A Compression-Loaded Double Lap Shear Test: Part 1, Theory

International Journal of Adhesion and Adhesives, submitted A Compression-Loaded Double Lap Shear Test: Part 1, Theor D.-A. Mendels, S.A. Page, J.-A. E. Månson Laboratoire de Technologie des Composites et Polmères LTC Ecole Poltechnique Fédérale de Lausanne EPFL J.A. Nairn Material Science & Engineering Department, Universit of Utah Salt Lake Cit, UT USA Abstract This paper introduces a new approach to adhesion tests b a compression-loaded, double lap shear specimen and presents stress analsis of that specimen. The geometr modifies conventional double lap shear tests to minimize peel stresses and facilitate specimen fabrication, and thereb increase test reproducibilit. The bonded part of the new specimen is identical to a conventional double lap shear specimen, but the specimen is end-loaded in compression instead of in tension. We analzed the stress state using a new shear-lag theor for multilaered sstems that includes residual stresses and friction on debonded surfaces. Axial stresses, shear stresses, and axial displacements were calculated for specimens stressed in the elastic regime, or beond where local damage or ielding of the adhesive are present. The all compared well with the results of finite element analsis. Failure models of the specimen were derived using both strength and energ release rate methods. The model captured all the essential features of the energ release rate, including the effects of internal stresses and of friction at the debonded interface, within a certain range of debond length and friction stress. The results in this paper will be used in the remaining articles of this series, demonstrating the effects of internal stresses and phsical aging of the adhesive on practical adhesion. 1. Introduction Advanced composite materials are widel used in applications in the aircraft and automotive industr due to their high strength-to-stiffness ratio and high corrosion resistance. Their use as structural materials often requires the polmer-based composite to be attached to another structure, b means of bolted joints or an adhesive laer. An advantage of an adhesive laer is that it can limit stress concentrations due to bolts [1, ]. A bonded adhesive structure consists of three components of different mechanical properties, namel the two adherents and the adhesive laer. Much attention has been paid to describing the mechanical response of such assembled sstems, including their behavior under bending [3], tension, and shear [4, 5]. The bending and tension tests of joints generall involve mixed modes of failures with unknown shear components [6]. In order to predict failure in a non-trivial geometr, it is of primar importance to determine the characteristics of shear failure alone, which is the subject of the present series. The problem of shear failure of interfaces is relevant to several applications, such as metal to polmer composite bonding, metal to metal brazing, and composite interlaminar failure [7]. Adhesive failure is generall modelled b one of four approaches, namel: a shear strength criterion [8, 9, 1, 11]; a local shear strain criterion [13]; a fracture mechanics approach, using either the stress intensit factor [14, 15], or the energ release rate [16]. All these methods rel first on determination of the stresses in the sstem. Because of the complexit of the multi-material geometr, however, an exact analtical treatment of stress is not possible [17]. All analtical models, therefore have used simplifing assumptions [18]. Alternativel, numerical methods, including finite element FEA and boundar element analzes BEA, have been used [13, 19,, 1]. Due to specific 1

limitations of FEA at interfaces, it is not possible to treat all three components of an adhesivel bonded structure as an elastic continua, and the problem is generall handled b using interface elements or b averaging stresses over a region close to the interface. The primar factors influencing the choice of stress-analsis approximations are the adhesive-to-adherents thickness ratios, and the ratio of the adherent thickness to the lateral joint dimensions [1, 11]. Most analtical models stem from the classical theor of beams or plates and shells [18]. These usuall consist of first-order approximations of displacements, which, however, are incompatible with high deformations. In addition, these approximations neglect free-edge effects, although some asmptotic methods have been used in homogenizing these solutions to satisf most boundar conditions [, 3, 4]. The failure of an adhesive joint is not a propert of the adhesive alone, but is a sstem propert depending on adherents, the adhesive, the joint geometr, preparation, and service or test conditions [5]. The most widel used test methods are the general lap-shear tests. Earl attempts, including those b Volkersen [1] or Goland and Reissner [11], to describe the stress field in simple lap-shear geometr, consisting of two plates bonded b a glue film, have led to inaccurate results essentiall due to the existence of a peel stress generated b flexure of the adherents due to misalignment. Its magnitude, as well as the geometr variations between samples, was found to strongl reduce the reliabilit of the test results [1]. It is difficult to manufacture specimens with a controlled bond thickness without leaving spacers inside the joint. The effects of a glue meniscus [14] and the initial stress state [9, 16] on the strength of the adhesive bond are generall ignored. Furthermore, internal stresses, such as thermal residual stresses, have been shown to pla an important role in adhesive performance [6, 7], but their effect has not been studied directl in lap joints. Moreover, several experiments have shown the need to account for local variations in the adhesive mechanical behavior. This includes possible damage to the adhesive, such as local interfacial debonding [8] or transverse cracking [9], and other effects such as viscoelasticit and plasticit [3]. In order to stud the above effects, a new lap-shear test geometr, which allows specimen preparation difficulties to be minimized and takes into account the development of complex stress states, is introduced The proposed modified double lap shear MDLS specimen enables tight control of the thickness of the adhesive laers with sharp edges, and introduces the possibilit of moulding several samples at the same time, with virtuall identical boundar conditions and thermal histor. The stress state in the MDLS specimen, including residual stresses, was calculated here using a recent shear-lag method [39]. This theor was extended to account for non-perfect interfaces and also local variations in the adhesive mechanical behavior. Two alternative methods are used to interpret failure properties of lap joints, namel the interfacial shear strength approach and the fracture mechanics or energ release rate ERR approach. While the strength approach is straightforward once the local stress state is known, the ERR approach requires further calculations. First, an exact expression of the energ release rate was obtained b adapting results for a general composite geometr [16] to that of the MDLS specimen. After some minor approximations, this expression gave the energ release rate in terms of onl axial stresses and displacements. Inserting the shear-lag results for these terms led to an approximate, analtical result for ERR. The local stress state and the approximate energ release rate compared well to FEA results. The theor developed in this article will be used in subsequent papers to analze the results of: local strain mapping experiments [31]; experiments where the internal stress state of the adhesive was tailored b modifing the cure ccle or b adding a chemical modifier to the adhesive [3]; experiments where the mechanical behavior and internal stress state of the adhesive were changed b aging at temperatures below the glass transition temperature of the adhesive [33]; interfacial crack propagation in laminates subjected to static or cclic loading [34].. Problems Addressed and Associated Sstem Geometr Experimental results [31 34] show that loading of a modified double lap shear specimen MDLS leads to several damage modes which necessitates several model geometries for analsis. Fig. 1 presents the geometr for a MDLS specimen with cracks of length a between the adhesive laer and the central adherent. The

3 Compression Double Lap Shear Test I σ σ σ σ σ Fig. 1. Geometr of the modified double lap-shear specimen with cracks of length a between the adhesive and the central adherent laer. sstem is shown with 3 constituents: an inner plate of thickness t 1, with Young s modulus E 1, Poisson s ratio ν 1, and thermal expansion coefficient α 1 ; two adhesive laers of thickness t and thermo-elastic properties E, ν, α ; and two outer plates of thickness t 3 and thermo-elastic properties E 3, ν 3, α 3. The boundar conditions are defined b a compression stress of positive magnitude σ spread over the entire inner plate while the two outer plates are simpl supported. The top an bottom surfaces of the adhesive laer are not loaded or supported. Additionall, two lateral pressures σ t can be applied on the outer adherents. The origin of the x,, z axis sstem is in the middle of the specimen and aligned with the crack tip. The x axis is in the thickness direction while the axis is in the loading direction. The specimen geometr is smmetrical about the midplane, which means onl half the specimen needs to be analzed see Fig.. The inner plate is labeled 1, the adhesive laer, and the outer plate 3. The width of the specimen W is assumed large enough to be in plane stress conditions in the z = plane of the specimen. This will be verified in the experimental part of this work b examining the fracture surfaces of the specimen [3 34]. The shear-lag analsis consists of determining the three averaged axial stresses σ 1, σ, σ 3 in the laers, and the interfacial shear stresses τ 1 at the laer 1/laer interface, τ at the laer /laer 3 interface. The shear stress at the midplane τ and on the lateral surface τ 3 are zero b smmetr or b boundar conditions. The calculation of the energ release rate required determination of the average displacements in the direction < v i > in the sstem. These could also be found b the shear-lag analsis. From experimental results [31 34], it was determined that four problems need to be addressed: i. the

4 σ σ I I I II II II II σ σ Fig.. Geometr of the MDLS specimen with a damaged zone of length d in the adhesive and zones I and II for the analsis. The damage zone can be a transversel cracked adhesive or a ielded adhesive specimen fails in the elastic regime, ii. the adhesive mechanical properties are degraded locall prior to failure, iii. the adhesive partiall ields prior to adhesive failure, and iv. a crack propagates at the interface between the inner plate and the adhesive. The first problem can be solved b a strength analsis using the shear-lag analsis. The three others are complicated if all local dissipative phenomena are to be accounted for. Instead, we used a global approach: the properties of a transversel damaged zone or of a ielded zone were averaged over the damage or ield length d Fig.. The two problems shown in Fig. are ver similar when a global approach is used. The both necessitate the introduction of two zones, labelled I upper zone with degraded mechanical properties and II zone in which the mechanical properties are intact connected at =, which is done using stress continuit. The case of the specimen with a crack is different because a frictional stress is present at the debonded interface. Therefore, three zones are required, as shown in Fig. 3. For validation purposes, material properties defined in Table I were used. The choice of materials was dictated b experimental work [31 34]. Although the theor is developed for orthotropic materials, the validation examples use isotropic materials, which is the most common application of lap-joints. Two adhesives are introduced, one having a brittle tensile behavior, the TGMDA-based epo, and one which can ield, the DGEBA-based epo, also denoted as a model epo network [38]. In these examples, thicknesses of t 1 = t 3 = 4 mm and t =.15 mm were chosen. Tpical meshes used in the finite element analses are shown in Fig. 4. This figure shows a crude mesh. For the calculations in this paper, these elements were refined until convergence was confirmed. For fracture calculations b FEA, the crack surfaces had to be prevented from overlap b using special interface elements. 3. Optimal Shear-Lag Analsis A new shear-lag theor for determining stresses in a multi-laered structure of n orthotropic laers was recentl derived [39]. The multilaered specimen is described b n laers extending from x i 1 to x i for i = 1 to n with thickness t i = x i x i 1. The mechanical properties for laer i include the axial and

5 Compression Double Lap Shear Test I Table I. Material properties used in the validations Material Modulus Poisson s Thermal expansion Shear ield Ref. MPa Ratio coefficient m/m/k stress MPa Aluminum 69.33.36E-5 - [35, 36] TGMDA epo - undamaged 317.35 6.7E-5 - [36] - damaged 317.35 6.7E-5 - DGEBA epo 185.33 6.67E-5 [38, 7] transverse tensile moduli E i and E i x, the shear modulus G i, the Poisson s ratio υ, i and the axial and transverse thermal expansion coefficients α i and α i x. Thus, this analsis is for orthotropic materials while a previous simplified analsis, involving isotropic materials onl, appeared elsewhere [35, 4]. From an analsis of the most common assumptions used in shear-lag models, it was proposed that the shear stress τ i x, in laer i can be written as a function of two unknown functions of x onl, L i x and R i x the left and right shape functions and the interfacial shear stress τ x i between laers i 1 and i which are a function onl of as: τ i x, = τ x i 1 L i x + τ x i R i x 1 σ σ II τ I II σ I I τ τ II σ σ II II III III τ III τ III σ III σ III σ σ σ III Fig. 3. Geometr of the MDLS specimen with a crack and definition of the three zones I, II, and III used in the analsis.

6 Undamage Interface Cracked Interface Fig. 4. Meshes used in the finite element analsis for specimens with an undamaged interface or ones with an interfacial crack. For clarit, the adhesive laer is shown thicker than in the actual specimens and not all elements are shown. The converged FEA calculations subdivided each of these elements about three times.

7 Compression Double Lap Shear Test I with { 1 at x = xi 1 L i x = a at x = x i { at x = xi 1 R i x = b 1 at x = x i Defining τ = τ x 1, τ x,..., τ x n 1, the shear-lag analsis can be reduced to a sstem of n 1 differential equations in terms of the n 1 unknown interfacial shear stresses [39]: where [A] and [B] are both tridiagonal matrices with elements: A i,i 1 = t iξ i L i G i, A i,i = t i+11 ξ i+1 L i+1 G i+1 1 B i,i 1 = 1 E i t i, B i,i = [A] τ [B]τ = τ 3 E i+1 Here, the dimensionless coordinate in laer i is defined b The right side of the equation is defined b: τ = + t iξ i R i G i + 1, B t i+1 E i i,i+1 = t i, A i,i+1 = t i+11 ξ i+1 R i+1 G i+1 1 E i+1 t i+1 ξ i = x x i 1 t i 5 τ E 1 t 1,,..., τ n E n t n where τ and τ n are the shear stresses in the middle of the specimen and on the free surface. For all calculations in this paper, except one, these shear stress boundar conditions are zero. For solution purposes, Eq. 3 is rewritten as: τ [M τ]τ = [M τ ][B] 1 τ 7 where [M τ ] = [A] 1 [B]. The solution of Eq. 7 is found b an eigen-analsis [39]: σ i = Ei τ x i = τ + i j=1 t j E j τ te n τ + 4 6 n 1 aj e λj b j e λj ω j,i 8 where t = n i=1 t i is the semi-thickness of the composite, E = 1 n t i=1 t ie i is the rule-of-mixtures effective composite modulus of an undamaged structure in the direction, λ j for j = 1 to n 1 are the eigenvalues of the [M τ ] matrix, ω j,i is the i th element of the eigenvector of [M τ ] associated with λ j, and a j and b j for j = 1 to n 1 are constants to be determined b boundar conditions. Note that internal stresses do not appear explicitl in this equation, but are included through a j and b j, calculated as shown below. Additionall, b making use of the equation of equilibrium, it was shown that the average axial stresses are obtained from: [ n 1 τ τ n + tσ is the rule-of-mixtures, effective -direction thermal expansion coefficient of an undamaged struc- where α ture given b: te + α j=1 α i T ] + j=1 aj e λj + b j e λj ω j,i 1 ω j,i t i λ j 9 α = n i=1 α i t i E i te 1

8 Note that σ = τ τ n + σ is the total applied axial stress in the -direction for problems with constant, non-zero shear stresses applied to the sides of the specimen. When there are no shear-stress boundar conditions, σ is constant and equal to the total applied axial stress. The average displacements in the -direction in each laer i can be obtained as a function of the axial displacement at the interface x i or at x i+1, using [39]: v i = vx i t iτ x i 1 G i The average displacement difference across laer i is thus [39]: ξ i L i t iτ x i ξ G i i R i 11 v i+1 v i = t i+11 ξ i+1 R i+1 τ x i+1 + t iξ i L i τ x i 1 1 + G i+1 t i+1 1 ξ i+1 L i+1 G i+1 G i + t iξ i R i τ G i x i In the following sections these equations are applied to several different analzes of the MDLS specimen. 4. Strength Approach 4.1. Specimen Failing in the Linear-Elastic Region 4.1.1. Stress Analsis Solution Method For a specimen failing in the linear-elastic region, we eliminated the crack and used the modified shear-lag theor with n = 3 laers. The interfacial shear τ i = τ x i and average axial stresses σ i are given b [39]: τ 1 = a 1 e λ1 b 1 e λ1 ω 1,1 + a e λ b e λ ω,1 τ = a 1 e λ1 b 1 e λ1 ω,1 + a e λ b e λ ω, σ 1 = C 1 a 1 e λ1 + b 1 e λ1 ω 1,1 λ 1 t 1 a e λ + b e λ ω,1 λ t 1 13 σ = C a 1 e λ1 + b 1 e λ1 ω 1, ω 1,1 a e λ + b e λ ω, ω,1 λ 1 t λ t σ 3 = C 3 + a 1 e λ1 + b 1 e λ1 ω 1, + a e λ + b e λ ω, λ 1 t 3 λ t 3 where the constants C i define the far-field stress state i.e., the stresses far awa from specimen ends and cracks and are given b: C i = E i t 1σ + α te α i T 14 Here σ = σ where σ is the positive magnitude of the applied compressive stress. The remaining constants, a j and b j, j = 1,, were determined b the boundar conditions on the average axial stress in the inner adherent and the adhesive laer. With the origin of the sstem at the middle of the specimen, the boundar conditions are: σ 1 = σ ; σ 1 = ; σ = 15 =L/ = L/ =±L/ The average axial stress in the laer 3 is given b force balance. Solving for the four unknowns a 1, a, b 1, b

9 Compression Double Lap Shear Test I as functions of C 1 and C gives: a 1 = σ e 4 sinh λ1l b 1 = σ e 4 sinh λ1l a = σ e 3 sinh λl b = σ e 3 sinh λl + cosh λ1l e 1 e 4 e e 3 sinh λ 1 L e C e 4 C 1 sinh λ 1L cosh λ1l e 1 e 4 e e 3 sinh λ 1 L e C e 4 C 1 sinh λ 1L + cosh λl e e 3 e 1 e 4 sinh λ L e1 C e 3 C 1 sinh λ L cosh λl e e 3 e 1 e 4 sinh λ L e1 C e 3 C 1 sinh λ L where e 1 = ω 1,1 ; e = ω,1 ; e 3 = ω 1, ω 1,1 ; e 4 = ω, ω,1 17 λ 1 t 1 λ t 1 λ 1 t λ t The obtained set of equations presents a full, closed form solution of both axial and shear stresses in the MDLS sample. This solution depends upon the choice of shape functions which enter in the definition of matrix [A]. For a three-laer problem, the final solution depends onl on six dimensionless averages of the shape function in the laers, namel ζ 1 R 1, 1 ζ L, 1 ζ R, ζ L, ζ R and 1 ζ 3 L 3. 4.1.. The Influence of Shape Functions The solution in the previous section was compared to FEA results for various shape functions. A common assumption in prior shear-lag methods is to take all shape function as linear, which implies L i = 1 ξ i and R i = ξ i. The required averages reduce to 16 ξ i L i = 1 6 ; 1 ξ il i = 1 3 ; ξ ir i = 1 3 ; 1 ξ ir i = 1 6 18 The plot shown in Fig. 5 shows reasonable agreement with FEA results, particularl in terms of stress integral, corresponding to the elastic energ stored in the sstem c.f. Ref. [39]. Like all shear-lag models, this analsis fails to give zero shear stress on the ends because of a lack of degrees of freedom in the shear-lag analsis. The solution makes use of all axial stress boundar conditions to find the unknown constants and can not therefore satisf additional boundar conditions such as zero shear stress on the end. Instead, the shear stress reaches a maximum value on the ends. The FEA results show peaks ver near the ends and then tend towards zero, although never become exactl zero. The FEA results show that there are indeed shear stress maxima ver close to the ends of the specimen. More accurate analtical or numerical results would require accounting for the singularit existing at the interface between two different media at a boundar. This problem, however, remains unsolved for such complex geometr, and approximate solutions are needed. The maximum shear stress at the end of the shear-lag solution can be regarded as an approximate representation of the real shear stress peak and therefore an approximate accounting for the singularit effects. It is possible to refine the shear-lag solution b selecting non-linear shape functions; the approach is similar to that used for a single fibre or a platelet embedded in an infinite elastic medium [37]. Because the adhesive laer is ver thin, it is likel that linear shape functions are sufficient. Even if the adhesive laer shear stress in non-linear, the effect on the final solution is expected to be negligible. Thus L and R are assumed be linear as in the previous calculations. The remaining shape function averages, ζ 1 R 1 and 1 ζ 3 L 3, were then regarded as unknowns; the best values were found b fitting FEA results. The best agreement with FEA for the average axial stresses σ 1 and σ 3 was obtained when the two eigenvectors were orthonormal. Such eigenvectors are 1/ 1/ and 1/ 1/. Next < ξ 1 R 1 > and < 1 ξ 3 L 3 > were obtained b an inverse method to that used to obtain these eigenvectors. Since the process also changes the eigenvalues, one repeats all the calculation steps. Interestingl, the best fitting values were found to be close to: < ξ 1 R 1 >= E1 t 1 E t ; < 1 ξ 3 L 3 >= E 3 t 3 E t 19

1 τ σ σ Fig. 5. Axial and shear stresses vs. distance along lap-joint, linear and optimal shape function solutions TGMDA-based epo adhesive, stresses are normalized vs. σ, and T = In other words, the optimal shape function averages were equal to the fraction of the specimen stiffness contributed b that laer. Figure 5 plots the stresses obtained using optimal shape function, in comparison to those obtained with linear shape functions. The solution is improved b the use of refined shape functions. The average axial stresses in the adherents are nearl exact to the FEA results. The shear stress calculation is improved but still has the limitation of all shear-lag models of maximum shear stress on the ends instead of zero shear stress. It should be noted that FEA results do not provide an exact solution either, since this method does not deal with interfaces in a continuous wa, i.e., stresses and their derivatives are non continuous across the interface. Therefore, the optimal shear-lag method presented here, representing the best shear-lag solution, together with the use of its shape functions, is believed to be accurate on the level of FEA results to serve as a basis for further investigations. The plots show average axial stresses and shear stresses. Shear lag can also find average axial displacements and the similarl agree with FEA results. Shear lag has used to find other terms, such as transverse stresses or displacements, bu such results are not accurate [39]. In fracture mechanics calculations, it is necessar to find total strain energ in the specimen. Because the strain energ in the MDLS specimen can be found using boundar integrals and reduced to a result that depends onl on axial stresses and displacements, this energ can be found accuratel b shear-lag analsis [39].

11 Compression Double Lap Shear Test I σ σ σ σ σ σ Fig. 6. Axial stresses vs. distance along a lap-joint, TGMDA-based epo adhesive, solution with hperbolic shape functions, including internal stresses as indicated on the figure b T, σ = 5 MPa. 4.1.3. Influence of Internal Stresses Internal stresses are included and enter the solution through the constants C i. Previous calculation used T = ; some more results for axial and shear stresses when T are given in Figs. 6 and 7, respectivel. Figure 6 compares the contributions of internal stresses to the axial stresses in the sstem. It is obvious that the stress state in the adhesive laer is greatl affected. The residual stresses have onl a small effect, however, on the axial stresses of the adherents, which agrees with experimental measurements [4]. Most of the experimental work found in the literature was derived b measuring the displacements in the adherents using either strain gages or interferometric methods. The minor influence of the adhesive internal stress state on the adherents axial stress probabl explains wh internal stresses have been neglected in the corresponding analsis of the test. This approach has led to crude approximations, since internal stresses are shown to generate high axial stress and shear stresses in the adhesive laer. As seen in Fig. 7, the shear stress at the adhesive-outer adherent interface, τ, is substantiall increased upon the introduction of internal stresses, while the shear stress at the adhesive-inner adherent interface, τ 1, is decreased down to a negative value. This formalism can be used to describe the onset of crack propagation at the interface of a double lap shear specimen [3, 35]. It was found interesting to compare several cure conditions or model sstems with additives, b which the internal stress level could be varied. To analze theses tests, it is important to isolate the residual stress terms. Defining K i, i = 1,, as the thermal stress terms in the constants C 1 and C b writing: K i = E i α α i T

1 τ τ τ τ Fig. 7. Shear stresses vs. distance along a lap-joint, TGMDA-based epo adhesive, solution with hperbolic shape functions, including internal stresses as indicated on the figure b T, σ = 5 MPa. The shear stress on the inner surface of the adhesive at = L/ can be written as: τ 1 = σ ζ 1 + χ 1 1 =L/ where and e 4 cosh λ 1 L te + t 1 e E e 4 E 1 sinh λ 1L ζ 1 = ω 1,1 e 1 e 4 e e 3 sinh λ 1 L te e 3 cosh λ L te + t 1 e 1 E e 3 E 1 ω,1 χ 1 = ω 1,1 e K e 4 K 1 e 1 e 4 e e 3 e 1 e 4 e e 3 sinh λ L te λ1 L tanh sinh λ L e 1 K e 3 K 1 + ω,1 tanh e 1 e 4 e e 3 λ L 3 The shear stress on the other surface of the adhesive is given b: τ = σ ζ + χ 4 =L/

13 Compression Double Lap Shear Test I where and e 4 cosh λ 1 L te + t 1 e E e 4 E 1 sinh λ 1L ζ = ω 1, e 1 e 4 e e 3 sinh λ 1 L te e 3 cosh λ L te + t 1 e 1 E e 3 E 1 ω, χ = ω 1, e K e 4 K 1 e 1 e 4 e e 3 e 1 e 4 e e 3 sinh λ L te λ1 L tanh sinh λ L e 1 K e 3 K 1 + ω, tanh e 1 e 4 e e 3 λ L This formulation is useful because it splits the apparent interfacial shear strength for applied mechanical loads into internal stress effects and the true shear stresses required for debonding. Writing the experimental shear stress at failure in terms of the applied compression stress as one obtains at failure: τ 1 or, if debonding starts on the other interface: τ exp = F W L = σ t 1 L 5 6 7 L = τ = τ exp ζ 1 + χ 1 8 =L/ t 1 τ exp = t 1 Lζ 1 τ χ 1 9 τ exp = t 1 Lζ τ χ 3 In this formalism, τ represents the true, local shear strength of the interface, which does not depend on internal stresses or on the geometr. In contrast, the observed experimental shear strength, τ exp, is affected both b interfacial properties and b residual stresses effects. In brief, τ is the more fundamental propert for characterization of the interface. This simplified analsis was found to be robust enough to analze experimental data for the onset of debonding, when the specimen failed in the linear-elastic regime. 4.. Stresses in a Transversel Cracked Specimen The problem of stresses when the adhesive is damaged b transverse cracks arises from experimental observations in Ref. [3]. It was observed that some brittle epo adhesives transversel crack prior to debonding. The extent of transverse cracking was reproducible. This section establishes the equations necessar for describing this phenomenon. Consider a damaged zone of length d as shown in Fig.. As a first approximation, we considered that the shear modulus of the adhesive, and thereb its Young s modulus too, were reduced b transverse cracking. This hpothesis was well supported b the stress-strain curves obtained during loading of the MDLS specimen, which showed a decrease of the effective modulus of the joint. Using this hpothesis, the stresses of the sstem could be found b analzing two zones with distinct mechanical properties for the adhesive, and making axial and shear stresses continuous across the two zones. The stress in the two zones are each given b Eq. 13 except that separate constants are needed for each zone or a j a k j, b j b k j, C j Cj k, ω j,i ωj,i k, and λ j λ k j where k = I, II. The constants Ck j and α ik for laer properties and rule-of-mixtures results redefined as are given b Eq. 14 using Eik α k E k t 1 + E k t + E 3 t 3 t t 1 + α E k t + α 3 E 3 t 3 te k = E1 = α1 E 1 31 3

14 Notice that onl the mechanical properties of laer differ between zones I and II. The eight unknown constants, a k j, bk j with j = 1, were obtained from the following the eight boundar conditions: σ 1II σ I =d = ; =d L = ; τ 1I σ I σ II = = = τ 1II ; = = =d L σ II = ; = ; τ I σ 1I σ 1I = τ II = =d = σ = = Figure 8 shows the distribution of stresses in the adhesive joint submitted to internal stresses and applied stress, considering a fragmented zone of length d = 3 mm in which E ii =.1E 1II or damage caused 9% degradation. This value was set arbitraril, based on those obtained from fitting the force vs. displacement curves of TGMDA-based specimens used in experiments detailed elsewhere [3]. It will be further shown that the experimental stress-strain curve is modelled accuratel b the present theor, the major difficult being the determination of the onset of transverse microcracking. On this plot the two zones connected at the origin = clearl appear, and the involve a local reduction of the axial tensile stress in the damaged adhesive which is essentiall due to Young s modulus reduction and the release of internal stresses it induces. The shear stress profiles are still influenced b internal stresses, which involve a stress increase at the loaded end of the specimen = 3 on Fig. 8. More information than available would be needed to accuratel calculate the modulus reduction in the fragmented zone: the angle of the transverse crack with the interface plane is usuall observed to be close to 45, but some variations were observed between specimens with different internal stress states; the distance between two consecutive fragments would also provide useful information on the rate of the damage propagation. As a consequence, the present analsis, which uses an average loss of mechanical properties, is a first approximation of the fracture mechanics involved in the process. Accurate modelling of the local stress state and fracture mechanics involving fragmentation nucleation, propagation, as well as the friction between fragments, would better describe the residual properties of the fragmented adhesive zone. Nevertheless, one of the attractive features of this model is to quantif the fracture process efficientl b describing initiation, propagation and first adhesive failure of the joint, as will be illustrated b experimental results in Ref. [3]. 4.3. Specimen with a Partiall Yielded Adhesive For this analsis, the adhesive was considered as a time-independent, elastic-perfectl-plastic material. Viscoelastic relaxation of internal stresses can be included b using an effective temperature step, as described in Ref. [33]. One can then model the MDLS specimen as two connected zones with different properties Fig. : a first zone in which the adhesive ields and which is characterized b a constant shear stress, connected to a second zone in which the adhesive has an elastic behavior. This case resembles the previous one, in which zone I had altered properties. One major difference arises: for a ielded interface, the ield stress can be evaluated experimentall while it is more difficult to determine the correct results to use for a degraded modulus. In this analsis, the shear stress in zone I is a constant τ p equal to the shear ield stress of the adhesive, which leads to zone I stress state of: = σ 1II = 33 τ 1I = τ I = τ p ; σ 1I = τ p t 1 d σ ; σ 3I = τ p d 34 t 3 Because the shear stress in laer is constant and also b force balance, the average stress in the adhesive in zone I σ I is zero. The zone II stress state is again given b Eq. 13. The boundar conditions for the four unknowns in zone II involve the continuit of axial stresses between the two zones, and the free end conditions: σ 1I = σ 1II ; σ 1II = 35 = = =d L σ I = σ II = ; σ II = = = =d L

15 Compression Double Lap Shear Test I σ τ τ σ σ Fig. 8. Stresses vs. distance along a lap-joint, with a damage zone of length d = 3 mm, TGMDA-based epo adhesive, Young s modulus E I =.1 E II, T = 1 C, and σ = 5 MPa Sample calculations obtained following this approach are shown in Fig. 9. Interestingl, the interfacial shear stresses τ 1 and τ can be superposed almost everwhere, except close to the unloaded end of the specimen at = d L, where the differ b less than one percent. In this model, τ 1 = τ throughout the ield zone and at the tip of the ield zone. This analsis can thus not predict which interface will debond first. On the other hand, this model is remarkabl accurate for describing the interfacial strength of specimens having similar interface characteristics but different shear ield stresses see Ref. [33]. 4.4. Determination of Stresses in a Cracked Specimen Three Zone Model To interpret experiments in which a crack propagates along the interface, we considered a specimen with a crack of length a at the interface between laer 1 and. The origin of the axis is located at the crack tip. The analsis proceeds b considered three zones as indicated in Fig. 3. The stresses in zone III are again given b Eq. 13. The two new zones were analzed as follows: 4.4.1. Zone I The stresses in zone I can be obtained b considering the shear stress as zero in the midplane and increasing linearl to a constant shear stress on the crack denoted b τ f the positive magnitude of the friction stress on the crack. Two singularities of the problem are neglected at = a and =, where the shear stress of the cracked interface should be zero and singular, respectivel. At = a, it is not necessar to refine the problem b introducing a zone of infinitel small length, as was done for other tpes of load and geometr

16 τ σ σ σ Fig. 9. Stresses vs. distance along a lap-joint, DGEBA-based epo adhesive, solution with hperbolic shape functions, including constant shear stress τ p = MPa in the ielded zone, σ = MPa, T = 15 C. [41, 4, 43], because the ERR does not depend on the shear stress τ a. The singularit at = is more of a problem, and the accurac of the model should be checked at that location. Force equilibrium on an element of length d leads to see Ref. [39] for details: Using the boundar condition: the zone I axial stress is: 4.4.. Zone II σ 1I σ 1I σ 1I = τ f t 1 36 =a = σ 37 = τ f a t 1 σ 38 In zone II, the friction stress in the debonded interface is of opposite sign to that of zone I, and therefore equals τ f. This case corresponds to a two-laer, shear-lag analsis with non-zero boundar shear stress. It can be solved using the following equation [39]: τ II β τ II = τ f 1 E t t + t3 3G 3G 3 39

17 Compression Double Lap Shear Test I where the optimal shear-lag parameter is Solving Eq. 39, one obtains: t t 3 σ II σ 3II β = 1 + 1 E t E 3 t 3 t 3G + t3 3G 3 = a II e β + b II e β + E τ f a t E II t + t 3 + = a II e β b II e β + E 3 t τ f a 3 E II t + t 3 + τ II E 3 t 3 τ f = βa II e β + βb II e β t + t 3 E II α II α T α II α 3 T 4 41 and E II andα II are the rule-of-mixtures properties for the sstem with laers, i.e.: 4.4.3. Boundar Conditions α II E II = E t + E 3 t 3 4 t + t 3 = α E t + α 3 E 3 t 3 E II t + t 3 The above stresses have six unknowns: a II, b II, a III 1, b III 1, a III, and b III. These unknowns were determined from the following six boundar conditions: σ II =a = ; σ 1III σ II =a L = ; = = σ III σ III =a L = ; σ 1III = ; τ III = = τ f a t 1 σ 43 = = τ II As shown in Figs. 1 and 11, the stresses found b this analsis agree well with FEA results for a cracked specimen. = 5. Energ Release Rate Using the stress state in the previous section, it is possible to calculate the energ release rate for growth of an interfacial crack. The calculation of the energ release rate proceeds in two steps. First, an exact expression for the energ release rate which can be reduced to a result that depends onl on axial displacements and average axial stresses is derived. Second, the shear-lag results for these displacements and stresses are used to derive an approximate energ release rate. 5.1. Exact Results A general result for energ release rate in composite fracture which includes residual stresses and traction loads on cracks e.g., friction was recentl derived as [16]: G = 1 T. uds + σα T dv 44 A S+S c where A is the cracked area, S denotes the outer surface, S c stands for the cracked surfaces, and V denotes the volume. T is the traction loads which are applied to S and possibl also on S c, and u = u, v, w where the components are the displacements in the x,, and z directions. V

18 σ σ σ Fig. 1. Normalized axial stresses vs. distance along a lap-joint, TGMDA-based epo adhesive, solution with hperbolic shape functions, including constant friction in debond zone τ f =.5σ, T =. Table II. Calculation of the traction-displacement integrals for all loaded boundar surfaces and tractionloaded crack surfaces. Surface T u S+S c T. uds top plate 1 σ u 1 v 1 x, a w 1 σ t 1 W v 1 a t bottom plate 3 σ 1 u 3 v 3 x, a w 3 σ t 1 W v 3 a L t3 crack surface, plate 1 σ t τ f u 1 t 1, v 1 t 1, w 1 crack surface, plate σ t τ f u t 1, v t 1, w crack surfaces W a τ f v 1 t 1, v t 1, d side surface σ t u 3 t, v 3 w 3 σ t LW u 3 t From the two preceding lines T u = σ t u 1 t 1 + τ f v 1 t 1, τ f v t 1, + σ t u t 1, but the transverse stress terms cancel because either u 1 t 1 = u t 1, when the crack surfaces are in contact, or σ t =, when the crack surfaces are not in contact.

19 Compression Double Lap Shear Test I τ τ Fig. 11. Normalized shear stresses vs. distance along a lap-joint, TGMDA-based epo adhesive, solution with hperbolic shape functions, including constant friction in debond zone τ f =.5σ, T =. The vectors T and u on loaded surfaces and crack surfaces of the MDLS specimen and their contributions to the surface integral are given in Table II. Here σ t is the transverse stress, as shown in Fig. 1. Using these results with the crack area da = W da, G can be recast as: G = 1 d W da +W [ σ t 1 W v 1 a + σ t 1 W v 3 a L σ t LW u 3 t 45 a ] τ f v 1 t 1, v t 1, d + σα T dv V Simplifing the volume integral [16] leads to: [ G = 1 d σ t 1 v 1 a + σ t 1 v 3 a L σ t L u 3 t 46 da ] a + τ f v 1 t 1, v t 1, d + L α 1 α T t 1 σ 1 + L α 3 α T t 3 σ 3 where the bars denote phase averaged stresses, calculated as: σ i = σ i dv = 1 V i L a a L σ i d 47

5.. Approximate Solution The expression for the energ release rate G determined b Eq. 46 is exact and depends onl on axial stresses and displacements in the three laers. The optimal shear lag analsis is used in this section to get a approximate solution for G. 5..1. Calculation of Displacements in the 3 Zones Zone I: In its most general form, v 1 = v 1 + ε 1 d = v 1 + σ 1 + α 1 E 1 T d 48 The onl approximation here is to ignore contributions of transverse stresses to the axial strain [39]. Taking as a reference for displacements v 1 =, and making use of Eq. 38, one obtains v 1 = τ t 1 E 1 f a 1 σ t 1 + α 1 T 49 and therefore: v 1 a = a 1 t 1 E 1 τ f a σ t 1 The edge displacement v 1 t 1, also needs to be calculated. From Eq. 11: v 1 t 1, = v 1 + τ f t 1 G 1 ξ 1 R 1 = t 1 E 1 + α 1 T a 5 τ f a 1 σ t 1 + α 1 T + τ f G 1 t 1 ξ 1 R 1 51 Zone II: B a similar analsis: v = v + σ E + α T d 5 With σ II given b Eq. 41: v = v + aii e β 1 bii e β 1 βe t βe t + τ f a t + t 3 E II + α II T 53 To obtain the v term, the edge displacement v t 1, needs to be calculated first. Again, from Eq. 11, and with accounting for shear stresses on the edges of zone II: v t 1, = v τ II t ξ R τ f G The continuit of axial displacement between zones I and II at = : G t ξ L 54 v t 1, = v 1 t 1, 55 ields and v = t ξ R G v + τ t ξ R G βa II βb II τ f G t ξ L = E 3 t 3 τ f E II t + t 3 + τ f G τ f G 1 t 1 ξ 1 R 1 56 t ξ L + τ f G 1 t 1 ξ 1 R 1 57

1 Compression Double Lap Shear Test I Assembling all contributions: v t 1, = t ξ R βa II βb II G + aii e β 1 bii βe t βe E 3 t 3 τ f t + t 3 + τ II a E II e β 1 t + τ f E II + τ f G 1 t + t 3 + αii T t 1 ξ 1 R 1 58 The difference of displacements is now: v 1 t 1, v t 1, = t 1 E 1 + βt ξ R G τ f a 1 σ t 1 + α 1 1 βe t α II T a II e β 1 b II e β 1 τ f a t + t 3 E II 59 Integrating this expression between and a gives: a τ f v 1 t 1, v t 1, d = τ f { τ f a 3 3σ t 1 a + 6t 1 E 1 βt ξ R G 1 βe t + α 1 α II T a + τ f a 3 6 3 t + t 3 E II [ e a II βa 1 e a + b II βa ] } 1 + a β β Zone III: Eq. 1: The last displacements that need to be evaluated in Eq. 46 are v 3 a L v 1 a. From v = v 1 + τ t 1 ξ R v 3 = v + τ G [ t 3 1 ξ 3 L 3 G 3 + τ 1 [ t 1 ξ 1 R 1 G 1 + t ξ R G ] ] + t 1 ξ L G + τ 1 t ξ L G 61a 61b and, b eliminating v results in: v 3 = v 1 + τ + τ 1 [ t 3 1 ξ 3 L 3 G 3 [ t 1 ξ 1 R 1 G 1 + t ξ R G + t 1 ξ L G ] + t 1 ξ R G + t ξ L G ] 6 It follows from Eq. 48 that: v 3 a L v 1 a = a L a + + σ 1 E 1 [ t 1 ξ 1 R 1 G 1 + α 1 T [ t 3 1 ξ 3 L 3 G 3 d + t 1 ξ L G + t ξ R G ] + t ξ L τ G 1 a L 63 ] + t 1 ξ R G τ a L

Making use of the phase-averaged stress definition Eq. 47: v 3 a l v 1 a = L σ1 E 1 + α 1 T L + [ t 3 1 ξ 3 L 3 G 3 [ t 1 ξ 1 R 1 G 1 + t ξ R G + t 1 ξ L G + t 1 ξ R G ] + t ξ L ] G τ 1 a L τ a l 64 5... Phase Averaged Stresses Combining all displacement results, the energ release rate can be written as: G = 1 d [σ t 1 da + L σ1 α E 1 1 T L + [ t 3 1 ξ 3 L 3 G 3 + t ξ R G [ t 1 ξ 1 R 1 G 1 + t 1 ξ L G ] + t 1 ξ R G α 1 { τf a 3 3σ t 1 a σ t Lu 3 t + τ f + 6t 1 E 1 [ βt ξ R 1 e + a II βa 1 G βe t β + L α 1 α ] + t ξ L τ G 1 a L τ a L α II T a e a + b II βa 1 T t 1 σ 1 + L α 3 α T t 3 σ 3 ] + τ f a 3 3 t + t 3 E II ] } + a β 65 This ke terms needed are the phase-averaged axial stresses that are obtained from: The following integrals are needed: σ 1 = 1 L σ 3 = 1 L a a L a L σ 1I σ 1III σ 3III d + d + a a σ 1I σ 3II d 66 d 67 d = τ f t 1 a σ a 68 The exponential coefficients and eigenvalues a k i, bk i, β and λ i do not depend upon the axial coordinate, which makes it possible to integrate the stresses; the result is a σ 3II d = 1 t 3 { a II 1 e βa β + E 3 t 3 a + bii e βa 1 β [ α II α 3 T ]} τ f a E II t + t 3 69

3 Compression Double Lap Shear Test I and a L a L σ 1III σ 3III [ d = a III 1 1 e λ1a L ] + b III 1 e λ1a L ω1,1 1 λ 1 t 1 [ + a III + E 1 a L e λa L 1 t 1 σ + α te α 1 T 1 e λa L + b III ] ω,1 λ t 1 [ d = a III 1 e λ1a L 1 + b III 1 1 e λ1a L] ω,1 λ 1 t 3 [ + a III + E 3 a L e λa L 1 + b III 1 e λa L] ω, λ t 3 t 1 σ + α te α 3 T The final energ release rate is found b substituting Eqs 68 71 into Eq. 65 and differentiating with respect to crack length a. Because the constants a k i and b k i are functions of a, the final differentiation step must be done numericall but the calculations are straightforward. 5.3. Results Figure 1 plots the energ release rate G calculated from the model solid curves compared to that obtained b FEA smbols, with and without internal stresses, and for various friction stresses. The calculations shown were obtained for a fixed value of σ = 5 MPa. For a friction stress of τ f =, the energ release rate increases rapidl to about 5 kj/m and then continues to increase, albeit slowl, to 3 kj/m. Because G increases with crack length, the prediction is that crack growth, once started, will be unstable until the end of the specimen. The situation changes dramaticall when friction is added. When τ f is not zero, the energ release rate rises to maximum for a certain crack length, and then decreases. In experiments, the prediction is that the crack will rapidl propagate to a length corresponding to the maximum G. Thereafter, because G decreases with crack length, the load will have to be increased to propagate the crack. In other words, friction stabilizes the crack propagation. The shear-lag calculations agree ver well with FEA at long crack length a > 3 mm, but agree less well at shorter crack length. In particular, the maximum in G when there is friction, is at a much shorter crack length in the shear-lag analsis than in the FEA analsis. The shear-lag analsis was derived from a shear-lag stress analsis that ignores all singularities in the stress state. The energ release rate was derived from a derivative of this stresses in this analsis. Such a derivative analsis will be accurate whenever the effects of the singularit are small or whenever the contribution of the singularities is a constant. When the effect is a constant, it will drop out in the differentiation step. Based on the results in Fig. 1 we claim that the singular effects at the crack tip are a constant provided the crack tip is not too close to the start of the specimen. In other words, the shear-lag analsis is accurate provided a is not too short. When a is too short, the crack tip complexities will be influenced b the free surface. The shear-lag analsis does not account for this effect and thus the accurac is lowered. Similar conclusions were reached in the analsis of debond growth and the fiber-matrix interface in the microbond specimen where a shear-lag analsis was shown to agree with FEA results provided the debond crack tip was not to close to either end of the specimen [44]. The effect of internal stresses, represented b the difference between the upper and lower curves obtained for a same friction stress but different T, agrees ver well with FEA calculations. This agreement follows because the approximation of average axial stresses b shear lag is ver accurate, and because σ 1 and σ 3 dominate the internal stress response of the sstem at long crack length. The accurate description of internal stress effects works for all values of friction stress. The model accurac is influenced b the choice of shape functions, and it is possible to improve the results b adjusting them. Unfortunatel, this approach onl provides a partial solution to an disagreements 7 71

4 τ τ σ τ σ Fig. 1. Energ release rate of a debonding interface: model solid curves vs. FEA results smbols, for various coefficients of friction and internal stress levels. For each friction coefficient, the upper curve has T = 1 C, and the lower curve T =. mentioned above. Another approach could be to fix the shape functions as linear, but to increase the number of laers. In other words, each phsical laer could be divided into additional laers and the resulting problem analzed b multilaer shear-lag methods [39]. The model s accurac would benefit from this approach, but the calculations would require more numerical work. To conclude, the model is applicable provided the friction stress is between and.5σ, and the crack length is greater than about 3 mm. The friction stress has a larger influence on the energ release rate than the internal stress level, and it should therefore be determined accuratel in an experimental stud. 6. Conclusions This stud introduced a new geometr for a lap-shear test, the modified double lap shear MDLS specimen. The stress state in the MDLS specimen, including residual stresses, was obtained from a modified shear-lag theor, extended to account for non-perfect interfaces and also local variations in the adhesive mechanical behavior. Two alternative theories were devised to interpret the test results, namel the interfacial shear strength approach and the energ release rate approach. Stress shape functions were used to refine the stress analsis, and average displacements were calculated accordingl. The local stress states and the approximate energ release rate were shown to compare well to FEA results. For energ release rate, it was shown that the model is valid provided the crack length is larger than a certain value. Also, the frictional stress greatl influences the value of G; thus friction greatl influences crack growth and must be well understood before interpreting an experimental results. The MDLS specimen was analzed with a generic multilaered theor. This theor could be applied to other sstems such as the modelling of transverse nano-indentation of multilaered sstems.

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