Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1312 https://internationalpubls.com Thermal Stress Analysis of Two Dimensional Thermoelastic Problem for Inhomogeneous Half Plane Shamal D. Nirde1, Kirtiwant P. Ghadle2, Aishwary K. Ghadle3 1,2Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Chhatrapati Sambhajinagar - 431004(M.S.)-India. 3CSMSS, Chhatrapati Shahu College of Engineering, Chhatrapati Sambhajinagar-431002(M.S.)-India, 1nirdeshamal@gmail.com, 2drkp.ghadle@gmail.com, 3aishwaryghadle@gmail.com. Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: This paper develops a method for analytically solving plane elasticity and thermo-elasticity problems for inhomogeneous half planes. With the direct integration of the equilibrium equation, the original problems can be reduced to a set of governing harmonic equations with corresponding boundary conditions. Distribution of Young's modulus, shearing modulus and dimensionless stresses are illustrated numerically and shown graphically. Keywords: Thermal stress, isotropic material, inhomogeneous half plane, simple iteration technique, integro-differential equation, Fourier transform 1. Introduction There exist various methods for analysis of thermoelastic problem for arbitrary inhomogeneous solids with boundary conditions in terms of stresses. Several analytical, semi-analytical approaches were developed for solving the heat conduction problems. Explicit analytical solutions are restricted to simple geometries but these are well organized computationally. Exact solutions of the inverse heat conduction problems are significant because they provide closed form solution for heat flux in terms of temperature measurements. In this paper we extend the technique to avoid latter complications to represents solution for an elastic isotropic material. A. Hamoud et al. [2] solved integro-differential equations by using numerical techniques. A. Yasinskyy and O. Ierokhova [3] gives optimization of nonstationary thermal displacements in a given cross section of a half space in the plane strain state. B. Kalynyak et al. [4] studied direct and inverse problems of thermomechanics concerning the optimization and identification of the thermal stressed state of deformed solids. Y. Tokovyy and Ma. Chien-Ching [9, 10,11] gives an explict form solution to the plane elasticity and thermoelasticity problems for anisotropic and inhomogeneous solids and find out analytical solutions to the 2D elasticity and thermoelasticity problems for inhomogeneous planes and half planes. This method was established by V. Vigak [7]. This method was already applied to solve some direct and inverse boundary value problems [8]. After integrating the differential equilibrium equations, we can determine the relationship between the stress tensor component. With this technique the governing equations are reduced to integro-differential equation for stress tensor component. With application of simple iteration method, derived integral equations has been solved for constructing the solution in explicit form expression with interdependence of elastic moduli Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1313 https://internationalpubls.com 2. Preliminary In this section, we collect some basic definitions that will be important to us in the sequel. 2.1 Definition A Fourier transform of function f(x) is defined as [6]: F(ω) = ∫ f(x)e−iωxdx ∞ −∞ . 2.2 Definition Inverse Fourier transform of function f(x) is defined as [12]: f(x) = 1 2π ∫ F(ω)eiωxdx ∞ −∞ . 3. Problem Formulation Consider an isotropic inhomogeneous half plane 𝐷 = {(𝑥, 𝑦) ∈ [0, ∞) × (−∞, ∞)}. The problem is governed by the equilibrium equation [8], ∂σxx ∂x + ∂σxy ∂y + X = 0, ∂σxy ∂x + ∂σyy ∂y + Y = 0, (x, y) ∈ D. (1) Strain-compatibility equations [7]: 𝜕2𝜏𝑥𝑥 𝜕𝑦2 + 𝜕2𝜏𝑦𝑦 𝜕𝑥2 = 𝜕2𝜏𝑥𝑦 𝜕𝑥𝜕𝑦 . (2) Stress- Strain relations [8]: 𝜏𝑥𝑥 = 1 𝐸∗ (𝜎𝑥𝑥 − 𝜗𝜎𝑦𝑦 − 𝑐∗ + 𝛼∗𝑇(𝑥, 𝑦)), (3) 𝜏𝑥𝑥 = 1 𝐸∗ (𝜎𝑥𝑥 − 𝜗𝜎𝑦𝑦 − 𝑐∗ + 𝛼∗𝑇(𝑥, 𝑦)), (4) 𝜏𝑥𝑦 = 1 𝐺 𝜎𝑥𝑦. (5) where σxx, σxy, σyy are the stress tensor components and τxx, τxy, τyy are the strain tensor components. G, E, ν are shear modulus, modulus of elasticity and Poisson’s ratio respectively and α is the coefficient of thermal expansion. X = X (x, y), Y = Y (x, y) are the stress - dimensional projections of body forces in the abscissa and co-ordinate. For plane strains 𝐸∗ = 1 1−𝜗2 , 𝜗∗ = 𝜗 1−𝜗2 , 𝛼∗ = 𝛼(1 − 𝜗), 𝑐∗ = 𝜗𝑐. (6) For plain strain, 𝐸∗ = 𝐸, 𝜗∗ = 𝜗, 𝛼∗ = 𝛼, 𝑐∗ = 𝑐. (7) According to Hook's law, 𝐸𝑐 = 𝜎𝑧𝑧 − 𝜗(𝜎𝑧𝑧 + 𝜎𝑧𝑧) + 𝛼𝐸𝑇, (8) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1314 https://internationalpubls.com c=constant is out of plane strain and 𝜎𝑧𝑧 is the out of plane strain and 𝑇 = 𝑇(𝑥, 𝑦) is the temperature distribution. We impose tractions at the boundary 𝜎𝑥𝑥|𝑥=0 = −𝑝1(𝑦) , 𝜎𝑥𝑦|𝑥=0 = 𝑞1(𝑦). (9) Assume that, as |𝒚| → ∞ the stresses are tending to 0. The steady state temperature T (x, y) can be found from the following heat conduction equation [5]: 𝜕2𝑇 𝜕𝑦2 + 𝜕2𝑇 𝜕𝑥2 = −𝑞(𝑥,𝑦) 𝐾 , (10) where k=constant under boundary conditions employed at boundary x=0. The imposed boundary conditions of the given problems are 𝑇(𝑥, 𝑦) = 𝑇0(𝑦) 𝑎𝑡 𝑥 = 0, 𝜕𝑇(𝑥,𝑦) 𝜕𝑥 + 𝑎0T(x, y) = 𝜑0(𝑦), at x=0, 𝜕𝑇(𝑥,𝑦) 𝜕𝑥 = 𝑏0, at x=0. Where 𝑎0, 𝑏0 are constant and 𝑇0(𝑦), 𝜑0(𝑦) are given functions. 4. Solution Formulations From the physical relation of (3, 4, 5) and the equilibrium equation (1) representing (2) as follows, ∆ [ 𝜎 𝐸∗ + 𝛼∗𝑇] = 𝜎𝑥𝑥 2 𝑑2 𝑑𝑥2 ( 1 𝐺 ) + 𝑑2𝑐∗ 𝑑𝑥2 + 𝑑 𝑑𝑥 ( 1 𝐺 ) + 1 2𝐺 [ 𝜕𝑋 𝜕𝑥 + 𝜕𝑌 𝜕𝑥 ]. (12) To compute the total stress 𝜎 = σxx + σyy in terms of 𝜎𝑦𝑦. We use the relation, ∆𝜎𝑥𝑥 = 𝜕2𝜎 𝜕𝑦2 − 𝜕𝑋 𝜕𝑥 + 𝜕𝑌 𝜕𝑥 , (13) ∆ denotes two -dimensional Laplace operator. To find out solution for problem (1) to (13), selecting one key stress out of three stress components. To find out the two -dimensional stressed state, the equation of continuity for these regions, written for the normal stresses 𝜎𝑦𝑦 Integrating equation (1) as in [7], express the stresses 𝜎𝑥𝑦 in terms of 𝜎𝑥𝑥, 𝜎𝑦𝑦. 4𝜎𝑦𝑦 = 𝑞1 − ∫ ( ∂σyy ∂y + Y) sgn(x − η)dη − ∫ ( ∂σxx ∂x + 𝑋) 𝑠𝑔𝑛(𝑦 − 𝜉)𝑑𝜉, ∞ −∞ ∞ 0 (14) 𝑠𝑔𝑛𝑥 = { −1 𝑥 < 0 0 𝑥 = 0 1 𝑥 > 0. (15) To find out the key stresses, using integral Fourier transform [10] for (12) - (13), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1315 https://internationalpubls.com ( 𝑑2 𝑑𝑥2 − 𝜔2) [ 𝜎 𝐸∗ + 𝛼∗𝑇] = − 𝜎𝑥𝑥 2 ( 𝜔2 𝐺 ) − 𝜔2𝑐∗2𝜋𝛿(𝜔) + 𝑖𝜔𝑋 ( 1 𝐺 ) + 1 2𝐺 (𝑖𝜔𝑋 + 𝑑𝑌 𝑑𝑦 ), (16) 𝑑2σxx 𝑑𝑥2 − 𝜔2σxx = −𝜔𝜎 − (𝑖𝜔𝑋 − 𝑑𝑌 𝑑𝑦 ), (17) 𝜎𝑥𝑥|𝑥=0 = −𝑝1 , 𝜎𝑥𝑦|𝑥=0 = 𝑞1, (18) here, δ(ω) is the Dirac delta function. This key stress σxx should satisfy the boundary condition ∂σxx ∂y |𝑥=0 = −𝑖𝜔𝑞1̅̅̅ + 𝑋(0). (19) Here, ω denotes integral transform parameter 𝑖 = √−1. Solving (16) -(17), particular solution for 𝜎xx from equation (17) obtained in the form, we obtain the expression as, �̅�xx = −𝑝1̅̅̅𝑒𝑥𝑝(−|𝜔|𝑥) + |𝜔| 2 ∫ 𝜎(𝜉) ∞ 0 [𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|) − 𝑒𝑥𝑝(−|𝜔||𝑥 + 𝜉|)]𝑑𝜉 − 1 2|𝜔| ∫ (𝑖�̅�(𝜉) − 𝑑�̅� 𝑑𝜉 ) ∞ 0 [𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|) − 𝑒𝑥𝑝(−|𝜔||𝑥 + 𝜉|)]𝑑𝜉, (20) which satisfy integral conditions, 𝜔2 ∫ 𝜎𝑒𝑥𝑝(−|𝜔|𝑥)𝑑𝜉 = −|𝜔|�̅� ∞ 0 − 𝑖𝜔�̅� − 𝑋(0) + ∫ (𝑖�̅�(𝜉) − 𝑑�̅� 𝑑𝜉 ) ∞ 0 𝑒𝑥𝑝(−|𝜔|𝑥)𝑑𝜉. (21) Analogously, we construct solution from equation (17) to equation (16) in the form, 𝜎 = 𝐸∗ [𝐴𝑒𝑥𝑝(−|𝜔|𝑥 − 𝛼∗�̅� − 𝜋 |𝜔| ∫ 𝑐∗(𝜉)𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|)𝑑𝜉 + 1 2𝜔 ∫ ( 𝑋(𝜉) 𝐺(𝜉) + 1 2𝐺(𝜉) (𝑖𝜔�̅�(𝜉) − ∞ 0 ∞ 0 𝑑�̅� 𝑑𝜉 ) − 1 4𝜔 𝜎𝑥𝑥 𝐺(𝜉) ) 𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|)] 𝑑𝜉, (22) where, A is the constant of integration. Substitution of the expression (20) into (22) yields expression of the form, 𝜎 = 𝐸∗ [𝜓 − 𝛼∗�̅� − 𝜙1 − 1 8 ∫ 𝜎(𝜉1 ∞ 0 )𝑁(𝑥, 𝜉1) ] 𝑑𝜉1 . (23) Where, 𝑁(𝑥, 𝜉1) = ∫ 1 𝐺(𝜉1) ∞ 0 𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|)𝑑𝜉1; 𝜓 = 𝜋 𝜔 ∫ 𝑐∗𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|)𝑑𝜉 ∞ 0 ; 𝜙1 = 1 2𝜔 ∫ 𝑋(𝜉) 𝐺(𝜉) 𝜔 0 + 1 𝐺(𝜉) ((𝑖𝜔𝑋(𝜉) + 𝑑�̅� 𝑑𝜉 ). Following the solution technique [7], construct the solution to the integral equation (23) as the limit, 𝜎 = lim 𝑛→∞ 𝜎𝑛. We can solve (23), by simple iteration method [7] as follows, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1316 https://internationalpubls.com 𝜎𝑛 = 𝐸∗ [𝜓 − 𝛼∗�̅� − 𝜙1 − 1 8 ∫ 𝜎𝑛−1(𝜉1 ∞ 0 )𝑁(𝑥, 𝜉1) ] 𝑑𝜉1; (24) 𝜎0 = 0, 𝑛 = 1, 2, 3. .. For n = 1 iteration can be calculated easily, let us represent the expression (24) in the form for the n th iteration, 𝜎𝑛 = 𝜎1 + �̅�𝑛−1. By the substitution of (20) into (23), we obtain, 𝜎 = 𝐸∗ [𝜓 + 𝑝1 + 𝐴𝑒𝑥𝑝(−|𝑠|𝑥 − 𝛼∗�̅� − 𝜙2 − 1 8 ∫ 𝜎(𝜉1 ∞ 0 )𝑁(𝑥, 𝜉1) ] 𝑑𝜉1, (25) 𝐻𝑒𝑟𝑒, 𝑁(𝑥, 𝜉1) = ∫ 1 𝐺(𝜉1) ∞ 0 [[𝑒𝑥𝑝(−|𝜔|(|𝜉 − 𝜉1| + |𝑥 − 𝜉1|) − 𝑒𝑥𝑝(−|𝜔||𝜉 + 𝜉1||𝑦 + 𝜉1|)]]. 5. Examples To find the exact solution, consider an example of inhomogeneity, Let 𝑋 = 𝑌 = 𝑇 = 0 , 𝐸 = 𝐸0𝑓(𝑥), 𝑓(𝑥) = 𝑒𝑥𝑝(𝑥4), 𝐸0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑏 > 0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 From (20), we obtain required thermal stresses 𝜎𝑥𝑥, as follows 𝜎xx = −𝑝1̅̅̅𝑒𝑥𝑝(−|𝜔|𝑥) + |𝜔| 2 ∫ 𝜎(𝜉) ∞ 0 [𝑒𝑥𝑝(−|𝜔||𝑥 − 𝜉|) − 𝑒𝑥𝑝(−|𝜔||𝑥 + 𝜉|)]𝑑𝜉. From (25) 𝜎 = 𝐸∗ [𝜓 + 𝑝1 + 𝐴𝑒𝑥𝑝(−|𝑠|𝑥 − 𝛼∗�̅� − 𝜙2 − 1 8 ∫ 𝜎(𝜉1 ∞ 0 )𝑁(𝑥, 𝜉1) ] 𝑑𝜉1 . Consider the Poisson’s ratio µ=const. By the relation between Young’s and Shear Modulus latter can represent in the form 𝐸 = 𝐸0𝑓(𝑥)where, 𝐺0 = 𝐸0 ( 1 1+𝜇 ). We can see that figure-1 indicates the distribution of µ(x) and figure-2 indicates the distribution of normal stress σyy, σxy respectively at α= 1, 1.5, 2 respectively. we conclude that σxx,,σxy, 𝜎are independent on G and depend on Poisson’s ratio ν which vary with x-coordinate. Figure 1: Distribution of 𝝁(𝒙) 𝒇𝒐𝒓 𝜶 = 𝟏 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1317 https://internationalpubls.com Figure 2: Distribution of 𝝈𝒙𝒙 𝒇𝒐𝒓 𝜶 = 𝟏, 𝟏. 𝟓, 𝟐 6. Conclusion In this manuscript, we studied an numerical approach to solve the two-dimensional problems of elasticity and thermoelasticity in terms of stresses for isotropic material in an inhomogeneous strip which is infinite. This approach is placed on the direct integration of differential equilibrium equations. This technique permits to construct analytical solution for interdependence between the elastic modulie of an isotropic material. We reduce the governing integro-differential equations with variable coefficients in accordance with compatibility and equilibrium equations. The calculation for constructing the solution can be also applied to solve optimization problems, comparable inverse thermoelasticity problems in terms of stresses. In this method we can easily calculate the stressed state in an infinite strip, as compare to solving such problem in terms of displacement. With the help of simple iteration method, we have solved these governing equations. This method gives exact analytical solutions if the shear modulus is reciprocal of linear function in Cartesian coordinate system for corresponding problems. Direct integration method is very useful technique to solve the boundary value problems. Since, application of this method depend on the direct integration of the equilibrium equations for efficient analysis of inhomogeneous solids. 7. Acknowledgement The authors are thankful to Council of Scientific and Industrial Research (CSIR), Delhi for awarding the research fellowship 09/809-0028(2021)-EMR. References [1] A. Rychahivskyy and Y. Tokovyy: Correct analytical solutions to the thermoelasticity problems in a semi plane, Journal of Thermal Stresses, Vol. 31, pp-1125-1145, 2008. [2] A. Hamoud, N. Mohammed, K. Ghadle and S. Dhondge: Solving integro-differential equations by using numerical techniques, International Journal of Applied Engineering Research, Vol. 14, pp. 3219-3225, 2019. [3] A. Yasinskyy and O. Ierokhova: Optimization of nonstationary thermal displacements in a given cross section of a half space in the plane strain state, Journal of Mathematical Sciences, Vol. 223, pp. 140-147, 2017. [4] B. Kalynyak, Y. Tokovyy and A. Yasinskyy: Direct and inverse problems of thermomechanics concerning the optimization and identification of the thermal stressed state of deformed solids, Journal of Mathematical Science, Vol. 236, pp. 21-34, 2019. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9(s) (2025) 1318 https://internationalpubls.com [5] K. Ghadle and A. Adhe: Steady state temperature analysis to 2D elasticity and thermoelasticity problems for inhomogeneous solids in half plane, The journal of the Korean Society for Industrial and Applied Mathematics, Vol. 24, pp. 93-102, 2020. [6] S. Nirde and K. Ghadle: Two dimensional thermoelasticity problems in an inhomogeneous strip with application of direct integration method, Journal of Fractional Calculus and Applications, Vol. 16, Issue 1, No-2, Jan. 2025. [7] V. Vigak: Correct solutions of plane elastic problems for a half-plane, Int. Applied Mechanics, Vol. 40, pp. 283-289, 2004. [8] Y. Tokovyy and A. Rychahivskyyi: Reduction of plane thermoelasticity problem in inhomogeneous strip to integral Volterra type equation, Mathematical Modelling and Analysis, Vol. 10, pp. 91-100, 2005. [9] Y. Tokovyy and Ma. Chien-Ching: Elastic analysis of inhomogeneous solids, Journal of Mechanics, Vol. 35, pp. 613-626, 2019. [10] Y. Tokovyy and Ma. 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