THERMOELASTICITY OF ELASTOMERIC CONSTRUCTIONS WITH INITIAL STRESSES

The article presents an algorithm for solving linked problems of thermoelasticity of elastomeric structural elements on the basis of a moment scheme of finite elements. To model the processes of thermoelastic deformation of structures with initial stresses the incremental theory of a deformed solid is used. At each step of deformation, the stiffness matrix is adjusted using an incremental geometric stiffness matrix. The use of triple approximation of displacements, deformations and volume change function allows to consider the weak compressibility of elastomers. The components of the stress tensor are calculated according to the Duhamel-Neumann law. To solve the problem of thermal conductivity, a thermal conductivity matrix considering the boundary conditions on the surface of a finite element is constructed. A sequential approximation algorithm is used to solve the thermoelasticity problem. At each stage of the solution, the characteristics of the thermal stress state are calculated. Based on the obtained components of stress and strain tensors, the intensity of internal heat sources is calculated as the dissipative energy averaged over the load cycle. To calculate the dissipative characteristics of the viscoelastic elastomer the parameters of the Rabotnov’s relaxation nucleus are used. Solving the problem of thermal conductivity considering the function of internal heat sources allows you to specify the heating temperature of the body. At each cycle of the algorithm, the values of the physical and mechanical characteristics of the thermosensitive material are refined. This approach to solving thermoelastic problems is implemented in the computing complex "MIRELA+". Based on the considered approach, the solutions of a number of problems are obtained. The results obtained satisfactorily coincide with the solutions of other authors. Considering the effect of preload and the dependence of physical and mechanical properties of the material on temperature leads to significant adjustments to the calculated values.


Introduction.
In solving related problems of thermoelasticity of elastomeric structural elements, various theories and approaches are used, which are based on the relations of the thermoelasticity obtained by many researchers [1][2][3][4][5][6] and others. One of the most important criteria for the study of viscoelastic bodies with a non-uniform temperature field is to take into account the temperature dependence of physical and thermophysical characteristics of the material: modulus of elasticity or shear modulus, coefficient of thermal expansion and coefficient of thermal conductivity.
In the case of a slight increase in temperature, its effect on the mechanical behaviour of the viscoelastic body can be neglected and an unrelated problem of thermovisoelasticity can be considered. This approach is reflected in [7] and others.
When using a coupled linear model, the temperature and thermoelastic states are determined by the solution of the system, which consists of the equations of thermal conductivity, classical equations of motion, Hooke's law equations and classical compatibility equations [8][9][10][11], etc.
It is obvious that all thermomechanical processes depend on time and for their research there are problems called non-stationary (dynamic). However, we can identify a number of processes, thermomechanical state, during which, although it changes over time, but from a certain point in time, the system comes to a stationary (static) state, which does not depend on time. In addition, it should be noted that elastomers are a nonlinear viscoelastic material and their stress-strain state depends on the load history.
1. Problem statement and its solution. The formulation of the related problem for a quasi-stationary formulation can be represented as the equation Biot and the equation of thermal conductivity [11].
The formulation of the initial provisions for the description of the deformation process considering the initial stresses will begin with the representation of this process as a sequence of equilibrium states: On the other hand, taking the initial state ( ) n  and using the rectangular coordinates of the increase in deformation can be determined [12]: The increments of deformations are connected by the relations: The components of the deformation gain tensor can be represented as linear and nonlinear components: Jacobian transformations: ( ) , , At the point Q (n+1) the stress can be described by a modified Kirchhoff tensor [12].
Variational statement of the problem for an elastic body with initial stresses with given additional mass and surface forces in the form of the principle of virtual work in the state ( 1) n   looks like: where p i , q i -vectors of surface and volumetric additional forces at n+1 step.
-equilibrium state, then in the equation the terms of the variation of elastic energy and taking into account the initial stress: The equilibrium equation takes the form:

 
To study the stress-strain state of spatial structures made of elastomers, consider the isoparametric finite element in the form of a hexagonal parallelepiped with an edge length of 2. The origin of the basic coordinate system Z i and an arbitrary local system x i , the axis of which coincides with the direction of its edges, placed in the centre of the cube. Consider the construction of matrices of stiffness of a finite element with initial stresses based on the moment scheme of finite elements. Under the pre-stresses we mean those stresses that arose in the structure in the initial state, i.e. before the deformation process under consideration, before the application of the working load.
The solvating relations in general case can be represented as: where 0 st K -incremental geometric matrix of stiffness, which taking into account the action of prestresses [13].
When forming a matrix st K for a weakly compressible elastomeric layer, a moment scheme of finite elements with triple approximation of displacement fields, deformations, and volume change functions is used. [14]. The components of the stress tensor are determined by the Duhamel -Neumann thermoelasticity law. To solve the problem of calculating the temperature of dissipative heating, it is necessary to solve the problem of thermal conductivity. To construct a thermal conductivity matrix for a layered finite element, the hypothesis of continuity of temperature fields and heat fluxes at the interface is used.
In matrix form, the system of equations for the layer takes the form: where H -thermal conductivity matrix, H (st) -matrix conditioned to boundary conditions of the 3rd kind on the surface of the construction, P -equivalent load vector conditioned to the internal heat generation source, S -equivalent load vector conditioned to heat fluxes and body surface temperature.
The heat generation function is calculated as averaged over the deformation cycle.
The solution of the linked problem is performed using the method of successive approximations.
2. Results of the calculation and analysis of solutions. Let us consider the process of determining the temperature of dissipative heating of elastomeric structures as a solution of the linked problem of thermoelasticity for a stable mode of cyclic deformation and heat exchange with the environment. In this case, the solution of the quasi-static thermoelasticity problem requires the solution of several problems: determination of the function of internal sources in an elastic -hereditary body (solution of the thermoelasticity problem) at the initial temperature; calculation of the temperature field under given boundary conditions (solving the problem of stationary thermal conductivity); solving the problem of thermoelasticity for the final temperature of self-heating. When constructing a mathematical model of the problem, it is assumed that the stress state significantly depends on the coordinates, as a result of which the field of heat and temperature sources is inhomogeneous. In addition, there is a need to take into account the dependence of physical and mechanical properties on temperature. The method of successive approximations is used to solve the related thermoelasticity problem.
The algorithm for solving the related problem is represented by the following sequence: where the first term is a vector due to elastic displacements, the second term is a vector of forces due to thermal displacements. 2. To determine the power of internal heat generation sources, it is necessary to determine the amount of scattered energy per load cycle.
The use of the simplest hypotheses about the homogeneity of the displacement field in the direction of reinforcement and the homogeneity of the field of generalized forces for shear stresses and normal stresses to the fibers, allows you to calculate the power of internal heat sources as an average, equal to dissipative energy. The power of internal sources for cyclic loading can also be determined by the formula: The temperature field is determined from the solution of the stationary thermal conductivity problem.
The thermal conductivity problem is nonlinear because the matrix [H] and the equivalent heat load vector {R} depends on temperature.
The system of solving equations of stationary thermal conductivity using the method of successive approximations is written in the form: -experimentally determined constants, which characterize the dependence of elastic modules and coefficients of thermal expansion on temperature.
The Poisson's ratio for the thermosensitive materials, which are under consideration, is almost independent of temperature, so we consider it constant. The components of the stress tensor arising from the temperature are determined by the Duhamel-Neumann law.
Physical and mechanical parameters are taken as follows: 10   The constructed system of equations in displacements with variable coefficients is solved using the proposed approach. The obtained results satisfactorily coincide with the results obtained in analytical and approximate solutions using shell theory by the authors [15].  Fig. 2 shows graphs of radial displacements, obtained using the MIRELA+ complex (curve 1) and the authors [15]. Thermal conductivity coefficient λ = 0,293 W/(m•К); the heat transfer coefficients with the metal fittings and the environment are respectively equals Н 1 = 5240 m -1 , Н 2 = 40 m -1 .
In fig. 3 shows graphs of the distribution of the steady-state temperature of self-heating with preliminary deformation (2, 3) and without pre-compression. Analysis of the results shows that the increase in the initial deformation significantly affects the temperature of dissipative heating.
Conclusions. An algorithm for solving the related problems of thermoelasticity of elastomeric elements is built on the basis of the moment scheme of finite elements. Incremental theory is used to model the processes of deformation of structures with initial stresses.
The analysis of the obtained results shows that the proposed approach allows to obtain satisfactory calculation results.
Considering the action of prestresses, as well as the dependence of the physical and mechanical properties of the material makes significant adjustments to the values of the calculated values.