Mechanics of Materials 42 (2010) 537–547

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Mechanics of Materials

journal homepage: www.elsevier .com/locate /mechmat

Viscoelastic adhesive interfacial model and experimentalcharacterization for interfacial parameters

J. Wang a,b, Q.H. Qin c, Y.L. Kang a,* , X.Q. Li a, Q.Q. Rong a

a Department of Mechanics, Tianjin University, Tianjin 300072, PR Chinab Department of Mechanics, Hebei University of Technology, Tianjin 300130, PR Chinac Department of Engineering, Australian National University, Canberra, ACT 0200, Australia

a r t i c l e i n f o

Article history:Received 1 July 2009Received in revised form 22 February 2010

Keywords:Adhesive interfacial modelViscoelasticityExperiment-based identification methodGenetic algorithm

0167-6636/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.mechmat.2010.03.002

* Corresponding author. Tel./fax: +86 22 2740361E-mail address: tju_ylkang@yahoo.com.cn (Y.L. K

a b s t r a c t

In this paper, a three-parameter interfacial model based on Needleman’s cohesive theory ispresented to characterize the viscoelastic mechanical properties of adhesive structures. Formost adhesive structures, the mechanical behavior of adhesive interface layer can be sim-ulated by the proposed adhesive interfacial model. To evaluate effectively the materialsparameters of the adhesive layer an improved experiment-based identification method isproposed including four major steps: (1) video-recorded experimental measurement, (2)numerical simulation based on the time-dependent adhesive interfacial model, (3) geneticalgorithm, and (4) independent experiment verification. Using the proposed experiment-based identification method, the viscoelastic interfacial mechanical parameters of metaladhesive structures and rubber adhesive structures under tension or shear loading aredetermined, respectively. Based on the identified parameters, the numerical computationalresults are in good agreement with the independent experimental measurement results. Itseems that the proposed adhesive interfacial model is effective to characterize the mechan-ical properties of the adhesive layer and the improved experiment-based identificationmethod is promising in solving parameter characterization problems of complex adhesivestructures.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Adhesive bonding is of interest in a variety of industrialand technological applications such as the automotive,construction, electronic packaging, and aeronautics sec-tors. In these adhesive structures, the performance of theadhesive interface layer is of crucial importance in provid-ing effective stress transfer. However, damage may easilyoccur due to stress concentration or bond imperfection un-der loading. Thus, the mechanical properties of the adhe-sive interface layer are critical in the design andapplication of adhesive structural components in generalengineering applications.

. All rights reserved.

0.ang).

Over the past decades a number of theoretical andexperimental methods have been developed with the aimof investigating the mechanical behavior of adhesive struc-tures under different loading. The embedded-process-zonemodel (Tvergaard and Hutchinson, 1993, 1996) was one ofthe approaches used to investigate interfacial fracture ofbi-material systems, where a three-parameter traction-separation law together with the opening and shear stresswas involved. Wei and Hutchinson (1998) developed anembedded cohesive-zone model for steady-state peelingof a thin elastic–plastic film bonded to an elastic substrateand then conducted parameter characterization usingnumerical approaches. Yang et al. (1999) used an embed-ded-process-zone model to study the coupling betweeninterface fracture and plastic strain of the adherend underT-peel test. For the adhesive, a traction-separation lawincluding plasticity was used. Later, Yang et al. (2001)

538 J. Wang et al. / Mechanics of Materials 42 (2010) 537–547

considered the same traction-separation law for elastic–plastic mode-II crack growth modeling in which Eng-Notched Flexure specimens subjected to a bending loadwere used to validate the model. Kafkalidis and Thouless(2002) performed a numerical analysis of single-lap shearjoints using a cohesive-zone approach. The cohesive-zonemodel allowed not only the influence of the geometry tobe considered, but also included in the analysis the cohe-sive properties of the interface and plastic deformation ofthe adherends. Li et al. (2005a) used a cohesive-zone mod-el previously developed by Li et al. (2005b) on adhesivelybonded joints to validate both two- and three-parameterlaws. Using this method, the strength and deformationswere accurately described, as well as the transition be-tween failure of the composite and failure of the interface.The Compact-Tension test was used to determine the prop-erties of the traction-separation laws. Thouless et al.(2006) used a cohesive-zone approach to model themixed-mode fracture of adhesive GFRP single-lap joints.Using Euler–Bernoulli beam theory, Alfredsson (2003) pre-sented an exact analytical inverse solution to determinethe constitutive properties of thin adhesive layers loadedin shear. Recently, Leffler et al. (2007) implemented adhe-sive layer theory to determine the complete shear stressversus deformation relation of adhesive layer. Gustafsonand Waas (2009) investigated the influence of adhesiveparameters on the outcome of cohesive-zone finite ele-ment simulation. However, the works mentioned abovewere used only for analyzing the time-independentmechanical properties of adhesive materials. It is especiallyworth noting that time-dependent mechanical propertiesare inevitable due to the viscoelasticity of the adhesivelayer (Jagota et al., 2000). For the study on dynamic behav-ior of the adhesive materials, Du et al. (2000) used the lin-ear fracture mechanics for analyzing the time-dependentfracture behavior of Rubber-modified epoxies in experi-mental tests. Landis et al. (2000) investigated the materialseparation in the fracture process zone, which is the sum ofan elastic and viscoplastic contribution, and cohesive trac-tions are time-dependent only if plastic opening exists.Liechti and Wu (2001) extracted the time-dependentparameters for the traction-separation law under mixed-mode loading, where the quasi-static debonding betweenrubber and metal was modeled by non-linear springs anddashpots. Xu et al. (2003a) provided the standard linear so-lid model to simulate the viscoelastic material behavior,and calibrated the relative parameters by comparing a ser-ies of numerical simulations with experimental curves todescribe the crack growth in a thermoplastic adhesive(Xu et al., 2003b).

To characterize mechanical properties of structuresmany experimental procedures were developed duringthe past decades (see, for example, the classical handbookby Kobayashi, 1987). Among these approaches, the hybridmethod (Laermann, 1981) is very effective in characteriz-ing mechanical properties as it relies on the combinationof experimental, analytical and numerical methods. Hy-brid methods introduced by Laermann are known as ‘‘di-rect method” because they are based on the combined useof experimental, numerical, and theoretical analysis.Experiments may be real experimental measurement or

numerical simulations of experiments (Kobayashi, 1983;Asundi, 1999). By introducing optimization methods intothe above mentioned hybrid framework, an effective hy-brid/inverse methods were developed. Using the hybrid/inverse method, a great of work were done for the identi-fication of material parameters (Chalal et al., 2004;Molimard et al., 2005; Hartmann et al., 2006; Cooremanet al., 2007; Jiang et al., 2008; Balakrishnan and Socrate,2008; Davendralingam and Doyle, 2008; Cosola et al.,2008). Considering the importance of experimental mea-surement, recently, Kang et al. (2004) and Wang et al.(2008) proposed the experiment-based hybrid/inversemethod to analyze the material mechanical properties.In this method, the three components say the experimen-tal measurement, numerical simulation and optimizationmethod, were combined integrally based on the experi-mental measurements.

The focus of this paper is to describe the viscoelasticadhesive interfacial mechanical properties by experimen-tal characterization method. Firstly, a developed three-parameter interfacial model based on Needleman’s cohe-sive model is presented to describe the interfacialmechanical properties of adhesive structures under ten-sion or shear loading. Subsequently, to quantitativelycharacterize the interfacial parameters of this model, animproved experiment-based identification method isdeveloped. In the process of improved experiment-basedidentification method, experimental analysis is veryimportant. Experimental data of real structure have beenused as the initial data for finite element calculation andthe parameter identification procedure. First, a video-re-corded experiment system under tension or shear loadingis designed and used to obtain the deformation anddamage information of the adhesive layer. Meanwhile,on the basis of the proposed viscoelastic adhesive interfa-cial model, numerical simulations are performed bymeans of the finite element software ABAQUS and itsuser-defined element subroutine. Combining the experi-mental, numerical, and identification technique with animproved genetic algorithm, an experiment-based identi-fication method is constructed for identifying viscoelasticmaterial properties. Numerical results are presented toshow the effectiveness and applicability of the proposedmethod. Finally, independent experimental verificationis performed and the results from experimental measure-ment are compared with those from numericalsimulation.

2. The viscoelastic adhesive interfacial layer model

To evaluate the viscoelastic interfacial properties ofadhesive structural components, an improved viscoelasticadhesive interfacial layer model under tension or shearloading is proposed in this section. In the analysis, theadhesive layer is represented by an improved cohesivezone (Kelvin element). To introduce the viscoelastic factorinto this model, a modified exponential cohesive model(Needleman, 1990) which is arranged in parallel with adashpot is used to characterize the behavior of the adhe-sive layer.

J. Wang et al. / Mechanics of Materials 42 (2010) 537–547 539

In the model (Fig. 1), material separation in the normalor tangential direction is considered, respectively. The cor-responding constitutive law can be expressed as

r ¼ Eeþ g _e ð1Þs ¼ Gcþ l _c ð2Þ

Taking the mode under shear loading for example, the vis-coelastic adhesive interfacial model can be deduced basedon the following three assumptions: (a) the stiffness of theNeedleman’s cohesive-zone model deT t=dDt is used to re-place the spring stiffness G in the Kelvin model. (b) The dis-placement jump Dt across the cohesive-zone and thecohesive-surface traction Tt are used to replace the strainc and stress s in the Kelvin model. (c) A constant gt witha unit of force per velocity per area (dashpot coefficientper area) is used to replace g in the Kelvin model with aunit of force per strain rate per area (viscosity). Thus, thecorresponding constitutive relation between stress andopening rate is

Tt ¼ eT t þ gtdDt=dt ð3Þ

In Eq. (3), dDt/dt is the opening rate, eT t is the modifiedexponential traction-separation law proposed by Needle-man (1990). The constitutive equations are as follows:

Fig. 1. The viscoelastic adhesiv

eT t ¼ �~smaxzeDt=dtc expð�zDt=dtcÞ ðDt 6 dtcÞ ð4ÞeT t ¼ 0 ðDt > dtcÞ ð5Þ

where z = 16e/9, e = exp(1), ~smax is the cohesive strengthunder shear loading and dtc is the critical tangential dis-placement jump. The hat over the cohesive-zone quantitiesrepresents interfacial parameters where viscoelasticity isnot considered.

In this model, the macroscopic mechanical property ofan adhesive layer can be modeled by a constitutive lawwhere the level of stress depends on the deformation ofthe adhesive layer and a time-dependent constant coeffi-cient is included. Under monotonically increasing loading,the stress first increases, to the maximum, ~smax, at whichdamage takes place, the stress then decreases. When thedeformation has increased to a critical value, dtc, the stressreaches zero. At this point, crack growth takes place andfailure of adhesive structure is formed. For adhesive struc-ture, it is convenient to enable the constitutive relationrepresenting the mechanical behavior of the entire adhe-sive layer. Such a constitutive relation describes activitiesin the adhesive layer before and at fracture.

e interfacial layer model.

540 J. Wang et al. / Mechanics of Materials 42 (2010) 537–547

Similar, the constitutive relation of viscoelastic adhe-sive interfacial layer model in normal direction can be ex-pressed as

Tn ¼ eT n þ gndDn

dtð6Þ

whereeT n ¼ �~rmaxzeDn=dnc expð�zDn=dncÞ ðDn 6 dncÞ ð7ÞeT n ¼ 0 ðDn > dncÞ ð8Þ

Then, ~rmax is the cohesive strength under tension loadingand dnc is the critical normal displacement jump.

The mechanical properties of the viscoelastic adhesiveinterfacial layer model in tangential or normal directioncan be characterized by three parameters, respectively,namely interfacial strength limit, ~smax=~rmax, displacementjump, dtc/dnc, and viscosity coefficient, gt/gn. In this model,the action of dashpot is time-dependent and its viscositycoefficient is assumed to be constant; the action of Needle-man’s model is time-independent and its strength limitand displacement jump are also constants. Thus, basedon the Needleman’s cohesive model, a three-parameterinterfacial constitutive model is constructed to investigatedeformation, damage and debonding behavior of the adhe-sive layer in adhesive structure. In addition, it is conve-nient to characterize the mechanical behavior of theadhesive layer from the experimental measurement duringengineering application through the parameterized inter-facial model. In the following analysis the vectors contain-ing the three unknown interfacial parameters, such as,rð~smax; dtc;gtÞ for shear loading and sð~rmax; dnc;gnÞ for ten-sion loading in adhesive layer of metal adhesive structureand rubber adhesive structure, are identified with the fol-lowing improved experiment-based identification method,respectively.

3. Improved experiment-based identification method

In this section the improved experiment-based identifi-cation method is developed for the determination of visco-elastic adhesive interfacial mechanical properties ofadhesive structures under tension and shear loading. Theinterfacial mechanical properties identification problemto be considered is based on the experimental measure-ment, to seek such an interfacial parameter vector in thesolution space that the deformation and failure informa-tion in adhesive layer obtained from the numerical simula-tion is in a good agreement with or convergent to therelated experimental results.

In this method, there are four main components: (1)numerical calculations based on the proposed viscoelasticinterfacial layer model; (2) experimental measurement un-der tension or shear loading in real time; (3) an appropri-ate optimization technique, and (4) the independentexperimental verification. To identify the unknown param-eter vector rð~smax; dtc;gtÞ and sð~rmax; dnc;gnÞ, the whole pro-cedure of the proposed method is performed iteratively:firstly, the initial interfacial properties, which are ex-pressed in terms of a set of unknown parameters, are cal-culated employing a finite element computer program.

The viscoelastic adhesive interfacial behavior can be repre-sented by the constitutive relation of the proposed adhe-sive interfacial model. While the deformation, damageand failure of the adhesive layer can be simulated numer-ically. At the same time, the corresponding experimentalresponse of the shear and tension specimens is recorded.To identify the viscosity parameter (gn or gt), a video-re-corded data acquisition system is designed to obtain thereal-time deformation and damage information of theadhesive layer. Using the results from experimental mea-surement and simulation calculation, an objective functionis constructed and minimization of the objective functionwith respect to the unknown vector r or s is performedby means of an improved genetic algorithm. The optimalinterfacial layer parameter vector r�ð~smax; dtc;gtÞ ors�ð~rmax; dnc;gnÞ can thus be obtained through the inverseprocess. Finally, an independent experiment is proposedand conducted to verify the identified results. The logicalflowchart of the experiment-based hybrid/inverse analysisis shown in Fig. 2.

4. Identification of viscoelastic adhesive interfacialmechanical properties

In this section, the viscoelastic adhesive interfacialmechanical properties of metal adhesive structure (elasticmaterial) and rubber adhesive structure (hyperelasticmaterial) under tension and shear loading, respectively,are quantitatively characterized using the proposed exper-iment-based identification method. In the analysis, identi-fication process of the rubber adhesive structure specimenunder tension loading is performed in detail. Furthermore,the rubber specimen under shear loading and the metaladhesive structure specimen under shear and tension load-ing are also considered.

4.1. Experimental test

The rubber adhesive specimens used in the tensionloading are shown schematically in Fig. 3. The specimenswere loaded through an Instron 3343 mechanical testingmachine at constant loading speeds of 0.3, 0.5 and1.0 mm/s under tension loading. In addition, to facilitatein-situ optical observations during the deformation pro-cess, a marker technique was adopted and uniform squaregrids were printed on the free surface of the specimen withthe density of 2 lines/mm. A high-resolution CCD (BaslerA202k with the resolution up to 1004 � 1004 and thelength of a pixel is 7.4 lm) coupled with a video-recordedimage data acquisition system (Matrox meteor II Cameralink) was used to measure the actual displacement of thespecimen. Thus, the local deformation and the processesof failure initiation and propagation in the adhesive layerwere recorded in real time.

Employing the video-recorded image acquisition sys-tem, the deformation and damage information of the adhe-sive interface were recorded for a rubber adhesivestructure specimen under tension loading (at constantloading speed of v = 1.0 mm/s), the deformation imagesof t = 4 s and 12 s are shown in Fig. 4. It can be seen from

Fig. 2. Flowchart of the improved experiment-based identification method.

J. Wang et al. / Mechanics of Materials 42 (2010) 537–547 541

these images that the failure of the specimen during thedeformation process is a kind of peel failure at the interfaceof the adhesive structure. By means of these deformationimages, the position of the failure at the adhesive interfacecan be recorded in real time, which is important for con-structing the objective function of the identificationproblem.

With the same apparatus, the shear experiment of rub-ber adherend was loaded at constant loading speeds of 1.0,1.5 and 2.0 mm/s. For metal specimen, two LY12-CZ alumi-num alloy adherends (Young’s modulus E = 71 GPa, Pois-son’s ratio v = 0.33) were coated with adhesive (WL-506)at room temperature. The specimen was loaded through

an Instron 3343 mechanical testing machine at constantloading speeds of 0.02, 0.1 and 0.2 mm/s under shear load-ing and at constant loading speeds of 0.02, 0.2, 1.0 and2.0 mm/s under tension loading, respectively.

4.2. Numerical simulation

Numerical simulation plays an important role in theexperiment-based identification method. Numerical pre-diction of the interfacial behavior and the correspondingexperimental results are two important components ofthe proposed identification method. The interfacial dam-age is numerically simulated by means of ABAQUS (2001)

Fig. 4. Deformation images of rubber specimen under tension loading at v = 1.0 mm/s.

Fig. 3. Tension specimen of rubber adhesive structure.

542 J. Wang et al. / Mechanics of Materials 42 (2010) 537–547

and its user-defined subroutine capability. The time-dependent interfacial elements have four nodes and thecorresponding integration is carried out at two integrationpoints. The continuum elements for the adherends arefour-node bilinear elements. Adjacent to the interfacialelements, representing the adhesive interfacial layer, onerow of square solid elements is located. Subsequently,the size of the elements is gradually increased with in-creased distance from the interfacial elements. The initialsize of the square elements and the size of the interfacialelements is 0.1 mm. In the numerical calculations, the pro-posed viscoelastic adhesive interfacial constitutive modelwas implemented into the finite element software ABA-QUS/Standard (2001) by means of a user-defined elementsubroutine, to define the interaction behavior of the adhe-sive interfacial layer. It should be mentioned that the pro-posed element is a two-dimensional element with fournodes and zero initial thickness. Before deformation theelement nodes of the upper face coincide with the corre-sponding nodes of the lower face. In this paper, only onedisplacement degree of freedom at each node is consid-

ered. It is the one along the normal direction under tensionloading or along the tangential direction under shear load-ing. As the continuum elements connected to the interfaceelement may deform, the displacement between the upperand lower faces increases from zero and follows the as-sumed traction-displacement relationship in Eq. (3) undershear loading or Eq. (6) under tension loading. The bulkelements for the adherend are four-node bilinear elements.The material properties used in the analysis for aluminumalloy are E = 71 GPa and v = 0.33. For rubber material sev-eral non-linear constitutive laws represented by hyper-elasticity have been incorporated in ABAQUS. A Van derWaals material model is used here and is defined by

U ¼ l �ðk2m � 3Þðlogð1� gÞ þ gÞ � 2

3a

I1 � 32

� �32

( )ð9Þ

where g ¼ I1�3km�3

� �12. km, l, and a are material parameters

which can be determined through experimental tests. Forthe specimen used in this work, the material parameters

J. Wang et al. / Mechanics of Materials 42 (2010) 537–547 543

of rubber are km ¼ 663:29, l = 0.877, and a = 4.72 � 10�2.Four different mesh densities are used to study the conver-gence of the numerical simulation.

4.3. Objective function

For a given identification problem, it is important toconstruct a reasonable objective function which is sensi-tive to the parameters to be identified. To reduce the ef-fects of the noise during the tests, the failure region andthe damage state of the interfaces are introduced to theobjective function as base variables. In the experiment,the grid technique was used to mark damage state of theinterfacial layer by the gap between the two surfaces. Con-struction of the objective function is as follows. First, anumber i ði ¼ 1;2; . . . ; pÞ is allocated to each of the adhe-sive layer elements. Let mi(e, t) and �miðtÞ ði ¼ 1;2; . . . ; pÞstand for the damage variable state of the ith adhesivelayer element in the simulation calculation and the exper-imental results at the same moment t, respectively. It is as-sumed that mi(e, t) or �miðtÞ equals 0, when the damageoccurs in the ith element, otherwise mi(e, t) or �miðtÞ equals1. At the same time, a new function Hðmiðe; tÞ; �miðtÞÞ is pre-sented, representing the difference in the damage variablestate obtained from numerical calculation and experimen-tal observation, with the assumption that Hðmiðe; tÞ; �miðtÞÞequals 0 in the case of miðe; tÞ ¼ �miðtÞ, otherwise it equals1. From the above discussion, considering a given periodof time, say q, the objective function and the optimizationproblem can, respectively, be shown as

FðeÞ ¼Xq

t¼1

Xp

i¼1

H½miðe; tÞ; �miðtÞ� ð10Þ

Under shear loading, e ¼ rð~smax; dtc;gtÞ, the objective func-tion can be expressed as

FðrÞ ¼Xq

t¼1

Xp

i¼1

H½miðr; tÞ; �miðtÞ� ð11Þ

r� ¼ arg minr2R

FðrÞ ð12Þ

where r* represents the optimal interfacial layer parame-ters vector of adhesive interfacial layer in three-dimen-sional space Rð~smax; dtc;gtÞ under shear loading.

Similarly, under tension loading noting that e ¼sð~rmax; dnc;gnÞ, the corresponding objective function canbe written as

FðsÞ ¼Xq

t¼1

Xp

i¼1

H½miðs; tÞ; �miðtÞ� ð13Þ

s� ¼ arg mins2S

FðsÞ ð14Þ

where s* represents the optimal interfacial layer parame-ters vector of adhesive interfacial layer in three-dimen-sional space Sð~rmax; dnc;gnÞ under tension loading.

4.4. Identification of viscoelastic interfacial layer parametersby genetic algorithm

It is noted that some challenging problems exist in solv-ing an inverse problem as inverse problems are usually

highly non-linear and ill-posed (Liu and Han, 2003). There-fore, in many cases, traditional inverse techniques such asthe gradient-based optimization method are not suitablefor solving a complex inverse problem. It can be seen fromEq. (10) that the objective function employed is a discretefunction for evaluating the fitness of individuals. For dis-crete function, it is very difficult to calculate its derivative.Such derivative is, however, necessary to the gradient-based search method (Liu and Han, 2003). To by pass thisproblem, an advanced inverse technique based on a genet-ic algorithm is used in the inverse analysis. The geneticalgorithm was introduced by Holland (1975) as a methodfor searching the global optimum of a complicated prob-lem. The concept of genetic algorithm originates from theprinciples of Charles Darwin’s theory of biological evolu-tion (Goldberg, 1989; Michalewicz, 1994). According tothe algorithm, individuals are produced through three sto-chastic operations: selection, crossover, and mutation. Theprimitive genes of individuals are selected randomlyaccording to the genetic operators. The crossover operatorrecombines a pair of individuals into two new individuals,and the mutation operator alters a single individual bychanging its genes with the given probability, respectively.Individuals are evaluated using fitness function which iscalled objective function in the inverse problem. The candi-dates with better fitness are selected for reproductionwhich replaces the solutions with worse fitness. The previ-ous studies showed that genetic algorithm is promising indealing with large, discrete, non-linear and poorly under-stood optimization problem, where expert knowledge isscarce or difficult to obtain. The method has been widelyaccepted as optimization methods in various fields, suchas flaw detection (Liu and Chen, 2001; Perera and Torres,2006), interfacial parameter identification (Lin et al.,2005), material properties determination (Cunha et al.,1999; Chakraborty et al., 2002; Vishnuvardhan et al.,2008) and parameter calibration (Koh et al., 2003; Salo-monsson and Andersson, 2008).

In this study, the optimization procedure for identify-ing parameters of viscoelastic adhesive interfacial struc-tures was programmed in MATLAB 7, which requiredthe preparation of a vast amount of input data from ABA-QUS solver. Finally, at given loading speed, through aseries of experiments and algorithm performance com-parisons, the population size of each generation is setas 30 and the probabilities of crossover and mutationwere set to 0.8 and 0.05, respectively. The optimum valuecould be obtained after 100 generations (the geneticparameter which represents the maximum number ofoperations).

The history of fitness value under tension loading inrubber material, normalized to the range of 0–1 by thenumber of interfacial elements of the model, against thenumber of generations for a genetic algorithm run is plot-ted in Fig. 5. It is observed from the convergence curve thatthe genetic algorithm converges very quickly at the begin-ning and slows significantly at the final stage of searching.

Finally, the identifying values of the three interfaciallayer parameters of different materials (rubber or alumi-num alloy) under tension loading and shear loading areshown in Table 1. In comparison with the interfacial layer

Fig. 5. History of the fitness value under tension loading in rubber material.

Table 1Identification results of viscoelastic adhesive interfacial model.

Tension Shear

Parameters rmax ðMPaÞ dnc ðmmÞ gn ðMPa s=mmÞ smax ðMPaÞ dtc ðmmÞ gt ðMPa s=mmÞ

Rubber 7.4 0.68 19.5 16.3 1.19 20.1Aluminum alloy 8.1 0.35 15.2 19.4 1.02 15.7

544 J. Wang et al. / Mechanics of Materials 42 (2010) 537–547

parameters of different materials (Table 1), it can be seenthat:

(a) The strength and stiffness of the interface are mode-dependent. For the interfacial mechanical propertyof the rubber adhesive structure, the interfacialshear strength, ~smax, is much higher than the interfa-cial tensile strength, ~rmax ð~smax � 2:3~rmax hereÞ. Theshear stiffness of the interface dtc is higher than itstensile stiffness dnc (dtc � 2dnc in this work). More-over, for the interfacial mechanical properties ofthe aluminum alloy adhesive structure, the similartrend can be observed from Table 1. It is noted thatthe fracture energy of the adhesive structure loadedin shear is higher than that loaded in tension. Thus,in engineering application, the structures are oftendesigned to be loaded in shear.

(b) The time-dependent viscosity coefficient of adhesivestructure is mode-independent. For the rubber adhe-sive structure, there is little difference between theviscosity coefficient gt (loaded in shear) and gn

(loaded in tension) within the adhesive interfaciallayer. The similar conclusion can be obtained fromthe results for aluminum alloy adhesive structure.Thus, for a given adhesive structure (either elasticmaterial or hyperelastic material), the time-depen-dent viscosity coefficient is insensitive to the loadingmode. Considering the mode-independent property,the viscosity coefficient under mixed loading can beobtained from the identification results under eithershear loading or tensile loading.

(c) The interfacial mechanical property is relative to thematerial property of adhesive structure. For theinterfacial mechanical property of different adhesivestructures under same loading condition (shearloading or tensile loading), the interfacial strengthof aluminum alloy adherend (elastic) is higher thanthe corresponding one of rubber adherend (hyper-elastic). However, the interfacial stiffness and vis-cosity coefficient of aluminum alloy adhesivestructure are lower than those of rubber adhesivestructure. This might be due to the fact that, duringthe loading process, the deformation of the adher-end might weaken the loading on the adhesive inter-face. For rubber adherend, there was moredeformability than aluminum alloy adherend.

5. Discussion and conclusions

Viscoelastic shear interfacial parameters of the adhesivelayer are identified by the improved experiment-basedidentification method based on the loading speed(v = 1.0 mm/s). However, for the experiment-based identi-fication analysis method, problems such as unstable solu-tion or multi-solution of the parameters may exists inthe inverse process. To overcome this problem and to iden-tify the viscoelastic interfacial parameters correctly, anindependent experimental verification method is devel-oped. In this method, the independent experimental mea-surement at other loading speeds and the correspondingnumerical simulation with the interfacial parameters iden-

J. Wang et al. / Mechanics of Materials 42 (2010) 537–547 545

tified at loading v = 1.0 mm/s are included. Firstly, twoother loading speeds (v = 0.3 mm/s and v = 0.5 mm/s) arecarried out as the verification experiments under the sameexperimental condition described in Section 4. Subse-

Fig. 6. Results of experiment versus numerical simula

quently, compared the experimental results at v = 0.3mm/s and v = 0.5 mm/s loading speeds with the numericalsimulation ones based on the interfacial parameters iden-tified above. The results are presented in Fig. 6(a).

tion of independent experimental verification.

546 J. Wang et al. / Mechanics of Materials 42 (2010) 537–547

It can be seen from Fig. 6(a) that the curves from experi-ment and simulation at the same loading speed provide sat-isfactory coincidence and the results of the identifiedparameters are effective at given loading speeds. The resultsfor interfacial layer parameters of adhesive layer identifiedat the loading speed v = 1.0 mm/s are in good agreementwith those from the other two loading conditions. Similarresults of rubber adhesive structure under shear loadingcan be observed from Fig. 6(b). For aluminum alloy adhesivestructure, the results of independent experimental verifica-tion under shear loading is shown in Fig. 6(c).

From the discussion above, it is reasonable for themechanical parameters identified to characterize the prop-erties of the adhesive layer. The three-parameter viscoelas-tic adhesive interfacial model developed in this paper isthus effective for analyzing time-dependent adhesivestructures and can be used to quantitatively characterizeviscoelastic mechanical properties of the adhesive layerunder shear loading and tension.

In this paper, a three-parameter interfacial model basedon Needleman’s cohesive theory was proposed to simulatethe viscoelastic interfacial mechanical property of adhesivestructure. In this model, the deformation, damage and deb-onding behavior of the adhesive layer can be characterizedin terms of interfacial strength limit, ~smax=~rmax, displace-ment jump, dtc/dnc, and viscosity coefficient, gt/gn. Anexperiment-based identification procedure including full-field real-time measurements, finite element simulation,global optimization and independent verification is pre-sented. The procedure is then used to identify quantita-tively interfacial layer mechanical parameters of thealuminum alloy adhesive structure or rubber adhesivestructure under tensile and shear loading conditions,which are difficult to measure directly from experiment.In particular, the improved experiment-based identifica-tion method can be further extended for solving complex,non-linear, real-time and multi-parameter identificationproblems of actual adhesive structures.

Acknowledgements

The authors would like to thank Professor K.H. Laer-mann (Bergische Universität-GH Wuppertal, D-42285Wuppertal, Germany) for his useful discussion in hybrid/inverse method and would like to acknowledge the sup-port from the National Science Foundation of China (No.10572102) and National Basic Research Program of China(No. 2007CB714000). J. Wang is also grateful to supportfrom the China Scholarship Council.

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