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Abstract <p>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.</p>

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Numerical Solution of the Equations of A.A. Ilyushin’s Coplanarity Hypothesis for a Loading Process in the Deviatoric Stress Space

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • S. L. Subbotin,
  • A. A. Alekseev

摘要

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.