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L. V. Stepanova

ASYMPTOTIC ANALYSIS OF THE STRESS FIELD AT A CRACK TIP IN A LINEARLY ELASTIC MATERIAL: EXPERIMENTAL DETERMINATION OF WILLIAMS EXPANSION COEFFICIENTS

The paper deals with analytical determination of the coefficients of the complete
Williams asymptotic expansion for the stress field at the tips of two collinear cracks in an infinite elastic plate under mixed-mode (Mode I and Mode II) loading. A method for the determination of the coefficients is presented, which is based on the classical complex representation of the Kolosov-Muskhelishvili solution and its series expansion in the vicinity of the crack tip. An analytical representation of the coefficients of the complete Williams asymptotic expansion (T-stresses and higher-order approximation coefficients) as functions of applied loads, crack lengths and inter-crack distances is found for a plate with two collinear cracks. The paper presents experimental results on the photoelastic study of the stress field at the crack tips of two collinear cracks in a plate made of an optically active material (epoxy resin). It is demonstrated that higher-order approximations must be kept in the Williams asymptotic expansion for the accurate description of the stress field and the correct processing of the interference fringe pattern. The longer the distance from the crack tip to the point on the isochromatic fringe, the more terms of the asymptotic expansion need to be kept.

Keywords: Williams asymptotic expansion, experimental determination of the coefficients of the Williams power series expansion, photo elasticity method, higher-order terms of the Williams asymptotic expansion

Bibliography:

1.Hello G., Tahar M.-B., Roelandt J.-M. Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium. International Journal of Solids and Structures, 2012, vol. 49, iss. 3–4, pp. 556–566. DOI: 10.1016/j.ijsolstr.2011.10.024.

2. Stepanova L.V. Matematicheskie metody mekhaniki razrusheniya [Mathematical methods of fracture mechanics]. Samara, Samarskii universitet Publ., 2006, 232 p. (In Russian).

3.Muskhelishvili N.I. Nekotorye osnovnye zadachi matematicheskoy teorii uprugosti [Some basic problems of the mathematical theory of elasticity]. Moscow, Nauka Publ., 1966, 708 p. (In Russian).

4.Gupta M., Alderliesten R.C., Benedictus R. A review of T-stress and its effects in fracture mechanics. Engineering Fracture Mechanics, 2015, vol. 134, 218–241. DOI: 10.1016/j.engfracmech.2014.10.013.

5.Matvienko Yu.G. The effect of the non-singular t-stress components on crack tip plastic zone under mode I loading. Procedia Materials Science, 2014, vol. 3, pp. 141–146. DOI: 10.1016/j.mspro.2014.06.026.

6.Stepanova L.V., Adylina E.M. Stress-strain state in the vicinity of a crack tip under mixed loading. Journal of Applied Mechanics and Technical, 2014, vol. 55, iss. 5, pp. 181–194. DOI: 10.1134/S0021894414050186.

7.Kobayashi A.S. Handbook on Experimental Mechanics, Society for Experimental Mechanics, Seattle, 1987.

         

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Founder:  Institute of Engineering Science, Russian Academy of Sciences (Ural Branch)
Chief Editor:  E.S. Gorkunov
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