The Stressed-Strained State of a Rod at Crystallization Considering the Mutual Influence of Temperature and Mechanical Fields

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.<br><br>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.


Introduction
Recently, 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.
It is known that thermomechanical processes are inherently non-linear. Examples include a series of problems related to modern technology. The aerodynamic heating and destruction of bodies when moving in the atmosphere are considered in [1]; high-temperature processes in rocket enginesin work [2]. Accounting for temperature stresses in the structural elements at high-temperature heat loads is necessary for the development of a new class of small solid-fuel rocket engines [3]. New technologies of thermocyclic destruction of rocks were considered in [4]; the radiation heating in metallurgical furnacesin [5]. Paper [6] examines technologies that use phase transitions; the movements of elastic waves in variable-length ropes for machines that lift and lower weights using ropesin [7].
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. Mathematical difficulties often force a researcher to approach the direct linearization of thermomechanics equations (thermal conductivity and thermoelasticity). In many cases, researchers turn to numerical methods that make it possible to derive approximate solutions to the non-linear problems of mechanics. In particular, work [8] considers the numerical modeling of nonlinear processes in the mechanics of continuous media; [9] solves a non-linear thermal problem in the presence of phase transitions. Paper [10] proposes a numerical procedure to solve the problem of deforming a polymeric crystallizing environment, taking into consideration large deformations. Work [11] reports the results of computer simulations of casting crystallization. Despite the versatility of numerical methods, the results of such studies are approximate, they often involve the linearization of equations and boundary conditions, and are not always convenient for analysis.
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. It is impossible to resolve it without careful and complete mathematical modeling of the crystallization process. This is due to the fact that experimental studies into the features of metal crystallization, for example, in the laypoint of a centrifugal machine are typically complicated (high melt temperatures, the presence of rotation, a gassed mold cavity, etc.).
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.

Literature review and problem statement
Study [12] addresses the construction of research methods only for contact problems involving the elements of structures for casting, taking into consideration phase transitions. The authors consider an associated problem of thermoelasticity in the form of a system of differential equations in particular derivatives and inequalities, which comes down to variational inequalities. However, the issues related to determining the change law of an interphase boundary and taking into consideration the viscoelastic properties of material remained unresolved. The problem of a semi-space with microstructural transformations, exposed to the influence of thermal and mechanical pulses, was considered in [13]. The problem is solved numerically by the method of step-by-step implicit integration over time. There are quantitative estimates of temperature effects of thermostructural mechanical bonding. However, the issue of phase transition was not addressed at all. The initial and boundary-value problems describing the thermomechanical behavior of alloys with a shape memory were investigated in work [14]. Some issues related to non-linearity problems in metallurgical thermomechanics were considered in [15]; the approaches to solving them were outlined. In addition, the cited paper notes the need to develop and construct analytical procedures to study the nonlinear thermal processes; however, no specific analytical methods were considered. Work [16] derived the defining equations for growing zero-thickness surfaces and their spatial discretization using a finite element method. Issues related to the reciprocal influence of mechanical and temperature fields were not addressed. Paper [17] tackles the construction of linear models for thin plates made from polymer gels. A multi-scale approach to modeling the hardening process in magnetic-sensitive polymer materials was proposed in [18]. Work [19] gives a one-dimensional formulation of the finite elements for temporal analysis of geometrically nonlinear frames associated with viscoelastic viscoplastic materials, including mechanical destruction. The issues of numerical modeling of the interaction among inclusions in a flat element of the elastic heterogeneous environment, depending on their orientation, shape, size, and rigidity characteristics, were considered in [20]. The safest options for their mutual location were established. In order to analyze significantly heterogeneous stresses [21], associated with the presence of foreign structures in bodies in the form of inclusions, cracks, pores, etc., the projection and iterative schemes of the implementation of network methods are effective [22].
Thus, one should note that the issue of determining the stressed-strained state of a body, which is in a phase transition, has not yet been considered or addressed comprehensively. Namely, there is no proposed approach, which would make it possible to determine not only the tense-deformed state of a body taking into consideration the mutual influence of temperature and mechanical fields but also to define, based on this relation, the motion law of a crystallization front.

The aim and objectives of the study
The aim of this study is to determine the stressedstrained state in a rod at crystallization, taking into consideration the mutual influence of temperature and mechanical fields.
To accomplish the aim, the following tasks have been set: -to build an approximate analytical method and an algorithm to solve the quasi-related problem of thermoviscoelasticity for growing bodies in the presence of a phase transition, taking into consideration the heat exchange with the environment; -to define, based on the constructed method, the motion law of an interphase boundary, a temperature field, and the stressed-strained state in a rod from the solution to the quasi-related problem of thermoviscoelasticity. configuration, with the x 3 =x axis directed along the rod axis. One end of the rod x=0 is tightly pinched, and the other, x=l, is free. Suppose that the rod substance exists in two phaseshigh-temperature and low-temperature. The behavior of a rod's material in the low-temperature phase is described by the equations of state of a non-uniformly aging viscoelastic body, and in the high-temperatureof an elastic body. Denote the temperature of a phase transition through θ 0 : at θ 2 <θ 0 , the material's element is in the low-temperature phase, at θ 1 >θ 0 -in the high-temperature one. For certainty, we believe that before a deformation, the substance of a rod in the low-temperature phase occupies region ω 2 (t)={l 0 ≤x≤l}, and the substance in the high-temperature phaseregion ω 1 (t)={0≤x≤l 0 }. The thermal interaction between the lateral and end surfaces of the low-temperature phase of the rod with the environment is characterized by convective heat release coefficients α 1, conv , α 2, conv (respectively), and the radiation. In this case, the high-temperature phase is supposed to be heat-isolated.
At time point t=0, a longitudinal distributed compressing load of intensity P(t) is applied to the rod; the heat is released into the environment from the free end of the rod x=l. After this, part of the rod material enters the low-temperature phase (material buildup) so that its length changes according to the law x=a=a(t).
It is required to define, under the assigned law of change in the external load P=P(t), the motion law of an interphase boundary a=a(t), the temperature θ i =θ i (t,x) (i=1,2) and the stressed-strained state in the rod for the case of a single-axial strained state.
A mathematical statement of the formulated problem of thermoviscoelasticity for a rod, growing under the phase transition conditions, implies finding the enumerated unknowns from the joint solution to the following equations: where μ is the heat of the phase transition.

2. The method and algorithm to solve the problem
Ratios (1) to (5) are the statement of the thermoviscoelasticity problem for a growing two-phase rod in a closed form. The difference between this statement and those considered in [24] is that the thermal conductivity problem (2), (4), (5) takes into consideration the non-linearity of the first kind due to the dependence of the thermal-physical characteristics of material on temperature. One approach to solving non-linear thermal conductivity problems is the method of successive intervals [25], which implies splitting the entire process into finite intervals, in each of which the material's characteristics and the internal heat sources are constant. In such a statement, the non-linear problem of thermal conductivity is reduced to solving a set of linear problems with different initial and boundary conditions for each time interval. Therefore, to solve problem (1) to (5), it is proposed to use the approximate analytical method, suggested in [24], in combination with the method of successive intervals and a Gibbs variation principle in the following interpretation.
The time interval [0, t fin ], within which we investigate the stressed-strained state in body Ω, is divided by points t k =kΔ, Δ=t fin /N, k=0, 1,…, N into sub-intervals, so that the continuous gain process is replaced with the following discrete process. At moment t k , the surface efforts f (k) , massive forces F (k) are applied to the body Ω; the amount of thermal energy equal to ω (k) is removed from the body. Let the quantities ū * (displacement vector), θ * (temperature field), and a=a(t k ) (an interphase boundary position) fully describe the true state of the body at each time period. When external conditions change, the body exits the thermodynamic equilibrium ( ) the state it was in at moment t k , and instantly enters a new equilibrium state is a kinematically possible equilibrium state of the body Ω at moment t k +0 (that is, at t>t k ). The true state of the body Ω (implemented in reality) at mo- The inner energy of body Ω in the states ( ) and V (k) , respectively. These quantities are bound by the law of energy conservation where A (k) is the work of external surface and mass forces is characterized by the fact that it delivers the maximum value of a body entropy among all possible equilibrium states determined by equality (6).
In order to fulfill the principle formulated above and thus ensure that mechanical and temperature fields are interconnected, the problem of thermal conductivity (2), (4), (5) must first be solved at each time t k . Its solution takes the following form: Next, we shall determine the field of displacement from the solution to the elastic contact problem of two bodies ω 1 (t k ) and ω 2 (t k ). In this case, let us assume that a substance in the liquid phase occupies region ω 1 (t k )={0≤x≤a(t k )}, in the solid phase -ω 2 (t k )={a(t k )≤x≤l}.
In this case, the contact problem for two bodies ω 1 (t k ) and ω 2 (t k ) at moment t k is reduced to solving the following system of equations (the bar means a derivative for coordinate x) under boundary conditions A solution to problem (7), (8) takes the following form [24]: Calculate the work of external forces A (k+1) when moving from a thermodynamical equilibrium state ( ) , , By substituting expressions (11), (12) in ratio (13) and integrating, we obtain a t a t a t a t Find the inner energy V (k) of a rod in a thermodynamical equilibrium state at moment t k (k=1,2,…,N). To this end, let us assume the following: 1. At overheating, there are intense streams of heat in the liquid phase due to convection; temperature changes in it are almost absent. Therefore, following [26], a temperature difference along the length of the liquid part of the rod is taken as Δθ 1 .
2. According to the law of energy conservation, the difference between the amount of heat that is released by the side surface and the end of the rod by convection and radiation to the environment is equal to the heat accumulated by a given body: where s is the cross-sectional area of a rod; p is perimeter; ( ) 2 m t θ  is the incremental temperature per unit of time in a body; -the amount of heat accumulated in a body when heat spreads in the axial direction; -the amount of heat that the side surface of the rod releases to the environment. Equation (15), taking into consideration the solution for θ 2 (t,x), can be converted to the following form Then, under the assumptions made, the internal energy will be calculated from the following formula ( ) where ( ) (0) 0 0 * 1 0 1 2 0 2 V l l l = ρ ψ + ρ − ψ is the inner energy of a body in its natural state; 0 i ψ (i=1,2) is the specific internal energy of a substance in the liquid and solid phases, respectively, in its natural state. In a given example, assume 0 0. i ψ = We substitute expressions (9), (10) into relation (17), and perform integration over the variable x. We obtain Determine the entropy S (k) in a thermodynamic equilibrium state The entropy ( 1) k S + in a transition from the thermodynamic equilibrium state ( ) ( ) ( ) at time t k to the thermodynamic equilibrium state at moment t k+1 will take the following form Then, in accordance with a Gibbs variation principle, the point a (k+1) =a(t k ), which characterizes the position of an interphase boundary at time t k+1 , delivers the maximum value of function F(a (k+1) )=S (k+1) -S (k) . It follows from the condition for an extremum of the function that ( 1 9 ) Substituting ratios (14), (18) and function θ 2 (t, x) into (19), we obtain Believing that the a(t), P(t), θ i (t, x) functions are continuous and are differentiated for t over the examined time period, we assume where the ellipsis denotes terms of the Δ 2 order. Substituting (21) into θ i (t,x) and (20), by moving to the limit at N→∞, Δ→0, t k →t, we obtain   (22) to (24) are a system of non-linear equations to determine the temperature field in a rod and the position of an interphase boundary.
Transform equations (16), (24). To this end, differentiate equation (24) for t; substitute ( ) 2 m t θ  from (16) into the result. As a result, we shall have the following system of differential equations to determine the motion law of a crystallization front and the mean temperature in a rod: Once the function a=a(t) is determined, a temperature field in the rod is determined from formulae (22), (23), and the stressed-strained statefrom formulae (9) to (12).
Thus, one can build the following algorithm to determine the stressed-strained state in a body Ω during the phase transition. At moment t k : 1) fix the interphase surface a(t k ); determine, from a solution to the stationary problem of thermal conductivity, the temperatures ( ) ( ) ) in a body Ω; 2) solve the contact problem for two bodies occupying the regions ω 1 (t k ) and ω 2 (t k ), and determine the displacement field ( ) ; k u 6) at the next time t k+1 , the initial temperature and the position of an interphase boundary correspond to those found at moment t k , that is In this case, the proposed algorithm for solving the associated problem of thermoviscoelasticity takes into consideration the backstory of the entire process at each point of time t k in the motion law of an interphase boundary, determined from a Gibbs variation principle.

Solving the quasi-related problem of thermoviscoelasticity for a rod at crystallization
To illustrate the applicability of the proposed method, let us consider specific examples: the crystallization (build-up) of a rod with a heat-insulated side surface; the problem of building up a rod when interacting with the environment taking into consideration a phase transition.

1. Solving the problem of crystallization (build-up) of a rod with a thermally insulated side surface
For the case of crystallizing a rod with a heat-insulated side surface(α 2,conv =α 2 =0) in problem (1) to (5) the expressions defining the motion law of an interphase boundary, as well as a temperature field in the solid phase, are simplified and take the following form   and the stressed-strained state is determined from formulae (9) to (12).

2. Solving the problem of building up a rod in thermal interaction with the environment, taking into consideration the phase transition
For the case of crystallization (build-up) of a rod in thermal interaction between the lateral and end surfaces and the environment taking into consideration the phase transition, the results of solving the equations (25), (26) are shown in Fig. 7, 8. The results of the problem solving showed that both the geometric dimensions of a body and the conditions of heat exchange with the environment exert a significant impact on the hardening process.

Discussion of results obtained in solving the quasi-related problem of thermoviscoelasticity for a rod at crystallization
An analysis of our results demonstrates that cooling intensification, first, reduces the hardening time, and, second, increases the size of the solid phase at a fixed time value (Fig. 1). Fig. 2 show that the increase in heat release factor leads to a sharp drop in the temperature θ 2 (t, l) at the initial mo-ment in time. Thereafter, this temperature changes not so dramatically and, throughout the entire hardening process, it is a monotonously descending function. At high heat release factor values, the end surface is instantly cooled to a temperature close to ambient temperature and remains constant until the end of the process.
The charts shown in Fig. 4 demonstrate that changing the initial length of the solid phase has a significant impact on the crystallization process. This is confirmed by the results shown in Fig. 5, 6.
Thus, it has been established that both the geometric dimensions of a body and the conditions of heat exchange with the environment have a determining influence on the hardening process, which is consistent with known experimental and estimated data [27,28]. Fig. 7 shows that if there is a heat exchange between the side surface and the environment, it significantly reduces the time of the hardening of the material.
The charts shown in Fig. 8 demonstrate that the heat exchange from the side of a rod has a greater impact on the crystallization process than that from the end surface, as evidenced by known experimental and estimated data [27,28].
The results obtained are explained by the statement of the problem and the method of solving it.
One of the features of the proposed method is that the motion law of an interphase boundary is determined analytically from the joint solution to the contact problem and the problem of thermal conductivity, rather than set from the outside or determined from the experiment. The second feature is that the combined application of the method of successive intervals and a Gibbs variation principle makes it possible to take into consideration the mutual influence of mechanical and temperature fields not only on each other but also on the motion law of an interphase boundary.
All functions that are part of the defining ratios should be continuous.
It should be noted that the issue of solving the related (rather than quasi-related) problem of thermoviscoelasticity remains open. Under a classical statement, the boundness of the fields requires that either the temperature should be part of the mechanical equations of state or vice versa. This point has not yet been taken into consideration in this paper but it is the subject of further research.
One of the options for advancing the current study could tackle the need to take into consideration the dependence of heat-physical characteristics on time. However, it might be difficult to derive an analytical solution.

Conclusions
1. An approximate analytical method and an algorithm have been constructed for solving the quasi-related problem of thermoviscoelasticity for growing bodies in the presence of a phase transition considering heat exchange with the environment. This method makes it possible to assess the influence of both geometric and physical parameters of the problem on the formation of the stressed-strained state of a body taking into consideration the connection between thermomechanical fields, as well as analytically determine the motion law of a crystallization front.
2. Based on the developed method, two applied problems have been solved: the crystallization (build-up) of a rod with a heat-insulated side surface; the problem of building up a rod when interacting with the environment, taking into consideration the phase transition. The results of solving these problems have shown that both the geometric dimensions of a body and the conditions of heat exchange with the environment have a significant impact on the hardening process. Namely, a change in the initial length of the solid phase affects the process of crystallization and, as a result, the stressed-strained state of the body. The presence of heat exchange between a side surface and the environment significantly reduces the hardening time of a material. In this case, the heat exchange from the side surface of the rod has a greater impact on the crystallization process than from the end surface.

Introduction
In recent years, in agricultural production for the purification, separation and processing of cereals, seeds, fruits and other bulk materials, technologies using electric and magnetic fields have spread [1][2][3][4][5]. On a large amount of experimental data, the effectiveness of the effect of an artificial electromagnetic field for stimulating the vital and growth processes of plants has been proved [4,5]. Therefore, the use of electrophysical methods of processing biological objects is a good alternative to using it in agricultural production to increase crop yields of chemicals and genetically