Frm: Steve Sctt, Jinsek K, Syun ichi Shiraiwa T: MSE enthusiasts Re: MSE mem 101b: allwable thickness f Vitn sheet Nvember 25, 2008 Update frm MSE Mem 101b Let s assume: Vitn thickness = 1 mm Vitn mdulus = 3.e6. The basis fr this number is that www.acmerubber.cm/vitsheet lists 0.020 thick Vitn (=0.5mm) that has a Shre hardness f 55, which crrespnds t abut 3.e6. Use a lens hlder fabricated frm SiC. SiC has a larger CTE than C-C, i.e. it is better matched t the SFL6. Then, by the same analysis as was dne belw: change in length fr ΔT = 30 Celsius is 0.0006 cm strain in Vitn = 0.003 stress in Vitn = 9,000 N This stress is abut a factr f TEN smaller than the value cmputed in MSE mem 101b, and it is nly 2.8% f the allwable stress f 3.2e5 (see last sectin f the mem). Even if we used nly 5mm thickness f this Vitn, the thermal stress wuld be acceptable. The thermal resistance fr a 1mm Vitn thickness wuld be acceptable if the thermal cnductivity is 0.0022. If the thermal cnductivity were t be 0.001, then we prbably need t g t 0.5mm thickness. The 1mm slutin : Ttal thickness f the 2 layers f Vitn wuld be 0.078. Accuracy f cutting lens is 0.004. If we culd machine the hle t an accuracy f 0.002 then we wuld be OK. Cnclusin: if we can get Vitn sheet with the fllwing prperties: 1 mm thick κ = 0.002 W / m K
elastic mdulus = 3.e6 Then a single-piece lens munt assembly is feasible and attractive. Intrductin In this mem, we address the questin f whether r nt we culd implement the strap lens that is munted inside a single piece f C-C r Si-C lens-munt. There will be Vitn between the lens and the lens munt. As temperature changes, the glass expands mre than the lens munt, and s the Vitn gets squished. This puts stress n the lens. The questin is: hw much stress is generated, and is this amunt acceptable? Hardness versus Mdulus There is a rugh relatinship between the Yung s mdulus f a material and its hardness, as measured by a Shre durmeter. There are several Shre scales fr hardness. The ne nrmally used fr measuring the hardness f sft materials like rubber is the Shre A scale. The relatinship between Shre A hardness and the Yung s mdulus is listed in Table 1. Shre A Mdulus (MPA) 10 0.27 20 0.61 30 0.97 40 1.54 50 2.45 60 3.45 70 5.47 80 8.68 Table 1. Relatinship between Yung s mdulus and the Shre A hardness scale. Frm http://www.pspglbal.cm/prperties-hardness.html
What we really want t knw is the behavir f the Vitn sheet under cmpressin. Rubber tends t be nnlinear, s fr example the stress required t stretch a piece f rubber 100% is nt necessarily 10 times larger than the stress required t stretch it 10%. Fr example, frm file VITON-01FEF1BE3AF035.PDF, the stress required fr 10% elngatin fr GBLT-600S is 0.8 MPa, but the stress required fr 100% elngatin is 4.2 MPa. S the numbers in Table 1 prvide nly a rugh guide. Nevertheless, they ffer sme hpe that we might be able t find sme Vitn with elastic mdulus clser t 5 MPa rather than 9 MPa. Accrding t http://www.warc.cm/pages/prducts/sheet/vitn/html#dupntgasketcdes, the range f Shre hardness is frm 60 t 75. CTE f Vitn Accrding t Wikipedia, the CTE f rubber is 77 e-6. I was able t lcate a value fr the CTE f nly ne type f Vitn (RX-55-AE), which is 60e-6. The value fr Vitn is abut 6.6 times larger than the value fr glass. But the thickness f the glass (40mm) is ging t be at least 40 times larger than the thickness f the Vitn. Therefre, the expansin f the Vitn itself will cntribute nly negligibly t the stress it experiences the stress is effectively set by the differential CTE f the glass and the C-C r SiC. Stress generated by differential CTE in Vitn sheet In a 4 cm x 10 cm slab f glass, there will bviusly be 2.5 times mre expansin in the lng directin than in the shrt directin. S let s assume that we put a generus amunt f Vitn n the ends f the aperture (i.e. alng the 4-cm sides). This will greatly restrict heat flw ut the ends f the glass, but that s OK, because the heat will flw pretty quickly f the sides f the glass. S let s nly wrry abut thermal expansin ver a distance f 4 cm. The CTE f SFL6 is 9e-6, and the CTE f C-C is abut 1e-6. S the difference is 8e-6. The distance D that the Vitn wuld be squished by a 30 Celsius temperature swing is D = 8e-6 x 30 x 4 = 0.00096 cm (nte: the situatin is a little better if we use Si-C rather than C-C. Si-C has a CTE f abut 5e-6, s the difference in CTE between glass and Si-C is nly abut 4e-6, i.e. a factr 2 smaller than the difference between glass and C-C). Let s assume that the Vitn n each side is 0.5 mm thick. The ttal Vitn thickness is then 1.0 mm. The strain is then
σ = 0.00096/0.1 = 0.0096 The mdulus fr Vitn is abut 6e6 t 9e6. Let s be cnservative and use the value f 9.e6. S the stress is Stress = 0.0096 x 9e6 = 86,400 Pascals. S nce again, temperature changes wuld lead t stress in the lens. The big questin is: is this amunt f thermal-induced stress acceptable r nt? Thermal resistance fr strap lens Let s assume that the lens is rectangular, 4 x 10 cm. The radiatin resistance is R rad = 1/4σ T 3 A with A = 40 cm 2 = 40.8 We can analyze the thermal resistance f the rectangle in the limit f a very lng rectangle r a fat rectangle, as was dne in MSE mem 95b. R lng = b/(8hκl) remember width=2b, length=2l R fat = (L/4hκb) ( (b/l) lg(1+(b/l)) S b=2, L=5, h=1.3, κ=0.01 We can evaluate these frmulas using the thickness at the center f the lens (1.3 cm) and at the edge f the lens (0.41 cm): center edge R lng 3.8 12.2 R fat 3.05 9.7 Nw, in principle ur gemetry is prbably clser t the fat than t the lng gemetry. But we are prbably ging t insert a thick Vitn sheet at the lng ends f the lens (i.e. the sides that are 4 cm wide), t reduce CTE-induced stresses in the Vitn that get transmitted back t the lens. S we ll assume that the resistance is 3.8 (center f lens) and 12.2 (edge f lens). Finally, we want t cmpute the resistance f the Vitn sheet at the side f the lens. It has area A=2 x 0.86 x 10 = 17.2 cm 2. Remember that there will be 1.3 cm thickness f Vitn cntacting the glass at the middle f the lens, but nly 0.41 at the edge, s the average is 0.86. The factr f 2 cmes frm having Vitn n bth sides f the lens. R vitn = w / (Area κ)
κ=0.001 κ=0.0022 w=0.25 mm 1.45 0.66 w=0.5mm 2.91 1.32 w=1.0mm 5.8 2.64 Table 2: thermal resistance f Vitn sheet at edge f 4 x 10 cm lens. Let s assume that we g with a Vitn sheet that has a resistance f 2.64. Then the fractin f temperature drp acrss the lens cmpared t the ttal temperature drp is: (3.8 + 2.6) / (3.8+2.6+40.8) = 0.135 (center f the lens) (12.2+2.6) / (12.2+2.6+40.8) = 0.27 (edge f lens) Cmparisn t existing lens: currently, the lens is circular, with mean thickness 0.86 cm. Its thermal resistance is (frm mem 95) R = 1/(4πhκ) = 9.2, and the radiatin resistance is 13.3. S the temperature drp acrss the existing lens is 9.2/(9.2+13.3) = 41% f the ttal temperature drp. This analysis suggests that ging t a strap lens, by itself, prvides nly marginal benefit, at least at the edge f the lens. Mst f the benefit in Syun ichi s simulatins must cme frm the fact that the C-C heats up abut the same as the lens itself. Cmparisn t internal stress in lenses In MSE mem 95, we argued that Jinsek s bench tests shwed that a temperature slew rate f 1.3 Celsius / hur is applied t the periphery f the lens generates a just-acceptable spurius change in plarizatin angle f 0.1 degrees. The mst apprpriate thermal simulatins f Jinsek s bench heating experiments are Unifrm_lens_heating_2.eps and Unifrm_lens_heating_stress_2.eps, which Syun ichi emailed t us n 10/5. The mdel ramped the edge temperature at 5.7 Celsius per hur. The maximum stress, at r=6.25 cm, was abut 1.4e6 Pascals. Assuming that the stresses vary linearly with the temperature slew rate, a maximum stress f 3.19e5 Pascals wuld be generated by a temperature slew rate f 1.3 Celsius per hur. S the stress due t cmpressing the Vitn (0.5 mm thick) due t a temperature rise f 30 Celsius wuld be abut 27% f the stress due t temperature gradients within the glass itself, when the edge temperature is ramped at 1.3 Celsius per hur.
If squishing the Vitn generates t much stress, then we will definitely need sme srt f mechanical system (such as springs) t create pressure that desn t change with temperature s much.