Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase Volume 4, Number 2, June 2023, pp.144-153 https://doi.org/10.5206/mase/16387 ANALYSIS OF THERMAL STRESSES TO 2D PLANE THERMOELASTIC INHOMOGENEOUS STRIP ABHIJEET B. ADHE, KIRTIWANT P. GHADLE, AND UDAY S. THOOL Abstract. This paper deals with study of the plane elasticity of thermoelastic problems for inho- mogenous strip. Here, the original problems are reduced to set the governing equations in the volterra integral equations by making the use of direct integration method. Further using the iteration tech- nique the numerical calculations have been performed. The stress distribution obtained and calculated numerically and shown graphically. 1. Introduction A large development of the subject, thermoelasticity is motivated by various fields of engineering sciences, during the last few decades. The main physical drawback in the theory of uncoupled ther- moelasticity is that an elastic body has no effect on the temperature and vice versa. The interest of researchers to study elasticity and thermo-elasticity problems has grown very fast due to wide applica- tions to real world. Biot [3] derived the equation of thermal conductivity by including the coupling between thermal fields and strain fields. A novel work done by Lord et.al. [10] introduced two generalizations to the coupled theory of thermoelasticity and given successful alternate to Fourier’s law in heat conduction. Tokovyy et.al. in [15] emphasized on analytical treatment of the one dimensional and two dimensional elasticity and thermoelasticity problems using direct integration method, for a long hollow cylinder and a long annular radially non-homogeneous cylinder respectively. Babich et al. [2] solved the plane problem of a horizontal concentrated load by using the linearized elasticity theory from an infinite inhomogeneous stringer to an elastic infinite strip with initial stresses clamped at one edge. The problem is reduced to system of integro-differential equations which then solved by means of Fourier Transform. Manthena et. al. [12] analysed the same problems for a mixture of metals like copper and zinc. Jafari et al. [6] discussed the stress analysis in an orthotropic infinite plate with a circular hole using complex variable technique to the two dimensional thermoelastic problem. Kalynyak et al.[7] focused on development by Prof. Vihak in the field of direct and inverse problems of heat conduction and thermomechanics which are important in investigating problems of thermal power engineering. By considering an inverse thermoelastic problem in [8] Prof. Kalyanyak discussed the presence of a stationary temperature field for a long rectangular beam of inhomogeneous nature. Mahakalkar et. al. [11] studied thermoelastic transient heat conduction problem with internal heat by using classical method. They investigated results on temperature distibution, thermal deflection and stresses by integral transform. Iqbal Kaur et.al.[5] studied recent thermoelastic theories and models related to micro-nano beams and bars, their uses and limitations. Received by the editors 3 April 2023; accepted 21 June 2023; published online 26 June 2023. 2020 Mathematics Subject Classification. Primary 74A10 ; Secondary 74A15, 74E05. Key words and phrases. 2D plane thermoelastic problems, inhomogeneous strip, thermal stresses, Volterra integral equation, analytical solution. 144 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/16387 ANALYSIS OF THERMAL STRESSES TO INHOMOGENEOUS STRIP 145 Tokovyy [16, 17] discussed an analytical solution of plane themroelasticity inhomogeneous problem for planes, half-planes and strips with the aid of direct integration method, here, the governing equations reduced to an integral equations which are then solved by using iteration method which produces a solution of the problem in explicit form. And extended their work in [18] in terms of stresses for a infinite strip for a case of inhomogeneous isotropic material. Solution is found using Fourier transforms and iteration method. Kushnir [9] used direct integration method for generalization of the original equations of solution of 2D problems of thermoelasticity for solids with corner points and they are reduced to a governing integrodifferential equations for a key function, an explicite form solution is found. Tianhu [19] investigated the magneto-thermoelastic response of a homogeneous and isotropic finite thin slim strip subjected to a moving heat source by using Lord-Shulman theory and Laplace transform. Vigak [20, 21, 22] has been developed a method to find solution of the elasticity problems in a semi-plane using the method of direct integration of equilibrium equation. Equilibrium conditions for tractions and compatibility equations for the displacements has been found correct. In [23] Vigak et. al. invented a new analytic method for solving quasi-static thermoelastic problem for stresses in rectangular region, the initial problem is reduced to governing integral differential equations for stress components. The solution is obtained as the series expansion according to Saint-Venant’s principle. Youssef et al.[24] developed a new model of three dimensional generalized thermoelasticity problem by using classic L- S model. The double Fourier transform and Laplace technique had been applied to the governing equations subjected to rectangular traction free surface, with the study of the temperature analysis, stresses, strain and displacement in a three dimensional half-space. Zhihe et. al. [25] emphasizes on characterization of FGM strip using thermoelastic problem. In the thermoelasticity one can determine the stresses produced due to the temperature field and moreover to find the temperature distribution by internal forces which vary with time. Our intent of this paper is to extend our own work [1, 4] for obtaining an analytical solutions to the thermoelastic problems which occures in isotropic and inhomogeneous strip under some thermal condition applied. 2. Problem Formulation Consider, 2D plane thermoelastic problem in the strip of inhomogeneous isotropic material with infinite width R = {(x,y) ∈ (−∞,∞) × (−a,a)}, where a > 0 is dimensionless parameter. Thermoelastic equillibrium of plane R is ruled by the equillibrium equations,  ∂σxx ∂x + ∂σxy ∂y + Fx = 0, ∂σxy ∂x + ∂σyy ∂y + Fy = 0, (2.1) strain-compability equations, ∂2�yy ∂x2 + ∂2�xx ∂y2 = ∂2�xy ∂x∂y , (2.2) stress strain relations,   �xx = σxx E1(x) − v1(x)σyy E1(x) + α(x)T(x,y), �yy = σyy E1(x) − v1(x)σx E1(x) + α(x)T(x,y), �xy = σxy G(x) , G1(x) = E1(x) 2(1 + v1(x)) . (2.3) 146 A. B. ADHE, K. P. GHADLE, AND U. S. THOOL Figure 1. Schematic diagram of strip under consideration. Here, σxx,σyy,σxy are stress-tensor components, �xx,�yy,�xy denotes strain components, Fx,Fy are stress dimensional projections of forces in dimensionless components respectively, and E1,G1,α,v1 denotes Young’s modulus, shear modulus, coefficient of thermal expansion and poisson’s ratio. Due to temperature distribution, the normal and shearing stresses arise on the boundarres y = ±a in the strip R, σyy(x,−a) = −p1(x), σyy(x,a) = p2(x), (2.4) σxy(x,−a) = −q1(x), σxy(x,a) = q2(x). The two dimensional steady-state temperature T(x,y) can be found from the heat conduction equation [13] ∂2T ∂x2 + ∂2T ∂y2 = −w(x,y), (2.5) under conditions imposed on the boundary in the region −∞≤ x ≤∞  ∂T ∂x = 0 at x = ±∞, T(x,y) + k1 ∂T ∂y = 0 at y = a, T(x,y) −k2 ∂T ∂y = 0 at y = −a, (2.6) where, w(x,y) = q(x,y) k and q(x,y) denoting the heat generated due to internal heat generated and k1,k2 are coefficient of thermal conductivity. Using equilibrium condition (2.1) we have ∂2σxx ∂x2 − ∂2σyy ∂y2 + ∂Fx ∂x − ∂Fy ∂y = 0. Differentiating first equation in (2.1) with respect to x and second equation with respect to y and subtracting, we get ∂2σxx ∂x2 + ∂Fx ∂x = ∂2σyy ∂y2 + ∂Fy ∂y . Adding ∂2σyy ∂x2 on both sides which yields ∆σyy = ∂2σ ∂x2 + ∂Fx ∂x − ∂Fy ∂y (2.7) where ∆ = ∂2 ∂x2 + ∂2 ∂y2 ANALYSIS OF THERMAL STRESSES TO INHOMOGENEOUS STRIP 147 denotes the two-dimensional Laplace differential operator. Putting stress-strain relations (2.3) and equillibrium conditions (2.1), equation (2.2) can be written as ∆ [ (1 −v1) 2G σ + α(1 + v1)T ] = σyy 2 d2 dy2 ( 1 G ) −Fy d dy ( 1 G ) − 1 2G ( ∂Fx ∂x + ∂Fy ∂y ) . (2.8) The equations (2.7) and (2.8) are bounded by two boundary conditions (2.4) for σyy and for their derivatives which satisfies equilibrium condition (2.1) at y = ±a:  σyy ∂y = − ∂q1 ∂x −Fy(x) at y = −a, σyy ∂y = − ∂q2 ∂x −Fy(x) at y = a. (2.9) The shear stress is found by integrating the equilibrium conditions which gives 2σxy = q1 + q2 − ∫ a −a ( ∂σxx ∂x + Fx)sgn(y − ξ)dξ − ∫ a −a ( ∂σyy ∂y + Fy)sgn(x−η)dη (2.10) where, sgn =   1, for x > 0, 0, for x = 0, −1, for x < 0. 3. Solution of Thermoelastic Problem To find the solution of the formulated problem, we apply the Fourier transform [14] with respect to x defined by f̄(y; ω) = ∫ ∞ −∞ f(x,y) exp(−iωx)dx (3.1) where f(x,y) is an arbitrary function, i2 = −1; ω is a parameter. We choose σyy and σ to be the governing functions. To calculate the key stresses, we apply the Fourier transform (3.1) to equation (2.7) to get( d2 dy2 −ω2 ) σ̄yy = −ω2σ̄ + iωF̄x − d dy F̄y. (3.2) Applying Fourier integral transform (3.1) to equation (2.8) with the conditions in (2.4), we obtain( d2 dy2 −ω2 )[ (1 −v1) 2G σ̄ + α(1 + v1)T̄ ] = σ̄yy 2 d2 dy2 ( 1 G ) − F̄y d dy ( 1 G ) − 1 2G ( iωF̄x ∂x + ∂F̄y ∂y ) , (3.3) σ̄yy(x,−a) = −p̄1, σ̄yy(x,a) = p̄2 (3.4) ∂σ̄yy ∂y (x,−a) = −iωq̄1 − F̄y(x,−a), ∂σ̄yy ∂y (x,a) = iωq̄2 + F̄y(x,a). The solution of differential equation (3.2) is σ̄yy = c1coshωy + c2sinhωy + 1 ω ∫ y −a ( iωF̄x − dF̄y dy −ω2σ̄ ) sinh(ω(y − ξ))dξ (3.5) 148 A. B. ADHE, K. P. GHADLE, AND U. S. THOOL where c1 and c2 are the constants of integration. Using the first two boundary conditions in (2.4), the solution can be expressed as σ̄yy = −p̄2cosh(ω(y + a)) − ( iq̄2 + F̄x(x,−a) ω ) sinh(ω(y + a)) + 1 ω ∫ y −a ( iωF̄x − dF̄y dy −ω2σ̄ ) sinh(ω(y − ξ))dξ. (3.6) It satisfies two integral conditions:∫ a −a σ̄sinhωξdξ = i(q̄1 + q̄2) sinhωa ω + (p̄2 − p̄1) coshωa ω + ( F̄y(x,a) + F̄y(x,−a) ) sinhωa ω2 + 1 ω ∫ a −a ( iF̄x − 1 ω dF̄y dξ ) sinhωξdξ, (3.7) and ∫ a −a σ̄coshωξdξ = i(q̄1 − q̄2) coshωa ω − (p̄2 + p̄1) sinhωa ω + ( F̄y(x,a) − F̄y(x,−a) ) coshωa ω2 + 1 ω ∫ a −a ( iF̄x − 1 ω dF̄y dξ ) coshωξdξ. (3.8) Hence, the solution of equation (3.3) can be found as σ̄ = 2G1 1 −v1 [ c1coshωy + c2sinhω −α(1 + v1)T̄ + Θ(y) + Φ(y) + Q(y) + Ψ(y) + ∫ y −a σ̄(η)K(y,η)dη ] (3.9) where, Θ(y) = − p̄2 2ω ∫ y −a d2 dξ2 ( 1 G(ξ) ) cosh(w(a + ξ))sinh(w(y − ξ))dξ, Φ(y) = − iq̄2 2ω ∫ y −a d2 dξ2 ( 1 G(ξ) ) sinh(w(a + ξ))sinh(w(y − ξ))dξ, Q(y) = 1 2ω ∫ y −a d2 dξ2 ( 1 G(ξ) ) sinh(ω(y − ξ)) ∫ ξ −a ( iF̄x − 1 ω dF̄y dξ ) sinh(ω(ξ −η))dηdξ, Ψ(y) = − 1 ω ∫ y −a ( F̄y(ξ) ( 1 G(ξ) ) + 1 2G1(ξ) (isF̄x + dF̄y dξ ) ) sinh(ω(y − ξ))dξ − F̄y(−a) q̄2 2ω2 ∫ y −a d2 dξ2 ( 1 G(ξ) ) sinh(w(a + ξ))sinh(w(y − ξ))dξ, K(y,η) = ∫ y −η d2 dξ2 ( 1 G(ξ) ) sinh(w(y − ξ))sinh(w(ξ −η))dξ. Different types of techniques can be used to determine the solution of equation (3.9). Here we use the method of resolvent kernel σ̄ = 2G1 1−v1 [ c1ncoshωy + c2nsinhω −α(1 + v1)T̄ + Θ(y) + Φ(y) + Q(y) + Ψ(y) + ∫y −a σ̄n(η)K(y,η)dη ] . The resolvent kernel is calculated as <(y,η) = ∞∑ n=0 Kn+1(y,η) (3.10) ANALYSIS OF THERMAL STRESSES TO INHOMOGENEOUS STRIP 149 where   K1(y,ξ) = K(y,ξ), Kn+1 = ∫ y −a K(y,ξ)Kn(ξ,η)dη, n = 1, 2, · · · . We have a fact that the recurring kernels Kn+1 −→ 0 as n −→∞ which shows the initial condition for convergence holds . Consequently, for a natural number N, <(y,ξ) ≈ 0. Numerical computations have been done using Python programming language. Let, G = G0 exp(ky), v1 = constant, where, k is constant and G0 = E0/2(1 + v0). Distribution of a function µ(x) for b = 2 is depicted in Figure 2. It shows equation (4.1) gives smooth curve and highest value for x → 0 and vanishes for x →±∞ rapidly, which makes the equation more useful to verify analytic solution. Introduce the parameter v0 = 1 + 1 1 − by Figure 3 demonstrates the distribution of dimensionless stresses in R for different valus of b. The solid line curve shows the case of the homogeneous material properties i.e. b=0. The case b = 0.5 and b = 1 corresponds to dotted and dashed lines respectively on stress distribution. As the expection the curves are symmetric about y = 0 which clearly shows an effect of inhomogeneity. Thus, transversal stress σ̄yy should have maximum value at x = 0 which is shifted towards the direction of inhomogeneity increase. 5. Conclusions This article develops an approach for solving an analytic solution of the plane two dimensional thermoelastic problems in terms of inhomogeneous isotropic strip. In this study we arrive at following conclusions. • The original thermoelastic problem is reduced to that of solution of integral equations using direct integration method. ANALYSIS OF THERMAL STRESSES TO INHOMOGENEOUS STRIP 151 Figure 2. Distribution of µ(x) Figure 3. Distribution of σ̄yy for b=0, 0.5, 1 • It provides the solution of volterra integral equation of second kind which is then solved by resolvent kernel method which provides an efficient technique for analysis of inhomogeneous thermoelastic problems in terms of stress components in the strip R. • The presesnted technique can be applied without any restrictions for material properties . • One can solve corresponding inverse thermoelastic problem in displacements using constructed solutions. • In the present article, analytical solution of thermal stresses is constructed by assuming the fact that, stresses are vanishing at infinity. We can see that, same technique can be used for problems with different loading conditions, instead of Fourier transform. 152 A. B. ADHE, K. P. GHADLE, AND U. S. THOOL References [1] A. Adhe and K. Ghadle , Thermal Stress Analysis of Inhomogeneous Infinite Solid to 2D Elasticity of Thermoe- lastic Problems, Springer Proceedings in Mathematics and Statistics (2022), 509–521. https://doi.org/10.1007/ 978-981-19-9307-7_41 [2] S.Babich and N. Dikhtyaruk, Load Transfer from an Infinite Inhomogeneous Stringer to a Prestressed Elastic Strip Clamped at One Edge, International Applied Mechanics, 56(6)(2020), 708-716. https://doi.org/10.1007/ s10778-021-01047-9. [3] M. Biot, Thermoelasticity and Irreversible Thermodynamics, Journal of Applied Physics, 27(3)(1956) 240-253. [4] K. Ghadle and A. Adhe, Steady-State Temperature Analysis to 2D Elasticity and Thermoelasticity Problems for Inhomogeneous Solids in Half-Plane, Journal of Korean Society for Industrial and Applied Mathematics, 24(1)(2020), 93-102. https://doi.org/10.12941/jksiam.2020.24.093. [5] I. Kaur, K. Singh and E. Cracium, Recent advances in the theory of thermoelasticity and the modified mod- els for the nanobeams: a review, Discover Mechanical Engineering, 2(2)(2023). https://doi.org/10.1007/ s44245-023-00009-4. [6] M. Jafari and M. Jafari, Thermal Stress Analysis of Otrhotropic Plate Containing a Rectangular Hole using Com- plex Variable Method, European Journal of Mechanics/ A Solids, 73(2019), 212-223. https://doi.org/10.1016/j. euromechsol.2018.08.001. [7] B. Kalynyak, Stationary Temperature Field Ensuring the Absence of Thermal Stresses in an Inhomoge- neous Rectangular Beam, Journal of Mathematical Sciences, 256(2022), 551-560. https://doi.org/10.1007/ s10958-022-06070-w. [8] B. Kalynyak, Y. Tokovyy and A. Yasinskyy, Direct and Inverse Problems of Thermo-mechanics Concerning Stressed State of Deformed Solids, Journal of Mathematical Sciences, 236(1)(2019), 21-34. https://doi.org/10.1007/ s10958-018-4095-3. [9] R. Kushnir, Y. Tokovyy, M. Yuzvyak and A. Yasinskyy, Reduction of the Two-Dimensional Thermoelasticity Problems for Solids with Corner Points to Key Integrodifferential Equations, Ukrainian Mathematical Journal, 73(2022),1566-1579. https://doi.org/10.1007/s11253-022-02014-4. [10] H. Lord and Y. Shulman, A Generalized Dynamical Theory of Thermoelasticity Journal Mech. Phys. Solids, 15(1967), 299-309. [11] A. Mahakalkar and V. Verghese, Thermoelastic Analysis of Annular Sector Plate Under Restricted Boundaries Amidst Elastic Reaction, Journal of Solid Mechanics, 13(3)(2021), 325-337. https://doi.org/10.22034/JSM.2020. 1892727.1547. [12] V. Manthena, N. Lamba and G. Kedar, Thermoelastic Analysis of a Rectangular Plate with Nonhomogeneous Ma- terial Properties and Internal Heat Source, Journal of Solid Mechanics, 10 (1)(2018), 200-215. [13] M. Ozisik, Boundary Value Problems of Heat ConductionDover Publications, INC Mineola, New York (1968). [14] I. Sneddon, Fourier Transform, McGraw-Hill Book Company, INC (1951). [15] Y. Tokovyy, B. Kalynyak and C. Ma, Nonhomogeneous Solids: Integral Equation Approach, Encyclopedia of Thermal Stresses, (2014). [16] Y. Tokovyy and C. Ma, Analytical Solutions to the 2D Elasticity and Thermoelasticity Problems for Inhomogeneous Planes and Half-planes Arch. Appl. Mech. 79(2009), 441-456. https://doi.org/10.1007/s00419-008-0242-5. [17] Y. Tokovyy and C. Ma, An Explicit form Solution to the Plane Elasticity and Thermoelasticity Problems for Anisotropic and inhomogeneous Solids, International Journal of Solids and Structures, 46(2009), 3850-3859. https: //doi.org/10.1016/j.ijsolstr.2009.07.007. [18] Y. Tokovyy and A. Rychahivskyy, Reduction of Plane Thermoelasticity Problem in Inhomogeneous Strip to Integral Volterra Type Equations, Mathematical Modeling and Analysis, 10(1)(2005), 91-100. https://doi.org/10.1080/ 13926292.2005.9637274. [19] Tianhu He and Li cao, A Problem of Generalized Magneto-thermoelastic Thin Slim Strip Subjected to Moving Heat Sources, Mathematical and Computing Modelling, 49(2009), 1710-1720 . https://doi.org/10.1016/j.mcm.2008.12. 004. [20] V. Vigak and A. Rychachivskyy, Bounded Solutions of Plane Elasticity Problems in Semi-plane, Journal of Compu- tational and Applied Mechanics, 2(2)(2001), 263-272. [21] V. Vigak, Correct Solutions of plane Elastic Problem for a Semi-plane, International Applied Mechanics, 40(2004), 283-289. [22] V. Vigak, The Solution of the Plane Thermoelasticity Problem for a Rectangular Domain, Journal of Thermal Stresses, 21(5)(1998), 254-261. https://doi.org/10.1007/978-981-19-9307-7_41 https://doi.org/10.1007/978-981-19-9307-7_41 https://doi.org/10.1007/s10778-021-01047-9 https://doi.org/10.1007/s10778-021-01047-9 https://doi.org/10.12941/jksiam.2020.24.093 https://doi.org/10.1007/s44245-023-00009-4 https://doi.org/10.1007/s44245-023-00009-4 https://doi.org/10.1016/j.euromechsol.2018.08.001 https://doi.org/10.1016/j.euromechsol.2018.08.001 https://doi.org/10.1007/s10958-022-06070-w https://doi.org/10.1007/s10958-022-06070-w https://doi.org/10.1007/s10958-018-4095-3 https://doi.org/10.1007/s10958-018-4095-3 https://doi.org/10.1007/s11253-022-02014-4 https://doi.org/10.22034/JSM.2020.1892727.1547 https://doi.org/10.22034/JSM.2020.1892727.1547 https://doi.org/10.1007/s00419-008-0242-5 https://doi.org/10.1016/j.ijsolstr.2009.07.007 https://doi.org/10.1016/j.ijsolstr.2009.07.007 https://doi.org/ 10.1080/13926292.2005.9637274 https://doi.org/ 10.1080/13926292.2005.9637274 https://doi.org/10.1016/j.mcm.2008.12.004 https://doi.org/10.1016/j.mcm.2008.12.004 ANALYSIS OF THERMAL STRESSES TO INHOMOGENEOUS STRIP 153 [23] V. Vigak, Method for Direct Integration of the Equations of an Axisymmetric Problem of Thermo-Elasticity in Stresses for Unbounded regions, International Applied Mechanics, 35(1999), 262-268. [24] H. Youssef and A. Al-Lehaibi,The Boundary Value Problem of a Three Dimensional Generalized Thermoelastic Half-space Subjected to Moving Rechangular Heat Source, Boundary Value Problems, 8(2019), 1-15. https://doi. org/10.1186/s13661-019-1119-y. [25] J. Zhihe and R. Batra,Thermal Shock Resistance of Functionally Graded Materials, Encyclopedia of Thermal Stresses, (2014), 5135-5146. https://doi.org/10.1007/978-94-007-2739-7. A. B. Adhe, corresponding author, CSMSS Chh. Shahu College of Engineering, Aurangabad - 431010 India. Email address: adhe.abhijeet@gmail.com K. P. Ghadle, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad - 431004 India. Email address: drkp.ghadle@gmail.com U. S. Thool, Department of Mathematics, Institute of Science, Nagpur - 440001 India. Email address: maths.thool@iscnagpur.ac.in https://doi.org/10.1186/s13661-019-1119-y https://doi.org/10.1186/s13661-019-1119-y https://doi.org/10.1007/978-94-007-2739-7 1. Introduction 2. Problem Formulation 3. Solution of Thermoelastic Problem 4. Numerical Results 5. Conclusions References