Strength of Glass from Hertzian Line Contat Wenrui Cai, Brian Cuerden, Robert E. Parks, James H. Burge College of Optial Sienes, University of Arizona, Tuson, AZ USA 85721 Phone: 52-626-486, 52-621-8182 wai@optis.arizona.edu, jburge@optis.arizona.edu ABSTRACT In optial lens assembly, metal retaining rings are often used to hold the lens in plae. If we mount a lens to a sharp metal edge using normal retention fore, high ompressive stress is loaded to the interfae and the alulated tensile stress near the ontat area from Hertzian ontat appears higher than allowable. Therefore, onservative designs are used to ensure that glass will not frature during assembly and operation. We demonstrate glass survival with very high levels of stress. This paper analyzes the high ontat stress between glass lenses and metal mounts using finite element model and to predit its effet on the glass strength with experimental data. We show that even though ontat damage may our under high surfae tensile stress, the stress region is shallow ompared to the existing flaw depth. So that glass strength will not be degraded and the omponent an survive subsequent applied stresses. Keywords: glass lenses, Hertzian ontat, tensile stress, strength of glass 1. INTRODUCTION The general rule of thumb for mounting lenses is that polished glass an withstand tensile stresses of about 1, psi (6.9 MPa) before failure 1.Tensile stress will our in the glass near the ontat, whih an be alulated using Hertzian ontat theory (shown in Fig. 1). The darkness of the ontour olor indiates the relative amount of tensile stress. There is more tensile stress in the darker region, while the white region inside the tensile stress region is the ompressive stress region. While this stress may indeed be present, we were unable to reate an effet large enough to break glass samples. This highly onentrated tensile stress may form raks in the subsurfae of the glass. However, shallow damage to the glass may not lead to failure. We need to answer the question: If damage does our, will the omponent still survive subsequent applied stresses? How does ontat damage affet the strength of glass? This paper analyzes this phenomenon using the finite element analysis (FEA) and predits its effet on glass strength with experimental data. More speifially, we exerted a load on B27 flat windows (1.15 mm thik, 5 mm in diameter) using a metal ring that simulates the lens mount. The objetive here was to show that with ommon sharp orner radii and large loads (R=.1 in = 254 um and R 5 um, F = 2 lb f = 89 N), the strength of the glass will not degrade (via double ring strength test 5, 6 ). (a) (b) Figure 1. (a) Retaining ring mount, (b) Priniple normal stress field 3. Region I is in tension, while region II is in ompression. The maximum tensile stress loates just outside the ontat area on the glass surfae.
2. BACKGROUND KNOWLEDGE 2.1 Hertzian Contat for Cylinder 2, 3 In sharp edge ring-mounted lenses, ontat loading with large stresses is applied over a highly loalized region. This type of onfiguration in the elasti range is alled Hertzian ontat. When the radius of the mount edge is small ompared to the radius of the ontat surfae. The glass-mount ontat an be approximated by Hertzian ontat of a ylinder on a flat glass surfae 2. The equation for ontat pressure is: 2P σ = π a, (1) where P (lb f /in) is Loading fore, a is the ontat half-width, whih an be expressed as: 1 a = 4PR* π E *, (2) 2 2 where 1 1 R* = + is the ontat radii and 1 ν 1 ν m g E* = Rm R + is the effetive modulus, whih is g Em E g omprised of Young s moduli E m and E g,and Poisson ratios ν m and ν g for metal and glass, respetively. The maximum tensile stress of Hertzian point ontat is 1 : 1 2 σ t = 3 ν g (3) The general form of the ontat stress field, espeially the tensile stress, is shown in gray olors in Figure 1b. The important feature of the indentation stress field for the initiation of a onial frature is the tensile region near the speimen surfae just outside the area of ontat. Hatheway 4 also developed a set of losed-form equations for the state of stress over the surfae of the lens with ring ontat. He stated that if radius of the ring ontat inrease to infinity, the tensile stress will beome zero. While if the radius of the ring ontat derease to from a point ontat, the equation will agree with equation (3). There are some limitations to these equations. First of all, the ontat is assumed to be purely elasti, but in the real ase, metals have plasti properties, whih redue the stress onentration. Seondly, frition is absent at the ontat interfae, but frition is an important fator affeting the amount of tensile stress inside the glass, this will be disussed in the simulation setion following. 2.2 Strength of Glass 5 Glass does not possess a single harateristi strength. The strength of the material is dependent on the distribution of raks or surfae flaws. The strength of a partiular element an be estimated using the following two methods. 2.2.1 Frature Mehanis Approah The maximum bending strength of a glass sample depends on the size and geometry of the surfae flaws. In the ase of a flaw with a small depth in a thik plate with tensile fores ating normal to the rak plane, one an define a stress intensity fator K I as: σ 1 σ a KI 2σ nominal tensile stress perpendiular to the stress plane depth of the flaw. a (4)
A flaw will result in a frature if K I is larger than a ritial value alled the frature toughness K IC. If exposed to onstant stress over time, a surfae flaw an grow to ritial size with the rate of growth depending on K I and the atmospheri moisture ontent. For example, the maximum rak depth is about.5 inh (1.27 mm) for a polished BK7 glass that an withstand tensile stresses of about 1, psi (6.9 MPa). 2.2.2 Statisti Approah For a partiular type of glass, it is reasonable to suppose that surfae fratures an be statistially haraterized by a funtion, whih an be related to the probability of failure as a two-parameter Weibull distribution. Based on laboratory test results obtained under well-defined onditions one an alulate design strengths for loads and onditions posed by speial appliation requirements. Equation (5) gives the probability that the sample will fail if it is loaded to stressσ. The quantities σ and m are model parameters that must be experimentally determined. These parameters for several glasses are given in referene 5. m F( σ) = 1 exp( ( σ / σ ) ) F( σ ) Probability of failure at tensile stress σ σ Charateristi strength (F( σ ) = 63,21 %) m Weibull fator (satter of the distribution.) (5) 3. ANSYS FEA SIMULATION Sine the ring-glass ontat is plane symmetri, a 2D model instead of a 3D one was used in ANSYS. In figure. 2, the left edge of the grid is the enter line of the ontat area. Just half of the stress field is shown beause of the symmetry. The ontour is the tensile stress field under a 5 lb f /in (8.7k N/m) line fore load with a.1 in (254um) ontat radius. From figure 1 and figure 2, we an see the high tensile stress field is just outside the ontat area. The depth of the stress field will not hange when the density of mesh grid is hanged. Figure 2. Half-plane tensile stress field on the glass ross setion using FEA in ANSYS. (a)fritionless ontat, (b) Contat with.5 frition oeffiient
Fig. 2a is the half ross-setion for a fritionless ontat, while Fig. 2b is a ontat with a frition oeffiient of.5. The maximum tensile stress is at the surfae at the loation marked MX. The maximum stresses are 19332 psi (133 MPa) for Fig. 2a and 966 psi (6.8 MPa) for Fig. 2b, respetively. The depths of stresses are about 4 um for Fig. 2a and 1 um for Fig. 2b, respetively. These depths are 2 to 3 orders of magnitude less than the allowable flaw depth for a polished BK7 glass that an withstand tensile stresses of about 1, psi (6.9 MPa). Herztian ontat (eq.1 and 2) assumes two materials ontat without any frition 3, whih is not the real ase 7. A statially loaded model with a frition oeffiient was evaluated. The tensile stress dereased when the frition oeffiient was inreased whih is shown in the omparison between Fig 2a and 2b. This is due to different Young s moduli and Poisson ratios between glass and steel. In this ase, stresses were redued beause the ontat zone is larger. But this may not hold true for ases where the steel omponent is replaed by a lower modulus omponent. In addition, more loads applied to the glass may yield the steel, so the sharp orner will be flattened and stress will be dereased. Beause of that, more loads will not neessarily make more degradation to the strength of glass. If we use aluminum instead of steel, we will have a smaller minimum load to yield the mount and flatten the sharp orner. In another ase, when we applied an additional shear fore to the indenter (sliding), tensile stress inreased on one side. Sliding may happen during both assembly and operation. We need suffiient torque on the retaining ring to hold the lens in plae. In the final turn, beside the perpendiular load, a tangential fore will be applied to the glass surfae. Also temperature hange will result in a shear fore between two materials having different CTEs (oeffiients of thermal expansion). 4. EXPERIMENTS A piee of glass breaks when two onditions oinide. The first is the presene of enough tensile stress at the surfae and the seond is the presene of a flaw in the region of the tensile stress. In the experimental proedure, we first made some flaws on the glass using sharp edge ring ontat. We then establish allowable load levels that applied tensile stress on the glass surfae and test the glass sample to failure. 4.1 Applied stress via line ontat 4.1.1 Stati load The setup is shown in Fig. 3. The INSTRON hardness testing mahine provides a ontrollable vertial load fore (manually) and a platform. A ball tip was used against the load ell to prevent a side fore. A Loadstar iload Mini load ell 11 was attahment to the indenter. A lamp fork and bamboo forks were used to onentrially align the indenter, glass sample and the supporting ring. A piee of rubber was plae between the glass plate and the supporting ring to prevent the irregular edges damaging the lower surfae of the plate. We exerted a load on B27 flat windows (1.15 mm thik, 5 mm in diameter). The load value was displayed instantaneously on a omputer sreen via a user interfae. The maximum indenting load was held for 5 seonds before release. We loaded up to 17.4k N / m (1 lb f / in), whih indiates about 16 ksi maximum ontat pressure and 26 ksi maximum tensile stress. Even at suh high loads, we did not observe any failure. Figure 3. Load the glass with a sharp edge indenter. Drawing is on the left side and the atual set up on the right side.
4.1.2 Shok Load 12 We used the benh handling proedure from MIL-STD 81D to simulate the shok load. Tape was used to lamp the indenter sample and an aluminum substrate together. To avoid damaging the sample due to irregularities on the metal surfae, a piee of paper was plaed between the glass and the aluminum blok. Using one edge as a pivot, we lifted the opposite edge and let, then let go the whole pakage with the lifted edge just below the point of perfet balane. The proedure was repeated, using the other edges for a total of four drops. Figure 4. Shok load 4.2 Double Ring Test of Strength of the Glass The test proedure was aording to DIN 52292-1 Double ring method 5. In this ring-on-ring bending test, the vertial load applied to the sample is read by a mini load ell 11, shown in figure 6. The load will transfer via moment to the glass, and then the tensile stress appears on the surfae. From Roark s 1, for the solid irular plate under uniform annular line load with simply supported edge restraint (shown in Fig. 5), we have the bending moment: M = wal9, (6) where w is the amount of annular line load (lb f / in), a is the outer radius where the support fore exert, 2 L9 is one of the general plate funtions: r 1+ v a 1 v r, L9 = ln + 1 a 2 r 4 a r is the radius of the annular line load from the enter line, v is the Poisson ratio of the plate, And the bending tensile stresses an be found on the onvex side of the plate from 6M 2 t where t is the thikness of the annular plate. M by the expression: σ =, (7) Figure 5. Plate under uniform annular line load We used both COSMOSWork in SolidWorks modeling and formulas from Roark 1 to verify the tensile stress alulation. The results agreed well with eah other.
The setup is shown in figure 6. Three lamping forks were used to align the double rings and the glass sample. Rubber films were added between the glass and metal rings. This helped to prevent the sharp edge on the metal adding more high loal stress to the glass. Gently apply the load until the glass breaks. The load ell software automatially reords the maximum load. Figure 6. Double ring strength test. Using the same load ell in a different set up, we tested the samples to failure in order to get the strength of the glass ompared to the ontrol group. Figure 7. A rak pattern of a sample from a double-ring-test. The dotted rings represent the relative radii among different ontat rings on the glass. From the raking pattern in figure 7, we an see that the initial rak was from the entral region of the sample, where the tensile stress was applied while bending. Beause the tensile stress is uniform inside the smaller ring, the initial rak will our at the loation where the deepest existing flaw was. The dotted rings shown in figure. 7 represent the relative radii among different ontat rings on the glass. Sine there was no tensile fore outside the outer ring, the rak paths in this region maybe from the rak propagation from the glass inside. 5.1 Fit the Weibull distribution 5. RESULTS AND ANALYSIS One we obtained a set of tensile stress data, a probability was assigned to eah data point using Harris method 1 and then fit the Weibull distribution 8. In order to fit a straight line, we rewrite eq. 4 into: 1 { } We applied a linear fitting to ln{ ln (1 ) } ln ln (1 F( σ )) = mlnσ mlnσ (8) F 1 vs. lnσ, obtained modulus m as the inverse of the slope and the harateristi strength σ from the interept mlnσ. Matlab is used to fit load data. Two examples of data sets are shown in figure 8. The major unertainties were from the thikness variation of the glass plates and the annular line load
fore we exerted on the samples. We an see from the eq. 7, the sample thikness is inverse square proportional to the tensile stress value and the load is linear. (a) (b) Figure 8. Weibull distribution in a linear fitting. (a) Before exposing to ontat stress, (b) After 1 lb f /in loading fore. The error bars are the unertainties in the alulated tensile stresses. To ompare the strength before and after exposing to ontat stress, we need a group of 25 samples to test the strength with any damage. Using this method we an get the satter of the distribution m and harateristi strength σ for different sample groups, whih were summarized in Table 1. There is no data for optially polished BK7, so we sale between optially polished Zerodur, D64 ethed Zerodur, and D64 ethed BK7 to obtain an estimated for optially polished BK7. The strength we obtained from our experiment is in the proper range. Situation Table 1 harateristi strength σ and satter of the distribution m Charateristi strength Quantity σ ksi Satter of the distribution m 1. Control group 25 25.9 4.4 2. 17.4k N/m, R=254 um 25 24.2 4.9 3. 17.4k N/m, R 5 um 1 21.3 4. 4. Shok load 1 27.3 3.8 5. Grind with 25um ompound 7 1.5 7.2 5.2 level of onfidene in the results The t-statistis may be used to test the hypothesis that two data sets have the same mean. We an verify that if the two sets of data arose from idential physial auses. We used the table of the student s distribution 13 to determine if the results of two sets of data are the same, exept for statistial error. For example, x is the average stresses before breaking in the double ring test (ontrol group); and 1 x is the average 2 stresses after 17.4k N/m (1 lb f /in) loading fore. Their sample quantities and standard deviations are N 1, S 1 ; N 2, S 2, respetively. Then value of the parameter t in the student distribution is:
t = x x N + N 2 1 2 1 2 2 2 1 1+ NS 1/ N 2 2 1+ 1/ N2 NS Using equation (9), we an alulate the t value for eah situation ompare with the ontrol group, whih is list in table 2. Then we an find the t value in appendix C 13. The number orresponds to a onfidene level. In ommon situation, we an rejet the hypothesis that the two sets of data are from the same ause, when the t value exeed the.95 onfidene level. The t value for 95% onfidene level is about t = 2.6. We an see that all situations listed are well within this range (t < t ), exept for situation 5. So we an aept that the situations 1 to 4 are from the same physial ause. The apparent degradation listed in table 1 are due to statistial issues. This means the strength of the glass will not signifiantly degrade at these levels of load. Table 2 t values, average and standard deviation of tensile stress to break the sample in eah situation Average tensile Standard deviation Situation Quantity stress (ksi) (ksi) 1. Control group 25 23.7 5.4 t value 2. 17.4k N/m, R=254 um 25 22.3 5.6.953 3. 17.4k N/m, R 5 um 1 21.8 6.1.879 4. Shok load 1 24.6 6.9.398 5. Grind with 25um ompound 7 9.9 1.6 6.468 (9) Figure 9. Histograms of the tensile stress values in eah situation
Another way to state the t-statistis is that the mean of the tensile stress in the ontrol group an be known no more aurately than σ t avg= 23.7 ± 2.1 ksi for a 95% onfidene level. The sample means of situation 2 to 4 are within this range. Again, we an aept that they are from the same ause, whih means that exposing to ontat stresses in situation 2 to 4 has little effet on the strength of glass. All the tensile stress values in eah situation are plotted in histograms in figure 9. 6. CONCLUSION The strength of glass is highly dependent on its surfae finish with optially polished glass being substantially stronger than oarse ground or srathed glass. Although maximum tensile stress on a surfae of a optially polished glass high under high stati annular line load (1 lb f / in) with sharp metal edge ontat, the shallow tensile stress region inside the glass helps to prevent deep raks and maintains the strength of the glass. It is safe to say the assumption, that polished glass an only withstand tensile stresses of about 1, psi, is too onservative. Adding a safety fator of 4 in our experiment, we an onlude that at 4.4k N/m (25lb/in) stati load with R=254 um (R=.1 in), the strength of glass will not degrade. Moreover, shok load seems do not have atastrophi effet to the glass ontating with sharp edge. REFERENCES [1] Yoder,P. R., [Opto-Mehanial Systems Design], 3rd ed.,spie Press, Bellingham, Washington, pp.733-794(26). [2] Johnson,K. L., [Contat Mehanis], Cambridge University Press, Cambridge, UK,pp.9-11(1985). [3] Lawn,B. R., Indentation of eramis with sphere: A entury after Hertz, J. Am. Ceram. So., 81[8] 1977-94 (1998) [4]Hatheway, A. E., Tensile stresses in Ring-mounted Glass Lenses, Pro. of SPIE Vol. 7424 74249-3 (29) [5] Shott Glass, TIE-33: Design strength of optial glass and ZERODUR, (24) [6] Doyle,K. B. and Kahan, M. A., Design strength of optial glass, SPIE Proeedings 5176, p. 14 (23). [7] Fisher-Cripps,A. C. and Collins, R. E., The Probability of Hertzian Frature, J. Mater. Si., 29, 2216 3 (1994). [8] Salamin, E., The Weibull distribution in the strength of glass, graduate lass assignment: Opti 521 Tutorial, College of Optial Sienes, University of Arizona. [9]Ahmad, A.,[Handbook of OptomehanialEngineering], CRC-Press; 1 st edition (1997). [1] Roark, R.J., [Formulas for Stress and Strain], 4 th edition, New York: MGraw-Hill (1965). [11] Load ell spes and software: http://www.loadstarsensors.om/iloadmini.html [12] MIL-STD-81D, military standard: environmental test methods and engineering guidelines (1983)., [13] Frieden, R. B., [Probability, Statistial Optis, and Data Testing], Springer-Verlag, 2 nd edition, pp.37-319 (1991)