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Numerical analysis and optimisation of cylindrical adhesive joints under tensile loads
International Journal of Adhesion & Adhesives 30 (2010) 706–719
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International Journal of Adhesion & Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
Numerical analysis and optimisation of cylindrical adhesive joints under tensile loads Jean Yves Cognard a,n, Herve´ Devaux b, Laurent Sohier c a
Brest Laboratory of Mechanics and Systems, ENSIETA, 29806 Brest Cedex 09, France HDS, 45 rue de l’Elorn, 29200 Brest, France c Brest Laboratory of Mechanics and Systems, UBO, 29285 Brest Cedex, France b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 20 July 2010 Available online 24 July 2010
This study is concerned with increasing the performance of adhesively bonded structures. Adhesively bonded joints offer many advantages, but stress singularities can contribute to the initiation and the propagation of crack in adhesive joints. Therefore, designing adhesively bonded joints which strongly limit the influence of edge effects can significantly increases the transmitted load by the assembly. Cylindrical joints are associated with high substrate strength in the radial direction, meaning that peel and cleavage forces have different effects compared to simple lap joints. But, for such assemblies, edge effects also exist. The aim of this paper is first to analyse stress concentrations in cylindrical joints in the case of axial loadings, starting from refined finite element computations under a linear elastic assumption. Thus as the stress state in the adhesive thickness can be complex associated to edge effects, simplified methods can lead to an overestimation of the transmitted load by the assembly as they generally are not able to represent effects of stress concentrations in the adhesive thickness. Secondly, geometries which strongly limit the influence of edge effects are proposed. An optimisation of the maximum transmitted load of cylindrical joints is proposed using a pressure-dependent elastic limit of the adhesive. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Finite element stress analysis Lap-shear Joint design Cylindrical joint
1. Introduction Using adhesive for the design of assemblies can reduce the cost and the weight of structures, especially when assembling dissimilar or composite materials. Adhesive joining techniques do not require holes, as riveted or bolted joints do, which can lead to stress concentrations. But adhesively bonded joints are also often characterised by significant edge effects associated with geometrical and material parameters. Adhesively bonded joints offer many advantages for the design of various industrial structures, but currently, use of this technology is limited by a lack of confidence in it [1]. Thus, optimising the design of adhesively bonded assemblies requires accurate numerical tools which must take the possible stress singularities due to edge effects into account [2]. Stress singularities can contribute to the initiation and propagation of cracks in the adhesive [3,4]. Thus, in order to respect safety design constraints, especially for high-tech applications, it can also be necessary to take into account the complex, non-linear behaviour of the adhesive (influence of viscous effects, influence of hydrostatic pressure, complex anelastic flow rule, etc. [5–8]). Various simplified
n
Corresponding author. Tel.: +33 2 98 34 88 16; fax: + 33 2 98 34 87 30. E-mail address: [email protected] (J.Y. Cognard).
0143-7496/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2010.07.003
models have been proposed in order to describe the behaviour of some bonded joints using 1D or 2D models [9,10], but often such models cannot describe the effect of stress singularities which often have an influence on the maximum load transmitted by the bonded assemblies. Therefore, understanding the stress distribution in an adhesive can lead to improvements in adhesively bonded assemblies; for instance, designing assemblies which strongly limit the edge effects can be very interesting. Several analytical studies have been proposed to analyse the influence of the geometry on stress singularities for bi-material joints [11,12]. In the case of elastic behaviour, the two main parameters are the relative elastic properties of the materials and the geometry of the substrates. For bi-materials, these studies have shown that use of beaks can limit the edge effects. However, with those approaches it is difficult to analyse the influence of the various parameters. For instance, the geometry of the bonded assembly, the interaction between the two interfaces of the thin joint, the non-linear behaviour of the adhesive and the external loading on the structure can have an influence on the stress distribution. Precise finite element computations are therefore useful to analyse the stress singularities in order to optimise the design of bonded assemblies [13,14]. In the case of single lap type joints, peel and cleavage forces strongly limit the transmitted load of the assembly despite various techniques proposed to limit the influence of edge effects.
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Many studies have been proposed to reduce the stress concentrations, especially in the case of lap joints, such as effects of spew and chamfer size [15,16], influence of slots [17,18], using a modulus graded bondline adhesive [19], etc. Cylindrical joints are associated with high strength of the substrates in the radial direction, meaning that the influence of peel and cleavage forces can be strongly limited. But, for such assemblies, edge effects also exist. Various studies exist on the behaviour of cylindrical joints under axial and torsion loading: stress analysis [15,20,21], influence of geometrical parameters (surface roughness and bond thickness) [22,23], use of a modulus graded bondline adhesive [19], influence of non-linear behaviour [24], influence of thermal residual stresses [25], fatigue life evaluation [26], etc. As one of the key points in the design of bonded assemblies is associated with the influence of edge effects, numerical and experimental results are presented using single lap shear specimens in order to underline the influence of stress concentrations on the behaviour of bonded assemblies. The aim of this paper is first to analyse the stress concentrations in the case of cylindrical joints under axial loadings, starting from refined finite element computations under an elastic assumption for the materials. Such problems can be solved using axisymmetric models which are less costly than 3D models. It has been shown for other geometries of bonded joints, on the one hand, that it is necessary to strongly limit the influence of edge effects, starting from computation under linear elasticity, in order to limit the risk of the first rupture near the edge of the joint; and on the other hand, that the nonlinear behaviour of the adhesive (plasticity) does not seem to limit the influence of the edge effects [4,27]. The influences of various geometrical parameters, such as joint thickness, overlap length and tubular thickness on the stress distribution are analysed. Secondly, geometries which strongly limit the influence of edge effects are proposed. An optimisation of the maximum transmitted load of cylindrical joints is proposed using a pressuredependent elastic limit of the adhesive defined from the two stress invariants, hydrostatic stress and von Mises equivalent stress; such models are often used to model the behaviour of polymers [5]. The finite element simulations were made with the code CAST3M (CEA, Saclay, France) [28].
2. Edge effects for single lap shear joints In order to determine the way stress evolves through the thickness of the adhesive, accurate finite element analyses must be performed, even when assuming that the components have a linear elastic behaviour. Since multi-material structures were being modelled, compliance with the mechanical properties of perfect interfaces was required. With the standard finite element method, based on the variational principal of minimum potential energy, whose single variable is the displacement field, the
707
continuity of the displacement field is satisfied but the continuity of the stress vector is not exactly verified. Therefore, refined meshes are also needed near the interfaces in order to obtain good numerical results, especially for significant material heterogeneity of the assemblies [14,27]. Various simulations have shown that good numerical results are obtained using meshes with 20 linear rectangular elements for a 0.1 mm thickness of adhesive. Fig. 1 presents the geometry of single lap shear specimen used in this section; Fig. 1c also shows the mesh of the adhesive close to the free edges and the mesh of the substrate close to the interface substrate–adhesive in order to respect the properties previously presented. Fig. 1c presents a part of the mesh associated with an adhesive thickness of e¼0.2 mm (only half of the adhesive is modelled). To facilitate the analysis of the numerical results, adhesive is meshed with rectangular elements, thus the evolution of the stresses through the thickness of the adhesive can be easily analysed. In order to limit the number of elements, in the model in the middle of the overlap length, the element size can be increased in the x direction without influencing the numerical results. Moreover it can be seen that the substrate is meshed using triangular elements which allow to quickly increase the element size in the substrate in order to limit the finite element model size. Computations were made in 2D (plane stresses) on half of the specimen by applying adequate boundary conditions. Results are presented for aluminium substrates (Young modulus: Ea ¼75 GPa, Poisson’s ratio: na ¼0.3). The material parameters for the adhesive are: Ej ¼2.2 GPa, nj ¼0.3. The parameters which define the geometry of the single lap shear specimen (Fig. 1) are such that: l1¼80 mm, h1¼5 mm, e¼0.2 mm. A so-called straight edge of the free edges of the adhesive is used to simplify the study. Results are presented for different overlap lengths (l ¼25, 50, 75 and 100 mm) and the average shear stress in the adhesive is normalised to 1 Mpa, in order to facilitate analysis of the results; thus, for the different assemblies the transmitted load is proportional to the overlap length. Moreover it is important to notice that results are only presented ob half of the joint thickness. Fig. 2a presents the way shear stress evolves in the middle of the adhesive with respect to the overlap length (i.e., along the x-axis with the origin at the centre of the joint). Fig. 2b shows the evolution of the shear stress close to the adhesive–substrate interface (y¼e/2, Fig. 1). It may be noted that there is only a limited evolution of the shear stress through the thickness of the adhesive. Fig. 2c and d present the evolution of the peel stress in the middle of the adhesive and close to the adhesive–substrate interface. Thus it can be noticed that the edge effects are mainly observed on the peel stress (yy component). Moreover with such single lap shear specimens, an increase in the overlap length leads to an increase in edge effects. Modification in the free edges of the adhesive (for instance, effects of spew and chamfer size [16] and influence of slots [18]) can reduce the influence of stress concentrations, but in order to
C
A
Fig. 1. Presentation of the single lap-shear joint (adhesive thickness of e¼ 0.2 mm for the mesh). (a) Geometry of the specimen; (b) close-up of the upper central part and (c) mesh (close-up view).
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12
12
10
10 l=100 mm
8
l=50 mm
shear stress (MPa)
shear stress (MPa)
l=100 mm l=75 mm
6 4
l=25 mm
8 l=50 mm 6 l=25 mm
4 2
2
0
0
-60
-40
l=75 mm
-20
0
20
40
60
-60
-40
-20
x (mm)
0
20
40
60
x (mm)
interface 50
von Mises equivalent stress (MPa)
peel stress (MPa)
40 30 20 l=100 mm l=50 mm
l=75 mm
10
l=25 mm
0 -60
-40
-20
l=100 mm
50
0
20
40
60
l=75 mm
40
l=50 mm
30 l=25 mm
20 10 0
-60
-40
-20
0
-10
-10
x (mm)
x (mm)
20
40
60
Fig. 2. Shear and peel stresses in the middle of the adhesive and close to the adhesive–substrate interface for an average shear stress of 1 MPa.
strongly limit the edge effects it seems also necessary to modify the geometries of the substrate close to the free edges of the adhesive [4]. For such specimens, the properties of the substrate (thickness, material, etc.) and the adhesive thickness have an influence on the stress distribution in the adhesive. Moreover, the computation assumptions can also have an influence on the numerical results. Plane stress conditions are representative of points close to the free edges of the single lap-shear specimen and plane strain conditions are nearly representative at points inside the specimen; a 3D computation can give better results, but the cost is higher. For such computations, plane strain and plane stress assumptions give nearly similar results. It can be noticed that for such studies, finite element analysis and results of asymptotic approaches give similar results [29]. Fig. 3 illustrates some influence of the stress concentrations on the behaviour of bonded assemblies. The thick adherend shear test (TAST) [30] is a widely used [31] and can be seen as an optimised single lap shear test, as thick substrates and a small overlap are used in order to limit the influence of stress singularities (Fig. 3a). Fig. 3c presents experimental results for the adhesive HuntsmanTM Araldites 420 A/B epoxy resin (Basel, Switzerland) and it can be seen that cracks exist at the beginning of the so-called non-linear behaviour of the adhesive. Dx represents the relative displacements of both ends of the adhesive in the x direction and the average shear stress is computed from the applied load and the adhesive section. It can be notice that the numerical analysis of such experimental results involving non-linear material behaviour and crack propagation can be complex. It has been numerically demonstrated and
experimentally verified that sharp beaks on the substrates, cleaned edges of the adhesive and using rigid supports can strongly limit edge effects; thus a modified TAST has been proposed which principle is presented in Fig. 3b [32], using the same adhesive cross-sectional area than the TAST. The small bonded samples (1) are mounted on a rigid support (2) and fixed with a fastening device (3). For this test, it can be seen that a homogeneous deformation of adhesive is observed (without crack propagation closed the free edges). Moreover the responses of the two tests are different; thus, limiting the influence of edge effects can lead to an increase of the transmitted load by the bonded assembly. Moreover, it has been shown that the plastic non-linear behaviour of the adhesive joint does not seem to limit the influence of the edge effects [4,27]; thus, the lower the equivalent stress (computed under linear elasticity), the lower the risk will be of first rupture near the edge of the joint.
3. Edge effects for cylindrical joints under tensile loadings This section presents a preliminary study, in order to analyse the influence of edge effects on cylindrical joints. Fig. 4 presents the geometry of the cylindrical bonded joint analysed in this section. As a prescribed tensile loading is employed, a usual axisymmetric model can be used in order to limit the computational cost of the various simulations. The geometry of the studied assembly is defined by the following parameters: r1, internal radius of the internal tube; r3, external radius of the external tube; r2, average radius of the joining; e, thickness of the adhesive (r2 and e give the external radius of the internal tube and the internal radius of the
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
(3)
709
(1)
(2)
x
Modified TAST
Average shear stress (MPa)
30
substrate
Modified TAST
adhesive
25 TAST
20
substrate
15
TAST 10
substrate 5
adhesive
0
0
0.1
0.2 0.3 Dx (mm)
0.4
0.5
substrate crack
crack
Fig. 3. Influence of edge effects, experimental results for TAST and modified TAST for an adhesive thickness of 0.4 mm. (a) TAST specimen (width: 25.4 mm); (b) Modified TAST principle and (c) experimental results and deformation of the adhesive.
Fig. 4. Presentation of the cylindrical joint studied with two local geometries of the adhesive (close-up). (a) Cylindrical joint; (b) straight edges and (c) cleaned edges.
external tube) and l, the overlap length. The first numerical results were obtained with r1¼10 mm, r2¼15 mm, r3¼ 20 mm, e¼0.2 mm and for different overlap lengths (l¼25, 50, 75 and 100 mm); moreover, it was assumed that the adhesive had straight free edges in a first stage. The behaviour of the different materials (aluminium substrate and adhesive) is assumed to be elastic and the same elastic parameters, which have been defined in the previous section, are used. For the various drawings, the average shear stress in the adhesive is normalised to 1, as in the previous section. Fig. 5a presents the evolution of the shear stress (component stress rz) in the middle of the adhesive with respect to the overlap length (i.e., along the z axis with the origin at the centre of the joint). For this example, there are two adhesive–substrate interfaces (here for the complete adhesive); thus, in order to simplify the presentation, only the evolution of the maximum values of the shear stress in the thickness of the adhesive with respect to the abscissa x. Fig. 5a and b present an idea of the stress evolution through the adhesive thickness as for this example, for a given
abscissa x, the minimum value is obtained in the middle of the joint and the maximum values is obtained in one of the adhesive– substrate interfaces (Fig. 5c and d). Fig. 5c and d describe, for an overlap length of 50 mm, at the two ends of the joint, trends in the shear stress in the adhesive on different lines with respect to the radius r, using an adimensional parameter l ¼ (r r2)/(2e). l ¼0 corresponds to the middle of the adhesive, l ¼ 1 corresponds to a line close to the interface between the adhesive and the internal tube and l ¼1 corresponds to a line close to the interface between the adhesive and the external tube. These results highlight the fact that quite large shear stress gradients are observed close to the adhesive–substrate interfaces. For this geometry, the stress state in the middle of the adhesive thickness (lA[ 0.5, 0.5]) is nearly constant for a given abscissa. Results of asymptotic analysis highlight the strong influence of the local geometry at the free ends of the adhesive, and can explain the different shear stress states close to the two interfaces. Fig. 6a and b shows the evolution of the minimum and maximum values of the radial stress in the thickness of the
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14
14
12
12
10
10
shear stress (MPa)
shear stress (MPa)
710
8 l=100 mm
6
l=75 mm l=50 mm
4
8
2
2
0 -40
-20
20
8
40
60
-60
-40
-20
λ=0
λ=1
λ=0.5
5
λ=1
4 λ=1
3 2
λ=−1
1
λ=0
-24.75 z (mm)
shear stress (MPa)
shear stress (MPa)
6
λ=−0.5
-25
λ=−1 λ=−0.5
3
λ=0.5
-24.5
60
λ=0
4
0 -25.25
40
7
λ=1
1
20
λ=−0.5 λ=0.5
5
0 z (mm)
8
λ=−1
λ=−1
6
2
l=25 mm
0 0
z (mm)
7
l=50 mm
4
l=25 mm
-60
l=100 mm l=75 mm
6
-24.25
0 24.25
λ=0.5
24.75
λ=−0.5 λ=0
25.25
z (mm)
Fig. 5. Shear stress through the thickness of the adhesive with straight edge for an average shear stress of 1 MPa. (a) Shear stress in the middle of the adhesive; (b) maximum values of shear stress; (c) shear stress through the adhesive thickness (zoom for an overlap of 50 mm) and (d) shear stress through the adhesive thickness (zoom for an overlap of 50 mm).
adhesive. These figures highlight the influence of edge effects on the peel stress. Moreover, quite similar results are observed for hoop and axial stresses. The equivalent von Mises stress also emphasizes this property as it takes into account the different stress components (Fig. 6c and d). Single lap shear specimens and cylindrical joints under axial loading are both associated with large edge effects. But the main difference is that single lap shear specimens generate tensile peel stresses and cylindrical joints generate compression peel stresses. This is an important point, since it has been shown that compression has a strong positive influence on bonded assembly strength [29]. Fig. 7 presents the evolution of the radial stress and the von Mises equivalent stress in the adhesive in the case of a cleaning radius r ¼0.75 e (defined on Fig. 4c). For this computation the mesh presented on Fig. 1 is deformed in the axial direction in order to respect the geometry of the adhesive free edges. The use of the so-called cleaned edges, associated with a cleaning of the adhesive before curing enables the influence of edge effects to be significantly reduced with respect to straight edges (Fig. 6) [27], a quite large reduction of the peel stress can be observed.
on the geometry of the bonded assembly, especially close to the free edges of the adhesive. Limiting the influence of edge effects can lead to an increase in the load transmitted by the assembly. Thus, it is necessary to develop accurate criteria, taking into account the real behaviour of the adhesive in order to optimise such assemblies. Various studies have shown that an accurate representation of the elastic yield domain of an adhesive in an assembly requires using a pressure-dependent constitutive model, i.e. a model taking the two stress invariants, hydrostatic stress and von Mises equivalent stress into account [7,8]. For this study, only an elastic behaviour of the adhesive was used. Starting from the experimental results obtained for the epoxy resin HuntsmanTM Araldites 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 (leaving out the viscous effects). This test allows us to determine the behaviour of an adhesive in an assembly under radial loadings (tensile/ compression-shear loadings) [33]. An exponential Drucker–Prager yield function provides good representation of the experimental data for the initial so-called elastic limit [34] F0 ¼ aðsvm Þb þ ph -pt0
4. Strength criteria of the adhesive in an assembly 4.1. Cylindrical joints with cleaned edges The numerical results presented in the previous section highlight the complex stress states in the adhesive which depend
ð1Þ
where svm is the von Mises equivalent stress and ph is the hydrostatic stress. a, b and pt0 are material parameters. The results of the identification are presented in Table 1 and the initial yield surface (F0 ¼0) is drawn in Fig. 8b. Fig. 8a gives the hydrostatic stress–von Mises equivalent stress diagram showing the stress in the adhesive for a thickness of
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
von Mises equivalent stress
15 l=75 mm 10 l=50 mm 5 l=25 mm 0 -40 -20 -5 0 20 40 -10 -15 -20 -25 -30 -35 -40 -45 z (mm)
radial stress
l=100 mm
-60
35
35
30
30
von Mises equivalent stress
-40
radial stress
-60
15 10 5 0 -20 -5 0 20 40 60 -10 l=25 mm l=50 mm -15 l=75 mm -20 l=100 mm -25 -30 -35 -40 -45 z (mm)
25 20 15 l=100 mm
10 l=50 mm
5
l=75 mm
-40
-20
20
l=100 mm
15
l=75 mm l=50 mm
10 5
l=25 mm
0 z (mm)
60
25
0
-60
711
l=25 mm
0
20
40
60
-60
-40
-20
0 z (mm)
20
40
60
Fig. 6. Minimum and maximum values of the stresses through the thickness of the adhesive with straight edges for an average shear stress of 1 MPa. (a) Minimum values of the radial stress; (b) maximum values of the radial stress; (c) minimum values of the von Mises stress and (d) maximum values of the von Mises stress.
0.1 mm, for cleaned edges of the adhesive and for the four overlap lengths. For an average shear stress equal to 1, the hydrostatic stress and the von Mises equivalent stress are computed for each integration point of each element of the finite element mesh. In the hydrostatic stress–von Mises equivalent stress diagram, only the envelope of those points is drawn here to simplify the presentation (all the points are inside the so-called envelope). Using the linear property of elastic problems, we can determine the maximal transmitted loading of the bonded assembly using the elastic limit, which can be defined with the maximal average shear stress in the assembly: taverage maxi. Results are presented in Table 2. It can be noted that the maximum average shear stress decreases with the overlap length; but the maximum transmitted load by the assembly, Pl, increases slowly with overlap length. Fig. 8b presents the hydrostatic stress–von Mises equivalent stress diagram with the stress in the adhesive for the maximum transmitted load and for the four overlap lengths. It can be noticed that the envelopes are similar for the different overlap lengths. Moreover, those results highlight the positive influence of the negative hydrostatic stress, associated with compression peel stress in the adhesive, on the maximum transmitted load by the assembly. Fig. 9 shows, for the maximum transmitted load of the assemblies, the evolution of the minimum and maximum values of the distance, in the hydrostatic stress–von Mises equivalent stress diagram, of a characteristic point to the elastic surface in the thickness of the adhesive with respect to the overlap length; for F0 ¼0, the elastic limit is reached for the considered point and the distance is equal to 0. This distance is computed using the
properties of radial loadings: it is determined in the direction of the stress path associated to an increase of the loading. Those results highlight the significant influence of edge effects and the similar stress state in the adhesive for the different overlap lengths.
4.2. Influence of geometric parameters Thin substrate thickness leads to a different behaviour of the assembly, in conjunction with the low radial rigidity of the tubes (Fig. 10a). A large evolution of the stress state can be seen from a thickness of 1–5 mm. Increasing the thickness of the tubes from 10 to 15 mm does not lead to significant variation of the stress distribution within the adhesive. An increase in the adhesive thickness is associated with an increase in the transmitted load (Fig. 10b) under elastic behaviour. Results of analytical methods are often compared with finite element results, with only very few elements (about four) in the adhesive thickness. They nearly approximate the behaviour of the adhesive in the middle of the joint [21]. Refined and coarse meshes of the adhesive can give nearly the same numerical results in the middle of the joint, but only refined meshes can give reliable information about the stress state in the thickness of the adhesive close to the free edges (Fig. 6a and b). Fig. 11 presents the hydrostatic stress–Mises equivalent stress diagram showing the stress state in the middle of the adhesive for an overlap length l¼50 mm, for an adhesive thickness of 0.2 mm, and for straight and cleaned edges of the adhesive. It can be seen that for the two
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
radial stress
-60
15 10 5 l=25 mm 0 -40 -20 -5 0 20 40 l=50 mm -10 l=75 mm -15 l=100 mm -20 -25 -30 -35 -40 -45
60
radial stress
712
35
35
30
30
von Mises equivalent stress
von Mises equivalent stress
z (mm)
15 10 l=100 mm l=50 mm 5 l=25 mm l=75 mm 0 -60 -40 -20 -5 0 20 40 -10 -15 -20 -25 -30 -35 -40 -45 z (mm)
25 20 15 l=100 mm
10 l=50 mm
l=75 mm
5
25 20 15
-40
-20
l=100 mm l=75 mm l=50 mm l=25 mm
10 5
l=25 mm
0 -60
60
0
0
20
40
60
-60
-40
z (mm)
-20
0
20
40
60
z (mm)
Fig. 7. Minimum and maximum values of the stresses through the thickness of the adhesive with cleaned edges for an average shear stress of 1 MPa. (a) Minimum values of the radial stress; (b) maximum values of the radial stress; (c) minimum values of the von Mises stress and (d) maximum values of the von Mises stress.
Table 1 Adhesive parameters for the initial yield surface (defined by the Eq. (1)). a (SI units)
b
pt0 (MPa)
1. E-6
4.87
31.7
l=75 mm l=50 mm
20
40
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
l=100 mm
15
10
l=25 mm
5
35 30 25 20 15 10 5
0 -15
-10
-5
0
Hydrostatic stress (MPa)
5
l=25mm l=50mm l=75mm l=100mm Elastic limit
-40
-30
0 -20 -10 0 10 20 hydrostatic stress (MPa)
30
40
Fig. 8. Stress in the adhesive with cleaned edges in the hydrostatic stress–von Mises equivalent stress diagram. (a) Average shear stress of 1 MPa; (b) maximum transmitted load.
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713
Table 2 Definition of the maximum load transmitted by the assembly (Pl is the maximum transmitted load by an assembly with an overlap length l; and Pl ¼ 25 is associated with l ¼25 mm). Overlap length: l
25 mm
50 mm
75 mm
100 mm
taverage maxi
6.93 MPa 1.00
3.88 MPa 1.12
2.59 MPa 1.12
1.95 MPa 1.13
40
40
35
35
30
30
distance (MPa)
distance (MPa)
Pl/Pl ¼ 25
25 20 l=25 mm
l=75 mm
15
25 20 l=100 mm
10 5
5
l=100 mm
0
0 -60
-40
-20
l=75 mm
15
10 l=50 mm
l=25 mm
l=50 mm
0 z (mm)
20
40
-60
60
-40
-20
0 z (mm)
20
40
60
Fig. 9. (a) Minimum and (b) maximum values of the distance to the elastic surface of the adhesive with cleaned edges for the maximum transmitted load of each joint.
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
40
30 th=5 mm
20
th=10 mm th=15 mm
10
e=0.4 mm
15 e=0.8 mm
-20
-10
0
10
5
-15
20
10
e=1.6 mm
0 -30
20
e=0.2 mm
th=1 mm
hydrostatic stress (MPa)
0 -10 -5 0 hydrostatic stress (MPa)
5
Fig. 10. (a) Influence of the substrate thickness and (b) adhesive thickness with cleaned edges for an overlap length of l ¼100 mm and for an average shear stress of 1 MPa.
16
12
8
4
-12
-8
-4
hydrostatic stress (MPa)
16
12
8
4
0
0 -16
envelope middle line
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
envelope middle line
0
4
-16
-12
-8
-4
0
4
hydrostatic stress (MPa)
Fig. 11. Stress state in the middle of the adhesive for an overlap length l ¼ 50 mm, for an adhesive thickness of 0.2 mm and for an average shear stress of 1 MPa. (a) Straight edges and (b) cleaned edges.
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geometries of the adhesive free edges, the stress states in the middle of the joint are nearly similar. Edge effects nearly only affect the stress state close to the substrate–adhesive interface near the free edges. Thus, using a simplified method for dimensioning adhesive joints with singular points can
overestimate the maximum transmitted load by the bonded assembly. Figs. 12 and 13 present the influence of substrate thickness for an adhesive thickness of 0.2 mm and of adhesive thickness for a substrate thickness of 5 mm on the maximal average shear stress,
16
16 l=25mm l=50mm l=75mm l=100mm
12 10 8 6 4
14
τaverage maxi (MPa)
τaverage maxi (MPa)
14
l=25mm l=50mm l=75mm l=100mm
12 10 8 6 4 2
2
0
0 0
5
10
0
15
5 10 substrate thickness (mm)
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Fig. 13. Influence of the adhesive thickness on the maximal average shear stress for a substrate thickness of 5 mm (—: taking only the stresses in the middle line of the adhesive into account). (a) Straight edges and (b) cleaned edges
Fig. 14. Presentation of the cylindrical joint studied with different local geometries (zoom). (a) Cylindrical joint; (b) straight substrates; (c) cleaned edges of the adhesive; (d) substrates with beaks and (e) substrates with inverse beaks.
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In the case of large edge effects, the adhesive thickness has very little influence on the maximal transmitted load, even though it has quite a large influence on the stress state at the middle line of the adhesive (Fig. 13a). However, for elastic behaviour, limiting the influence of edge effects leads to an increase in the maximum transmitted load along with an increase in the adhesive thickness (Fig. 13b). It is important to note that for some planar joints, an increase in the adhesive thickness leads to an increase in the influence of edge effects [27].
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i.e., on the maximum transmitted load of the bonded assembly. Results, starting from the stress state in the adhesive, are presented for the four overlap joints studied, using large symbols. The results obtained using only the stress state in the middle of the adhesive are also presented with small symbols linked by thin lines. For the example studied, a substrate thickness greater than about 5 mm does not lead to an increase in the maximal transmitted load, but rather to a slight change in the stress state of the middle line of the adhesive (Fig. 12).
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beaks were defined with the parameters: h3¼2 mm and d ¼e. The radius R0 was defined to obtain a tangent circle to the adhesive– substrate interface in order to prevent angular points which can also lead to stress singularities. Those geometries (inverse beaks) have been used to analyse long term strength of adhesively bonded joints in sea water [36]. The machining of inverse beaks can be easier than the creating beaks on this kind of cylindrical parts. For inverse beaks the mesh presented on Fig. 1 is deformed in the radial direction in order to respect the adhesive geometry.
5. Proposal of geometries to limit the influence of edge effects 5.1. Geometrical parameters The numerical results presented in the previous section highlight the strong influence of stress concentrations on the maximal transmitted load of cylindrical joints. Thus, in this section, the geometries used in order to strongly limit the influence of edge effects in the case of a planar joint [27], are evaluated for cylindrical joints, taking into account some manufacturing constraints. Fig. 14 presents the different geometries used; cylindrical parts with shoulders are used to facilitate the machining, and these shoulders modify the local geometry near the adhesive free edges and can lead to a reduction in the influence of the edge effects [35]. Computations were made for an average radius of the adhesive joint of r2¼ 15 mm and an external radius of the external tube of r3¼30 mm. The shoulders were defined with the parameter dr¼1 mm. The cleaned edges were defined with the radius r ¼1.5e, e being the adhesive thickness. The beaks were defined with the following parameters: h1 ¼0.1 mm, h2 ¼1 mm and a ¼301, in order to strongly limit the influence of edge effects, while limiting the space associated with modifying the geometry. The beaks are used to analyse the behaviour of adhesive in assemblies with an Arcan device [27,34]. The so-called inverse
5.2. Influence of geometrical parameters for a given joint thickness Fig. 15 presents the hydrostatic stress–von Mises equivalent stress diagram showing the stress state of the adhesive for the maximum transmitted load for the three local geometries, for straight edges and cleaned edges of the adhesive, for the two overlap lengths (l¼25 mm and l¼100 mm), and for an adhesive thickness of 0.2 mm. It can be seen in this diagram that the overlap length has only slight influence on the dimensioning of the assembly for a given local geometry. It can be noticed that the overlap length has only a little influence of the stress state. But the local geometry of the substrate close to the adhesive and the cleaning of the free edge of the adhesive has a quite large influence on the stress state. Those results highlight the major influence of edge effects.
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Fig. 18. Influence of the adhesive thickness for different geometries of the substrates (—: taking only the stresses in the middle line of the adhesive into account). (a) Straight substrates – straight edges; (b) straight substrates – cleaned edges; (c) beaks – straight edges; (d) beaks – cleaned edges; (e) inverse beaks – straight edges and (f) inverse beaks – cleaned edges
Fig. 16 shows, in the case of beaks with cleaned edges, for the four overlap lengths, the evolution of the minimum and maximum values of the distance, in the hydrostatic stress–von Mises equivalent stress diagram, of a characteristic point to the elastic surface in the adhesive thickness with respect to the overlap length. For this example, it can be seen that the stress state in the thickness of the adhesive is nearly constant for a given position z in the overlap length. It is important to notice the difference between the results presented on Fig. 9 (associated with large influence of edge effects) and the results presented on Fig. 16 (associated with very low influence of edge effects).
5.3. Influence of joint thickness Fig. 17 presents the hydrostatic stress–Mises equivalent stress diagram with the stress state in the whole joint and in the middle of the adhesive for an overlap length of l¼50 mm for inverse beaks and cleaned edges of the adhesive. It can be seen that for an adhesive thickness of 0.2 mm, the stress state in the middle of the adhesive is close to the maximal stress state in the whole joint (Fig. 17a). For a given average shear stress in the joint, an increase in the adhesive thickness leads to a decrease in the maximal equivalent stress in the middle of the joint stress. Therefore, an increase in the maximal transmitted load can be obtained if the
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influence of edge effects is required to be low. It can be noted that for a joint thickness of 0.8 mm, the dimensioning points are not in the middle of the joint for this geometry (Fig. 17b). Fig. 18 presents the influence of adhesive thickness on the maximal average shear stress, i.e. on the maximum transmitted load of the bonded assembly. Results, starting from the stress
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state in the adhesive, are presented for the six different geometries and for the four overlaps studied using large symbols. The results obtained using only the stress state on the middle line of the adhesive, are also presented using small symbols linked by thin lines. Fig. 18a and b underline the positive influence of shoulders in the case of so-called straight substrates (Fig. 13). In the problem studied, using beaks or inverse beaks increases the possibilities for a bonded assembly with respect to straight substrates. It can be noted that the stress state in the middle of the joint is nearly independent of the local geometry of the assembly close to the free edges of the adhesive. The use of cleaned edges with beaks or with inverse beaks allows the adhesive loading close to the free edges to be limited, but better results are obtained with beaks. The same holds true for the use of straight edges with beaks or with inverse beaks. Moreover, quite similar results are obtained with straight substrates associated with adhesive cleaned edges and with inverse beaks associated with adhesive straight edges. An optimisation of the geometries of the beak and of the inverse beak can lead to some slight changes in the results. A coupling between the effects of the joint thickness and the effects of the local geometry of the assembly can be seen close to the free edges of the adhesive. To show this coupling’s effect, Fig. 19 presents some results associated with the shear stress distribution in the thickness adhesive in the case of beaks with straight and cleaned edges of the adhesive, for three joint thicknesses (0.1, 0.4 and 1.6 mm) and for an average shear stress of 1 MPa. On the middle line of the joint, the shear stress distributions are quite similar for straight and cleaned edges of the adhesive (thus only one drawing is presented, Fig. 19a), and increasing the adhesive thickness leads to a reduction in the maximum shear stress. In the case of cleaned edges, associated with a weak influence by edge effects, the maximum values of the shear stress through the joint thickness are quite similar to the shear stress on the middle line of the adhesive; but an increase in the adhesive thickness leads to a decrease in the maximum value of the shear stress in the whole joint (Fig. 19b). The use of adhesive straight edges, which are associated with a greater influence of edge effects, leads to an increase in the shear stress close to free edges of the adhesive, especially when the adhesive thickness increases (Fig. 19c). But it is important to remember that the influence of edge effects is greater for normal stresses than for shear stress (Figs. 4–6). Those results highlight the fact that for a given local geometry of the assembly, the choice of the adhesive thickness allows us to optimise the maximal transmitted load of the assembly under an elastic assumption for the different parts (Fig. 18). Strongly limiting the influence of edge effects allows a significant increase in the maximum transmitted load of the assembly, taking only an elastic behaviour of the adhesive into account. Moreover, it is important to note that such geometries can strongly limit the loading of the adhesive close to the adhesive–substrate interface near the free edges, which is often the weakest part of the assembly (Fig. 19).
0.4 6. Conclusions
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Fig. 19. Shear stress in the adhesive for an overlap length of l ¼25 mm and for an average shear stress of 1 MPa in the case of substrates with beaks. (a) Stress in the middle joint for cleaned and straight edges of the adhesive; (b) maximum stress through the adhesive thickness for cleaned edges of the adhesive and (c) maximum stress through the adhesive thickness for straight edges of the adhesive.
Adhesively bonded assemblies offer many advantages, but analysing the mechanical behaviour is made difficult, in particular, by the stress singularities due to edge effects. The first part of the paper presents differences between the stress states for usual single lap shear and cylindrical joints under tensile loadings. However, both types of bonded assemblies can be associated with large stress concentrations. The use of refined finite element computations highlights significant changes in the stress state
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within the thickness of the joint, especially close to the free edges of the adhesive. Therefore, it can be difficult to obtain precise dimensioning of adhesively bonded assemblies using simplified methods which generally cannot analyse large stress concentrations. Under elastic assumption of the materials, the use of a pressure-dependent elastic limit of the adhesive can provide an accurate analysis of the load transmitted by a bonded assembly. Such models, defined from the hydrostatic and von Mises equivalent stresses, accurately represent the behaviour of various polymers. Secondly, geometries which strongly limit the influence of edge effects have been proposed in order to increase the maximum transmitted load of the assembly, taking into account only an elastic behaviour of the adhesive. It is also important to note that these geometries can strongly limit the stress state within the adhesive close to the adhesive–substrate interface near free edges, which is often the weakest part of the assembly. A precise analysis of the stress state in the adhesive highlights the fact that there is a coupling of the effects of the joint thickness and the effects of the local geometry of the assembly, close to the free edges of the adhesive. Those numerical results provide some rules for manufacturing conditions to optimise bonded cylindrical joints under tensile loadings. Under the elastic assumption, the key point is to strongly limit the influence of edge effects in order to increase the transmitted load. An optimisation of the geometrical parameters of the two proposed geometries (beaks and inverse beaks) can improve the behaviour the assemblies. Experimental tests must be developed to analyse the real behaviour of the proposed geometries. In particular, for inverse beaks, the influence of increasing the adhesive thickness close to the free edges should be investigated. Numerical and experimental studies are also required to analyse the behaviour of the proposed geometries in industrial-type applications. More complex loadings, such as torsion-tensile or bending, must be taken into account, and the influence of the non-linear behaviour of the adhesive should also be analysed. References [1] Adams RD. Adhesive bonding: science, technology and applications. Bristol: Woodhead Publishing Ltd.; 2005. ¨ chsner A. Modeling of adhesive bonded joints. Berlin: [2] Da Silva LFM, O Springer; 2008. [3] Dean G, Crocker L, Read B, Wright L. Prediction of deformation and failure of rubber-toughened adhesive joints. Int J Adhes Adhes 2004;24:295–306. [4] Cognard JY, Cre´ac’hcadec R, Sohier L, Davies P. Analysis of the non linear behaviour of adhesives in bonded assemblies. Comparison of TAST and ARCAN tests. Int J Adhes Adhes 2008;28:393–404. [5] Raghava RS, Cadell RM. The macroscopic yield behaviour of polymers. J Mater Sci 1973;8:225–32. [6] Wang CH, Chalkley P. Plastic yielding of a film adhesive under multiaxial stresses. Int J Adhes Adhes 2000;20:155–64. [7] Mahnken R, Schlimmer M. Simulation of strength difference in elastoplasticity for adhesive materials. Int J Num Methods Eng 2005;63:1461–77. ¨ [8] Rolfres R, Volger M, Ernst G, Huhne C. Strength of textile composites in multiscale simulation, in trends in computational structures technology. Stirlingshire, Scoltland: Saxe-Coburg Publications; 2008. Chapter 7, 151–171. [9] Volkersen O. Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten. Luftfahrtforschung 1938;15:41–7.
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[10] Lubkin JL, Reissner E. Stress distribution and design data for adhesive lap joints between circular tubes. Trans ASME 1956;78:1213–21. [11] Qian ZQ. On the evaluation of wedge stress intensity factor of bi-material joints with surface tractions. Comput Struct 2001;79:53–64. [12] Kotousov A. Effect of a thin plastic adhesive layer on the stress singularity in a bi-material wedge. Int J Adhes Adhes 2007;27:647–52. [13] Pandley PC, Narasimhan S. Three-dimensional nonlinear analysis of adhesively bonded lap joints considering viscoplasticity in adhesives. Comput Struct 2001;79:769–83. [14] Cheikh M, Coorevits P, Loredo M. Modelling the stress vector continuity at the interface of bonded joints. Int J Adhes Adhes 2001;22:249–57. [15] Adams RD, Peppiat NA. Stress analysis of adhesive bonded tubular joints. J Adhes 1977;9:1–18. [16] Belingardi G, Goglio L, Tarditi A. Investigating the effect of spew and chamfer size on the stresses in metal/plastics adhesive joints. Int J Adhes Adhes 2002;22:273–82. [17] Sancaktar E, Nirantar P. Increasing strength of single lap joints of metal adherends by taper minimization. J Adhes Sci Technol 2003;17:655–75. [18] Yan ZM, You M, Yi XS, Zheng XL, Li Z. A numerical study of parallel slot in adherend on the stress distribution in adhesively bonded aluminium single lap joint. Int J Adhes Adhes 2007;27:687–95. [19] Kumar S. Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads. Int J Adhes Adhes 2009;29: 785–95. [20] Kawamura H, Sawa T, Yoneno M, Nakamura T. Effect of fitted position on stress distribution and strength of a bonded shrink fitted joint subjected to torsion. Int J Adhes Adhes 2003;23:131–40. [21] Nemes O, Lachaud F, Mojtabi A. Contribution to the study of cylindrical adhesive joining. Int J Adhes Adhes 2006;26:474–80. [22] Beevers A. Critical assessment coaxial joints in engineering assembly. Mater Sci Technol 1986;2:97–102. [23] Kwon LW, Lee DG. The effects of surface roughness and bond thickness on the fatigue life of adhesively bonded tubular single lap joints. J Adhes Sci Technol 2000;14(8):1085–102. [24] Hosseinzadeh R, Taheri F. Non-linear investigation of overlap length effect on torsional capacity of tubular adhesively bonded joints. Compos Struct 2009;91:186–95. [25] Apalak MK, Gunes R, Eroglu S. Thermal residual stresses in an adhesively bonded functionally graded tubular single lap joint. Int J Adhes Adhes 2007;27:26–48. [26] Nayeb-Hashemi H, Rossettos JN, Melo AP. Multiaxial fatigue life evaluation of tubular adhesively bonded joints. Int J Adhes Adhes 1997;17:55–63. [27] Cognard JY. Numerical analysis of edge effects in adhesively-bonded assemblies. Application to the determination of the adhesive behaviour. Comput Struct 2008;86:1704–17. [28] Cast3m documentation, /www-cast3m.cea.fr/cast3mS. [29] Cognard JY, Cre´ac’hcadec R, Sohier L, Leguillon D. Influence of adhesive thickness on the behaviour of bonded assemblies under shear loadings using a modified TAST fixture. Int J Adhes Adhes 2010;30:257–66. [30] ASTM D5656-95, Standard test method for thick-adherend metal lap-shear joints for determination of the stress-strain behavior of adhesives in shear by tension loading, 1995. [31] Zgoul M, Crocombe AD. Numerical modeling of lap joints bonded with ratedependent adhesive. Int J Adhes Adhes 2004;24:355–66. [32] Cognard JY, Creac’hcadec R. Analysis of the non linear behaviour of an adhesive in bonded assemblies under shear loadings. Proposal of an improved TAST. J Adhes Sci Technol 2009;23:1333–55. [33] Cognard JY, Davies P, Sohier L, Cre´ac’hcadec R. A study of the non-linear behavior of adhesively-bonded composite assemblies. Compos Struct 2006;76:34–46. [34] Cognard JY, Cre´ac’hcadec R, Maurice J, Davies P, Peleau M, Da Silva LFM, Analysis of the influence of hydrostatic stress on the behaviour of an adhesive in an assembly, J Adhes Sci Technol, in press, doi:10.1163/ 016942410X507696. [35] Leguillon D, Sanchez-Palancia E. Computation of singular solutions in elliptic problems and elasticity. Paris: Editions Masson; 1987. [36] Bordes M, Davies P, Cognard JY, Sohier L, Sauvant-Moynot V, Galy J. Prediction of long term strength of adhesively bonded joints in sea water. Inter J Adhes Adhes 2009;29:595–608.
Contents lists available at ScienceDirect
International Journal of Adhesion & Adhesives journal homepage: www.elsevier.com/locate/ijadhadh
Numerical analysis and optimisation of cylindrical adhesive joints under tensile loads Jean Yves Cognard a,n, Herve´ Devaux b, Laurent Sohier c a
Brest Laboratory of Mechanics and Systems, ENSIETA, 29806 Brest Cedex 09, France HDS, 45 rue de l’Elorn, 29200 Brest, France c Brest Laboratory of Mechanics and Systems, UBO, 29285 Brest Cedex, France b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 20 July 2010 Available online 24 July 2010
This study is concerned with increasing the performance of adhesively bonded structures. Adhesively bonded joints offer many advantages, but stress singularities can contribute to the initiation and the propagation of crack in adhesive joints. Therefore, designing adhesively bonded joints which strongly limit the influence of edge effects can significantly increases the transmitted load by the assembly. Cylindrical joints are associated with high substrate strength in the radial direction, meaning that peel and cleavage forces have different effects compared to simple lap joints. But, for such assemblies, edge effects also exist. The aim of this paper is first to analyse stress concentrations in cylindrical joints in the case of axial loadings, starting from refined finite element computations under a linear elastic assumption. Thus as the stress state in the adhesive thickness can be complex associated to edge effects, simplified methods can lead to an overestimation of the transmitted load by the assembly as they generally are not able to represent effects of stress concentrations in the adhesive thickness. Secondly, geometries which strongly limit the influence of edge effects are proposed. An optimisation of the maximum transmitted load of cylindrical joints is proposed using a pressure-dependent elastic limit of the adhesive. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Finite element stress analysis Lap-shear Joint design Cylindrical joint
1. Introduction Using adhesive for the design of assemblies can reduce the cost and the weight of structures, especially when assembling dissimilar or composite materials. Adhesive joining techniques do not require holes, as riveted or bolted joints do, which can lead to stress concentrations. But adhesively bonded joints are also often characterised by significant edge effects associated with geometrical and material parameters. Adhesively bonded joints offer many advantages for the design of various industrial structures, but currently, use of this technology is limited by a lack of confidence in it [1]. Thus, optimising the design of adhesively bonded assemblies requires accurate numerical tools which must take the possible stress singularities due to edge effects into account [2]. Stress singularities can contribute to the initiation and propagation of cracks in the adhesive [3,4]. Thus, in order to respect safety design constraints, especially for high-tech applications, it can also be necessary to take into account the complex, non-linear behaviour of the adhesive (influence of viscous effects, influence of hydrostatic pressure, complex anelastic flow rule, etc. [5–8]). Various simplified
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Corresponding author. Tel.: +33 2 98 34 88 16; fax: + 33 2 98 34 87 30. E-mail address: [email protected] (J.Y. Cognard).
0143-7496/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2010.07.003
models have been proposed in order to describe the behaviour of some bonded joints using 1D or 2D models [9,10], but often such models cannot describe the effect of stress singularities which often have an influence on the maximum load transmitted by the bonded assemblies. Therefore, understanding the stress distribution in an adhesive can lead to improvements in adhesively bonded assemblies; for instance, designing assemblies which strongly limit the edge effects can be very interesting. Several analytical studies have been proposed to analyse the influence of the geometry on stress singularities for bi-material joints [11,12]. In the case of elastic behaviour, the two main parameters are the relative elastic properties of the materials and the geometry of the substrates. For bi-materials, these studies have shown that use of beaks can limit the edge effects. However, with those approaches it is difficult to analyse the influence of the various parameters. For instance, the geometry of the bonded assembly, the interaction between the two interfaces of the thin joint, the non-linear behaviour of the adhesive and the external loading on the structure can have an influence on the stress distribution. Precise finite element computations are therefore useful to analyse the stress singularities in order to optimise the design of bonded assemblies [13,14]. In the case of single lap type joints, peel and cleavage forces strongly limit the transmitted load of the assembly despite various techniques proposed to limit the influence of edge effects.
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Many studies have been proposed to reduce the stress concentrations, especially in the case of lap joints, such as effects of spew and chamfer size [15,16], influence of slots [17,18], using a modulus graded bondline adhesive [19], etc. Cylindrical joints are associated with high strength of the substrates in the radial direction, meaning that the influence of peel and cleavage forces can be strongly limited. But, for such assemblies, edge effects also exist. Various studies exist on the behaviour of cylindrical joints under axial and torsion loading: stress analysis [15,20,21], influence of geometrical parameters (surface roughness and bond thickness) [22,23], use of a modulus graded bondline adhesive [19], influence of non-linear behaviour [24], influence of thermal residual stresses [25], fatigue life evaluation [26], etc. As one of the key points in the design of bonded assemblies is associated with the influence of edge effects, numerical and experimental results are presented using single lap shear specimens in order to underline the influence of stress concentrations on the behaviour of bonded assemblies. The aim of this paper is first to analyse the stress concentrations in the case of cylindrical joints under axial loadings, starting from refined finite element computations under an elastic assumption for the materials. Such problems can be solved using axisymmetric models which are less costly than 3D models. It has been shown for other geometries of bonded joints, on the one hand, that it is necessary to strongly limit the influence of edge effects, starting from computation under linear elasticity, in order to limit the risk of the first rupture near the edge of the joint; and on the other hand, that the nonlinear behaviour of the adhesive (plasticity) does not seem to limit the influence of the edge effects [4,27]. The influences of various geometrical parameters, such as joint thickness, overlap length and tubular thickness on the stress distribution are analysed. Secondly, geometries which strongly limit the influence of edge effects are proposed. An optimisation of the maximum transmitted load of cylindrical joints is proposed using a pressuredependent elastic limit of the adhesive defined from the two stress invariants, hydrostatic stress and von Mises equivalent stress; such models are often used to model the behaviour of polymers [5]. The finite element simulations were made with the code CAST3M (CEA, Saclay, France) [28].
2. Edge effects for single lap shear joints In order to determine the way stress evolves through the thickness of the adhesive, accurate finite element analyses must be performed, even when assuming that the components have a linear elastic behaviour. Since multi-material structures were being modelled, compliance with the mechanical properties of perfect interfaces was required. With the standard finite element method, based on the variational principal of minimum potential energy, whose single variable is the displacement field, the
707
continuity of the displacement field is satisfied but the continuity of the stress vector is not exactly verified. Therefore, refined meshes are also needed near the interfaces in order to obtain good numerical results, especially for significant material heterogeneity of the assemblies [14,27]. Various simulations have shown that good numerical results are obtained using meshes with 20 linear rectangular elements for a 0.1 mm thickness of adhesive. Fig. 1 presents the geometry of single lap shear specimen used in this section; Fig. 1c also shows the mesh of the adhesive close to the free edges and the mesh of the substrate close to the interface substrate–adhesive in order to respect the properties previously presented. Fig. 1c presents a part of the mesh associated with an adhesive thickness of e¼0.2 mm (only half of the adhesive is modelled). To facilitate the analysis of the numerical results, adhesive is meshed with rectangular elements, thus the evolution of the stresses through the thickness of the adhesive can be easily analysed. In order to limit the number of elements, in the model in the middle of the overlap length, the element size can be increased in the x direction without influencing the numerical results. Moreover it can be seen that the substrate is meshed using triangular elements which allow to quickly increase the element size in the substrate in order to limit the finite element model size. Computations were made in 2D (plane stresses) on half of the specimen by applying adequate boundary conditions. Results are presented for aluminium substrates (Young modulus: Ea ¼75 GPa, Poisson’s ratio: na ¼0.3). The material parameters for the adhesive are: Ej ¼2.2 GPa, nj ¼0.3. The parameters which define the geometry of the single lap shear specimen (Fig. 1) are such that: l1¼80 mm, h1¼5 mm, e¼0.2 mm. A so-called straight edge of the free edges of the adhesive is used to simplify the study. Results are presented for different overlap lengths (l ¼25, 50, 75 and 100 mm) and the average shear stress in the adhesive is normalised to 1 Mpa, in order to facilitate analysis of the results; thus, for the different assemblies the transmitted load is proportional to the overlap length. Moreover it is important to notice that results are only presented ob half of the joint thickness. Fig. 2a presents the way shear stress evolves in the middle of the adhesive with respect to the overlap length (i.e., along the x-axis with the origin at the centre of the joint). Fig. 2b shows the evolution of the shear stress close to the adhesive–substrate interface (y¼e/2, Fig. 1). It may be noted that there is only a limited evolution of the shear stress through the thickness of the adhesive. Fig. 2c and d present the evolution of the peel stress in the middle of the adhesive and close to the adhesive–substrate interface. Thus it can be noticed that the edge effects are mainly observed on the peel stress (yy component). Moreover with such single lap shear specimens, an increase in the overlap length leads to an increase in edge effects. Modification in the free edges of the adhesive (for instance, effects of spew and chamfer size [16] and influence of slots [18]) can reduce the influence of stress concentrations, but in order to
C
A
Fig. 1. Presentation of the single lap-shear joint (adhesive thickness of e¼ 0.2 mm for the mesh). (a) Geometry of the specimen; (b) close-up of the upper central part and (c) mesh (close-up view).
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12
12
10
10 l=100 mm
8
l=50 mm
shear stress (MPa)
shear stress (MPa)
l=100 mm l=75 mm
6 4
l=25 mm
8 l=50 mm 6 l=25 mm
4 2
2
0
0
-60
-40
l=75 mm
-20
0
20
40
60
-60
-40
-20
x (mm)
0
20
40
60
x (mm)
interface 50
von Mises equivalent stress (MPa)
peel stress (MPa)
40 30 20 l=100 mm l=50 mm
l=75 mm
10
l=25 mm
0 -60
-40
-20
l=100 mm
50
0
20
40
60
l=75 mm
40
l=50 mm
30 l=25 mm
20 10 0
-60
-40
-20
0
-10
-10
x (mm)
x (mm)
20
40
60
Fig. 2. Shear and peel stresses in the middle of the adhesive and close to the adhesive–substrate interface for an average shear stress of 1 MPa.
strongly limit the edge effects it seems also necessary to modify the geometries of the substrate close to the free edges of the adhesive [4]. For such specimens, the properties of the substrate (thickness, material, etc.) and the adhesive thickness have an influence on the stress distribution in the adhesive. Moreover, the computation assumptions can also have an influence on the numerical results. Plane stress conditions are representative of points close to the free edges of the single lap-shear specimen and plane strain conditions are nearly representative at points inside the specimen; a 3D computation can give better results, but the cost is higher. For such computations, plane strain and plane stress assumptions give nearly similar results. It can be noticed that for such studies, finite element analysis and results of asymptotic approaches give similar results [29]. Fig. 3 illustrates some influence of the stress concentrations on the behaviour of bonded assemblies. The thick adherend shear test (TAST) [30] is a widely used [31] and can be seen as an optimised single lap shear test, as thick substrates and a small overlap are used in order to limit the influence of stress singularities (Fig. 3a). Fig. 3c presents experimental results for the adhesive HuntsmanTM Araldites 420 A/B epoxy resin (Basel, Switzerland) and it can be seen that cracks exist at the beginning of the so-called non-linear behaviour of the adhesive. Dx represents the relative displacements of both ends of the adhesive in the x direction and the average shear stress is computed from the applied load and the adhesive section. It can be notice that the numerical analysis of such experimental results involving non-linear material behaviour and crack propagation can be complex. It has been numerically demonstrated and
experimentally verified that sharp beaks on the substrates, cleaned edges of the adhesive and using rigid supports can strongly limit edge effects; thus a modified TAST has been proposed which principle is presented in Fig. 3b [32], using the same adhesive cross-sectional area than the TAST. The small bonded samples (1) are mounted on a rigid support (2) and fixed with a fastening device (3). For this test, it can be seen that a homogeneous deformation of adhesive is observed (without crack propagation closed the free edges). Moreover the responses of the two tests are different; thus, limiting the influence of edge effects can lead to an increase of the transmitted load by the bonded assembly. Moreover, it has been shown that the plastic non-linear behaviour of the adhesive joint does not seem to limit the influence of the edge effects [4,27]; thus, the lower the equivalent stress (computed under linear elasticity), the lower the risk will be of first rupture near the edge of the joint.
3. Edge effects for cylindrical joints under tensile loadings This section presents a preliminary study, in order to analyse the influence of edge effects on cylindrical joints. Fig. 4 presents the geometry of the cylindrical bonded joint analysed in this section. As a prescribed tensile loading is employed, a usual axisymmetric model can be used in order to limit the computational cost of the various simulations. The geometry of the studied assembly is defined by the following parameters: r1, internal radius of the internal tube; r3, external radius of the external tube; r2, average radius of the joining; e, thickness of the adhesive (r2 and e give the external radius of the internal tube and the internal radius of the
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
(3)
709
(1)
(2)
x
Modified TAST
Average shear stress (MPa)
30
substrate
Modified TAST
adhesive
25 TAST
20
substrate
15
TAST 10
substrate 5
adhesive
0
0
0.1
0.2 0.3 Dx (mm)
0.4
0.5
substrate crack
crack
Fig. 3. Influence of edge effects, experimental results for TAST and modified TAST for an adhesive thickness of 0.4 mm. (a) TAST specimen (width: 25.4 mm); (b) Modified TAST principle and (c) experimental results and deformation of the adhesive.
Fig. 4. Presentation of the cylindrical joint studied with two local geometries of the adhesive (close-up). (a) Cylindrical joint; (b) straight edges and (c) cleaned edges.
external tube) and l, the overlap length. The first numerical results were obtained with r1¼10 mm, r2¼15 mm, r3¼ 20 mm, e¼0.2 mm and for different overlap lengths (l¼25, 50, 75 and 100 mm); moreover, it was assumed that the adhesive had straight free edges in a first stage. The behaviour of the different materials (aluminium substrate and adhesive) is assumed to be elastic and the same elastic parameters, which have been defined in the previous section, are used. For the various drawings, the average shear stress in the adhesive is normalised to 1, as in the previous section. Fig. 5a presents the evolution of the shear stress (component stress rz) in the middle of the adhesive with respect to the overlap length (i.e., along the z axis with the origin at the centre of the joint). For this example, there are two adhesive–substrate interfaces (here for the complete adhesive); thus, in order to simplify the presentation, only the evolution of the maximum values of the shear stress in the thickness of the adhesive with respect to the abscissa x. Fig. 5a and b present an idea of the stress evolution through the adhesive thickness as for this example, for a given
abscissa x, the minimum value is obtained in the middle of the joint and the maximum values is obtained in one of the adhesive– substrate interfaces (Fig. 5c and d). Fig. 5c and d describe, for an overlap length of 50 mm, at the two ends of the joint, trends in the shear stress in the adhesive on different lines with respect to the radius r, using an adimensional parameter l ¼ (r r2)/(2e). l ¼0 corresponds to the middle of the adhesive, l ¼ 1 corresponds to a line close to the interface between the adhesive and the internal tube and l ¼1 corresponds to a line close to the interface between the adhesive and the external tube. These results highlight the fact that quite large shear stress gradients are observed close to the adhesive–substrate interfaces. For this geometry, the stress state in the middle of the adhesive thickness (lA[ 0.5, 0.5]) is nearly constant for a given abscissa. Results of asymptotic analysis highlight the strong influence of the local geometry at the free ends of the adhesive, and can explain the different shear stress states close to the two interfaces. Fig. 6a and b shows the evolution of the minimum and maximum values of the radial stress in the thickness of the
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14
14
12
12
10
10
shear stress (MPa)
shear stress (MPa)
710
8 l=100 mm
6
l=75 mm l=50 mm
4
8
2
2
0 -40
-20
20
8
40
60
-60
-40
-20
λ=0
λ=1
λ=0.5
5
λ=1
4 λ=1
3 2
λ=−1
1
λ=0
-24.75 z (mm)
shear stress (MPa)
shear stress (MPa)
6
λ=−0.5
-25
λ=−1 λ=−0.5
3
λ=0.5
-24.5
60
λ=0
4
0 -25.25
40
7
λ=1
1
20
λ=−0.5 λ=0.5
5
0 z (mm)
8
λ=−1
λ=−1
6
2
l=25 mm
0 0
z (mm)
7
l=50 mm
4
l=25 mm
-60
l=100 mm l=75 mm
6
-24.25
0 24.25
λ=0.5
24.75
λ=−0.5 λ=0
25.25
z (mm)
Fig. 5. Shear stress through the thickness of the adhesive with straight edge for an average shear stress of 1 MPa. (a) Shear stress in the middle of the adhesive; (b) maximum values of shear stress; (c) shear stress through the adhesive thickness (zoom for an overlap of 50 mm) and (d) shear stress through the adhesive thickness (zoom for an overlap of 50 mm).
adhesive. These figures highlight the influence of edge effects on the peel stress. Moreover, quite similar results are observed for hoop and axial stresses. The equivalent von Mises stress also emphasizes this property as it takes into account the different stress components (Fig. 6c and d). Single lap shear specimens and cylindrical joints under axial loading are both associated with large edge effects. But the main difference is that single lap shear specimens generate tensile peel stresses and cylindrical joints generate compression peel stresses. This is an important point, since it has been shown that compression has a strong positive influence on bonded assembly strength [29]. Fig. 7 presents the evolution of the radial stress and the von Mises equivalent stress in the adhesive in the case of a cleaning radius r ¼0.75 e (defined on Fig. 4c). For this computation the mesh presented on Fig. 1 is deformed in the axial direction in order to respect the geometry of the adhesive free edges. The use of the so-called cleaned edges, associated with a cleaning of the adhesive before curing enables the influence of edge effects to be significantly reduced with respect to straight edges (Fig. 6) [27], a quite large reduction of the peel stress can be observed.
on the geometry of the bonded assembly, especially close to the free edges of the adhesive. Limiting the influence of edge effects can lead to an increase in the load transmitted by the assembly. Thus, it is necessary to develop accurate criteria, taking into account the real behaviour of the adhesive in order to optimise such assemblies. Various studies have shown that an accurate representation of the elastic yield domain of an adhesive in an assembly requires using a pressure-dependent constitutive model, i.e. a model taking the two stress invariants, hydrostatic stress and von Mises equivalent stress into account [7,8]. For this study, only an elastic behaviour of the adhesive was used. Starting from the experimental results obtained for the epoxy resin HuntsmanTM Araldites 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 (leaving out the viscous effects). This test allows us to determine the behaviour of an adhesive in an assembly under radial loadings (tensile/ compression-shear loadings) [33]. An exponential Drucker–Prager yield function provides good representation of the experimental data for the initial so-called elastic limit [34] F0 ¼ aðsvm Þb þ ph -pt0
4. Strength criteria of the adhesive in an assembly 4.1. Cylindrical joints with cleaned edges The numerical results presented in the previous section highlight the complex stress states in the adhesive which depend
ð1Þ
where svm is the von Mises equivalent stress and ph is the hydrostatic stress. a, b and pt0 are material parameters. The results of the identification are presented in Table 1 and the initial yield surface (F0 ¼0) is drawn in Fig. 8b. Fig. 8a gives the hydrostatic stress–von Mises equivalent stress diagram showing the stress in the adhesive for a thickness of
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
von Mises equivalent stress
15 l=75 mm 10 l=50 mm 5 l=25 mm 0 -40 -20 -5 0 20 40 -10 -15 -20 -25 -30 -35 -40 -45 z (mm)
radial stress
l=100 mm
-60
35
35
30
30
von Mises equivalent stress
-40
radial stress
-60
15 10 5 0 -20 -5 0 20 40 60 -10 l=25 mm l=50 mm -15 l=75 mm -20 l=100 mm -25 -30 -35 -40 -45 z (mm)
25 20 15 l=100 mm
10 l=50 mm
5
l=75 mm
-40
-20
20
l=100 mm
15
l=75 mm l=50 mm
10 5
l=25 mm
0 z (mm)
60
25
0
-60
711
l=25 mm
0
20
40
60
-60
-40
-20
0 z (mm)
20
40
60
Fig. 6. Minimum and maximum values of the stresses through the thickness of the adhesive with straight edges for an average shear stress of 1 MPa. (a) Minimum values of the radial stress; (b) maximum values of the radial stress; (c) minimum values of the von Mises stress and (d) maximum values of the von Mises stress.
0.1 mm, for cleaned edges of the adhesive and for the four overlap lengths. For an average shear stress equal to 1, the hydrostatic stress and the von Mises equivalent stress are computed for each integration point of each element of the finite element mesh. In the hydrostatic stress–von Mises equivalent stress diagram, only the envelope of those points is drawn here to simplify the presentation (all the points are inside the so-called envelope). Using the linear property of elastic problems, we can determine the maximal transmitted loading of the bonded assembly using the elastic limit, which can be defined with the maximal average shear stress in the assembly: taverage maxi. Results are presented in Table 2. It can be noted that the maximum average shear stress decreases with the overlap length; but the maximum transmitted load by the assembly, Pl, increases slowly with overlap length. Fig. 8b presents the hydrostatic stress–von Mises equivalent stress diagram with the stress in the adhesive for the maximum transmitted load and for the four overlap lengths. It can be noticed that the envelopes are similar for the different overlap lengths. Moreover, those results highlight the positive influence of the negative hydrostatic stress, associated with compression peel stress in the adhesive, on the maximum transmitted load by the assembly. Fig. 9 shows, for the maximum transmitted load of the assemblies, the evolution of the minimum and maximum values of the distance, in the hydrostatic stress–von Mises equivalent stress diagram, of a characteristic point to the elastic surface in the thickness of the adhesive with respect to the overlap length; for F0 ¼0, the elastic limit is reached for the considered point and the distance is equal to 0. This distance is computed using the
properties of radial loadings: it is determined in the direction of the stress path associated to an increase of the loading. Those results highlight the significant influence of edge effects and the similar stress state in the adhesive for the different overlap lengths.
4.2. Influence of geometric parameters Thin substrate thickness leads to a different behaviour of the assembly, in conjunction with the low radial rigidity of the tubes (Fig. 10a). A large evolution of the stress state can be seen from a thickness of 1–5 mm. Increasing the thickness of the tubes from 10 to 15 mm does not lead to significant variation of the stress distribution within the adhesive. An increase in the adhesive thickness is associated with an increase in the transmitted load (Fig. 10b) under elastic behaviour. Results of analytical methods are often compared with finite element results, with only very few elements (about four) in the adhesive thickness. They nearly approximate the behaviour of the adhesive in the middle of the joint [21]. Refined and coarse meshes of the adhesive can give nearly the same numerical results in the middle of the joint, but only refined meshes can give reliable information about the stress state in the thickness of the adhesive close to the free edges (Fig. 6a and b). Fig. 11 presents the hydrostatic stress–Mises equivalent stress diagram showing the stress state in the middle of the adhesive for an overlap length l¼50 mm, for an adhesive thickness of 0.2 mm, and for straight and cleaned edges of the adhesive. It can be seen that for the two
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
radial stress
-60
15 10 5 l=25 mm 0 -40 -20 -5 0 20 40 l=50 mm -10 l=75 mm -15 l=100 mm -20 -25 -30 -35 -40 -45
60
radial stress
712
35
35
30
30
von Mises equivalent stress
von Mises equivalent stress
z (mm)
15 10 l=100 mm l=50 mm 5 l=25 mm l=75 mm 0 -60 -40 -20 -5 0 20 40 -10 -15 -20 -25 -30 -35 -40 -45 z (mm)
25 20 15 l=100 mm
10 l=50 mm
l=75 mm
5
25 20 15
-40
-20
l=100 mm l=75 mm l=50 mm l=25 mm
10 5
l=25 mm
0 -60
60
0
0
20
40
60
-60
-40
z (mm)
-20
0
20
40
60
z (mm)
Fig. 7. Minimum and maximum values of the stresses through the thickness of the adhesive with cleaned edges for an average shear stress of 1 MPa. (a) Minimum values of the radial stress; (b) maximum values of the radial stress; (c) minimum values of the von Mises stress and (d) maximum values of the von Mises stress.
Table 1 Adhesive parameters for the initial yield surface (defined by the Eq. (1)). a (SI units)
b
pt0 (MPa)
1. E-6
4.87
31.7
l=75 mm l=50 mm
20
40
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
l=100 mm
15
10
l=25 mm
5
35 30 25 20 15 10 5
0 -15
-10
-5
0
Hydrostatic stress (MPa)
5
l=25mm l=50mm l=75mm l=100mm Elastic limit
-40
-30
0 -20 -10 0 10 20 hydrostatic stress (MPa)
30
40
Fig. 8. Stress in the adhesive with cleaned edges in the hydrostatic stress–von Mises equivalent stress diagram. (a) Average shear stress of 1 MPa; (b) maximum transmitted load.
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
713
Table 2 Definition of the maximum load transmitted by the assembly (Pl is the maximum transmitted load by an assembly with an overlap length l; and Pl ¼ 25 is associated with l ¼25 mm). Overlap length: l
25 mm
50 mm
75 mm
100 mm
taverage maxi
6.93 MPa 1.00
3.88 MPa 1.12
2.59 MPa 1.12
1.95 MPa 1.13
40
40
35
35
30
30
distance (MPa)
distance (MPa)
Pl/Pl ¼ 25
25 20 l=25 mm
l=75 mm
15
25 20 l=100 mm
10 5
5
l=100 mm
0
0 -60
-40
-20
l=75 mm
15
10 l=50 mm
l=25 mm
l=50 mm
0 z (mm)
20
40
-60
60
-40
-20
0 z (mm)
20
40
60
Fig. 9. (a) Minimum and (b) maximum values of the distance to the elastic surface of the adhesive with cleaned edges for the maximum transmitted load of each joint.
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
40
30 th=5 mm
20
th=10 mm th=15 mm
10
e=0.4 mm
15 e=0.8 mm
-20
-10
0
10
5
-15
20
10
e=1.6 mm
0 -30
20
e=0.2 mm
th=1 mm
hydrostatic stress (MPa)
0 -10 -5 0 hydrostatic stress (MPa)
5
Fig. 10. (a) Influence of the substrate thickness and (b) adhesive thickness with cleaned edges for an overlap length of l ¼100 mm and for an average shear stress of 1 MPa.
16
12
8
4
-12
-8
-4
hydrostatic stress (MPa)
16
12
8
4
0
0 -16
envelope middle line
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
envelope middle line
0
4
-16
-12
-8
-4
0
4
hydrostatic stress (MPa)
Fig. 11. Stress state in the middle of the adhesive for an overlap length l ¼ 50 mm, for an adhesive thickness of 0.2 mm and for an average shear stress of 1 MPa. (a) Straight edges and (b) cleaned edges.
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geometries of the adhesive free edges, the stress states in the middle of the joint are nearly similar. Edge effects nearly only affect the stress state close to the substrate–adhesive interface near the free edges. Thus, using a simplified method for dimensioning adhesive joints with singular points can
overestimate the maximum transmitted load by the bonded assembly. Figs. 12 and 13 present the influence of substrate thickness for an adhesive thickness of 0.2 mm and of adhesive thickness for a substrate thickness of 5 mm on the maximal average shear stress,
16
16 l=25mm l=50mm l=75mm l=100mm
12 10 8 6 4
14
τaverage maxi (MPa)
τaverage maxi (MPa)
14
l=25mm l=50mm l=75mm l=100mm
12 10 8 6 4 2
2
0
0 0
5
10
0
15
5 10 substrate thickness (mm)
substrate thickness (mm)
15
Fig. 12. Influence of the substrate thickness on the maximal average shear stress for an adhesive thickness of 0.2 mm (—: taking only the stresses in the middle line of the adhesive into account). (a) Straight edges and (b) cleaned edges.
l=25mm l=50mm l=75mm l=100mm
25
l=25mm l=50mm l=75mm l=100mm
20
τaverage maxi (MPa)
τaverage maxi (MPa)
20
25
15 10 5
15 10 5
0
0 0.0
0.4
0.8
1.2
1.6
2.0
2.4
adhesive thickness (mm)
2.8
3.2
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
adhesive thickness (mm)
Fig. 13. Influence of the adhesive thickness on the maximal average shear stress for a substrate thickness of 5 mm (—: taking only the stresses in the middle line of the adhesive into account). (a) Straight edges and (b) cleaned edges
Fig. 14. Presentation of the cylindrical joint studied with different local geometries (zoom). (a) Cylindrical joint; (b) straight substrates; (c) cleaned edges of the adhesive; (d) substrates with beaks and (e) substrates with inverse beaks.
J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 30 (2010) 706–719
l=25mm l=100mm elastic limit
von Mises equivalent stress (MPa)
40 35 30 25 20 15 10 5
In the case of large edge effects, the adhesive thickness has very little influence on the maximal transmitted load, even though it has quite a large influence on the stress state at the middle line of the adhesive (Fig. 13a). However, for elastic behaviour, limiting the influence of edge effects leads to an increase in the maximum transmitted load along with an increase in the adhesive thickness (Fig. 13b). It is important to note that for some planar joints, an increase in the adhesive thickness leads to an increase in the influence of edge effects [27].
-30
-20
-10
25 20 15 10 5 0
0
10
20
30
40
-40
-30
-10
l=100mm
35
elastic limit
30 25 20 15 10 5 -10
10
20
30
l=100mm elastic limit
25 20 15 10 5 -30
-20
-10
0
10
20
30
40
25 20 15 10 5
l=25mm
40 von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
l=25mm
hydrostatic stress (MPa)
30
l=100mm
35
elastic limit
30 25 20 15 10 5
0 -10
40
30
-40
40
l=25mm l=100mm elastic limit
35
-20
30
0 0
40
-30
20
35
hydrostatic stress (MPa)
-40
10
40
0 -20
0
hydrostatic stress (MPa)
von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
-20
l=25mm
40
-30
elastic limit
30
hydrostatic stress (MPa)
-40
l=100mm
35
0 -40
l=25mm
40 von Mises equivalent stress (MPa)
i.e., on the maximum transmitted load of the bonded assembly. Results, starting from the stress state in the adhesive, are presented for the four overlap joints studied, using large symbols. The results obtained using only the stress state in the middle of the adhesive are also presented with small symbols linked by thin lines. For the example studied, a substrate thickness greater than about 5 mm does not lead to an increase in the maximal transmitted load, but rather to a slight change in the stress state of the middle line of the adhesive (Fig. 12).
715
0 0
10
20
hydrostatic stress (MPa)
30
40
-40
-30
-20
-10
0
10
20
30
40
hydrostatic stress (MPa)
Fig. 15. Stress state in the adhesive in the hydrostatic stress–von Mises equivalent stress diagram for the maximum transmitted load and for an adhesive thickness of 0.2 mm. (a) Straight substrates – straight edges; (b) straight substrates – cleaned edges; (c) beaks - straight edges; (d) beaks - cleaned edges; (e) inverse beaks – straight edges and (f) inverse beaks – cleaned edges.
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beaks were defined with the parameters: h3¼2 mm and d ¼e. The radius R0 was defined to obtain a tangent circle to the adhesive– substrate interface in order to prevent angular points which can also lead to stress singularities. Those geometries (inverse beaks) have been used to analyse long term strength of adhesively bonded joints in sea water [36]. The machining of inverse beaks can be easier than the creating beaks on this kind of cylindrical parts. For inverse beaks the mesh presented on Fig. 1 is deformed in the radial direction in order to respect the adhesive geometry.
5. Proposal of geometries to limit the influence of edge effects 5.1. Geometrical parameters The numerical results presented in the previous section highlight the strong influence of stress concentrations on the maximal transmitted load of cylindrical joints. Thus, in this section, the geometries used in order to strongly limit the influence of edge effects in the case of a planar joint [27], are evaluated for cylindrical joints, taking into account some manufacturing constraints. Fig. 14 presents the different geometries used; cylindrical parts with shoulders are used to facilitate the machining, and these shoulders modify the local geometry near the adhesive free edges and can lead to a reduction in the influence of the edge effects [35]. Computations were made for an average radius of the adhesive joint of r2¼ 15 mm and an external radius of the external tube of r3¼30 mm. The shoulders were defined with the parameter dr¼1 mm. The cleaned edges were defined with the radius r ¼1.5e, e being the adhesive thickness. The beaks were defined with the following parameters: h1 ¼0.1 mm, h2 ¼1 mm and a ¼301, in order to strongly limit the influence of edge effects, while limiting the space associated with modifying the geometry. The beaks are used to analyse the behaviour of adhesive in assemblies with an Arcan device [27,34]. The so-called inverse
5.2. Influence of geometrical parameters for a given joint thickness Fig. 15 presents the hydrostatic stress–von Mises equivalent stress diagram showing the stress state of the adhesive for the maximum transmitted load for the three local geometries, for straight edges and cleaned edges of the adhesive, for the two overlap lengths (l¼25 mm and l¼100 mm), and for an adhesive thickness of 0.2 mm. It can be seen in this diagram that the overlap length has only slight influence on the dimensioning of the assembly for a given local geometry. It can be noticed that the overlap length has only a little influence of the stress state. But the local geometry of the substrate close to the adhesive and the cleaning of the free edge of the adhesive has a quite large influence on the stress state. Those results highlight the major influence of edge effects.
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Fig. 17. Influence of the adhesive thickness with cleaned edges and inverse beaks for an overlap length of l ¼50 mm and for an average shear stress of 1 MPa. (a) adhesive thickness of 0.2 mm and (b) adhesive thickness of 0.8 mm.
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Fig. 18. Influence of the adhesive thickness for different geometries of the substrates (—: taking only the stresses in the middle line of the adhesive into account). (a) Straight substrates – straight edges; (b) straight substrates – cleaned edges; (c) beaks – straight edges; (d) beaks – cleaned edges; (e) inverse beaks – straight edges and (f) inverse beaks – cleaned edges
Fig. 16 shows, in the case of beaks with cleaned edges, for the four overlap lengths, the evolution of the minimum and maximum values of the distance, in the hydrostatic stress–von Mises equivalent stress diagram, of a characteristic point to the elastic surface in the adhesive thickness with respect to the overlap length. For this example, it can be seen that the stress state in the thickness of the adhesive is nearly constant for a given position z in the overlap length. It is important to notice the difference between the results presented on Fig. 9 (associated with large influence of edge effects) and the results presented on Fig. 16 (associated with very low influence of edge effects).
5.3. Influence of joint thickness Fig. 17 presents the hydrostatic stress–Mises equivalent stress diagram with the stress state in the whole joint and in the middle of the adhesive for an overlap length of l¼50 mm for inverse beaks and cleaned edges of the adhesive. It can be seen that for an adhesive thickness of 0.2 mm, the stress state in the middle of the adhesive is close to the maximal stress state in the whole joint (Fig. 17a). For a given average shear stress in the joint, an increase in the adhesive thickness leads to a decrease in the maximal equivalent stress in the middle of the joint stress. Therefore, an increase in the maximal transmitted load can be obtained if the
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influence of edge effects is required to be low. It can be noted that for a joint thickness of 0.8 mm, the dimensioning points are not in the middle of the joint for this geometry (Fig. 17b). Fig. 18 presents the influence of adhesive thickness on the maximal average shear stress, i.e. on the maximum transmitted load of the bonded assembly. Results, starting from the stress
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state in the adhesive, are presented for the six different geometries and for the four overlaps studied using large symbols. The results obtained using only the stress state on the middle line of the adhesive, are also presented using small symbols linked by thin lines. Fig. 18a and b underline the positive influence of shoulders in the case of so-called straight substrates (Fig. 13). In the problem studied, using beaks or inverse beaks increases the possibilities for a bonded assembly with respect to straight substrates. It can be noted that the stress state in the middle of the joint is nearly independent of the local geometry of the assembly close to the free edges of the adhesive. The use of cleaned edges with beaks or with inverse beaks allows the adhesive loading close to the free edges to be limited, but better results are obtained with beaks. The same holds true for the use of straight edges with beaks or with inverse beaks. Moreover, quite similar results are obtained with straight substrates associated with adhesive cleaned edges and with inverse beaks associated with adhesive straight edges. An optimisation of the geometries of the beak and of the inverse beak can lead to some slight changes in the results. A coupling between the effects of the joint thickness and the effects of the local geometry of the assembly can be seen close to the free edges of the adhesive. To show this coupling’s effect, Fig. 19 presents some results associated with the shear stress distribution in the thickness adhesive in the case of beaks with straight and cleaned edges of the adhesive, for three joint thicknesses (0.1, 0.4 and 1.6 mm) and for an average shear stress of 1 MPa. On the middle line of the joint, the shear stress distributions are quite similar for straight and cleaned edges of the adhesive (thus only one drawing is presented, Fig. 19a), and increasing the adhesive thickness leads to a reduction in the maximum shear stress. In the case of cleaned edges, associated with a weak influence by edge effects, the maximum values of the shear stress through the joint thickness are quite similar to the shear stress on the middle line of the adhesive; but an increase in the adhesive thickness leads to a decrease in the maximum value of the shear stress in the whole joint (Fig. 19b). The use of adhesive straight edges, which are associated with a greater influence of edge effects, leads to an increase in the shear stress close to free edges of the adhesive, especially when the adhesive thickness increases (Fig. 19c). But it is important to remember that the influence of edge effects is greater for normal stresses than for shear stress (Figs. 4–6). Those results highlight the fact that for a given local geometry of the assembly, the choice of the adhesive thickness allows us to optimise the maximal transmitted load of the assembly under an elastic assumption for the different parts (Fig. 18). Strongly limiting the influence of edge effects allows a significant increase in the maximum transmitted load of the assembly, taking only an elastic behaviour of the adhesive into account. Moreover, it is important to note that such geometries can strongly limit the loading of the adhesive close to the adhesive–substrate interface near the free edges, which is often the weakest part of the assembly (Fig. 19).
0.4 6. Conclusions
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Fig. 19. Shear stress in the adhesive for an overlap length of l ¼25 mm and for an average shear stress of 1 MPa in the case of substrates with beaks. (a) Stress in the middle joint for cleaned and straight edges of the adhesive; (b) maximum stress through the adhesive thickness for cleaned edges of the adhesive and (c) maximum stress through the adhesive thickness for straight edges of the adhesive.
Adhesively bonded assemblies offer many advantages, but analysing the mechanical behaviour is made difficult, in particular, by the stress singularities due to edge effects. The first part of the paper presents differences between the stress states for usual single lap shear and cylindrical joints under tensile loadings. However, both types of bonded assemblies can be associated with large stress concentrations. The use of refined finite element computations highlights significant changes in the stress state
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within the thickness of the joint, especially close to the free edges of the adhesive. Therefore, it can be difficult to obtain precise dimensioning of adhesively bonded assemblies using simplified methods which generally cannot analyse large stress concentrations. Under elastic assumption of the materials, the use of a pressure-dependent elastic limit of the adhesive can provide an accurate analysis of the load transmitted by a bonded assembly. Such models, defined from the hydrostatic and von Mises equivalent stresses, accurately represent the behaviour of various polymers. Secondly, geometries which strongly limit the influence of edge effects have been proposed in order to increase the maximum transmitted load of the assembly, taking into account only an elastic behaviour of the adhesive. It is also important to note that these geometries can strongly limit the stress state within the adhesive close to the adhesive–substrate interface near free edges, which is often the weakest part of the assembly. A precise analysis of the stress state in the adhesive highlights the fact that there is a coupling of the effects of the joint thickness and the effects of the local geometry of the assembly, close to the free edges of the adhesive. Those numerical results provide some rules for manufacturing conditions to optimise bonded cylindrical joints under tensile loadings. Under the elastic assumption, the key point is to strongly limit the influence of edge effects in order to increase the transmitted load. An optimisation of the geometrical parameters of the two proposed geometries (beaks and inverse beaks) can improve the behaviour the assemblies. Experimental tests must be developed to analyse the real behaviour of the proposed geometries. In particular, for inverse beaks, the influence of increasing the adhesive thickness close to the free edges should be investigated. Numerical and experimental studies are also required to analyse the behaviour of the proposed geometries in industrial-type applications. More complex loadings, such as torsion-tensile or bending, must be taken into account, and the influence of the non-linear behaviour of the adhesive should also be analysed. References [1] Adams RD. Adhesive bonding: science, technology and applications. Bristol: Woodhead Publishing Ltd.; 2005. ¨ chsner A. Modeling of adhesive bonded joints. Berlin: [2] Da Silva LFM, O Springer; 2008. [3] Dean G, Crocker L, Read B, Wright L. Prediction of deformation and failure of rubber-toughened adhesive joints. Int J Adhes Adhes 2004;24:295–306. [4] Cognard JY, Cre´ac’hcadec R, Sohier L, Davies P. Analysis of the non linear behaviour of adhesives in bonded assemblies. Comparison of TAST and ARCAN tests. Int J Adhes Adhes 2008;28:393–404. [5] Raghava RS, Cadell RM. The macroscopic yield behaviour of polymers. J Mater Sci 1973;8:225–32. [6] Wang CH, Chalkley P. Plastic yielding of a film adhesive under multiaxial stresses. Int J Adhes Adhes 2000;20:155–64. [7] Mahnken R, Schlimmer M. Simulation of strength difference in elastoplasticity for adhesive materials. Int J Num Methods Eng 2005;63:1461–77. ¨ [8] Rolfres R, Volger M, Ernst G, Huhne C. Strength of textile composites in multiscale simulation, in trends in computational structures technology. Stirlingshire, Scoltland: Saxe-Coburg Publications; 2008. Chapter 7, 151–171. [9] Volkersen O. Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten. Luftfahrtforschung 1938;15:41–7.
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