Calculation Of The Heat-Stressed State Of The Disk Using Free Software Code_aster

A numerical calculation of the axisymmetric heat-stress state of the turbine disk was performed using the free Code_Aster software. Distributions of temperature, radial displacements, and also radial and axial stresses in the cross section of the disk are presented.


Introduction
An important stage in the design of transport and stationary power plants and many other machines in various industries is the calculation of temperature and stress elds in structural elements operating at high temperatures [1]. For example, special requirements are imposed on disks of turbines of engines. Moreover, in stationary modes, the temperature and stress elds remain at a constant, but rather high level, which leads to the accumulation of damage [1,2].
All over the world, the nite element method (FEM) has been successfully used for calculating the heat-stressed state of structural elements. One of the options for calculating mechanical engineering components, using the FEM, is the usage of ready-made libraries of nite element programs written in compiled high-level programming languages, for example deal.II. This approach involves writing your own code to implement a specic version of the FEM in the language of C + + using ready-made tested subroutines with its further compilation and launch.
An alternative to the previous approach is the usage of software nite element systems. There are a large number of commercial systems: ANSYS, Abaqus, MSC.Nastran, LS-Dyna, and free software (Code_Aster, CalculiX, Elmer, etc.). The usage of free software for calculating the heat-stressed state of structural elements is relevant, because, rstly, dependency on owners of commercial licenses is eliminated, secondly, software costs are reduced and, thirdly, the possibilities of adapting the code for solving specic problems are opened. However, the choice of a free software package requires a large amount of validation test calculations.
This paper presents the construction of the temperature and stress elds for the test problem of a thick-walled pipe, as well as for a turbine disk using the free software package Code_Aster (acronym for Analysis of Structures and Thermomechanics for Studies and Research solving strength and thermomechanics problems for scientic purposes and education) [3,4]. The classical problem of determining thermal stresses in a thick-walled pipe in a state of stationary uneven heating is considered. The pipe inner radius is a = 0.1 m, and outer radius is b = 0.25 m. The temperature on the inner surface is T 1 = 543 K, on the outside T 2 = 393 K. Pipe material is a non-hardened steel, the following parameter values were taken in the calculations: α (T ) = 1.76 · 10 −5 K −1 , E = 1.96 · 10 11 Pa, ν = 0.3.
The formulation of the associated stationary thermomechanical problem in the region V , corresponding to a homogeneous body of isotropic material has the form [2]: where T (M ) is a temperature;σ andε are stress and strain tensors, respectively, u is a displacement vector; ε = ε 11 + ε 22 + ε 33 ;Î is a unit tensor of the second rank; E is a Young's modulus, ν is a Poisson's ratio, and α (T ) is a coecient of thermal deformation. The boundary conditions for such a problem have the form: are known function and vector functions.
The geometric model of the pipe ( Fig. 1) was built in the Geometry module of the Salome-Meca. The mesh was created in the MESH module, using quadrangular quadratic elements with a side of 0.01 m.   Modelling of the problem is carried out in the AsterStudy module. To calculate the temperature eld, the previously constructed grid is read (LIRE_MAILLAGE), and by using the AFFE_MODELE command the modelling type is set. The thermal conductivity problem is dened by the keyword THERMIQUE. For the problem, which is symmetric with respect to the axis, the MODELISATION parameter takes the value AXIS. The concept of the material is determined by setting the thermal conductivity coecient λ and the product ρC p : DEFI_MATERIAU > LAMBDA = 27.0, RHO_CP = 5250000.0. The indicated material properties are applied everywhere. The boundary conditions are dened by the command AFFE_CHAR_THER, At the second stage of modeling the grid is re-read to calculate the mechanics problem, which is dened on the AFFE_MODELE tab: PHENOMENE > MECANIQUE, MODELISATION > AXIS. Then the temperature eld is projected. This option requires setting the following parameters: RESULTAT is the concept of the result created at the rst stage, MODELE_1 is the model of the projected temperature eld, MODELE_2 is the model on which the projection is performed.
To calculate the stationary temperature eld, the parameters λ and ρC p are set, and the boundary conditions are imposed: on the right boundary of the geometric model T 1 = 773 K, on the left T 2 = 573 K. Previously, the rst stage of modelling is generated, the mesh is read and the type of model is determined. Due to the symmetry of the disk with respect to the axis Ox, a zero ux q = 0 is set at the lower boundary of the model.
In addition to the thermal loads on the disk there are also power loads: inertial rotation around the axis Oy with an angular velocity ω = πn/30, where n is the number of disk revolutions per minute (in the calculations n = 11200 rpm), as well as tensile stresses equal to σ = 200 MPa [6].
The simulation of the problem of determining the stress-strain state begins with reading the mesh constructed earlier. The problem of mechanics is directly solved at the third stage of modelling. Using constant functions, the characteristics of the material ρ, ν, α (T ) and the boundary conditions ω, σ are specied; the dependence of Young's modulus on temperature is determined. The concept of the result of projecting the temperature eld is added to the denition of material: AFFE_VARC > EVOL > NOM_VARC = TEMP. The boundary conditions are taken into account using the function AFFE_CHAR_MECA. The ROTATION command sets inertial boundary conditions and the pressure PRES_REP > PRES = σ is applied to the right side. The DDL_IMPO function xes the movements of the lower bound along the Oy axis, then the type of analysis is determined and the parameters for calculating the temperature stresses in the grid nodes are set. Figures 10-12 below show the distribution of temperature, radial displacements, and also radial and axial stresses in the cross section of the disk.     Using free software, the stationary problem of thermoelasticity was solved, and the heat-stressed state of the turbine disk was calculated. The distributions of temperature, radial displacements, and also radial and axial stresses in the cross section of the disk are given.