Inverse quasi–static unsteady–state thermal stresses in a thick circular plate

Abstract In this article, an attempt has been made to discuss the inverse thermoelastic problem of a thick circular plate defined as 0 ≤ r ≤ a, –h ≤ z ≤ h subjected to the arbitrary heat supply at interior point while circular edge of the thick circular plate at the outer surface and at the lower surface is maintained at zero temperature. The conductivity equation and the corresponding initial and boundary conditions have been solved using finite Hankel and Laplace integral transform techniques. Goodier’s and Michell’s functions are used to obtain the displacement components and its associated stresses. The results are obtained in a form in terms of Bessel’s function. The results for unknown temperature, displacement, and stresses have been computed numerically considering special functions and illustrated graphically.


Introduction
As a result of the increased usage of industrial and construction materials, the interest in the inverse thermal stress problems has grown considerably, typified by main shaft of lathe and the role of the rolling mill due to the elementary geometry involved. As a result of this, a number of theoretical studies concerning them have been reported so far. However, to simplify this, almost all the studies were conducted on the assumption that the upper and lower surfaces of the circular plate are insulated or that the heat is dissipated with uniform heat transfer coefficients throughout the surfaces as direct problems. For example, Sabherwal (1965) investigated the inverse problem of transient ABOUT THE AUTHORS Ishaque Khan is an assistant professor in Mathematics at M. G. College, Armori, Gadchiroli (MS), India and pursuing research work in the field of Thermoelasticity BVP. Lalsingh Khalsa has published 21 research papers and 5 books at international level, besides being an excellent supervisor. Vinod Varghese has also extended his full support in assisting the work and has lent a helping hand in providing it with a complete touch.

PUBLIC INTEREST STATEMENT
The results are useful to scientists working on two-dimensional thick plate theories, especially when circumstances are of inverse quasi-static transient conditions. Inverse problems are very crucial, especially in view of its relevance to various industrial machines and it can be the theoretical base for several geophysical and industrial applications. The physical applications are encountered in the context of problems such as ground explosions and oil industries, where the interest is in various phenomena occurring during the measurement of displacements, stresses, and temperature field due to the presence of certain sources.
heat conduction in circular plate. Grysa, Ciałkowski, and Kamiński (1981) discussed on an inverse temperature filled problem of the theory of thermal stresses. Noda (1989) discussed an analytical method for an inverse problem of three-dimensional transient thermoelasticity in a transversely isotropic solid by integral transform technique with newly designed potential function and illustrated practical application of the method in engineering problem. Ashida, Choi, and Noda (1996) investigated an inverse thermoelastic problem in an isotropic structural plate onto which a piezoelectric ceramic plate is perfectly bonded. When unknown heating temperature acts on the free surface of the isotropic structural plate, an electric potential is induced in the piezoelectric ceramic plate. Deshmukh and Wanhede (1997, 1998a, 1998b discussed the inverse transient problem of quasistatic thermal deflection in these clamped circular plates and axisymmetric inverse steady-state problem of thermoelastic deformation of finite length hollow cylinder and inverse quasi-static transient thermoelastic problem in a thin annular disk. Again Ashida et al. (2002) emphasized on the inverse transient thermoelastic problem for a composite circular disk. Yang, Chen, Chang (2002) studied inverse boundary value problem of coupled thermoelasticity in an infinitely long annular cylinder using simulated exact and inexact measurements. Patil and Prasad (2013) studied inverse steady-state thermoelastic problem of a thin rectangular plate using operational methods. From the previous literatures regarding thick plate as considered, it was observed by the author that no analytical procedure has been established for thick circular plate, considering inverse quasi-static thermoelastic analysis.
In this problem, we consider some new interesting results of the inverse heat conduction problem of thick circular plate occupying the space D = {(x, y, z) ∊ R 3 :0 ≤ (x 2 + y 2 ) 1/2 ≤ a, −h ≤ z ≤ h}, where r = (x 2 + y 2 ) 1∕ 2 . In a condition wherein a thick circular plate is subjected to arbitrary heat supply at interior point while the circular edge of the thick circular plate at the outer surface and at the lower surface is maintained at zero temperature, the governing heat conduction equation has been solved using integral transform method. The results are obtained in series form in terms of Bessel's functions. The mathematical model of final thick circular plate has been constructed with the help of numerical illustrations.

Formulation of the problem
Consider a thick circular plate of thickness 2 h occupying space D defined by 0 ≤ r ≤ a, −h ≤ z ≤ h, as shown in Figure 1. Let the plate be subjected to an arbitrary known interior temperature f(r, t) within the region −h ≤ z ≤ h. With lower surface and circular surface, r = a at zero temperature. Under this more realistic prescribed condition, the unknown temperature g(r, t) which is at the upper surface of the plate z = h and quasi-static thermal stresses due to unknown temperature g(r, t) are to be determined.

Temperature distribution
The transient heat conduction equation of the plate is given as follows: where κ is the thermal diffusivity of the material of the disk (which is assumed to be constant), subjected to the initial and boundary conditions: Following Noda et al. (2003), we assume that the Navier's equations in the absence of body forces for axisymmetric two-dimensional thermoelastic problem as:

Thermal displacements and thermal stresses
where U r and U z are the displacement components in the radial and axial directions, respectively, and the dilatation e as: The displacement function in the cylindrical coordinate system is represented by Goodier's thermoelastic displacement potential ϕ and Michel's function M: in which Goodier's thermoelastic potential must satisfy: and the Michel's function M must satisfy: in which K is the restraint coefficient and temperature change = T − T i, T i is the initial temperature, and: The component of the stresses is represented as: where G and ν are shear modulus and Poisson's ratio, respectively; for the traction-free surface, the stress function is: Equations (1) to (17) constitute the mathematical formulation of the problem.

Solution for temperature distribution
In order to solve Equation (1) under the boundary condition (3), we firstly introduce the finite Hankel transform of order m over the variable r; the integral transform and its inversion theorem (Sneddon, 1972) can be written as: in which α 1 , α 2 , α 3 , … are the roots of the transcendental equation J 0 (α m a) = 0.
Applying the finite Hankel transform and Laplace integral transform, and its inversion theorems, results in the final temperature distribution as: The function given in Equation (23) represents the temperature at every instant and at all points of the circular thick plate of finite height.
The unknown temperature g(r, t) can be obtained by substituting z = h in the equation.
g(r, t) = T (r z, t) at z = h, one obtains: in which:

Goodier thermoelastic displacement potential ( )
Referring to the fundamental Equation (1) and its solution (19) for the heat conduction problem, the solution for the displacement function is represented by Goodier's thermoelastic displacement potential governed by Equation (11); this is represented by:

Michell's function M
Similarly, the solution for Michell's function M is assumed so as to satisfy the governed condition of Equation (12) as: in which H mn and R mn are arbitrary functions.

Displacement and thermal stresses
In this manner, two displacement functions in the cylindrical coordinate system ϕ and M are fully formulated. Now in order to obtain the displacement components, we substitute the values of thermoelastic displacement potential φ and Michell's function M in Equations (9) and (10), which results in:

Determination of unknown arbitrary function H mn and R mn
Applying boundary condition Equation (17) to Equations (25) and (28), one obtains:

Concluding remarks
In this problem, a thick circular plate is considered which is kept traction-free as well as subjected to arbitrary known interior temperature and determined for the expressions of unknown temperature, displacements, and stress functions due to the unknown temperature. As a special case, mathematical model is constructed for f (r) = (r 2 − a 2 ) 2 (1 − e t ) and numerical calculations were performed.
The thermoelastic behaviors such as temperature, displacements, and stresses are examined with the help of arbitrary known interior temperature along the radial direction as a → −0.2, b → −0.1, c → 0, d → 0.1. Figure 2 indicates that the unknown temperature decreases from r = 0 to r = 0.3 and increases from 0.3 to 1 with the thickness of the circular plate. As the source of known temperature varies from a negative to positive value, the unknown temperature decreases its magnitude along the radial direction.
As shown in Figure 3, the source of known temperature varies from bottom to top, the radial displacement decreases at r = 0, and the radial displacement vanishes; otherwise, its existence would have been visible.
As shown in Figure 4, the source of known temperature varies from bottom to top; the axial displacement increases along radial direction; and it shows its existence. Figure 5 shows that the radial stress decreasing from bottom to (lower surface to upper surface) top. Stress at r = 0 and r = a is zero; otherwise, it shows its existence. Figure 6 indicates that the stress function σ θθ decreases with the thickness of the circular plate. It shows the existence for small thickness. Also, it develops tensile stresses in the radial direction.
In this article, we analyzed an inverse thermoelastic problem of a thick circular plate and determined the expressions of unknown temperature, displacement, and thermal stresses. The heat conduction differential equation is solved using finite Hankel and Laplace integral transform techniques, and their inversion theorems. Goodier's and Michell's functions are used to obtain the displacement components. As a special case, a mathematical model is constructed for steel (SN 50C) thick plate, with the material properties specified as above, and examined for the thermoelastic behaviors in unsteady-state field for unknown temperature change, displacement, and thermal stresses. We conclude that the displacement and stress components occur near the heat source region. As the temperature increases, the circular plate will tend to expand in the radial direction as well as in the axial direction. Also any particular case of special interest can be derived by assigning values to the parameters and functions in the expressions (19)-(28).