LOCAL STRESS AND STRAIN STATE IN THE REGION OF CRACK FOR DIFFERENT GLOBAL STRESS STATES IN A PLATE UDC : :

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1 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol., N o 7, 000, pp LOCAL STRESS AND STRAIN STATE IN THE REGION OF CRACK FOR DIFFERENT GLOBAL STRESS STATES IN A PLATE UDC :59.:6.07.(05) Dragan B. Jovanović, Milena B. Jovanović Faculty of Mechanical Engineering, University of Niš Beogradska, 8000 Niš, Yugoslavia, e-ail: jdragan@asfak.asfak.ni.ac.yu Vojvode Tankosića 9, 8000 Niš, Yugoslavia e-ail: jovanovic@asfak.asfak.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 geoetry and global stress state. By taking in consideration of elliptically shaped crack (Griffith's crack), which can degenerate to slit, by giving one sei-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 assued that global stress state is plane and threediensional stress and displaceent state at the vicinity of crack was obtained. Finite eleent ethod and three-diensional odel 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 noralization of local stresses related to global stresses was done. The conventional plate theories assue that the stress variations in ters of the thickness coordinate are known as a priori. For plane stress state, the transversal noral stress coponent σ z is assued to be zero throughout the plate, and in-plane stresses σ x, σ y, and τ xy are independent of thickness coordinate. These assuptions are valid under vary liited circustances, and in the very narrow fraework. 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 σ x, σ y, σ z, τ xy, τ yz, τ zx, as well as displaceents u x, u y and u z are given in this paper. Evidently varying of stress tensor coponents is related to the coordinate z perpendicular to the iddle plane of the plate, and σ z is different fro zero up to certain distance fro the crack tip. Based on obtained diagras of stress distribution, and displaceents distribution, conclusions about influence of global stress state on local distribution of stresses and displaceents at surrounding of the crack were drawn out. Received August 0, 000 Presented at 5th YUSNM Niš 000, Nonlinear Sciences at the Threshold of the Third Milleniu, October -5, 000, Faculty of Mechanical Engineering University of Niš

2 96 D. B. JOVANOVIĆ, M. B. JOVANOVIĆ. INTRODUCTION. FRAMEWORK. This paper is analyzing plate with Griffith's elliptical crack [], [], [], [], [5], and it is assued that aterial is ideal elastic [6], with aterial constants for polyester Palatal P-6. Young's elasticity odulus is E = 60 N/, and Poisson's odulus is ν = 0.8 for Palatal P-6. It is assued that global stress state is plane (two-diensional) and it is given by unifor distributed loading forces σ = N/ on the edge sides of the plate. Finite eleent ethod is used for deterining of the stress tensor coponents and displaceent coponents [0]. Three loading cases are taken in consideration in this paper: - plate loaded by unifor distributed tension forces in x direction, as it is presented on Fig., - plate loaded by unifor distributed tension forces in y direction, as it is presented on Fig. 0, - plate loaded by unifor distributed tension forces in x, and y direction, as it is presented on Fig. 9. It is visible that third case is superposition of first and second loading case.. LOCAL STRESS AND STRAIN STATE FOR DIFFERENT GLOBAL STRESS STATES Plate loaded in x direction is shown on Fig.. Distributions of stress coponents σ x, σ y, σ z, τ xy and τ zx on the plate surface, and for section y = 0 are presented on Fig.,,, 5 and 6 respectively. It is visible that at the ost part of plate, plane stress state is approved. Also, it is visible on Fig. that stress coponent σ z is different than zero at region close to crack tip, and that stress coponents σ x and σ y are nonlinearly distributed across the thickness in z direction, in the region close to crack tip. Loads in this case are causing closure of the crack. Figures 7, 8, and 9 are presenting displaceents u x, u y, and u z. It is visible that crack didn't ake influence on displaceents in the plate. Plate loaded in y direction is shown on Fig. 0. Distributions of stress coponents σ x, σ y, σ z, τ xy and τ zx on the plate surface, and for section y = 0 are presented on Fig.,,, and 5 respectively. It is visible that at the ost part of plate plane stress state is approved. Also, it is visible on Fig. that stress coponent σ z is different than zero at region close to crack tip, and that stress coponents σ x and σ y are nonlinearly distributed across the thickness in z direction, in the region close to crack tip. crack front σ Fig.. Fig.. σ X

3 Local Stress and Strain State in the Region of Crack for Different Global Stress States in a Plate 97 σ Y Fig.. Fig.. σ Z τ xy Fig. 5. Fig. 6. τ zx u x Fig. 7. Fig. 8. u y

4 98 D. B. JOVANOVIĆ, M. B. JOVANOVIĆ σ u z crack front Fig. 9. Fig. 0. σ x Fig.. Fig.. σ y σ z Fig. Fig.. τ xy

5 Local Stress and Strain State in the Region of Crack for Different Global Stress States in a Plate 99 τ zx Fig. 5. Fig. 6. u x u y Fig. 7. Fig. 8 Loads in this case are causing opening of the crack. It is visible on Fig. 6, 7, and 8, which are presenting displaceents u x, u y, and u z, that crack influenced on displaceents in the plate. Contraction in z direction is higher on the crack tip then in the rest of the plate, and it is lower on the iddle of crack contour than in the rest of the plate. It sees like contour line on the crack tip goes down, and iddle of the crack contour x = 0 goes up. Plate loaded in x and y direction is shown on Fig. 9. Distributions of stress coponents σ x, σ y, σ z, τ xy and τ zx on the plate surface, and for section y = 0 are presented on Fig. 0,,, and respectively. It is visible that at the ost part of plate plane stress state is ruling stress state. Also, it is visible on Fig. that stress coponent σ z is different than zero at region close to crack tip, and that stress coponents σ x and σ y are nonlinearly distributed across the thickness in z direction, in the crack tip vicinity. Loads in x direction are causing closure, and loads in y direction are causing opening of the crack contour. It is visible on Fig. 5, 6, and 7, which are presenting displaceents u x, u y, and u z, that crack influenced on displaceents in the plate, in sae way like when loads in y direction are acting alone. Consequently, contraction in z direction is higher on the crack tip then in the rest of the plate, and it is lower on the iddle of crack contour than in the rest of the plate. It sees like contour line on the crack tip goes down, and iddle of the crack contour x = 0 goes up. u z

6 90 D. B. JOVANOVIĆ, M. B. JOVANOVIĆ σ crack front σ Fig. 9. Fig. 0. σ x σ y Fig.. Fig.. σ z τ xy Fig.. Fig.. τ zx

7 Local Stress and Strain State in the Region of Crack for Different Global Stress States in a Plate 9 u x Fig. 5. Fig. 6. Using Kolosov-Muskhelishivili forulas [9], [] and conforal apping ethod can solve previously presented probles. The stress functions for an infinite plate under biaxial loads is []: Z = σ z z a where: z = x + iy. Then the stresses can be deterined by using relations []: u z σ x = Re Z y I Z, Fig. 7. σ y = Re Z + y I Z, () τ = y Re Z. Solutions for stress coponents, based on coplex function and conforal apping for elliptical contours and elliptical-hyperbolic coordinate syste are given in the paper [], [], [5], and [6]: σ (, ϕ ) = σ = gr + cos(k ) ϕ A + cos(k + ) ϕ A + cos(k + ) ϕ A h k k k k k 6 { ( cosk ϕ A k + + B k g g gr (k + )g k + ϕ B cos(k ) A k k + g g gr g(k ) g k B k g g gr g(k + ) + Bk + )} 6 6 g gr g(k + ) x y u y ()

8 9 D. B. JOVANOVIĆ, M. B. JOVANOVIĆ k k σϕ(, ϕ) = σe = Ak { [ coskϕ cos(k ) ϕ + cos(k + ) ϕ cos(k + ) ϕ ] 6 gr g [ + cos(k ) ϕ cos(k ) ϕ + cosk ϕ + + cos(k + ) ϕ g + ϕ } ] + k { ϕ + + ϕ cos(k ) B cosk cos(k ) + cos(k + ) ϕ + 6 k g R + [ ϕ ϕ + ϕ + + ϕ ] } + cosk cos(k ) + + cos(k ) cos(k ) g(k ) () k k τϕ(, ϕ) = τhe = { Ak { [ sinkϕ sin(k ) ϕ sin(k ) ϕ + sin(k + ) ϕ 6 gr g ] [ sin(k + ) ϕ + sin(k ) ϕ + sin(k ) ϕ + + sinkϕ g + ϕ + + ϕ ] }} + { { [ + k sin(k ) sin(k ) B + sin(k + ) ϕ 6 k g R sinkϕ sin(k + ) ϕ ] + [ sin(k ) ϕ g(k + ) ϕ + + sink + sin(k + ) ϕ sin(k + ) ϕ ] }} Previously entioned analytical solutions are disregarding known reality that stress state is three-diensional at the crack tip vicinity. According to that atter of fact, known stress intensity factors deterined by using two-diensional solutions should be corrected [], [], [], [], [], [5]. (5).THREE-DIMENSIONAL STRESS AND STRAIN STATE AT THE VICINITY OF THE CRACK TIP It is evidently fro presented figures, that in all three loading cases, for plane global stress state, three-diensional stress state is local ruling in the region close to crack tip. Strain state is undisturbed except at region of the crack. It is observed that front and back surface off the plate is not flat at the vicinity of the crack, or in other words contour line of elliptical crack does not lie in plane, and contour line is three-diensional.. CONCLUSION AND DISCUSSION For previously assued plane global (general) stress state three-diensional stress state at the vicinity of crack tip is obtained. It is known in literature that local stress state at crack tip region is three-diensional, and that question is discussed in soe papers [0], []. Knowledge of the fact that stress state is locally three-diensional gives to us reason to develop ore accurate analytical solutions and ore reliable experiental ethods for deterining of stress and strain state at the neighbourhood of the crack [6], [7], [8].

9 Local Stress and Strain State in the Region of Crack for Different Global Stress States in a Plate 9 REFERENCES. Gdoutos E. E., Fracture Mechanics, Kluwer Acadeic Publishers, Dordrecht, 99. Broek David, Eleentary Engineering Fracture Mechanics, Martinus Nijhoff Publishers, The Hague, 98. Hedrih K., Jovanović D., (990), Priena funkcije kopleksne proenljive i konfornof preslikavanja za određivanje eliptično-hiperboličkih koordinata tenzora napona ravno napregnutih ploča, XIX Jugoslovenski kongres teorijske i prienjene ehanike, Zbornik radova, pp -8, Ohrid. Hedrih K., Jovanović D., The stress state of the elliptical-annular plate by the coplex variable function and conforal apping ethod, 5 str., 7-87, časopis, Teorijska i prienjena ehanika, br. 7, Jugoslovensko društvo za ehaniku, Beograd, Hedrih K., Jovanović D., The strain state of the elliptical - annular plate by the coplex variable function and conforal apping ethod, 7 str., 9-5, časopis, Teorijska i prienjena ehanika, br., Jugoslovensko društvo za ehaniku, Beograd, Hedrih K., Jovanović D., Stress state around elliptical cracks in elastic plates, International inisyposiu MF Recent Advances in Fracture and Daage Mechanics, Zbornik radova, Str. 5-57, YUCTAM Vrnjačka Banja, Hedrih K., Jovanović D., Fenoeni nelinearne dinaike, saznanja ehanike loa i teorije slučajnih oscilacija i projektovanje ašinskih sistea sa sigurnošću od rizika i loa, 6. JUPITER KONFERENCIJA sa eđunarodni učešće, Zbornik radova, Str , Beograd, februar, Jovanović D., (990), Priena etode fotoelastičnosti za ispitivanje naponskog stanja eliptičnoprstenaste ploče opterećene koncentrisani silaa, XIX Jugoslovenski kongres teorijske i prienjene ehanike, Zbornik radova, Ohrid 9. Mushelishvili N.I., (95), Soe Basic probles of the Matheatical Theory of Elasticity, P. Noordhof LTD., Groningen - Holland. 0. Parsons I. D., Hall J. F., Rosakis A. J., A Finite Eleent Investigation of the Elastostatic State Near a Three Diensional Edge Crack, SM Report 86-9, Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California, 986. Sedak S. Uvod u ehaniku loa i konstruisanje sasigurnošću od loa, onografija, Sederevska Palanka, 980. Sih G. C., Mechanics of Fracture Initiation and Propagation, Kluwer Acadeic Publishers, Dordrecht, 990. Sih G. C., Lee Y. D., Review of Triaxial Crack Border Stress and Energy Behavior, Theroretical and Applied Fracture Mechanics, No., pp. -7, Elsevier Science Publishers, B. V., 989. Sneddon, I. N., Lowengrub, M., Crack Probles in the Classical Theory of Elasticity John Wiley & Sons, Inc., New York, Šuarac D., Krajčinović D., Osnovi ehanike loa, Naučna knjiga, Graževinski fakultet, Beograd, Tioshenko S. P. and Goodier J. N., (95), Theory of Elasticity, New York LOKALNO STANJE NAPONA I STANJE DEFORMACIJA U OBLASTI PRSLINE ZA RAZLIČITA GLOBALNA STANJA NAPONA U PLOČI Dragan B. Jovanović, Milena B. Jovanović U ploči u kojoj vladaju različiti slučajevi globalnog ili opšteg stanja napona, javiće se u okolini prsline različita lokalna stanja napona i deforacija, koja su posledica eđusobnog uticaja geoetrije prsline i globalnog stanja napona. Uziajući u razatranje prslinu eliptičnog oblika (Griffith-ovu prslinu), koja se ože svesti na zasek zadavanje jedne polu-ose jednake nuli i uvodeći različita opšta naponska stanja, dobijeni su odgovarajući lokalni rasporedi napona i poeranja u okolini takve prsline. Pošlo se od pretpostavke da je globalno stanje napona ravno, a dobijeno je trodienzionalno stanje napona i poeranja u okoloni prsline. Prienjena je etoda konačnih eleenata i načinjen je trodienzionalni odel ploče sa prslino. Globalna opterećenja

10 9 D. B. JOVANOVIĆ, M. B. JOVANOVIĆ su zadata tako da su naponi u delu ploče bez poreećaja usled uticaja prsline jediničnih vrednosti i izvršena je noralizacija lokalnih napona u odnosu na globalne napone. Konvencionalna teorija ploča pretpostavlja da je unapred poznata proena napona u funkciji od koordinate z u pravcu debljine ploče. Za ravno stanje napona pretpostavlja se da je koponentni noralni napon σ z jednak nuli svuda u ploči, a da su naponi u ravni ploče σ x, σ y, i τ xy nezavisni od koordinate z. Ovakve pretpostavke se ogu prihvatiti pod vrlo ograničeni uslovia. Razatrani su slučajevi globalnog stanja napona kada aksijalne zatežuće sile deluju u pravcu upravno na ravan prsline ili u pravcu prsline. U radu su dati grafički prikazi rasporeda napona σ x, σ y, σ z, τ xy, τ yz, τ zx, kao i poeranja u x, u z i u z. Pokazalo se da su koponente tenzora napona u funkciji od koordinate z u pravcu debljine ploče i da je σ z različito od nule do određenog rastojanja od vrha prsline. Na osnovu dobijenih dijagraa rasporeda napona i poeranja, doneti su zaključci o uticaju globalnog stanja napona na lokalni raspored napona i poeranja u okolini prsline.