The article presents calculation formulas and a technique for the practical determination of the functionals of the theory of elastic-plastic processes based on experimental data for plane strain trajectories. To obtain the calculation formulas, a vector representation of stresses and strains is used, and a constitutive relation in the form of A.A. Ilyushin’s coplanarity hypothesis is applied. In the presented method for calculating stress vector components, experimental data are recalculated to equally spaced points along the strain trajectory using linear interpolation. The resulting data are smoothed to reduce random experimental errors using the least-squares method and known formulas. Using the smoothed data and numerical differentiation formulas, the derivatives of the stress vector components along the strain trajectory are found, and from these, the values of the plasticity functionals along the given strain trajectory are determined. The accuracy of the result using the applied methodology is assessed by the proximity of the experimental values of the stress vector components to the coplanarity hypothesis relations calculated by numerical integration. The results of processing an experiment involving complex loading in the deviatoric stress space with axial force and torque (P + M experiment) on a thin-walled tubular specimen made of 45 steel are presented. The results of calculating the plasticity functionals along a plane curved strain trajectory consisting of eight successive semicircles are presented in clear graphical form. The functionals are often complex, which complicates their direct use in mathematical models of plasticity theory. Therefore, approximations of the functionals are often used, allowing them to be replaced with simpler expressions while ensuring sufficient accuracy. The presented method for determining plasticity functionals can be used to process experimental data and construct approximations of the functionals.
The paper presents experimental data on complex deformation of thin-walled cylindrical shells. The material of the samples is structural steel with a carbon content of 0.45
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.
