Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Lomakin E. V., Minaev N. G. Axisymmetric Stress Field Near a Circular Cut in a Solid with Stress State Dependent Plastic Properties. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 3, pp. 317-325. DOI: 10.18500/1816-9791-2019-19-3-317-325

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2019
Full text:
(downloads: 98)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
539.3

Axisymmetric Stress Field Near a Circular Cut in a Solid with Stress State Dependent Plastic Properties

Autors: 
Lomakin Evgenii Viktorovich, Lomonosov Moscow State University
Minaev Nikita G., Lomonosov Moscow State University, Institute of Mechanics, Russia
Abstract: 

The paper analyzes the properties of the constitutive relations of the theory of plasticity for a continuum, which plastic properties depend on the type of stress state. The plasticity condition presented in the corresponding generalized form is used, where the parameter of the type of stress state is introduced, which is the ratio of the hydrostatic stress component to the equivalent von Mises stress, named in the literature the stress triaxiality. For a particular type of plasticity condition, an analytical solution of the problem for a solid with a circular hole under plane strain is obtained. The stress distributions corresponding to the obtained solution are compared with the ones for a solid whose plastic properties are invariant to the stress state using the Huber – Mises plasticity condition. The influence of the degree of sensitivity of materials properties to the type of stress state on the stress distributions in plastic areas are studied.

References: 
  1. Drucker D. C., Prager W. Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics, 1952, vol. 10, no. 2, pp. 157–165.
  2. Mohr O. Welche Umstande begingen bie Elastizitatsgrenze und den bruch eines Materiales? Zeitschrift des Vereins deutscher Ingenieure, 1900, vol. 44 (45), pp. 1524–1530.
  3. Radaev Yu. N. Instantaneously not Elongated Directors in Three-dimensional Kinematics of the Coulomb – Mohr Medium. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 4, pp. 467–483 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2018-18-4-467-483
  4. Lomakin E. V. Dependence of the limit state of composite and polymer materials on the type of the stress state. 1. Experimental dependences and determining equations. Mech Compos Mater, 1988, vol. 24, iss. 1, pp. 1–7. DOI: https://doi.org/10.1007/BF00611327
  5. Rudnicki J. W., Rice J. R. Conditions for the localization of deformation in pressuresensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 1975, vol. 23, iss. 6, pp. 371–394. DOI: https://doi.org/10.1016/0022-5096(75)90001-0
  6. Alexandrov S. Geometry of plane strain characteristic fields in pressure-dependent plasticity. ZAMM: Zeitschrift fur Angewandte Mathematik und Mechanik, 2015, vol. 95, iss. 11, pp. 1296–1301. DOI: https://doi.org/10.1002/zamm.201400017
  7. Stavrogin A. N., Protosenya A. G. Plastichnost’ gornykh porod [Plasticity of rocks]. Moscow, Nedra, 1979. 301 p. (in Russian).
  8. Deshpande V. S., Fleck N. A. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids, 2000, vol. 48, iss. 6–7, pp. 1253–1283. DOI: https://doi.org/10.1016/S0022-5096(99)00082-4
  9. Green R. J. A plasticity theory for porous solids. International Journal of Mechanical Sciences, 1972, vol. 14, iss. 4, pp. 215–224. DOI: https://doi.org/10.1016/0020-7403(72)90063-X
  10. Miller R. E. A continuum plasticity model for the constitutive and indentation behaviour of foamed metals. International Journal of Mechanical Sciences, 2000, vol. 42, iss. 4, pp. 729–754. DOI: https://doi.org/10.1016/S0020-7403(99)00021-1
  11. Onck P. R. Application of a continuum constitutive model to metallic foam DEN-specimens in compression. International Journal of Mechanical Sciences, 2001, vol. 43, iss. 12, pp. 2947–2959. DOI: https://doi.org/10.1016/S0020-7403(01)00060-1
  12. Vatulyan A. O., Lyapin A. A., Kossovich E. L. Studying of Elastoplastic Properties of Coal Specimens Using Indentation Technique. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 4, pp. 412–420. DOI: https://doi.org/10.18500/1816-9791-2018-18-4-412-420
  13. Seltzer R., Cisilino A. P., Frontini P. M., Yiu-Wing Mai Determination of the Drucker – Prager parameters of polymers exhibiting pressure-sensitive plastic behaviour by depthsensing indentation. International Journal of Mechanical Sciences, 2011, vol. 53, iss. 6, pp. 471–478. DOI: https://doi.org/10.1016/j.ijmecsci.2011.04.002
  14. Alexandrov S., Jeng Y-R., Lomakin E. An exact semi–analytic solution for residual stresses and strains within a thin hollow disc of pressure-sensitive material subject to thermal loading. Meccanica, 2014, vol. 49, iss. 4, pp. 775–794. DOI: https://doi.org/10.1007/s11012-013-9826-4
  15. Alexandrov S., Jeng Y-R., Lomakin E. Effect of Pressure-Dependency of the Yield Criterion on the Development of Plastic Zones and the Distribution of Residual Stresses in Thin Annular Disks. ASME Journal of Applied Mechanics, 2011, vol. 78, iss. 3, pp. 031012–1–031012–5. DOI: https://doi.org/10.1115/1.4003361
Received: 
19.05.2019
Accepted: 
16.06.2019
Published: 
31.08.2019