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On the interfacial strength of bonded scarf joints
Engineering Fracture Mechanics 131 (2014) 142–149
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
On the interfacial strength of bonded scarf joints Zhixue Wu ⇑, Shuai Tian, Yajun Hua, Xiang Gu Mechanical Engineering College, Yangzhou University, Yangzhou 225127, PR China
a r t i c l e
i n f o
Article history: Received 17 August 2013 Received in revised form 21 July 2014 Accepted 22 July 2014 Available online 1 August 2014 Keywords: Stress singularity Interfacial strength Bonded joint Finite element analysis Interface corner
a b s t r a c t The interfacial strength of bi-material bonded scarf joints is evaluated by an integrated experimental and numerical investigation. The effects of the scarf angle, interface dimension and the stress singularity on the interfacial strength were examined using a series of aluminum/PMMA and aluminum/polycarbonate bi-material specimens under static bending moments. The experimental results showed that the maximum principal tensile stress at failure did not vary significantly with interface dimension. The measured maximum principal tensile stress of scarf joints at failure decreases as the scarf angle increases. We correlated fracture initiation with critical values of the stress intensities of the singular fields near the three-dimensional interface corner. The order of the singularity at threedimensional interface corners and edges was obtained numerically using common finite element methods. Finally, an empirical equation based on interface fracture mechanics was determined by curve fitting of the experimental results to predict the interfacial strength of bonded scarf joints. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Bonded components are widely used in the automotive, aerospace, and electronic industries. Measurement of bi-materials interface bonding strength is crucial for the design and application of structures with two or more materials since bi-material interface debonding is one of the major modes of failures for these structures. However, accurate determination of the interface bonding strength is not an easy task due to the discontinuity of material properties at the interface. There may exist a stress singularity at the edges of bi-material interface, where the debonding usually initiates. In addition, other critical factors may also affect the interface bonding strength and the reliability of adhesive joints, including bond thickness, interface dimension, joint geometry, etc. Steven and Stefan [1] investigated the interfacial strength of butt joints with different interface angles, and employed a simple point-stress criterion in combination with highly accurate finite element calculations to predict the strength of the interfaces. Paul and Martin [2] designed and fabricated a series of aluminum/epoxy bi-material specimens with interface edges and corners to investigate fracture initiation at three-dimensional bi-material interface corners. Their experimental results showed that the nominal stress at failure varied significantly with specimen size. They proposed an approach to characterizing fracture initiation at three-dimensional material interface corners using critical values of the stress intensities considering the effect of interface dimension. The effect of bond thickness upon the strength of adhesive joint has been investigated extensively by numerous researchers [3–6]. In general, the strength of adhesive joints increases as the bond thickness decreases. An important factor that is crucial in evaluating the joints strength is the stress concentration or the ⇑ Corresponding author. Tel.: +86 514 7851796; fax: +86 514 7887937. E-mail address: [email protected] (Z. Wu). http://dx.doi.org/10.1016/j.engfracmech.2014.07.026 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
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Z. Wu et al. / Engineering Fracture Mechanics 131 (2014) 142–149
Nomenclature 3-D K KC PC PMMA W
w k
rf
three-dimensional intensity of the singularity fracture toughness polycarbonate polymethylmethacrylate interface width scarf angle the order of singularity the maximum failure stress
stress singularity near the vicinity of interface corners and edges. Many researchers have investigated these features by adopting scarf joints with various scarf angles [1,4–11]. While some investigations have been carried out on the stress analysis and strength prediction of adhesive scarf joints, the relationship between the stress singularity and strength is not clear yet due to the complexity of stress field at various three-dimensional (3-D) bi-material interface corners and edge geometries. Some results showed that the order of stress singularity at a vertex of 3-D joints was different from those near the vicinity of interface edges of 3-D joints or at a vertex of two-dimensional joints [2,12–18]. In this work, the interfacial strength of bi-material bonded scarf joints is investigated by an integrated experimental and numerical method. The objectives are threefold. First is to examine the effect of interface dimension on the interfacial strength since it has been reported that the nominal stress at failure varied significantly with specimen size [2]. In order to validate this, we designed and fabricated a series of aluminum/PMMA and aluminum/polycarbonate bi-material butt adhesive joints with varying interface dimensions. The specimens were loaded in four-point flexure until fracture. Second is to correlate fracture initiation with critical values of the stress intensities of the singular fields near the 3-D interface corner using bonded scarf joints with different scarf angles. The order of the singularity at 3-D interface corners was determined from a rigorous asymptotic analysis of the 3-D interface corner stress state. Finally, an empirical equation based on interface fracture mechanics was determined by curve fitting of the numerical results to predict the interfacial strength of bonded scarf joints. 2. 3-D Finite element analysis for stress singularity Consider the bi-material bonded scarf joint shown schematically in Fig. 1. In general, the interface corner consists of intersecting faces (planes) and edges (lines) of arbitrary orientation. The joint consists of two elastic, isotropic, homogeneous and perfectly bonded materials, each material (denoted by A and B) occupies a part of the joint. The joint is loaded at remote boundaries by tractions or displacements. In the sense of linear elasticity, for most material combinations stress singularity may exist both at the interface corners and along the interface edges. We assume the existence of a separable solution in spherical coordinates so that the asymptotic stress and displacement fields near the three-dimensional interface corner can be expressed in each material as [2] in the case of a real stress singularity:
rMij ¼
X Mk r kk f ij ð/; h; kk Þ
ð1Þ
k¼1 Mk
where r is the radial distance from the point with singularity; M = A, B; f ij ð/; h; kk Þ is a function of angles /, h, representing the intensity of the stress field in material M; / and h are angles in the spherical coordinates shown in Fig. 2, describing the angular variations of the stress; kk is the kth eigenvalue. Let k1 < k2 < < kk < < 0, Eq. (2) can be re-arranged as follows: M2
M ij
r ¼
M1 rk1 f ij ð/; h; k1 Þ
1þr
k2 k1
f ij
M1
f ij
M3
þr
k3 k1
f ij
M1
f ij
!
Mk
þ þ r
kk k1
f ij
M1
f ij
ð2Þ
þ :
c d A
Interface corner
Interface edge
B
b
ψ a
Interface edge
Interface corner Fig. 1. General configuration of three-dimensional bi-material bonded joints.
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z
y
A
r
x
φ θ O
B
Fig. 2. Spherical coordinate system showing coordinates axes.
When the radial distance from the point with singularity, r, is sufficiently small, the second and subsequent terms are approximated to zero because of > kk k1 > > k3 k1 > k2 k1 > 0. In this case, Eq. (2) can be written as M1
logðrM ij Þ k1 logðrÞ þ logðf ij ð/; h; k1 ÞÞ:
ð3Þ
Obviously, the plots of stress distribution against r will become straight in a log–log scale, and the order of singularity k1 can be obtained from the estimation of the slope of the stress distribution near the point with singularity. In the following sections, k1, the lowest order of singularity, will be represented by k for the sake of simplicity. We use the models shown in Fig. 3(a) to obtain the k at the interface corner (point a or b in Fig. 1) and that at the interface edge (line ab and ad in Fig. 1), respectively. Common finite element (FEM) method is used to obtain the k at a point of interest. All finite element models are solid spherical sectors with the same radius R. Ten-node tetrahedral structural solid elements are used in all models. The average side length of the elements near the point of interest is nearly 2 106R in all finite element models, and the total number of elements is in the range of 18643–54742. An example of the FEM model for interface corner and its mesh division is given in Fig. 3(b). See reference [18] for the details and validation of the numerical method. The effects of variation of the scarf angle of w upon the k at interface corner and that at the interface edge are examined using two groups of material combinations: (i) polymethylmethacrylate (PMMA) and aluminum; (ii) polycarbonate (PC) and aluminum. The material data is listed in Table 1. The material B in Fig. 1 is always aluminum. The variation of k at interface corner and interface edge with scarf angle of w is shown in Fig. 4. The order of singularity k at interface edge is the value along ab, see Fig. 1. The k at interface edge ad is constant, which equals the value of w = 90°. In fact, the k at interface edge agrees with that obtained under plane strain assumption. It can be seen from Fig. 4 that, with a given scarf angle of w, the k at interface corner is always larger than that at interface edge, which means that stress singularity at interface corner is stronger than that at interface edge for the scarf joints. The k decreases monotonously as scarf angle of w reduces when w 6 90°. The stress singularity will disappear (k = 0) when the scarf angle of w is 50.6° (at interface corner) or 53.2° (at interface edge) for PMMA/aluminum scarf joint, and 47.9° (at interface corner) or 49.6° (at interface edge) for PC/aluminum scarf joint. The stress singularity is zero means that the effects of stress singularity on the joints strength will disappear. These results can help us to determine the smallest scarf angle for bi-material bonded scarf joints. 3. Experimental details and results A schematic of the test bonded scarf joint and the load configuration is shown in Fig. 5, where, the material B is always aluminum. The bonded structures are prisms with a square cross section and an overall length of 220 mm. These joints were tested under quasistatic conditions using the four-point flexure configuration shown in Fig. 6. Similar experiments were also
Fig. 3. An example of the model for interface corner and its mesh division.
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Z. Wu et al. / Engineering Fracture Mechanics 131 (2014) 142–149 Table 1 Material data. Aluminium
PMMA
PC
Elastic modulus, E (MPa) Poisson ratio, v
70.3 103 0.33
4.0 103 0.35
2.1 103 0.42
Order of stress singularity (-λ )
Property
0.5 PMMA/Al. PC/Al. At corner
0.4
At edge
0.3 0.2 0.1 0
0
30
60
90
120
150
180
ψ (degree) Fig. 4. Variation of k with scarf angle of w for PMMA/aluminum and PC/aluminum scarf joints.
conducted in references [1,2]. For the configuration here considered the stress field is singular at both lower and upper corners. Due to the oblique cut, the application of a pure tensile load would be difficult due to skew deformation. In addition, all the stress singularities at these corners would play significant roles in the initiation of material failure if a pure tensile load is applied. Instead, the joints were subjected to a pure bending moment in this investigation. Failure of the joints was expected to occur in tension at lower corners. Since subjected to compression, the stress singularities at upper corners were not expected to play significant roles in the initiation of material failure. Specimens were prepared as follows. Prior to bonding, bonding surfaces were uniformly polished with #2000 waterproof abrasive paper and afterward degreased with acetone. Each joint halves was joined by use of a thin layer of epoxy adhesive (Araldite 2011). The adhesive had two components. They were mixed before bonding. The adhesive was pasted at the interfaces of the adherends and the joints were pushed with some compressive forces. All specimens were cured at room temperature at least 36 h. After specimens were totally cured, the excess adhesive was removed by knife and portable grinder. The thickness of the adhesive layer was measured after experiments, which varied between 0.09 and 0.11 mm. Thus it could be considered that the experiments were carried out under the same thickness of bonding layer. To examine the effect of interface dimension on the interfacial strength, we designed square specimens with the scarf angle 90° (butt joint specimens) and with cross section dimension W W, where W = 28.0, 18.0, 14.0, and 8.0 mm. A set of scarf joints with a cross-section of 18.0 by 18.0 mm was used to examine the effect of the stress singularity on the interfacial strength at different scarf angles. The scarf angle is chosen as 90°, 75°, 60°, 53° and 90°, 75°, 60°, 49° for PMMA/aluminum scarf joints and PC/aluminum scarf joints, respectively. In all the tests, the load-displacement response was linear until brittle fracture occurred. The applied load deceased suddenly when fracture initiated. Figs. 7 and 8 show the measured maximum principal tensile stresses at failure plotted as a function of interface width W for PMMA/aluminum and PC/aluminum scarf joints, respectively. As can be seen from the figures, the scatter in results was somewhat substantial, but the trends were consistent. Note that the maximum principal tensile stresses occurs at the interface edge (line ab) and the interface corner (point a and b), see Fig. 1. In all the tests, it was found that all the brittle fracture was characterized by a crack that initiated at the interface on the tensile side of the specimen followed by unstable crack propagation along the PMMA/adhesive layer or PC/adhesive layer. It can be note that, despite the different material combinations used for the specimens, the average of maximum principal tensile stresses at failure does not differ much from each other. It can also be found that the measured maximum principal tensile stresses
P/2
P/2 90
A
B
ψ 180
Fig. 5. Four-point bending test set-up and specimen dimension.
W
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Fig. 6. Photo of four-point bending test set-up.
Maximum principal tensile stress at failure (MPa)
10 8 6 4
Experimental results The average
2 0
0
10
20
30
W (mm) Fig. 7. Maximum principal tensile stresses at failure versus interface width W for PMMA/aluminum butt joints.
Maximum principal tensile stress at failure (MPa)
10 8 6 4
Experimental results The average
2 0
0
10
20
30
W (mm) Fig. 8. Maximum principal tensile stresses at failure versus interface width W for PC/aluminum butt joints.
at failure do dot depend on the interface width W. These results differ from those in reference [2], where significant dependence of the nominal stress at failure on specimen size was reported. In reference [2], the aluminum/epoxy specimens were fabricated by casting epoxy to an aluminum block, i.e. no adhesive agent was used for the specimens. More work is required to reveal the effect of interface dimension on interfacial strength since little investigation on this topic can be found in literature. The measured maximum principal tensile stresses at failure plotted as a function of scarf angle w for PMMA/aluminum and PC/aluminum scarf joints are shown in Fig. 9. It can be found for all joint configurations that, in despite of the scatter of data, the maximum principal tensile stresses at failure decreases as the scarf angle increases. Similar results can also be found in reference [1,4,6]. A larger scarf angle was apparently related to a stronger singular stress field in the range of 45° < w 6 90°, see Fig. 4. This means that a lower loading is required to induce a critical failure stress in the joint.
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4. Strength prediction of scarf joints Various methods exist in the literature for predicting the strength of bonded scarf joints, among which the following two methods are the most widely used: (1) the maximum principal stress criterion [1,5]; (2) linear elastic fracture mechanics where failure is assumed to occur at a critical value of mode-dependent interfacial fracture toughness [2–4,6]. In general, a stress singularity of the form K kr may develop at the interface corner. Here r is the distance from the interface corner, k is the order of the singularity and K is the intensity of the singularity, which is thereafter referred to as the free-edge intensity factor. Based on this, the intensity of the singularity K takes the form
K ¼ r0 C k Q ða; b; kÞ;
ð4Þ
where Q is a non-dimensional constant function of the material elastic parameters (a, b) and the order of the singularity k, r0 is the remotely applied uniaxial load, C is a characteristic length scale, which is chosen as the adhesive layer thickness in [3– 4,6] or as the interface dimension in [2]. The intensity K may be used as a valid parameter for characterizing the fracture strength of bonded scarf joints, however, the use of this fracture criterion seems not to be convenient enough from the point of engineering application. First, the units of K, which are of the form (MPa) (mm)k, do not allow a direct comparison of the magnitude of K between two different scarf joints unless both joints have the same magnitude of k. Second, besides on combination of materials, the Q also depends on the singularity k, and numerical evaluation of its value is not easy. In all the experiments, it was found that failure occurred at the material interface in the immediate vicinity of the lower bi-material corner. It is reasonable to assume that the failure of these joints, with cracks originating at the material interface, would be related to the order of stress singularity at the corner. For a given combination of materials and a constant C value, it can be found that Eqn.(4) shows the relationship between the failure stresses and the singularity k when fracture occurs if fracture toughness KC is constant. Here, we propose a simple empirical equation to establish the relationship between the maximum failure stresses and the singularity k to predict the interfacial strength of bonded scarf joints, which takes the form
rf C k ¼ K C ;
ð5Þ
where rf is the maximum failure stress, k is the order of stress singularity at the corner, C and KC are constant determined by the combination of materials. This equation can be rewritten in the form
logðrf Þ ¼ logðK C Þ k logðCÞ;
ð6Þ
A plot of the average failure stress log (rf) versus the order of stress singularity k at corner is shown in Fig. 10 for the PMMA/aluminum and PC/aluminum scarf joints. It can be seen that the failure stress decreases with the increase of the order of stress singularity k, i.e. decreasing the stress singularity at corner can increase effectively the strength of adhesive joints. The constants C and KC can be obtained by curve fitting of the experimental data in Fig. 10 and the values of C and KC are 6.5989 and 11.2008, respectively, for PMMA/aluminum scarf joints, and C = 10.1494 and KC = 15.0326 for PC/aluminum scarf joints. Fig. 11 shows the predicted and measured failure stress rf versus the order of stress singularity k at corner for the PMMA/aluminum and PC/aluminum scarf joints. It can be seen that the prediction is in good agreement with the measured data for all investigated material combinations. The results also reveal that reducing the order of stress singularity k can increase the interface bonding strength. Our previous results [18] showed that the order of stress singularity at interface corner can be reduced to the level of the edge singularity after smoothing the intersection of interface edges by generating a circular-arc fillet at the intersection of two side free surfaces. This means that the order of stress singularity k can be reduced, thus the interface bonding strength can be increased according to the results shown in Fig. 4 if we just generate a circular-arc fillet at the intersection of two side free surfaces for the scarf joints. We have designed scarf joints with circular-arc fillet and experiments are under way.
Maximum principal tensile stress at failure (MPa)
20 Exp. Average PC/aluminum PMMA/aluminum
15
10
5
0 40
50
60
70
80
90
100
scarf angle ψ (degree) Fig. 9. Maximum principal tensile stresses at failure versus scarf angle w for PMMA/aluminum and PC/aluminum scarf joints.
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Z. Wu et al. / Engineering Fracture Mechanics 131 (2014) 142–149
Order of stress singularity -λ
0.5
PC/aluminum 0.4
PMMA/aluminum
0.3 0.2 0.1 0 0.7
0.8
0.9
1
1.1
1.2
The average failure stress log (σ f )
Maximum failure stress σ f (MPa)
Fig. 10. The average failure stress log (rf) vs the order of stress singularity k at corner. The scarf angle is 90° (0.3306), 75° (0.2626), 60° (0.1331), 53° (0.0407) and 90° (0.3977), 75° (0.3197), 60° (0.1854), 49° (0.0203) for PMMA/aluminum and PC/aluminum scarf joints, respectively. The value in brackets is the corresponding order of stress singularity k.
18 16
PC/aluminum PMMA/aluminum
14
Exp.
Eqn.(5)
12 10 8 6 4 2 0
0
0.1
0.2
0.3
0.4
Order of stress singularity (-λ) Fig. 11. The predicted and measured failure stress rf versus the order of stress singularity k at corner for the PMMA/aluminum and PC/aluminum scarf joints.
5. Conclusions In this work, we have investigated experimentally and numerically the effects of the scarf angle, interface dimension and the stress singularity on the interfacial strength using a series of aluminum/PMMA and aluminum/PC bi-material scarf joints. Despite the scatter of data, experimental results show that the average of maximum principal tensile stresses at failure does not vary significantly with interface dimension. Similar to the reports in [1,4–6], the measured maximum principal tensile stresses of scarf joints at failure decreases as the scarf angle w increases when w 6 90°. We obtained numerically the order of the singularity at three-dimensional interface corners and edges and correlated fracture initiation with critical values of the stress intensities of the singular fields near the three-dimensional interface corner. Here, the criterion of fracture initiation is analogous to the interface corner failure criterion based on linear elastic fracture mechanics. Based on an analysis of the intensity of the singularity, an empirical equation was determined by curve fitting of the experimental data to predict the interfacial strength of bonded scarf joints. The prediction is in good agreement with the measured data. Acknowledgements The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51075354). References [1] [2] [3] [4] [5]
Steven R-A, Stefan H. Strength prediction of beams with bi-material butt-joints. Engng Fract Mech 2003;70:1491–507. Paul EWL, Martin LD. Fracture initiation at three-dimensional bimaterial interface corners. J Mech Phys Solids 2001;49:609–43. Akisanya AR, Meng CS. Initiation of fracture at the interface corner of bi-material joints. J Mech Phys Solids 2003;51:27–46. Qian Z, Akisanya AR. An experimental investigation of failure initiation in bonded joints. Acta Mater 1998;46(14):4895–904. Nakano N, Sekiguchi Y, Sawa T. FEM stress analysis and strength prediction of scarf adhesive joints under static bending moments. Int J Adhes Adhes 2013;44:166–73.
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[6] Afendi M, Teramoto T, Bakri HB. Strength prediction of epoxy adhesively bonded scarf joints of dissimilar adherends. Int J Adhes Adhes 2011;41:402–11. [7] Qian Z, Akisanya AR. An investigation of the stress singularity near the free edge of scarf joints. Eur J Mech A/Solids 1999;46(14):443–63. [8] Chaudhuri RA, Chiu SJ. Three-dimensional asymptotic stress field in the vicinity of an adhesively bonded scarf joint interface. Compos Struct 2009;89:475–83. [9] Chiu SJ, Chaudhuri RA. A three-dimensional eigenfunction expansion approach for singular stress field near an adhesively-bonded scarf joint interface in a rigidly-encased plate. Engng Fract Mech 2011;78:2220–34. [10] Afendi M, Majid AMS, Daud R, Rahman AA, Teramoto T. Strength prediction and reliability of brittle epoxy adhesively bonded dissimilar joint. Int J Adhes Adhes 2013;45:21–31. [11] Kimiaeifar A, Toft H, Lund E, Thomsen OT, Sorensen JD. Reliability analysis of adhesive bonded scarf joints. Eng Struct 2012;35:281–7. [12] Picu CR, Gupta V. Three-dimensional stress singularities at the tip of a grain triple junction line intersecting the free surface. J. Mech. Phys. Solids 1997;45:1495–520. [13] Koguchi H. Stress singularity analysis in three-dimensional bonded structure. Int J Solids Struct 1997;34:461–80. [14] Koguchi H, Muramoto T. The order of stress singularity near the vertex in three-dimensional joints. Int J Solids Struct 2000;37:4737–62. [15] Koguchi H, Costa JA. Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface. Int J Solids Struct 2010;47:3131–40. [16] Xie M, Chaudhuri RA. Three-dimensional asymptotic stress field at the front of a bimaterial wedge of symmetric geometry under antiplane shear loading. Compos Struct 2001;54:509–14. [17] Lee Y, Im S. On the computation of the near-tip stress intensities for three-dimensional wedges via two-state M-integral. J Mech Phys Solids 2003;51:825–50. [18] Wu ZX. A method for eliminating the effect of 3-D bimaterial interface corner geometries on stress singularity. Engng Fract Mech 2006;73:953–62.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
On the interfacial strength of bonded scarf joints Zhixue Wu ⇑, Shuai Tian, Yajun Hua, Xiang Gu Mechanical Engineering College, Yangzhou University, Yangzhou 225127, PR China
a r t i c l e
i n f o
Article history: Received 17 August 2013 Received in revised form 21 July 2014 Accepted 22 July 2014 Available online 1 August 2014 Keywords: Stress singularity Interfacial strength Bonded joint Finite element analysis Interface corner
a b s t r a c t The interfacial strength of bi-material bonded scarf joints is evaluated by an integrated experimental and numerical investigation. The effects of the scarf angle, interface dimension and the stress singularity on the interfacial strength were examined using a series of aluminum/PMMA and aluminum/polycarbonate bi-material specimens under static bending moments. The experimental results showed that the maximum principal tensile stress at failure did not vary significantly with interface dimension. The measured maximum principal tensile stress of scarf joints at failure decreases as the scarf angle increases. We correlated fracture initiation with critical values of the stress intensities of the singular fields near the three-dimensional interface corner. The order of the singularity at threedimensional interface corners and edges was obtained numerically using common finite element methods. Finally, an empirical equation based on interface fracture mechanics was determined by curve fitting of the experimental results to predict the interfacial strength of bonded scarf joints. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Bonded components are widely used in the automotive, aerospace, and electronic industries. Measurement of bi-materials interface bonding strength is crucial for the design and application of structures with two or more materials since bi-material interface debonding is one of the major modes of failures for these structures. However, accurate determination of the interface bonding strength is not an easy task due to the discontinuity of material properties at the interface. There may exist a stress singularity at the edges of bi-material interface, where the debonding usually initiates. In addition, other critical factors may also affect the interface bonding strength and the reliability of adhesive joints, including bond thickness, interface dimension, joint geometry, etc. Steven and Stefan [1] investigated the interfacial strength of butt joints with different interface angles, and employed a simple point-stress criterion in combination with highly accurate finite element calculations to predict the strength of the interfaces. Paul and Martin [2] designed and fabricated a series of aluminum/epoxy bi-material specimens with interface edges and corners to investigate fracture initiation at three-dimensional bi-material interface corners. Their experimental results showed that the nominal stress at failure varied significantly with specimen size. They proposed an approach to characterizing fracture initiation at three-dimensional material interface corners using critical values of the stress intensities considering the effect of interface dimension. The effect of bond thickness upon the strength of adhesive joint has been investigated extensively by numerous researchers [3–6]. In general, the strength of adhesive joints increases as the bond thickness decreases. An important factor that is crucial in evaluating the joints strength is the stress concentration or the ⇑ Corresponding author. Tel.: +86 514 7851796; fax: +86 514 7887937. E-mail address: [email protected] (Z. Wu). http://dx.doi.org/10.1016/j.engfracmech.2014.07.026 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
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Nomenclature 3-D K KC PC PMMA W
w k
rf
three-dimensional intensity of the singularity fracture toughness polycarbonate polymethylmethacrylate interface width scarf angle the order of singularity the maximum failure stress
stress singularity near the vicinity of interface corners and edges. Many researchers have investigated these features by adopting scarf joints with various scarf angles [1,4–11]. While some investigations have been carried out on the stress analysis and strength prediction of adhesive scarf joints, the relationship between the stress singularity and strength is not clear yet due to the complexity of stress field at various three-dimensional (3-D) bi-material interface corners and edge geometries. Some results showed that the order of stress singularity at a vertex of 3-D joints was different from those near the vicinity of interface edges of 3-D joints or at a vertex of two-dimensional joints [2,12–18]. In this work, the interfacial strength of bi-material bonded scarf joints is investigated by an integrated experimental and numerical method. The objectives are threefold. First is to examine the effect of interface dimension on the interfacial strength since it has been reported that the nominal stress at failure varied significantly with specimen size [2]. In order to validate this, we designed and fabricated a series of aluminum/PMMA and aluminum/polycarbonate bi-material butt adhesive joints with varying interface dimensions. The specimens were loaded in four-point flexure until fracture. Second is to correlate fracture initiation with critical values of the stress intensities of the singular fields near the 3-D interface corner using bonded scarf joints with different scarf angles. The order of the singularity at 3-D interface corners was determined from a rigorous asymptotic analysis of the 3-D interface corner stress state. Finally, an empirical equation based on interface fracture mechanics was determined by curve fitting of the numerical results to predict the interfacial strength of bonded scarf joints. 2. 3-D Finite element analysis for stress singularity Consider the bi-material bonded scarf joint shown schematically in Fig. 1. In general, the interface corner consists of intersecting faces (planes) and edges (lines) of arbitrary orientation. The joint consists of two elastic, isotropic, homogeneous and perfectly bonded materials, each material (denoted by A and B) occupies a part of the joint. The joint is loaded at remote boundaries by tractions or displacements. In the sense of linear elasticity, for most material combinations stress singularity may exist both at the interface corners and along the interface edges. We assume the existence of a separable solution in spherical coordinates so that the asymptotic stress and displacement fields near the three-dimensional interface corner can be expressed in each material as [2] in the case of a real stress singularity:
rMij ¼
X Mk r kk f ij ð/; h; kk Þ
ð1Þ
k¼1 Mk
where r is the radial distance from the point with singularity; M = A, B; f ij ð/; h; kk Þ is a function of angles /, h, representing the intensity of the stress field in material M; / and h are angles in the spherical coordinates shown in Fig. 2, describing the angular variations of the stress; kk is the kth eigenvalue. Let k1 < k2 < < kk < < 0, Eq. (2) can be re-arranged as follows: M2
M ij
r ¼
M1 rk1 f ij ð/; h; k1 Þ
1þr
k2 k1
f ij
M1
f ij
M3
þr
k3 k1
f ij
M1
f ij
!
Mk
þ þ r
kk k1
f ij
M1
f ij
ð2Þ
þ :
c d A
Interface corner
Interface edge
B
b
ψ a
Interface edge
Interface corner Fig. 1. General configuration of three-dimensional bi-material bonded joints.
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Z. Wu et al. / Engineering Fracture Mechanics 131 (2014) 142–149
z
y
A
r
x
φ θ O
B
Fig. 2. Spherical coordinate system showing coordinates axes.
When the radial distance from the point with singularity, r, is sufficiently small, the second and subsequent terms are approximated to zero because of > kk k1 > > k3 k1 > k2 k1 > 0. In this case, Eq. (2) can be written as M1
logðrM ij Þ k1 logðrÞ þ logðf ij ð/; h; k1 ÞÞ:
ð3Þ
Obviously, the plots of stress distribution against r will become straight in a log–log scale, and the order of singularity k1 can be obtained from the estimation of the slope of the stress distribution near the point with singularity. In the following sections, k1, the lowest order of singularity, will be represented by k for the sake of simplicity. We use the models shown in Fig. 3(a) to obtain the k at the interface corner (point a or b in Fig. 1) and that at the interface edge (line ab and ad in Fig. 1), respectively. Common finite element (FEM) method is used to obtain the k at a point of interest. All finite element models are solid spherical sectors with the same radius R. Ten-node tetrahedral structural solid elements are used in all models. The average side length of the elements near the point of interest is nearly 2 106R in all finite element models, and the total number of elements is in the range of 18643–54742. An example of the FEM model for interface corner and its mesh division is given in Fig. 3(b). See reference [18] for the details and validation of the numerical method. The effects of variation of the scarf angle of w upon the k at interface corner and that at the interface edge are examined using two groups of material combinations: (i) polymethylmethacrylate (PMMA) and aluminum; (ii) polycarbonate (PC) and aluminum. The material data is listed in Table 1. The material B in Fig. 1 is always aluminum. The variation of k at interface corner and interface edge with scarf angle of w is shown in Fig. 4. The order of singularity k at interface edge is the value along ab, see Fig. 1. The k at interface edge ad is constant, which equals the value of w = 90°. In fact, the k at interface edge agrees with that obtained under plane strain assumption. It can be seen from Fig. 4 that, with a given scarf angle of w, the k at interface corner is always larger than that at interface edge, which means that stress singularity at interface corner is stronger than that at interface edge for the scarf joints. The k decreases monotonously as scarf angle of w reduces when w 6 90°. The stress singularity will disappear (k = 0) when the scarf angle of w is 50.6° (at interface corner) or 53.2° (at interface edge) for PMMA/aluminum scarf joint, and 47.9° (at interface corner) or 49.6° (at interface edge) for PC/aluminum scarf joint. The stress singularity is zero means that the effects of stress singularity on the joints strength will disappear. These results can help us to determine the smallest scarf angle for bi-material bonded scarf joints. 3. Experimental details and results A schematic of the test bonded scarf joint and the load configuration is shown in Fig. 5, where, the material B is always aluminum. The bonded structures are prisms with a square cross section and an overall length of 220 mm. These joints were tested under quasistatic conditions using the four-point flexure configuration shown in Fig. 6. Similar experiments were also
Fig. 3. An example of the model for interface corner and its mesh division.
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Z. Wu et al. / Engineering Fracture Mechanics 131 (2014) 142–149 Table 1 Material data. Aluminium
PMMA
PC
Elastic modulus, E (MPa) Poisson ratio, v
70.3 103 0.33
4.0 103 0.35
2.1 103 0.42
Order of stress singularity (-λ )
Property
0.5 PMMA/Al. PC/Al. At corner
0.4
At edge
0.3 0.2 0.1 0
0
30
60
90
120
150
180
ψ (degree) Fig. 4. Variation of k with scarf angle of w for PMMA/aluminum and PC/aluminum scarf joints.
conducted in references [1,2]. For the configuration here considered the stress field is singular at both lower and upper corners. Due to the oblique cut, the application of a pure tensile load would be difficult due to skew deformation. In addition, all the stress singularities at these corners would play significant roles in the initiation of material failure if a pure tensile load is applied. Instead, the joints were subjected to a pure bending moment in this investigation. Failure of the joints was expected to occur in tension at lower corners. Since subjected to compression, the stress singularities at upper corners were not expected to play significant roles in the initiation of material failure. Specimens were prepared as follows. Prior to bonding, bonding surfaces were uniformly polished with #2000 waterproof abrasive paper and afterward degreased with acetone. Each joint halves was joined by use of a thin layer of epoxy adhesive (Araldite 2011). The adhesive had two components. They were mixed before bonding. The adhesive was pasted at the interfaces of the adherends and the joints were pushed with some compressive forces. All specimens were cured at room temperature at least 36 h. After specimens were totally cured, the excess adhesive was removed by knife and portable grinder. The thickness of the adhesive layer was measured after experiments, which varied between 0.09 and 0.11 mm. Thus it could be considered that the experiments were carried out under the same thickness of bonding layer. To examine the effect of interface dimension on the interfacial strength, we designed square specimens with the scarf angle 90° (butt joint specimens) and with cross section dimension W W, where W = 28.0, 18.0, 14.0, and 8.0 mm. A set of scarf joints with a cross-section of 18.0 by 18.0 mm was used to examine the effect of the stress singularity on the interfacial strength at different scarf angles. The scarf angle is chosen as 90°, 75°, 60°, 53° and 90°, 75°, 60°, 49° for PMMA/aluminum scarf joints and PC/aluminum scarf joints, respectively. In all the tests, the load-displacement response was linear until brittle fracture occurred. The applied load deceased suddenly when fracture initiated. Figs. 7 and 8 show the measured maximum principal tensile stresses at failure plotted as a function of interface width W for PMMA/aluminum and PC/aluminum scarf joints, respectively. As can be seen from the figures, the scatter in results was somewhat substantial, but the trends were consistent. Note that the maximum principal tensile stresses occurs at the interface edge (line ab) and the interface corner (point a and b), see Fig. 1. In all the tests, it was found that all the brittle fracture was characterized by a crack that initiated at the interface on the tensile side of the specimen followed by unstable crack propagation along the PMMA/adhesive layer or PC/adhesive layer. It can be note that, despite the different material combinations used for the specimens, the average of maximum principal tensile stresses at failure does not differ much from each other. It can also be found that the measured maximum principal tensile stresses
P/2
P/2 90
A
B
ψ 180
Fig. 5. Four-point bending test set-up and specimen dimension.
W
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Fig. 6. Photo of four-point bending test set-up.
Maximum principal tensile stress at failure (MPa)
10 8 6 4
Experimental results The average
2 0
0
10
20
30
W (mm) Fig. 7. Maximum principal tensile stresses at failure versus interface width W for PMMA/aluminum butt joints.
Maximum principal tensile stress at failure (MPa)
10 8 6 4
Experimental results The average
2 0
0
10
20
30
W (mm) Fig. 8. Maximum principal tensile stresses at failure versus interface width W for PC/aluminum butt joints.
at failure do dot depend on the interface width W. These results differ from those in reference [2], where significant dependence of the nominal stress at failure on specimen size was reported. In reference [2], the aluminum/epoxy specimens were fabricated by casting epoxy to an aluminum block, i.e. no adhesive agent was used for the specimens. More work is required to reveal the effect of interface dimension on interfacial strength since little investigation on this topic can be found in literature. The measured maximum principal tensile stresses at failure plotted as a function of scarf angle w for PMMA/aluminum and PC/aluminum scarf joints are shown in Fig. 9. It can be found for all joint configurations that, in despite of the scatter of data, the maximum principal tensile stresses at failure decreases as the scarf angle increases. Similar results can also be found in reference [1,4,6]. A larger scarf angle was apparently related to a stronger singular stress field in the range of 45° < w 6 90°, see Fig. 4. This means that a lower loading is required to induce a critical failure stress in the joint.
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4. Strength prediction of scarf joints Various methods exist in the literature for predicting the strength of bonded scarf joints, among which the following two methods are the most widely used: (1) the maximum principal stress criterion [1,5]; (2) linear elastic fracture mechanics where failure is assumed to occur at a critical value of mode-dependent interfacial fracture toughness [2–4,6]. In general, a stress singularity of the form K kr may develop at the interface corner. Here r is the distance from the interface corner, k is the order of the singularity and K is the intensity of the singularity, which is thereafter referred to as the free-edge intensity factor. Based on this, the intensity of the singularity K takes the form
K ¼ r0 C k Q ða; b; kÞ;
ð4Þ
where Q is a non-dimensional constant function of the material elastic parameters (a, b) and the order of the singularity k, r0 is the remotely applied uniaxial load, C is a characteristic length scale, which is chosen as the adhesive layer thickness in [3– 4,6] or as the interface dimension in [2]. The intensity K may be used as a valid parameter for characterizing the fracture strength of bonded scarf joints, however, the use of this fracture criterion seems not to be convenient enough from the point of engineering application. First, the units of K, which are of the form (MPa) (mm)k, do not allow a direct comparison of the magnitude of K between two different scarf joints unless both joints have the same magnitude of k. Second, besides on combination of materials, the Q also depends on the singularity k, and numerical evaluation of its value is not easy. In all the experiments, it was found that failure occurred at the material interface in the immediate vicinity of the lower bi-material corner. It is reasonable to assume that the failure of these joints, with cracks originating at the material interface, would be related to the order of stress singularity at the corner. For a given combination of materials and a constant C value, it can be found that Eqn.(4) shows the relationship between the failure stresses and the singularity k when fracture occurs if fracture toughness KC is constant. Here, we propose a simple empirical equation to establish the relationship between the maximum failure stresses and the singularity k to predict the interfacial strength of bonded scarf joints, which takes the form
rf C k ¼ K C ;
ð5Þ
where rf is the maximum failure stress, k is the order of stress singularity at the corner, C and KC are constant determined by the combination of materials. This equation can be rewritten in the form
logðrf Þ ¼ logðK C Þ k logðCÞ;
ð6Þ
A plot of the average failure stress log (rf) versus the order of stress singularity k at corner is shown in Fig. 10 for the PMMA/aluminum and PC/aluminum scarf joints. It can be seen that the failure stress decreases with the increase of the order of stress singularity k, i.e. decreasing the stress singularity at corner can increase effectively the strength of adhesive joints. The constants C and KC can be obtained by curve fitting of the experimental data in Fig. 10 and the values of C and KC are 6.5989 and 11.2008, respectively, for PMMA/aluminum scarf joints, and C = 10.1494 and KC = 15.0326 for PC/aluminum scarf joints. Fig. 11 shows the predicted and measured failure stress rf versus the order of stress singularity k at corner for the PMMA/aluminum and PC/aluminum scarf joints. It can be seen that the prediction is in good agreement with the measured data for all investigated material combinations. The results also reveal that reducing the order of stress singularity k can increase the interface bonding strength. Our previous results [18] showed that the order of stress singularity at interface corner can be reduced to the level of the edge singularity after smoothing the intersection of interface edges by generating a circular-arc fillet at the intersection of two side free surfaces. This means that the order of stress singularity k can be reduced, thus the interface bonding strength can be increased according to the results shown in Fig. 4 if we just generate a circular-arc fillet at the intersection of two side free surfaces for the scarf joints. We have designed scarf joints with circular-arc fillet and experiments are under way.
Maximum principal tensile stress at failure (MPa)
20 Exp. Average PC/aluminum PMMA/aluminum
15
10
5
0 40
50
60
70
80
90
100
scarf angle ψ (degree) Fig. 9. Maximum principal tensile stresses at failure versus scarf angle w for PMMA/aluminum and PC/aluminum scarf joints.
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Order of stress singularity -λ
0.5
PC/aluminum 0.4
PMMA/aluminum
0.3 0.2 0.1 0 0.7
0.8
0.9
1
1.1
1.2
The average failure stress log (σ f )
Maximum failure stress σ f (MPa)
Fig. 10. The average failure stress log (rf) vs the order of stress singularity k at corner. The scarf angle is 90° (0.3306), 75° (0.2626), 60° (0.1331), 53° (0.0407) and 90° (0.3977), 75° (0.3197), 60° (0.1854), 49° (0.0203) for PMMA/aluminum and PC/aluminum scarf joints, respectively. The value in brackets is the corresponding order of stress singularity k.
18 16
PC/aluminum PMMA/aluminum
14
Exp.
Eqn.(5)
12 10 8 6 4 2 0
0
0.1
0.2
0.3
0.4
Order of stress singularity (-λ) Fig. 11. The predicted and measured failure stress rf versus the order of stress singularity k at corner for the PMMA/aluminum and PC/aluminum scarf joints.
5. Conclusions In this work, we have investigated experimentally and numerically the effects of the scarf angle, interface dimension and the stress singularity on the interfacial strength using a series of aluminum/PMMA and aluminum/PC bi-material scarf joints. Despite the scatter of data, experimental results show that the average of maximum principal tensile stresses at failure does not vary significantly with interface dimension. Similar to the reports in [1,4–6], the measured maximum principal tensile stresses of scarf joints at failure decreases as the scarf angle w increases when w 6 90°. We obtained numerically the order of the singularity at three-dimensional interface corners and edges and correlated fracture initiation with critical values of the stress intensities of the singular fields near the three-dimensional interface corner. Here, the criterion of fracture initiation is analogous to the interface corner failure criterion based on linear elastic fracture mechanics. Based on an analysis of the intensity of the singularity, an empirical equation was determined by curve fitting of the experimental data to predict the interfacial strength of bonded scarf joints. The prediction is in good agreement with the measured data. Acknowledgements The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51075354). References [1] [2] [3] [4] [5]
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