Stationary thermoelasticity problem for a solid cylinder with a disc-shaped crack enclosed in a thin shell

Аuthors

, Kirichek V. A.^*

Taganrog institute named after A.P. Chekhov (Branch of the Rostov State University of Economics), Initsiativnaya str., 48, Taganrog, 347936, Russia

*e-mail: Zhornik_Victoria@mail.ru

Abstract

The axisymmetric stationary thermoelasticity problem for a continuous cylinder of unlimited length with a transverse coaxial disc-shaped crack surrounded by a thin extremely «soft» or extremely «hard» shell with a sliding seal on the contact is considered. A flat heat source of a given intensity acts in front of the top of the crack, and the banks of the crack are maintained at a known temperature; a normal load is applied to them. Thermal, temperature and force effects depend on the radius. There is heat exchange between the outer cylindrical shell’s surface and the medium of zero temperature. Based on the symmetry of the problem with respect to the crack plane, the thermal elasticity problem was reduced to the one for a semi-infinite solid cylinder with mixed boundary conditions at the end. The heat transfer problem solution was found in the form of integral representation, which included multiplications of Bessel function for the radial variable and exponential and trigonometric ones for the axial variable. The thermostressed state problem solution for the cylinder was found by the Goodyear method and reduced to the Fredholm integral equation of the second kind solution with respect to certain function determining the thermostressed state of the cylinder with a disc-shaped crack, in particular, the axial stress before it’s tip. Since the most important cracks’ characteristic — the stress factor controlling the cracks’ development — is determined in the present problem by the axial stress before the crack tip, that is why it depends on the required function, which is the solution of the Fredholm integral equation of the second kind mentioned above. The solution of this equation was found by the successive approximations’ method. The obtained solution of the thermoelasticity problem for a cylinder with a disc-shaped crack covers all the special cases for the cylinder and space authors know (for this purpose, the cylinder radius should be directed to infinity). The special case is considered in the paper: numerical calculation and analysis is carried out for a flat heat source of finite width of constant power acting before the crack’s tip, the constant temperature is maintained on its’ banks and constant load is applied to them; the heat exchange coefficient is accepted equal to infinity. The dependences of the breaking load applied to the crack banks on the crack banks’ temperature at different powers of the flat heat source acting in front of the crack tip in case the stress factor reaches the critical value (material fracture toughness) have been determined. Dependencies for different cases of influence are analyzed.

References

1. Zhornik V.A. Razvitie diskoobraznoj treshchiny v sploshnom cilindre s istochnikami tepla [Development of a discshaped crack in a solid cylinder with heat sources]. Teplovye processy v tekhnike — Thermal processes in engineering, 2009, vol. 1, no. 4, pp. 152–158. In Russ

2. Zhornik A.I., Kirichek V.A. Termouprugost’ sploshnogo cilindra s tonkim pokrytiem na cilindricheskoj poverhnosti [Thermoelasticity of a solid cylinder with a thin covering on the cylinder surface]. Teplovye processy v tekhnike — Thermal processes in engineering, 2016, vol. 8, no. 7, pp. 301–311. In Russ

3. Kartashov E.M., Kudinov V.A. Analiticheskaya teoriya teploprovodnosti i prikladnoj termouprugosti [Analytical theory of thermal conductivity and applied thermoelasticity]. Moscow: URSS, 2012. 251 p. In Russ

4. Sneddon I.N. The distibution of stress in the neighbourhood of a crack in an elastic solid. Proceedings of the Royal Society of London, 1946, vol. 187A, pp. 229–260.

5. Zhornik A.I. Ob odnoj stacionarnoj zadache termouprugosti dlya zaklyuchennogo v tonkuyu obolochku sploshnogo neogranichennogo cilindra s diskoobraznoj treshchinoj [On one stationary problem of thermoelasticity for a solid unbounded cylinder with a disk-shaped crack enclosed in a thin shell]. Prochnost’ i zhivuchest’ konstrukcij. Tez. dokl. Vserossijskoj nauch.-tekhn. konf. [Strength and survivability of structures. Theses of reports. All-Russian scientific and technical. conf.]. Vologda, 1993, pp. 47–48. In Russ

6. Das B.R. A note on thermal stresses in a long circular cylinder containing a penny-shaped crack. Int. J. Engng. Sci., 1968, vol. 6, pp. 667–676.

7. Sneddon I.N., Welch J.T. A note on the distribution of stress in a cylinder conntaining a penny-shaped crack. Int. J. Engng. Sci., 1963, vol. 1, pp. 411–419.

8. Coj B., Kartashov E.M., Shevelev V.V. Prochnost’ i razrushenie polimernyh plenok i volokon [Strength and destruction of polymer films and fibers]. Moscow: Himiya, 1999. 495 p. In Russ