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Tensile, bending and shear strength distributions of adhesive-bonded butt joint specimens
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 1421–1427 www.elsevier.com/locate/compscitech
Tensile, bending and shear strength distributions of adhesive-bonded butt joint specimens Do Won Seo a, Jae Kyoo Lim a
b,*
Materials and Fracture Laboratory, Department of Mechanical Design, Chonbuk National University, Deokjin 1-664-14, Jeonju, JB561-756, Korea b Faculty of Mechanical and Aerospace System Engineering, Automobile Hi-Technology Research Institute, Chonbuk National University, Deokjin 1-664-14, Jeonju, JB561-756, Korea Available online 8 January 2005
Abstract Adhesive bonding is becoming one of the popular joining techniques in metal industries since it offers different options over other techniques such as welding and diffusion bonding. Any dissimilar metals are easily adhesive-bonded together. In this study, experiments were carried out in order to provide the statistical data with strength evaluation methods: tensile, shear and four-point bending tests for thermosetting epoxy resin based adhesive-bonded butt joints. The certification of the probability in the adhesive strength that shows the tendency of brittle fracture is studied. The effect of the adhesive sectional area on the adhesive strength is evaluated. For the tensile and shear tests, the strength distributions of 5 · 6 mm2 sectional area specimens show higher probability than the 2 · 3, 3 · 4 and 4 · 5 mm2 area specimens, and Weibull modulus of 5 · 6 mm2, m = 4.049 and 4.412 for the tensile and shear tests respectively. For the four-point bending test, 3 · 4 mm2 specimens have the highest probability value than others. And the shear test is less affected by sectional area comparing with other test methods. The adhesive bonding strength distribution between 304 stainless steels with thermosetting adhesives shows the highest probability in four-point bending test. The strength testing methods that have higher probability of strength are four-point bending, shear and tensile tests in order. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Adhesive joints; Sectional area
1. Introduction Recently, several types of high strength adhesives have become available for assembling structural elements, so, adhesive bonding is used increasingly in many industrial fields [1–3]. Adhesive bonding has several advantages over other joining techniques such as welding and diffusion bonding, e.g., any dissimilar metals are easily bonded together, and there is no need to care about the thermal strain which is often induced by welding, and moreover, the adhesive layer can play the role of vibration damping [4,5]. In the case of brittle fracture, *
Corresponding author. Tel.: +82 63 270 2321; fax: +82 63 270 2460. E-mail address: [email protected] (J.K. Lim). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.12.013
the strengths have to be used with a statistical distribution [6]. Because the size of defect in each specimen of brittle fracture materials is different from each other, the severe level of defect is not uniform in specimens. So, the fracture behavior of brittle materials has to express with Weibull distribution [7–9]. This situation is not for the ductile fracture material like almost of metals. The plastic deformation by the movement of dislocation relieves the density of severity of defect under the stress, even though metals have the defects in the distribution [10]. Therefore, the same metal specimens have almost the same strength distribution, but the strength of conventional Al2O3 has the range of distribution from 220 to 400 MPa [11]. Hence, the lowest strength by the severest defect is to be the fracture strength of brittle materials, if the specimen has the same stress distribution
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during testing. So, in the statistical data from a large number of specimens, the strength distribution can be treated as the extreme value statistics [12–14]. This research consists of statistical analysis of brittle fracture and probability analysis based on statistics. In this study, the epoxy-based adhesive metal butt joints were used to evaluate the strength and failure probability. The thermosetting epoxy-based adhesives have a tendency of brittle fracture such as ceramic material. The conventional test methods such as tensile, shear and four-point bending tests were conducted to evaluate the probability distribution of adhesive strength. And the effect of the adhesive sectional area on the adhesive strength was evaluated.
SðrÞ ¼ ½1 expðqDV Þ½1 P ðrÞ þ expðqDV Þ
ð5Þ
and DBðrÞ ¼ lnf1 P ðrÞ½1 expðqDV Þg:
ð6Þ
Now DV can be made as small as we like and for an infinitesimal volume dV, we have dBðrÞ ¼ P ðrÞqdV :
ð7Þ
As shown by Weibull [15], the risk of rupture B(r) of a solid is the sum of the risks of rupture of all the elements and hence for a specimen of volume V under a uniform stress BðrÞ ¼ P ðrÞqV :
ð8Þ
The probability of failure Pv(r) of a tensile specimen of volume V at a stress r or less is therefore given by 2. Strength distribution for small specimens
P v ðrÞ ¼ 1 exp½BðrÞ:
Fracture in a single-phase brittle solid subjected to a uniform tensile stress in an inert atmosphere will initiate at the largest flaw. In inhomogeneous or multiphase solids the first crack to initiate may be arrested but if the solid is homogeneous the first crack will propagate unstably and lead to complete fracture. Hence, the strength of a tensile specimen formed in a homogeneous brittle solid will be determined by the size of the largest flaw. This weakest link concept of brittle fracture was originally proposed by Weibull [15] who assumed that the probability of failure P(r) of an elemental volume containing a single flaw at a stress r or less is given by
For the elemental strength distribution given by Eq. (1), we have from Eqs. (8) and (9)
m
P ðrÞ ¼ 1 exp½ðr=r0 Þ :
ð1Þ
If the specimenÕs volume V is large, then the number of flaws will be very close to the expected number N given by N ¼ qV ;
ð2Þ
where q is the density of the flaws. Hence, the probability of failure in a large specimen is given by P v ðrÞ ¼ 1 exp½qV ðr=r0 Þm ;
ð3Þ
which is the usual Weibull strength distribution. However, if the specimen is small then the actual number of flaws in any particular specimen can deviate significantly from the expected number N and the probability of failure will not be given by Eq. (3). Following Weibull [15] the risk of rupture DB(r) is defined for an elemental volume DV small enough to contain at most one flaw by the equation DBðrÞ ¼ ln SðrÞ;
ð4Þ
where S(r) is the probability of survival of the element at a stress r. The probability of the survival of any element is the joint probability of it containing a flaw and that the flaw is small enough for its strength to be greater than r plus the probability that there exists no flaw in the element. Hence,
ð9Þ
m
P v ðrÞ ¼ 1 expfqV f1 exp½ðr=r0 Þ gg:
ð10Þ
If the expected number of flaws N is large, the well known limit equation expðxÞ ¼ lim ð1 x=N Þ N !1
N
ð11Þ
can be used to show that the limit of Eq. (10) as N ! 1 is the Weibull strength distribution given by Eq. (3). If N is not large, Eq. (10) tends to the Weibull distribution only if (r/r0)m 1. If the stress in a specimen is not uniform then Eq. (7) can be integrated to give the risk of rupture Z BðrÞ ¼ P ðrÞqdV : ð12Þ For pure bending of rectangular cross-sectioned specimens, assuming that fractures only initiate from flaws in the tension half of the beam, the probability of failure Pv (r) when the maximum stress is r is given from Eq. (9), by m Z 1 qV r P v ðrÞ ¼ 1 exp 1 exp um du 2 0 r0 ð13Þ and the corresponding expression for three-point bending of beams where the span is much larger than the depth of the beam so that the shear stresses are small and can be neglected is Z 1Z 1 qV P v ðrÞ ¼ 1 exp 2 0 0 m r m m 1 exp u v dudv : ð14Þ r0
D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
The integrals in Eqs. (13) and (14) have to be evaluated numerically. However, the probabilities given by Eqs. (13) and (14) have the usual Weibull strength distributions m qV r P v ðrÞ ¼ 1 exp ð15Þ 2 ð m þ 1 Þ r0
w
20 20
t
80 (a) Tensile test
ð16Þ
for three-point bending when qV ! 1 or when r/r0 ! 0. If a specimen has a Weibull strength distribution then there is a straight-line relationship between ln ln [1 Pv(r)]1 and ln r with a slope equal to the Weibull modulus m whether the specimen is under tension or bending. The probability of failure can be found experimentally by testing a number (n) of specimens and ranking their strengths in ascending order. If the strength of the i th order specimen is ri, the probability is given by P v ðri Þ ¼ i=ðn þ 1Þ:
Ø8
R10
for pure bending and "
m # qV r P v ðrÞ ¼ 1 exp 2ðm þ 1Þ2 r0
1423
w 60 (b) Four-point bending test
w 20
3. Experimental procedures A commercial ANSI 304L stainless steel was used as the specimens for tensile test, four-point bending test and shear test in this study. The adhesive sectional areas were varied as 2 · 3, 3 · 4, 4 · 5 and 5 · 6 mm2 to evaluate the effect of area on strength distribution. The thickness of the specimen was changed namely as 2, 3, 4 and 5 mm. And 20 specimens were tested for the each condition. The specimens are shown in Fig. 1 with various test methods. The contact surfaces of cross-section were uniformly polished with 400 grit SiC abrasive papers under running water. The two-part Technicoll 8266/8267 (a general purpose adhesive manufactured by H.B. Fuller, Germany), thermosetting epoxy-based adhesives, was used as the adhesive. Technicoll 8266 component A is an epoxy resin and Technicoll 8267 component B is an amine-setting agent (thixotropic, pasty). 8266 and 8267 were mixed by 100:100 vol% as a reference ratio and cured in the electric furnace at 125 C within 30 min. Applied pressure on the speci-
5
t
unit: mm
(c) Shear test
ð17Þ
The Weibull modulus is then found from the slope of the best straight-line relationship. Hu et al. [13] show that if a straight-line relationship is found for large specimens it may be curved for small specimens. Only for small values of (r/r0) will the plot of ln ln [1 Pv(r)]1 against ln r generally give a straight line of slope m. If a straight line of best fit is forced through the data for the small specimen, an effective modulus m that decreases with specimen size will be obtained.
t
Fig. 1. Specimen configurations.
mens was 10kPa and the same pressure was applied to every condition. The adhesive thickness was about 0.18–0.23 mm. The adhesive strength was observed after 120 h aging at room temperature. Table 1 shows the properties of adhesive and the reaction product. Tensile, shear and four-point bending tests were conducted with a hydraulic material testing machine (8516, Instron Co., USA) and the load cell of 5 kN was used. The crosshead speed was maintained constant at 3 mm/min to make the same condition for three kinds
Table 1 Properties of adhesives Description Product data Component Mixing ratio (v/v) Density Viscosity Reaction product Hardness Weight/volume Peeling strength
Bonding strength
Technicoll 8266 comp. A
Technicoll 8267 comp. B
Epoxy resin 100 1.30 ± 0.05 g/cm3 496 ± 160 Pa (25 C)
Amine setting agent 100 1.10 ± 0.05 g/cm3 400 ± 96 Pa (25 C) 70 ± 5 Shore D (set 30 min at 120 C) 1.20 ± 0.05 g/cm3 ca. 4.5 N/mm2 (setting 1 d/RT and 2 h/ 100 C) ca. 20–22 N/mm2 after 7 d/22 C ca. 30–32 N/mm2 after 30 min/125 C
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of tests. The data were obtained with 50 Hz. The equipment of shear test was specially designed for small specimens such as ceramic or metal adhesive joints. The micrometer was attached for better accuracy.
2
ln l n [ 1 / (1 - P f ) ]
1
4. Results and discussion Fig. 2 shows the fracture behaviors in tensile tests with various sectional areas. As shown in the figure, the statistical analysis of Weibull can be used [15–17], because the fracture behaviors of the specimens show as the brittle fracture. From Fig. 2(b), the specimen of 5 · 6 mm2 has the lowest standard deviation; in other words the wider the sectional area becomes, the lower stress the specimen has. Because the defects directly affected on fracture strength, the part of incomplete adhesive structure may have relatively uniform distribution under stress as sectional area increases. 4.1. Effect of sectional area Fig. 3 shows the diagram of the tensile test results according to the sectional area of specimen by Weibull theory. The symbol m stands for Weibull modulus, and the degree of inclination obtained, using the least square method under each condition. The m value of 5 · 6 mm2 was the highest, 4.049, in the case of tensile test. Higher m value stands for higher probability of strength compared with the other conditions. The result of Fig. 2(b) about the standard deviation shows the same tendency as shown in Fig. 3. The distribution of the defects becomes uniform in the sectional area of specimens as the adhesive sectional area increases. However, Weibull modulus has little difference among the specimens; that is, the sectional area does not greatly affect the probability distribution.
1000
400
200
0
0.05
0.1
-1 m = 3.011 (3x4) m = 3.591 (2x3) -2 m = 3.996 (4x5)
-3
m = 4.049 (5x6) -4 2.5
3
3.5
4
0.15
Displacement (mm)
(a) Load vs. displacement
0.2
0.25
4.5
ln σ (Pa)
Fig. 3. Weibull plot of tensile test with various sectional area.
The Weibull plot of four-point bending test is shown in Fig. 4. Relatively small specimens, 2 · 3 and 3 · 4 mm2, have higher m values, and the 3 · 4 mm2 specimen has the highest probability. There is tensile stress between outer spans and compressive stress between inner spans, because of the geometrical characteristics of four-point bending test. The inner compressive stress makes the crack growth stop instantly due to structural reason. In other words, this causes the increase of strength. Fracture behavior like this is observed when the sectional area is wider, and it is the main reason to make the failure probability of the adhesive low. The specimens with a small sectional area show the increased deflection and the restraint of structural crack decreases. Fig. 5 shows the Weibull plot of the shear test results and the specimen of 5 · 6 mm2 had the highest value of m, 4.412. Therefore, the widest specimen had the highest probability distribution of strength in the shear test.
Tensile stregth distribution (Pa)
Load in tensile test (N)
600
0
0
120
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
800
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6 Mean strength
100
Std Dev. = 13.42
80
Std Dev. = 12.28 60
Std Dev. = 9.40 Std Dev . = 8.42
40
20
0 0
10
20
30
Sectional area (mm2)
(b) Strength vs. sectional area
Fig. 2. Fracture behavior in tensile test with various sectional area.
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D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
2
ln ln [ 1 / (1 - P f ) ]
1 0 -1 -2
m = 11.121 (2x3) m = 5.374 (5x6)
-3 m = 11.247 (3x4)
m = 8.929 (4x5) -4 3
3.5
4
4.5
5
ln σ4p (Pa) Fig. 4. Weibull plot of four-point bending test with various sectional area.
2 Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
ln ln [ 1 / (1 - P f ) ]
1
0 m = 3.488 (2x3) -1
1425
lue. For tensile and shear tests, the m values were almost same. Four-point bending test has the highest probability distribution in this study, because the eccentricity does not affect the strength distribution and the test equipment is relatively simple. The non-straight line of applied load causes the eccentricity in adhesive bonded joints. Table 2 shows the results of probabilistic analysis according to various test methods and sectional areas. The mean strengths cannot be compared with those of other test methods because their equations are different. The strength probability increases according to the increase of sectional area in tensile and shear tests, because there is a possibility to have a relatively uniform distribution of defect in the adhesive bonding layer. But, four-point bending test does not show the highest probability of strength at the widest specimen because the compressive stress of inner span is a potent influence on the fracture behaviors than the defect distribution in the adhesive bonding layer. From the results of four-point bending test, 3 · 4 mm2 specimens have the highest probability of strength. The shear test has the distribution of the lowest strength, however, it shows the least standard deviation among the three kinds of tests. That is, the shear test is less affected by sectional area as compared with others. And we can confirm that each sectional area has a relatively uniform deviation and probability of strength.
m = 3.9687 (4x5) -2
4.3. Classification of fracture surface
m = 4.412 (5x6) -3 m = 3.443 (3x4) -4 1.5
2
2.5
3
3.5
ln τ (Pa) Fig. 5. Weibull plot of shear test with various sectional area.
This result may have been influenced by becoming uniform in the distribution of the severe defect as the sectional area increases. In this shear test, the Weibull moduli of specimens for each condition showed little difference. Therefore, the shear test had a probability of adhesive strength equivalent to conventional tensile test. 4.2. Effect of test method Fig. 6 shows the strength distributions and the Weibull moduli according to various test methods with the same sectional areas. In this figure, a four-point bending test has the highest probability of failure in all test conditions, i.e., area 2 · 3, 3 · 4, 4 · 5 and 5 · 6 mm2. And tensile and shear tests have almost the same value of Weibull modulus. Fig. 6(a) shows that the Weibull modulus of four-point bending test has also the highest m va-
After failure, the fracture shape of adhesive surface had different features according to the strength distribution. Fig. 7 shows the fracture surfaces of specimens with 50· magnification and they are classified according to the range of the applied strength and sectional area. The bright field is the surface of metal, and the dark field is the adhesive left, not torn out. It can be observed that there is a large part of adhesive torn out from metal surface when specimens failed in the relatively lower strength in all test methods. And the adhesive left on metal surface has an irregular shape. Comparatively, the crack grew along the inside of epoxy adhesive in the case of higher strength, and the fracture surface has the condition of relatively uniform surface roughness. In the case of the mean strength, the fracture behaviors are shown as the mixed conditions of the maximum strength and minimum strength. There is less adhesive torn from the metal surface compared with the case of minimum strength, and the surface is rougher than the case of maximum strength. However, the shapes of fracture surfaces are not changed according to sectional area. And the bright fields are observed where the stress concentrates for four-point bending and shear tests.
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D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
Area 2x3 mm2
2
Area 3x4 mm2
2
m = 3.443 (Shear)
m = 3.488 (Shear) 1
ln ln [ 1 / (1 - P f ) ]
ln ln [ 1 / (1 - P f ) ]
1 0 -1 -2
0 -1 -2 m = 3.011 (Tension)
m = 3.591 (Tension)
-3
-3
m = 11.121 (Bending) -4
1
2
3
4
m = 11.247 (Bending) -4
5
1
2
2
m = 4.412 (Shear) 1
ln ln [ 1 / (1 - P f ) ]
ln ln [ 1 / (1 - P f ) ]
Area 5x6 mm2
m = 3.969 (Shear)
1 0 -1 -2 m = 3.996 (Tension)
-3
0 -1 -2 m = 4.049 (Tension)
-3
m = 5.374 (Bending)
m = 8.929 (Bending) -4 1.5
5
(b) Area 3×4 mm2
(a) Area 2×3 mm2
Area 4x5 mm2
4
ln stress, ln σ (Pa)
ln stress, ln σ (Pa)
2
3
-4 2
2.5
3
3.5
4
4.5
2
2.4
ln stress, ln σ (Pa)
2.8
3.2
3.6
4
ln stress, ln σ (Pa)
(d) Area 5×6 mm2
(c) Area 4×5 mm2
Fig. 6. Weibull plot of various test methods in same sectional area. Table 2 Weibull data of tensile, bending and shear tests Test method
Area (mm2)
Strength (Pa) Mean
Max.
Standard deviation (Pa)
Weibull modulus
No. of spec.
Min.
Tensile
2·3 3·4 4·5 5·6
46.59 35.36 37.12 33.33
67.62 59.74 49.78 46.41
28.21 21.50 23.17 19.10
13.42 12.28 9.40 8.42
3.59 3.01 4.00 4.05
20 19 20 20
Four-point bending
2·3 3·4 4·5 5·6
91.92 94.66 50.97 41.18
111.1 105.3 68.97 52.91
70.95 76.68 40.16 24.70
8.39 8.71 6.02 7.41
11.12 11.25 8.93 5.37
20 20 20 20
Shear
2·3 3·4 4·5 5·6
10.35 14.07 16.74 15.12
17.55 25.46 32.24 24.62
5.30 6.60 10.86 10.17
3.07 4.32 4.76 3.75
3.49 3.44 3.97 4.41
19 20 19 20
Fig. 7. Fracture surfaces with strength fields.
D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
5. Conclusions In this study, the epoxy-based adhesive metal joints with relatively small sectional area were used to evaluate the strength and failure probability. The thermosetting epoxy-based adhesives have a tendency of brittle fracture such as ceramic material. The conventional test methods such as tensile, shear and fourpoint bending tests were conducted to evaluate the probability of adhesive strength. And the effect of the adhesive sectional area on the adhesive strength was evaluated. The following conclusions were obtained from this study. The shear test has the equivalent probability of strength to conventional tensile test. The adhesive joint between metals with thermosetting epoxy resin based adhesive has the best strength probability in four-point bending test. The strength probability in the conventional strength test methods is like the following: fourpoint bending test > shear test P tensile test. The specimen of sectional area 5 · 6 mm2 has the best probability in the tensile and shear tests, and the 3 · 4 mm2 specimen has the best one in the four-point bending test. The strength probability in the shear test is less affected by the sectional area compared with others.
References [1] Pe´richaud MG, Dele´tage JY, Fre´mont H, Danto Y, Faure C. Reliability evaluation of adhesive bonded SMT components in industrial applications. Microelectron Reliab 2000;40:1227–34.
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[2] Higgins A. Adhesive bonding of aircraft structures. Int J Adhes Adhes 2000;20:367–76. [3] Lavery M. Sealants in the automotive industry. Int J Adhes Adhes 2002;22:443–5. [4] Takiguchi M, Yoshida F. Plastic bending of adhesive-bonded sheet metals. J Mater Process Tech 2001;113:743–8. [5] Chester RJ, Walker KF, Chalkley PD. Adhesively bonded repairs to primary aircraft structure. Int J Adhes Adhes 1999;19:1–8. [6] Meyers MA, Chawla KK. Mechanical behavior of materials. New Jersey: Prentice-Hall; 1999. [7] Phrukkanon S, Burrow MF, Tyas MJ. Effect of cross-sectional surface area on the bond strength between resin and dentin. Dent Mater 1998;14:120–8. [8] Towse A, Potter K, Wisnom MR, Adams RD. Specimen size effects in the tensile failure strain of an epoxy adhesive. J Mater Sci 1998;33:4307–14. [9] Cardoso P, Braga R, Carrilho M. Evaluation of micro-tensile, shear and tensile tests determining the bond strength of three adhesive systems. Dent Mater 1998;14:394–8. [10] Towse A, Potter K, Wisnom MR, Adams RD. The sensitivity of a Weibull failure criterion to singularity strength and local geometry variations. Int J Adhes Adhes 1999;19:71–82. [11] Kubel Jr EJ. Structural ceramics: materials of the future. Adv Mater Process 1988;134(2):25–7. [12] Hu XZ, Cotterell B, Mai YW. A statistical theory of fracture in a two-phase brittle material. Proc R Soc Lond A 1985;401:251–65. [13] Hu XZ, Cotterell B, Mai YW. The statistics of brittle fracture of small specimens. Phil Mag Lett 1988;57(2):69–73. [14] Hu XZ, Mai YW, Cotterell B. A statistical theory of timedependent fracture for brittle materials. Phil Mag 1988;58:299–324. [15] Weibull WA. Statistical distribution function of wide applicability. J Appl Phys 1951;18:293. [16] Hadj-Ahmed R, Foret G, Ehrlacher A. Probabilistic analysis of failure in adhesive bonded joints. Mech Mater 2001;33:77–84. [17] Lu C, Danzer R, Fischer FD. Fracture statistics of brittle materials: Weibull or normal distribution. Phys Rev E 2002;65:067102.
Tensile, bending and shear strength distributions of adhesive-bonded butt joint specimens Do Won Seo a, Jae Kyoo Lim a
b,*
Materials and Fracture Laboratory, Department of Mechanical Design, Chonbuk National University, Deokjin 1-664-14, Jeonju, JB561-756, Korea b Faculty of Mechanical and Aerospace System Engineering, Automobile Hi-Technology Research Institute, Chonbuk National University, Deokjin 1-664-14, Jeonju, JB561-756, Korea Available online 8 January 2005
Abstract Adhesive bonding is becoming one of the popular joining techniques in metal industries since it offers different options over other techniques such as welding and diffusion bonding. Any dissimilar metals are easily adhesive-bonded together. In this study, experiments were carried out in order to provide the statistical data with strength evaluation methods: tensile, shear and four-point bending tests for thermosetting epoxy resin based adhesive-bonded butt joints. The certification of the probability in the adhesive strength that shows the tendency of brittle fracture is studied. The effect of the adhesive sectional area on the adhesive strength is evaluated. For the tensile and shear tests, the strength distributions of 5 · 6 mm2 sectional area specimens show higher probability than the 2 · 3, 3 · 4 and 4 · 5 mm2 area specimens, and Weibull modulus of 5 · 6 mm2, m = 4.049 and 4.412 for the tensile and shear tests respectively. For the four-point bending test, 3 · 4 mm2 specimens have the highest probability value than others. And the shear test is less affected by sectional area comparing with other test methods. The adhesive bonding strength distribution between 304 stainless steels with thermosetting adhesives shows the highest probability in four-point bending test. The strength testing methods that have higher probability of strength are four-point bending, shear and tensile tests in order. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Adhesive joints; Sectional area
1. Introduction Recently, several types of high strength adhesives have become available for assembling structural elements, so, adhesive bonding is used increasingly in many industrial fields [1–3]. Adhesive bonding has several advantages over other joining techniques such as welding and diffusion bonding, e.g., any dissimilar metals are easily bonded together, and there is no need to care about the thermal strain which is often induced by welding, and moreover, the adhesive layer can play the role of vibration damping [4,5]. In the case of brittle fracture, *
Corresponding author. Tel.: +82 63 270 2321; fax: +82 63 270 2460. E-mail address: [email protected] (J.K. Lim). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.12.013
the strengths have to be used with a statistical distribution [6]. Because the size of defect in each specimen of brittle fracture materials is different from each other, the severe level of defect is not uniform in specimens. So, the fracture behavior of brittle materials has to express with Weibull distribution [7–9]. This situation is not for the ductile fracture material like almost of metals. The plastic deformation by the movement of dislocation relieves the density of severity of defect under the stress, even though metals have the defects in the distribution [10]. Therefore, the same metal specimens have almost the same strength distribution, but the strength of conventional Al2O3 has the range of distribution from 220 to 400 MPa [11]. Hence, the lowest strength by the severest defect is to be the fracture strength of brittle materials, if the specimen has the same stress distribution
1422
D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
during testing. So, in the statistical data from a large number of specimens, the strength distribution can be treated as the extreme value statistics [12–14]. This research consists of statistical analysis of brittle fracture and probability analysis based on statistics. In this study, the epoxy-based adhesive metal butt joints were used to evaluate the strength and failure probability. The thermosetting epoxy-based adhesives have a tendency of brittle fracture such as ceramic material. The conventional test methods such as tensile, shear and four-point bending tests were conducted to evaluate the probability distribution of adhesive strength. And the effect of the adhesive sectional area on the adhesive strength was evaluated.
SðrÞ ¼ ½1 expðqDV Þ½1 P ðrÞ þ expðqDV Þ
ð5Þ
and DBðrÞ ¼ lnf1 P ðrÞ½1 expðqDV Þg:
ð6Þ
Now DV can be made as small as we like and for an infinitesimal volume dV, we have dBðrÞ ¼ P ðrÞqdV :
ð7Þ
As shown by Weibull [15], the risk of rupture B(r) of a solid is the sum of the risks of rupture of all the elements and hence for a specimen of volume V under a uniform stress BðrÞ ¼ P ðrÞqV :
ð8Þ
The probability of failure Pv(r) of a tensile specimen of volume V at a stress r or less is therefore given by 2. Strength distribution for small specimens
P v ðrÞ ¼ 1 exp½BðrÞ:
Fracture in a single-phase brittle solid subjected to a uniform tensile stress in an inert atmosphere will initiate at the largest flaw. In inhomogeneous or multiphase solids the first crack to initiate may be arrested but if the solid is homogeneous the first crack will propagate unstably and lead to complete fracture. Hence, the strength of a tensile specimen formed in a homogeneous brittle solid will be determined by the size of the largest flaw. This weakest link concept of brittle fracture was originally proposed by Weibull [15] who assumed that the probability of failure P(r) of an elemental volume containing a single flaw at a stress r or less is given by
For the elemental strength distribution given by Eq. (1), we have from Eqs. (8) and (9)
m
P ðrÞ ¼ 1 exp½ðr=r0 Þ :
ð1Þ
If the specimenÕs volume V is large, then the number of flaws will be very close to the expected number N given by N ¼ qV ;
ð2Þ
where q is the density of the flaws. Hence, the probability of failure in a large specimen is given by P v ðrÞ ¼ 1 exp½qV ðr=r0 Þm ;
ð3Þ
which is the usual Weibull strength distribution. However, if the specimen is small then the actual number of flaws in any particular specimen can deviate significantly from the expected number N and the probability of failure will not be given by Eq. (3). Following Weibull [15] the risk of rupture DB(r) is defined for an elemental volume DV small enough to contain at most one flaw by the equation DBðrÞ ¼ ln SðrÞ;
ð4Þ
where S(r) is the probability of survival of the element at a stress r. The probability of the survival of any element is the joint probability of it containing a flaw and that the flaw is small enough for its strength to be greater than r plus the probability that there exists no flaw in the element. Hence,
ð9Þ
m
P v ðrÞ ¼ 1 expfqV f1 exp½ðr=r0 Þ gg:
ð10Þ
If the expected number of flaws N is large, the well known limit equation expðxÞ ¼ lim ð1 x=N Þ N !1
N
ð11Þ
can be used to show that the limit of Eq. (10) as N ! 1 is the Weibull strength distribution given by Eq. (3). If N is not large, Eq. (10) tends to the Weibull distribution only if (r/r0)m 1. If the stress in a specimen is not uniform then Eq. (7) can be integrated to give the risk of rupture Z BðrÞ ¼ P ðrÞqdV : ð12Þ For pure bending of rectangular cross-sectioned specimens, assuming that fractures only initiate from flaws in the tension half of the beam, the probability of failure Pv (r) when the maximum stress is r is given from Eq. (9), by m Z 1 qV r P v ðrÞ ¼ 1 exp 1 exp um du 2 0 r0 ð13Þ and the corresponding expression for three-point bending of beams where the span is much larger than the depth of the beam so that the shear stresses are small and can be neglected is Z 1Z 1 qV P v ðrÞ ¼ 1 exp 2 0 0 m r m m 1 exp u v dudv : ð14Þ r0
D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
The integrals in Eqs. (13) and (14) have to be evaluated numerically. However, the probabilities given by Eqs. (13) and (14) have the usual Weibull strength distributions m qV r P v ðrÞ ¼ 1 exp ð15Þ 2 ð m þ 1 Þ r0
w
20 20
t
80 (a) Tensile test
ð16Þ
for three-point bending when qV ! 1 or when r/r0 ! 0. If a specimen has a Weibull strength distribution then there is a straight-line relationship between ln ln [1 Pv(r)]1 and ln r with a slope equal to the Weibull modulus m whether the specimen is under tension or bending. The probability of failure can be found experimentally by testing a number (n) of specimens and ranking their strengths in ascending order. If the strength of the i th order specimen is ri, the probability is given by P v ðri Þ ¼ i=ðn þ 1Þ:
Ø8
R10
for pure bending and "
m # qV r P v ðrÞ ¼ 1 exp 2ðm þ 1Þ2 r0
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w 60 (b) Four-point bending test
w 20
3. Experimental procedures A commercial ANSI 304L stainless steel was used as the specimens for tensile test, four-point bending test and shear test in this study. The adhesive sectional areas were varied as 2 · 3, 3 · 4, 4 · 5 and 5 · 6 mm2 to evaluate the effect of area on strength distribution. The thickness of the specimen was changed namely as 2, 3, 4 and 5 mm. And 20 specimens were tested for the each condition. The specimens are shown in Fig. 1 with various test methods. The contact surfaces of cross-section were uniformly polished with 400 grit SiC abrasive papers under running water. The two-part Technicoll 8266/8267 (a general purpose adhesive manufactured by H.B. Fuller, Germany), thermosetting epoxy-based adhesives, was used as the adhesive. Technicoll 8266 component A is an epoxy resin and Technicoll 8267 component B is an amine-setting agent (thixotropic, pasty). 8266 and 8267 were mixed by 100:100 vol% as a reference ratio and cured in the electric furnace at 125 C within 30 min. Applied pressure on the speci-
5
t
unit: mm
(c) Shear test
ð17Þ
The Weibull modulus is then found from the slope of the best straight-line relationship. Hu et al. [13] show that if a straight-line relationship is found for large specimens it may be curved for small specimens. Only for small values of (r/r0) will the plot of ln ln [1 Pv(r)]1 against ln r generally give a straight line of slope m. If a straight line of best fit is forced through the data for the small specimen, an effective modulus m that decreases with specimen size will be obtained.
t
Fig. 1. Specimen configurations.
mens was 10kPa and the same pressure was applied to every condition. The adhesive thickness was about 0.18–0.23 mm. The adhesive strength was observed after 120 h aging at room temperature. Table 1 shows the properties of adhesive and the reaction product. Tensile, shear and four-point bending tests were conducted with a hydraulic material testing machine (8516, Instron Co., USA) and the load cell of 5 kN was used. The crosshead speed was maintained constant at 3 mm/min to make the same condition for three kinds
Table 1 Properties of adhesives Description Product data Component Mixing ratio (v/v) Density Viscosity Reaction product Hardness Weight/volume Peeling strength
Bonding strength
Technicoll 8266 comp. A
Technicoll 8267 comp. B
Epoxy resin 100 1.30 ± 0.05 g/cm3 496 ± 160 Pa (25 C)
Amine setting agent 100 1.10 ± 0.05 g/cm3 400 ± 96 Pa (25 C) 70 ± 5 Shore D (set 30 min at 120 C) 1.20 ± 0.05 g/cm3 ca. 4.5 N/mm2 (setting 1 d/RT and 2 h/ 100 C) ca. 20–22 N/mm2 after 7 d/22 C ca. 30–32 N/mm2 after 30 min/125 C
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of tests. The data were obtained with 50 Hz. The equipment of shear test was specially designed for small specimens such as ceramic or metal adhesive joints. The micrometer was attached for better accuracy.
2
ln l n [ 1 / (1 - P f ) ]
1
4. Results and discussion Fig. 2 shows the fracture behaviors in tensile tests with various sectional areas. As shown in the figure, the statistical analysis of Weibull can be used [15–17], because the fracture behaviors of the specimens show as the brittle fracture. From Fig. 2(b), the specimen of 5 · 6 mm2 has the lowest standard deviation; in other words the wider the sectional area becomes, the lower stress the specimen has. Because the defects directly affected on fracture strength, the part of incomplete adhesive structure may have relatively uniform distribution under stress as sectional area increases. 4.1. Effect of sectional area Fig. 3 shows the diagram of the tensile test results according to the sectional area of specimen by Weibull theory. The symbol m stands for Weibull modulus, and the degree of inclination obtained, using the least square method under each condition. The m value of 5 · 6 mm2 was the highest, 4.049, in the case of tensile test. Higher m value stands for higher probability of strength compared with the other conditions. The result of Fig. 2(b) about the standard deviation shows the same tendency as shown in Fig. 3. The distribution of the defects becomes uniform in the sectional area of specimens as the adhesive sectional area increases. However, Weibull modulus has little difference among the specimens; that is, the sectional area does not greatly affect the probability distribution.
1000
400
200
0
0.05
0.1
-1 m = 3.011 (3x4) m = 3.591 (2x3) -2 m = 3.996 (4x5)
-3
m = 4.049 (5x6) -4 2.5
3
3.5
4
0.15
Displacement (mm)
(a) Load vs. displacement
0.2
0.25
4.5
ln σ (Pa)
Fig. 3. Weibull plot of tensile test with various sectional area.
The Weibull plot of four-point bending test is shown in Fig. 4. Relatively small specimens, 2 · 3 and 3 · 4 mm2, have higher m values, and the 3 · 4 mm2 specimen has the highest probability. There is tensile stress between outer spans and compressive stress between inner spans, because of the geometrical characteristics of four-point bending test. The inner compressive stress makes the crack growth stop instantly due to structural reason. In other words, this causes the increase of strength. Fracture behavior like this is observed when the sectional area is wider, and it is the main reason to make the failure probability of the adhesive low. The specimens with a small sectional area show the increased deflection and the restraint of structural crack decreases. Fig. 5 shows the Weibull plot of the shear test results and the specimen of 5 · 6 mm2 had the highest value of m, 4.412. Therefore, the widest specimen had the highest probability distribution of strength in the shear test.
Tensile stregth distribution (Pa)
Load in tensile test (N)
600
0
0
120
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
800
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6 Mean strength
100
Std Dev. = 13.42
80
Std Dev. = 12.28 60
Std Dev. = 9.40 Std Dev . = 8.42
40
20
0 0
10
20
30
Sectional area (mm2)
(b) Strength vs. sectional area
Fig. 2. Fracture behavior in tensile test with various sectional area.
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Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
2
ln ln [ 1 / (1 - P f ) ]
1 0 -1 -2
m = 11.121 (2x3) m = 5.374 (5x6)
-3 m = 11.247 (3x4)
m = 8.929 (4x5) -4 3
3.5
4
4.5
5
ln σ4p (Pa) Fig. 4. Weibull plot of four-point bending test with various sectional area.
2 Area 2x3 mm2 Area 3x4 Area 4x5 Area 5x6
ln ln [ 1 / (1 - P f ) ]
1
0 m = 3.488 (2x3) -1
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lue. For tensile and shear tests, the m values were almost same. Four-point bending test has the highest probability distribution in this study, because the eccentricity does not affect the strength distribution and the test equipment is relatively simple. The non-straight line of applied load causes the eccentricity in adhesive bonded joints. Table 2 shows the results of probabilistic analysis according to various test methods and sectional areas. The mean strengths cannot be compared with those of other test methods because their equations are different. The strength probability increases according to the increase of sectional area in tensile and shear tests, because there is a possibility to have a relatively uniform distribution of defect in the adhesive bonding layer. But, four-point bending test does not show the highest probability of strength at the widest specimen because the compressive stress of inner span is a potent influence on the fracture behaviors than the defect distribution in the adhesive bonding layer. From the results of four-point bending test, 3 · 4 mm2 specimens have the highest probability of strength. The shear test has the distribution of the lowest strength, however, it shows the least standard deviation among the three kinds of tests. That is, the shear test is less affected by sectional area as compared with others. And we can confirm that each sectional area has a relatively uniform deviation and probability of strength.
m = 3.9687 (4x5) -2
4.3. Classification of fracture surface
m = 4.412 (5x6) -3 m = 3.443 (3x4) -4 1.5
2
2.5
3
3.5
ln τ (Pa) Fig. 5. Weibull plot of shear test with various sectional area.
This result may have been influenced by becoming uniform in the distribution of the severe defect as the sectional area increases. In this shear test, the Weibull moduli of specimens for each condition showed little difference. Therefore, the shear test had a probability of adhesive strength equivalent to conventional tensile test. 4.2. Effect of test method Fig. 6 shows the strength distributions and the Weibull moduli according to various test methods with the same sectional areas. In this figure, a four-point bending test has the highest probability of failure in all test conditions, i.e., area 2 · 3, 3 · 4, 4 · 5 and 5 · 6 mm2. And tensile and shear tests have almost the same value of Weibull modulus. Fig. 6(a) shows that the Weibull modulus of four-point bending test has also the highest m va-
After failure, the fracture shape of adhesive surface had different features according to the strength distribution. Fig. 7 shows the fracture surfaces of specimens with 50· magnification and they are classified according to the range of the applied strength and sectional area. The bright field is the surface of metal, and the dark field is the adhesive left, not torn out. It can be observed that there is a large part of adhesive torn out from metal surface when specimens failed in the relatively lower strength in all test methods. And the adhesive left on metal surface has an irregular shape. Comparatively, the crack grew along the inside of epoxy adhesive in the case of higher strength, and the fracture surface has the condition of relatively uniform surface roughness. In the case of the mean strength, the fracture behaviors are shown as the mixed conditions of the maximum strength and minimum strength. There is less adhesive torn from the metal surface compared with the case of minimum strength, and the surface is rougher than the case of maximum strength. However, the shapes of fracture surfaces are not changed according to sectional area. And the bright fields are observed where the stress concentrates for four-point bending and shear tests.
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Area 2x3 mm2
2
Area 3x4 mm2
2
m = 3.443 (Shear)
m = 3.488 (Shear) 1
ln ln [ 1 / (1 - P f ) ]
ln ln [ 1 / (1 - P f ) ]
1 0 -1 -2
0 -1 -2 m = 3.011 (Tension)
m = 3.591 (Tension)
-3
-3
m = 11.121 (Bending) -4
1
2
3
4
m = 11.247 (Bending) -4
5
1
2
2
m = 4.412 (Shear) 1
ln ln [ 1 / (1 - P f ) ]
ln ln [ 1 / (1 - P f ) ]
Area 5x6 mm2
m = 3.969 (Shear)
1 0 -1 -2 m = 3.996 (Tension)
-3
0 -1 -2 m = 4.049 (Tension)
-3
m = 5.374 (Bending)
m = 8.929 (Bending) -4 1.5
5
(b) Area 3×4 mm2
(a) Area 2×3 mm2
Area 4x5 mm2
4
ln stress, ln σ (Pa)
ln stress, ln σ (Pa)
2
3
-4 2
2.5
3
3.5
4
4.5
2
2.4
ln stress, ln σ (Pa)
2.8
3.2
3.6
4
ln stress, ln σ (Pa)
(d) Area 5×6 mm2
(c) Area 4×5 mm2
Fig. 6. Weibull plot of various test methods in same sectional area. Table 2 Weibull data of tensile, bending and shear tests Test method
Area (mm2)
Strength (Pa) Mean
Max.
Standard deviation (Pa)
Weibull modulus
No. of spec.
Min.
Tensile
2·3 3·4 4·5 5·6
46.59 35.36 37.12 33.33
67.62 59.74 49.78 46.41
28.21 21.50 23.17 19.10
13.42 12.28 9.40 8.42
3.59 3.01 4.00 4.05
20 19 20 20
Four-point bending
2·3 3·4 4·5 5·6
91.92 94.66 50.97 41.18
111.1 105.3 68.97 52.91
70.95 76.68 40.16 24.70
8.39 8.71 6.02 7.41
11.12 11.25 8.93 5.37
20 20 20 20
Shear
2·3 3·4 4·5 5·6
10.35 14.07 16.74 15.12
17.55 25.46 32.24 24.62
5.30 6.60 10.86 10.17
3.07 4.32 4.76 3.75
3.49 3.44 3.97 4.41
19 20 19 20
Fig. 7. Fracture surfaces with strength fields.
D.W. Seo, J.K. Lim / Composites Science and Technology 65 (2005) 1421–1427
5. Conclusions In this study, the epoxy-based adhesive metal joints with relatively small sectional area were used to evaluate the strength and failure probability. The thermosetting epoxy-based adhesives have a tendency of brittle fracture such as ceramic material. The conventional test methods such as tensile, shear and fourpoint bending tests were conducted to evaluate the probability of adhesive strength. And the effect of the adhesive sectional area on the adhesive strength was evaluated. The following conclusions were obtained from this study. The shear test has the equivalent probability of strength to conventional tensile test. The adhesive joint between metals with thermosetting epoxy resin based adhesive has the best strength probability in four-point bending test. The strength probability in the conventional strength test methods is like the following: fourpoint bending test > shear test P tensile test. The specimen of sectional area 5 · 6 mm2 has the best probability in the tensile and shear tests, and the 3 · 4 mm2 specimen has the best one in the four-point bending test. The strength probability in the shear test is less affected by the sectional area compared with others.
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