 Advances in Pure Mathematics, 2013, 3, 430-437 http://dx.doi.org/10.4236/apm.2013.34061 Published Online July 2013 (http://www.scirp.org/journal/apm) Quasi-Static Problem of Thermoelasticity for Thermosensitive Infinite Circular Cylinder of Complex Heat Exchange Halyna Harmatij1, Marta Król2, Vasyl Popovycz1 1Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine, Lviv, Ukraine 2Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, Rzeszow, Poland Email: dept19@iapmm.lviv.ua, krolmb@prz.edu.pl Received January 20, 2013; revised March 10, 2013; accepted April 13, 2013 Copyright © 2013 Halyna Harmatij et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Nonlinear nonstationary heat conduction problem for infinite circular cylinder under a complex heat transfer taking into account the temperature dependence of thermophysical characteristics of materials is solved numerically by the method of lines. Directing it to the Cauchy’s problem for systems of ordinary differential equations studied feature which takes place on the cylinder axis. Taken into account the dependence on the temperature coefficient of heat transfer that the different interpretation of its physical content makes it possible to consider both convective and convective-ray or heat ray. Using the perturbation method, the corresponding thermoelasticity problem taking into account the temperature dependence of mechanical properties of the material is construed. The influence of the temperature dependence of the material on the distribution of temperature field and thermoelastic state of infinite circular cylinder made of titanium alloy Ti-6Al-4V by radiant heat transfer through the outer surface has been analyzed. Keywords: Thermoelastic State; Heat Transfer; Boundary Value Problem 1. Introduction It is of interest to consider the condition of convective heat transfer coefficient depending on the temperature of heat transfer  tttt0cSttntSnS. (1) Here t—coefficient of thermal conductivity, which depends on the temperature; ct—temperature environ- ment, which washes the body surface ; -external nor- mal to the surface . That kind of heat transfer condi- tions also considers heating or cooling the body by ra- diation, or heat when carried out simultaneously by con- vection and radiation. Thus the physical content of Equ- ation (1) is quite different. For radiation heat transfer co- efficient t is called the coefficient of radiation heat transfer, it has the same dimension as the coefficient of convective heat transfer is: 22ccttttt , (2) — Stefan-Boltzmann constance, —coefficient of black- ness. The coefficient  is the total coefficient of con- vective heat transfer k and coefficient of radiation tkt r,,rz in the case of convective heat transfer beam where . (3) Transient heat transfer body will be called convective heat transfer with temperature depending on the heat transfer coefficient. The mechanism of that heat transfer is described by the Relation (1). In the study of processes of heat conduction in solids, which are shaped circular cylinder radius 0 using a cy- lindrical coordinate system . If the temperature is independent of  and (the case of axial symme- try), the heat equation has the form: z—reduced rate of radiation, 01,0tvttrct rrrr r  (4)  ctwhere vThe solution of Equation (4) satisfies the boundary condition of the third kind, the described types of heat transfer are as follows: —volumetric heat capacity. Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. 4310,ttrttr0,0ctrt0r, (5) and at satisfies the condition of boundedness of solution, which is equivalent to the condition 0m 0tttrlir,0. (6) Besides, the solution of Equation (4) satisfies the ini- tial condition ptr t, (7) where ptt tr—the initial temperature of the cylinder. Thus, the mathematical model to determine the tem- perature field in a circular cylinder has the form of non- linear boundary value Problems (4)-(7). For the convenience of calculations in constructing the solution to the Problems (4)-(7) we pass to dimensionless variables. To do this, take the temperature of the heating medium c by counting the temperature 0, and radius 0 for the typical size. We introduce the dimensionless temperature 0 and coordinate Ttt0rr. The growth temperature pT denoted by TT, where 0tppTt ,tct. 2. Preliminaries Specifications of the material tv and heat transfer coefficient are represented as: t000,,,tttvvvtTccTtT,,cct (8) where 000tv—basic coefficients that are relevant dimension, and,   ,,TcT T Ttv —functions that describe the dependence of these characteristics on di- mensionless temperature , and 1vpTcTtp. We introduce the dimensionless time 2000 00tFoa racand Bios criterion’ 00 0tBi r. That Problems (4)-(7) to increase the temperature in the dimensionless quantity becomes a  1tTTc  ,0 1TTFo, (9) 1, 1,TFoTBiT0tcTFoT (10) 0lim0tTT, (11) 00FoT, (12) where ccpTTT. Nonlinear boundary value Problems (9)-(12) will be solved numerically using the method of lines. We intro- duce a uniform grid ;0,, 1hiih iNhN 01 in the interval . The differential operator 1tTT , (13) has been replaced by the difference operator  at the points of the grid,1, 1iiN 12, ,1iiiiiT, (14) where12, 12,ii iiihTTTh , ,1iiiTTTh i, and the value  determined by the formula 12iiitTT (15) or 12ti tiiTTh. (16) The difference operator (14) approximates the dif- ferential operator (13) to the second order in . We will approach these functions ,TT Foii by using the functions,0,yyFoi Nii , which are based on Equation (9) satisfying the system of ordinary differ- ential equations 1112 112dd1,1, 1iii iiii iiiiyFoyy yyhc hhiN   (17) where ccT1iviWe construct the difference boundary condition for the second-order approximation for N. . The expan- sion of these values in series Taylor in neighborhood of the point N is written as follows:  122,122222NNNNNNNNNt NtTThThTTOhhTTOh     Then considering the differential Equation (9) at the point N we receive Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. 432   222.NNNNNtTTOhTOhOh12 ,222122NNNNNNttttNtNtTThTTTThTTThTTcTFo      Hence, taking into account that NNTy, we obtain 12 ,NN Ny d2dNNNNNyFo h c, (18) where 11NcT TBi T . The difference analog of condition is constructed (11). The values ,0T і 1 decompose in the Taylor series:  220,.TOhOh2,0 2001022ttThThTT  Then   00220TTOhTOh1,00220022tttttTThTTTThTT (19) In view of Equation (9) we write   ttTTTTTcTFo  . (20) Since 0lim 0T, then we have uncertainty 0tTT of type 00. We now turn to the border and reveal uncertainty using the rule of de L’Hospitala, we get 00lim ttTTTT  . Thus from (20) we obtain  00012tTTTcTFo   . (21) Taking into account (21), we write (19) as 21,0 004vhTTcT OhFo. (22) Since the value of 0 approaches 0, then from (22) we obtain the following differential equation at the point Ty0 01,0d4dyy, (23) Fohcwhere . 00vccTThus, we obtain the Cauchy problem for systems of ordinary differential equations approximating the partial derivatives of the space variables with the second-order boundary-value Problems (8)-(11) 011 020d4dyyyFoc h, (24)  12 111212d1,d1,1 ,iiiii iiiiiiyyy yyFo hciN   (25) 112d2dNNNNN NNNNyyyFo h ch 00iy, (26) . (27) Thus, the dependence forms of the heat transfer co- efficient tt, thermal conductivity t and volu- metric heat capacity ctv of the selected material are dependent on temperature, then we solve the system numerically (24)-(27). As a result, we obtain the value of temperature increase in grid points ,1,iNi along the radius of the cylinder for a given time Fo,,rr zz. 3. Thermoelastic State of a Cylinder The thermoelastic cylinder state with the activity of the found axisymmetric temperature field are defined by three non-zero components of stress tensor , which in dimensionless form is rewritten as: 00,,,, 2rr zztoGt 0G0t, where і —supporting values of the shear mo- Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. 433dulus and coefficient of linear thermal expansion. However, they are expressed through the dimension- less radial displacement 00touurt, so *1,uuT eTT1GT T (28)  1,T eTT1uuGT T T  (29)  *e T1uuGT TT(30) and satisfy the equation of balance 0. (31) Here 0zzeetot, where zze—was the axial strain;  01,d12 1TtGT TGT TTT, ( ,,TTGTt—the function describing the de- pendence of the Poisson coefficient of the linear thermal expansion and modulus of the dimensionless temperature increase. If we substitute the dependence (28)-(30) into equilibrium Equation (31), we get the differential equa- tion  ,T1uTuunTeTmT   (32) where    ln 1,11TGTTmTGT TGTGTnT GT T1,.TT The solution of Equation (32), which satisfies the con-dition of the limited movement of the cylinder axis, found by perturbation [1-3] has the form of: 0kkuu, (33) where 0201000211 1212nnuc HHHeHH   (34)  02102111211,12kkkn nkkeucH HHHk   (35)     000110111dddd.mmmmnmmkkkkkHTHTHnHfuufTmT    ,, According to (33) the thermal stress  0,,,,, is found by the formulas: ppk kkk  , (36) where 0102012,nTHGT cHeTH (37)   11,1,pkkknkGT ceTHHk  (38) 001 201212,nTHGT cTTHevTH   (39) Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. 434 1kkk nGT ceTHH1,1kk (40)    001002121,nGTT cTTevTvTH0T H  (41)   10121kkkkGTT ceTTH0,1nT Hk (42) where 02,,, 1.Hnk10k0kek21212THH The sustainable integration ck and parts of the development of axial strain determine the conditions on the outer surface 10k10d0k  and relations for unmounted ends of the cylinder , by solving system of linear equations ,0kkB kAX , (43) where   1111111 1221 221112 1121022010 11120 101110210,,1,2d1,kkkknkkkkcaaeaaaa HaGTTaGTTbHHbGT TbHbGTTH1200ddd.kknbbTHTH  AXB Here 111 11111,,1 2 ,TGGT.   0eFor fixed ends of the cylinder we have . If the thermomechanical properties of the material the cylinder does not depend on temperature, then transfer and thermal stresses are calculated by the formulas: 01,d ,1itiiiiiuTFoc (44) 2011,d ,112iitiii iiiiiGTFone  (45)  2011,d ,11,12iitiiiiiiiiiGTFoTFone   (46)  12121 ,,12 1iiiiii iitiiiiGce TFo   (47) where 121013 ,d1itiiiicTFo, 122100111 2,d12iitiiiieTFo , ,TFo*,Gtrr300 Ktti—increase of temperature in the cylinder for constant thermophysical characteristics; i—steel coefficient of linear thermal expansion and modulus ta- ken at the initial or maximum temperature, or mean inte- gral value of the selected temperature range. 4. Numerical Study This section investigated the temperature field and ther- moelastic state of thermosensitive infinite circular cylin- der. Radiant heat exchange with the environment at a temperature c is due to the surface 0 of the cylin- der, which is free from power loads. In that case, the ra- diant heat exchange has been formed to convective heat transfer coefficient in the form of (2). The initial tem- perature of the cylinder is p. The medium tem- perature c is equal to 1100 K and it is selected by sup- porting. Titanium alloy Ti-6Al-4V taken by the material. Temperature dependence of thermal and mechanical properties of the alloy are the form  in the temperature range from 300 K to 1100 K. Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. 435 1.1 0.017,tvpttct cttWm K (48) 3kJ kgK ,2142 73.510 8.78109.74104.43 10pct ttt   (49) 3.0kgm4420.0 1.0300ttt, (50) 122101 К,tt697.43 105.56 102.69t t (51) 122.7 0.0565 GEt tPa68832.010tt, (52) 0.28. (53) For the purpose of comparison, we studied the tem- perature field and caused it thermoelastic stable in the same noetherian sensitive cylinder under constant ther- mal and mechanical characteristics of titanium alloy. This steel coefficient of thermal conductivity tctGt300 Kt, volumetric heat capacity v, modulus , coeffi- cient of linear thermal expansion and coefficient of Poisson , we take as: a) the initial temperature p, and b) the maximum temperature of tc = 1100 K, c) mean-integral temperature range from 300 - 1100 K, under tttt1dnptnptctt. The results of numerical studies of the temperature T0.1Fo 1 growth, which were obtained in dimensionless form, are presented in graphs in Figures 1-2. The distribution rates of temperature along the radial coordinate cylinder for values of the Fourier criterion are shown in Figure 1. The dependence of the temperature growth on the criterion of Fourier on the surface  of the cylinder is illustrated in Figure 2. Here curve 1 correspond to the results of calculations taking into account the temperature dependence of ther- Figure 1. Temperature dependence of the radial coordinate. mophysical characteristics of titanium alloy (48)-(51), curve 2—at maxttttvvct ct, max ; curve 3 —at ttmaxtt, ct ctmaxvv; curve 4—for sustainable medium-integer values of material from a selected range of temperature. The analysis of the re- search shows that the largest discrepancy between the temperature growth in the cylinder taking into account the temperature dependence of thermophysical charac- teristics of materials and for stable values of characteris- tics is in the case ,ttct ctmax maxtt vvAs it is visible from the graphs in Figure 2 the ma- ximum difference between the increases of temperature . T1 on the surface  of the cylinder by taking into account the temperature dependence of thermophysical of characteristics material (curve 1) and stable medium- integer values of thermal conductivity and volumetric heat capacity (curve 4) does not exceed 10% (for ). 0.1Fo Figures 3-9 show graphs of distributions of displace- ment and stress tensor component along the radial coor- dinate  for the Fourier criterion and on the surface of the cylinder 0.1Fo 1Fo.  depending on Here curve 1 correspond to the results of calculations taking into account the temperature dependence of ther- mal and mechanical characteristics of titanium alloy (48)-(53). Figure 2. Temperature dependence of the criterion of the Fourier. Figure 3. Temperature dependence of the radial coordinate. Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. 436 Figure 4. Temperature dependence of the criterion of the Fourier. Figure 5. Temperature dependence of the radial coordinate. Figure 6. Temperature dependence of the criterion of the Fourier. The curves 2 - 4 correspond to displacement or stress, found by constant values of thermal and mechanical properties of the material: curve 2—initially under the temperature curve 3—the maximum temperature curve 4—for sustainable medium-integer values of titanium alloy with a temperature range 300 - 1100 K in Figures 3-9. In the form of curves 5 - 7 there are the data distri- butions for the temperature-dependent thermal properties shown tv, and constant values of mechanical properties (which are taken for the initial (curve 5), and the maximum temperature (curve 6)) and sustainable me- Figure 7. Temperature dependence of the radial coordinate. Figure 8. Temperature dependence of the criterion of the Fourier. Figure 9. Temperature dependence of the radial coordinate. dium integer values of mechanical characteristics of tita- nium alloy with temperature range 300 - 1100 K (curve 7). The analysis of differences between the values of both displacements and stresses, which are calculated for all dependent on the temperature characteristics of the mate- rial (heat-sensitive body), and their values calculated by the constant characteristics (not heat-sensitive body) in- dicates that:  they exceed 60%, if the properties take non heat-sen- sitive value for thermo-sensitive characteristics at ma- ximum temperature;  ,tct within 12% - 40% when take on characteristics non Copyright © 2013 SciRes. APM H. HARMATIJ ET AL. Copyright © 2013 SciRes. APM 437heat-sensitive body thermo-sensitive properties at the initial temperature;  they are within 4% - 20%, when the characteristics non heat-sensitive body take mid-integral values of thermal and mechanical characteristics. Then the ma- ximum difference between them exceeds 60%. Thus, studies show that by ignoring the temperature dependence of thermal and mechanical characteristics of the material, the distribution of temperature field and de- fined by its thermo-elastic state of the body that differ significantly from the true can be achieved. The likely thermoelastic state of structural elements of modern technology, which in the process of their manu- facture and operation exposed to high heat or cooled to low temperatures, preferably determined, based on the model of thermosensitive bodies [5,6]. In that model we consider the temperature dependence of thermal and mechanical properties of the material. That mathematical model to determine the thermoelastic parameters is sig- nificantly more complicated in comparison with the same model by neglecting thermo-sensitive material. The tem- perature is determined from the nonlinear problem, which is not only nonlinear heat equation and boundary condi- tions and in case of default on the body surface heat flux, convective or convective-radiation heat transfer [7,8]. The corresponding thermoelasticity problem is the same boundary value problem for differential equations with variable coefficients . For these mathematical models the construction of solution is usually carried out by nu-merical methods . REFERENCES  J. J. Burak and R. M. Kushnir, “Modeling and Optimiza- tion in Termomehanitsi Electrically Inhomogeneous Bod- ies,” In: J. J. Burak and R. M. Kushnir, Eds., Thermoelas- ticity Thermosensitive Bodies, Vol. 3, Lviv, 2009, p. 412.  V. A. Lomakin, “The Theory of Elasticity of Inhomogene- ous Bodies,” Moscow State University Press, Moscow, 1976, p. 367.  N. H. Moiseev, “The Asymptotic Methods of Nonlinear Mechanics,” Nauka, Moscow, 1981, p. 400.  R. M. Kushnir and V. 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Popovich, “Nu- merical Solution of Unsteady Heat Conduction Problems with Temperature-Sensitive Body Convective Heat Trans- fer,” Engineering, No. 1, 2002, pp. 21-25.  A. Samarskiy, “The Theory of Difference Schemes,” Nau- ka, Moscow, 1977, p. 656.