A Thermal and Thermal Stress Analysis in Thermoelectric Solid ...

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 2013In terms of the non-dimensional quantities defined in equation(43), the above governing equations reduce to (dropping thedashed for convenience)2 2 2 2 2,x ,x 1 1 2 ,t ,tt(β 1) e u β θ β αβ (β E u ) β u , (44) 2 s s 2 θ ε 21+ o s ε1s e where all the initial state of the material is at rest(59)2 2 2 2 2,y ,y 1 1 1 ,t ,tt(β 1) e v β θ β αβ (β E v ) β v , (45)2 θ (θ τθ ) ε (e τ e ) e , (46),t ,tt 1 ,t ,tt oh E1v= β1E1 ε2 , (47) y t th E2u= β1E2 ε2 x t t, (48) E1 E2h = y x t. (49)The constitutive equations reduce to2 2x x ,xσ (β 2)e 2 u β θ , (50)2 2y y ,yσ (β 2)e 2 v β θ , (51)zz2 2σ (β 2)e β θ , (52)σx y u,y v,x. (53)Differentiating equation (44) with respect to x, and equation (45)with respect to y, and then sum, we obtain22 2 2 h β1α e θ β21 α 0t t t(54)Differentiating equation (47) with respect to y, and equation (48)with respect to x, and then sum, we obtain22 e β1 ε2 h 2t t , (55) tThe heat equation will take the form2 22 1 o 2 12θ ε + ε e . (56) tt ttIII. FORMULATION IN THE LAPLACE TRANSFORM DOMAINWe will apply Laplace transform defined asst f s f t e d t ,0hence, the above equations will take the forms2 2 2 21 1 β αs s e θ β αsh 0(57)2 21 2 β s ε s h se(58)IV. FORMULATION IN THE FOURIER TRANSFORM DOMAINWe will apply Fourier transform defined as1 iqyf (q) e f (y) d y2 ,where1 iqyf (y) e f (q) dq2 .we get2d edx22d θdx2 2 2 q β1α+ε 1+ o s 1 ε1s e2 2 s s θ + β1αsh 2 2 21 o 1, (60) ε + s ε s e q s s θ , (61)2dh2 2se q +β21s ε2s hdx . (62)We can write the above system as follows:2d 2 1 e 2θ + 3hdx , (63) 2ddx2 5 θ 4e, (64)2d 2 7 h 6edx where2 2 q β α+ε + s 1 ε s , 1 1 1 o 123 14 1 o 16 s , 7 q+β1s ε2s .2 β αs , ε + s ε s ,(65)22 s s ,2 25 q s s ,The above system gives the following ordinary differentialequations6 4 2d d d L M N θ 0 6 4 2d x d x d x , (66)6 4 2d d d L M N e 0 6 4 2d x d x d x , (67)and6 4 2d d d L M N h 0 6 4 2d x d x d x , (68)whereL= ,1 5 7M= ,1 5 1 7 5 7 2 4 3 67