odre\ivanje napona u materijalu primjenom polarizacije

Ma{instvo 3(6), 149 – 156, (2002)S.Su}eska: ODRE\IVANJE NAPONA U MATERIJALU…gdje su: c= a-b - naponsko-opti~ki koeficijentn i - index prelamanja koji ima svjetlosnitalas polariziran u pravcu glavne ose iσ I, II, III - glavni naponiUticaj napona na promjenu indexa prelamanjamaterijala se naziva elasto-opti~ki efekat. Zrak kojipada normalno na transparentnu plo~u debljine d,napregnute naprezanjima u ravni plo~e σ x i σ y(σ I = σ x , σ II = σ y , σ III =0) usljed djelovanje napona }e serastaviti na dvije komponente, koje }e na izlasku izplo~e relativno kasniti za:r xy = (n x - n y )d= c(σ x - σ y )d (2)gdje su: r xy – relativna retardacijac= a-b - naponsko-opti~ki koeficijentd – debljina plo~eOvaj zakon se naziva osnovni zakon fotoelasti~nosti.Glavna primjena stati~kog elasto-opti~kog efekta jeu analizi napona. Za primjenu postupka je potrebnopripremiti transparentni model, napregnuti ga ipostaviti izme|u ukr{tenih polarizatora i analizatora,a zatim posmatrati izlaznu svjetlost. Specijaliziraniure|aji pomo}u kojih se vr{i posmatranje dvojnogprelamanja materijala nazivaju se polariskopi. Upolariskopima se mogu promatrati prozirna tijela uravanski ili kru`no polariziranoj svjetlosti. U slu~ajuda samo posmatrano tijelo na neki na~in nepolarizira svjetlost koja kroz njega prolazi na izlazuiz analizatora se ne bi pojavila svjetlost ako supolarizator i analizator postavljeni tako da im ravnipolarizacije zatvaraju ugao od 90°. Ako je prozirnotijelo, napravljeno od dvolomnog materijala,napregnuto onda }e ono polarizirati svjetlost udvije okomite ravni ~iji se smjerovi poklapaju sasmjerovima glavnih napona u materijalu, a izlaznipolarizovani zraci }e se fazno razlikovati za2πcd∆ = ( σ1−σ2) = K(σ1−σ2)λgdje je: ∆- fazna razlika redovnog i vanrednogzrakaλ- talasna du`ina svjetlostic- konstanta materijala modela (povr{ina/sila)d- debljina plo~e od dvolomnog materijalaσ 1 ; σ 2 - glavni naponiK=2 πcd/ λ – konstanta materijala za datu talasnudu`inu izvora.Ako je izvor svjetlosti bijeli onda se iza analizatoradobiva slika tijela sa spektralnim bojama, a ako jeizvor svjetlosti monohromatski onda se dobiva slikatijela sa tamnim i svjetlim prstenovima ili prugama. Naizgled dobivenih slika uti~e vi{e faktora kao {to su:oblik tijela, stanje naprezanja, opti~ke karakteristikamaterijala, temperatura…where: c= a-b - stress-optical coefficientn i - refractive index having light wavepolarised in direction of principal axis iσ I, II, III - principal stressesEffect of stresses on change refractive index ofmaterial is called elasto-optical effect. Wave fallingperpendicular on transparent plate having thicknessd, stressed by stresses in plane σ x i σ y (σ I = σ x , σ II = σ y ,σ III =0) due to stress effect will be broken into twocomponents, which will relatively lag for:r xy = (n x - n y )d= c(σ x - σ y )d (2)where: r xy - relative retardationc= a-b - stress-optical coefficientd - plate thicknessThis rule is called fundamental law of photoelasticity.The main application of statical elasto-optical effectis in stress analysis. To use the method it isnecessary to prepare transparent model, load itand put it between crossed polariser and analyser,then the observe outcoming light. Specialisedapparatus, by which observations of birefringentmaterial are made, is called polariscope.Transparent bodies in plane or circularly polarisedlight can be observed by the polariscopes. In casethe observed body, in any way, does not polarisethe light passing through it, light would not appearfrom analyser outlet if polariser and analyser areplaced in such a way that its planes ofpolarisation make an angle of 90°. If transparentbody, made of birefringent material, is loaded thenit polarises light in two perpendicular planeswhose directions coincide with directions ofprincipal stresses in the material, and outcomingpolarised beams will have the difference in phase:2πcd∆ = ( σλ1−σ2) = K(σ1−σ2)(3)where: ∆- phase difference between regular andextraordinary waveλ- wavelength of lightc- stress-optical coefficientd- thickness of plate made of birefringentmaterialσ 1 ; σ 2 - principal stressesK=2 πcd/ λ - constant of material for givenwavelength of source.If light source is white then picture of body made ofspectrum colours is produced behind analyser, but iflight source is monohromatic then picture of boduwith dark and shine rings or lines is produced. Manufactors affect on appearance produced pictures suchas: shape of body, stress conditions, opticalcharacteristics of material, temperature…- 151 -

Ma{instvo 3(6), 149 – 156, (2002)S.Su}eska: ODRE\IVANJE NAPONA U MATERIJALU…Iz predhodne jednakosti se mo`e zaklju~iti da unekom dijelu modela od razlike glavnih napona zavisiboja (kod bijelog izvora svjetlosti), odnosno intenzitetsvjetlosti (kod monohromatskog izvora svjetlosti).Konstanta K plo~e modela se odre|uje u polariskopuna uzorku materijala plo~e napregnutom na poznatastanja naprezanja.Ako se model ~iju raspodjelu napona treba odreditipostavi u polariskop izme|u polarizatora ianalizatora ~ije su ravni polarizacije postavljene pod90°, tada }e napregnuti model dati dvijeortogonalno polarizirane zrake (redovnu i vanrednu)~ije se oscilovanje mo`e izraziti slijede}imjedna~inama:r = aK 0 sin(φ+∆)cosθ (4)v = aK 0 sin φsinθ (5)gdje su: r- redovna zrakav- vanredna zrakaa- amplituda zraka koji izlazi izpolarizatoraK 0 - faktor apsorpcije modela (neka je K 0jednak i za redovni i za vanredni zrak)φ - ugao kojim se opisuje oscilatornokretanje zraka (φ=2π/T)∆- fazni pomakIt can be concluded from the previous equationthat in any part of model the colour (white lightsource), or light intensity (monohromatic lightsource) depends on difference of principalstresses. The constant of model plate K isdetermined in the polariscope on sample plate ofmaterial loaded with known stress conditions.If the model, whose stress distribution is to bedetermined, is placed in polariscope betweenpolariser and analyser whose polarisation planesmake an angle of 90°, then the loaded model willproduce two orthogonal polarised waves (regularand extraordinary) whose oscillations can beexpressed by the following equations:r = aK 0 sin(φ+∆)cosθ (4)v = aK 0 sin φsinθwhere: r- regular wavev- extraordinary wavea- amplitude of wave coming out ofpolariserK 0 - factor absorption of the model (letthe K 0 be equal for regular and extraordinarywave)φ- angle describing the oscillatory motionof wave (φ=2π/T)∆- phase differenceNakon prolaska kroz analizator redovna ivanredna zraka }e se ravanski polarizirati i po{tose fazno razlikuju za ∆ slo`it }e se u zrak koji semo`e opisati slijede}om jedna~inom:w = rsinθ- vcosθ = aK 0 sinθcos[sin(φ+∆) -sinφ] = aK 0 sin2θsin(½∆)cos(φ+½∆) (6) w = rsinθ- vcosθ = aK 0 sinθcos[sin(φ+∆)- (6)sinφ] = aK 0 sin2θsin(½∆) cos(φ+½∆) (6)gdje je w zrak nastao slaganjem polarizovanihredovnog i vanrednog zraka na izlazu iz analizatora.Svjetlost iza analizatora nestaje (w=0) u dvaposebna slu~aja:where the w is a wave produced by superimposingof the regular and extraordinary wavescoming out of the analyser. Behind analyser, lightextinguishes (w=0) in two special cases:1) sin2θ=0 tj. 2θ=nπ tj. θ=½nπ (n=1, 2, …) (7) (7)1) sin2θ=0 => 2θ=nπ => θ=½nπ (n=1, 2, …) (7)2) sin(½∆)=0 tj. ½∆= nπ tj. ∆=2nπ (n=1, 2, …)(8)2) sin(½∆)=0 => ½∆= nπ => ∆=2nπ (n=1, 2, .) (8)Tamne linije (w=0) koje se dobiju prvim slu~ajem(θ=nπ/2) se nazivaju izokline, linije koje spajajuta~ke sa jednakim smjerovima glavnih napona. Zacrtanje rasporeda izoklina u modelu uzima sesredina polja izoklina kao linija izoklina. Ako seu~vrsti polo`aj polarizatora i analizatora i zajednose okrenu za izvjestan ugao slika polja izoklina }ese promijeniti. Izokline se vra}aju u isti polo`ajnakon obrtanja za 90°.After passing through the analyser, the regular andthe extraordinary wave will be plane polarised andbecause they have phase difference of ∆ they willsuperimpose in wave which can be described bythe following equation:Dark lines (w=0) produced in the first case(θ=nπ/2) are called isoclinics, lines connecting thepoints with same slope of principal stresses. Todraw distribution of isoclinics in the model, themiddle of the isoclinic’s field is to be used as theisoclinic line. If the position of the polariser andanalyser is locked and then they turned togetherby any angle, the picture of isoclinic’s fields willbe changed. Isoclinics come back in the sameposition after turning for the angle of 90°.- 152 -

Ma{instvo 3(6), 149 – 156, (2002)S.Su}eska: ODRE\IVANJE NAPONA U MATERIJALU…Linije drugog sistema (∆=2nπ) se nazivajuizohrome. To su linije koje spajaju ta~ke sajednakom razlikom glavnih napona (σ 1 -σ 2 ). Ako jeizvor svjetlosti monohromatski sve ta~ke sa istomrazlikom napona su na jednoj liniji, a ako je izvorbijele svjetlosti onda sve ta~ke sa istom razlikomglavnih napona imaju iste spektralne boje. Uprakti~nom radu na polariskopu se i u ovomslu~aju dobiva polje izohroma, pa se za linijuizohroma uzima sredina polja izohroma. Kodpolarizatora sa monohramatskim izvorom svjetlostiokretanjem spojenih polarizatora i analizatora sedobiva razlikovanje izoklina i izohroma, jer seizokline mijenjaju sa zakretanjem unakrsnihpolarizatora i analizatora, dok se izohrome nemijenjaju. Ako se u polariskopu primjeni bijeli izvorsvjetlosti vrlo je lako uo~iti razliku izme|u izohromai izoklina: izohrome su obojena polja (po ~emu sui dobile naziv), izokline ne.Izokline su tamne linije, ili bolje re}i polja,koje ~ine ta~ke modela u kojima su smjeroviglavnih napona jednaki, a koje zadovoljavaju uslovθ=nπ/2 (n=1, 2, …), tj. pravac glavnih naponamodela se poklapa sa pravcem polarizacijepolarizatora {to dovodi do ga{enja svjetlosti priizlasku iz analizatora. Dakle, pravci glavnih naponau svim ta~kama na izoklini su jednaki (θ). Kako sukomponente glavnog napona me|usobno normalnepoznavanjem pravca jednog glavnog napona,poznat je i pravac drugog u nekoj ta~ci na izoklini.Izohrome su linije jednake boje nastalega{enjem boje ~iji je cjelobrojni proizvod talasnedu`ine jednak retardaciji redovnog i vanrednogzraka dobivenoj razlikom glavnih napona u modelu.Pove}anjem optere}enja pove}ava se brojizohroma. Izvori{ta izohroma su mjesta na kojimase generiraju izohrome (mjesta na kojima djelujukoncentrirane sile, gornja i donja vlakna {tapaoptere}enog ~istim savijanjem).Fotografiranjem napregnutog polariziranog modeladobiju se dva podatka: raspored ta~aka sajednakom razlikom glavnih napona u modelu iraspored ta~aka u modelu sa istim nagibompravaca glavnih napona. Za rje{avanje ravanskihproblema potrebna su tri podatka: dva glavnanapona i smjer glavnih napona. Zatvaranje sistemajedna~ina za odre|ivanje pravca i smjera glavnihnapona jo{ nije rije{eno na prikladan na~in. Slijedeneki od na~ina zatvaranja sistema jedna~ina:pomo}u jedna~ine deformacije u smjeru debljinemodela, pomo}u interferometra, rotiranjem modelau polariskopu, diferencni metod tangencijalnihnapona koji koristi jedna~ine ravnote`e i grani~neuslove.The lines of the second system (∆=2nπ) are calledisochromatics. These are the lines connecting thepoints with equal difference of the principal stresses(σ 1 -σ 2 ). If light source is monochromatic, all pointswith equal difference of principal stresses are on oneline, but if it is used source of white colour then allpoints with equal difference of principal stresses havethe same spectrum colour. In fact, polariscopeproduces isochromatic field in this case too, andisochromatic line is presented by the middle of theisochromatic field. In the case of the polariser withmonochromatic light source, turning the locked polariserand analyser produces discrimination of isoclinicsand isochromatics, because the isoclinics are changedby turning the crossed polariser and analyser, but theisochromatics are not changed. If the polariscope useswhite light source then it is very easy to perceive thedifference between the isochromatics and isoclinics:the isochromatics are coloured fields (which arenamed by), the isoclinics not.The isoclinics are dark lines, or it is better to sayfields, which are made of points of the modelhaving equal slopes of principal stresses, and whichfulfil the condition θ=nπ/2 (n=1, 2, …), that is thedirection of principal stresses of the modelcoincides with the direction of polarisation of thepolariser, which causes extinguishing of light comingout from analyser. So, the directions of the principalstresses are equal in all points on the isoclinic (θ).As the components of the principal stress aremutually perpendicular, by knowing direction of oneprincipal stress, the direction of the second is alsoknown in any point on the isoclinic.Isochromatics are lines with the same colourproduced by extinction of colour having integermultiple of its wavelength equal to the retardationof the regular and extraordinary waves producedby difference of principal stresses in the model.Increase of load produces increase in the numberof isochromatics. Origins of isochromatics areplaces where they are generated (places whereconcentrated forces act, upper and lower fibres ofthe beam loaded by pure bending).Taking the photo of a loaded polarised model givestwo data: distribution of points having equaldifference of principal stresses in the model, anddistribution of points in the model having equalslopes of principal stresses. Three data arenecessary to solve planar problem: two principalstresses and slopes of principal stresses. Closing ofthe equation system to determine the line and slopeof the principal stresses has not been properlysolved so far. Some ways of closing the equationsystem are the following: by the strain equation inthe direction of model thickness, by interferometer,by rotation of the model in the polariscope, methoddifferences of share stresses utilizing the equationsof balance and boundary conditions.- 153 -

Ma{instvo 3(6), 149 – 156, (2002)S.Su}eska: ODRE\IVANJE NAPONA U MATERIJALU…1) Izokline 2) Izohrome1) Isoclinics 2) IsochromaticsSlika 1: Slike izoklina i izohroma dobivene pomo}u polariskopaFigure 1: Photos o f isoclinics and isochromatics made by polariscopePosljednji postupak, koji se naj~e{}e koristi, neodre|uje experimentalno tre}i podatak nego gara~una diferencnim jedna~inama, koje su rje{enjadiferencijalne jedna~ine ravnote`e za ravanskonaponsko stanje uz poznate grani~ne uslove.Aproximiranjem diferencnim jedna~inama iteoremom srednje vrijednosti uvodi se kumulativnagre{ka koja raste sa udaljavanjem od konture. Naovaj na~in je mogu}e odrediti tenzor napona uproizvoljnoj ta~ci napregnutog uzorka.Procedura ra~unanja napona u proizvoljnoj ta~ci jeslijede}a:1) Odrediti polja izohroma i izoklina2) Nacrtati osnovni pravac du` ose x i dvapomo}na pravca na rastojanju ∆y inapraviti podjelu sa korakom ∆x3) U presje~nim ta~kama dobivene mre`eodrediti parametre izohrome i izokline4) U ta~kama ½(x i +x i+1 ) na pravcima a i bizra~unati tangencijalne napone po formuli:τ xy =½(σ 1 - σ 2 )sin2θ’ (9)5) Izra~unati ∆τ xy /∆y po formuli: ∆τ xy /∆y =(τ xy(a) - τ xy(b) )/∆y (10)6) Izra~unati napon σ x(i) po formuli: σ x(i) = σ x(i-1)- (∆τ xy /∆y)∆x (11)7) Izra~unati σ y(n) po formuli: σ y(n) = σ x(n) -(σ 1 -σ 2 ) (n) cos2 θ’ (n) (12)The last method, the most commonly used, doesnot determine the third data by experiment, but itcalculates the data by difference equations, beingsolutions of the differential equation of balance forplanar stress conditions with known boundaryconditions. By approximation with differenceequations and the middle value theorem, cumulativeerror is introduced and it increases when thedistance from the contour increases. The tensor ofstresses in any point of loaded model can bedetermined by this method.The calculation procedure of the stresses in anypoint is the following:1) Determine the fields of the isochromatics andisoclinics2) Draw main direction along x axis, and twoarbitrary directions on distance of ∆y andmake division with step of ∆x3) Determine the parameters of the isochromaticsand the isoclinics in cross points of the mesh4) Calculate share stresses in points ½(x i +x i+1 )on arbitrary directions utilising equation:τ xy =½(σ 1 - σ 2 )sin2θ’ (9)5) Calculate ∆τ xy /∆y by equation: ∆τ xy /∆y =(τ xy(a) - τ xy(b) )/∆y (10)6) Calculate stress σ x(i) by equation: σ x(i) = σ x(i-1) -(∆τ xy /∆y)∆x (11)- 154 -