National University of Science and Technology “MISIS” (Moscow, Russia)

A. S. Budnikov, Cand. Eng., Associate Prof., Dept. of Metal Forming
G. P. Zhigulev, Cand. Eng., Associate Prof., Dept. of Metal Forming
V. A. Fadeev, Cand. Eng., Associate Prof., Dept. of Metal Forming, fdv_viktor@mail.ru
T. Y. Sidorova, Senior Lecturer, Dept. of Metal Forming

The energy model of a deformation object, which is a densely packed system of material particles, provides a fundamental basis for applying the mathematical apparatus of continuous functions. This approach allows for an adequate description of the complex behavior of materials in metal forming processes. When an external force is applied by the tool to the system of particles in the workpiece, the equations of motion formulated in lagrangian coordinates play a key role. Their main advantage lies in the ability to account for the individual loading history of each fixed material particle, which is crucial for analyzing the deformation state of the object. The lagrangian equations not only determine the trajectory and specific motion of each particle but also allow for establishing its precise position in space, described by eulerian coordinates. Within the framework of the model, computational relations were proposed for determining the partial derivatives of the Eulerian coordinates with respect to the Lagrangian variables and time. Based on these, an effective methodology for calculating the kinematic invariants of the motion of material particles was developed. These invariants objectively characterize the deformation state, independent of the reference frame. The use of the Lagrangian description is methodologically preferential because fundamental physical laws are formulated precisely for material particles, not for points in space. The practical significance of the model is confirmed by obtaining invariant characteristics of the deformed state, using examples of plane and axisymmetric upsetting. It is important to note that the equations of motion in lagrangian coordinates open up the possibility of applying the powerful method of superposition. This is especially relevant for modeling complex, combined processes where it is necessary to account for the superposition of several types of deformation. This approach significantly improves the accuracy of prediction and the optimization of technological parameters.

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