VERIFICATION OF BALL-ON-RING TEST USING FINITE ELEMENT ANALYSIS Seung-Hyun Chae1,3, Jie-Hua Zhao2, Darvin R. Edwards2, and Paul S. Ho1 1 The University of Texas at Austin Austin, TX 78758, U.S.A. 2 Texas Instruments, Inc. Dallas, TX 75243, U.S.A. 3 Phone: (512) 471-8995 Fax: (512) 471-8969 Email:
[email protected]
ABSTRACT The ball-on-ring (BOR) test is an effective technique used to characterize the biaxial fracture strength of brittle materials. In particular, damages induced by wafer backgrinding process can be evaluated using the BOR test. It is difficult to measure directly the radius of contact area between the loading ball and a specimen, which is needed for stress determination by an analytic solution. In this study, parametric finite element analyses were performed to compare with known closed-form solutions. It was found that the effect of small loading area must not be ignored and that the radius of contact could be precisely determined using Hertz’s contact theory. This can serve as a guideline to accurately obtain the fracture strength of a BOR specimen. KEY WORDS: ball-on-ring (BOR), biaxial strength, Si die fracture, equivalent radius, contact radius, finite element method.
flaws which are induced by the backgrinding process, whereas the uniaxial strength determined by other tests such as the 3point or 4-point bending test is sensitive to edge flaws as well as surface flaws [8-9]. A schematic of the BOR test is illustrated in Fig. 1.
Steel punch with ball head
Specimen diameter, 2R Load, P
Contact diameter, 2z
Specimen thickness, t
Specimen support Circular hole diameter, 2a
E P R a b beq r t z
NOMENCLATURE Young’s modulus, MPa load, N specimen radius, mm radius of specimen support, mm radius of uniform loading, mm equivalent radius, mm radius of loading ball, mm specimen thickness, mm radius of contact area between the loading ball and the specimen, mm
Greek symbols ν Poisson’s ratio σmax maximum tensile stress, MPa INTRODUCTION The ball-on-ring (BOR) test has been widely used to characterize the biaxial strength of brittle materials [1-4]. In particular, the BOR test is usually employed when a wafer backgrinding process needs to be evaluated [5-8]. This is because the biaxial strength is influenced only by surface
Fig. 1. Schematic of the BOR test. The maximum tensile stress at the bottom of a specimen in the BOR test is determined by a closed-form solution. One solution was developed by Kirstein and Woolley originally for a uniform concentric load [10], which was later adapted for the BOR test by Shetty [1]. Hu proposed another solution which was derived using Hertz’s contact theory [5]. Both are functions of the geometry of a specimen and test setup as well as an applied load. Among them the radius of contact area is difficult to be measured accurately. Accordingly, a simple approximation has been often used without further verification [1-2]. Also, the effect of small loading area is sometimes ignored in the determination of the maximum stress [5]. In this paper, we emphasize obtaining an accurate value of the biaxial strength from the BOR test by applying the contact theory and considering the effect of small loading area. Parametric finite element method (FEM) calculation was performed to compare the maximum tensile strength with those from the two analytic solutions. First, both closed-form solutions were compared with each other as a function of the contact radius with and
without the treatment of the effect of small loading area. Then FEM was carried out to obtain the maximum stress as a function of loading ball radius and load. Critical factors necessary to determine the biaxial strength accurately could be identified by comparing the analytical and numerical solutions.
σ max =
3P (1 + ν ) ⎡ a 1 −ν ⎢1 + 2 ln + b 1 +ν 4πt 2 ⎣
⎛ b2 ⎞ a2 ⎤ ⎜⎜1 − 2 ⎟⎟ 2 ⎥ . ⎝ 2a ⎠ R ⎦
(1)
for b < 1.724t , for b > 1.724t
(2)
which is depicted in Fig. 2. It should be emphasized that the effect of small loading area on σmax in Eq. (1) shall be significant especially when b is smaller than 0.5t. Accordingly, Eq. (1) is rewritten as
σ max =
3P (1 + ν ) ⎡ a 1 −ν + ⎢1 + 2 ln 2 beq 1 + ν 4πt ⎢⎣
⎛ b ⎞a ⎤ ⎜⎜1 − 2 ⎟⎟ 2 ⎥ . ⎝ 2a ⎠ R ⎥⎦ 2
2
1.5 1 0.5 0 0
σmax was found to be independent of the number of support points. In case that the load is concentrated within a relatively small area, the “special theory of slabs” should be considered instead of the “ordinary theory”, according to Westergaard [11]. The special theory takes into account the deformations due to the vertical stresses while the ordinary theory assumes that a straight line which is drawn through a slab and perpendicular to the slab remains straight and perpendicular to the neutral surface. When a concentrated load is applied at the top of a slab, the ordinary theory resulted in an infinite bending moment at the point of application of the load. The tensile stresses at the bottom, however, are not infinite. In this case, the special theory is required. Since the special theory is much more complicated than the ordinary theory, it is convenient to express the results of the special theory in terms of the ordinary theory. This can be done by finding an equivalent radius beq in a way that the stresses at the bottom based on the special theory with b are equal to those by the ordinary theory with beq. In accordance with Nádai’s analysis [12], Westergaard proposed the relation between b, t, and beq given by [11] ⎧⎪ 1.6b 2 + t 2 − 0.675t , beq = ⎨ ⎪⎩b,
2
b eq /t
THEORIES Kirstein and Woolley presented the stress distribution at the bottom of a thin elastic plate supported at several points in 1967 [10]. For a uniform concentric load, the maximum radial and tangential stresses at the center, which are equal, are given by
2.5
(3)
0.5
1
1.5
2
2.5
b/t Fig. 2. Relation proposed by Westergaard [11] between the equivalent radius beq, the true radius of uniform loading b, and the thickness t. Shetty applied the solution by Kirstein and Woolley for the BOR test in 1980 [1]. Yet it is difficult to determine the value for b, that is, the effective radius of contact between the loading ball and the specimen where the loading can be considered uniform. The contact radius can be calculated using the Hertz’s elastic contact theory [2,13]: ⎡ 3P ⋅ r ⎛ 1 − ν 12 1 − ν 22 ⎜⎜ z=⎢ + E2 ⎢⎣ 4 ⎝ E1
⎞⎤ ⎟⎟⎥ ⎠⎥⎦
1/ 3
,
(4)
where the subscripts 1 and 2 denote the specimen and the loading ball, respectively. Shetty and de With suggested using beq ≈ t/3 from Eq. (2) under typical BOR test conditions for ceramic or glass materials where z < ~0.1t, even though the effective radius of contact is not accurately known [1-2]. de With also calculated beq for the 2.1 mm-thick glass plate as an example by using Eqs. (4) and ⎧⎪ 1.6 z 2 + t 2 − 0.675t , beq = ⎨ ⎪⎩ z,
for z < 1.724t . for z > 1.724t
(5)
In Eq. (5), b in Eq. (2) was simply substituted by z. The result proves that beq fell within the range of 0.68-0.70 mm (≈ t/3) for 1.5 mm < r < 5 mm and 10 N < P < 200 N. In contrast, Table 1 lists beq for Si dies with different thicknesses. If a Si die is as thick as 2.1 mm, the approximation of beq ≈ t/3 would be still valid. Practically, however, for thinner dies, beq deviates from t/3 and becomes more dependent on the loading ball radius and the applied load. Therefore, Eq. (4) should be used along with Eq. (5) to determine beq in the BOR test for Si die strength.
2.1 0.7 0.37 0.1
Loading ball Equivalent radius, beq (mm) radius, r P = 10 N P = 50 N P = 100 N (mm) 1 0.683 0.685 0.686 5 0.685 0.689 0.693 1 0.230 0.234 0.238 5 0.234 0.247 0.257 1 0.124 0.133 0.140 5 0.133 0.155 0.174 1 0.047 0.071 0.089 5 0.071 0.125 0.162
Hu developed a separate closed-form solution for the BOR test of Si wafers in 1982 [5]. In his analysis, the axisymmetric load distribution was applied according to Hertz’s contact theory. With rearranging the equation in [5], the maximum tensile stress at the bottom of the specimen is given by
σ max
3P(1 + ν ) ⎡ 8 a 1 −ν = + ⎢ + 2 ln 2 2z 1 +ν 4πt ⎣3
⎛ 2z 2 ⎜⎜1 − 2 ⎝ 5a
⎞ a2 ⎤ , ⎟⎟ 2 ⎥ ⎠R ⎦
(6)
σ max
⎛ 2b ⎞ a ⎤ ⎜1 − ⎟ ⎥, 2 ⎟ 2 ⎜ 5 a ⎝ ⎠ R ⎥⎦ 2 eq
2
(7)
where beq is defined in Eq. (5). Fig. 3 shows the maximum tensile stress calculated from Eqs. (3) and (7) as a function of the contact radius and the applied load. The geometric parameters chosen are R = 4.875 mm, a = 3.675 mm, and t = 0.37 mm. These values are consistently used throughout this study. As Eqs. (3) and (7) have a similar form, both closed-form solutions lead to comparable results. Note that when the equivalent radius beq is not taken into account the maximum stress deviates substantially and becomes singular as z → 0. This effect is prominent especially for z < 0.5t = 0.185 mm. The difference between the two closed-form solutions is expressed as
σ max,Hu − σ max,K & W
2 3P(1 + ν ) ⎡ 5 1 − ν beq ⎤ . = ⎢ − 2 ln 2 + ⎥ 1 + ν 10 R 2 ⎥⎦ 4πt 2 ⎢⎣ 3
1200 1000 800
w/ b_eq w/o b_eq w/ b_eq w/o b_eq w/ b_eq w/o b_eq w/ b_eq w/o b_eq
600 400 200 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Radius of contact, z (mm) 1600
P=20N, P=20N, P=30N, P=30N, P=40N, P=40N, P=50N, P=50N,
(b)
1400 1200 1000 800
w/ b_eq w/o b_eq w/ b_eq w/o b_eq w/ b_eq w/o b_eq w/ b_eq w/o b_eq
600 400 200
where z is determined by Eq. (4). Although Eq. (6) was derived by directly solving the contact problem it failed to consider the effect of small loading area. Accordingly, Eq. (6) should be modified to a 3P (1 + ν ) ⎡ 8 1 −ν = + ⎢ + 2 ln 2 2beq 1 + ν 4πt ⎢⎣ 3
P=20N, P=20N, P=30N, P=30N, P=40N, P=40N, P=50N, P=50N,
(a)
1400
σ max (MPa)
Specimen thickness, t (mm)
1600
σ max (MPa)
Table 1. Equivalent radius beq in the BOR test of Si dies according to Hertz’s contact theory (Eq. (4)). E1 = 200 GPa, ν1 = 0.29 for steel, and E2 = 130 GPa, ν2 = 0.28 for Si are used in calculation. In this paper, finite element analyses are performed with the 0.37 mm-thick Si die.
(8)
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Radius of contact, z (mm) Fig. 3. Maximum tensile stress as a function of the contact radius and the applied load in the BOR test calculated from (a) the equation by Kirstein and Woolley (Eq. (3)), and (b) the equation by Hu (Eq. (7)). The bold solid lines denote that the effect of small loading area is taken into account by introducing the equivalent radius, beq. The dotted lines denote that the effect of small loading area is ignored by replacing beq by z in Eqs. (3) and (7). With the aforementioned experimental parameters, Eq. (8) reduces to
σ max,Hu − σ max,K&W ≈ (0.626 + 0.00237beq2 ) P ≈ 0.626P ,
(9)
where σ is in MPa and P is in N. Both solutions by Kirstein and Woolley and by Hu are depicted together in Fig. 4. Hu’s solution leads to a slightly higher stress than the solution by Kirstein and Woolley. For z = 0.1 mm, for example, the difference is about 3.6%.
P=20N, P=20N, P=30N, P=30N, P=40N, P=40N, P=50N, P=50N,
1100 1000
σ max (MPa)
900 800 700
K& Hu K& Hu K& Hu K& Hu
W
Uniform pressure
W W W
Si die
600 500 400 300 200
(a)
100 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Radius of contact, z (mm)
Loading ball
Fig. 4. Comparison between the closed-form solutions for the maximum tensile stress by Kirstein and Woolley and by Hu.
In Fig. 7, FEM results are compared with the analytic solution by Kirstein and Woolley (Eq. (2)) for the case of uniform loading. The FEM results agree very well with the analytic solution where the effect of small loading area was suitably taken into account by introducing beq except for b < 0.05 m. For b < 0.05 mm, FEM resulted in σmax deviated from the analytic solution or did not converge. It could be attributed to the nonlinear excessive deformation at the point of concentrated load, which could not be captured by the analytic solution. Fig. 7(b) provides a normalized plot, where the axes are dimensionless.
Si die
(b) Fig. 5. Finite element models for (a) uniform pressure loading, and (b) ball loading.
900 895
σ max (MPa)
FINITE ELEMENT METHOD CALCULATION Parametric FEM was performed to investigate the stress distribution during the BOR test as well as to verify the closed-form solutions. First, uniform pressure loading problems were solved to compare with Eq. (3) by Kirstein and Woolley. Then ball loading cases were simulated with different loading ball diameters. 2D axisymmetric analyses were conducted, and the large-deformation option was turned on. Contact elements were deployed on the regions of contact between the ball and the specimen and between the specimen and the support. The geometric dimensions were R = 4.875 mm, a = 3.675 mm, and t = 0.37 mm, and the applied load ranged from 20 N to 50 N. These parameters were chosen based on the typical experimental conditions for Si die strength tests [8]. The finite element models for uniform loading and ball loading cases are shown in Fig. 5. The mesh size is 0.005-0.01 mm for uniform loading cases and 0.01 mm for ball loading cases. Fig. 6 shows that the mesh sensitivity is pretty low with this range of mesh size.
890 885
0.02%
880
0.055%
875 870 0
0.01
0.02
0.03
0.04
0.05
Mesh size (mm) Fig. 6. Mesh sensitivity for a ball loading case with ball radius of 1 mm.
P P P P P P P P
1200
800
20 N (FEM) 20 N (K & W) 30 N (FEM) 30 N (K & W) 40 N (FEM) 40 N (K & W) 50 N (FEM) 50 N (K & W)
(a) 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Radius of uniform load, b (mm) 2.6 FEM
2.4
Kirstein & Woolley
2.2
Radius of contact, z (mm)
400
0
2
0.12
600
200
Normalized max. stress, σ max t / P
strength using the closed-form solutions. The normalized plot is given in Fig. 9(b), with which one can readily estimate the maximum stress at a certain level of load with geometrical information.
(a) 0.10 0.08 0.06
r= r= r= r= r= r= r= r= r= r=
0.04 0.02
2.0
15
20
25
1.6
1000
1.4
900
1.0 0.00
30
35
40
45
50
55
Load (N)
1.8
1.2
1 mm, Hertz 1 mm, FEM 1.5 mm, Hertz 1.5 mm, FEM 2 mm, Hertz 2 mm, FEM 2.5 mm, Hertz 2.5 mm, FEM 3 mm, Hertz 3 mm, FEM
0.00
(b) 0.05
0.10
0.15
0.20
0.25
Normalized radius of uniform load, b / a
Fig. 7. Maximum tensile stress by analytical (Kirstein and Woolley) and numerical (finite element method) calculations for the case of uniform loading. (a) σmax as a function of the loading radius, b, and applied load, P. (b) Normalized plot with dimensionless axes.
σ max (MPa)
σ max (MPa)
1000
= = = = = = = =
(b)
800 700 600 500
K & W, r = 2 mm Hu, r = 2 mm FEM, r = 2mm
400 300 15
20
25
30
35
40
45
50
55
Load (N) Figs. 8 and 9 represent the results of ball loading cases. The radius of ball varied from 1 to 3 mm. Fig. 8(a) compares the radius of contact area obtained from FEM with that from Hertz’s contact theory (Eq. (4)). Considering the former is limited by the mesh size of 0.01 mm, it is verified that Hertz’s contact theory accurately predicts the contact radius of the loading ball. σmax from the closed-form solutions (Eqs. (3) and (7)) and FEM is plotted as a function of the applied load for r = 2 mm in Fig. 8(b). The FEM result is located between the two analytic solutions, of which the difference is ~0.626P as explained in Eq. (9). σmax is close to but not exactly a linear function of load because the contact radius also varies with load, which changes the equivalent radius in Eqs. (3) and (7). The maximum tensile stress is depicted in terms of the contact radius in Fig. 9. For the various ball radii, the results of FEM match the analytic calculations very well. It confirms that the use of Hertz’s contact theory and Westergaard’s equivalent radius are required in order to accurately determine the biaxial
Fig. 8. (a) Comparison of the contact radii determined by Hertz’s contact theory (Eq. (4)) and FEM. (b) Comparison of σmax obtained from analytic and numerical calculations. CONCLUSIONS The known closed-form solutions for the BOR test were comprehensively reviewed, compared, and verified with FEM. The maximum tensile stress at the bottom of a specimen determined according to Kirstein and Woolley is slightly lower than that based on Hu’s equation. The difference, which is dependent on the applied load, is typically only a few percent. The results calculated by FEM were located between those two solutions. To avoid a stress singularity and achieve this good agreement between analytic and numerical solutions, the introduction of the equivalent radius attributed to Westergaard was found to be critical. The equivalent radius is dependent on the area of contact area which varies with the applied load. The contact radius can be precisely predicted by
Hertz’s contact theory. This paper provides a straightforward guideline to accurately determine the biaxial strength using the BOR test.
[2] [3]
1200 1100 1000
FEM for r = 1 mm FEM for r = 1.5 mm FEM for r = 2 mm FEM for r = 2.5 mm FEM for r = 3 mm
[4] P = 50 N
σ max (MPa)
900
[5]
800 700
P = 40 N
600 500
[6]
P = 30 N
400
P = 20 N
300 200 0.05
[7] 0.06
0.07
0.08
0.09
0.10
0.11
0.12
Radius of contact, z (mm)
2
Normalized max. stress, σ max t / P
(a) 2.8
[8]
2.6
2.6
2.5
2.4
2.4
2.2
[9]
2.3 0.00
2.0
0.02
0.04
1.8
[10]
1.6 1.4 1.2 1.0 0.00
Kirstein & Woolley Hu FEM
0.05
0.10
[11] 0.15
0.20
0.25
[12]
Normalized radius of contact, z / a
(b) Fig. 9. Maximum tensile stress by analytical and numerical calculations for the case of ball loading. In analytical calculations, beq was determined based on Eqs. (4) and (5). (a) σmax as a function of the contact radius, z, which depends on the radius of loading ball, r, and applied load, P. Thin solid lines denote the solution by Kirstein and Woolley (Eq. (3)), and dash-dot lines denote the solution by Hu (Eq. (7)). (b) Normalized plot with dimensionless axes. Inset: magnified view of the region of the FEM results. REFERENCES [1]
D. K. Shetty et al., “Biaxial flexure test for ceramics,” Am. Ceram. Soc. Bull., vol. 59, pp. 1193-1197, 1980.
[13]
G. de With and H. Wagemans, “Ball-on-ring test revisited,” J. Am. Ceram. Soc., vol. 72, no. 8, pp. 15381541, Aug. 1989. A. Simpatico, W. R. Cannon, and M. J. Matthewson, “Comparison of hydraulic-burst and ball-on-ring tests for measuring biaxial strength,” J. Am. Ceram. Soc., vol. 82, no. 10, pp. 2737-2744, Oct. 1999. C. H. Hsueh, M. J. Lance, and M. K. Ferber, “Stress distribution in thin bilayer discs subjected to ball-on-ring tests,” J. Am. Ceram. Soc., vol. 88, no. 6, pp. 1687-1690, Jun. 2005. S. M. Hu, “Critical stress in silicon brittle fracture, and effect of ion implantation and other surface treatments,” J. Appl. Phys., vol. 53, no. 5, pp. 3576-3580, May 1982. J.-H. Zhao, J. Tellkamp, V. Gupta, and D. R. Edwards, “Experimental evaluations of the strength of silicon die by 3-point-bend versus ball-on-ring tests,” IEEE Trans. Electron. Packag. Manuf., vol. 32, no. 4, pp. 248-255, Oct. 2009. J. Brueckner, R. Dudek, S. Rzepka, and B. Michel, “An integrated experimental and theoretical approach to evaluate Si strength dependent on the processing history,” in Proc. ASME InterPACK, San Francisco, CA, Jul. 2009. S.-H. Chae, J.-H. Zhao, D. R. Edwards, and P. S. Ho, “Effect of dicing technique on the fracture strength of Si dies with emphasis on multimodal failure distribution,” IEEE Trans. Dev. Mater. Reliab., in press. B. Cotterell, Z. Chen, J.-B. Han, and N.-X. Tan, “The strength of the silicon die in flip-chip assemblies,” J. Electron. Packag., vol. 125, no. 1, pp. 114-119, 2003. A. F. Kirstein and R. M. Woolley, “Symmetrical bending of thin circular elastic plates on equally spaced point supports,” J. Res. Natl. Bur. Stand. C, vol. 71C, no. 1, pp. 1-10, 1967. H. M. Westergaard, “Stresses in concrete pavements computed by theoretical analysis,” Public Roads, vol. 7, no. 2, pp. 25-35, Apr. 1926. A. Nádai, “Die biegungsbeanspruchung von platten durch einzelkräfte,” Schweizerische Bauzeitung, vol. 76, p. 257, 1920. K. L. Johnson, Contact Mechanics, Cambridge, U.K.: Cambridge University Press, 1985.
