Abstract
This paper reports a solution to the problem of determining the motion law of the crystallization front and the thermomechanical state of a two-phase rod for the case of mutual influence of the temperature and mechanical fields. An approximate analytical method has been used to solve the problem, combined with the method of successive intervals and a Gibbs variation principle. This method should indicate what is more beneficial to nature under the assigned external influences ‒ to change the temperature of the fixed element of a body or to transfer this element from one aggregate state to another. It is this approach that has made it possible, through the defined motion law of an interphase boundary, to take into consideration the effect of temperature on the tense-deformed state in the body, and vice versa. The ratios have been obtained to define the motion law of an interphase boundary, the temperature field, and the tense-deformed state in the rod. The results are shown in the form of charts of temperature and stress dependence on time and a coordinate. An analysis of the results shows that changes in the conditions of heat exchange with the environment and geometric dimensions exert a decisive influence on the crystallization process, and, consequently, on temperature and mechanical fields. The principal result is the constructed approximate analytical method and an algorithm for solving the problem on thermoviscoelasticity for growing bodies (bodies with a moving boundary) in the presence of a phase transition considering the heat exchange with the environment. Based on the method developed, the motion law of an interphase boundary, a temperature field, and the tense-deformed state are determined while solving the so-called quasi-related problem of thermoviscoelasticity. An approximate analytical solution has been obtained, which could be used by research and design organizations in modeling various technological processes in machine building, metallurgy, rocket and space technology, and construction
Highlights
The linear theory of thermal conductivity can no longer meet the requirements of both new fields of technology and traditional industrial sectors, such as heat generation, machine building, and, especially, metallurgy
Modern practical tasks require taking into consideration the essential non-stationarity, heterogeneity, non-linearity, and other features to which mathematical methods developed in the classical theory of thermal conductivity are hardly applicable
Study [12] addresses the construction of research methods only for contact problems involving the elements of structures for casting, taking into consideration phase transitions
Summary
The linear theory of thermal conductivity can no longer meet the requirements of both new fields of technology and traditional industrial sectors, such as heat generation, machine building, and, especially, metallurgy. Modern practical tasks require taking into consideration the essential non-stationarity, heterogeneity, non-linearity, and other features to which mathematical methods developed in the classical (linear) theory of thermal conductivity are hardly applicable Such problems can only be solved through the use of non-linear mathematical modeling. The specified processes are associated with a significant change in heat-physical characteristics over a wide range of temperature changes All these are non-linearities of the I, II, and III kinds; it is impossible to further improve modern technologies without considering them. An effective way to study the non-linear problems of mechanics is the use of approximate-analytical approaches The relevance of this scientific issue is predetermined by that the improvement of thermal technologies in order to preserve energy resources, the design of better structures of industrial machines, the choice of their optimal operational modes is an important industrial task. The proposed scientific research should be carried out to ensure that their results could be used in research and design organizations in the simulation of various technological processes in machine building, metallurgy, rocket and space technology, construction, as well as in the training process
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