On axisymmetric adhesive joints with graded interface stiffness

International Journal of Adhesion & Adhesives 41 (2013) 57–72

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On axisymmetric adhesive joints with graded interface stiffness S. Kumar a,n, J.P. Scanlan b a b

Materials Engineering Department, University of California Santa Barbara, CA 93106-5050, USA Computational Engineering Design Centre, School of Engineering Sciences, University of Southampton, Highfield SO17 1BJ, UK

a r t i c l e i n f o

abstract

Article history: Accepted 5 August 2012 Available online 5 October 2012

An improved analytical model is presented for the stress analysis of interface stiffness graded axisymmetric adhesive joints. The governing integro-differential equation of the problem is obtained through a variational method which minimizes the complementary energy of the bonded assembly. The joint is composed of similar or dissimilar polar anisotropic and/or isotropic adherends and a functionally modulus graded bondline (FMGB) adhesive. The elastic modulus of the adhesive is functionally graded along the bondlength by assuming smooth modulus profiles which reflect the behavior of practically producible graded bondline. Influence of non-zero radial stresses in the bonded system on shear and normal stresses is evaluated. The stress distribution predicted by this refined model is compared with that of mono-modulus bondline (MMB) model for the same axial tensile load in order to estimate reduction in shear and normal stress peaks in the bondline and the adherends. A systematic parametric study indicates that an optimum joint strength can be achieved by employing a stiffness graded bondline with an appropriate combination of geometrical and material properties of the adherends. This model can also be applied to examine the effects of loss of interface stiffness due to an existing defect and/or damage in the bondline. & 2012 Published by Elsevier Ltd.

Keywords: Adhesive joint Graded interface Stress analysis Material tailoring

1. Introduction Adhesive bonding is almost ubiquitous and is increasingly being used in a wide spectrum of industries to realize more efficient, cost-effective structural connections involving a variety of material combinations. Plane transmission of forces yields homogenous distribution of stresses in the bonded region. However, steep stress gradients exist at the ends of the overlap. Stiffness mismatch between the adherends can further amplify the stress concentrations at the ends of overlap as it changes the distribution of load. Accurate estimation of stresses is the key to optimal design and longevity assessment of these structural systems. Because of the complexities associated with modeling of these multi-material systems, exact analytical treatment is hopelessly complicated. The existing analytical solutions have, therefore, been developed under certain simplifying assumptions on stress fields, treating the adherend–adhesive interface as either strong (both stresses and displacements are continuous across the interface) or weak (the displacement may or may not be continuous across interface). The elastic weak interfaces, sometimes called as spring-layer models, assume that the stresses are the functions of

n

Corresponding author. Tel.: þ1 805 893 2739; fax: þ1 805 893 8486. E-mail address: [email protected] (S. Kumar).

0143-7496/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijadhadh.2012.09.001

displacement jump across interface [1]. A few researchers, neglected the adhesive in the system and developed models treating the adherends as membranes [2,3]. Volkersen [4] and Erdogan and Ratwani [5] assumed that the adherends are membranes and the adhesive is a shear spring. Subsequently, models have been developed assuming the adherends as plates and the adhesive as a tension-shear spring [6–9]. A purely elastic analysis of Hart-Smith [8] represents a considerable improvement over classical solution of Goland and Reissner [6]. Hart-Smith, also examined elastic–plastic adhesive behavior in shear in his study [8]. Chen and Nelson [10] and Hart-Smith [8] proposed models to predict stress distributions in bonded materials owing to coefficient of thermal expansion (CTE) mismatch between these materials following Goland and Reissner [6]. On the other hand, finite element (FE) methods have been used increasingly for the past four decades as it has the ability to cope with any complex geometry and material models in order to capture the stress gradients both along and through the adhesively bonded systems (see, e.g., [11,12]). Wooley and Carver conducted a geometrically linear finite element analysis on bonded lap joints to predict stress concentration factors [13]. Alwar and Nagaraja conducted axisymmetric FE analysis of tubular joints under axial load and demonstrated the significance of viscoelastic behavior of adhesive on structural response [14]. Adams and Peppiatt [15] carried out axisymmetric FE studies on tubular joints considering both axial and torsional loads and

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Nomenclature E1, n1

Young’s modulus and Poison’s ratio of an isotropic inner adherend respectively E2, n2 Young’s modulus and Poison’s ratio of an isotropic outer adherend respectively E, n Young’s modulus and Poison’s ratio of the MMB adhesive Eil, Eit, nitl , nitt , Gitl elastic properties of polar anisotropic adherends (i¼ 1 for inner adherend and i¼ 2 for outer adherend) Ef1, Ef2, Ef3, Ef4, Ef5 modulus functions of the FMGB adhesive Em, Eo maximum and minimum values of Young’s modulus of the FMGB adhesive respectively E(z) generic modulus function of the FMGB adhesive a, b inner and outer radii of the inner adherend respectively c, d inner and outer radii of the outer adherend respectively t1, t2 thickness of inner and outer adherends respectively

compared their predictions with that of Goland and Reissner [6] for axial load case. Alwar and Nagaraja subsequently continued FE study on tubular joints, incorporating nonlinear elastic behavior of adhesive and found that the linear elastic behavior of adhesive underestimates the stresses [16]. Delale et al. [17] proposed a plane strain closed form solution for single lap joints under various loading conditions by treating the adherends to be plates and neglecting thickness variation of stresses in the adhesive layer following Goland and Reissner [6] and verified their analytical predictions with FE results treating both adherends and adhesive to be elastic continua. Pickett and Hollaway performed theoretical and finite element studies on lap joints with elastic– perfectly plastic adhesive [18]. Reddy and Roy used an updated Lagrangian formulation to develop a 2D finite element for the analysis of adhesively bonded joints accounting for geometric nonlinearity and investigated the effect of boundary conditions and mesh on the stress distributions in lap joints [19]. Bigwood and Crocombe proposed simple elastic design formulae for bonded joints ensuring strain continuity at the adherend– adhesive interface and assuming a 2D plane strain state [20]. Oplinger analyzed the effects of adherend deflection in single lap joints [21]. Tsai and Morton evaluated the single-lap joint analytically and compared with nonlinear finite element analysis results [22]. Tsai et al. proposed improved theoretical solutions for adhesively bonded single- and double-lap joints [23]. Pandey’s group conducted 2D [24] and 3D [25] geometrically nonlinear FE studies on single lap joints, considering viscoplastic constitutive behavior of the adhesive and identified a decrease of peel and shear stresses in the adhesive interlayer. Zou et al. [26] adopted a similar approach of Delale et al. [17] but considered thickness variation of shear and peal stresses to develop an analytical model and compared the results with FE studies. Luo and Tong presented a geometrically nonlinear analytical solution for composite singlelap adhesive joints treating both adherends and adhesive as beams and compared their predictions with geometrically nonlinear FE results [27]. Recently, Kumar and Pandey performed nonlinear FE studies on single-lap joints to predict fatigue crack initiation life [28]. Most of the analytical studies cited above failed to satisfy traction-free boundary conditions (BCs) at the ends of the overlap and/or omitted thickness variation of the stresses. Several researchers have proposed 2D analytical solutions for lap joints, overcoming these limitations (see, e.g., [29–33]). Comprehensive

t P L r, y, z

thickness of the adhesive layer axial tensile load bondlength of the joint radial, circumferential and axial coordinates of the tubular system respectively Z ¼ z=L normalized axial distance over bondlength of the joint q, f axial edge stresses in the inner and outer adherends of the jointed portion respectively ðiÞ ðiÞ ðiÞ sðiÞ rr , syy , szz , trz stress components in the bonded assembly (i¼1 for inner adherend, i¼2 for outer adherend and i ¼a for adhesive) ðiÞ ðiÞ ðiÞ eðiÞ rr , eyy , ezz , grz strain components in the bonded assembly (i¼1 for inner adherend, i¼2 for outer adherend and i ¼a for adhesive) P1 , P2 , P3 complementary energy in the inner adherend, outer adherend and adhesive respectively P complementary energy of the bonded system

review on analytical models of adhesively bonded systems by da Silva et al. indicates that almost all the analytical models reported thus far in the literature are 2D implying that the 2D solutions are generally sufficient because the stresses in the width direction are significantly lower than those of the loading direction [34]. Nonetheless, there are practical situations, for instance, bonded patches under in-plane loading (see, e.g., [35–37]) can experience significant stresses in the width direction. Various techniques have been used to minimize the stress concentrations at the ends of the overlap in lap joints in order to maximize their structural capacity. These include modifying the adherend geometry (see, e.g., [5,38,39]), the adhesive geometry (see, e.g., [40]) and the spew geometry (see, e.g., [41–44]). A few researchers (e.g., [45]) employed a stiff bondline adhesive in place of a compliant one to enhance the lap-shear strength of the joints. Nonetheless, in this case, adhesives are prone to interfacial and/or cohesive brittle failure (see Fig. 1) owing to high peel stresses they experience. For composite laminates, resistance to peel stresses may be considerably lower, so even greater care must be taken with these materials to minimize peel stresses. Currently a lot of research effort has been focussed on the design and development of adhesive materials at various length scales in order to enhance/ optimise their macroscopic mechanical properties (see, e.g., [46]). Low stiffness and strength of the adhesive interlayer make it structurally weaker than the adherends and hence the design objective is to minimize the stresses at the interface so as to maximize the structural efficiency of the system. Stress estimation in the adhesive interlayer is complicated by its nonlinear stress– strain relationship (see, e.g., [47]), time dependent behavior (see, e.g., [48–50]), and its sensitivity to temperature and humidity [51]. Interfacial failure in adhesive joints generally originate either from the stress singular corners (see, Fig. 1) or from a pre-existing flaw. Physically these singularities correspond to regions of high stress at which plastic flow or even crack initiates [52]. Singular stress field has been used to predict crack initiation (see, e.g., [52–55]). If singular stress field considering plastic behavior of the material is of interest to the reader, the paper by Hutchinson [56] gives ample information to start. Initiated crack may grow during service (see, e.g., [57–62]), leading to loss of structural integrity of the system. However, we do not focus on singular stress fields in this study. In general interfacial failure is difficult to predict. A few researchers (see, e.g., Peretz [63]) claim that there is

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59

Fig. 1. Potential failure modes in an adhesively bonded tubular joint. S1 , S2 , S3 and S4 are stress singular points.

effectively an interlayer of adhesive between the adherend and bulk of the adhesive owing to different cure conditions close to the adherend and has different material properties from the bulk which can affect the results of stress analysis and can potentially lead to different failure properties. However, recent experimental study clarifies that there is neither an evidence of an interphase at the adherend–adhesive interface nor any change in modulus for different thicknesses of the adhesive, though there may be slight variations in composition close to the substrates [64]. Generally, failure tends to be mixed mode (see Fig. 1), i.e., a combination of interfacial and cohesive failure [65,66]. Research work on adhesively bonded structural systems with stiffness graded interfaces was pioneered by Raphael [67], HartSmith [68] and Srinivas [69], with an objective of increasing the structural capacity by redistributing the bondline stresses. Recently, there is growing interest in this area. Sancaktar and Kumar predicted optimum lap joint strength by use of rubber toughening in epoxy adhesive layer at the ends of the overlap [70]. Subsequently, it has been demonstrated experimentally and/ or numerically that the joint strength can be increased by employing more than one adhesive in the bondline [71–76]. All these investigators have considered only one-step variation in adhesive modulus over the bondlength. Recent study on cylindrical lap joints, considering a multi-step variation of modulus of adhesive along the bondlength, was the first attempt to provide an analytical framework for the stress analysis of interface stiffness graded bonded systems [77,78]. In this current study, following the analytical work of Kumar [77], we develop improved analytical models to accurately predict the stress distribution in the members of the bonded assembly as a function of geometrical and mechanical properties of the system under axial tensile load. These analytical models are based on simplifying assumptions on the behavior of adhesive and adherends that lead to treatable mathematical formulations. Alternatively, one can adopt a material surface treatment of the adhesive and the joined components to study the behavior of the bonded system as the thickness and the stiffness of the adhesive layer tend to zero. This is commonly referred to as asymptotic approach. Extensive body of literature is available on asymptotic analysis of thin layered structures (see, e.g., [79–85]). Asymptotic approach could be used as a check to ascertain the validity of the assumptions adopted and the accuracy of the solution reported in the work. Nevertheless, this is left to a subsequent study.

2. Axisymmetric model

stresses (sðiÞ rr , here, i denotes each sub-system in the bonded system) in order to develop a simple model. In this present study, we account for the non-zero radial stresses (sðiÞ rr a 0) in the bonded assembly and develop a refined theoretical framework to determine the stress state in the bonded system while using an adhesive whose modulus varies along the bondlength of the joint. Consider two tubes of different materials (polar anisotropic and/or isotropic) and dimensions as shown in Fig. 2a. The two tubes are lap-jointed by a FMGB adhesive. The joint is subjected to an axial tensile load P. Fig. 2b shows the coordinate system with coordinates r and z and the edge stresses (q and f) of the bonded portion whose length is L. The task here is to determine the stress distribution in the graded adhesive layer and the adherends of the joint under the action of tensile load P. The following assumptions have been adopted in this axisymmetric model.

 The bending load experienced by the adherends due to



 

eccentric load path is neglected and hence the longitudinal stresses in the inner and outer adherends are assumed not to depend on the transverse coordinate r and hence they are ð2Þ functions of the axial coordinate z only, i.e., sð2Þ zz ¼ szz ðzÞ and ð1Þ ð1Þ szz ¼ szz ðzÞ. The radial stresses both in inner and outer adherends are ð1Þ ð2Þ ð2Þ functions of the radius r only. i.e., sð1Þ rr ¼ srr ðrÞ; srr ¼ srr ðrÞ and the radial stress in the adhesive is assumed to be constant, i.e., srrðaÞ ¼ w. Axisymmetric condition implies that the following shear stresses are zero. i.e., tðiÞ ¼ 0, tðiÞ ¼ 0 in all three domains. ry zy For a thin adhesive, the thickness variation of shear stress is ðaÞ very small and, hence, the longitudinal stress szz in the adhesive may be neglected as compared with shear stress ðaÞ trz . This assumption has been justified by a detailed study, employing a high order semi-elastic adhesive layer model and found that the longitudinal tensile stress in the adhesive is negligible [86].

The non-zero stress components in the bonded system are: ð1Þ ð1Þ ð1Þ  Inner adherend: sð1Þ rr ðrÞ, trz ðr,zÞ, syy ðr,zÞ, szz ðzÞ. ðaÞ ðaÞ ðaÞ  Adhesive: srr , trz ðr,zÞ, syy ðzÞ. ð2Þ ð2Þ ð2Þ  Outer adherend: sð2Þ rr ðrÞ, trz ðr,zÞ, syy ðr,zÞ, szz ðzÞ.

Accommodating the assumptions stated above, the continuum differential equations of equilibrium [87] are reduced to the following: ðiÞ

It has been demonstrated that the static load carrying capacity of the adhesively bonded cylindrical joints can be significantly improved by employing a functionally modulus graded bondline (FMGB) adhesive in lieu of a mono-modulus bondline (MMB) adhesive [77,78]. In their studies, authors omitted the radial

s 1@ @tðiÞ rz ðr sðiÞ  yy ¼ 0 rr Þ þ r @r @z r

ð1Þ

1@ @sðiÞ zz ðr tðiÞ ¼0 rz Þ þ r @r @z

ð2Þ

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Fig. 2. (a) Adhesively bonded tubular joint. (b) Coordinate system ðr, y,zÞ and edge stresses on jointed portion.

Considering equilibrium of an elemental length dz of the inner tube, the shear stress tð1Þ rz can be given by

tð1Þ rz ðr,zÞ ¼

ðr 2 a2 Þ dsð1Þ zz 2r dz

ð7Þ

ð1Þ rz

ð1Þ rr

Using t given by Eq. (7) and s given by Eq. (6) in the equilibrium equation (1), we obtain the tangential stress in the inner adherend as 2

2 2 sð1Þ yy ðr,zÞ ¼ w1 ða 3r Þ þ

Fig. 3. Variation of radial stress across the radius of the bonded system: i¼ 1 for inner adherend, i¼2 for outer adherend and i¼ a for adhesive.

ðr 2 a2 Þ d sð1Þ zz 2 2 dz

ð8Þ

2.1.2. Adhesive The radial stress in the adhesive is assumed to be constant in the adhesive as it is very thin compared to adherends and it can be expressed by

srrðaÞ ¼ w Here, i ¼1 for inner adherend, i¼2 for outer adherend and i¼a for adhesive. The stress field in the axisymmetric system should satisfy the equations of equilibrium, the traction boundary conditions prescribed at z ¼ 0; z ¼ L, the traction-free boundary conditions and the conditions of stress continuity across the dividing surfaces ðr ¼ b; r ¼ cÞ. The equilibrium of the bonded system gives the following relationship between q and f: 2

2

2

2

2

2 ð2Þ 2 ðaÞ 2 qðb a2 Þ ¼ f ðd c2 Þ ¼ sð1Þ zz ðb a Þ þ szz ðd c Þ þ szz ðc b Þ

ð3Þ

ðaÞ zz

Noting that the longitudinal stress in the adhesive, s is zero, the longitudinal stress in the outer adherend is given by ð2Þ zz

ð1Þ zz

s ¼ f þ rs

ð4Þ

where

ð9Þ

where w is another constant. Similarly considering equilibrium of the elemental length dz of the inner adherend and adhesive ðaÞ together, we can express trz as 2

tðaÞ rz ðr,zÞ ¼

ðb a2 Þ dsð1Þ zz 2r dz

ð10Þ ðaÞ rz

ðaÞ rr ðaÞ

Again, using expressions for t and s in the equilibrium equation (1), the circumferential stress syy in the adhesive is obtained as 2 2 sðaÞ yy ðzÞ ¼ w1 ða b Þ þ

2

2

ðb a2 Þ d sð1Þ zz 2 2 dz

ð11Þ

Note that the circumferential stress in the adhesive is independent of r since we assumed that szz is negligible.

2

r¼

ðb a2 Þ

ð5Þ

2

ðc2 d Þ

2.1. Stress fields in the bonded assembly

2.1.3. Outer adherend The radial stress in the outer adherend sð2Þ rr varies nonlinearly with r (see Fig. 3) and is given by 2 2 sð2Þ rr ¼ w2 ðd r Þ

ðiÞ rr )

ð12Þ

In this refined model, radial stresses in the adherends (s are assumed to vary as a nonlinear function of the radius r, while the radial stress in the adhesive (sðaÞ rr ) is assumed to be constant in the adhesive layer as shown in Fig. 3.

w2 is yet another constant. To ensure the continuity of radial stress at the interfaces (r ¼ b,r ¼ c), the following condition needs to be satisfied:

2.1.1. Inner adherend The radial stress in the inner adherend is assumed to be of the form

Considering equilibrium of elemental length dz of the outer adherend, the shear stress can be expressed by

2 2 sð1Þ rr ¼ w1 ða r Þ

ð6Þ

where w1 is a constant. w1 depends upon material and geometrical properties as well as the loading condition of the joint.

w2 ¼ w1 r

ð13Þ

2

tð2Þ rz ðr,zÞ ¼

ðr 2 d Þ dsð2Þ zz 2r dz

ð14Þ

Applying the shear stress continuity condition at the adhesive– ðaÞ adherend outer interface (tð2Þ rz at r ¼c is equal to trz at r ¼c), we

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61

can relate the longitudinal stress gradient of adherends as given by dsð2Þ dsð1Þ zz ¼ r zz dz dz

ð15Þ

Using the above in Eq. (14), we get 2

tð2Þ rz ðr,zÞ ¼ r

ðr 2 d Þ dsð1Þ zz 2r dz

ð16Þ

ð2Þ Now, using expressions for sð2Þ rr , w2 and trz ðr,zÞ in equilibrium equation (1), the tangential stress in adherend 2 can be written as 2 2 sð2Þ yy ðr,zÞ ¼ rw1 ðd 3r Þ þ r

2

2

ðr 2 d Þ d sð1Þ zz 2 2 dz

ð17Þ

Note that the stress components (radial and shear) are continuous across the adherend–adhesive interfaces. The stress ð1Þ ð1Þ components in the inner adherend [sð1Þ rr ðrÞ, trz ðr,zÞ, syy ðr,zÞ], in ðaÞ ðaÞ ðaÞ the adhesive [srr , trz ðr,zÞ, syy ðr,zÞ] and in the outer adherend ð2Þ ð2Þ ð2Þ [sð2Þ rr ðrÞ, trz ðr,zÞ, syy ðr,zÞ, szz ðzÞ] are expressed in terms of a single unknown stress function sð1Þ zz ðzÞ. This statically determinate problem is solved by applying the traction boundary conditions prescribed at the ends of overlap. The boundary conditions are

sð1Þ sð1Þ zz ð0Þ ¼ q, zz ðLÞ ¼ 0 ðaÞ rz ðr,0Þ ¼ 0,

t

t

ðaÞ rz ðr,LÞ ¼

ð18Þ 0, r A ½b,c

Fig. 5. Variation of Young’s modulus of the adhesive over the bondlength.

ð19Þ

Note that the stress fields defined above also satisfy the traction boundary conditions at r ¼a and r ¼d. 2.2. Stiffness graded adhesive interlayer The concept of stiffness grading of interface in adhesive joints was initially pursued by Raphael [67] and Hart-Smith [68]. Recently, it has been demonstrated experimentally and/or numerically that the joint strength can be increased by grading the elastic properties of the bondline (see, e.g., [72,88,75,73]). All these investigators have considered single-step, discontinuous variation of adhesive modulus over the bondlength. In this study, following the analytical work of Kumar [77], we consider continuous variation of modulus along the bondlength as shown in Fig. 4. In this figure, the interval of modulus chosen reflects the

range of modulus of structural adhesives being used in bonding applications. The smooth variation of bondline modulus can be obtained by applying a number of rings of adhesive of different moduli in the bondline. The brittle ones are applied in the middle portion of the bondline while the compliant ones are applied at the overlap end zones where steep stress gradients are expected. As the thickness of individual rings tend to zero, the multimodulus bondline exactly represents the continuously varying modulus function. The smoothly varying modulus function given by Ef2 is shown in Fig. 4. The modulus function is approximated such that Z L Ef ðzÞ dz  2E0 L0 þ 2E1 L1 þ    þ2Em1 Lm1 þ Em Lm ð20Þ 0

A recent paper [89] gives an overview of achieving a multimodulus bondline in practice. Various modulus profiles examined in the analysis are given below and are shown in Fig. 5 in normalized form. These modulus functions are arbitrarily chosen and they reflect the behavior of practically producible graded bondline: 2

Ef 1 ¼ Em e4 lnðEm =Eo Þðz=L1=2Þ

ð21Þ

 2  z z þ E0 Ef 2 ¼ 4ðE0 Em Þ 2  L L

ð22Þ

    ! z 1 4 z 1 2 Ef 3 ¼ 8ðEm E0 Þ 2    þ Em L 2 L 2 64 ðE0 Em Þ Ef 4 ¼ 5 Ef 5 ¼ Em



   ! z 1 6 z 1 4   þ þ Em L 2 L 2

ð23Þ

ð24Þ

ð25Þ

3. Constitutive models of the adherends and FMGB adhesive Fig. 4. Approximation of multi-modulus bondline as a functionally modulus graded bondline over bondlength.

Polar anisotropic, axisymmetric constitutive relationship for the adherends [90] is given below. Here, i¼1 for inner adherend

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and i¼2 for outer adherend. 2 nitt Eit Eit i i 2 ðiÞ 3 6 1nilt nitl 1nitt 2 nlt Et srr 6 6 ðiÞ 7 6 nitt Eit Eit i i 6s 7 6 6 yy 7 6 1nitt 2 1nilt nitl nlt Et ¼ 6 ðiÞ 7 6 i 6 szz 7 6 i i 4 5 6 nlt Et nilt Eit 1Enli ni 6 lt tl 4 tðiÞ rz 0 0 0

P3 ¼ p

3

Also nilt Eil ¼ nitl Eit . Note that there are five independent constants. For a FMGB elastic adhesive the axisymmetric constitutive model is given by 2 32 ðaÞ 3 2 ðaÞ 3 1n n n 0 err srr 6 76 ðaÞ 7 6 ðaÞ 7 1n n 0 6 n 76 eyy 7 6 syy 7 EðzÞ 6 76 6 7 7 76 eðaÞ 7 6 sðaÞ 7 ¼ ð1 þ nÞð12nÞ 6 n n 1n 0 4 54 zz 5 4 zz 5 0 0 0 1=2ð1nÞ gðaÞ tðaÞ rz rz Plastic behavior of FMGB adhesive is not considered in this study.

4. Variational principle Variational method in linear elasticity could be based either on potential energy or complimentary energy of the system. In potential energy formulation, we consider a family of kinematically admissible displacement fields, and define the potential energy U so that the elastostatic state minimizes U. Alternatively, we may consider a family of statically admissible stress fields, and define complementary energy P so that the elastostatic stress field minimizes P (e.g., [91–93]). In the current work, we use the principle of minimum complementary energy, following the analysis developed by Kumar [77] for lap joints comprising multi-modulus bondline. The problem can be defined as obtaining a true solution for an unknown stress function sð1Þ zz by minimizing the complementary energy of the bonded system where the stress components for the adherends and the FMGB adhesive have been defined in terms of a single stress function sð1Þ zz . The admissible stress states are those which satisfy continuum differential equations of equilibrium, stress boundary conditions, tractionfree BCs of the joint and stress continuity at the adherend– adhesive interfaces. Among all the possible stress states, the true solution (real stress state) results in smallest complimentary energy of the system. Once sð1Þ zz has been obtained, then all the stress components both in the adhesive and in the adherends can be obtained. 4.1. Case I: FMGB1 ðsðiÞ rr a 0 and w1 a0Þ

ð26Þ

P1 is the complementary energy of the inner adherend, P2 is the complementary energy of the outer adherend and P3 is the complementary energy of the FMGB adhesive. P1 , P2 and P3 are given by Z LZ b ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ P1 ¼ p ½sð1Þ ð27Þ rr err þ szz ezz þ syy eyy þ trz grz r dr dz

P2 ¼ p

Z

a

LZ 0

d c

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ½sð2Þ rr err þ szz ezz þ syy eyy þ trz grz r dr dz

Z

c

b

ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ ½srr err þ sðaÞ yy eyy þ trz grz r dr dz

ð29Þ

P1 and P2 for polar anisotropic adherends can be evaluated by ðiÞ ðiÞ ðiÞ using the strains (eðiÞ rr , eyy , ezz , grz ) given by the anisotropic constitutive model described earlier. For an isotropic system, P1 , P2 and P3 become Z Z p L b ð1Þ2 ð1Þ2 ð1Þ2 P1 ¼ fsrr þ szz þ syy E1 0 a ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ2 2n1 ðsð1Þ rr szz þ srr syy þ szz syy Þ þ2ð1 þ n1 Þtrz gr dr dz

P2 ¼

p

Z

E2

L

Z

0

d

c ð2Þ rr

ð2Þ2 ð2Þ2 fsð2Þ2 rr þ szz þ syy

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ2 2n2 ðs sð2Þ zz þ srr syy þ szz syy Þ þ2ð1 þ n2 Þtrz gr dr dz

P3 ¼ p

Z

L 0

Z b

c

ð30Þ

ð31Þ

1 ðaÞ ðaÞ ðaÞ2 fsðaÞ2 þ sðaÞ2 yy 2nsrr syy þ 2ð1 þ nÞtrz gr dr dz EðzÞ rr ð32Þ

Introducing expressions (Eqs. (6), (7) and (8)) for stresses (sð1Þ rr , ð1Þ sð1Þ yy and trz Þ in Eq. (30) and integrating the resulting expression over the radius r, the energy functional P1 reduces to 0 Z L 2 2 d sð1Þ d sð1Þ zz zz P1 ¼ p @A 1 þ A 2 sð1Þ þA10 sð1Þ zz þA 3 zz 2 2 0 dz dz !2 !2 1 2 d sð1Þ dsð1Þ zz zz Adz þA4 þ A11 2 dz dz

ð33Þ

The explicit expressions for the constants A 1 , A 2 , A 3 , A10, A4 and A11 are detailed in Appendix A. Similarly, plugging expressions for ð2Þ ð2Þ ð2Þ stresses (sð2Þ rr , szz , syy and trz ) in Eq. (31) and integrating the resulting expression over the radius r, the energy functional P2 reduces to 0 !2 Z L 2 2 d sð1Þ d sð1Þ zz zz ð1Þ ð1Þ2 @ P2 ¼ p C 1 þC 2 szz þ C 3 szz þ C 3 þ C6 2 2 0 dz dz !2 1 2 d sð1Þ dsð1Þ zz zz Adz ð34Þ þ C 16 þC 15 sð1Þ zz 2 dz dz The explicit expressions for the material and geometric parameters C 1 , C 2 , C3, C 3 , C6, C15 and C16 are detailed in Appendix A. ðaÞ ðaÞ Again introducing expressions for stresses (sðaÞ yy , srr and trz ) in Eq. (32) and integrating the resulting expression over the radius r, the energy functional for an FMGB adhesive becomes 0 ! ! 1 Z L 2 ð1Þ 2 ð1Þ 2 ð1Þ 2 d s d s d s zz zz zz @B 1 ðzÞ þ B 2 ðzÞ Adz P3 ¼ p þ B7 ðzÞ þB3 ðzÞ 2 2 dz 0 dz dz ð35Þ

The complementary energy of the joint comprising polar anisotropic adherends and a functionally modulus graded adhesive is given by P, where

0

L 0

0 72 ðiÞ 3 7 err 76 ðiÞ 7 6 7 0 7 76 eyy 7 76 ðiÞ 7 76 ezz 7 4 5 0 7 7 ðiÞ 5 grz Gitl

P ¼ P1 þ P2 þ P3

Z

ð28Þ

The parameters B 1 ðzÞ, B 2 ðzÞ, B7 ðzÞ and B3 ðzÞ vary along the bondline and the expressions for these variable parameters are given in Appendix A. Now combining Eqs. (33)–(35), the complimentary energy in the whole assembly is given by 0 ! !2 Z L 2 ð1Þ 2 2 ð1Þ dsð1Þ zz ð1Þ d szz @b1 sð1Þ2 þ b2 ðzÞ d szz P¼p þ b s þ b ðzÞ 3 zz 4 zz 2 2 dz 0 dz dz ! 2 d sð1Þ zz ð1Þ 2 þ ð b þ w kÞ s þð b þ w mðzÞ þ w sÞ dz þðb5 þ w1 hðzÞÞ 6 7 1 1 zz 1 2 dz ð36Þ In the above functional, the constant coefficients b1 , b3 , b5 , b6 , b7 , k, s and the variable coefficients b2 ðzÞ, b4 ðzÞ, h(z) and m(z) depend on geometrical and material properties as well as the loading conditions of the bonded joint and are given in Appendix A.

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

The optimal value of w1 is given by ! 2 L d sð1Þ zz ð1Þ p hðzÞ þ kszz þ 2w1 mðzÞ þ s dz ¼ 0 2 0 dz Z

ð37Þ

The functional given by Eq. (36) can be expressed as ! Z L 2 ð1Þ ð1Þ ð1Þ dszz d szz , P¼ j szz , ,z dz 2 dz 0 dz

ð38Þ

We now need the differential equation satisfied by the function

sð1Þ zz which minimizes the above functional. Performing variational calculus on the above functional yields ! ! 2 @j d @j d @j  þ 2 ¼0 00ð1Þ dz @s0ð1Þ @sð1Þ dz @szz zz zz 2

2

ð1Þ 0ð1Þ ð1Þ where s00ð1Þ zz ¼ d szz =dz , szz ¼ dszz =dz. Explicit form of the above differential equation is expressed by 4

b2 ðzÞ

d sð1Þ zz 4

dz

b04 ðzÞ

3

þ2b02 ðzÞ

d sð1Þ zz 3

dz

2

þ ðb3 b4 ðzÞ þ b002 ðzÞÞ

d sð1Þ zz 2

dz

dsð1Þ w1 00 b þ w1 k zz ¼0 þ b1 sð1Þ h ðzÞ þ 6 zz þ 2 dz 2

ð40Þ

The integral equation (37) and the differential equation (40) are to be solved simultaneously to find the actual sð1Þ zz and its derivatives to predict the stresses in the bonded system using the traction boundary conditions given by Eqs. (18) and (19). The solution procedure is described in the next section. We refer this model henceforth by the name ‘FMGB1’. 4.2. Case II: FMGB ðsðiÞ rr ¼ 0 and w1 -0Þ If we neglect the radial stress in the assembly, i.e., sðiÞ rr ¼ 0, the constant w1 -0. Setting w1 ¼ 0 in the functional, P given by Eq. (36) and performing variational calculus of the resulting functional, we recover the nonlinear differential equation of the FMGB model of Kumar [77] as given below: 4

b2 ðzÞ

d sð1Þ zz 4

dz

3

þ2b02 ðzÞ

d sð1Þ zz 3

dz

2

þ ðb3 b4 ðzÞ þ b002 ðzÞÞ

In this case, the optimal value of w1 is obtained by differentiating Eq. (42) with respect to w1 and equating the resulting expression to zero. The optimal value of w1 is given by the following integral equation: ! Z L 2 d sð1Þ zz ð1Þ p h þ k s þ 2 w m þ s dz ¼ 0 ð43Þ 1 zz 2 0 dz We now need the differential equation satisfied by the function sð1Þ zz which minimizes the functional given by Eq. (42). Performing variational calculus on the functional given by Eq. (42) yields 4

b2 ð39Þ

d sð1Þ zz

þ b1 sð1Þ zz þ

b6 þ w 1 k 2

¼0

ð44Þ

2

d sð1Þ zz

þðb3 b4 Þ

4

dz

d sð1Þ zz 2

dz

þ b1 sð1Þ zz þ

b6 2

¼0

ð46Þ

This ODE is solved in Matlab using BVP4C program and using the boundary conditions given by Eqs. (18) and (19). Henceforth we refer this model by the name ‘MMB’.

5. Solution procedure Differentiating the functional given by Eq. (36) with respect to

w1 and setting that zero yields optimal value of w1 and is given by

When we have a mono-modulus bondline (MMB) adhesive, the modulus function of the adhesive becomes a constant, i.e., EðzÞ-E and all the parameters of the model which are function of the bondlength z become constant, i.e., b2 ðzÞ-b2 , b4 ðzÞ-b4 , hðzÞ-h and mðzÞ-m. Accordingly the complementary energy functional of the system, P reduces to the following: 0 ! !2 Z L 2 ð1Þ 2 2 ð1Þ dsð1Þ zz ð1Þ d szz @b1 sð1Þ2 þ b2 d szz P¼p þ b s þ b 3 zz 4 zz 2 2 dz 0 dz dz

dz

2

dz

Performing variational calculus on the above functional yields the following fourth order linear ODE. Note that, in the limit of sðiÞ rr ¼ 0 and w1 -0, we recover the MMB model of Kumar [77]:

p

d s

d sð1Þ zz

Omission of radial stresses i.e., sðiÞ rr ¼ 0 makes w1 ¼ 0. For monomodulus bondline (MMB) adhesive, the modulus function of the adhesive becomes a constant, i.e., EðzÞ-E and all the parameters of the model which are function of the bondlength z become constant, i.e., b2 ðzÞ-b2 and b4 ðzÞ-b4 . For this case, complementary energy functional of the system P given by Eq. (42) reduces to the following: 0 !2 !2 Z L 2 2 ð1Þ d sð1Þ dsð1Þ zz zz ð1Þ2 ð1Þ d szz @ P¼p b1 szz þ b2 þ b3 szz þ b4 2 2 dz 0 dz dz ! 2 d sð1Þ zz ð45Þ þ b5 þ b6 sð1Þ zz þ b7 dz 2 dz

ð41Þ

4.3. Case III: MMB1 ðsðiÞ rr a 0 and w1 a 0Þ

þ ðb5 þ w1 hÞ

dz

4

In this model we have only a single fourth order differential equation with four traction boundary conditions given by Eqs. (18) and (19) and it is numerically solved in Matlab using the function BVP4C. The solution sð1Þ zz and its derivatives can be used to predict the stresses in the bonded system but noting that w1 ¼ 0. We refer this model henceforth by the name ‘FMGB’.

ð1Þ zz 2

þðb3 b4 Þ

4

4.4. Case IV: MMB ðsðiÞ rr ¼ 0 and w1 -0Þ

b2

ds b6 þ b1 sð1Þ ¼0 b04 ðzÞ zz þ dz 2

2

2

d sð1Þ zz

Now the fourth order linear ordinary differential equation (ODE) given by Eq. (44) and the integral equation given by Eq. (43) are simultaneously solved in Matlab using the function BVP4C and using the boundary conditions given by Eqs. (18) and (19). The solution procedure is same as that of the case I which is detailed in the next section. Henceforth we refer this model by the name ‘MMB1’.

2

dz

ð1Þ zz

63

! 2 þ ðb6 þ w1 kÞsð1Þ zz þ ðb7 þ w1 m þ w1 sÞ dz

ð42Þ

Z

!

2

L

hðzÞ

d sð1Þ zz

0

2

dz

þksð1Þ zz þ2w1 mðzÞ þs dz ¼ 0

ð47Þ

Performing variational calculus of the functional (Eq. (36)) gives the following nonlinear fourth order ODE: 4

b2 ðzÞ

d sð1Þ zz 4

dz

b04 ðzÞ

3

þ 2b02 ðzÞ ð1Þ zz

d sð1Þ zz 3

dz

2

þ ðb3 b4 ðzÞ þ b002 ðzÞÞ

d sð1Þ zz 2

dz

ds w1 00 b þ w1 k ¼0 þ b1 sð1Þ h ðzÞ þ 6 zz þ 2 dz 2

ð48Þ

The integral equation (47) and the differential equation (48) are simultaneously solved to get the solution sð1Þ zz in Matlab using bvp4c program with traction boundary conditions given by Eqs. (18) and (19). Eq. (48) can be solved only if we know the

64

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

Table 1 Geometric and material properties of adhesive and adherends [32]. Item

Tube 1 Tube 2

Material

AU 4G G0969/ M18 Adhesive AV119

E (GPa)

n

a (mm)

75 0.3 44.8 44.080 0.325 – 2.7

0.35

–

b (mm)

c (mm)

d (mm)

f (MPa)

47.8 –

– 48

– 50

– 100

47.8

48

–

–

Fig. 7. Shear stress distribution at the midsurface of the adhesive layer, inner adherend and outer adherend using FMGB model in which sðiÞ rr ¼ 0.

Fig. 6. Shear stress distribution at the midsurface of the adhesive layer, inner adherend and outer adherend using FMGB1 model in which sðiÞ rr a 0.

value of w1 . But w1 can be evaluated only if we know the stress function sð1Þ zz . However, the crux of the problem is to determine the actual sð1Þ zz which minimizes the complementary energy of the bonded system. Therefore, we initially find the approximate value of w1 by fitting a cubic polynomial for the stress function sð1Þ zz since ð1Þ we know four boundary conditions (sð1Þ zz and dszz =dz) at the ends of the overlap. Then we use this approximate value of w1 together with traction BCs given by Eqs. (18) and (19) to find the solution of the differential equation (48). Now we have the numerical approximate solution of sð1Þ zz and its derivatives over the entire bondlength. Now we use this solution set to evaluate a new w1 solving the integral equation (Eq. (47)). Again use this current value of w1 to solve the differential equation. This process is repeated until the value of w1 attains a constant value, i.e., ði þ 1Þ ðwði1 þ 1Þ wðiÞ is the optimal value and the sð1Þ zz 1 Þ  0. This w1 ði þ 1Þ corresponding to this w1 is the actual stress state. Once we ð1Þ know the actual distribution of szz and its derivatives, we can get the complete stress state in the entire system.

6. Results and discussion Initially, analysis of the joint under axial tensile was performed using FMGB1 model in which sðiÞ rr a0, considering adherends whose geometrical and mechanical properties are given in Table 1, with a graded adhesive of modulus function Ef 1 (Em ¼2700 MPa, E0 ¼280 MPa and L¼80 mm). In Table 1, AU 4G is aluminum alloy and G0969/M18 is carbon/epoxy. Analysis was also performed using FMGB model in which sðiÞ rr ¼ 0, considering

Fig. 8. Shear stress distribution at the midsurface of adhesive layer based on graded- and mono-modulus models.

the same geometrical and material properties as well as loading condition. The results were then compared to study the influence of non-zero radial stresses sðiÞ rr on peak shear and normal stresses and their distribution. Figs. 6 and 7 show the shear stress distribution in the members of the joint over the bondlength using FMGB1 and FMGB models at their midsurface respectively. From these figures, we can see that the shear stress in the graded adhesive changes whereas the shear stresses in the adherends do not change appreciably. Fig. 8 shows the shear stress distribution at the midsurface of the adhesive layer based on these two graded models and also based on their respective mono-modulus counterparts (i.e., MMB1 model in which sðiÞ rr a 0 and MMB model in which sðiÞ rr ¼ 0). The mono-modulus adhesive properties are given in Table 1. Inclusion of radial stress components in the

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

Fig. 9. Normal stress distribution at the midsurface of the adhesive layer, inner adherend and outer adherend using FMGB1 model in which sðiÞ rr a 0.

Fig. 10. Normal stress distribution at the midsurface of the adhesive layer, inner adherend and outer adherend using FMGB model in which sðiÞ rr ¼ 0.

functionally modulus graded model changes the shear stress peaks in the adhesive (8.7% in this case) and its distribution over the bondlength appreciably and so does the mono-modulus model. It can also be seen from Fig. 8 that shear stress peaks and its distribution in the adhesive layer predicted by the graded models are less severe than those of mono-modulus models. The shear stress peak reduces by 19% by employing FMGB1 model in lieu of MMB1 model whereas it reduces by 17% by employing FMGB model in lieu of MMB model. Figs. 9 and 10 show the normal stress distribution in the members of the joint over the bondlength using FMGB1 and FMGB models at their midsurface respectively. From these figures, we can see that the normal stresses both in the graded adhesive and in the adherends change significantly unlike the shear stresses. Peak normal stress in the adhesive layer increases

65

Fig. 11. Normal stress distribution at the midsurface of adhesive layer based on graded- and mono-modulus models.

by 19% whereas peak normal stress in the inner adherend and in the outer adherend increases by 150% and 180% respectively. Inclusion of radial stress components in modulus graded model changes the normal stress peak in the adhesive and adherends and its distribution over the bondlength appreciably and so does the mono-modulus model. Fig. 11 shows normal stress distribution at the midsurface of adhesive layer based on these two functionally modulus graded models and also based on their respective mono-modulus counterparts (i.e., MMB1 model in ðiÞ which sðiÞ rr a 0 and MMB model in which srr ¼ 0). Drastic increase in normal stress in the adhesive may lead to cohesive or interfacial failure within the adhesive whereas higher normal stress in the adherends will lead to failure within adherends if composite adherends having low through-thickness transverse strength are employed. It can also be seen from Fig. 11 that normal stress intensity and its distribution in the adhesive layer predicted by the graded models are less severe than those of mono- modulus models. The normal peak stress in the adhesive decreases by 73% by employing FMGB1 model in lieu of MMB1 model whereas it reduces by 70% by employing FMGB model in lieu of MMB model. The shear and normal stress intensities both at the interface and at the midsurface of adhesive predicted by FMGB1 model are much smaller and their distribution along the bondline is more uniform than those of a MMB1 adhesive joint. The peak normal stress in the FMGB adhesive joints appear close to the overlap ends, while it appears exactly at the overlap ends in a MMB adhesive joints. This is because the stiffness jump in graded joints is more gradual than the stiffness jump in mono-modulus joints. 6.1. Influence of bondlength (L) Stress analyses have been performed by selectively varying the bondlength from 40 to 250 mm, i.e., L¼(40, 50, 80, 100, 120, 150, 200, 250) and adopting exponential modulus function profile Ef1 for the adhesive using FMGB1 model in order to study the effect of bondlength on stress distribution. The prediction using FMGB1 model was also compared with the predictions of MMB1 model. Figs. 12 and 13 show the shear stress distribution at the midsurface of the adhesive as a function of bondlength. At small bondlengths, the shear stress distribution in both FMGB1 and MMB1 adhesives are parabolic, with stress peaks at mid-bondlength. For L r50 mm, the shear stress in the graded adhesive is much severe than that of

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

10

8

8

6

[MPa] τ(a) rz

τ(a) [MPa] rz

66

6 FMGB1 MMB1

4

FMGB1 MMB1

2

2 0

4

0

10

20 z [mm]

30

0

40

7

0

10

20 30 z [mm]

40

50

80

100

8 FMGB1 MMB1

6

FMGB1 MMB1

6 τ(a) [MPa] rz

τ(a) [MPa] rz

5 4 3

4 2

2 0

1 0

0

20

40 z [mm]

60

80

−2

0

20

40 60 z [mm]

Fig. 12. Shear stress distribution at the midsurface of the adhesive using FMGB1 model compared with that of MMB1 model as a function of bondlength.

Fig. 13. Shear stress distribution at the midsurface of the adhesive using FMGB1 model compared with that of MMB1 model as a function of bondlength.

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

67

Fig. 14. Normal stress distribution at the midsurface of the adhesive using FMGB1 model compared with that of MMB1 model as a function of bondlength.

the mono-modulus adhesive. Shear stress peaks predicted by both FMGB1 and MMB1 models decrease and their distribution becomes more uniform with increase of bondlength. Shear stress peak in the adhesive of MMB1 model decreases with increase of bondlength from L¼50 mm up to L¼ 80 mm and increases for L 480 mm but up to L¼150 mm. Beyond a certain bondlength (L¼150 mm), the increase of bondlength does not reduce shear stress peak in MMB1 adhesive appreciably. On the other hand, peak shear stress in FMGB1 adhesive decreases with increase of bondlength up to L¼250 mm. In both models the shear stress peaks move close to overlap ends with increase of bondlength. Figs. 14 and 15 show the normal stress distribution at the midsurface of the adhesive along the bondlength for selected values of bondlength based on both FMGB1 and MMB1 models. For any value of bondlength, normal stress peak in the graded bondline adhesive is much less and its distribution is more uniform than those of mono-modulus bondline adhesive. An increase of bondlength reduces the normal stress peak up to a certain bondlength (L¼100 mm) and increases thereafter in FMGB1 adhesive whereas the shear stress peak in the MMB1 adhesive decreases up to L¼ 80 mm and increases with further increase of bondlength up to L¼120 mm and remains constant thereafter. Therefore, the bondlength at which the normal stress starts to increase with increase of bondlength is considered to be an optimum bondlength. The optimum bondlength in this case for FMGB1 model is L¼100 mm. Both shear stress and normal stress peaks move towards the overlap ends with an increase of bondlength. However, the stress distribution does not change in the MMB1 joint after L¼120 mm, for the variables used here. 6.2. Influence of modulus function Different modulus function profiles have been examined to reduce the intensity of stresses and their gradients in the FMGB1

adhesive and also compared with adhesive stresses predicted using MMB1 model, keeping all other geometrical and material parameters constant. The shear and normal stress distributions for different modulus functions are shown in Figs. 16 and 17 respectively. The shear stress intensity is less for modulus function Ef3 while the normal stress intensity is less for modulus function Ef2. If we choose a stiff MMB1 adhesive to have maximum shear strength, it will fail due to high normal stresses. Unlike the MMB1 adhesive, the modulus function of the FMGB1 adhesive can be so tailored simultaneously to achieve both shear and normal strengths. Analysis also indicates that an optimized joint performance can be achieved by grading the modulus of the bondline adhesive. However, optimal choice of modulus profile only leads to a local minimum since we do not perturb the parameters of adherends. 6.3. Influence of stiffness mismatch Shear and normal stresses have been estimated for several values of stiffness ratio of adherends, keeping all other parameters constant. Figs. 18 and 19 show the shear and normal stress distribution respectively, at the midsurface of the adhesive as a function of stiffness mismatch between two adherends. For the balanced joint, the shear stress distribution is symmetric and the normal stress distribution is anti-symmetric about the mid bondlength. Note that the shear stress distribution loses its symmetry and normal stress distribution loses its antisymmetry about mid-bondlength when E1 A1 a E2 A2 . Here A1 and A2 are the area of cross section of inner and outer adherends respectively. The stress distribution is compared with the MMB1 adhesive model and found that the stress distribution in FMGB1 adhesive is much less than that of MMB1 adhesive. Therefore, these models yield more accurate results for thin balanced joints suffering negligible bending deformations.

68

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

Fig. 15. Normal stress distribution at the midsurface of the adhesive using FMGB1 model compared with that of MMB1 model as a function of bondlength.

8 7 FMGB1:Ef1 FMGB1:Ef2

Shear stress τ(a) [MPa] rz

6

FMGB1:Ef3 FMGB1:Ef4

5

MMB1:Ef5

4 3 2 1 0

0

20 40 60 Axial distance over bondlength, z [mm]

80

Fig. 16. Shear stress distribution at the midsurface of the adhesive layer based on FMGB1 model for different modulus functions compared with that of MMB1 model.

7. Conclusions A refined theoretical model is presented to investigate the influence of a variable stiffness adhesive interlayer on intensity of stresses and their distribution in an axisymmetric joint, based on a variational principle, which minimizes the complimentary energy of a multi-material bonded system. This model accounts for the stress-free BCs and through-thickness variation of shear

Fig. 17. Normal stress distribution at the midsurface of the adhesive layer based on FMGB1 for different modulus functions compared with that of MMB1 model.

stresses in the interlayer. It has been observed that the inclusion of radial stresses significantly influences both intensity and distribution of normal stresses in the joint. Previous studies [77,78] underestimate the normal stresses although predict the shear stresses accurately. However, this model accurately predicts both shear and normal stresses in the entire assembly. This model is particularly useful to capture accurate normal stresses both in the adherends and in the adhesive, which are prone to brittle failure (e.g., composite adherends having low through thickness transverse strength, stiffer and thick-bondline adhesives which

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

69

Fig. 18. Shear stress distribution at the midsurface of the adhesive as a function of stiffness mismatch.

exhibit higher stress concentrations at the ends of overlap). It has been observed that the shear and normal stress concentrations at the overlap ends in the FMGB1 adhesive joints are much less than those of MMB1 adhesive joints under the same axial load. Reduced shear and normal stress concentrations can potentially lead to improved joint strength and service life. It has been observed through parametric evaluation that the shear and normal stress peaks and their gradients in the bondline can be significantly reduced by selectively perturbing the geometrical and material properties of the bonded system. This simple analytical treatment not only allows us to predict the stresses in a stiffness graded bonded systems but also permits to examine the effect of loss of interface stiffness due to an existing defect and/or damage, on structural response.

A 1 , A 2 , A 3 , A10, A4 and A11 are given below:

x1 ¼ 1=6b6 1=6a6 1=2a2 ðb4 a4 Þ þ 1=2a4 ðb2 a2 Þ, A1 ¼

Explicit expressions for the constants (function of material and geometrical properties as well as loading condition of the joint)

x1

2

x2 ¼ 1=2

b a2 , E1

A2 ¼ x2

ðA:3Þ

x3 ¼ 3=2b6 3=2a6 3=2a2 ðb4 a4 Þ þ 1=2a4 ðb2 a2 Þ, A3 ¼

w21 E1

x3 ðA:4Þ

Appendix A. Case I: FMGB1 ðsðiÞ rr a 0 and w1 a 0Þ

The complementary energy in the inner adherend is given by Z L 2 2 d sð1Þ d sð1Þ zz zz P1 ¼ p A 1 þ A 2 sð1Þ þA10 sð1Þ zz þA 3 zz 2 2 0 dz dz !2 !2 1 2 d sð1Þ dsð1Þ zz zz Adz ðA:1Þ þ A11 þ A4 2 dz dz

E1

ðA:2Þ

x4 ¼ 1=6b6 1=6a6 1=2a2 ðb4 a4 Þ þ 1=2a4 ðb2 a2 Þ, A4 ¼

A.1. Inner adherend

w21

x5 ¼ 1=2b6 þ 1=2a6 þ a2 ðb4 a4 Þ1=2a4 ðb2 a2 Þ, A5 ¼

1 x 4E1 4 ðA:5Þ

w1 E1

x5 ðA:6Þ

x6 ¼ 1=4b4 þ 1=4a4 þ 1=2a2 ðb2 a2 Þ, A6 ¼

2n1 w1 x6 E1

x7 ¼ 1=2b6 1=2a6 a2 ðb4 a4 Þ þ1=2a4 ðb2 a2 Þ, A7 ¼

ðA:7Þ

2n1 w21 x7 E1 ðA:8Þ

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S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

Fig. 19. Normal stress distribution at the midsurface of adhesive as a function of stiffness mismatch.

n1 w1 x8 E1 ðA:9Þ

C 2 ¼ 1=2

2n1 w1 A9 ¼ x9 E1

ðA:10Þ

C 3 ¼ 1=2

n1 x E1 10

ðA:11Þ

C4 ¼

x8 ¼ 1=6b6 þ 1=6a6 þ 1=2a2 ðb4 a4 Þ1=2a4 ðb2 a2 Þ, A8 ¼

4

4

2

2

2

x9 ¼ 3=4b þ3=4a þ 1=2a ðb a Þ,

x10 ¼ 1=4b4 1=4a4 1=2a2 ðb2 a2 Þ, A10 ¼

x11 ¼ 3=4a4 a4 lnðaÞ þ 1=4b4 b2 a2 þ a4 lnðbÞ, A11 ¼

A 1 ¼ A1 þ A3 þ A7 ,

A 2 ¼ A6 þ A9 ,

2

2

f ðd c2 Þ E2

ðA:16Þ

r2 ðd2 c2 Þ

ðA:17Þ

E2 2

ð1þ n1 Þ x11 2E1 ðA:12Þ

A 3 ¼ A5 þ A8

ðA:13Þ

f rðd c2 Þ E2

ðA:18Þ

Z2 ¼ 3=2d6 3=2c6 3=2d2 ðd4 c4 Þ þ 1=2d4 ðd2 c2 Þ, C 5 ¼ w21

Z3 ¼ 1=6d6 1=6c6 1=2d2 ðd4 c4 Þ þ 1=2d4 ðd2 c2 Þ, C 6 ¼

r2

r2 4E2

Z3 ðA:20Þ

A.2. Outer adherend The complementary energy in the outer adherend is given by 0 ! Z L 2 ð1Þ 2 ð1Þ 2 @C 1 þ C 2 sð1Þ þ C 3 sð1Þ2 þ C 3 d szz þ C 6 d szz P2 ¼ p zz zz 2 2 0 dz dz 1 !2 2 d sð1Þ dsð1Þ zz zz Adz ðA:14Þ þ C 15 sð1Þ þC 16 zz 2 dz dz Explicit expressions for the constants C 1 , C 2 , C3, C 3 , C6, C15 and C16 are given below:

Z1 ¼ 1=6d6 1=6c6 1=2d2 ðd4 c4 Þ þ1=2d4 ðd2 c2 Þ, C 1 ¼ w21

Z

E2 2 ðA:19Þ

2

r

Z4 ¼ 1=2d6 þ1=2c6 þ d2 ðd4 c4 Þ1=2d4 ðd2 c2 Þ, C 7 ¼ w1

E2

Z4 ðA:21Þ

Z5 ¼ 1=4d4 þ1=4c4 þ 1=2d2 ðd2 c2 Þ, C 8 ¼ 2n2 w1 r C 9 ¼ 2n2 w1

r2 E2

Z5

f Z E2 5

ðA:22Þ

ðA:23Þ

Z6 ¼ 1=2d6 1=2c6 d2 ðd4 c4 Þ þ 1=2d4 ðd2 c2 Þ

Z

E2 1 ðA:15Þ

r2

C 10 ¼ 2n2 w21

r2 E2

Z6

ðA:24Þ

S. Kumar, J.P. Scanlan / International Journal of Adhesion & Adhesives 41 (2013) 57–72

71

!

2

6

2

6

4

4

4

2

2

þðb5 þ w1 hðzÞÞ

Z7 ¼ 1=6d þ 1=6c þ1=2d ðd c Þ1=2d ðd c Þ C 11 ¼ n2 w1

r2 E2

Z7

4

2

2

2

Z8 ¼ 3=4d þ 3=4c þ1=2d ðd c Þ, C 12 ¼ 2n2 w1 f

r

E2

E2

Z8

Z8

r2 E2

w

r E2

Z9

Z9

ðA:28Þ

ðA:29Þ

Z10 ¼ 1=4c4 þ c2 d2 d4 lnðcÞ3=4d4 þ d4 lnðdÞ, C 16 ¼ ð1 þ n2 Þ

ðA:41Þ

depend on geometrical and material properties as well the loading conditions of the bonded joint and they are given below:

b1 ¼ A2 þC 3 , b2 ðzÞ ¼ A4 þ B3 ðzÞ þ C 6 , b3 ¼ A10 þC 15

ðA:42Þ

b4 ðzÞ ¼ A11 þ B7 ðzÞ þ C 16 , b5 ¼ C 14 , b6 ¼ C 4 , b7 ¼ C 2

ðA:43Þ

r

k ¼ 2



n1

E1

s ¼ 2n2 r

ðx6 þ x9 Þ þ

r2 ðZ5 þ Z8 Þ

E1

ðA:44Þ

ðA:45Þ

2

2E2

Z10

C 1 ¼ C 1 þ C 2 þ C 5 þ C 8 þ C 10 þ C 12

mðzÞ ¼

hðzÞ ¼

C 3 ¼ C 7 þ C 11 þ C 14



n1

f ðZ þ Z 8 Þ E2 5

x1 þ x3 2n1 x7 E1

ðA:30Þ

C 2 ¼ C 4 þ C 9 þ C 13 ,

w1 sÞ dz

ðA:27Þ

Z9 ¼ 1=4d4 1=4c4 1=2d2 ðd2 c2 Þ, C 14 ¼ n2 f C 15 ¼ n2

þ ðb6 þ w1 kÞs

2 1 mðzÞ þ

In the above functional, the constant coefficients b1 , b3 , b5 , b6 , b7 , k, s and the variable coefficients b2 ðzÞ, b4 ðzÞ, h(z) and m(z) ðA:26Þ

r2

2

dz

ð1Þ zz þðb7 þ

ðA:25Þ 4

C 13 ¼ 2n2 w1

d sð1Þ zz

x5 n1 x8 E1

þ1=2

þ

l1 ð1nÞ EðzÞ

l4 ð1nÞ EðzÞ

þ

þ

r2 ðZ1 þ Z2 2n2 Z6 Þ E2

r2 ðZ4 n2 Z7 Þ E2

ðA:46Þ

ðA:47Þ

ðA:31Þ References

A.3. FMGB adhesive The complementary energy functional for an FMGB adhesive is given by 0 !2 1 !2 Z L 2 2 d sð1Þ dsð1Þ d sð1Þ zz zz zz @ Adz P3 ¼ p B 1 ðzÞ þ B 2 ðzÞ þ B7 ðzÞ þB3 ðzÞ 2 2 dz 0 dz dz ðA:32Þ The parameters B 1 ðzÞ, B 2 ðzÞ, B7 ðzÞ and B3 ðzÞ vary along the bondline and the expressions for these variable parameters are given below:

l1 ¼ ða2 b2 Þ2 ðc2 b2 Þ, B1 ðzÞ ¼ w21

1 l1 2EðzÞ

ðA:33Þ

l2 ¼ l1 , B2 ðzÞ ¼ B1 ðzÞ

ðA:34Þ

2

l3 ðzÞ ¼ 1=8

2

ðb a2 Þ2 ðc2 b Þ , EðzÞ

B3 ðzÞ ¼ l3 ðzÞ

l4 ¼ ða2 b2 Þðb2 a2 Þðc2 b2 Þ, B4 ðzÞ ¼ l5 ¼ l2 , B5 ðzÞ ¼ n l6 ¼ l4 , B6 ðzÞ ¼ n

w21 EðzÞ

w1 2EðzÞ

ðA:35Þ

l4

l5

w1 2EðzÞ

ðA:37Þ

l6

ðA:38Þ

l7 ¼ 1=4ðb2 a2 Þ2 ðlnðbÞ þ lnðcÞÞ, B7 ðzÞ ¼ B 1 ðzÞ ¼ B1 ðzÞ þ B2 ðzÞ þB5 ðzÞ,

ðA:36Þ

2ð1 þ nÞ l7 EðzÞ

B 2 ðzÞ ¼ B4 ðzÞ þ B6 ðzÞ

ðA:39Þ ðA:40Þ

A.4. Complementary energy of bonded system Now combining Eqs. (A.1), (A.14) and (A.32), the complementary energy in the whole assembly is given by 0 !2 !2 Z L 2 2 ð1Þ d sð1Þ dsð1Þ zz zz ð1Þ2 ð1Þ d szz @ P¼p b1 szz þ b2 ðzÞ þ b3 szz þ b4 ðzÞ 2 2 dz 0 dz dz

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