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Analytical solution of the harmonic response of visco-elastic adhesively bonded single-lap and double-lap joints
Author’s Accepted Manuscript Analytical solution of the harmonic response of visco-elastic adhesively bonded single-lap and double-lap joints Khalid H. Almitani, Ramzi Othman www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(16)30157-9 http://dx.doi.org/10.1016/j.ijadhadh.2016.08.004 JAAD1887
To appear in: International Journal of Adhesion and Adhesives Received date: 16 December 2015 Accepted date: 3 August 2016 Cite this article as: Khalid H. Almitani and Ramzi Othman, Analytical solution of the harmonic response of visco-elastic adhesively bonded single-lap and doublelap joints, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2016.08.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analytical solution of the harmonic response of viscoelastic adhesively bonded single-lap and double-lap joints Khalid H. Almitani, Ramzi Othman* Mechanical Engineering Department, Faculty of Engineering, King Abdul-Aziz University, P.O. Box 80248, Jeddah 21589, Saudi Arabia *Corresponding
author: [email protected] (Tel.: +966 2 640 0000.)
Abstract This paper describes a new analytical solution for adhesively-bonded single-lap and double-lap joints based on an improved shear-lag model. The derived equations assume that all substrates are of the same material. In the case of single-lap joints, the substrates have equal thicknesses. In the case of the double-lap joints, the outer substrates’ thickness is half the thickness of the inner substrate. The proposed solution takes into account the visco-elastic behavior of the adhesive and substrates. The analytical solutions of aluminum and polymer single-lap joints are compared to finite element solutions. Good agreement is then observed. Subsequently, a parametric study is undertaken and shows that the natural frequencies are sensitive to the material and geometrical properties of the adhesive layer and the substrates. Mainly, it is reported that substrates’ damping attenuates more the resonances heights than the adhesive damping does. Keywords: lap joint; adhesives; visco-elastic; harmonic response; analytical solution.
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Nomenclature General parameters : Time : Axial coordinate : Vertical coordinate : Angular frequency of the harmonic load ( ): Real part of the complex number ̃ : Complex magnitude of such as (̃
)
Geometrical parameters : Overlap length : Thickness of the substrates : Thickness of the adhesive layer Material parameters : Young’s modulus of the substrates : Shear modulus of the substrates : Poisson ratio of the substrates : Material density of the substrates : Loss factor of the substrates : Complex Young’s modulus of the substrates : Complex shear modulus of the substrates : Shear modulus of the adhesive : Loss factor of the adhesive : Complex shear modulus of the adhesive Displacements : Lower substrate axial displacement : Upper substrate axial displacement : Axial displacement at the interface between the adhesive and the lower substrate : Axial displacement at the interface between the adhesive and the upper substrate 〈
〉
∫
: The through-the-thickness average axial displacement in a cross-section of
lower substrate 〈
〉
∫
: The through-the-thickness average axial displacement in a cross-
section of upper substrate Strains : Lower substrate axial strain : Lower substrate shear strain : Upper substrate axial strain : Upper substrate shear strain : Adhesive shear strain
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Stresses : Lower substrate axial stress : Lower substrate shear stress : Upper substrate axial stress : Upper substrate shear stress : Adhesive shear stress Forces : External load applied at the right end of the lower substrate : Reaction at the left end of the upper substrate : Normal or axial force applied on a cross-section of the lower substrate : Normal or axial force applied on a cross-section of the upper substrate Constants , Waves ( ), ( ),
, ( ) and
,
( ): four waves to be determined in terms of the boundary
conditions
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1. Introduction Structural bonding using adhesives is increasingly used in multiple engineering applications. It has the main advantage of assembling dissimilar materials. It can help reducing the overall weight of structures. Thus, it is employed in terrestrial, railway, flight and marine vehicles. Adhesive bonding induces no thermal alteration of substrates, which then conserve their original mechanical properties. Moreover, it induces less stress concentration. It needs no pre-bonding operation except for surface preparation. The dynamic behavior of adhesively bonded joints is a main concern as transportation vehicles can be submitted to vibration, crash and impact. Moreover, the vibration response can be used to find out and analyze fatigue damage in adhesive joints [1-2]. This dynamic behavior can either be analyzed analytically [34], investigated numerically [5-11] or characterized experimentally [11-16]. Deriving analytical solutions mostly needs important assumptions and simplifications. Nevertheless, they have the advantage of giving closed-form solutions which can be used to carry out a first analysis of the physical problem. Since the pioneer work of Volkersen [17], multiple analytical solutions have been derived to predict the static response of adhesive joints. The reader is referred to Da Silva et al. [18-19] for a comprehensive review on the topic. Vibration response of adhesive joints can involve transverse and longitudinal motion. Most of works have dealt with the single-lap joint. Rao and Crocker [20] considered substrates as beams. He and Rao [21-22] used the energy method and Hamilton’s principle to derive vibration equations of single-lap joints. Using an Euler-Bernoulli free-free beam theory, Ingole and Chatterjee [23] derived natural frequencies and modes for longitudinal and transverse vibration of single-lap joints. Yuceoglu et al. [24] were dealing with free bending vibration of adhesively bonded orthotropic plates. Thus, they applied Mindlin plate theory and a Levy-type solution was derived. Vaziri and co-workers [25-29] were interested in void effects on the harmonic response of some adhesive joints. They assumed an elastic behavior for the 4|Page
adherends and a viscoelastic behavior for the adhesive. Vaziri et al. [25-26] investigated the effect of voids on the response of single-lap joints subjected to a peeling force. The substrates were modeled as Euler-Bernoulli beams. This model was later used by Vaziri and Nayeb-Hashemi [27] to predict the dynamic response of composite beams repaired with adhesively bonded patches. Vaziri and NayebHashemi [28] dealt also with annular void effects on torsional response of tubular joints. They assumed that the adhesive shears in the circumferential direction while the adherends deform in the axial direction. Moreover, they were interested in tubular joints with annular voids subjected to axial harmonic loads [29]. To this aim, they extended the basic shear-lag model of Volkersen [17] to dynamics by including the adhesive and adherends inertia. The solution was expressed in terms of Bessel functions. In terms of the impact response, Pang et al. [30] have considered single-lap composite joints. They first derived the impact force using a mass-spring model. The impact response of the composite joint was deduced by considering a first-order laminate plate theory for the substrates. Helms et al. [31] derived an equivalent stiffness for a hybrid laminate-metal joint. Dealing with the impact response of single-lap joints, Sato [32-34] derived the stress at the edge of the adhesive layer. He extended the simple shear-lag model [17] to include the inertia of adherends. However, the adhesive inertia was neglected. Tsai et al. [32] have developed a modified shear-lag analytical model which takes into account the shear deformation in the substrates. This model was validated under static loads applied on single- and double-lap joints. Based on this model, Challita and Othman [3] have derived the harmonic response of a double-lap joint. Moreover, Hazimeh et al. [4] obtained the impact stress response at the edge of the adhesive layer of a double-lap joint of semi-infinite length. Challita and Othman [3] and Hazimeh et al. [4] have assumed an elastic behavior for both, the adherends and the adhesive. They have neglected inertia effects in the adhesive layer. Compared to simple shear-lag solutions, the solutions derived by Challita and Othman [3] and Hazimeh et al. [4] are closer to the stress response obtained by
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finite element method, mainly, if the adhesive shear stiffness cannot be neglected when compared to the adherends’ shear stiffness. In this paper, we are interested in the harmonic response of single-lap and double-lap joints including viscoelastic properties of the adhesive and the adherends. This is motivated by the fact that adhesives are polymer materials for which the viscoelastic behavior is more appropriate than the elastic behavior. This is also the case of polymer-based substrates. The analytical solution will be compared to a numerical solution obtained by finite element analysis.
2. Methodlogy 2.1. Analytical model This work aims at predicting the harmonic response of single-lap and doublelap adhesively bonded joints. Only the overlap region will be considered in the analysis (Figure 1). The same material is used for the two substrates. In the case of the single-lap joint, the upper and lower substrates have the same thickness. In the case of the double-lap joints, the central adherend thickness is twice the thickness of the outer adhrends. The left extremities of the substrates are cantilevered; whereas, a harmonic axial loading is applied on the lower adherend’s right end of the singlelap joint or on the central adherend’s right end of the double-lap joint (Figure 2). It is worth noting that for the double-lap geometry and the load applied to it a mid-plane symmetry exists in the -direction. Thus the problem can be solved for half of the geometry, i.e., for the positive
. Besides, solving the double-lap joint problem in
region is equivalent to solving the simple-lap joint problem.
Therefore, only the single-lap joint problem will be solved. The results of the double-lap joint problem can be deduced from the results of the single-lap joint problem using symmetry. Henceforth, we will focus the analysis on the single-lap joint problem (Figures 1(a) and 2(a)). While interested in the quasi-static response, Tsai et al. [35] assumed that: (i) the adherends and adhesive move in the axial ( -)direction; (ii) the shear stress in the adherends is linear through the thickness ( -direction); and (iii) the shear stress 6|Page
in the adhesive is constant through the thickness; thus, it is proportional to the difference between the displacements at the two adhesive-adherend interfaces. Moreover, we assume here that the adhesive and the adhrends have a viscoelastic behavior, which is represented by the use of complex moduli of the materials [36]. These complex moduli can be written using a Kelvin-Voigt model [20] or simply by using a Loss factor [22-26]. The latter will be followed here. Thus, the (
complex shear modulus of the adhesive is written:
). Likewise, the
complex Young’s moduli and the complex shear moduli of the substrates are written:
(
) and
lower substrate and
(
), respectively, where
for the upper substrate.
elastic moduli and Poisson’s ratios (
,
The load is assumed harmonic:
,
and (̃
and
for the
are deduced from the
) using Hooke’s law. ), where ̃ and
are the
magnitude and angular frequency. Therefore, any time-dependent parameter (displacements, strains, stresses and forces) is assumed to be the real part of a (̃
complex number, i.e.,
). The harmonic solution is derived in terms of
the complex magnitudes of displacements, strains, stresses and forces. The analytical solution are derived here assuming a width of one unit. Considering the visco-elastic behavior of adhrends and substrates, axial stress in substrates and, the shear stress in substrates and adhesive write: ̃(
)
̃(
),
,
(1a)
̃(
)
̃(
),
,
(1b)
and ̃ (
)
̃ (
).
(1c)
The shear stress in the substrates is assumed linear through-the-thickness [35]. The shear stress at the lower surface of the lower adherend and the shear stress at the upper surface of the upper adherend are zero, i.e., ̃ ( ̃ (
)
)
and
, respectively. Consequently, the adherend’s shear stresses
write: ̃ (
)
̃ (
),
(2a)
and 7|Page
̃ (
)
(
(
)
) ̃ (
).
(2b)
The substrates are assumed to only move along the axial ( -) direction. Thus the shear strain writes ̃ (
)
̃(
). ̃ (
) can also be calculated
using Eqs. (1b) and (2). In order to obtain the displacement in the adherends, the shear strain is integrated along . After simplifications, we obtain: ̃ (
)
̃ (
)
)
̃ (
)
̃ (
),
(3a)
and ̃ (
(
)
̃ (
).
The axial strain can be deduced using ̃ (
(3b) )
̃(
). Subsequently,
Eq. (1a) is applied to deduce the axial stresses: ̃(
)
(
̃
)
(
̃
̃
),
(4a)
and ̃(
(
)
̃
).
(4b)
Later the axial forces are calculated by integrating the axial stresses throughthe-thickness of the substrates assuming a width of one unit. After simplifications, we obtain: ̃(
)
̃
̃
,
(5a)
)
̃
̃
.
(5b)
and ̃(
Considering the free-body diagrams drawn in Figure 3 and applying the second Newton’s law to parts of substrates of length ̃
give after simplification:
̃ (
)
(
) ̃ (
),
(6a)
̃ (
)
(
) ̃ (
).
(6b)
and ̃
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The shear stress in the adhesive is constant through-the-thickness. Hence, it is proportional to the difference between the displacements of the adhesiveadherends interfaces. More precisely, ̃ (
)
(̃ (
)
̃ (
))
(7)
In order to obtain a differential equation for ̃ , we first subtract Eq. (5a) from Eq. (5b) and then we differentiate with respect to . Second, we subtract Eq. (6a) from Eq. (6b). Equating the two results yields: ̃
̃
(8)
which also means that (̃
̃ )
(̃
̃ )
(9)
Likewise, we add Eq. (5a) to Eq. (5b) and then we differentiate with respect to . Besides, we add Eq. (6a) to Eq. (6b). Equating the two results gives: (̃
̃ )
(̃
̃ )
(10)
Eqs. (9) and (10) are classical second-order linear differential equations. Their solutions write: ̃ (
)
̃ (
)
̃ (
)
̃ (
)
( )
)
( ( )
( )
( )
( )
)
(11a)
and ̃ (
( )
( )
( )
,
respectively. If there is no damping (elastic behavior), are non-propagating damped waves. Besides,
(11b) is real and, ( ) and ( )
is also real and, ( ) and ( ) are
propagating non-damped waves. In this work, we are dealing with visco-elastic adhesives and substrates. Thus,
( ),
( ),
( ) and
( ) are propagating
damped waves. ( ) and ( ) propagate rightwards (increasing ), while ( ) and ( ) move leftwards (decreasing ). Eqs. (11) can be used to obtain separately ̃ and ̃ . Subsequently, the results can be substituted in Eqs. (5) to get the axial forces and in Eqs. (3) to get the displacements. After simplifications, the axial forces read: ̃(
)
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( )
( )
(
( )
( )
(
( )
( ) ( )
( )
)
),
(12a)
and ̃(
) ( )
( )
(
( )
( )
(
( )
( ) ( )
( )
)
).
(12b)
Besides, the through-the-thickness average displacements write: 〈̃ 〉 (
)
( ( )
( )
( )
( )
)
( ( )
( )
( )
( )
),
(13a)
( )
( )
( )
)
( ( )
( )
( )
( )
).
(13b)
and 〈̃ 〉 (
)
( ( )
( ),
The four waves
( ),
( ) and
( ) are determined using the boundary
conditions at the extremities of the two substrates. Namely, the right end of the upper substrate and the left end of the lower substrate are free; consequently, ̃(
)
̃(
)
. Besides, the right end of the lower adherend receives the
external load; thus, ̃ (
̃ ( ). Finally, the left end of the upper adherend is
)
cantilevered; hence, ̃ (
)
. However, we prefer not writing the boundary
condition in terms of displacements but in terms of forces. More precisely, we will consider that a force equal to the reaction of the support ̃ is rather applied to the left end of upper substrate (Figure 4), i.e., ̃ (
)
̃ ( ). This highly helps in
calculating a close-form solution. The reaction ̃ can later be determined using the boundary condition ̃ (
)
. Substituting the boundary conditions in Eqs. (12)
gives a system of four ordinary linear equations with four unknowns; the solution of which writes after simplifications: ( ) ( )
̃( )
̃ ( ) (
̃( )
)
,
(14a)
̃ ( ) (
)
,
(14b)
10 | P a g e
( )
̃( )
̃ ( ) (
,
(14c)
)
and ( )
̃( )
̃ ( ) (
)
.
(14d)
We would like now to consider the boundary condition ̃ ( )
. Actually, the
improved shear-lag model assumes that the displacement is varying with the thickness. Thus a boundary condition where the displacement is constant is impossible to write [4]. Therefore, we will consider that the average displacement instead, i.e., 〈 ̃ 〉 (
. More precisely, Eq.(13b) is considered where we
)
. Later we substitute ( ), ( ), ( ) and ( ) by their expressions
consider
in the Eqs. (14). Considering, that this average displacement is null, and solving gives the reaction at the support, which reads: ̃( )
(
)
(
)
(
)
(
)
̃( )
(15)
Eq. (15) closes the solution as it gives the reaction at the support ̃ ( ) in terms of the applied load ̃ ( ). Once ̃ ( ) is calculated, it is possible to calculate ( ), ( ), ( ) and ( ) using Eqs. (14). Subsequently it is possible to compute axial forces (Eqs. (12)), average displacements (Eqs. (13)) and the shear stress in the adhesive (Eq. (11a)). It is also possible to derive the transfer function of the system as the ratio of the average displacement at the right end of the lower substrate to the applied load. More precisely, ̃( )
〈̃ 〉 ( ) ̃ ( ) .
(16)
3. Validation ANSYS finite element package was used to validate the accuracy of the proposed mathematical model. As the out of plane dimension in both models is much greater than the other two dimensions, the out-of-plane normal and shear strains are small compared to the cross-sectional strains. As a result, the current problem can be
11 | P a g e
approximated to a plane strain problem and a two-dimensional model is sufficient to catch the dynamics of the structure. Figure 5(a) shows the 2D surfaces used to model the adherent and adhesive layers as well as the applied load and boundary conditions. The upper adherent surface is kept fixed at the left edge. A unit force applied to the right edge of the lower adherent is used as a source of excitation. All surfaces were meshed using SURF153 elements which are applicable to twodimensional structural analysis. This element has three nodes and two degrees of freedom per node which are the axial and transverse displacements. A convergence analysis was performed on the fundamental frequencies of the two models. It yields that the three-first natural frequencies are insensitive to a mesh size ranging between 0.2 and 2mm. Accordingly, the mesh size is chosen 1mm substrates and 0.5mm
1mm for the
1mm for the adhesive layer as shown in Figure 5(b).
Two simulations have been undertaken. Each time the transfer function is calculated and compared to the transfer function predicted by the analytical model. In the first simulation, the substrates are considered of aluminum. Thus, the Young’s modulus, Poisson’ ratio and density are considered equal to 69 GPa, 0.34 and 2800 kg/m3, respectively. A very low loss factor is considered: 0.001. In the second simulation, the substrates are assumed made of a polymer-like material. Hence, the Young’s modulus, Poisson’ ratio and density are considered equal to 3 GPa, 0.4 and 1800 kg/m3[3], respectively. The loss factor is 0.1. For both simulations, the adhesive has a shear modulus of 0.71 GPa, a Poisson’s ratio of 0.4, a density of 1200 kg/3 and a loss factor of 0.5. Figure 6 shows a comparison between the finite element transfer function and the analytical transfer function. In the case of aluminum substrates, the analytical model is in very good agreement with the finite element model. The magnitude of the transfer function and the natural frequency positions are well predicted. In the case of polymer-like materials, the analytical model gives a good approximation of the overall behavior. All natural frequencies are caught with a good accuracy, though the analytical model assumes a simple deformation state and 12 | P a g e
the adherends and adhesive loss factors are high. Considering the finite element results as a reference, the analytical model works better with aluminum substrates than polymer-like substrates. Two possible reasons can explain this fact. The analytical model would work better for joints with high substrates stiffness-toadhesive stiffness ratio. It is also possible that the analytical model work better for low values of damping. This is being said, the analytical model gives satisfactory results with the polymer-like substrates. This means that the analytical model has good accuracy though the substrates stiffness-to-adhesive stiffness ratio is not high and the damping is important.
4. Results and discussion In this section, we are interested in illustrating the capabilities of the analytical model derived in section 2. To this aim, several possible materials were considered for the substrates (Table 1) as well as for the adhesive layer (Table 2). First, we are interested in depicting the effects of adhesive and adherends damping. Two adhesive joints were considered: aluminum-aluminum and polymerpolymer joints (Table 1). The two joints are bonded with the same adhesive (second adhesive of Table 2). Besides, the same geometry is chosen. More precisely, the substrates and the adhesive layer thicknesses are 2 and 0.5 mm, respectively, and the overlap length is 50 mm. Figure 7 depicts the effect of the adhesive loss factor on the harmonic response of the of the aluminum-aluminum joint. Several loss factor values were considered for the adhesive: 0.001, 0.1, 0.5 and 1. The loss factor of aluminum adherends is assumed 0.001.
The harmonic response is plotted in Figure 7(a). Multiple
resonances are recorded. The first natural frequency is around 21 kHz. The natural frequencies are slightly pushed to the higher frequencies as the adhesive loss factor increases. Besides, the heights of all resonances are increasingly damped as the loss factor increases. The average adhesive shear stress shows also peaks which occur at the natural frequencies of the joint (Figure 7(b)). As for the transfer function, the peaks heights are decreasing for increasing loss factor. The dynamic average shear 13 | P a g e
stress is almost equal to the static average shear stress for a wide low frequency range (up to ~10 kHz). In the neighborhood of the natural frequencies, the dynamic average stress is around 500 times the static stress. Likewise, the maximum adhesive shear stress has peaks at the resonant frequencies (Figure 7(c)). These peaks also lose their height as the loss factor increases. The dynamic maximum stress is almost equal to the static maximum stress for the low frequency range (up to ~10 kHz). In the neighborhood of the natural frequencies, the maximum shear stress can be up to 1000 times as high as the static maximum shear stress, for the same amplitude of the input force. Similar behavior is also observed for the reaction at the cantilevered end (Figure 7(d)). The effect of damping is also investigated on the harmonic response of polymerpolymer joints. The transfer function, the adhesive average shear stress, the adhesive maximum shear stress and the reaction are plotted in Figure 8. Multiple resonances are observed. The first natural frequency is ~6.2 kHz. The effect of the damping is localized in the neighborhood of the natural frequencies. Almost no effect is observed elsewhere. Around the natural frequencies, the amplitude of the signals is reduced as the loss factor increases. Compared to the aluminum-aluminum joint, the effect of damping in the polymer-polymer joint is lower. The transfer function, the adhesive average shear stress, the adhesive maximum shear stress and the reaction are almost constant and equal to their quasi-static values, respectively, in the low frequency range (up to ~3kHz). As polymers are visco-elastic materials, the effects of adherends’ loss factor is also studied and depicted in Figure 9. Similar tendencies are observed for the transfer function, the adhesive average shear stress, the adhesive maximum shear stress and the reaction. Considering the same value of the adherends and adhesive loss factors, the peaks around the natural frequencies are more damped due to adherends’ damping than due to adhesive damping. These peaks are highly damped for adherends’ loss factor of 0.25. They are almost erased for 0.5 of loss factor. Whereas, no such effect is observed for an adhesive loss factor as high as unity. Actually, the adherends’ damping is preponderant on the adhesive damping (Figure
14 | P a g e
10). Though the adhesive loss factor is increased from 0.001 to 0.5, the same response is observed by considering adherends’ loss factor of 0.2. In addition to damping effects, the effects of materials and geometrical properties on the first three natural frequencies are also investigated and showed in Figures 11 and 12. More precisely, four adhesives are considered (Table 2) and their effects are depicted in Figure 11(a). Likewise, five materials were considered for the substrates (Table 1) and their effects are depicted in Figure 12(b). It appears, that the natural frequencies slightly increase as the adhesive shear modulus increases. When the shear stiffness is multiplied by 4, the three natural frequencies increase by less than 10%. The increase of the natural frequencies can be interpreted in terms of the shear stiffness of the adhesive joint. Actually, Higher adhesive’s shear modulus leads to higher joint’s shear stiffness and then higher natural frequencies because the natural frequencies of a mechanical system depend on the square-root of the stiffness divided by the inertia. As the analytical model neglects the adhesive inertia, no effect of this parameter can be revealed. However, the effects of substrates material reveals the competition between stiffness and inertia (Figure 11(b)). The highest natural frequencies are obtained with CFRP adherends which are ranked second in terms of Young’s modulus. The steel adherends, which have the highest Young’s modulus, are only ranked four in terms of natural frequencies. This can be interpreted in terms of the steel density. Actually, the inertia of steel adherends is about 4 times higher than the inertia of CFRP, using the same geometry. In order to consider both stiffness and inertia effects, it is more appropriate to consider √ ⁄ , which is the sound speed in the case of homogeneous material. Indeed, the natural frequencies increase as the wave speed increases, except for steel. CFRP substrates have the highest wave speed and the highest natural frequencies. Moreover, aluminum and GFRP have comparable values for the wave speed and the natural frequencies. Finally, polymer-like substrates have the lowest wave speed and the lowest natural frequencies. However, steel substrates have higher wave speed than aluminum and GFRP substrates but lower natural frequencies. Consequently, the natural frequencies are significantly 15 | P a g e
dependent on √ ⁄ . In the analytical solution, this is included in the complex wavenumber
. However, a second important parameter can influence the natural
frequencies of adhesive joint:
. This complex number parameter depend on
several parameters including the shear stiffness of the substrates, which can explain the behavior of steel joints. However, the effects of this behavior are less obvious to interpret. In terms of geometrical parameters, an increase in the overlap length, the adhesive thickness or the adherends’ thickness leads to a decrease in the natural frequencies (Figure 12). Actually, an increase in the overlap length decreases the axial stiffness of the adherends and an increase of the substrates’s thickness reduces the their shear stiffness. Likewise, an increase of the adhesive thickness decreases the shear stiffness of the adhesive layer. Therefore, an increase of the three geometrical parameters leads to a reduction of the joint’s stiffness. As this increase yields an augmentation of the joint’s inertia. Consequently, the natural frequencies are decreased.
5. Conclusion Using an improved shear-lag model, the harmonic response of single-lap and double-lap was derived accounting for the substrates’ and adhesive damping. The analytical solution was validated taking a finite element solution as a reference. A parametric study was undertaken to assess the influence of some geometrical and material parameters. It is then possible to draw the following conclusions:
The substrates and adhesive damping attenuates the resonances of the joint.
The substrates damping has bigger influence on the harmonic response of the joint than the adhesive damping.
The 1st, 2nd and 3rd natural frequencies increase as the shear stiffness of the adhesive increases. They also increase as the sound speed in the substrates increases.
16 | P a g e
These natural frequencies decrease as the overlap length, the adhesive thickness and the adherends thickness increase.
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Challita G., Othman R, Casari P, Khalil K. Experimental investigation of the shear dynamic behavior of double-lap adhesively bonded joints on a wide range of strain rates. Int J Adhes Adhes 2011; 31: 146-153.
[13]
García-Barruetabeña J, Cortés F. Experimental analysis of the vibrational response of an adhesively bonded beam. Measurement 2014; 55: 238-245.
[14]
Du Y, Shi L. Effect of vibration fatigue on model properties of single lap adhesive joints. Int J Adhes Adhes 2014; 53: 72-79.
[15]
Hazimeh R, Challita G, Khalil K, Othman R. Experimental investigation of the influence of substrates’ fibers orientations on the impact response of composite double-lap joints. Compos Struct 2015; 134: 82-89.
[16]
He X. Study on the forced vibration behavior of adhesively bonded single-lap joints. J Vibroeng 2013; 15: 169-175.
[17]
Volkersen O. Die Nietkraftverteilung in zugbeanspruchten Nietverbindungen mit konstanten Laschenquerscnitten. Luftfahrtforschung 1938; 20: 41-47.
[18]
Da Silva L, Das Neves P, Adams R, Spelt J. Analytical models of adhesively bonded joints – part I: literature survey. Int J Adhes Adhes 2009; 29: 319– 330.
[19]
Silva L, Das Neves P, Adams R, Wang A, Spelt J. Analytical models of adhesively bonded joints – part II: Comparative study. Int J Adhes Adhes 2009; 29: 331-341.
[20]
Rao MD, Crocker MJ. Analytical and Experimental Study of the Vibration of Bonded Beams With a Lap Joint. J Vib Acoust 1990; 112: 444-451. 18 | P a g e
[21]
He S, Rao MD. Vibration analysis of adhesively bonded lap joint, Part I: Theory. J Sound Vib 1992; 152: 405–416.
[22]
He S, Rao MD. Longitudinal Vibration and Damping Analysis of Adhesively Bonded Double-Strap Joints. J Vib Acoust 1992; 114: 330-337.
[23]
Ingole SB, Chatterjee A. Vibration analysis of single lap adhesive joint: Experimental and analytical investigation. J Vib Control 2010; 17: 1547– 1556.
[24]
Yuceoglu U, Toghi F, Tekinalp O. Free Bending Vibrations of Adhesively Bonded Orthotropic Plates With a Single Lap Joint. J Vib Acoust 1996; 118, 122-134.
[25]
Vaziri A, Hamidzadeh, HR, Nayeb-Hashemi H. Dynamic response of adhesively bonded single-lap joints with a void subjected to harmonic peeling loads. Proc Inst Mech Eng K: J Multi-Body Dyn 2001; 215: 199-206.
[26]
Vaziri A, Nayeb-Hashemi H, Hamidzadeh HR. Experimental and analytical investigation of the dynamic response of a lap joint subjected to harmonic peeling loads. J Vib Acoust 2004; 126: 84–91.
[27]
Vaziri A, Nayeb-Hashemi H. Dynamic response of a repaired composite beam with an adhesively bonded patch under a harmonic peeling load. Int J Adhes Adhes 2006; 26: 314-324.
[28]
Vaziri A, Nayeb-Hashemi H. Dynamic response of the tubular joint with an annular void subjected to a harmonic torsional loading. Proc Inst Mech Eng K: J Multi-Body Dy 2002; 216: 361–370.
[29]
Vaziri A, Nayeb-Hashemi H. Dynamic response of tubular joints with an annular void subjected to a harmonic axial load. Int J Adhes Adhes 2002; 22, 367-373.
[30]
Pang SS, Yang C, Zhao Y. Impact response of single lap composite joints. Compos Eng 1995; 5: 1011-1027.
[31]
Helms JE, Li G, Pang SS. Impact Response of a Composite Laminate Bonded to a Metal Substrate. J Compos Mater 2001; 35: 237-252.
19 | P a g e
[32]
Sato C. Impact behaviour of adhesively bonded joints. In: Adams RD, editor. Adhesive bonding, science, technology and applications. Cambridge: Woodhead Publishing, pp. 164–87, 2005.
[33]
Sato C. Impact. In: Da Silva LFM, Ochsner A, editors. Modeling of adhesively bonded joints. Berlin: Springer, pp. 279–304, 2008.
[34]
Sato C. Dynamic stress responses at the edges of adhesive layers in lap strap joints of half-infinite length subjected to impact loads. Int J Adhes Adhes 2009; 29: 670-677.
[35]
Tsai MY, Oplinger, DW, Morton J. Improved theoretical solutions for adhesive lap joints. Int J Solids Struct 1998; 35: 1163-1185.
[36]
Othman R. On the use of complex Young's modulus while processing polymeric Kolsky-Hopkinson bars' experiments. Int J Impact Eng 2014; 73: 123-134.
[37]
Yokoyama T. Experimental determination of impact tensile properties of adhesive butt joints with the split Hopkinson bar. J Strain Analysis 2003; 38: 233-245.
[38]
Ghasemnejad H, Argentiero Y, Tez TA, Barrington PE. Impact damage response of natural stitched single lap-joint in composite structures. Mater Des 2013; 51:552-560.
[39]
Liao L, Sawa T, Huang C. Experimental and FEM studies on mechanical properties of single-lap adhesive joint with dissimilar adherends subjected to impact tensile loadings. Int J Adhes Adhes 2013; 44: 91-98.
[40]
Higuchi I, Sawa T, Okuno H, Kato S. Three-dimensional finite element analysis of stress response in adhesive butt joints subjected to impact bending moments. J Adhes 2003; 79:1017-1039.
[41]
Khashaba UA, Aljinaidi AA, Hamed MA. Nanofillers modification of Epocast 50-A1/946 epoxy for bonded joints. Chinese J Aeronaut 2014; 27: 12881300.
20 | P a g e
Figures
Figure 1: Schematic of the unloaded (a) single-lap and (b) double-lap bonded joint.
21 | P a g e
Figure 2: Schematic of the loaded (a) single-lap and (b) double-lap bonded joint.
22 | P a g e
Figure 3: Free-body diagram of (a) upper adherend, (b) adhesive layer, and (c) lower adherend.
Figure 4: Schematic of the equivalently loaded single-lap joint.
23 | P a g e
Figure 5: ANSYS 2D model for the adherent and adhesive layers. (a) applied loads and boundary conditions, (b) mesh.
Figure 6: Validation of the analytical solution: (a) aluminum substrates, and (b) polymer-like substrates.
24 | P a g e
Figure 7: Effect of adhesive damping on aluminum-aluminum joint’s harmonic response: (a) transfer function, (b) average shear stress in the adhesive, (c) maximum shear stresses ratio, and (d) reaction at the cantilevered end. (Substrates’ damping:
)
25 | P a g e
Figure 8: Effect of adhesive damping on polymer-polymer joint’s harmonic response: (a) transfer function, (b) average shear stress in the adhesive, (c) maximum shear stresses ratio, and (d) reaction at the cantilevered end. (Substrates’ damping:
)
26 | P a g e
Figure 9: Effect of adherends’ damping on polymer-polymer joint’s harmonic response: (a) transfer function, (b) average shear stress in the adhesive, and (c) reaction at the cantilevered end. (adhesive damping:
)
27 | P a g e
Figure 10: Effect of adhesive and adherends’ damping on polymer-polymer joint’s harmonic response: (a) transfer function, and (b) average shear stress.
28 | P a g e
Figure 11: Effect of materials on the first three natural frequencies: (a) effect of the adhesive, and (b) effect of substrates’ material. (refer to Tables 1 and 2 for materials properties)
29 | P a g e
Figure 12: Effect of geometrical parameters on the first three natural frequencies: (a) effect of the overlap length, (b) effect of adhesive thickness, and (c) effect of substrates’ thickness.
30 | P a g e
List of Tables Table 1: Substrates’ material properties √ ⁄ Substrate
Material
(GPa)
(GPa)
(kg/m3)
(m/s)
Ref.
1
Steel
209
80.4
0.3
7550
5261
[34]
2
Aluminum
69
25.7
0.34
2800
4964
[26]
3
CFRP
138
6.3
1800
8755
[35]
4
GFRP
35
4
1600
4677
[35]
5
Polymer-like
3
1.07
1800
1290
[3]
0.4
Table 2: Adhesives’ properties Adhesive
(GPa)
Ref.
1
0.41
[36]
2
0.71
[3]
3
1.31
[37]
4
1.60
[38]
31 | P a g e
PII: DOI: Reference:
S0143-7496(16)30157-9 http://dx.doi.org/10.1016/j.ijadhadh.2016.08.004 JAAD1887
To appear in: International Journal of Adhesion and Adhesives Received date: 16 December 2015 Accepted date: 3 August 2016 Cite this article as: Khalid H. Almitani and Ramzi Othman, Analytical solution of the harmonic response of visco-elastic adhesively bonded single-lap and doublelap joints, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2016.08.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analytical solution of the harmonic response of viscoelastic adhesively bonded single-lap and double-lap joints Khalid H. Almitani, Ramzi Othman* Mechanical Engineering Department, Faculty of Engineering, King Abdul-Aziz University, P.O. Box 80248, Jeddah 21589, Saudi Arabia *Corresponding
author: [email protected] (Tel.: +966 2 640 0000.)
Abstract This paper describes a new analytical solution for adhesively-bonded single-lap and double-lap joints based on an improved shear-lag model. The derived equations assume that all substrates are of the same material. In the case of single-lap joints, the substrates have equal thicknesses. In the case of the double-lap joints, the outer substrates’ thickness is half the thickness of the inner substrate. The proposed solution takes into account the visco-elastic behavior of the adhesive and substrates. The analytical solutions of aluminum and polymer single-lap joints are compared to finite element solutions. Good agreement is then observed. Subsequently, a parametric study is undertaken and shows that the natural frequencies are sensitive to the material and geometrical properties of the adhesive layer and the substrates. Mainly, it is reported that substrates’ damping attenuates more the resonances heights than the adhesive damping does. Keywords: lap joint; adhesives; visco-elastic; harmonic response; analytical solution.
1|Page
Nomenclature General parameters : Time : Axial coordinate : Vertical coordinate : Angular frequency of the harmonic load ( ): Real part of the complex number ̃ : Complex magnitude of such as (̃
)
Geometrical parameters : Overlap length : Thickness of the substrates : Thickness of the adhesive layer Material parameters : Young’s modulus of the substrates : Shear modulus of the substrates : Poisson ratio of the substrates : Material density of the substrates : Loss factor of the substrates : Complex Young’s modulus of the substrates : Complex shear modulus of the substrates : Shear modulus of the adhesive : Loss factor of the adhesive : Complex shear modulus of the adhesive Displacements : Lower substrate axial displacement : Upper substrate axial displacement : Axial displacement at the interface between the adhesive and the lower substrate : Axial displacement at the interface between the adhesive and the upper substrate 〈
〉
∫
: The through-the-thickness average axial displacement in a cross-section of
lower substrate 〈
〉
∫
: The through-the-thickness average axial displacement in a cross-
section of upper substrate Strains : Lower substrate axial strain : Lower substrate shear strain : Upper substrate axial strain : Upper substrate shear strain : Adhesive shear strain
2|Page
Stresses : Lower substrate axial stress : Lower substrate shear stress : Upper substrate axial stress : Upper substrate shear stress : Adhesive shear stress Forces : External load applied at the right end of the lower substrate : Reaction at the left end of the upper substrate : Normal or axial force applied on a cross-section of the lower substrate : Normal or axial force applied on a cross-section of the upper substrate Constants , Waves ( ), ( ),
, ( ) and
,
( ): four waves to be determined in terms of the boundary
conditions
3|Page
1. Introduction Structural bonding using adhesives is increasingly used in multiple engineering applications. It has the main advantage of assembling dissimilar materials. It can help reducing the overall weight of structures. Thus, it is employed in terrestrial, railway, flight and marine vehicles. Adhesive bonding induces no thermal alteration of substrates, which then conserve their original mechanical properties. Moreover, it induces less stress concentration. It needs no pre-bonding operation except for surface preparation. The dynamic behavior of adhesively bonded joints is a main concern as transportation vehicles can be submitted to vibration, crash and impact. Moreover, the vibration response can be used to find out and analyze fatigue damage in adhesive joints [1-2]. This dynamic behavior can either be analyzed analytically [34], investigated numerically [5-11] or characterized experimentally [11-16]. Deriving analytical solutions mostly needs important assumptions and simplifications. Nevertheless, they have the advantage of giving closed-form solutions which can be used to carry out a first analysis of the physical problem. Since the pioneer work of Volkersen [17], multiple analytical solutions have been derived to predict the static response of adhesive joints. The reader is referred to Da Silva et al. [18-19] for a comprehensive review on the topic. Vibration response of adhesive joints can involve transverse and longitudinal motion. Most of works have dealt with the single-lap joint. Rao and Crocker [20] considered substrates as beams. He and Rao [21-22] used the energy method and Hamilton’s principle to derive vibration equations of single-lap joints. Using an Euler-Bernoulli free-free beam theory, Ingole and Chatterjee [23] derived natural frequencies and modes for longitudinal and transverse vibration of single-lap joints. Yuceoglu et al. [24] were dealing with free bending vibration of adhesively bonded orthotropic plates. Thus, they applied Mindlin plate theory and a Levy-type solution was derived. Vaziri and co-workers [25-29] were interested in void effects on the harmonic response of some adhesive joints. They assumed an elastic behavior for the 4|Page
adherends and a viscoelastic behavior for the adhesive. Vaziri et al. [25-26] investigated the effect of voids on the response of single-lap joints subjected to a peeling force. The substrates were modeled as Euler-Bernoulli beams. This model was later used by Vaziri and Nayeb-Hashemi [27] to predict the dynamic response of composite beams repaired with adhesively bonded patches. Vaziri and NayebHashemi [28] dealt also with annular void effects on torsional response of tubular joints. They assumed that the adhesive shears in the circumferential direction while the adherends deform in the axial direction. Moreover, they were interested in tubular joints with annular voids subjected to axial harmonic loads [29]. To this aim, they extended the basic shear-lag model of Volkersen [17] to dynamics by including the adhesive and adherends inertia. The solution was expressed in terms of Bessel functions. In terms of the impact response, Pang et al. [30] have considered single-lap composite joints. They first derived the impact force using a mass-spring model. The impact response of the composite joint was deduced by considering a first-order laminate plate theory for the substrates. Helms et al. [31] derived an equivalent stiffness for a hybrid laminate-metal joint. Dealing with the impact response of single-lap joints, Sato [32-34] derived the stress at the edge of the adhesive layer. He extended the simple shear-lag model [17] to include the inertia of adherends. However, the adhesive inertia was neglected. Tsai et al. [32] have developed a modified shear-lag analytical model which takes into account the shear deformation in the substrates. This model was validated under static loads applied on single- and double-lap joints. Based on this model, Challita and Othman [3] have derived the harmonic response of a double-lap joint. Moreover, Hazimeh et al. [4] obtained the impact stress response at the edge of the adhesive layer of a double-lap joint of semi-infinite length. Challita and Othman [3] and Hazimeh et al. [4] have assumed an elastic behavior for both, the adherends and the adhesive. They have neglected inertia effects in the adhesive layer. Compared to simple shear-lag solutions, the solutions derived by Challita and Othman [3] and Hazimeh et al. [4] are closer to the stress response obtained by
5|Page
finite element method, mainly, if the adhesive shear stiffness cannot be neglected when compared to the adherends’ shear stiffness. In this paper, we are interested in the harmonic response of single-lap and double-lap joints including viscoelastic properties of the adhesive and the adherends. This is motivated by the fact that adhesives are polymer materials for which the viscoelastic behavior is more appropriate than the elastic behavior. This is also the case of polymer-based substrates. The analytical solution will be compared to a numerical solution obtained by finite element analysis.
2. Methodlogy 2.1. Analytical model This work aims at predicting the harmonic response of single-lap and doublelap adhesively bonded joints. Only the overlap region will be considered in the analysis (Figure 1). The same material is used for the two substrates. In the case of the single-lap joint, the upper and lower substrates have the same thickness. In the case of the double-lap joints, the central adherend thickness is twice the thickness of the outer adhrends. The left extremities of the substrates are cantilevered; whereas, a harmonic axial loading is applied on the lower adherend’s right end of the singlelap joint or on the central adherend’s right end of the double-lap joint (Figure 2). It is worth noting that for the double-lap geometry and the load applied to it a mid-plane symmetry exists in the -direction. Thus the problem can be solved for half of the geometry, i.e., for the positive
. Besides, solving the double-lap joint problem in
region is equivalent to solving the simple-lap joint problem.
Therefore, only the single-lap joint problem will be solved. The results of the double-lap joint problem can be deduced from the results of the single-lap joint problem using symmetry. Henceforth, we will focus the analysis on the single-lap joint problem (Figures 1(a) and 2(a)). While interested in the quasi-static response, Tsai et al. [35] assumed that: (i) the adherends and adhesive move in the axial ( -)direction; (ii) the shear stress in the adherends is linear through the thickness ( -direction); and (iii) the shear stress 6|Page
in the adhesive is constant through the thickness; thus, it is proportional to the difference between the displacements at the two adhesive-adherend interfaces. Moreover, we assume here that the adhesive and the adhrends have a viscoelastic behavior, which is represented by the use of complex moduli of the materials [36]. These complex moduli can be written using a Kelvin-Voigt model [20] or simply by using a Loss factor [22-26]. The latter will be followed here. Thus, the (
complex shear modulus of the adhesive is written:
). Likewise, the
complex Young’s moduli and the complex shear moduli of the substrates are written:
(
) and
lower substrate and
(
), respectively, where
for the upper substrate.
elastic moduli and Poisson’s ratios (
,
The load is assumed harmonic:
,
and (̃
and
for the
are deduced from the
) using Hooke’s law. ), where ̃ and
are the
magnitude and angular frequency. Therefore, any time-dependent parameter (displacements, strains, stresses and forces) is assumed to be the real part of a (̃
complex number, i.e.,
). The harmonic solution is derived in terms of
the complex magnitudes of displacements, strains, stresses and forces. The analytical solution are derived here assuming a width of one unit. Considering the visco-elastic behavior of adhrends and substrates, axial stress in substrates and, the shear stress in substrates and adhesive write: ̃(
)
̃(
),
,
(1a)
̃(
)
̃(
),
,
(1b)
and ̃ (
)
̃ (
).
(1c)
The shear stress in the substrates is assumed linear through-the-thickness [35]. The shear stress at the lower surface of the lower adherend and the shear stress at the upper surface of the upper adherend are zero, i.e., ̃ ( ̃ (
)
)
and
, respectively. Consequently, the adherend’s shear stresses
write: ̃ (
)
̃ (
),
(2a)
and 7|Page
̃ (
)
(
(
)
) ̃ (
).
(2b)
The substrates are assumed to only move along the axial ( -) direction. Thus the shear strain writes ̃ (
)
̃(
). ̃ (
) can also be calculated
using Eqs. (1b) and (2). In order to obtain the displacement in the adherends, the shear strain is integrated along . After simplifications, we obtain: ̃ (
)
̃ (
)
)
̃ (
)
̃ (
),
(3a)
and ̃ (
(
)
̃ (
).
The axial strain can be deduced using ̃ (
(3b) )
̃(
). Subsequently,
Eq. (1a) is applied to deduce the axial stresses: ̃(
)
(
̃
)
(
̃
̃
),
(4a)
and ̃(
(
)
̃
).
(4b)
Later the axial forces are calculated by integrating the axial stresses throughthe-thickness of the substrates assuming a width of one unit. After simplifications, we obtain: ̃(
)
̃
̃
,
(5a)
)
̃
̃
.
(5b)
and ̃(
Considering the free-body diagrams drawn in Figure 3 and applying the second Newton’s law to parts of substrates of length ̃
give after simplification:
̃ (
)
(
) ̃ (
),
(6a)
̃ (
)
(
) ̃ (
).
(6b)
and ̃
8|Page
The shear stress in the adhesive is constant through-the-thickness. Hence, it is proportional to the difference between the displacements of the adhesiveadherends interfaces. More precisely, ̃ (
)
(̃ (
)
̃ (
))
(7)
In order to obtain a differential equation for ̃ , we first subtract Eq. (5a) from Eq. (5b) and then we differentiate with respect to . Second, we subtract Eq. (6a) from Eq. (6b). Equating the two results yields: ̃
̃
(8)
which also means that (̃
̃ )
(̃
̃ )
(9)
Likewise, we add Eq. (5a) to Eq. (5b) and then we differentiate with respect to . Besides, we add Eq. (6a) to Eq. (6b). Equating the two results gives: (̃
̃ )
(̃
̃ )
(10)
Eqs. (9) and (10) are classical second-order linear differential equations. Their solutions write: ̃ (
)
̃ (
)
̃ (
)
̃ (
)
( )
)
( ( )
( )
( )
( )
)
(11a)
and ̃ (
( )
( )
( )
,
respectively. If there is no damping (elastic behavior), are non-propagating damped waves. Besides,
(11b) is real and, ( ) and ( )
is also real and, ( ) and ( ) are
propagating non-damped waves. In this work, we are dealing with visco-elastic adhesives and substrates. Thus,
( ),
( ),
( ) and
( ) are propagating
damped waves. ( ) and ( ) propagate rightwards (increasing ), while ( ) and ( ) move leftwards (decreasing ). Eqs. (11) can be used to obtain separately ̃ and ̃ . Subsequently, the results can be substituted in Eqs. (5) to get the axial forces and in Eqs. (3) to get the displacements. After simplifications, the axial forces read: ̃(
)
9|Page
( )
( )
(
( )
( )
(
( )
( ) ( )
( )
)
),
(12a)
and ̃(
) ( )
( )
(
( )
( )
(
( )
( ) ( )
( )
)
).
(12b)
Besides, the through-the-thickness average displacements write: 〈̃ 〉 (
)
( ( )
( )
( )
( )
)
( ( )
( )
( )
( )
),
(13a)
( )
( )
( )
)
( ( )
( )
( )
( )
).
(13b)
and 〈̃ 〉 (
)
( ( )
( ),
The four waves
( ),
( ) and
( ) are determined using the boundary
conditions at the extremities of the two substrates. Namely, the right end of the upper substrate and the left end of the lower substrate are free; consequently, ̃(
)
̃(
)
. Besides, the right end of the lower adherend receives the
external load; thus, ̃ (
̃ ( ). Finally, the left end of the upper adherend is
)
cantilevered; hence, ̃ (
)
. However, we prefer not writing the boundary
condition in terms of displacements but in terms of forces. More precisely, we will consider that a force equal to the reaction of the support ̃ is rather applied to the left end of upper substrate (Figure 4), i.e., ̃ (
)
̃ ( ). This highly helps in
calculating a close-form solution. The reaction ̃ can later be determined using the boundary condition ̃ (
)
. Substituting the boundary conditions in Eqs. (12)
gives a system of four ordinary linear equations with four unknowns; the solution of which writes after simplifications: ( ) ( )
̃( )
̃ ( ) (
̃( )
)
,
(14a)
̃ ( ) (
)
,
(14b)
10 | P a g e
( )
̃( )
̃ ( ) (
,
(14c)
)
and ( )
̃( )
̃ ( ) (
)
.
(14d)
We would like now to consider the boundary condition ̃ ( )
. Actually, the
improved shear-lag model assumes that the displacement is varying with the thickness. Thus a boundary condition where the displacement is constant is impossible to write [4]. Therefore, we will consider that the average displacement instead, i.e., 〈 ̃ 〉 (
. More precisely, Eq.(13b) is considered where we
)
. Later we substitute ( ), ( ), ( ) and ( ) by their expressions
consider
in the Eqs. (14). Considering, that this average displacement is null, and solving gives the reaction at the support, which reads: ̃( )
(
)
(
)
(
)
(
)
̃( )
(15)
Eq. (15) closes the solution as it gives the reaction at the support ̃ ( ) in terms of the applied load ̃ ( ). Once ̃ ( ) is calculated, it is possible to calculate ( ), ( ), ( ) and ( ) using Eqs. (14). Subsequently it is possible to compute axial forces (Eqs. (12)), average displacements (Eqs. (13)) and the shear stress in the adhesive (Eq. (11a)). It is also possible to derive the transfer function of the system as the ratio of the average displacement at the right end of the lower substrate to the applied load. More precisely, ̃( )
〈̃ 〉 ( ) ̃ ( ) .
(16)
3. Validation ANSYS finite element package was used to validate the accuracy of the proposed mathematical model. As the out of plane dimension in both models is much greater than the other two dimensions, the out-of-plane normal and shear strains are small compared to the cross-sectional strains. As a result, the current problem can be
11 | P a g e
approximated to a plane strain problem and a two-dimensional model is sufficient to catch the dynamics of the structure. Figure 5(a) shows the 2D surfaces used to model the adherent and adhesive layers as well as the applied load and boundary conditions. The upper adherent surface is kept fixed at the left edge. A unit force applied to the right edge of the lower adherent is used as a source of excitation. All surfaces were meshed using SURF153 elements which are applicable to twodimensional structural analysis. This element has three nodes and two degrees of freedom per node which are the axial and transverse displacements. A convergence analysis was performed on the fundamental frequencies of the two models. It yields that the three-first natural frequencies are insensitive to a mesh size ranging between 0.2 and 2mm. Accordingly, the mesh size is chosen 1mm substrates and 0.5mm
1mm for the
1mm for the adhesive layer as shown in Figure 5(b).
Two simulations have been undertaken. Each time the transfer function is calculated and compared to the transfer function predicted by the analytical model. In the first simulation, the substrates are considered of aluminum. Thus, the Young’s modulus, Poisson’ ratio and density are considered equal to 69 GPa, 0.34 and 2800 kg/m3, respectively. A very low loss factor is considered: 0.001. In the second simulation, the substrates are assumed made of a polymer-like material. Hence, the Young’s modulus, Poisson’ ratio and density are considered equal to 3 GPa, 0.4 and 1800 kg/m3[3], respectively. The loss factor is 0.1. For both simulations, the adhesive has a shear modulus of 0.71 GPa, a Poisson’s ratio of 0.4, a density of 1200 kg/3 and a loss factor of 0.5. Figure 6 shows a comparison between the finite element transfer function and the analytical transfer function. In the case of aluminum substrates, the analytical model is in very good agreement with the finite element model. The magnitude of the transfer function and the natural frequency positions are well predicted. In the case of polymer-like materials, the analytical model gives a good approximation of the overall behavior. All natural frequencies are caught with a good accuracy, though the analytical model assumes a simple deformation state and 12 | P a g e
the adherends and adhesive loss factors are high. Considering the finite element results as a reference, the analytical model works better with aluminum substrates than polymer-like substrates. Two possible reasons can explain this fact. The analytical model would work better for joints with high substrates stiffness-toadhesive stiffness ratio. It is also possible that the analytical model work better for low values of damping. This is being said, the analytical model gives satisfactory results with the polymer-like substrates. This means that the analytical model has good accuracy though the substrates stiffness-to-adhesive stiffness ratio is not high and the damping is important.
4. Results and discussion In this section, we are interested in illustrating the capabilities of the analytical model derived in section 2. To this aim, several possible materials were considered for the substrates (Table 1) as well as for the adhesive layer (Table 2). First, we are interested in depicting the effects of adhesive and adherends damping. Two adhesive joints were considered: aluminum-aluminum and polymerpolymer joints (Table 1). The two joints are bonded with the same adhesive (second adhesive of Table 2). Besides, the same geometry is chosen. More precisely, the substrates and the adhesive layer thicknesses are 2 and 0.5 mm, respectively, and the overlap length is 50 mm. Figure 7 depicts the effect of the adhesive loss factor on the harmonic response of the of the aluminum-aluminum joint. Several loss factor values were considered for the adhesive: 0.001, 0.1, 0.5 and 1. The loss factor of aluminum adherends is assumed 0.001.
The harmonic response is plotted in Figure 7(a). Multiple
resonances are recorded. The first natural frequency is around 21 kHz. The natural frequencies are slightly pushed to the higher frequencies as the adhesive loss factor increases. Besides, the heights of all resonances are increasingly damped as the loss factor increases. The average adhesive shear stress shows also peaks which occur at the natural frequencies of the joint (Figure 7(b)). As for the transfer function, the peaks heights are decreasing for increasing loss factor. The dynamic average shear 13 | P a g e
stress is almost equal to the static average shear stress for a wide low frequency range (up to ~10 kHz). In the neighborhood of the natural frequencies, the dynamic average stress is around 500 times the static stress. Likewise, the maximum adhesive shear stress has peaks at the resonant frequencies (Figure 7(c)). These peaks also lose their height as the loss factor increases. The dynamic maximum stress is almost equal to the static maximum stress for the low frequency range (up to ~10 kHz). In the neighborhood of the natural frequencies, the maximum shear stress can be up to 1000 times as high as the static maximum shear stress, for the same amplitude of the input force. Similar behavior is also observed for the reaction at the cantilevered end (Figure 7(d)). The effect of damping is also investigated on the harmonic response of polymerpolymer joints. The transfer function, the adhesive average shear stress, the adhesive maximum shear stress and the reaction are plotted in Figure 8. Multiple resonances are observed. The first natural frequency is ~6.2 kHz. The effect of the damping is localized in the neighborhood of the natural frequencies. Almost no effect is observed elsewhere. Around the natural frequencies, the amplitude of the signals is reduced as the loss factor increases. Compared to the aluminum-aluminum joint, the effect of damping in the polymer-polymer joint is lower. The transfer function, the adhesive average shear stress, the adhesive maximum shear stress and the reaction are almost constant and equal to their quasi-static values, respectively, in the low frequency range (up to ~3kHz). As polymers are visco-elastic materials, the effects of adherends’ loss factor is also studied and depicted in Figure 9. Similar tendencies are observed for the transfer function, the adhesive average shear stress, the adhesive maximum shear stress and the reaction. Considering the same value of the adherends and adhesive loss factors, the peaks around the natural frequencies are more damped due to adherends’ damping than due to adhesive damping. These peaks are highly damped for adherends’ loss factor of 0.25. They are almost erased for 0.5 of loss factor. Whereas, no such effect is observed for an adhesive loss factor as high as unity. Actually, the adherends’ damping is preponderant on the adhesive damping (Figure
14 | P a g e
10). Though the adhesive loss factor is increased from 0.001 to 0.5, the same response is observed by considering adherends’ loss factor of 0.2. In addition to damping effects, the effects of materials and geometrical properties on the first three natural frequencies are also investigated and showed in Figures 11 and 12. More precisely, four adhesives are considered (Table 2) and their effects are depicted in Figure 11(a). Likewise, five materials were considered for the substrates (Table 1) and their effects are depicted in Figure 12(b). It appears, that the natural frequencies slightly increase as the adhesive shear modulus increases. When the shear stiffness is multiplied by 4, the three natural frequencies increase by less than 10%. The increase of the natural frequencies can be interpreted in terms of the shear stiffness of the adhesive joint. Actually, Higher adhesive’s shear modulus leads to higher joint’s shear stiffness and then higher natural frequencies because the natural frequencies of a mechanical system depend on the square-root of the stiffness divided by the inertia. As the analytical model neglects the adhesive inertia, no effect of this parameter can be revealed. However, the effects of substrates material reveals the competition between stiffness and inertia (Figure 11(b)). The highest natural frequencies are obtained with CFRP adherends which are ranked second in terms of Young’s modulus. The steel adherends, which have the highest Young’s modulus, are only ranked four in terms of natural frequencies. This can be interpreted in terms of the steel density. Actually, the inertia of steel adherends is about 4 times higher than the inertia of CFRP, using the same geometry. In order to consider both stiffness and inertia effects, it is more appropriate to consider √ ⁄ , which is the sound speed in the case of homogeneous material. Indeed, the natural frequencies increase as the wave speed increases, except for steel. CFRP substrates have the highest wave speed and the highest natural frequencies. Moreover, aluminum and GFRP have comparable values for the wave speed and the natural frequencies. Finally, polymer-like substrates have the lowest wave speed and the lowest natural frequencies. However, steel substrates have higher wave speed than aluminum and GFRP substrates but lower natural frequencies. Consequently, the natural frequencies are significantly 15 | P a g e
dependent on √ ⁄ . In the analytical solution, this is included in the complex wavenumber
. However, a second important parameter can influence the natural
frequencies of adhesive joint:
. This complex number parameter depend on
several parameters including the shear stiffness of the substrates, which can explain the behavior of steel joints. However, the effects of this behavior are less obvious to interpret. In terms of geometrical parameters, an increase in the overlap length, the adhesive thickness or the adherends’ thickness leads to a decrease in the natural frequencies (Figure 12). Actually, an increase in the overlap length decreases the axial stiffness of the adherends and an increase of the substrates’s thickness reduces the their shear stiffness. Likewise, an increase of the adhesive thickness decreases the shear stiffness of the adhesive layer. Therefore, an increase of the three geometrical parameters leads to a reduction of the joint’s stiffness. As this increase yields an augmentation of the joint’s inertia. Consequently, the natural frequencies are decreased.
5. Conclusion Using an improved shear-lag model, the harmonic response of single-lap and double-lap was derived accounting for the substrates’ and adhesive damping. The analytical solution was validated taking a finite element solution as a reference. A parametric study was undertaken to assess the influence of some geometrical and material parameters. It is then possible to draw the following conclusions:
The substrates and adhesive damping attenuates the resonances of the joint.
The substrates damping has bigger influence on the harmonic response of the joint than the adhesive damping.
The 1st, 2nd and 3rd natural frequencies increase as the shear stiffness of the adhesive increases. They also increase as the sound speed in the substrates increases.
16 | P a g e
These natural frequencies decrease as the overlap length, the adhesive thickness and the adherends thickness increase.
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Figures
Figure 1: Schematic of the unloaded (a) single-lap and (b) double-lap bonded joint.
21 | P a g e
Figure 2: Schematic of the loaded (a) single-lap and (b) double-lap bonded joint.
22 | P a g e
Figure 3: Free-body diagram of (a) upper adherend, (b) adhesive layer, and (c) lower adherend.
Figure 4: Schematic of the equivalently loaded single-lap joint.
23 | P a g e
Figure 5: ANSYS 2D model for the adherent and adhesive layers. (a) applied loads and boundary conditions, (b) mesh.
Figure 6: Validation of the analytical solution: (a) aluminum substrates, and (b) polymer-like substrates.
24 | P a g e
Figure 7: Effect of adhesive damping on aluminum-aluminum joint’s harmonic response: (a) transfer function, (b) average shear stress in the adhesive, (c) maximum shear stresses ratio, and (d) reaction at the cantilevered end. (Substrates’ damping:
)
25 | P a g e
Figure 8: Effect of adhesive damping on polymer-polymer joint’s harmonic response: (a) transfer function, (b) average shear stress in the adhesive, (c) maximum shear stresses ratio, and (d) reaction at the cantilevered end. (Substrates’ damping:
)
26 | P a g e
Figure 9: Effect of adherends’ damping on polymer-polymer joint’s harmonic response: (a) transfer function, (b) average shear stress in the adhesive, and (c) reaction at the cantilevered end. (adhesive damping:
)
27 | P a g e
Figure 10: Effect of adhesive and adherends’ damping on polymer-polymer joint’s harmonic response: (a) transfer function, and (b) average shear stress.
28 | P a g e
Figure 11: Effect of materials on the first three natural frequencies: (a) effect of the adhesive, and (b) effect of substrates’ material. (refer to Tables 1 and 2 for materials properties)
29 | P a g e
Figure 12: Effect of geometrical parameters on the first three natural frequencies: (a) effect of the overlap length, (b) effect of adhesive thickness, and (c) effect of substrates’ thickness.
30 | P a g e
List of Tables Table 1: Substrates’ material properties √ ⁄ Substrate
Material
(GPa)
(GPa)
(kg/m3)
(m/s)
Ref.
1
Steel
209
80.4
0.3
7550
5261
[34]
2
Aluminum
69
25.7
0.34
2800
4964
[26]
3
CFRP
138
6.3
1800
8755
[35]
4
GFRP
35
4
1600
4677
[35]
5
Polymer-like
3
1.07
1800
1290
[3]
0.4
Table 2: Adhesives’ properties Adhesive
(GPa)
Ref.
1
0.41
[36]
2
0.71
[3]
3
1.31
[37]
4
1.60
[38]
31 | P a g e