13th International Conference on Fracture June 16–21, 2013, Beijing, China -7In accordance with the above assumptions, under the adiabatic conditions, the first approximation of heat conduction equation of the piezoelectric materials is t T c p ∂ ∂ = 0 (44) where 0c denotes specific heat capacity. For there is no external heat source, according to the ref. [7] of the electric shock which can retard effectively crack propagation, the heat source power can be introduced to the function of equivalent external point heat source J E p= ⋅ (45) where Jdenotes the electric current density vector field of the dielectric materials. Current density vector of the dielectric material can be expressed as t D J ∂ ∂ = (46) Functionally graded piezoelectric material belongs to the dielectric material. Similarly, introducing power of heat source, temperature field of piezoelectric medium is obtained from the time integration of expression which can be obtained by substituting (45-46) into expressions of (44), τ τ τ τ d y x D y x E c t y x T t ∫ ∂ ∂ = 0 0 ( , , ) ( , , ) 1 ( , , ) (48) For the geometric Eq. (3), constitutive Eq. (2) and fundamental solution (17-19), we get ds e epsA epsA e is p y x E isx y t y t x − +∞ −∞ ∫ + ⋅ = ( , ) ] [ ( , ) 2 1 ( , , ) 2 1 2 1 0 0 * κ π ds e epsA epsA is isx y t y t − +∞ −∞ ∫ + ⋅ + ( , ) ] [ ( , ) 2 1 4 3 4 3 π (49) ds e e p s A t e p s A t e p y x E isx y t y t y − +∞ −∞ ∫ + =− ( , ) ] [ ( , ) 2 1 ( , , ) 2 1 2 2 1 1 0 0 * κ π ds e e p s A t e p s A t isx y t y t − +∞ −∞ ∫ + − ( , ) ] ( , ) [ 2 1 4 3 4 4 3 3 π (50) ds e epsA epsA is e e p y x D isx y t y t x − +∞ −∞ ∫ + − ⋅ ⋅ = − ( , ) ] ( ) [ ( , ) 2 1 ) ( , , ) ( 2 1 2 1 0 0 11 15 * π κ κ ds e epsA epsA is isx y t y t − +∞ −∞ ∫ + − ⋅ ⋅ − ( , ) ] ( ) [ ( , ) 2 1 4 3 4 3 11 π κ (51) ds e e p s A t e p s A t e e p y x D isx y t y t y − +∞ −∞ ∫ + ⋅ = − ( , ) ] [ ( , ) 2 1 ) ( , , ) ( 2 1 2 2 1 1 0 0 11 15 * π κ κ ds e e p s A t e p s A t isx y t y t − +∞ −∞ ∫ + ⋅ − ( , ) ] ( , ) [ 2 1 4 3 4 4 3 3 11 π κ (52) Furthermore, the solution of electric field strength and electric displacement intensity in the Laplace transform domain is ] ( , ) ( , , , ) ( , ) ( , , , ) ( , , ) [ 1 1 2 1 * ∑ ∑ = = + = N l N l l l ex l l ex x N p u S p y x u K N p u R pyxu K c pyx E (53) ] ( , ) ( , , , ) ( , ) ( , , , ) ( , , ) [ 1 1 2 1 * ∑ ∑ = = + = N l N l l l ey l l ey y N p u S p y x u K N p u R pyxu K c pyx E (54) ] ( , ) ( , , , ) ( , ) ( , , , ) ( , , ) [ 1 1 2 1 * ∑ ∑ = = + = N l N l l l dx l l dx x N p u S p y x u K N p u R pyxu K c pyx D (55)
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