Note for OPTI 521 at UA
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Date Created: 02/06/15
Houssine Makhlouf To Opti521 Class College of Optical Sciences College of Optical Sciences University of Arizona University of Arizona hmakemailarizonaedu November 14 2008 Technical Analysis Strength of glass using Weibull statistics Motivation Glassfailure is primarily caused by small defects and takes place by crack propagation It is important to know the maximum level of stress that can be applied on a given type of glass in order to design mountings for optics in a relevant manner The strength of glass can be quantified statistically using the Weibull method 1 Weibull statistics A glass subjected to a stress of has a probability of failure Pf given by V m A E P 1 exp it 1 where on and m are material constants called scale factor and Weibull modulus respectively For a given probability of failure the failure stress of of a glass part of area A and flaw depth s is scaled to the failure stress of of a part of area A and flaw depth s according to 17f Aquot 15m sf 19 F El El 2 This scaling equation is useful as measurements can be made on several smaller test samples and the failure stress for any actual glass component can be deduced The use of this two equations can then tell us how likely a glass is to fail Practically one needs to have the knowledge of on and m for the glass of interest Equation 1 can be rewritten as in in 11 2110 mining 3 One can then make sequential measurements of the left hand term of equation 3 when applying a known stress of on a series of test samples One can finally plot the lefthand term as a function of lnof and perform a linear regression in order to determine the slope m and the offset lnou and thus 00 The two material constants are then known and the probability of failure can be computed subsequently using equation 1 2 Example of the method In page 740 of OptoMechanical Systems Design Yoder cites an example that was worked out by Harris 1999 A series of N 13 samples of ZnS were tested to failure under stress by the ringonring method The results are shown in Table 1 and the linear regression curve is reproduced in Figure 1 where the equation for the linear fit is written down In this example we find m 53764 and mlnou 24737 thus 00 996 MPa These values are slightly different from Harris values m 54485 and on 1014 Page 1 of2 MP2 pmsz becauseuf sums dWErEnCESm mp ementmgthe hnezr rEgressmn 1 used an Exce sheet t5 an my regressmn samplei Failum mess In In 1 lna MP2 1 P 1 52 722275 41271 2 59 7217927 42241 3 73 715422 429175 4 75 711595 422177 5 27 7172552 44559 5 29 475977 44225 7 9L7 472555 44992 2 93 4715177 45225 9 1m7 L7L759L7 451752 10 1L77 L72712 45722 11 1117 L75L7m7 4717175 12 125 L77592 42222 13 125 112175 42252 Table 1 Measured data fur Wetbuu deterrmnztmn uchs strength 77m 4 77 1w rigquote 1 met at data gwen bythe zsltwu cu umnsuszb e 1 and curregmhdtrgtu equztmn a Conclus Knuwmg the 592 s femur and the Wetbuu mudums uhe can predtct the prubzbmty uffz urefur the gas at nteresl ZHSWIHE Examp e abuve accurdmgtu equztmn 1 and muuntthe uptmszppmphztew W 2 gwen system Reveyences r Vukzbrztuwch h5tes2m72 r Vader 22 ch 15 Anz yas at the OptuMechzmcz Destgh Optomedymrca systems Design 3quot EdeH T2y urampFr2nms 2L7L75 r Hams DC Malenasfornfaed Mndows md Domes 5915 Press Benghzm 1999 Pagel 5n
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