Vol.3, No 11, 2001 pp.223-230
UDC 531.22:539.421:624.073(045)
STRESS STATE AND STRAIN ENERGY DISTRIBUTION
AT THE VICINITY OF ELLIPTICAL CRACK
WITH COMPRESSION FORCES ACTING ON IT'S CONTOUR
Dragan B. Jovanović1, Milena B. Jovanović2
1Faculty of Mechanical Engineering, University of Niš,
Beogradska 14, 18000 Niš, Yugoslavia, e-mail: jdragan@masfak.masfak.ni.ac.yu
2Vojvode Tankosića 9, 18000 Niš, Yugoslavia
e-mail: mjovanovic@masfak.masfak.ni.ac.yu
Abstract. Model of elliptical crack with contour pressurized by continual uniform forces is of interest and has found application in mechanical and civil engineering, as well as in geomechanics and geology. At solving of this problem stress functions in elliptical coordinate system can be used (Timoshenko, Goodier), or the region outside the elliptical contour can be transformed at simpler shape and find appropriate stress function, accordingly by application of complex variable function and conformal mapping method (Muskhelishvili). Stress functions for the case of plane crack suggested by Westergaard and Sneddon can be applied. All this solutions are related on the problem of plane stress state in a plate, where three-dimensional stress state at the vicinity of crack is neglected.
Analytical solutions for problem of elliptical shaped crack in an infinite plate by applying of complex variable function and conformal mapping method are presented in this paper. Crack is subjected to uniform pressure forces on it's contour, and plane stress state in all points of the plate is assumed. Comparable three-dimensional model of crack in the plate of finite dimensions is done. By application of finite element method, diagrams off stress and deformation distribution at the vicinity of crack, as well as at whole plate, are done.
Diagrams of stress components, in selected sections are presented. Strain energy for characteristic directions y = 0, z = 0, ... zk and z = 0, x = 0, ... xk is calculated by using well-known relations of theory of elasticity. Then, surface of the strain energy for points in the middle plane z = 0, and plane perpendicular to it y = 0, in front of the crack tip, by using best fitting curve, and best fitting surface, and iteration procedure is reconstructed. Conclusion on three-dimensional stress state at the vicinity of crack tip, is derived from obtained stress diagrams, and estimation up to which distance from crack tip three-dimensional stress state exist is done. Also, from reconstructed strain energy surfaces, it's concentration and three-dimensional distribution is visible. On certain distance from the crack strain energy gets constant value, and at the most part of the plate is "undisturbed state".
STANJE NAPONA I STANJE ENERGIJE DEFORMACIJE
U OKOLINI ELIPTIČNE PRSLINE
PRI DEJSTVU SILA PRITISKA NA NJENOJ KONTURI
Model prsline eliptičnog oblika na čijoj konturi dejstvuju kontinualne sile pritiska, od interesa je i ima primenu u mašinskom, građevinskom inženjeringu, kao i u mehanici tla i geologiji. U rešavanju ovog problema mogu se koristiti naponske funkcije u eliptičnom koordinatnom sistemu (Timoshenko, Goodier) ili se može transformisati oblast izvan eliptične konture u jednostavniji oblik i pronaći odgovarajuća naponska funkcija, odnosno primeniti metoda funkcije kompleksne promenljive i konformnog preslikavanja (Muskhelishvili). Mogu se primeniti i naponske funkcije predložene od Westergaard-a i Sneddon-a za slučaj ravne prsline. Sva ova rešenja odnose se na problem ravnog stanja napona i ploči, pri čemu je zanemareno lokalno trodimenzionalno stanje napona u okolini prsline.
U radu su prikazana analitička rešenja problema prsline eliptičnog oblika u beskonačnoj ploči, primenom funkcije kompleksne promenljive i konformnog preslikavanja. Prslina je izložena kontinualno jednako raspodeljenim silama pritiska po konturi i pretpostavljeno je ravno stanje napona u svim tačkama ploče. Načinjen je takođe uporedni prostorni model prsline u ploči konačnih dimenzija i primenom metode konačnih elemenata dobijeni su dijagrami rasporeda napona i deformacija u okolini prsline, kao i u čitavoj ploči.
Prikazani su dijagrami komponentnih napona u izabranim ravnim presecima ploče. Specifična energija deformacije je izračunata preko poznatih relacija iz teorije elastičnosti za karakteristične pravce y = 0, z = 0, ... zk i z = 0, x = 0, ... xk. Zatim je korišćenjem best- fitting krivih i best-fitting površina, kroz postupak uzastopnih iteracija, izvršena rekonstrukcija površine energije deformacije za tačke u središnjoj ravni ploče z = 0 i za ravan upravnu na nju y = 0. Na osnovu dobijenih dijagrama napona, donet je zaključak o trodimenzionalnom stanju napona u blizini vrha prsline, sa procenom do kog rastojanja od vrha prsline se javlja trodimenzionalno stanje napona. Takođe se iz rekonstruisanih površina energije deformacije vidi njena koncentracija i njen trodimenzionalni raspored ispred vrha prsline. Na određenom rastojanju od prsline ona dobija konstantnu vrednost i u najvećem delu ploče vlada "neporemećeno stanje".