Mathematical modeling of thermomechanical hardening in deformable solids based on a generalized Riccati equation

Abstract

The paper proposes a mathematical model of the thermomechanical hardening process in deformable solids based on a generalized Riccati equation with time-dependent coefficients. The relevance of the study is determined by the need for compact mathematical models capable of adequately describing the evolution of hardening processes under coupled mechanical loading and thermal effects while avoiding the complexity of large-scale multiparameter thermomechanical models. The aim of the study is to develop a mathematical model of thermomechanical hardening based on a generalized Riccati equation, investigate its qualitative properties, and develop an efficient numerical solution algorithm. The proposed model is derived using a local quadratic approximation of generalized hardening accumulation and relaxation functions, resulting in a non-autonomous Riccati equation whose coefficients possess a clear physical interpretation. A qualitative analysis of the model is performed to investigate the influence of the governing parameters on the evolution of the state function and to determine the conditions leading to stabilization of the hardening process. The initial value problem is solved numerically using the classical fourth-order Runge–Kutta method, which provides high computational accuracy and efficiency for parametric studies. A series of numerical experiments confirms the applicability of the proposed mathematical model and demonstrates good agreement between the theoretical analysis and numerical simulations. The results show that variations in the coefficients of the generalized Riccati equation make it possible to reproduce different evolution regimes of the thermomechanical hardening process, whereas the nonlinear coefficient has the greatest influence on the solution behavior by controlling the intensity of nonlinear self-regulation. The scientific novelty of the study consists in deriving a generalized Riccati equation from a physically justified balance between hardening accumulation and relaxation processes using a local quadratic approximation rather than introducing it phenomenologically. The proposed model can be used for parametric analysis, numerical simulation, and prediction of thermomechanical hardening processes under various technological operating conditions.