Vol.3, No 2, 2005 pp. 145 - 153
UDC 624.191.8(045)=20
ANALYTICAL APPROACH FOR RESOLVING STRESS STATES AROUND ELLIPTICAL CAVITIES
Dragan Lukić1, Petar Anagnosti2
1University of Novi Sad, Faculty of Civil Engineering Subotica, Serbia and Montenegro
2University of Belgrade, Faculty of Civil Engineering, Belgrade, Serbia and Montenegro
Abstract. The determination of stress states around cavities in the stressed elastic body, regardless of cavity shapes, that may be spherical, cylindrical, elliptical etc. in its analytical approach has to be based on selection of a stress function that will satisfy biharmonic equation ? 2? 2? = 0, under given boundary conditions. This paper is concerned with formulation and solution of the cited differential equation using elliptical coordinates in conformity with the cavity shape of oblong ellipsoid [1]. It is therefore considered that the formulation of the stress tensor will be done in conformity to the cited coordinates.
The paper describes basic statements and definitions in connection to harmonic functions used for determination of stress states around cavities formed in the stressed homogeneous space. The particular attention has been paid to the use of Legendre's functions, with definitions and derivation of recurrent formulas, that have been used for determination of stress states around an oblong ellipsoidal cavity, [1]. The paper also includes the description of procedures used in forming series based on Legendre's functions of the first order.
Key words: coordinates, biharmonic differential equation, stress state, stress function, harmonic functions, recurrent formulas
MATEMATIČKE OSNOVE ODREĐIVANJA
NAPONSKIH STANJA OKO ELIPTIČNIH OTVORA
Određivanje naponskih stanja oko otvora predstavlja veoma složen matematički problem. Zbog toga, pri razmatranju ovog problema potrebno je najpre definisati pojedine oblasti matematičke analize koje se pri tome koriste.
Prvi deo rada razmatra rešavanje biharmonijskih diferencijalnih jednačina ? 2? 2? = 0 uzimajući u obzir rešenje Papkovič ? Neubera ?1?.
U drugom delu rada definišu se odredjene klase specijalnih harmonijskih funkcija (tipa Ležandra) kao posebno značajnih za analizu naponskih stawa.Pored toga, u radu se prikazuju rekurentne formule definisane u radu ?1? kao i predstavljanje funkcija u obliku reda po Ležandrovim polinomima.