ANALYSIS OF THERMOELASTIC STRESSES IN LAYERED PLATES

a 2 2uz=− ( x + y ) +2z(15)⎡ 2μ1+μ ⎤+ ∫ ⎢− ( az + b)+ α T .1−1−⎥dz0 ⎣ μμ ⎦For completeness of the solution, we determinethe deformation characteristics ofthe reference surface (z = 0). Using equations(14) and (13) we have for strain onthe reference surface:ε( 0) ≡ ε0=b . (16)In the case of small deflections we can usefor curvatures the approximations:22∂ uz∂ uzæx = ; æ ∂2 y= .2x ∂ yThen, using equation (15) we find the curvaturesof the reference surfaceæx = æy ≡ æ0=−a. (17)Thus, the plate subjected to a uniform temperaturechange is bent to a spherical surface.2.2. Plate with sliding edgesFor a plate with sliding edges (Fig. 2), wecan suppose that∂uz∂uzuz= 0 , = = 0∂x∂yfor every point of the reference surface.Fig. 2. Plate with sliding edgesThen a = 0 and from equations (10) it followsthatNb = T(18)Band expression (8) for stresses is reduced to*σ = E − α T + b . (19)( )2.3. Plate with fixed edgesFor a plate with fixed edges (Fig. 3), wecan start from the assumptions that∂uz∂uzux = uy = uz=0 , = = 0∂x∂yon the reference surface.Fig. 3. Plate with fixed edgesThen a = b = 0 and equation (8) for stressesis reduced to*σ =−E α T . (20)3. LAYERED PLATE SUBJECTED TOA UNIFORM TEMPERATURECHANGE3.1. Plate with free edgesConsider a layered plate (Fig. 4) consistingof an arbitrary number n of layers with differentconstant thicknesses h i .Fig. 4. Layered plateThe location of an arbitrary layer i of theplate is specified by the coordinate z i ,which is the distance from the bottomplane of the plate to the top plane of the ithlayer. Assume that the coefficient of thermalexpansion α i , the modulus of elasticityE i and the Poisson’s ratio μ i do not changethrough the thickness of the ith layer. Then,assuming that the plate is subjected to auniform temperature change ΔT, and usingpiecewise integration, we can write the parametersN T , M T , B, C and D in equations(11) and (12) asnn* * ⎫NT =ΔT∑Eiαi( zi− zi−1)=ΔT∑Eiαihi⎪ i= 1 i=1 ⎪⎬;(21)nΔT* 2 2M = ( −−1)⎪T ∑Eiαizi zi2 i=1⎪⎭