Vol.1, No 7, 2000 pp. 925 - 934
UDC 539.4.014+539.37:539.421:624.073.1(045)
LOCAL STRESS AND STRAIN STATE IN THE REGION
OF CRACK FOR DIFFERENT GLOBAL STRESS STATES IN A PLATE
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. In a plate with different ruling cases of global or general stress state, different local stress, and strain states will appear at vicinity of the crack, as a result of interaction between crack geometry and global stress state. By taking in consideration of elliptically shaped crack (Griffith's crack), which can degenerate to slit, by giving one semi-axis equal to zero, and by introducing different general stress states, corresponding local stress and dislocation distributions at the vicinity of such crack were obtained. Previously was assumed that global stress state is plane and three-dimensional stress and displacement state at the vicinity of crack was obtained. Finite element method and three-dimensional model of the plate with crack was applied. Global loads were given in this way that stress in region of the plate without disturbances caused by crack are unit values, and normalization of local stresses related to global stresses was done.
The conventional plate theories assume that the stress variations in terms of the thickness coordinate are known as a priori. For plane stress state, the transversal normal stress component ?z is assumed to be zero throughout the plate, and in-plane stresses ?x, ?y, and ?xy are independent of thickness coordinate. These assumptions are valid under vary limited circumstances, and in the very narrow framework.
Cases of global stress state, when axial tension forces are acting in direction perpendicular to the crack plane, or in direction of the crack, are considered. Graphical presentations of stress distribution of ?x, ?y, ?z, ?xy, ?yz, ?zx, as well as displacements ux, uy and uz are given in this paper.
Evidently varying of stress tensor components is related to the coordinate z perpendicular to the middle plane of the plate, and ?z is different from zero up to certain distance from the crack tip. Based on obtained diagrams of stress distribution, and displacements distribution, conclusions about influence of global stress state on local distribution of stresses and displacements at surrounding of the crack were drawn out.
LOKALNO STANJE NAPONA I STANJE DEFORMACIJA
U OBLASTI PRSLINE ZA RAZLIČITA GLOBALNA STANJA NAPONA U PLOČI
U ploči u kojoj vladaju različiti slučajevi globalnog ili opšteg stanja napona, javiće se u okolini prsline različita lokalna stanja napona i deformacija, koja su posledica međusobnog uticaja geometrije prsline i globalnog stanja napona. Uzimajući u razmatranje prslinu eliptičnog oblika (Griffith-ovu prslinu), koja se može svesti na zasek zadavanjem jedne polu-ose jednake nuli i uvodeći različita opšta naponska stanja, dobijeni su odgovarajući lokalni rasporedi napona i pomeranja u okolini takve prsline. Pošlo se od pretpostavke da je globalno stanje napona ravno, a dobijeno je trodimenzionalno stanje napona i pomeranja u okoloni prsline. Primenjena je metoda konačnih elemenata i načinjen je trodimenzionalni model ploče sa prslinom. Globalna opterećenja su zadata tako da su naponi u delu ploče bez poremećaja usled uticaja prsline jediničnih vrednosti i izvršena je normalizacija lokalnih napona u odnosu na globalne napone.
Konvencionalna teorija ploča pretpostavlja da je unapred poznata promena napona u funkciji od koordinate z u pravcu debljine ploče. Za ravno stanje napona pretpostavlja se da je komponentni normalni napon ?z jednak nuli svuda u ploči, a da su naponi u ravni ploče ?x, ?y, i ?xy nezavisni od koordinate z. Ovakve pretpostavke se mogu prihvatiti pod vrlo ograničenim uslovima.
Razmatrani su slučajevi globalnog stanja napona kada aksijalne zatežuće sile deluju u pravcu upravnom na ravan prsline ili u pravcu prsline. U radu su dati grafički prikazi rasporeda napona ?x, ?y, ?z, ?xy, ?yz, ?zx, kao i pomeranja ux, uz i uz. Pokazalo se da su komponente tenzora napona u funkciji od koordinate z u pravcu debljine ploče i da je ?z različito od nule do određenog rastojanja od vrha prsline. Na osnovu dobijenih dijagrama rasporeda napona i pomeranja, doneti su zaključci o uticaju globalnog stanja napona na lokalni raspored napona i pomeranja u okolini prsline.