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3D models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives
Author’s Accepted Manuscript 3D models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives Nicolas Carrere, Claudiu Badulescu, Jean-Yves Cognard, Dominique Leguillon www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(15)00106-2 http://dx.doi.org/10.1016/j.ijadhadh.2015.07.005 JAAD1680
To appear in: International Journal of Adhesion and Adhesives Received date: 27 February 2015 Accepted date: 12 June 2015 Cite this article as: Nicolas Carrere, Claudiu Badulescu, Jean-Yves Cognard and Dominique Leguillon, 3D models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2015.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
3D models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives Nicolas CARRERE1*, LEGUILLON2
Claudiu
BADULESCU1,
Jean-Yves
COGNARD1,
Dominique
1
Laboratoire Brestois de Mécanique et des Systèmes, ENSTA Bretagne, Brest, France. Institut Jean Le Rond d'Alembert, CNRS UMR 7190, Sorbonne Universités, UPMC Université Paris 06, F-75005 Paris, France. 2
* Corresponding author: [email protected]
Summary In this paper, accurate numerical analyses of the stress distributions within the adhesive in scarf joints under elastic assumption using 3D models are developed. An elastic limit model that takes into account the hydrostatic stress and von Mises equivalent stress, permit to define the more stressed parts of the adhesive with respect to the scarf angle. It is shown that high stress concentrations are generated near the edges and the corners of the specimen at the interface between the substrate and the adhesive. The use of ronded substrates does not lead to decrease significantly these stress concentrations. It is shown that cleaning the free edge of the adhesive changes the location of edge effects without completely erasing them. A modification of the geometry, has been proposed. This modification leads to a quasi-homogeneous distribution of the stresses in the thickness of the adhesive with stresses that tend to 0 near the edges.
Keywords: Scarf joint, Bonded Assembly, Stress analysis, Finite element analysis
1.
Introduction
Thanks to their capabilities to assemble a large range of materials, adhesively-bonded joints appear to be an interesting technique to reduce the weight and the cost of structures in various industrial domains [1][2]. However, due to the lack of confidence in the non-destructive evaluation of quality, adhesives are drastically under-used. Another key point lies on the necessity to propose tests to determine the reliability of assembly options that can be considered for the design of adhesivelybonded structures. Due to the significant edge effects (associated with geometrical and material mismatch between the adhesive and the substrates) and the complex failure mechanics that could be generated, the experimental characterization of the feasible assemblies is not an easy task. Some tests have be normalized in order to facilitate this experimental characterization such as the singlelap joint test [3]. However, due to the stress concentrations near the ends of the overlap, the determination of a reliable strength is not trivial (even if some studies have proposed some solutions to reduce this edge effects [4]). Moreover, this test is devoted to the determination of the strength for a given ratio peel/shear stress assumed to be a quasi-pure shear although high peel stresses are generated at the ends of the overlap (some studies have proposed some modification to submit the adhesive to variable combinations of peel and shear stresses [5]). To overcome these difficulties, some authors have proposed specific test devices such as the modified Arcan test [6][7] device or the use of cylindrical specimens subjected to tensile-torsion loads [8][9]. These experimental devices permit to reduce the edge effects and submit the adhesive to variable multi-axial loadings. However, such devices seem difficult to use in an industrial environment. The use of specimens *
Corresponding author [email protected] - Tel : +33 2 98 34 88 67 - Fax : +33 2 98 34 87 30 Address: ENSTA Bretagne, LBMS, 2 rue François Verny, 29806 Brest Cedex 9, France
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with a scarf joint represents an interesting alternative. Indeed, this kind of specimen has been widely used in the literature. Indeed, it does not necessitate a specific experimental apparatus and could be used with a classical tensile machine. By adjusting the bond-line angle of the adhesive, different tensile-shear loadings can be obtained [10][11]. However, high stress concentrations are generated near the edges of the specimen [[11][12][13]] and the stresses are maximum at the interface [13][11]. It has also been recently shown that, using 2D [13] or 3D [12] Finite Element calculation, that an optimal angle leading to no edge effects could be determined. The aim of the present study is to know, using a refined 3D FE calculation (as compared to what could be found in the literature), if it is possible to find a geometrical configuration (in terms of the shape of the substrates, local geometry of the adhesive) (i) that modifies the location of the maximal stress (at the interface to determine the adhesion strength or in the middle plane of the adhesive to determine the strength of the adhesive) and (ii) that decreases the edge effects to facilitate the analysis of the experimental results. The first section is devoted to the presentation of the material and of the model. The behavior and the failure of adhesives being influenced by the hydrostatic pressure, the maximum load transmitted by the scarf joints will be determined using a pressure-dependent elastic limit of the adhesive defined from the two stress invariants, hydrostatic stress and von Mises equivalent stress. In the second section, the properties of the singularity exponent, obtained from an asymptotic analysis are presented. This approach permits understanding the effect of the bond line angle on the edge effects. The singularity exponent allows determining if the geometry could lead to singular stress fields (i.e. edge effects) or regular stresses (i.e. without edge effects). However, the knowledge of this exponent is not sufficient. This is the reason why 3D finite element calculations are performed in the third section. Firstly, the influence of the mesh size especially in the thickness direction is proposed. Different geometrical configurations are studied to know if it is possible to choose the location of the maximal stress and to reduce the edge effects. Finally, in order to obtain a configuration without edge effects and with maximal stresses located in the middle plane, a modified geometry is proposed in the last section.
2. Material and model 2.1. Definition of the geometry of the bonded specimens Figure 1 presents the main geometrical parameters of the scarf joint which are used herein. The width and the thickness of the substrate, denoted respectively by d and h, are constant, therefore the overlap length, denoted by l, depends only on the angle of the scarf joint. The adhesive thickness, denoted by e, is defined in the normal direction of the middle plane of the adhesive. The displacements of the lower part of the specimen are blocked and a tension is applied on the upper part. Moreover, the length of the bonded assembly has been chosen to avoid interaction between the end conditions (top and bottom ends of the adherends).
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Figure 1: 3D Model used for the scarf joint (the drawing is done with negative in order to facilitate the presentation).
2.2. Numerical model of the adhesive under tensile/compression-shear 3D computations were performed under small displacement and elastic assumptions. Results are presented for aluminum substrates (Young’s modulus: Es = 75 GPa, Poisson's ratio: s = 0.3) and the elastic material parameters for the adhesive are: Ea = 2.0 GPa, a = 0.4. For this study, only the elastic behavior of the adhesive is considered, so it is necessary to know its yield limit in order to remain well within the elastic range. A precise determination of the maximum transmitted load (i.e. the applied load leading to the onset of the non-linearity) of adhesively-bonded joints requires taking into account an accurate limit criterion in order to represent the real behavior of the adhesive in an assembly. Various studies underline that an accurate representation of the elastic yield surface of an adhesive requires the use of a pressure-dependent constitutive model, i.e. a model taking into account the two stress invariants, hydrostatic stress and von Mises equivalent stress. Starting from the experimental results obtained for the epoxy resin Huntsman Araldite® 420 A/B, with the modified Arcan device (adhesive thickness of 0.4 mm and displacement rate of the tensile machine crosshead of 0.5 mm/min) it is possible to define the initial yield function for the adhesive (neglecting viscous effects). This test allows defining the behavior of an adhesive in an assembly under radial loadings (tensile/compression-shear loadings) [14][15]. An exponential Drucker-Prager yield function leads to a good representation of the experimental data for the socalled ―initial‖ elastic limit [16]: F0=a(vm)b+ph-p0=0
(1)
Where vm is the von Mises equivalent stress and ph is the hydrostatic stress; a, b and p0 are material parameters. The results of the identification are presented in Table 1. Figure 2 presents the elastic limit. The determination of the maximum transmitted load under elastic assumption and proportional loads is detailed in [16].
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a (MPa-4) b p0 (MPa) 6. E-7 5. 31.7 Table 1: Material parameters for the initial yield surface.
60
von Mises stress (MPa)
50 40
30 20 10 0 -30
-20
-10
0
10
20
30
hydrostatic stress (MPa) Figure 2: Elastic limit of the adhesive in the von Mises equivalent stress - hydrostatic stress diagram.
3. Asymptotic analysis of the scarf joint The location where the adhesive interfaces meet the free surface is a site of stress concentrations that can be described using the singularity theory [17] within the framework of linear elasticity. This theory involves the so-called ―Williams’expansion‖ in the vicinity of the points under consideration It provides a description of the local displacement and stress fields written as a sum components that are function of the distance to the edge and of the singularity exponents (see eq. 7 and 8). The asymptotic analysis is a powerful tool to determine if a geometrical configuration could lead to edge effect or not. Indeed, if the singularity exponents are lower than 1, the solution is singular (and thus edge effects could be generated). In the present paper, this approach is used to study if, at the edges or at the corners of the specimen, edge effects could take place as a function of the angle of the bondline. For the scarf joint, modeled in 3D, there are different corners which can be reduced by two, using the symmetry with respect to the axis Oz (Figure 1). Half of the scarf joint is presented in order to present the different properties of the different corners.
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(a)
(b)
(c)
(d)
Figure 3: Different configurations of corners for a given scarf joint angle (the drawing is done with δ negative in order to facilitate the presentation); (a) one half of the scarf joint (Fig. 1); (b) case 1: 2D model of the scarf joint (plane x-y); (c) case 2: geometry in the plane y-z; (d) case 3: edges of the parallelepiped specimen.
For case 1, the two points A and B represent the two corners which have to be taken into account using 2D modeling of the scarf joint (plane x-y, Figure 3b). In 2D, close to a corner, two angles s and a can be used to define the local geometry of the substrate and of the adhesive. Two different types of local geometry exist for a given scarf joint angle ; they are characterized by sx + ax = (with X=A or X=B) ; thus two configurations have to be analyzed:
aA = ; sA =
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aB = ; sB =
For such a model, it is sufficient to consider 0 < aX < with sX + aX = (with X=A or X=B) For case 2, in the plane (y-z) two other type of corners can be analyzed using 2D models. The two points (C and C’), presented in Figure 3c, are characterized with the same properties:
aC = ; sC =
For case 3, the four wedges of the parallelepiped specimen are the last type of corners. Using the symmetry with respect to plane (O, x-y), only two corners have to be taken into account (points D and E, Figure 3d). These two points, which require a 3D analysis, are characterized with the following parameters:
aD = ; sD = ; = aE = ; sE = ; =
In the case of 2D problems under elasticity assumption close to a corner, the relevant parts of the displacement field U and the stress tensor , using the polar coordinates (r: radius, : angle) are given by [17] U(r, ) = C + k1 r
(r, ) = k1 r
u1() + k2 r
s1() + k2 r
u2() + k3 r
s2() + k3 r
u3() + …
(7)
s3() + …
(8)
The constant C is irrelevant and is present for consistency. The terms i are positive, they are socalled ―singularity‖ exponents (singular at least if i < 1); taking into account the definition of the stress (Eqn. (8)). They are the solution of an eigenvalue problem with the ui as eigenvectors. They depend only on the local properties, i.e. the elastic coefficients of the substrate and the adhesive and the angle a. In some simple cases they are analytically known and otherwise can be numerically determined. It can be seen that the closer the is to 0, the higher the singularity; thus a i close to 1 can be considered as a weak singular exponent. The coefficients ki are the generalized stress intensity factors, they are mainly functions of the global geometry, the loading mode and its intensity. They can be computed using path independent integrals in the orthogonal plane [17]. Here we are interested mainly in the values of the i: they allow the classification of the stress concentrations in terms of harmfulness (the smaller the exponent, the more harmful the stress concentration). In the case of 3D corners, the displacement field V can be written in the following form [17]
V(r, ) = C + K1 r
v1() + K2 r v2() + …
(9)
where r, are the spherical coordinates with origin at the corner point. The i are still solution of an eigenvalue problem with the vi as eigenvectors and the only way of determination is now by a numerical procedure [18]. The generalized stress intensity factors Ki are constants, their computation can be carried out using surface independent integrals or by identification [19]. -6-
Figure 4 shows the most singular exponent for the three different cases presented before, function of the local adhesive angleaX. At the top of Figure 4, the corresponding values of the scarf joint angle are plotted with respect to the adhesive angle a. 60
50
40
30
20
10
0
10
20
30
40
50
60
(°) scarf angle
1
1.25
0
Case 1
Case 3
Point B
Point A
Point D
0.75
singular solution
Exponent
1.00
Case 2 0.50
Point E Points C and C’
0.25
0.00 30
40
50
60
70
80
90
100
110
120
130
140
150
Angle of the adhesive a (°) Figure 4: Evolution of the most singular exponent with respect to the angle of the adhesive a, for the three cases illustrated in Fig. 3.
It can be seen that the singularity exponents are slightly smaller (i.e. more harmful) for case 3 than for case 2. Moreover, for case 2, the most singular exponent is lower than for case 1 in the case of a < 90°. Therefore a 3D analysis of the scarf joint is necessary because the most singular exponent is always obtained either in case 2 (2D) or in case 3 (3D) and not in case 1 (2D). A more precise analysis requires the computation of the generalized stress intensity factors ki or Ki in order to have a precise idea of the stress state. But, it is important to notice that the presence of singular exponents can lead to large stress concentrations, further aggravated taking into account geometrical or loading defects. In the following, accurate 3D finite element computations are carried out to analyze the stress state within the adhesive.
3. Finite element analysis of the specimen with a scarf joint All the Finite Element analyses are performed using Abaqus [20] under 3D assumptions.
3.1. Geometries of the adhesive and of the substrates and boundary conditions In order to study the stress state in the adhesive, it is necessary to perform 3D Finite Element calculations. Different geometries of the substrates and of the joint could be investigated. Two of them are considered: one with a rectangular section (noted geometry A) and one with round corners (noted geometry B). The adhesive could have different shapes: straight edge (noted with a lowercase letter a), cleaned edge (noted with a lowercase letter b). The different geometry are shown in Figure 5.
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(a) geometry Aa
(b) geometry Ab
(c) geometry Ba
(d) geometry Bb
Figure 5: Scarf joint with a rectangular section of the specimens (geometry A (a) and (b)) and a round shape of the substrates (geometry B (c) and (d)). (u,v) is the local coordinate system and (x,y) the global one (see Fig. 3). Only one quarter of the adhesive is shown (see Figure 6): (O,x,y) is a symmetry plane and (O,u,z) an anti-symmetry one. Some regions of interest are evidenced: C is the center part of the adhesive (no confusion with the corner C of the previous section), RB and LB respectively the right and the left borders, EB the edge border RC and LC respectively the right and left corners. The boundary conditions are shown in Figure 6. On the symmetry plane (O,x,y) displacement Uz is blocked. An antisymmetric condition is applied on the (O,u,z) plane. A tension is applied on the top of the specimen.
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Figure 6: Bonndary conditions applied on the quarter of the specimen with a scarf joint
3.2. Influence of the mesh size It has already been shown that the mesh size could have a great influence on the determination of the stress state in presence of stress concentrations [3]. This is the reason why, the influence of the mesh size has been studied, especially in the thickness of the adhesive. It has been studied in the case that leads to the higher stress concentrations: a rectangular section of the substrates and straight edges with a scarf joint angle of 0° (butt joint). In this case it is possible to model only one eighth of the specimen. The mesh in the plane of the adhesive near the right corner is shown in Figure 7. Near the edges linear brick elements (regions RC, LC, EB, LC, RB) are used while linear wedge elements are used in the center part. In the corners (RC, LC), the size of the elements is equal to 10m×10m (closed to the one proposed in [12]) while it is equal to 10m×100m near the edges (EB, LC, RB).
Figure 7: Mesh of the adhesive joint near the right corner.
Figure 8 shows the evolution of the maximum traction p0 that could be transmitted before to reach the elastic limit in the adhesive. The maximum traction is calculated in each part of the adhesive -9-
(i.e. the center and the edges) as a function of the number of elements in the half-thickness of the adhesive. In order to analyze this influence in the different parts of the joint, the results are presented for each zone of interest. These results show: (i) that obviously the maximum load transmitted in the middle part of the adhesive is independent of the number of elements and (ii) that it decrease in the edges as a function of the number of elements. It appears that 15 elements in the half-thickness is a good compromise between the computational cost and the accuracy of the result.
Figure 8: Influence of the number of elements in the half thickness of the adhesive on the maximum transmitted load taking into account the different parts of the joint for a rectangular substrate and straight edges with a scarf joint angle of 0° (butt joint). C et Edges represent respectively the results in the center of the adhesive and in the zone EB+LC+RB. Finally, the mesh used for the modelling presented in the next sections considers only a quarter of the scarf joint and is composed by 400000 elements in the adhesive and 150000 in the substrate.
3.3. Influence of the scarf joint angle for straight edges and rectangular substrates Figure 9 shows the maximum values, over the thickness of the adhesive joint (direction v), of the Mises equivalent stress and hydrostatic pressure, in the plane of the adhesive joint (plane u,z). The results are presented for three scarf joint angles: 0° (Figure 9a and b), 30° (Figure 9c and d) and 50° (Figure 9e and f). Results are presented for a traction p0 of 1 MPa. The maximum value is located near the interface between the adhesive and the substrate. For the different scarf joint angles, the stress state is homogeneous in a large part on both sides of the center of the joint. It can be seen that stress concentrations occur near the adhesive free edges and especially at the corners. These stress concentrations, computed with a given mesh definition, depend on the value of the scarf joint angle . Moreover, the location of the stress concentrations is also a function of the scarf joint angle (right edges for left edges for
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(a)
(b)
(c)
(d)
(e)
(f)
Figure 9: Influence of the scarf joint angle on the maximum Mises and hydrostatic stress through the thickness of the adhesive represented in the plane of the adhesive (u,z): scarf joint angle of 0° (a and b), scarf angle of 30° (c and d), scarf joint angle of 50° (e and f). The section of the specimen is rectangular and the adhesive has straight edges. The envelopes of the stresses (in the hydrostatic pressure-von Mises stress plane) in the adhesive for the maximum load transmitted by the joint under elastic assumption is represented in Figure 10 a and b (for scarf joint angles equal to 0° and 40°). The triaxiality ratio (Tx i.e. the ratio between the hydrostatic pressure and the von Mises stress) in the center of the specimen is very different for the two angles. The results presented in Figure 10 also allow determining the points of the adhesive, which first reach the elastic limit. It can be seen that the elastic limit is first reached in a zone that is not the center of the specimen (it is in the corner) for a triaxiality ratio very different of the one obtained in the center of the specimen.
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(a)
(b)
Figure 10: Influence of the scarf joint angle on the distribution of the stress state in the plane hydrostatic pressure/von Mises stress. Results for a scarf joint angle equal to 0° (a) and 40° (b). Each point corresponds to the stress state at a Gauss-Point in the adhesive. The Red square are the stress in the center part of the adhesive, the blue circle correspond to the envelop of the stresses in the other part of the adhesive. Figure 11 presents the evolution of the maximum traction p0 that could be transmitted before to reach the elastic limit in the adhesive with respect to the angle . In order to analyze the influence of the different parts of the joint, the results are presented for each zone of interest (the center correspond to CE, the edge to EB, the corners to RC+LC and the borders to RB+LB). The maximum transmitted load of the specimen with a scarf joint corresponds, for each scarf joint angle, to the minimum over the whole specimen (i.e. the solid line). The result in 2D corresponds to the dotted line (2D is only considered for the three zones of interest LB, C and RB).
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p0 (MPa)
angle (°) Figure 11: Influence of the scarf joint angle on the maximum transmitted load into the different parts of the joint for a rectangular substrate and straight edges. Edge refer to zone EB, Border refer to RB+LB and Corners refer to RC+LC.
Under, 2D assumptions, an optimal scarf joint angle (i.e. an angle leading to no –or low- stress concentration near the edges) close to 40° could be evidenced. Contrary to the results obtained with a 2D assumption, it is not possible to define an optimal angle than minimize the edge effects in the 3D case. This result shows that for a rectangular section of a scarf joint, a 3D model is necessary to describe in a correct manner the stress state in the adhesive. Indeed, the 3D finite element models permit to shown that stress concentrations occur in the corners of the specimen. The maximum transmitted load is always lower using 3D model than the ones determined using a 2D model. The results presented in this section show that the stresses in the adhesive have a highly nonhomogeneous distribution. Moreover, they are maximum at the interface between the adhesive and the substrates. It means that this geometry is more dedicated to study the adhesion between the adhesive and the substrate than to study the behavior of the adhesive itself. However, the present geometry necessitates 3D calculations to compute the stress in a correct manner. The aim of the following sections is to determine if a change of the geometry of the substrates reduce the stress concentration in the corner thus allowing the use of 2D calculations.
3.4. Influence of the geometry of the substrate I in the previous section, it has been evidenced that high stress concentrations are generated near the corners of the specimen. In order to decrease these stress concentrations, it is possible to use a specimen with a scarf joint and a round shape of the substrates (see Figure 5c). Figure 12 shows the maximum values over the thickness of the adhesive joint (direction v) of the Mises equivalent stress and hydrostatic pressure, in the mid-plane of the adhesive joint (plane u,z). This maximum is still located at the interface. Two angles of the scarf joint are illustrated: 0° (Figure 12a and b) and 50° (Figure 12c and d). Results are presented for a traction p0 of 1 MPa. The results are much closed to those obtained with a rectangular shape specimen. Indeed, some stress concentrations still occur near the adhesive free edges and especially in the rounded part of the - 13 -
specimen. Even with a smoothed geometry, the regions denoted RC and LC remain the critical zones for the strength of the adhesive.
(b) (a)
(c)
(d)
Figure 12: Influence of the scarf joint angle on the maximum Mises and hydrostatic stress through the thickness of the adhesive represented in the plane of the adhesive (u,z): scarf joint angle of 0° (a and b); scarf joint angle of 50° (c and d). The section of the specimen has rounded angle and the adhesive has straight edges. These results could also be observed in Fig. 13 that presents the evolution of the maximum traction p0 to reach the elastic limit in the adhesive with respect to the angle for a round shape substrate. These results must be compared to those presented with a rectangular section (previous section). Very similar results are obtained with both geometries i.e. the region near the corners are the most critical. The used of a rounded geometry does not lead to decrease the stress concentration near the corners which is necessary if one wants to use a simpler 2D approach to analyze the stress state in the specimen.
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p0 (MPa) angle (°) Figure 13: Influence of the scarf joint angle on the maximum transmitted load taking into the different parts of the joint for a rectangular substrates and straight edges. Edge refer to zone EB, Border refer to RB+LB and Corners refer to RC+LC
3.5. Influence of the geometry of the joint One other possible way to reduce the edge effects is to modify the local geometry of the adhesive joint. It has already been shown that a little modification of the geometry of the joint could change drastically the stress concentration near the edge [7]. For some specimens, using cleaned edges permits to reduce these stress concentrations. This is the reason why the geometry with rounded angle and cleaned edges (see Figure 5d) has been studied in the following. Figure 10 presents the Von Mises stress at the interface and in the mid-plane of the adhesive in the plane u,z, for two scarf joint angles ( = 0° and 50°). The use of cleaned edges allows a large reduction in the stresses at the interface as compared with the straight edges geometry. However, stresses are still much larger near the edges than in the center of the specimen.
(a)
(b) - 15 -
(c)
(d)
Figure 14: Influence of the scarf joint angle on the Mises stress at the interface and in the middle of the adhesive: scarf joint angle of 0° (a and b) and scarf joint angle of 50° (c and d). The section of the specimen has rounded angle and the adhesive has cleaned edges.
Figure 15 presents the evolution of the maximum traction p0 to reach the elastic limit in the adhesive with respect to the angle . These results must be compared to those obtained with a straight edge.
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p0 (MPa)
angle (°)
Figure 15: Influence of the scarf joint angle on the maximum transmitted load taking into the different parts of the joint for a rectangular substrates and cleaned edges. Edge refer to zone EB, Border refer to RB+LB and Corners refer to RC+LC.
The specimen with a scarf joint with rounded corners and cleaned edges reduces the stress concentration near the edges but also modifying the location of the maximum stress in the thickness of the adhesive for some scarf joint angles. In some cases, the stresses are maximal in the middle of the adhesive but stress concentrations are still observed near the border. In conclusion of this section dedicated to the classical specimen with a scarf joint, it has been shown that this specimen leads to stresses maximum at the interface between the adhesive and the substrate. It means that this test is well adapted to the characterization of the adhesion. The specimen induces high stress concentrations near the corner that necessitate the use of 3D calculations to compute the stresses in a correct manner. Modifying the specimen geometry (rounded substrates and/or cleaned edges) does not permit to reduce significantly the stress concentration near the corners keeping the maximum stresses at the interface.
4. Proposition of a modified specimen with a scarf joint to reduce the edge effects in the tests of the behavior of adhesives The results presented in the section 3 have shown that the classical specimen with a scarf joint is well adapted to test the adhesion between the adhesive and the substrates. It could be interesting to determine a geometry close to the classical one that allows testing preferentially the behavior of the adhesive. Indeed, in this case the same test (in terms of geometry, bonding system, experimental device) could allow testing both the interface and the adhesive. It is worth mentioning that geometry adapted to test the behavior of the adhesive means geometry without edge effects and with stresses maximum in the middle plane of the adhesive.
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4.1 Definition of the geometry It has been shown that a good way to reduce the stress singularities near the edges consists in machining a groove (also called beak) in the substrate as proposed by Cognard for the Arcan device [7]. The geometry of this groove could be optimized as a function of the nature of the substrate and of the adhesive. However, it is not possible to manufacture such a groove for a specimen with a scarf joint angle different of 0°. This is the reason why, a modified geometry of the substrate is proposed in this study in order to allow the machining of these beaks. The proposed geometry is shown in Figure 16. y y x
y x
Substrate
x
z
w
z
Middle branch L t
d
Adhesive
zoom
Substrate
Outer part
groove groove
l Face
Figure 16: geometry of the modified specimen with a scarf joint. For a scarf joint angle of 0°, the specimen has the shape of a cross. The middle branch is inclined for the other scarf joint angles. In order to decrease the cost of such specimens the cross like shape specimen could be machined using water jet cutting machine. The grooves are machined only on two faces of the specimen near the bonding surface (see zoom in Figure 16). This geometry has been chosen in order to be simple (non specific apparatus is needed as compare to Arcan tests device) with a stress field in the adhesive without edge effects and homogeneous in the thickness. The aim of the following section is to demonstrate that this geometry leads to this kind of stress field. The length of the bonded assembly has been chosen to avoid interaction between the end conditions (top and bottom ends of the substrates). The width l=50 mm and the thickness w=9.5 mm of the specimen (see Figure 16) have been chosen in order to have the same overlap length at 0° than the one classically used in the laboratory for an Arcan test. Therefore the overlap length, denoted by L, depends on the angle δ of the scarf joint. It is important to note that the dimensions of the specimen (especially the thickness t) have not been yet optimized. This optimization will be made later if the stress field has the expected form.
4.2. Analysis of the stress state in the adhesive Figure 17 show the hydrostatic pressure and the Von-Mises stress in the middle of the adhesive and at the interface along the bond surface for a scarf joint angle =0°. It can be seen, as expected, that there is no stress concentration thanks to the presence of the grooves and to the outer-part of the specimen (see Figure 16). Indeed, the stress state tends to 0 near the edges and is maximal in the - 18 -
center part of the adhesive. With the present specimen, a cohesive failure is expected since (i) there is no edge effects (that can lead to a premature failure of the specimen) and (ii) the stress is quasihomogenoeous in the thickness of the adhesive.
(a)
(b)
(c)
(d)
Figure 17: Von-Mises stress near the interface (a) and in the mid-plane of the adhesive (b). Hydrostatic pressure near the interface (c) and in the mid-plane of the adhesive (d). Case of a scarf joint angle =0°.
Figure 18 presents in the hydrostatic stress – von Mises stress diagram, the stress state in the adhesive for the loading that is necessary to reach the elastic limit in one point of the adhesive. The results are presented for two values of the scarf joint angle ( = 0 and 50°). These results must be compared with Figure 10 and Figure 12 obtained with the classical scarf joint geometry. The stress state obtained with this geometry is close to the one obtained with a modified Arcan test device but the experimental device is easier to achieve in the present case. With this modified geometry of the specimen with a scarf joint, the elastic limit is first reached in the center part of the adhesive. Some further works will be carried out to optimize this geometry and to perform some tests.
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(a)
(b)
Figure 18: Influence of the scarf joint angle δ on the distribution of the stress state in the hydrostatic pressure/von Mises stress plane for a scarf joint angle equal to 0° (a) and 50° (b) in the modified specimen geometry. Each point corresponds to the stress state at a Gauss-Point in the adhesive. The Red square are the stress in the center part of the adhesive, the blue circle correspond to the envelop of the stresses in the other part of the adhesive. Finally, Figure 19 shows hydrostatic pressure/von Mises stress for the loading that is necessary to reach the elastic limit in one point of the adhesive as a function of the scarf angle. This results show that varying the angle of the scarf joint permit to submit the adhesive to different triaxiality ratios which is necessary to identify an elastic limit in a complex material behavior [21]
- 20 -
50°
40°
30°
20°
10° 0°
Figure 19: Evolution of the stress state in the plane Hydrostatic pressure/von Mises stress for the loading necessary to reach the elastic limit in one point of the adhesive as a function of the angle of scarf joint (from 0° to 50°)
5. Conclusions The present paper deals with the 3D Finite Element modelling of scarf joints. The aim of this paper consists in determining the configuration (in term of shape of the substrate, scarf joint angle, local geometry of the adhesive) in order to reduce the edge effects and to initiate the failure at the interface (adhesive failure) or in the adhesive (cohesive failure). The main conclusions are the followings: For classical rectangular substrates and straight edges, stress concentrations are always generated in one of the corner of the specimen. Moreover, the triaxiality ratio is very different near the edges as compared with the center of the specimen. It means that a 3D model is necessary to study this kind of specimens and that premature failure could be generated near the edges or the corner with a ratio of triaxiallity very different form that obtained in the center of the specimen. Using substrates with rounded corners does not modify substantially the stress concentration leading to the same previous conclusions. Cleaning the adhesive increases the maximum transmitted load, however, premature failure could always arise near the corners of the specimen. For these three cases, failure occurs at the interface. It means that this test could be used to determine the strength of the interface but not that of the behavior (and the strength) of an adhesive in an assembly. It could also be used as a validation test for numerical models developed to predict the non-linear behavior up to the failure of adhesively bonded specimens. It is worth mentioning that the present paper has focused on the stress analysis. The results in term of maximal transmitted load correspond to the load that is necessary to reach the elastic limit in one point of the adhesive. Of course, this load is not equivalent to the failure load. Indeed, it has been shown that in presence of a stress singularity it is necessary to consider both stress and energy to predict the failure load [22]. This kind of approach, called coupled criterion, shows that the crack is initiated in a finite length [23] and that the mode mixity (i.e. the ratio between the stresses) must be taken into account - 21 -
[24]. This is the reason why, in a future work, the failure of specimens with a scarf joint will be studied using this coupled criterion. As previously explained, one objective of this paper was to determine the configuration of a specimen with a scarf joint that permits to locate the maximal stress in the mid-plane of the adhesive to attain a cohesive failure. In order to achieve this goal, a modification of the specimen with a scarf joint has been proposed. This modified geometry introduce an outer part that is not solicited and beaks the in the front and back faces. With this geometry, the stress is quasihomogeneous in the thickness of the adhesive and the elastic limit is first reach in the center part of the adhesive, which means there is not more edge effects. This scarf like specimen permits to subject the adhesive to multiaxial loading up to the failure. This failure is expected to occur in the adhesive (since there is no edge effects). This preliminary geometry is, in our point of view, a good compromise between the quality of the stress field in the adhesive (i.e. non edge effects and stress homogeneous in the thickness) and the experimental difficulty. Indeed, it does not necessitate any specific apparatus (as compared to the Arcan test device). The shape of the specimen is more difficult to manufacture as compare to other ones (classical Scarf joint specimens, single-lap joint specimens), but this difficulty remains reasonable. Moreover, each specimen could be used several times. First specimens have been manufacture using high-pressure water cut tool (to manufacture the global shape) and classical machining tools (to insure the planarity of the surface to bond and to manufacture the beaks). A bonding system has also been proposed. Some furthers works will be performed to optimize this geometry and perform some test with this kind of specimen.
Acknowledgements One of the authors (N. Carrere) would like to thank the financial support of the UIMM (Union des Industriels des Métiers de la Métallurgie) and ―La Région Bretagne‖.
References [1] Adams R.D., Adhesive bonding: Science, technology and applications, (Woodhead Publishing Ltd, Bristol, 2005). [2] da Silva L.F.M., Öchsner A., Modeling of Adhesive Bonded Joints, (Springer, Heidelberg, 2008). [3] ASTM D1002-10, Standard Test Method for Apparent Shear Strength of Single-Lap-Joint Adhesively Bonded Metal Specimens by Tension Loading (Metal-to-Metal), ASTM International, West Conshohocken, PA, 2010.
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[4] Cognard J.Y, Créac’hcadec R, Maurice. Numerical analysis of the stress distribution in single-lap shear tests under elastic assumption—Application to the optimisation of the mechanical behaviour. International Journal of Adhesion and Adhesives 2008;31:715-724 [5] Dragoni E., Goglio L., Kleiner F. Designing bonded joints by means of the JointCalc software. International Journal of Adhesion and Adhesives 2010;30:267–280 [6] Arcan L, Arcan M, Daniel I. SEM fractography of pure and mixed mode interlaminar fracture in graphite/epoxy composites. ASTM Tech 1987;948:41–67. [7] Cognard, J.Y. Numerical analysis of edge effects in adhesively-bonded assemblies application to the determination of the adhesive behavior. Computer and Structues 2008;86:1704–1717 [8] Spagiari A., Castagnetti D., Dragoni E. The Journal of Adhesion 2012;88:499-512 [9] N. Arnaud, R. Créac'hcadec, J.Y. Cognard,A tension/compression–torsion test suited to analyze the mechanical behaviour of adhesives under non-proportional loadings. International Journal of Adhesion and Adhesives 2014; 53:3-14 [10] Dan H, Sawan T, Iwamoto T , Hirayama Y. Stress analysis and strength evaluation of scarf adhesive joints subjected to static tensile loadings. International Journal of Adhesion and Adhesives 2010 ; 30 :387–392. [11] Abdel Wahab MM , Hilmy I , Ashcroft IA, Crocombe AD. Damage Parameters of Adhesive Joints with General Triaxiality Part I: Finite Element Analysis. Journal of Adhesion and Adhesives 2011 ; 25 :903-934 [12] Nakano H, Omiya Y, Sekiguchi Y, Sawa T. Three-dimensional FEM stress analysis and strength prediction of scarf adhesive joints with similar adherends subjected to static tensile loadings. International Journal of Adhesion and Adhesives 2014; 54:40–50 [13] Cognard JY, Leguillon D, Carrere N. Analysis of the influence of geometric parameters on the stress distributions in adhesively-bonded scarf joints using 2D models under elastic assumption. The Journal of Adhesion 2014; 90(11):877-898 [14] Cognard J.Y., Créac’hcadec R., Maurice J., Davies P., Peleau M., da Silva L.F.M.. Analysis of the influ- ence of hydrostatic stress on the behaviour of an adhesive in a bonded assembly. Journal of Adhesion and Adhesives 2010; 24:1977-1994 [15] Maurice J, Cognard JY, Créac’hcadec R, Davies P, Sohier L, Mahdi S. Characterization and modelling of the 3D elastic–plastic behaviour of an adhesively bonded joint under monotonic tension/compression-shear loads: influence of three cure cycles. Journal of Adhesion Science and Technology 2013 ; 27(2): 165-181 [16] Cognard J.Y., Sohier L., Créac’hcadec R., Lavelle F., Lidon N.. Influence of the geometry of coaxial adhesive joints on the transmitted load under tensile and compression loads. International Journal of Adhesion and Adhesives 2012 ; 37 :37–49 - 23 -
[17] Leguillon D., Sanchez- Palancia E., Computation of singular solutions in elliptic problems and elasticity, (Editions Masson, Paris, 1987). [18] Leguillon D. Computation of 3D-singularities in elasticity, in Boundary Value Problems and Integral Equations on Non-Smooth Domains, M. Costabel, M. Dauge, S. Nicaise eds., Lect. Notes in Pure and Applied Math., 167, Marcel Dekker, New York, 1995, 161-170. [19] Leguillon D. An attempt to extend the 2D coupled criterion for crack nucleation in brittle materials to the 3D case. Theor. Appl. Fract. Mech. 2013;74:7-17 [20] Abaqus, User Manual, 2014 [21] C. Badulescu, C. Germain, J.-Y. Cognard, N. Carrere, Characterisation and modelling of the viscous behaviour of adhesives using the modified arcan device. Journal of Adhesion Science and Technology 2015;29(5): 443-46 [22] Leguillon D. Strength or toughness ? A criterion for crack onset at a notch. European Journal of Mechanics – A/Solids 2002; 21:61-72. [23] Moradi A, Carrere N, Leguillon D, Martin E, Cognard JY. Strength prediction of bonded assemblies using a coupled criterion under elastic assumptions: Effect of material and geometrical parameters. International Journal of Adhesion and Adhesives 2013;47:73-82. [24] Carrere N, Martin E, Leguillon D. Comparison between models based on a coupled criterion for the prediction of the failure of adhesively bonded joints. Subjected for publication in Engineering Fracture Mechanics, 2015.
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PII: DOI: Reference:
S0143-7496(15)00106-2 http://dx.doi.org/10.1016/j.ijadhadh.2015.07.005 JAAD1680
To appear in: International Journal of Adhesion and Adhesives Received date: 27 February 2015 Accepted date: 12 June 2015 Cite this article as: Nicolas Carrere, Claudiu Badulescu, Jean-Yves Cognard and Dominique Leguillon, 3D models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2015.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
3D models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives Nicolas CARRERE1*, LEGUILLON2
Claudiu
BADULESCU1,
Jean-Yves
COGNARD1,
Dominique
1
Laboratoire Brestois de Mécanique et des Systèmes, ENSTA Bretagne, Brest, France. Institut Jean Le Rond d'Alembert, CNRS UMR 7190, Sorbonne Universités, UPMC Université Paris 06, F-75005 Paris, France. 2
* Corresponding author: [email protected]
Summary In this paper, accurate numerical analyses of the stress distributions within the adhesive in scarf joints under elastic assumption using 3D models are developed. An elastic limit model that takes into account the hydrostatic stress and von Mises equivalent stress, permit to define the more stressed parts of the adhesive with respect to the scarf angle. It is shown that high stress concentrations are generated near the edges and the corners of the specimen at the interface between the substrate and the adhesive. The use of ronded substrates does not lead to decrease significantly these stress concentrations. It is shown that cleaning the free edge of the adhesive changes the location of edge effects without completely erasing them. A modification of the geometry, has been proposed. This modification leads to a quasi-homogeneous distribution of the stresses in the thickness of the adhesive with stresses that tend to 0 near the edges.
Keywords: Scarf joint, Bonded Assembly, Stress analysis, Finite element analysis
1.
Introduction
Thanks to their capabilities to assemble a large range of materials, adhesively-bonded joints appear to be an interesting technique to reduce the weight and the cost of structures in various industrial domains [1][2]. However, due to the lack of confidence in the non-destructive evaluation of quality, adhesives are drastically under-used. Another key point lies on the necessity to propose tests to determine the reliability of assembly options that can be considered for the design of adhesivelybonded structures. Due to the significant edge effects (associated with geometrical and material mismatch between the adhesive and the substrates) and the complex failure mechanics that could be generated, the experimental characterization of the feasible assemblies is not an easy task. Some tests have be normalized in order to facilitate this experimental characterization such as the singlelap joint test [3]. However, due to the stress concentrations near the ends of the overlap, the determination of a reliable strength is not trivial (even if some studies have proposed some solutions to reduce this edge effects [4]). Moreover, this test is devoted to the determination of the strength for a given ratio peel/shear stress assumed to be a quasi-pure shear although high peel stresses are generated at the ends of the overlap (some studies have proposed some modification to submit the adhesive to variable combinations of peel and shear stresses [5]). To overcome these difficulties, some authors have proposed specific test devices such as the modified Arcan test [6][7] device or the use of cylindrical specimens subjected to tensile-torsion loads [8][9]. These experimental devices permit to reduce the edge effects and submit the adhesive to variable multi-axial loadings. However, such devices seem difficult to use in an industrial environment. The use of specimens *
Corresponding author [email protected] - Tel : +33 2 98 34 88 67 - Fax : +33 2 98 34 87 30 Address: ENSTA Bretagne, LBMS, 2 rue François Verny, 29806 Brest Cedex 9, France
-1-
with a scarf joint represents an interesting alternative. Indeed, this kind of specimen has been widely used in the literature. Indeed, it does not necessitate a specific experimental apparatus and could be used with a classical tensile machine. By adjusting the bond-line angle of the adhesive, different tensile-shear loadings can be obtained [10][11]. However, high stress concentrations are generated near the edges of the specimen [[11][12][13]] and the stresses are maximum at the interface [13][11]. It has also been recently shown that, using 2D [13] or 3D [12] Finite Element calculation, that an optimal angle leading to no edge effects could be determined. The aim of the present study is to know, using a refined 3D FE calculation (as compared to what could be found in the literature), if it is possible to find a geometrical configuration (in terms of the shape of the substrates, local geometry of the adhesive) (i) that modifies the location of the maximal stress (at the interface to determine the adhesion strength or in the middle plane of the adhesive to determine the strength of the adhesive) and (ii) that decreases the edge effects to facilitate the analysis of the experimental results. The first section is devoted to the presentation of the material and of the model. The behavior and the failure of adhesives being influenced by the hydrostatic pressure, the maximum load transmitted by the scarf joints will be determined using a pressure-dependent elastic limit of the adhesive defined from the two stress invariants, hydrostatic stress and von Mises equivalent stress. In the second section, the properties of the singularity exponent, obtained from an asymptotic analysis are presented. This approach permits understanding the effect of the bond line angle on the edge effects. The singularity exponent allows determining if the geometry could lead to singular stress fields (i.e. edge effects) or regular stresses (i.e. without edge effects). However, the knowledge of this exponent is not sufficient. This is the reason why 3D finite element calculations are performed in the third section. Firstly, the influence of the mesh size especially in the thickness direction is proposed. Different geometrical configurations are studied to know if it is possible to choose the location of the maximal stress and to reduce the edge effects. Finally, in order to obtain a configuration without edge effects and with maximal stresses located in the middle plane, a modified geometry is proposed in the last section.
2. Material and model 2.1. Definition of the geometry of the bonded specimens Figure 1 presents the main geometrical parameters of the scarf joint which are used herein. The width and the thickness of the substrate, denoted respectively by d and h, are constant, therefore the overlap length, denoted by l, depends only on the angle of the scarf joint. The adhesive thickness, denoted by e, is defined in the normal direction of the middle plane of the adhesive. The displacements of the lower part of the specimen are blocked and a tension is applied on the upper part. Moreover, the length of the bonded assembly has been chosen to avoid interaction between the end conditions (top and bottom ends of the adherends).
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Figure 1: 3D Model used for the scarf joint (the drawing is done with negative in order to facilitate the presentation).
2.2. Numerical model of the adhesive under tensile/compression-shear 3D computations were performed under small displacement and elastic assumptions. Results are presented for aluminum substrates (Young’s modulus: Es = 75 GPa, Poisson's ratio: s = 0.3) and the elastic material parameters for the adhesive are: Ea = 2.0 GPa, a = 0.4. For this study, only the elastic behavior of the adhesive is considered, so it is necessary to know its yield limit in order to remain well within the elastic range. A precise determination of the maximum transmitted load (i.e. the applied load leading to the onset of the non-linearity) of adhesively-bonded joints requires taking into account an accurate limit criterion in order to represent the real behavior of the adhesive in an assembly. Various studies underline that an accurate representation of the elastic yield surface of an adhesive requires the use of a pressure-dependent constitutive model, i.e. a model taking into account the two stress invariants, hydrostatic stress and von Mises equivalent stress. Starting from the experimental results obtained for the epoxy resin Huntsman Araldite® 420 A/B, with the modified Arcan device (adhesive thickness of 0.4 mm and displacement rate of the tensile machine crosshead of 0.5 mm/min) it is possible to define the initial yield function for the adhesive (neglecting viscous effects). This test allows defining the behavior of an adhesive in an assembly under radial loadings (tensile/compression-shear loadings) [14][15]. An exponential Drucker-Prager yield function leads to a good representation of the experimental data for the socalled ―initial‖ elastic limit [16]: F0=a(vm)b+ph-p0=0
(1)
Where vm is the von Mises equivalent stress and ph is the hydrostatic stress; a, b and p0 are material parameters. The results of the identification are presented in Table 1. Figure 2 presents the elastic limit. The determination of the maximum transmitted load under elastic assumption and proportional loads is detailed in [16].
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a (MPa-4) b p0 (MPa) 6. E-7 5. 31.7 Table 1: Material parameters for the initial yield surface.
60
von Mises stress (MPa)
50 40
30 20 10 0 -30
-20
-10
0
10
20
30
hydrostatic stress (MPa) Figure 2: Elastic limit of the adhesive in the von Mises equivalent stress - hydrostatic stress diagram.
3. Asymptotic analysis of the scarf joint The location where the adhesive interfaces meet the free surface is a site of stress concentrations that can be described using the singularity theory [17] within the framework of linear elasticity. This theory involves the so-called ―Williams’expansion‖ in the vicinity of the points under consideration It provides a description of the local displacement and stress fields written as a sum components that are function of the distance to the edge and of the singularity exponents (see eq. 7 and 8). The asymptotic analysis is a powerful tool to determine if a geometrical configuration could lead to edge effect or not. Indeed, if the singularity exponents are lower than 1, the solution is singular (and thus edge effects could be generated). In the present paper, this approach is used to study if, at the edges or at the corners of the specimen, edge effects could take place as a function of the angle of the bondline. For the scarf joint, modeled in 3D, there are different corners which can be reduced by two, using the symmetry with respect to the axis Oz (Figure 1). Half of the scarf joint is presented in order to present the different properties of the different corners.
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(a)
(b)
(c)
(d)
Figure 3: Different configurations of corners for a given scarf joint angle (the drawing is done with δ negative in order to facilitate the presentation); (a) one half of the scarf joint (Fig. 1); (b) case 1: 2D model of the scarf joint (plane x-y); (c) case 2: geometry in the plane y-z; (d) case 3: edges of the parallelepiped specimen.
For case 1, the two points A and B represent the two corners which have to be taken into account using 2D modeling of the scarf joint (plane x-y, Figure 3b). In 2D, close to a corner, two angles s and a can be used to define the local geometry of the substrate and of the adhesive. Two different types of local geometry exist for a given scarf joint angle ; they are characterized by sx + ax = (with X=A or X=B) ; thus two configurations have to be analyzed:
aA = ; sA =
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aB = ; sB =
For such a model, it is sufficient to consider 0 < aX < with sX + aX = (with X=A or X=B) For case 2, in the plane (y-z) two other type of corners can be analyzed using 2D models. The two points (C and C’), presented in Figure 3c, are characterized with the same properties:
aC = ; sC =
For case 3, the four wedges of the parallelepiped specimen are the last type of corners. Using the symmetry with respect to plane (O, x-y), only two corners have to be taken into account (points D and E, Figure 3d). These two points, which require a 3D analysis, are characterized with the following parameters:
aD = ; sD = ; = aE = ; sE = ; =
In the case of 2D problems under elasticity assumption close to a corner, the relevant parts of the displacement field U and the stress tensor , using the polar coordinates (r: radius, : angle) are given by [17] U(r, ) = C + k1 r
(r, ) = k1 r
u1() + k2 r
s1() + k2 r
u2() + k3 r
s2() + k3 r
u3() + …
(7)
s3() + …
(8)
The constant C is irrelevant and is present for consistency. The terms i are positive, they are socalled ―singularity‖ exponents (singular at least if i < 1); taking into account the definition of the stress (Eqn. (8)). They are the solution of an eigenvalue problem with the ui as eigenvectors. They depend only on the local properties, i.e. the elastic coefficients of the substrate and the adhesive and the angle a. In some simple cases they are analytically known and otherwise can be numerically determined. It can be seen that the closer the is to 0, the higher the singularity; thus a i close to 1 can be considered as a weak singular exponent. The coefficients ki are the generalized stress intensity factors, they are mainly functions of the global geometry, the loading mode and its intensity. They can be computed using path independent integrals in the orthogonal plane [17]. Here we are interested mainly in the values of the i: they allow the classification of the stress concentrations in terms of harmfulness (the smaller the exponent, the more harmful the stress concentration). In the case of 3D corners, the displacement field V can be written in the following form [17]
V(r, ) = C + K1 r
v1() + K2 r v2() + …
(9)
where r, are the spherical coordinates with origin at the corner point. The i are still solution of an eigenvalue problem with the vi as eigenvectors and the only way of determination is now by a numerical procedure [18]. The generalized stress intensity factors Ki are constants, their computation can be carried out using surface independent integrals or by identification [19]. -6-
Figure 4 shows the most singular exponent for the three different cases presented before, function of the local adhesive angleaX. At the top of Figure 4, the corresponding values of the scarf joint angle are plotted with respect to the adhesive angle a. 60
50
40
30
20
10
0
10
20
30
40
50
60
(°) scarf angle
1
1.25
0
Case 1
Case 3
Point B
Point A
Point D
0.75
singular solution
Exponent
1.00
Case 2 0.50
Point E Points C and C’
0.25
0.00 30
40
50
60
70
80
90
100
110
120
130
140
150
Angle of the adhesive a (°) Figure 4: Evolution of the most singular exponent with respect to the angle of the adhesive a, for the three cases illustrated in Fig. 3.
It can be seen that the singularity exponents are slightly smaller (i.e. more harmful) for case 3 than for case 2. Moreover, for case 2, the most singular exponent is lower than for case 1 in the case of a < 90°. Therefore a 3D analysis of the scarf joint is necessary because the most singular exponent is always obtained either in case 2 (2D) or in case 3 (3D) and not in case 1 (2D). A more precise analysis requires the computation of the generalized stress intensity factors ki or Ki in order to have a precise idea of the stress state. But, it is important to notice that the presence of singular exponents can lead to large stress concentrations, further aggravated taking into account geometrical or loading defects. In the following, accurate 3D finite element computations are carried out to analyze the stress state within the adhesive.
3. Finite element analysis of the specimen with a scarf joint All the Finite Element analyses are performed using Abaqus [20] under 3D assumptions.
3.1. Geometries of the adhesive and of the substrates and boundary conditions In order to study the stress state in the adhesive, it is necessary to perform 3D Finite Element calculations. Different geometries of the substrates and of the joint could be investigated. Two of them are considered: one with a rectangular section (noted geometry A) and one with round corners (noted geometry B). The adhesive could have different shapes: straight edge (noted with a lowercase letter a), cleaned edge (noted with a lowercase letter b). The different geometry are shown in Figure 5.
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(a) geometry Aa
(b) geometry Ab
(c) geometry Ba
(d) geometry Bb
Figure 5: Scarf joint with a rectangular section of the specimens (geometry A (a) and (b)) and a round shape of the substrates (geometry B (c) and (d)). (u,v) is the local coordinate system and (x,y) the global one (see Fig. 3). Only one quarter of the adhesive is shown (see Figure 6): (O,x,y) is a symmetry plane and (O,u,z) an anti-symmetry one. Some regions of interest are evidenced: C is the center part of the adhesive (no confusion with the corner C of the previous section), RB and LB respectively the right and the left borders, EB the edge border RC and LC respectively the right and left corners. The boundary conditions are shown in Figure 6. On the symmetry plane (O,x,y) displacement Uz is blocked. An antisymmetric condition is applied on the (O,u,z) plane. A tension is applied on the top of the specimen.
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Figure 6: Bonndary conditions applied on the quarter of the specimen with a scarf joint
3.2. Influence of the mesh size It has already been shown that the mesh size could have a great influence on the determination of the stress state in presence of stress concentrations [3]. This is the reason why, the influence of the mesh size has been studied, especially in the thickness of the adhesive. It has been studied in the case that leads to the higher stress concentrations: a rectangular section of the substrates and straight edges with a scarf joint angle of 0° (butt joint). In this case it is possible to model only one eighth of the specimen. The mesh in the plane of the adhesive near the right corner is shown in Figure 7. Near the edges linear brick elements (regions RC, LC, EB, LC, RB) are used while linear wedge elements are used in the center part. In the corners (RC, LC), the size of the elements is equal to 10m×10m (closed to the one proposed in [12]) while it is equal to 10m×100m near the edges (EB, LC, RB).
Figure 7: Mesh of the adhesive joint near the right corner.
Figure 8 shows the evolution of the maximum traction p0 that could be transmitted before to reach the elastic limit in the adhesive. The maximum traction is calculated in each part of the adhesive -9-
(i.e. the center and the edges) as a function of the number of elements in the half-thickness of the adhesive. In order to analyze this influence in the different parts of the joint, the results are presented for each zone of interest. These results show: (i) that obviously the maximum load transmitted in the middle part of the adhesive is independent of the number of elements and (ii) that it decrease in the edges as a function of the number of elements. It appears that 15 elements in the half-thickness is a good compromise between the computational cost and the accuracy of the result.
Figure 8: Influence of the number of elements in the half thickness of the adhesive on the maximum transmitted load taking into account the different parts of the joint for a rectangular substrate and straight edges with a scarf joint angle of 0° (butt joint). C et Edges represent respectively the results in the center of the adhesive and in the zone EB+LC+RB. Finally, the mesh used for the modelling presented in the next sections considers only a quarter of the scarf joint and is composed by 400000 elements in the adhesive and 150000 in the substrate.
3.3. Influence of the scarf joint angle for straight edges and rectangular substrates Figure 9 shows the maximum values, over the thickness of the adhesive joint (direction v), of the Mises equivalent stress and hydrostatic pressure, in the plane of the adhesive joint (plane u,z). The results are presented for three scarf joint angles: 0° (Figure 9a and b), 30° (Figure 9c and d) and 50° (Figure 9e and f). Results are presented for a traction p0 of 1 MPa. The maximum value is located near the interface between the adhesive and the substrate. For the different scarf joint angles, the stress state is homogeneous in a large part on both sides of the center of the joint. It can be seen that stress concentrations occur near the adhesive free edges and especially at the corners. These stress concentrations, computed with a given mesh definition, depend on the value of the scarf joint angle . Moreover, the location of the stress concentrations is also a function of the scarf joint angle (right edges for left edges for
- 10 -
(a)
(b)
(c)
(d)
(e)
(f)
Figure 9: Influence of the scarf joint angle on the maximum Mises and hydrostatic stress through the thickness of the adhesive represented in the plane of the adhesive (u,z): scarf joint angle of 0° (a and b), scarf angle of 30° (c and d), scarf joint angle of 50° (e and f). The section of the specimen is rectangular and the adhesive has straight edges. The envelopes of the stresses (in the hydrostatic pressure-von Mises stress plane) in the adhesive for the maximum load transmitted by the joint under elastic assumption is represented in Figure 10 a and b (for scarf joint angles equal to 0° and 40°). The triaxiality ratio (Tx i.e. the ratio between the hydrostatic pressure and the von Mises stress) in the center of the specimen is very different for the two angles. The results presented in Figure 10 also allow determining the points of the adhesive, which first reach the elastic limit. It can be seen that the elastic limit is first reached in a zone that is not the center of the specimen (it is in the corner) for a triaxiality ratio very different of the one obtained in the center of the specimen.
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(a)
(b)
Figure 10: Influence of the scarf joint angle on the distribution of the stress state in the plane hydrostatic pressure/von Mises stress. Results for a scarf joint angle equal to 0° (a) and 40° (b). Each point corresponds to the stress state at a Gauss-Point in the adhesive. The Red square are the stress in the center part of the adhesive, the blue circle correspond to the envelop of the stresses in the other part of the adhesive. Figure 11 presents the evolution of the maximum traction p0 that could be transmitted before to reach the elastic limit in the adhesive with respect to the angle . In order to analyze the influence of the different parts of the joint, the results are presented for each zone of interest (the center correspond to CE, the edge to EB, the corners to RC+LC and the borders to RB+LB). The maximum transmitted load of the specimen with a scarf joint corresponds, for each scarf joint angle, to the minimum over the whole specimen (i.e. the solid line). The result in 2D corresponds to the dotted line (2D is only considered for the three zones of interest LB, C and RB).
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p0 (MPa)
angle (°) Figure 11: Influence of the scarf joint angle on the maximum transmitted load into the different parts of the joint for a rectangular substrate and straight edges. Edge refer to zone EB, Border refer to RB+LB and Corners refer to RC+LC.
Under, 2D assumptions, an optimal scarf joint angle (i.e. an angle leading to no –or low- stress concentration near the edges) close to 40° could be evidenced. Contrary to the results obtained with a 2D assumption, it is not possible to define an optimal angle than minimize the edge effects in the 3D case. This result shows that for a rectangular section of a scarf joint, a 3D model is necessary to describe in a correct manner the stress state in the adhesive. Indeed, the 3D finite element models permit to shown that stress concentrations occur in the corners of the specimen. The maximum transmitted load is always lower using 3D model than the ones determined using a 2D model. The results presented in this section show that the stresses in the adhesive have a highly nonhomogeneous distribution. Moreover, they are maximum at the interface between the adhesive and the substrates. It means that this geometry is more dedicated to study the adhesion between the adhesive and the substrate than to study the behavior of the adhesive itself. However, the present geometry necessitates 3D calculations to compute the stress in a correct manner. The aim of the following sections is to determine if a change of the geometry of the substrates reduce the stress concentration in the corner thus allowing the use of 2D calculations.
3.4. Influence of the geometry of the substrate I in the previous section, it has been evidenced that high stress concentrations are generated near the corners of the specimen. In order to decrease these stress concentrations, it is possible to use a specimen with a scarf joint and a round shape of the substrates (see Figure 5c). Figure 12 shows the maximum values over the thickness of the adhesive joint (direction v) of the Mises equivalent stress and hydrostatic pressure, in the mid-plane of the adhesive joint (plane u,z). This maximum is still located at the interface. Two angles of the scarf joint are illustrated: 0° (Figure 12a and b) and 50° (Figure 12c and d). Results are presented for a traction p0 of 1 MPa. The results are much closed to those obtained with a rectangular shape specimen. Indeed, some stress concentrations still occur near the adhesive free edges and especially in the rounded part of the - 13 -
specimen. Even with a smoothed geometry, the regions denoted RC and LC remain the critical zones for the strength of the adhesive.
(b) (a)
(c)
(d)
Figure 12: Influence of the scarf joint angle on the maximum Mises and hydrostatic stress through the thickness of the adhesive represented in the plane of the adhesive (u,z): scarf joint angle of 0° (a and b); scarf joint angle of 50° (c and d). The section of the specimen has rounded angle and the adhesive has straight edges. These results could also be observed in Fig. 13 that presents the evolution of the maximum traction p0 to reach the elastic limit in the adhesive with respect to the angle for a round shape substrate. These results must be compared to those presented with a rectangular section (previous section). Very similar results are obtained with both geometries i.e. the region near the corners are the most critical. The used of a rounded geometry does not lead to decrease the stress concentration near the corners which is necessary if one wants to use a simpler 2D approach to analyze the stress state in the specimen.
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p0 (MPa) angle (°) Figure 13: Influence of the scarf joint angle on the maximum transmitted load taking into the different parts of the joint for a rectangular substrates and straight edges. Edge refer to zone EB, Border refer to RB+LB and Corners refer to RC+LC
3.5. Influence of the geometry of the joint One other possible way to reduce the edge effects is to modify the local geometry of the adhesive joint. It has already been shown that a little modification of the geometry of the joint could change drastically the stress concentration near the edge [7]. For some specimens, using cleaned edges permits to reduce these stress concentrations. This is the reason why the geometry with rounded angle and cleaned edges (see Figure 5d) has been studied in the following. Figure 10 presents the Von Mises stress at the interface and in the mid-plane of the adhesive in the plane u,z, for two scarf joint angles ( = 0° and 50°). The use of cleaned edges allows a large reduction in the stresses at the interface as compared with the straight edges geometry. However, stresses are still much larger near the edges than in the center of the specimen.
(a)
(b) - 15 -
(c)
(d)
Figure 14: Influence of the scarf joint angle on the Mises stress at the interface and in the middle of the adhesive: scarf joint angle of 0° (a and b) and scarf joint angle of 50° (c and d). The section of the specimen has rounded angle and the adhesive has cleaned edges.
Figure 15 presents the evolution of the maximum traction p0 to reach the elastic limit in the adhesive with respect to the angle . These results must be compared to those obtained with a straight edge.
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p0 (MPa)
angle (°)
Figure 15: Influence of the scarf joint angle on the maximum transmitted load taking into the different parts of the joint for a rectangular substrates and cleaned edges. Edge refer to zone EB, Border refer to RB+LB and Corners refer to RC+LC.
The specimen with a scarf joint with rounded corners and cleaned edges reduces the stress concentration near the edges but also modifying the location of the maximum stress in the thickness of the adhesive for some scarf joint angles. In some cases, the stresses are maximal in the middle of the adhesive but stress concentrations are still observed near the border. In conclusion of this section dedicated to the classical specimen with a scarf joint, it has been shown that this specimen leads to stresses maximum at the interface between the adhesive and the substrate. It means that this test is well adapted to the characterization of the adhesion. The specimen induces high stress concentrations near the corner that necessitate the use of 3D calculations to compute the stresses in a correct manner. Modifying the specimen geometry (rounded substrates and/or cleaned edges) does not permit to reduce significantly the stress concentration near the corners keeping the maximum stresses at the interface.
4. Proposition of a modified specimen with a scarf joint to reduce the edge effects in the tests of the behavior of adhesives The results presented in the section 3 have shown that the classical specimen with a scarf joint is well adapted to test the adhesion between the adhesive and the substrates. It could be interesting to determine a geometry close to the classical one that allows testing preferentially the behavior of the adhesive. Indeed, in this case the same test (in terms of geometry, bonding system, experimental device) could allow testing both the interface and the adhesive. It is worth mentioning that geometry adapted to test the behavior of the adhesive means geometry without edge effects and with stresses maximum in the middle plane of the adhesive.
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4.1 Definition of the geometry It has been shown that a good way to reduce the stress singularities near the edges consists in machining a groove (also called beak) in the substrate as proposed by Cognard for the Arcan device [7]. The geometry of this groove could be optimized as a function of the nature of the substrate and of the adhesive. However, it is not possible to manufacture such a groove for a specimen with a scarf joint angle different of 0°. This is the reason why, a modified geometry of the substrate is proposed in this study in order to allow the machining of these beaks. The proposed geometry is shown in Figure 16. y y x
y x
Substrate
x
z
w
z
Middle branch L t
d
Adhesive
zoom
Substrate
Outer part
groove groove
l Face
Figure 16: geometry of the modified specimen with a scarf joint. For a scarf joint angle of 0°, the specimen has the shape of a cross. The middle branch is inclined for the other scarf joint angles. In order to decrease the cost of such specimens the cross like shape specimen could be machined using water jet cutting machine. The grooves are machined only on two faces of the specimen near the bonding surface (see zoom in Figure 16). This geometry has been chosen in order to be simple (non specific apparatus is needed as compare to Arcan tests device) with a stress field in the adhesive without edge effects and homogeneous in the thickness. The aim of the following section is to demonstrate that this geometry leads to this kind of stress field. The length of the bonded assembly has been chosen to avoid interaction between the end conditions (top and bottom ends of the substrates). The width l=50 mm and the thickness w=9.5 mm of the specimen (see Figure 16) have been chosen in order to have the same overlap length at 0° than the one classically used in the laboratory for an Arcan test. Therefore the overlap length, denoted by L, depends on the angle δ of the scarf joint. It is important to note that the dimensions of the specimen (especially the thickness t) have not been yet optimized. This optimization will be made later if the stress field has the expected form.
4.2. Analysis of the stress state in the adhesive Figure 17 show the hydrostatic pressure and the Von-Mises stress in the middle of the adhesive and at the interface along the bond surface for a scarf joint angle =0°. It can be seen, as expected, that there is no stress concentration thanks to the presence of the grooves and to the outer-part of the specimen (see Figure 16). Indeed, the stress state tends to 0 near the edges and is maximal in the - 18 -
center part of the adhesive. With the present specimen, a cohesive failure is expected since (i) there is no edge effects (that can lead to a premature failure of the specimen) and (ii) the stress is quasihomogenoeous in the thickness of the adhesive.
(a)
(b)
(c)
(d)
Figure 17: Von-Mises stress near the interface (a) and in the mid-plane of the adhesive (b). Hydrostatic pressure near the interface (c) and in the mid-plane of the adhesive (d). Case of a scarf joint angle =0°.
Figure 18 presents in the hydrostatic stress – von Mises stress diagram, the stress state in the adhesive for the loading that is necessary to reach the elastic limit in one point of the adhesive. The results are presented for two values of the scarf joint angle ( = 0 and 50°). These results must be compared with Figure 10 and Figure 12 obtained with the classical scarf joint geometry. The stress state obtained with this geometry is close to the one obtained with a modified Arcan test device but the experimental device is easier to achieve in the present case. With this modified geometry of the specimen with a scarf joint, the elastic limit is first reached in the center part of the adhesive. Some further works will be carried out to optimize this geometry and to perform some tests.
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(a)
(b)
Figure 18: Influence of the scarf joint angle δ on the distribution of the stress state in the hydrostatic pressure/von Mises stress plane for a scarf joint angle equal to 0° (a) and 50° (b) in the modified specimen geometry. Each point corresponds to the stress state at a Gauss-Point in the adhesive. The Red square are the stress in the center part of the adhesive, the blue circle correspond to the envelop of the stresses in the other part of the adhesive. Finally, Figure 19 shows hydrostatic pressure/von Mises stress for the loading that is necessary to reach the elastic limit in one point of the adhesive as a function of the scarf angle. This results show that varying the angle of the scarf joint permit to submit the adhesive to different triaxiality ratios which is necessary to identify an elastic limit in a complex material behavior [21]
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50°
40°
30°
20°
10° 0°
Figure 19: Evolution of the stress state in the plane Hydrostatic pressure/von Mises stress for the loading necessary to reach the elastic limit in one point of the adhesive as a function of the angle of scarf joint (from 0° to 50°)
5. Conclusions The present paper deals with the 3D Finite Element modelling of scarf joints. The aim of this paper consists in determining the configuration (in term of shape of the substrate, scarf joint angle, local geometry of the adhesive) in order to reduce the edge effects and to initiate the failure at the interface (adhesive failure) or in the adhesive (cohesive failure). The main conclusions are the followings: For classical rectangular substrates and straight edges, stress concentrations are always generated in one of the corner of the specimen. Moreover, the triaxiality ratio is very different near the edges as compared with the center of the specimen. It means that a 3D model is necessary to study this kind of specimens and that premature failure could be generated near the edges or the corner with a ratio of triaxiallity very different form that obtained in the center of the specimen. Using substrates with rounded corners does not modify substantially the stress concentration leading to the same previous conclusions. Cleaning the adhesive increases the maximum transmitted load, however, premature failure could always arise near the corners of the specimen. For these three cases, failure occurs at the interface. It means that this test could be used to determine the strength of the interface but not that of the behavior (and the strength) of an adhesive in an assembly. It could also be used as a validation test for numerical models developed to predict the non-linear behavior up to the failure of adhesively bonded specimens. It is worth mentioning that the present paper has focused on the stress analysis. The results in term of maximal transmitted load correspond to the load that is necessary to reach the elastic limit in one point of the adhesive. Of course, this load is not equivalent to the failure load. Indeed, it has been shown that in presence of a stress singularity it is necessary to consider both stress and energy to predict the failure load [22]. This kind of approach, called coupled criterion, shows that the crack is initiated in a finite length [23] and that the mode mixity (i.e. the ratio between the stresses) must be taken into account - 21 -
[24]. This is the reason why, in a future work, the failure of specimens with a scarf joint will be studied using this coupled criterion. As previously explained, one objective of this paper was to determine the configuration of a specimen with a scarf joint that permits to locate the maximal stress in the mid-plane of the adhesive to attain a cohesive failure. In order to achieve this goal, a modification of the specimen with a scarf joint has been proposed. This modified geometry introduce an outer part that is not solicited and beaks the in the front and back faces. With this geometry, the stress is quasihomogeneous in the thickness of the adhesive and the elastic limit is first reach in the center part of the adhesive, which means there is not more edge effects. This scarf like specimen permits to subject the adhesive to multiaxial loading up to the failure. This failure is expected to occur in the adhesive (since there is no edge effects). This preliminary geometry is, in our point of view, a good compromise between the quality of the stress field in the adhesive (i.e. non edge effects and stress homogeneous in the thickness) and the experimental difficulty. Indeed, it does not necessitate any specific apparatus (as compared to the Arcan test device). The shape of the specimen is more difficult to manufacture as compare to other ones (classical Scarf joint specimens, single-lap joint specimens), but this difficulty remains reasonable. Moreover, each specimen could be used several times. First specimens have been manufacture using high-pressure water cut tool (to manufacture the global shape) and classical machining tools (to insure the planarity of the surface to bond and to manufacture the beaks). A bonding system has also been proposed. Some furthers works will be performed to optimize this geometry and perform some test with this kind of specimen.
Acknowledgements One of the authors (N. Carrere) would like to thank the financial support of the UIMM (Union des Industriels des Métiers de la Métallurgie) and ―La Région Bretagne‖.
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