Vol.3, No 13, 2003 pp. 599-612
UDC 532.59 539.3 539.421 530.145.6

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NEW CLASS OF SOLUTIONS OF THE REDUCED WAVE EQUATION APPLICABLE TO CRACK PROBLEMS
RELATED TO GRADIENT ELASTICITY
David J. Unger¹, Elias C. Aifantis^2
1Department of Mechanical and Civil Engineering, University of Evansville,
1800 Lincoln Avenue, Evansville, IN 47722, USA
²Center for Mechanics of Materials and Instabilities, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA

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Using complex variables, the two-dimensional reduced wave equation is transformed into an equation composed of two Hankel operators. Due to the symmetry of these operators, solutions of the transformed equation appear as products of Bessel functions, where each individual function's argument is associated with one of the newly defined independent variables. When a particular class of these solutions is identified with linear elastic displacement for finite length cracks, stresses are generated at the tips possessing the characteristic inverse square root singularity. As an application of this special class of solutions, an internal crack problem subject to the constitutive assumptions of the Aifantis strain gradient elastic theory is posed and solved. In this particular gradient elasticity theory, the analogous linear elastic displacement and stress is incorporated into the solution of the corresponding gradient elasticity problem.

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NOVA KLASA REŠENJA REDUKOVANE TALASNE JEDNAČINE PRIMENLJIVE NA PROBLEME PRSLINA
U VEZI SA GRADIJENT ELASTIČNOŠĆU Korišćenjem kompleksnih promenljivih, dvodimenzionalna redukovana talasna jednačina transformiše se u jednačinu koja se sastoji od dva Hankel operatora. Usled simetričnosti ovih operatora, rešenja transformisane jednačine javljaju se kao proizvodi Besel funkcija, gde je svaki pojedinačni argument funkcije vezan za jedan od novoodredjenih nezavisnih promenljivih. Kada se odredjena klasa ovih rešenja identifikuje sa linearnim elastičnim pomeranjem za prsline konačne dužine, stvaraju se naponi na vrhovima koji su karakteristični po obrnutoj vrednosti kvadratnog korena. Kao primena ove posebne klase rešenja, postavlja se i rešava problem unutrašnje prsline koji podleže sustinskim pretpostavkama Aifantis elastične teorije gradijenta napona. Unutar ove gradijentske teorije elastičnosti, analogno linearno elastično pomeranje i napon uračunati su u rešenje odgovarajućeg problema gradijenta elastičnosti.