Effect of Internal Heat Generation and Temperature Asymmetry in a Hollow Sphere and its Thermal Stresses

Transcription

1 Effect of Internal Heat Generation and Temperature Asymmetry in a Hollow Sphere and its Thermal Stresses S.P. Pawar, K.C. Deshmukh and G.D. Kedar Abstract-The thermal stresses in a hollow sphere with internal heat source and asymmetric surface are obtained with quasi-static approach. The heat generation and asymmetric boundary temperature factors affect the resulting thermal stresses in nature and location. There are certain radii of the sphere where the stresses are independent of the heat generation and temperature asymmetry. Independent stresses are discussed due to effect of heat generation and temperature asymmetry for certain radii of sphere. Keywords- Thermal stresses, hollow sphere, temperature asymmetry, heat generation. I. INTRODUCTION The study of thermal stresses in spherical objects is an important problem in engineering practice. Earlier literature on Themoelasticity found to be on the homogeneous material with constant and uniform thermo physical properties in the monographs of Noda [1], Nowacki [2], Carslaw [3], Boley and Weiner [4].With the developments in the functionally graded materials, it becomes the topic of study at large [5, 6, 7].In FGM, material properties like modulus of elasticity, coefficient of thermal expansion and thermal conductivity are considered as the function of radius in different forms. During the last two decades increased attention has been given to the problems of thermal stresses, where the heat generation and temperature asymmetry factors are taken into account, especially to those involving cylindrical and spherical geometries. The generation of heat has a significant effect on the temperature profile and thermal deformations in solids in the fields like Engineering and life sciences. H. Singh and A. Singh [8] studied the quasi-static temperature and stress distributions set up in an elastic sphere by radiation from a point source at a finite distance from the centre of the sphere and outside it, and the solutions are obtained in terms of series involving Legendre polynomials. S.P. Pawar, S.N. Mor and Smt. G.D. Saraf Science College Tumsar India ( subhashpawar98@rediffmail.com) K.C. Deshmukh, Dept of Mathematics, RTM Nagpur University, Nagpur, India ( kcdeshmukh2000@rediffmail.com) G.D. Kedar, Department of Mathematics, Kamala Nehru Mahavidyalaya, Nagpur, India. ( gdkedar@rediffmail.co.in) U. Güven and O Altay [9] have investigated the elastic-plastic stress distribution of a solid disk due to nonuniform heat source under exrenal pressure. Nasser M. EI- Maghraby [10] studied the two dimensional problem for thick plate with heat sources in generalized thermoelasticity. Peter John Heggs and John Dare [11] studied the effect of constant and uniform heat generation on the thermal behaviour of a porous solid with asymmetric boundary conditions. Jadiwiga Kidawa-Kukla [12] determined the temperature distribution in an annular plate with a moving discrete heat generation source. Kulkarni et al. [13] studied the determination of displacement and thermal stresses in a thin hollow circular disk due to internal heat generation and integral transform is used to solve the heat distribution and stresses are obtained in the form of Bessel s functions. Huseyin Yapici, Gulsah Ozisik and M. Serdar [14] presented numerical analysis of transient temperature and thermally induced stress distribution in a hollow steel sphere heated by a moving uniform heat source on outer surface. Deshmukh et al. [15] studied the thermal deflection which is built-inedge in a thin hollow disk subjected to the activity of heat source which changes its place on the plate surface with time. Recently Kedar and Deshmukh [17] studied the determination of thermal stresses in a thin clamped hollow disk under unsteady temperature field due to point heat source. The observations and study of all above cited papers and other referred literature on sphere and other geometries reveals that results appearing in the different articles are with complexities such as space and time dependent properties. Therefore it is to clarify, that how internal heat generation and temperature asymmetry affect the thermal stresses. In view of these findings, there is a need to quantify the conclusions regarding the effect of internal heat generation and temperature asymmetry in fundamental problems where the hollow sphere is homogeneous and thermo-physical properties are constant. This paper deals with an exact analytical solution for radial and tangential stresses in a steady state asymmetric boundary temperature with internal heat generation. The solutions are obtained and the effects of thermal stresses due to heat generation and temperature asymmetry on the sphere are analyzed. The results are illustrated graphically. This is a novel work to study the thermal stresses under changing source and temperature asymmetry in a hollow sphere. The results presented here 1

2 could not be found in the open literature despite an at, (9) extensive search. outer radius II. FORMULATION OF THE PROBLEM Consider a hollow sphere of inner radius and with uniform internal heat generation maintained at temperature on inner surface and on the outer surface. The sphere is homogeneous and isotropic. Assume one dimensional steady state radial temperature field. The heat conduction for temperature field is given as [18], Subjected to the following boundary condition (1), (2) at (3) The thermal stresses induced due to one dimensional steady state heat conduction field in a hollow sphere are given by the following integral equation [1, page 297], Dimensionless outer radius = (10) The equations (1-5) may be rendered dimensionless and written as (11) Where heat generation in dimensionless form is expressed as The boundary conditions in dimensionless form as, (12) at (13) at (14) (15) (4) (16) The solution of Temperature distribution from (11-14) is obtained as, (5) Where and are radial and tangential stresses (MPa), is the coefficient of thermal expansion of hollow sphere in, E is the modulus of elasticity of sphere (MPa), υ is Poisson s ratio for the hollow sphere. The boundary condition on the traction free surface is at and (6) Using (17) in (15) and (16) the stresses are obtained as (17) The equations (1-6) constitutes the Mathematical formulation of the problem III. ANALYTICAL SOLUTIONS By using the following dimensionless coordinates as [8], Dimensionless temperature at (7) (18) Dimensionless inner radius (8) Dimensionless outer temperature 2

3 IV. NUMERICAL AND GRAPHICAL ANALYSIS (19) The numerical calculation and graphs are obtained by using the MATLAB software. Fig 1 shows the radial stresses for different values of heat generation with and small values of temperature asymmetry. From the graph it is very much clear that the stresses on the inner and outer surface are null as per the assumed mechanical condition induced for radial stresses on surfaces. As the internal heat generation increases from to 20, the stresses change. For the nature of stresses are compressive throughout the sphere. Along the radii after the compression increases. For when the heat generation increases, it is observed that the nature of stresses change from compressive to tensile. For the region near the inner surface up to is in tension and the remaining part remains under compression. The region under tension expands further for in this part. For =0.5 and 5 the stresses are compressive and after the region of compression extends. Thus the internal heat generation affects the nature of the radial stress, its maximum value and the location inside the sphere. There is a tendency of radial stresses to change from compression to tension for, while remaining part is under compression with increasing function and constant value of temperature asymmetry In Fig 2 the effects of internal heat generation on the tangential stresses are illustrated with and and varying value of heat generation for, the stresses are tensile in nature and its location of maximum is towards outer surface. For hoop stresses are compressive and for percentage of compression on radial direction are increasing. the the Fig. 3 shows radial stresses along radial direction satisfying traction free boundary conditions. The stresses are zero on inner and outer boundary surface due to assumed boundary conditions. The effect of temperature asymmetry on the radial stress distribution is illustrated with constant heat generation. The graphs are plotted for 3 0.3, 0.5, 0.7, 0.9, the stresses are compressive throughout the sphere and the maximum compression occurs about and normal to radial direction. For 0.3 compressions is maximum and it decreasing as the temperature asymmetry increases. Lower the value of, larger the compression.thus the conclusion is that the temperature asymmetry function decides the nature of radial stresses. Fig 4 shows the effect of temperature asymmetry on the tangential stresses in the sphere. The stress is compressive with much higher stresses at the inner surface compared with the outer surface for the values of. At the common point occur where the stress level is the same irrespective of the temperature asymmetry. Fig 5 shows Radial stress distribution along the radial direction for. This graph is a mirror image of Fig 3 about R axis, the radial stress distribution in the interior of the sphere is tensile in nature. The tension increases with increase in the temperature asymmetry. Thus the nature of stress changes with change in the temperature asymmetry (either less than or greater than 1), compressive or tensile. In Fig 6, it seen that the inside surface of the sphere is in tension along and out surface is in compression. This again the mirror image of Fig 4, here the temperature asymmetry is greater than 1 and nature of stresses is reversed V. CONCLUSION In this article the exact analytical solution have been obtained for the hollow sphere with internal heat generation and subjected to asymmetric temperature on its curved boundary surfaces. The effects of thermal stresses due to heat generation and temperature asymmetry on the sphere are studied. In this study it is found that the thermal stresses can be tensile or compressive depending upon the value of heat generation inside the sphere and boundary temperature on the surfaces. Some point are obtained where the stresses and there nature is independent of these parameters. Depending upon the values of internal heat generation and temperature asymmetry, there are some neutral radii where the stress changes from compression to tension or vice versa. It is observed that the internal heat generation and temperature asymmetry values (either less or greater than 1) affects the nature and location of thermal stresses in a hollow sphere. This mathematical model is in

4 dimensionless coordinates therefore easily applicable in different engineering problems and solids. Figure 4: Tangential stress distribution due to temperature asymmetry (θ'<1) Figure 1: Radial stresses for different values of heat generation Figure 5: stress distribution due to temperature asymmetry (θ'>1) Figure 2: Tangential stresses for different values of heat generation Figure 6: Tangential stress distribution due to temperature asymmetry (θ'>1) Figure 3: Radial stress distribution due to temperature asymmetry (θ'<1) 4 ACKNOWLEDGEMENT The authors are thankful to University Grants Commission, New Delhi to provide the partial financial assistance under major/minor research project scheme. REFERENCES [1] N. Noda., R.B. Hetnarski and Y. Tanigawa, Thermal Stresses, 2ndEd,Taylor &Francis, [2] W. Nowacki, Thermoelasticity, Addison Wesley Publishing company, Inc,Reading, Massechusrtts,Palo Alto,London, 1962.

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