 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 13653 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8223
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| In English (Mech. Solids): | | 5430 |
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| S.L. Subbotin and A.A. Alekseev, "Numerical Solution of the Equations of A.A. Ilyushin’s Coplanarity Hypothesis for a Loading Process in the Deviatoric Stress Space," Mech. Solids. 60 (8), 6660-6666 (2025) |
| Year |
2025 |
Volume |
60 |
Number |
8 |
Pages |
6660-6666 |
| DOI |
10.1134/S0025654425605518 |
| Title |
Numerical Solution of the Equations of A.A. Ilyushin’s Coplanarity Hypothesis for a Loading Process in the Deviatoric Stress Space |
| Author(s) |
S.L. Subbotin (Tver State Technical University, Tver, 170026 Russia, sbtn@yandex.ru)
A.A. Alekseev (Tver State Technical University, Tver, 170026 Russia, alexeew@bk.ru) |
| Abstract |
The article presents calculation formulas and an algorithm for the numerical solution of the
equations of the theory of elastoplastic processes in the form of a coplanarity hypothesis when specifying a stress loading process. The plasticity functionals in the calculations must correspond to the
specified stress trajectory and the experimental response in the form of a strain trajectory, regardless
of the form of presentation of the experimental results — either depending on the arc length of the
stress trajectory or on the arc length of the strain trajectory. To assess the reliability of the specified
plasticity functionals, formulas expressing these functionals in terms of the loading process parameters
are presented. It is important to note that these formulas cannot be used in the calculation algorithm
due to the occurrence of feedback, leading to divergence in the calculation process. Suitable approximations of the plasticity functionals must be specified instead. The algorithm for integrating the equations of the theory of elastoplastic processes is based on the second-order Runge-Kutta method with
the calculation of all parameters using a single-step “prediction-correction” computational scheme
(the Euler-Cauchy method). The calculation formulas of the theory of elastoplastic processes in their
direct (kinematic or hard loading) and inverse (force or soft loading) forms are theoretically equivalent. However, a practical solution requires a sufficiently precise specification of stress trajectories.
Such calculations have their own peculiarities, and in some cases cannot be implemented at all. It is
shown that in the case of a constant stress deviator modulus (for example, in the absence of hardening
in the stress–strain diagram; passing through a yield plateau; loading along a circular arc centered at
the origin of the stress space coordinate system), the numerical solution becomes indeterminate due
to the vanishing of the principal determinant of the system of calculation equations. When approximating stress trajectories, for example, by circular arcs with a displaced center, the solution does not
suffer from this uncertainty. |
| Keywords |
plasticity, stress space loading, coplanarity hypothesis, numerical integration |
| Received |
10 May 2025 | Revised |
11 August 2025 | Accepted |
24 December 2025 |
| Link to Fulltext |
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