno (2004) analyzed a new type of single lap joints where theadherends have a ‘wavy’ configuration along the overlap. An FE analysis and exper-iments show that great strength improvements can be obtained with this technique.Fessel et al. (2007) also studied this type of joint as well as joints with bent substratesalong the overlap. An FE analysis and experiments show great joint strength im-provements. Dvorak et al. (2001) show with a FE analysis and experimentally thatadhesively bonded tongue-and-groove joints between steel and composite platesloaded are stronger than conventional strap joints.

Nakagawa and Sawa (2001) studied scarf joints using photoelasticity and a 2DFE model. One of the conclusions is that under a static tensile load, an optimumscarf angle exists where the stress singularity vanishes and the stress distributionsbecome flat near the edge of the interface, but under thermal loads, the optimumscarf angle is not found.

Apalak and Davies (1993, 1994) studied different types of corner joints using alinear FE analysis and proposed design guidelines based on the overall static stiff-ness and stress analysis. Apalak (1999, 2000) developed the previous study takinginto account the non-linear geometry. Feih and Shercliff (2005) modelled simplecorner joints with composites.


Fig. 5.2 Types of adhesive joints

Apalak (2002) studied the geometrical non-linear response of T-joints. Apalaket al. (2003) studied T-joints in terms of thermal and mechanical loads. da Silvaand Adams (2002) used the a FE model to predict the joint strength of T-jointsbased on the plasticity of the adherend. Marcadon et al. (2006) studied T-jointfor marine applications and found that both the overlap length and also the dis-tance between the T plywood and the base have an influence on the tearingstrength.

Kim et al. (1992) studied various configurations of tubular joints (single over-lap, double overlap, adherends tapering) and found with a FE analysis and ex-perimentally that the double lap configuration is the strongest joint. Kim andLee (2001, 2004) used FE models to study the effect of temperature on tubu-lar joints. Kim et al. (1999) modelled composite tubular joints. Oh (2007) useda FE computation to analyze tubular composite adhesive joints under torsion.Good agreement was obtained between the predicted joint strength and the avail-able experimental data. Serrano (2001) used a nonlinear 3D FE model to studyglued-in rods for timber structures. The model could simulate experimentalbehaviour.

Peel testing is a very common test for adhesive properties. The problem is highlynon-linear in terms of geometry and material and the FE method has been widelyused, as for example in the work of Du et al. (2004).


5.3 Adhesive Fillet

Various authors have shown that the inclusion of a spew fillet at the ends of theoverlap reduces the stress concentrations in the adhesive and the substrate (Adamsand Peppiatt, 1974; Crocombe and Adams, 1981; Adams et al., 1986; Adams andHarris, 1987; Dorn and Liu, 1993; Tsai and Morton, 1995; Lang and Mallick, 1998;Belingardi et al., 2002; Apalak and Engin, 2004; Andreassi et al., 2007; da Silva andAdams, 2007a; Deng and Lee, 2007). The adhesive fillet permits a smoother loadtransfer and decreases the peak stresses.

Adams and Peppiatt (1974) found that the inclusion of a 45◦ triangular spew filletdecreases the magnitude of the maximum principal stress by 40% when compared toa square end adhesive fillet. Adams and Harris (1987) tested aluminium/epoxy sin-gle lap joints with and without fillet and found an increase of 54% in joint strengthfor the filleted joint. Adams et al. (1986) tested aluminium/CFRP single lap jointsand found that the joint with a fillet is nearly two times stronger than the joint with-out a fillet. Crocombe and Adams (1981) did similar work but included geometric(overlap length, adhesive thickness and adherend thickness) and material (modulusratio) parameters. The reduction in peel and shear stresses is greatest for a low mod-ulus ratio (low adhesive modulus), a high adhesive thickness and a low adherendthickness.

Dorn and Liu (1993) investigated the influence of the spew fillet in plastic/metaljoints. The study includes a FE analysis and experimental tests and they concludethat the spew fillet reduces the peak shear and peel adhesive stresses and decreasesstress and strain concentrations in the adherends in the most critical regions. Theyalso studied the influence of different adhesive and different metal adherends. A duc-tile adhesive and a more balanced joint (aluminium/plastic instead of steel/plastic)give a better stress distribution.

Tsai and Morton (1995) studied the influence of a triangular spew fillet in lami-nated composite single lap joints. The FE analysis and the experimental tests (Moireinterferometry) proved that the fillet helps to carry part of the load thus reducing theshear and peel strains.

The above analyses are limited to triangular geometry. Lang and Mallick (1998)investigated eight different geometries: full and half triangular, full and half rounded,full rounded with fillet, oval and arc. They showed that shaping the spew to providea smoother transition in joint geometry significantly reduces the stress concentra-tions. Full rounded with fillet and arc spew fillets give the highest percent reductionin maximum stresses whereas half rounded fillet gives the less. Furthermore, in-creasing the size of the spew also reduces the peak stress concentrations.

Andreassi et al. (2007) used a two-dimensional computational fluid dynamicsmodel to study spew fillet formation considering the actual adhesive flow producedduring joint assembly. This is an interesting approach that allows tuning the adhesivejoint strength by changing the adhesive flow parameters.

The spew fillet is not beneficial in all situations. In effect, the spew fillet tendsto generate more thermal stresses than a square end geometry when used at lowtemperatures (da Silva and Adams, 2007a).


5.4 Adherend Rounding

Adhesive single lap joints may have several singular (infinite) stresses. The stressestend to infinite at crack tips and at bi-material junctions. Fracture mechanics canbe applied to treat the singular points at crack tips. As regards the bi-materialsingularities, there can be two or three, depending on the joint geometry (seeFig. 5.3). For both situations, there is an abrupt change in slope giving a sharp cor-ner. The stress field in the vicinity of a sharp corner depends on the mesh used. Astress limit criterion will therefore lead to arbitrary results. The first FE analyses 30years ago used coarse meshes due to computer memory limitations and thereforedid not capture the infinite stress at singular points. The joint strength predictionsobtained with stress or strain limit based criteria compared well with the experimen-tal results (Harris and Adams, 1984). With the computer power development, finermeshes can be used which lead to a better stress distribution description but also tostresses that tend to infinite at singular points. The stress or strain limit failure cri-teria give in this circumstance more conservative predictions and are therefore veryarguable.

Groth (1988) used a fracture mechanics approach without considering a pre-existing crack. He formulated a fracture criterion based on an equivalent generalisedstress intensity factor similar to that in classical fracture mechanics. Comparing it toa critical value, joint fracture may be predicted. However, the critical stress intensityfactor needs first to be tuned with an experimental test which makes this approachquestionable.

Another approach for dealing with the singularity points is to use a cohesivezone model (CZM). This approach is associated to interface elements and enablesto predict crack initiation and crack growth. It is a combination of a stress limit andfracture mechanics approach and relatively mesh insensitive. Many researchers areusing this tool with accurate results (Blackman et al., 2003; de Moura et al., 2006;Liljedahl et al., 2007). However, the parameters associated to the CZM require pre-vious experimental ‘tuning’ and the user needs to know beforehand where the failureis likely to occur. This subject is discussed in detail in Chap. 6.

In practical joints the sharp corners are always slightly rounded during manu-facture and the singular points will not necessarily exist (see Fig. 5.4). Adams andHarris (1987) studied the influence of the geometry of the corners. Using a simpli-fied model, the effect of rounding on the local stress was investigated. Rounding thecorner removes the singularity. Therefore small local changes in geometry have asignificant effect. They also performed an elastic-plastic analysis by calculating the

Fig. 5.3 Singular points in adhesive joints


Fig. 5.4 Rounding of adhesive and adherend

plastic energy density. They found that the stress distribution is more uniform with amaximum some distance from the corner. The reason they give is that as the corneris approached, although the normal stress increases, the rigid adherend restrains theadhesive in the transverse direction. The net hydrostatic component is increased andyield is suppressed so that close to the corner there is a reduction in plastic energydensity. It should be born in mind that the effects are local: far away from the cornerthe stress distribution is unaffected. Apart from the simplified model, Adams andHarris (1987) tested three types of aluminium/rubber-toughened epoxy single lapjoints so that local changes at the end of the joint could be assessed. They found ex-cellent agreement between the predicted joint strengths, with the modified models,and the experimental values.

5.5 Adherend Shaping

Adherend shaping is a powerful way to decrease the stress concentration at the endsof the overlap. Figure 5.5 presents typical geometries used for that purpose. Someanalytical models were proposed to have a more uniform stress distribution alongthe overlap (Cherry and Harrison, 1970; Adams et al., 1973; Groth and Nordlund,1991). However, the FE method is more appropriate for the study of adherends shap-ing. The concentrated load transfer at the ends of the overlap can be more uniformlydistributed if the local stiffness of the joint is reduced. This is particularly relevantfor adhesive joints with composites due to the low transverse strength of composites.Adams et al. (1986) addressed this problem. They studied various configurations of

Fig. 5.5 Adherend shaping


double lap joints where the central adherend is CFRP and the outer adherends aremade of steel. They found with FE and experiments that the inclusion of an inter-nal taper and an external fillet can triplicate the joint strength. The same geometrywas studied at low temperatures and it was found that the thermal stresses reducesubstantially the joint strength (da Silva and Adams, 2007a).

Hildebrand (1994) did similar work on SLJs between fibre reinforced plastic andmetal adherends. The optimisation of the SLJs was done by modifying the geometryof the joint ends. Different shapes of adhesive fillet, reverse tapering of the adherend,rounding edges and denting were applied in order to increase the joint strength. Theresults of the numerical predictions suggest that, with a careful joint-end design, thestrength of the joints can be increased by 90–150%.

The use of internal tapers in adherends in order to minimize the maximum trans-verse stresses at the ends of bonded joints has also been studied by Towse (1999),Rispler et al. (2000), Guild et al. (2001), Belingardi et al. (2002) and Kaye andHeller (2002). An evolutionary structural optimisation method (EVOLVE) was usedby Rispler et al. (2000) to optimise the shape of adhesive fillets. EVOLVE consistsof an iterative FE analysis and a progressive removal of elements in the adhesivewhich are low stressed.

Other examples of joint end modifications for joint transverse stress reductionbut using external tapers are those of Amijima and Fujii (1989), Sancaktar andNirantar (2003), Kaye and Heller (2005) and Vallee and Keller (2006). Kaye andHeller (2005) used numerical optimization techniques in order to optimize the shapeof the adherends. This is especially relevant in the context of repairs using compos-ite patches bonded to aluminium structures (see Sect. 5.9) due to the highly stressededges.

The FE method is a convenient technique for the determination of the optimumadherend geometry, however the complexity of the geometry achieved is not alwayspossible to realise in practice.

5.6 Other Forms of Geometric Complexity

The FE method can be used to study other complex geometrical features such asvoids in the bondline, surface roughness, notches in the adherend, etc. (see Fig. 5.6).Nakagawa et al. (1999) studied the effect of voids in butt joints subjected to thermalstresses and found that stresses around defects in the centre of the joint are moresignificant than those near the free surface of the adhesive. Lang and Mallick (1999),Olia and Rossettos (1996), de Moura et al. (2006) and You et al. (2007) studied theinfluence of gaps in the adhesive and found that a gap in the middle of the overlaphas little effect on joint strength.

Kim (2003) proposed an analytical model supported by FE modelling to studythe effect of variations of adhesive thickness along the overlap. The author showedthat the variations found in practice have little effect on the adhesive stresses alongthe overlap.


Fig. 5.6 Various forms of geometric complexity

Kwon and Lee (2000) studied the influence of surface roughness on the strengthtubular joints by modelling the stiffness of the interfacial layer between the ad-herends and the adhesive as a normal statistical distribution function of the surfaceroughness of the adherends. The authors found that the optimum surface roughnesswas dependent on the bond thickness and applied load.

Sancaktar and Simmons (2000) investigated the effect of adherend notching onthe strength and deformation behavior of single lap joints. The experimental resultsshowed a 29% increase in joint strength with the introduction of the notches, whichcompared well with the FE analysis results. Yan et al. (2007) studied a similar idea,where the notch is in the middle part of the overlap. A FE analysis showed that amore uniform stress distribution along the overlap can be obtained.

5.7 Three Dimensional Analyses

Most of the FE analyses of adhesive joints are two dimensional. Since the widthis much larger than the joint thickness, a plane strain analysis is generally accept-able. A generalised plane strain analysis where the joint is allowed to have a uni-form strain along two parallel planes in the width direction gives more realisticresults. However, for some problems, a two dimensional analysis might lead to er-roneous results. Richardson et al. (1993) show that applying the average load ontwo-dimensional models resulted in errors in the adhesive stresses as high as 20%.The authors propose solutions to limit this problem. Three dimensional effects in thewidth direction such as lateral straining and the anticlastic bending (see Fig. 5.7)are especially important when analysing composites. Adams and Davies (1996)analysed composite lap joints in three dimensions and showed that the Poisson’sratio effects are significant in the behavior of composite lap joints.


Fig. 5.7 Three dimensional effects in lap joints

The main problem of 3D analyses is the computational time due to the largenumber of elements. One way of overcoming this problem is to use the submodellingoption, where the results from a coarse mesh of the global model are applied asboundary conditions to a much finer mesh of a localised region of particular interest.Bogdanovich and Kizhakkethara (1999) applied this technique to composite doublelap joints and concluded that the submodelling approach is an efficient tool althoughconvergence of stresses along certain paths in the joint were not satisfied. They alsoused the same approach in two dimensions to study the effect of the spew fillet.

5.8 Hybrid Joints

Joints with different methods of joining are increasingly being used. The idea is togather the advantages of the different techniques leaving out their problems. Anotherpossibility is to use more than one adhesive along the overlap or varying the adhesiveand/or adherend properties. All theses cases have been grouped here under a section


Fig. 5.8 Hybrid joints

called ‘hybrid joints’ (see Fig. 5.8). Such joints are particularly difficult to simulateusing analytical models for obvious reasons. The FE method is the preferred tool toinvestigate the application of such techniques.

5.8.1 Mixed Adhesive Joints

Mixed modulus joints have been proposed in the past (Semerdjiev, 1970; Srinivas,1975; Patrick, 1976) to improve the stress distribution and increase the joint strength


of high modulus adhesives. The stiff, brittle adhesive should be in the middle of theoverlap, while a low modulus adhesive is applied at the edges prone to stress concen-trations. Sancaktar and Kumar (2000) used rubber particles to toughen the part of theadhesive located at the ends of the overlap and increase the joint strength. The con-cept was studied with the FE method and proved experimentally. Pires et al. (2003)and Fitton and Broughton (2005) also proved with a FE analysis and experimentallywith two different adhesives that the mixed adhesive method gives an improvementin joint performance. Temiz (2006) used a FE analysis to study the influence of twoadhesives in double lap joints under bending and found that the technique decreasesgreatly the stresses at the ends of the overlap. Bouiadjra et al. (2007) used the mixedmodulus technique for the repair of an aluminium structure with a composite patch.The use of a more flexible adhesive at the edge of the patch increases the strengthperformance of the repair. The technique of using multi-modulus adhesives has beenextended to solve the problem of adhesive joints that need to withstand low and hightemperatures by da Silva and Adams (2007b, 2007c). At high temperatures, a hightemperature adhesive in the middle of the joint retains the strength and transfers theentire load while a low temperature adhesive is the load bearing component at lowtemperatures, making the high temperature adhesive relatively lightly stressed. Theauthors studied various configurations with the FE method and proved experimen-tally that the concept works, especially with dissimilar adherends.

5.8.2 Adhesive Joints with Functionally Graded Materials

Functionally graded materials are increasingly being used in various applicationsincluding adhesive joints. For example, Apalak and Gunes (2007) studied the effectof a functionally gradient layer between a pure ceramic layer (Al2O3) and a puremetal layer (Ni). Gannesh and Choo (2002) studied the effect of spatial grading ofadherend elastic modulus on the peak stress and stress distribution in the single-lap bonded joint. The adherends in the overlap length was divided into ten equalregions and material properties assign as a function of the grading. The peak shearstresses could be reduced by 20%. The study previously referred of Sancaktar andKumar (2000) is effectively a functionally graded adhesive with the use of rubberparticles. This is an area that is being intensively studied and where modelling atdifferent scales is essential.

5.8.3 Rivet-Bonded Joints

Liu and Sawa (2001) investigated, using a three-dimensional FE analysis, rivet-bonded joints and found that for thin substrates bonded, riveted joints, adhesivejoints and rivet-bonded joints gave similar strengths whilst for thicker substrates therivet-bonded joints were stronger. They proved this experimentally. Later, the same


authors (Liu and Sawa, 2003; Liu et al., 2004) proposed another technique similarto rivet-bonded joints: adhesive joints with adhesively bonded columns. Strengthimprovements are also obtained in this case. The advantage of this technique ofthat the appearance is the joint is maintained in relation to an adhesive joint. Grassiet al. (2006) studied through-thickness pins for restricting debond failure in joints.The pins were simulated by tractions acting on the fracture surfaces of the debondcrack.

5.8.4 Bolted-Bonded Joints

Chan and Vedhagiri (2001) studied the response of various configurations of singlelap joints, namely bonded, bolted and bonded-bolted joints by three-dimensional FEmethod and the results were validated experimentally. The authors found that for thebonded-bolted joints, the bolts help to reduce the stresses at the edge of the overlap,especially after the initiation of failure. The same type of study was carried out byLin and Jen (1999).

5.8.5 Weld-Bonded Joints

Al-Samhann and Darwish (2003) demonstrated with the FE method that the stresspeaks typical of adhesive joints can be reduced by the inclusion of a weld spotin the middle of the overlap. They studied later the effect of adhesive modulus andthickness (Darwish and Al-Samhann, 2004). Darwish (2004) also investigated weld-bonded joints between dissimilar materials.

5.9 Repair Techniques

Adhesively bonded repairs are generally associated to complex geometries and theFE method has been extensively used for the optimization of the repair, especiallywith composites. The literature review of Odi and Friend (2002) about repair tech-niques illustrates clearly this point. Typical methods and geometries are presentedin Fig. 5.9. Among the various techniques available, bonded scarf or stepped re-pairs are particularly attractive because a flush surface is maintained which permitsa good aerodynamic behavior. Odi and Friend (2004) show that an improved 2Dplane stress model is sufficient to get reliable joint strength predictions. Gunnionand Herszberg (2006) studied scarf repairs and carried out a FE analysis to assesthe effect of various parameters. They found that the adhesive stress is not muchinfluenced by mismatched adherend lay-ups and that there is a huge reduction inpeak stresses with the addition of an over-laminate. Campilho et al. (2007) studiedscarf repairs of composites with a cohesive damage model and concluded that thestrength of the repair increased exponentially with the scarf angle reduction.


Fig. 5.9 Repair techniques

Bahei-El-Din and Dvorak (2001) proposed new design concepts for the repairof thick composite laminates. The regular butt-joint with a patch on both sides wasmodified by the inclusion of pointed inserts or a ‘zigzag’ interface in order to in-crease the area of contact and improve the joint strength.

Soutis and Hu (1997), Okafor et al. (2005) and Sabelkin et al. (2007) studiednumerically and experimentally bonded composite patch repairs to repair crackedaircraft aluminum panels. The authors concluded that the bonded patch repair pro-vides a considerable increase in the residual strength.

Tong and Sun (2003) developed a pseudo-3D element to perform a simplifiedanalysis of bonded repairs to curved structures. The analysis is supported by a full3D FE analysis. The authors found that external patches are preferred when theshell is under an internal pressure while internal patches are preferred when underan external pressure.

5.10 Special Techniques in Finite Element Simulation

One of the advantages of closed form analyses in relation to the FE method is thatparametric studies are much easier to perform. With conventional FE techniques,every time a geometric parameter of the model (adhesive or adherend thicknesses,


angle of the spew fillet, etc.) with a different value is considered, a new model isrequired which is time consuming. However, recent FE libraries include connectionsthat can be described in a parametric form. Similar families of connections needto be meshed only once and the users need only enter the parameters. Actis andSzabo (2003) have used parametric models for the study of bonded and fastenedrepairs. These models have associated p-type meshing, where convergence of thesolution is achieved by increasing the polynomial order of the element rather thanincreasing the number of elements in the model (known as h-type meshing). Morerecently, Kilic et al. (2006) present a finite element technique utilizing a globalelement coupled with traditional elements. The global element includes the singularbehaviour at the junction of dissimilar materials.

Other special techniques in FE simulation for the significant reduction of themodelling (mesh generation) and computation time are presented next, which iscritical when complex geometries are involved. The main focus is on techniqueswhich do not require any new code at all or are achievable by very simple rou-tines consisting of a few lines of computer code. Advanced topics which require thedevelopment of comprehensive new routines, for instance as in the case of new ele-ment formulation (e.g. to better approximate singularities), are not considered. Thepresented techniques are realisable by actual versions of several commercial finiteelements codes.

The techniques will be applied to the example of a single lap joint problemtaken from Zhao (1991) where the influence of different degrees of rounding on thejoint strength was experimentally and numerically investigated. The general prob-lem with geometric dimensions and boundary conditions is shown in Fig. 5.10.

The adherends were made of 3.2 mm thick aluminium sheets (Young’s modulus:70 GPa; Poisson’s ratio: 0.33) and the brittle adhesive epoxy resin Ciba MY750with hardener HY906 (Young’s modulus 2.8 GPa; Poisson’s ratio: 0.4) was used.A 2D plane strain problem was modelled because of the large joint width and the

Fig. 5.10 Single lap joint: (a) joint geometry and applied boundary conditions (dimensions in mm);(b) different degrees of rounding (the grey colour represents the adhesive) (Zhao, 1991)


ends of the joints were constrained without any rotation to account for the testingconditions. It should be mentioned here that Zhao (1991) used a two steps approachbecause of limitations in computer power: A first computation was based on a coarsemesh for the entire specimen, followed by a refined mesh in the corner region withthe displacement result from the first analysis as the new boundary condition. Thefollowing examples which were realised with the commercial code Marc� (MSCSoftware Corporation, Santa Ana, CA, USA) may be considered as an alternativefor such a two step approach.

5.10.1 Consideration of Point Symmetry

Commercial finite element codes allow to consider certain types of symmetryconditions. Common examples are symmetry about an axis (so-called reflectivesymmetry) or cyclic symmetry (structures with a geometry and a loading varyingperiodically about a symmetry axis). Point symmetry (or origin symmetry, rota-tional symmetry by 180◦) with respect to the origin of the coordinate system occursfor instance in the case of single lap joints under tensile load (cf. Fig. 5.11a). Thistype of symmetry is not covered by the standard symmetry conditions which arementioned above. If one can consider the point symmetric deformation behaviourof a single lap joint (cf. Fig. 5.11b), it is obvious that the mesh size and the resultingsystem of equations can be reduced by 50% which results in a significant reductionof the calculation time.

Looking at the deformation of a full single lap specimen (cf. Fig. 5.12) in thecentre plane, i.e. x = 0, one can see that a node at a distance +a from the glueline (y = 0) moves under load for the shown arrangement to the negative x and ydirection. The movement in the negative y direction is about two orders of magnitudesmaller than the displacement in the x direction and difficult to observe in the scaleof Fig. 5.12. On the other hand, a node at distance −a from the glue line moves withthe same magnitude, however, in the positive coordinate directions. This relationshipfor a pair of nodes at (0,+ a) and (0,– a) can be expressed by the following constraintcondition

Fig. 5.11 Deformation of a single lap joint under tensile load: (a) deformed S-shape of the entirespecimen; (b) consideration of the S-shape for the half specimen due to appropriate point symmetrycondition


Fig. 5.12 Details of a deformed single lap specimens to illustrate node translation in the centreplane x = 0

{u

v

}

x=0;y=−a

=

[−1.0 0

0 −1.0

]·{

u

v

}

x=0;y=+a

, (5.1)

where the vector of displacement at node y = −a is referred to as the tied (“slave”)node while the node on the right-hand side is referred to as the retained node(“master”). The connecting matrix is called the constrained matrix. Some com-mercial finite element codes allow the definition of arbitrary homogeneous con-straints between nodal displacements by user subroutines (in the case of MSC Marc:UFORMS routine). When such a routine is supplied, the user is simply replacing theone which exists in the program using appropriate control setup. In the consideredcase of point symmetry, it must be mentioned that the constraint condition must bedefined for each pair of nodes in the centre plane x = 0.

5.10.2 Connecting Dissimilar Meshes

Modern adhesives can be applied in films of several micron of thickness while thesurrounding adherends may extend to a much larger scale in the range of millime-tres, centimetres etc. If accuracy in the adhesive layer is requested, several ele-ments must be introduced over the adhesive thickness and this mesh density mustbe coarsened in order to limit the total number of elements to account for limita-tions in computer hardware (in particular the random access memory, RAM). Inaddition, regions with high stress gradients require a fine and regular mesh if thesechanges should be evaluated. Such constraints may result in a quite complicatedmesh which is difficult to generate and a huge system of equations whose solution istime-consuming. In addition, it might be necessary to introduce transition elementswhich may have poor performance for specific loading conditions, e.g. bending. Asan alternative, the approach of dissimilar meshes (cf. Fig. 5.13) may be introduced.


Fig. 5.13 Example of a finite element model composed of two dissimilar meshes (nodes are notcoincident at the interface)

The major idea is to generate different types of meshes which are not connected inthe classical way, i.e. by common nodes which belong to both touching elements.Regions where high accuracy of the analysis is requested may be composed of avery fine and regular mesh (Fig. 5.13, right part) while other parts can be modelledby coarse and irregular element representations (Fig. 5.13, left part).

One method of connecting these dissimilar meshes is to use interface elements(Schiermeier et al., 2001). Nowadays, actual versions of commercial finite elementcodes allow such an application in the scope of novel contact options, i.e. to “glue”dissimilar contacting meshes without the need for interface elements. In such acase, by specifying that the glue motion is activated, the constraint equations areautomatically written between the two meshes and the contact region is not allowedto separate.

In the case of the corner rounding influence, a basic single lap joint was meshedwith a quite coarse mesh (adherend length: 96.8 mm) and several refined inlays(cf. Fig. 5.10b) were separately generated and successively “glued” to the basicjoint. Figure 5.14 shows as an example the refined inlay with the sharp corner whichis “glued” at the dissimilar interfaces to the basic joint. Similar meshes for the inlayswith different degree of rounding were obtained.

To present some numerical results, the normalised von Mises stress along thebond line for the different configurations of corners is shown in Fig. 5.15. Itcan be clearly seen that the introduction of a rounding decreases the stress peak(as shown in Peppiatt (1974) and Chen (1985)) and thus, results in a higherstrength of the adhesive joint. The difference in the peak values are presented inTable 5.1 and compared to experimental findings taken from Zhao (1991). It canbe seen that this numerical simulations reveals the same tendency as the experi-mental values. It must be noted here that the presented numerical results are basedon a pure linear elastic analysis and small deformations and an additional con-sideration of the non-linear material behaviour and appropriate yield conditionsof both components can improve the obtained results compared to experimentalvalues.


Fig. 5.14 Dissimilar meshes in the case of the sharp corner: the corner region consists of a muchfiner mesh

Fig. 5.15 von Mises stress k normalised by average shear stress τave along the normalised bondline (y = 0). Stress distributions are obtained or the same external load F

Table 5.1 Comparison between joint strength prediction and experiments

Corners FE(Strength increase in %)

Experiment, Zhao (1991)(Strength increase in %)

sharp 0 (ref.) 0 (ref.)r = 0.25 8.54 16.50r = 1.6 20.55 20.00r = 3.2 24.28 40.15


5.11 Conclusions

This chapter describes studies that deal with adhesive joints of complex geometryusing the FE method. The main conclusions are:

1. The FE method can give the stress distribution in the whole bonded structure andis an efficient tool to identify the stress concentrations.

2. For this reason, the FE method is the most adequate to develop and optimizeadhesive joints.

3. FE studies of different complexity were discussed: lap joints with irregularshapes, rounding of adherend ends, spew fillet, hybrid joints and repair tech-niques.

4. The stress concentrations at sharp corners, where the failure is likely to occur, aredifficult to handle using traditional stress-strain approaches because the resultsare mesh dependent.

5. This problem can be solved, to some extend, rounding the edges, using a strengthsingularity approach or a cohesive zone model. However, even these methodsrequire some kind of experimental ‘tuning’.

6. The optimization of joint strength by the use of functionally graded adhesives isone the main challenges in adhesive joints modelling.

7. Parametric studies are more difficult to perform with the FE method due to there-meshing problem and the computation time. However, recent FE programsinclude routines for automatic re-meshing. Submodelling is another techniquefor saving computation time and change geometric parameters in an easier way.This has been performed in the present chapter and validated with experimentalresults.

References

Actis RL, Szabo BA (2003) Analysis of bonded and fastened repairs by the p-version of the finite-element method. Comput Math Appl 46: 1–14

Adams RD, Atkins RW, Harris JA, Kinloch AJ (1986) Stress analysis and failure properties ofcarbon-fibre-reinforced-plastic/steel double-lap joints. J Adhes 20: 29–53

Adams RD, Chambers SH, Del Strother PJA, Peppiatt NA (1973) Rubber model for adhesive lapjoints. J Strain Anal 8: 52–57

Adams RD, Davies R (1996) Strength of joints involving composites. J Adhes 59: 171–182Adams RD, Davies R (2002) Strength of lap shear joints. In: The mechanics of adhesion (Dillard

DA and Pocius AV, ed.). Amsterdam: Elsevier, pp. 111–144.Adams RD, Harris JA (1987) The influence of local geometry on the strength of adhesive joints.

Int J Adhes Adhes 7: 69–80Adams RD, Peppiatt NA (1974) Stress analysis of adhesive-bonded lap joints. J Strain Anal 9:

185–196Al-Samhann A, Darwish SMH (2003) Strength prediction of weld-bonded joints. Int J Adhes Ad-

hes 23: 23–28Amijima S, Fujii T (1989) A simple stress analysis method for adhesive bonded tapered joints. Int

J Adhes Adhes 9: 155–160


Andreassi L, Baudille R, Biancolini ME (2007) Spew formation in a single lap joint. Int J AdhesAdhes 27: 458–468

Apalak MK (1999) Geometrically non-linear analysis of adhesively bonded corner joints. J AdhesSci Technol 13: 1253–1285

Apalak MK (2000) Geometrically non-linear analysis of an adhesively bonded modified doublecontainment corner joint — I. J Adhes Sci Technol 14: 1159–1177

Apalak MK (2002) On the non-linear elastic stresses in an adhesively bonded T-joint with doublesupport. J Adhes Sci Technol 16: 459–491

Apalak MK, Apalak Z, Gunes R (2003) Thermal non-linear elastic stress analysis of an adhesivelybonded T-joint. J Adhes Sci Technol 17: 995–1016

Apalak MK, Davies R (1993) Analysis and design of adhesively bonded comer joints. Int J AdhesAdhes 13: 219–235

Apalak MK, Davies R (1994) Analysis and design of adhesively bonded corner joints: fillet effect.Int J Adhes Adhes 14: 163–174

Apalak MK, Engin A (2004) Effect of adhesive free-end geometry on the initiation and propagationof damaged zones in adhesively bonded lap joints. J Adhes Sci Technol 18: 529–559

Apalak MK, Gunes R (2007) Elastic flexural behaviour of an adhesively bonded single lap jointwith functionally graded adherends. Mater Des 28: 1597–1617

Avila AF, Bueno PO (2004) Stress analysis on a wavy-lap bonded joint for composites. Int J AdhesAdhes 24: 407–414

Bahei-El-Din YA, Dvorak GJ (2001) New designs of adhesive joints for thick composite laminates.Compos Sci Technol 61: 19–40

Belingardi G, Goglio L, Rossetto M (2002) Impact behaviour of bonded built-up beams: experi-mental results. Int J Adhes Adhes 25: 173–180

Blackman BRK, Hadavinia H, Kinloch AJ, Paraschi M, Williams JG (2003) The calculation ofadhesive fracture energies in mode I: revisiting the tapered double cantilever beam (TDCB)test. Eng Fract Mech 70: 233–248

Bogdanovich AE, Kizhakkethara I (1999) Three-dimensional finite element analysis of double-lap composite adhesive bonded joint using submodeling approach. Compos Part B-Eng 30:537–551

Bouiadjra BB, Fekirini H, Belhouari M, Boutabout B, Serier B (2007) Fracture energy for repairedcracks with bonded composite patch having two adhesive bands in aircraft structures. CompMater Sci 40: 20–26

Campilho RDSG, de Moura MFSF, Domingues JJMS (2007) Stress and failure analyses of scarfrepaired CFRP laminates using a cohesive damage model. J Adhes Sci Technol 21: 855–870

Chan WS, Vedhagiri S (2001) Analysis of composite bonded/bolted joints used in repairing. J Com-pos Mater 35: 1045–1061

Chen Z (1985) The failure and fracture analysis of adhesive bonds, PhD thesis, Department ofMechanical Engineering, University of Bristol, UK

Cherry BW, Harrison NL (1970) The optimum profile for a lap joint. J Adhes 2: 125–128Crocombe AD, Adams RD (1981) Influence of the spew fillet and other parameters on the stress

distribution in the single lap joint. J Adhes 13: 141–155da Silva LFM, Adams RD (2002) The strength of adhesively bonded T-joints. Int J Adhes Adhes

22: 311–315da Silva LFM, Adams RD (2007a) Techniques to reduce the peel stresses in adhesive joints with

composites. Int J Adhes Adhes 27: 227–235da Silva LFM, Adams RD (2007b) Joint strength predictions for adhesive joints to be used over a

wide temperature range. Int J Adhes Adhes 27: 362–379da Silva LFM, Adams RD (2007c) Adhesive joints at high and low temperatures using similar and

dissimilar adherends and dual adhesives. Int J Adhes Adhes 27: 216–226Darwish SM (2004) Analysis of weld-bonded dissimilar materials. Int J Adhes Adhes 24: 347–354Darwish SM, Al-Samhann A (2004) Design rationale of weld-bonded joints. Int J Adhes Adhes

24: 367–377


de Moura MFSF, Daniaud R, Magalhaes AG (2006) Simulation of mechanical behaviour of com-posite bonded joints containing strip defects. Int J Adhes Adhes 26: 464–473

Deng J, Lee MMK (2007) Effect of plate end and adhesive spew geometries on stresses inretrofitted beams bonded with a CFRP plate. Compos Part B-Eng, doi:10.1016/j.compositesb.2007.05.004 <http://dx.doi.org/10.1016/j.compositesb.2007.05.004>

Dorn L, Liu W (1993) The stress state and failure properties of adhesive-bonded plastic/metaljoints. Int J Adhes Adhes 13: 21–31

Du J, Lindeman DD, Yarusso DJ (2004) Modeling the peel performance of pressure-sensitive ad-hesives. J Adhes 80: 601–612

Dvorak GJ, Zhang J, Canyurt O (2001) Adhesive tongue-and-groove joints for thick compositelaminates. Compos Sci Technol 61: 1123–1142

Feih S, Shercliff HR (2005) Adhesive and composite failure prediction of single-L joint structuresunder tensile loading. Int J Adhes Adhes 25: 47–59

Fessel G, Broughton JG, Fellows NA, Durodola JF, Hutchinson AR (2007) Evaluation of differentlap-shear joint geometries for automotive applications. Int J Adhes Adhes 27: 574–583

Fitton MD, Broughton JG (2005) Variable modulus adhesives: an approach to optimized jointperformance. Int J Adhes Adhes 25: 329–336

Gannesh VK, Choo TS (2002) Modulus graded composite adherends for single-lap bonded joints.J Compos Mater 36: 1757–1767

Goland M, Reissner E (1944) The stresses in cemented joints. J Appl Mech 66: A17–A27Grassi M, Cox B, Zhang X (2006) Simulation of pin-reinforced single-lap composite joints. Com-

pos Sci Technol 66: 1623–1638Groth HL (1988) Stress singularities and fracture at interface corners in bonded joints. Int J Adhes

Adhes 8: 107–113Groth HL, Nordlund P (1991) Shape optimization of bonded joints. Int J Adhes Adhes 11: 204–212Guild FJ, Potter KD, Heinrich J, Adams RD, Wisnom MR (2001) Understanding and control of ad-

hesive crack propagation in bonded joints between carbon fibre composite adherends II. Finiteelement analysis. Int J Adhes Adhes 21: 435–443

Gunnion AJ, Herszberg I (2006) Parametric study of scarf joints in composite structures. ComposStruct 75: 364–376

Harris JA, Adams RD (1984) Strength prediction of bonded single lap joints by non-linear finiteelement methods. Int J Adhes Adhes 4: 65–78.

Hart-Smith LJ (1973) Adhesive bonded double lap joints. NASA CR-112235Hildebrand M (1994) Non-linear analysis and optimization of adhesively bonded single lap joints

between fibre-reinforced plastics and metals. Int J Adhes Adhes 14: 261–267Kaye R, Heller M (2005) Through-thickness shape optimisation of typical double lap-joints includ-

ing effects of differential thermal contraction during curing. Int J Adhes Adhes 25: 227–238Kaye RH, Heller M (2002) Through-thickness shape optimisation of bonded repairs and lap-joints.

Int J Adhes Adhes 22: 7–21Kilic B, Madenci E, Ambur DR (2006) Influence of adhesive spew in bonded single-lap joints. Eng

Fract Mech 73: 1472–1490Kim H (2003) The influence of adhesive bondline thickness imperfections on stresses in composite

joints. J Adhes 79: 621–642Kim JK, Lee DG (2001) Thermal characteristics of tubular single lap adhesive joints under axial

loads. J Adhes Sci Technol 15: 1511–1528Kim JK, Lee DG (2004) Effects of applied pressure and temperature during curing operation on

the strength of tubular single-lap adhesive joints. J Adhes Sci Technol 18: 87–107Kim KS, Kim WT, Lee DG, Jun EJ (1992) Optimal tubular adhesive-bonded lap joint of the carbon

fiber epoxy composite shaft. Compos Struct 21: 163–176Kim YG, Oh JH, Lee DG (1999) Strength of adhesively-bonded tubular single lap carbon/epoxy

composite-steel joints. J Compos Mater 33: 1897–1917Kwon JW, Lee DG (2000) The effects of surface roughness and bond thickness on the fatigue life

of adhesively bonded tubular single lap joints. J Adhes Sci Technol 14: 1085–1102


Lang TP, Mallick PK (1998) Effect of spew geometry on stresses in single lap adhesive joints. IntJ Adhes Adhes 18: 167–177

Lang TP, Mallick PK (1999) The e!ect of recessing on the stresses in adhesively bonded single-lapjoints. Int J Adhes Adhes 19: 257–271

Liljedahl CDM, Crocombe AD, Wahab MA, Ashcroft IA (2007) Modelling the environmentaldegradation of adhesively bonded aluminium and composite joints using a CZM approach. IntJ Adhes Adhes 27: 505–518

Lin W-H, Jen M-HR (1999) The strength of bolted and bonded single-lapped composite joints intension. J Compos Mater 33: 640–666

Liu J, Liu J, Sawa T (2004) Strength and failure of bulky adhesive joints with adhesively-bondedcolumns. J Adhes Sci Technol 18: 1613–1623

Liu J, Sawa T (2001) Stress analysis and strength evaluation of single-lap adhesive joints combinedwith rivets under external bending moments. J Adhes Sci Technol 15: 43–61

Liu J, Sawa T (2003) Strength and finite element analyses of single-lap joints with adhesively-bonded columns. J Adhes Sci Technol 17: 1773–1784

Marcadon V, Nadot Y, Roy A, Gacougnolle JL (2006) Fatigue behaviour of T-joints for marineapplications. Int J Adhes Adhes 26: 481–489

Nakagawa F, Sawa T (2001) Photoelastic thermal stress measurements in scarf adhesive jointsunder uniform temperature changes. J Adhes Sci Technol 15: 119–135

Nakagawa F, Sawa T, Nakano Y, Katsuo M (1999) Two-dimensional finite element thermal stressanalysis of adhesive butt joints containing some hole defects. J Adhes Sci Technol 13: 309–323

Odi RA, Friend CM (2002) A comparative study of finite element models for the bonded repair ofcomposite structures. J Reinf Plast Comp 21: 311–332

Odi RA, Friend CM (2004) An improved 2D model for bonded composite joints. Int J AdhesAdhes 24: 389–405

Oh JH (2007) Strength prediction of tubular composite adhesive joints under torsion. Compos SciTechnol 67: 1340–1347

Okafor AC, Singh N, Enemuoh UE, Rao SV (2005) Design, analysis and performance of adhe-sively bonded composite patch repair of cracked aluminum aircraft panels. Compos Struct 71:258–270

Olia M, Rossettos JN (1996) Analysis of adhesively bonded joints with gaps subjected to bending.Int J Solids Struct 33: 2681–2693

Patrick RL (ed.) (1976) Treatise on adhesion and adhesives – Structural adhesives with emphasison aerospace applications, Vol. 4. Marcel Dekker, Inc., New York

Peppiatt NA (1974) Stress analysis of adhesiv joints, PhD thesis, Department of Mechanical Engi-neering, University of Bristol, UK

Pires I, Quintino L, Durodola JF, Beevers A (2003) Performance of bi-adhesive bonded aluminiumlap joints. Int J Adhes Adhes 23: 215–223

Renton WJ, Vinson JR (1975) The efficient design of adhesive bonded joints. J Adhes 7: 175–193Richardson G, Crocombe AD, Smith PA (1993) Comparison of two- and three-dimensional finite

element analyses of adhesive joints. Int J Adhes Adhes 13: 193–200Rispler AR, Tong L, Steven GP, Wisnom MR (2000) Shape optimisation of adhesive fillets. Int

J Adhes Adhes 20: 221–231Sabelkin V, Mall S, Hansen MA, Vandawaker RM, Derriso M (2007) Investigation into cracked

aluminum plate repaired with bonded composite patch. Compos Struct 79: 55–66Sancaktar E, Kumar S (2000) Selective use of rubber toughening to optimize lap-joint strength.

J Adhes Sci Technol 14: 1265–1296Sancaktar E, Nirantar P (2003) Increasing strength of single lap joints of metal adherends by taper

minimization. J Adhes Sci Technol 17: 655–675Sancaktar E, Simmons SR (2000) Optimization of adhesively-bonded single lap joints by adherend

notching. J Adhes Sci Technol 14: 1363–1404Schiermeier JE, Kansakar R, Mong D, Ransom JB, Aminpour MA, Stroud WJ (2001) p-Version

interface elements in global/local analysis. Int J Numer Meth Eng 53: 181–206Semerdjiev S (1970) Metal to metal adhesive bonding. Business Book Limited, London, UK


Serrano E (2001) Glued-in rods for timber structures – a 3D model and finite element parameterstudies. Int J Adhes Adhes 21: 115–127

Soutis C, Hu FZ (1997) Design and performance of bonded patch repairs of composite structures.Prc Instn Mech Engrs Part G 211: 263–271

Srinivas S (1975) Analysis of bonded joints. NASA TN D-7855Temiz S (2006) Application of bi-adhesive in double-strap joints subjected to bending moment.

J Adhes Sci Technol 20: 1547–1560Tong L, Sun X (2003) Nonlinear stress analysis for bonded patch to curved thin-walled structures.

Int J Adhes Adhes 23: 349–364Towse A (1999) The use of Weibull statistics for predicting cohesive failure in double lap joints.

PhD Dissertation, University of Bristol, UKTsai MY, Morton J (1995) The effect of a spew fillet on adhesive stress distributions in laminated

composite single-lap joints. Compos Struct 32: 123–131Vallee T, Keller T (2006) Adhesively bonded lap joints from pultruded GFRP profiles. Part III:

Effects of chamfers. Compos Part B-Eng 37: 328–336Volkersen O (1938) Die nietkraftoerteilung in zubeanspruchten nietverbindungen mit konstanten

loschonquerschnitten. Luftfahrtforschung 15: 41–47Yan Z-M, You M, Yi X-S, Zheng X-L, Li Z (2007) A numerical study of parallel slot in adherend

on the stress distribution in adhesively bonded aluminum single lap joint. Int J Adhes Adhes27: 687–695

You M, Yan Z-M, Zheng X-L, Yu H-Z, Li Z (2007) A numerical and experimental study of gaplength on adhesively bonded aluminum double-lap joint. Int J Adhes Adhes 27: 696–702

Zhao X (1991) Stress and failure analysis of adhesively bonded lap joints, PhD thesis, Departmentof Mechanical Engineering, University of Bristol, UK

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