The heat treatment is one of the main stages of the production of metal devices. The quality and resource of these devices depend on their correct implementation. The main factors that ensure the quality of heat treatment are temperature and time. Thus, the above properties of these devices are determined by the cooling rate of the metal [1–4]. There are different approaches that give the possibility to control the cooling rate of metal products, which were elaborated. The manufacturing of different structures in a metallic material is accompanied by the redistribution of various structural defects (atoms, dislocations, ...) [1], [2], [4], [5]. The final fixed structural state of the metal is characterized by an internal stress consisting of the interaction of elastic fields of lattice stresses and defects. Residual stresses arising during heat treatment, combined with workloads, have a negative impact on operational reliability and service life. The main aim of the present paper is the formulation and analysis of a model of changes of stress in metals during heat treatment. The accompanying aim of the present paper is to elaborate on an analytical approach to analyze the considered model.

To solve the considered aim, we determine the distribution of temperature T (x,y,z,t) in space and time by solving the second Fourier's law [6], [7].

with boundary and initial conditions

T (x, y, z, 0) = Tr.

Here  is the coefficient of thermal conductivity. The value of the coefficient depends on the properties of the considered materials and temperature. Dependence of the coefficient of thermal conductivity could be approximated by the following function: (x,y, z,T)=ass(x,y,z)[1+ Td/T(x,y,z,t)]. c (T) = cass[1- exp(-T (x,y,z,t)/Td)] is the heat capacitance; Td is the Debye temperature. In the case when the current temperature T (x,y,z, t) is equal to or larger than the Debye temperature Td, then we can use the following approximation c (T)  cass. This issue is of most interest for our case. p (x,y,z,t) is the volumetric density of power, which is released in the considered sample. First of all, we estimate the distribution of temperature. To make the estimation, we solve Eq. (1) by the method of averaging of function corrections [8–10]. In the framework of this method, we replace the distribution of temperature in space and time by an unknown average value 1T on the right side of Eq. (1). The replacement gives a possibility to obtain the following equation to determine the first-order approximation of the considered temperature.

Integration of the left and right sides of Eq. (3) in time gives a possibility to obtain the required approximation in the following form.

Not yet known average value 1T was calculated by the following standard relation [8–10].

The second-order approximation of temperature was calculated by standard replacing of the required function T (x,y,z,t) on sum of average value of the considered approximation and approximation with the previous order (i.e. T (x,y,z,t)T2+T1(x,y,z, t)) in the right side of Eq. (1). The replacement gives a possibility to obtain equation to calculate the considered second-order approximation in the following form.

Termwise differentiation gives a possibility to transform the equation to the following form.

(6a)

Integration of the above equation over time gives a possibility to obtain an equation to determine the second-order approximation of temperature in the following form.

(6b) Average value T2 was calculated by using the following standard relation [8–10].

Substitution of the first- and the second-order approximations of the considered temperature gives a possibility to obtain the following equation to determine the average value T2.

Solution of the above equation (i.e., average value T2) depends on the value of the parameter . The solution of the above equation could be obtained by standard approaches [11]. Stress in metals was estimated by solving the following system of equations [12].

Here ij is the stress tensor, which equal to

;  is the density of metal; ij is the

Kronecker symbol; E is the tensile modulus (Young modulus); ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector ; xi, xj are the coordinate x, y, z;  is Poisson coefficient; 0 = (as-aEL)/aEL is the mismatch parameter; as, aEL are lattice distances in different areas of the considered material; K is the modulus of uniform compression;  is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for our case) with room temperature. With account of the above relation for the stress tensor, the system of Eq. (9) could be written as

Conditions for the above equations could be written as Now we determine solutions of Eq. (10). To determine the first-order approximations of the considered components in the framework of the method of averaging of functions corrections [8–10], we substitute not yet known average values i of the considered components instead of the required functions. The substitution gives a possibility to obtain the following equations to determine the considered approximations.

Integration of the left and right sides of the above equations at time t leads to the following results.

Approximations of the considered components of the displacement vector with the second order and approximations with higher order could be calculated by standard replacement of the required functions in Eq. (8) on the following sum i+ui-1(x, y,z,t) [8–10]. The replacement gives a possibility to obtain equations to calculate the second-order approximations in the following form.

Integration of the left and right sides of the above equations at time t leads to the following results.

In the framework of this paper, we calculate components of the displacement vector and the spatio-temporal distribution of temperature by using the second-order approximation in the framework of the method of averaging of function corrections. The approximation is usually good enough to make qualitative analysis and to obtain some quantitative results. All obtained results have been checked by comparison with the results of numerical simulations.

In this section, we analyzed the change in stress in metals during thermal treatment. Based on the analysis, we estimate residual stress after finishing the thermal treatment. Fig. 1 shows typical dependences of components of the displacement vector on the coordinate. In this figure, as an example, we consider the dependencies of component uz on coordinate z. An increase in the number of curves and a decrease in the coordinate correspond to an increase in the defects of metals (dislocations, pores). The defects became a reason for residual stress in metals.

In this paper, we introduce an approach for the estimation of stresses in metals during heat treatment and residual stresses in these materials after the treatment. In the framework of the considered approach, we introduce a model of stress in metals during heat treatment as well as an analytical approach for its analysis.

All aspects of the research and manuscript preparation were carried out by the author. The author has read and approved the final version of the manuscript.

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This article does not contain any studies with human participants performed by the author.