Abstract
A particular solution of a linear variant of the dynamic thermal elasticity problem is considered in application to modeling the conditions of surface hardening of metal products by an energy pulse. The authors determined the equation of medium motion with the model of temperature pulse tested earlier for compatibility with special cases of the equations of parabolic and hyperbolic thermal conductivity. The problem of loading a flat plane of a short circular cylinder (disk) with a temperature pulse is presented. Pulse is a consequence of adopted structure of the volumetric power density of the heat flux, the time multiplier of which has the form of a single wave of the Heaviside function. Classical thermoelastic displacement potential and the method of its division into the product of independent variables functions were used to construct the thermal stress tensor. Differential equations for multiplier functions and their general solutions were found. Natural boundary conditions were set for the components of thermal stress tensor, and their tasks were solved. The obtained solutions are in the form of segments of functional series (the Bessel function in radial coordinate and the exponential function in axial coordinate). The article considers a numerical example of loading a disk made of 40KhN steel which has the mechanical properties sensitive to temperature treatment. Maple computer mathematics package was used in the calculations. Approximate solutions take into account the first 24 terms of the functional series. Estimation of the example makes it possible to explain the presence of stress peaks and stress intensity as a consequence of mutually inverse processes of temperature stress growth and reduction of elasticity coefficients with temperature rise. The numerical example warns against relying only on estimates of solutions to thermoelasticity problems without taking into account the plastic and viscous properties of the material.
Highlights
A particular solution of a linear variant of the dynamic thermal elasticity problem is considered in application to modeling the conditions
The authors determined the equation of medium motion with the model
Pulse is a consequence of adopted structure of the volumetric power density
Summary
Моделирование термических напряжений при упрочнении поверхности изделия тепловым импульсом. Для построения тензора термических напряжений авторы использовали классический термоупругий потенциал перемещений и метод его разделения на произведение функций независимых переменных. Получены дифференциальные уравнения для функций-сомножителей, найдены их общие решения. Для компонент тензора термических напряжений поставлены естественные граничные условия. Полученные решения имеют форму отрезков функциональных рядов (функции Бесселя по радиальной координате и экспоненциальной функции по осевой координате). Рассмотрен численный пример нагружения диска из стали марки 40ХН, механические свойства которой чувствительны к температурной обработке. При расчетах авторы использовали пакет компьютерной математики Maple. Приближенные решения учитывают первые 24 члена функциональных рядов. Ключевые слова: круглый цилиндр, поверхностное упрочнение, термические напряжения, термоупругий потенциал перемещений, температурный импульс, граничные условия, сталь марки 40ХН, закалка, отпуск. Моделирование термических напряжений при упрочнении поверхности изделия тепловым импульсом // Известия вузов.
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