Using the Green function, the boundary integral formula and natural boundary integral equation for thermal elastic problems are obtained. Then based on bending solutions to circular plates subjected to the non-axi- symmetrical load, by utilizing the Fourier series and convolution formulae, the bending solutions under non-axisymmetrical thermal conditions have been obtained. The calculating process is simple. Examples show the discussed methods are effective.

Thermal Bending Problems Circular Plate Boundary Integral Formula Natural Boundary Integral Equation
1. Introduction

Due to the complexity of the thermoelasticity problems, analytic solutions can be obtained only for axisymmetrical problems and simply problems [1-6]. For general non-axisymmetrical loads and general non-axisymmetrical boundary conditions, the numerical computation is the main method [7,8,9]. For bending problems of solid circular plates, Fu Bao-lian adopted the reciprocal theorem and took the solution of the clamped circular plate as the basic solution to discuss some bending solutions under axis-symmetrical loads . Wang An-wen introduced the point source function to discuss the nonsymmetrical bending problems under the concentrated force [11,12]. Yu De-hao discussed bending problems of plates with the natural boundary element method [13,14]. Using the above methods, Li Shun-cai discussed the bending problems of solid circular plates under the boundary loads [15-17]. On the basis of the same method, using Fourier series and several convolution formulae, the boundary integral formula and natural boundary integral equation for the thermal bending of circular plates are obtained. The calculating process is simple. Examples show that the discussed methods are effective.

2. Boundary Integral Formula and Natural Boundary Integral Equation

The differential equation of elastic plate bending problems is

where, is the Laplacian operator, u is the deflection of the plate, q is the surface density of external loads, D is the bending rigidity of the plate, is the plate in a circle domain. For convenient, suppose the circle is a unit circle.

Using the Green formula of the bending problems for thin plates, we get

Let , which is the Green function of the biharmonic equation in , and then the Poisson integral equation of the bending problem of the plate can be found

where , , , , is the Laplacian operator related to . The Green function in the unit circular domain can be obtained from the basic solution of the biharmonic equation

where, and represent the polar coordinate and respectively. Thus  Hence, the Poisson integral formula of the bending circular plates can be obtained as where, * is the convolution with regard to , , denote the deflection and slope at the edge respectively. For the supported edge, , the above equation will be educed to

Suppose M is the differential boundary operator in the polar coordinate system, the bending moment Mu

where, is Poisson ratio. Let the boundary operator acts on Equation (5), and use the limit formula of generalized function The natural boundary integral equation of the bending problems can be obtained as

here 3. Thermoelasticity Equation and Boundary Conditions where q* is the surface distribution density of the thermoelasticity equivalent load over the plate. Suppose h is the thickness of the plate, E is elastic modulus, α is the thermal expansion coefficient and D is the bending rigidity of the plate. In general, suppose the thermal distribution is linear along the plate thickness, the equivalent load  where T(r, θ) is the thermal distribution function on the surface of the plate.

The equivalent boundary conditions of the clamped bending plate are , . The equivalent boundary conditions of the simply bending plate are If in the plate there are no internal heat sources, then , q* = 0, for the simply plate, , Equations (5) and (6) will be reduced to

4. Heat Sources on the Plate

Firstly consider internal heat sources in the plate. The solution process is discussed through some examples.

Example 1 For comparison, suppose the thermal distribution function on the surface of the plate is axisymmetrical,  For the clamped plate, from (5) This solution is according to the axisymmetrical solution For the simply plate, firstly from the Equation (7) to get un which is a constant in axisymmetrical problems. Using the convolution formula [18,19] if k = 0, we have From (7), we get So that Substituting it into (8) and using the convolution formula We get The solution is according to the axisymmetrical solution.

Example 2 Suppose the center of the thermal distribution function is in the point . This is a non-axisymmetrical problem. and For the clamped plate, from (5) Suppose μ = 0.3, D = 1, α/h = 1, by the numerical calculation, the deflections of the plate are shown as Figure 1.

The center deflection of the plate is 0.046 and the maximum deflection is 0.048.

For the simply plate, firstly using Equation (7) to get un , suppose Then suppose μ = 0.3, D = 1, α/h = 1, the left expression of (7) is expanded to Fourier series Substituting it into (8) which is an integral with a strongly singular Poisson kernel, and using the convolution formula, we get Then The deflections of the plate are shown as Figure 2.

The center deflection of the plate is 0.669 and the maximum deflection is 0.67.

5. No Heat Sources on the Plate

When there are no heat sources on the plate, T(r,θ) satisfies harmonic equation, , thus, q* = 0. For the clamped plate, there is no bending deflections on the plate. For the simply plate, (5) and (7) will be reduced to  Example 3 On the boundary of a simply plate, T(1,θ) = sin2θ, in the plate T(r, θ) = r2 sin2θ.

Suppose Then using the convolution formula s We get Substituting it into (8) and using the convolution formula We have From above equation, we get Suppose μ = 0.3, D = 1, α/h = 1, the bending deflections and the bending resultants are as Figure 3 and Figure 4:

6. Conclusions

Based on the Green function method, the boundary integral formula and natural boundary integral equation with

the strongly singular kernel are educed for the thermal bending problem of the plate supported at the boundary. he convolution formulae are utilized to get the solutions of deflection and slope directly for simple problems. As to complex problems, the Fourier series will be used to get the solutions with nice convergence velocity and computational accuracy. The calculating process is simple. accuracy. The calculating process is simple. The problems of other complicated loads can be solved with the similar method or by the superposition with the solutions of the above examples.

7. Acknowledgements

This work is supported by grants of National Basic Research Program of China, No. 2007CB209400 and Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), also supported by State Key Laboratory of Coal Resources and Safe Mining (CUMT) (SKLCRS08X04), supported by Foundation for National Doctoral Dissertation author of China (200760) and Program for New Century Excellent Talents in University (NCET-07-804).

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