13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) 1/4 2 (1) 2 0 0 0 1 1 1 1 2 2 1 1 1 1 1 1 ( ) ( ) ( ) ( , ) ( ) exp ( ) ( ) ( ) 2 4 2 1 ( ) ( ) ( ) ( ) exp , 2 4 2 a a a a a c k x x x T x x erfc x c x k x x x x x erfc ρ τ τ τ τ ρ τ τ π τ τ τ τ τ τ τ τ π ⎡ ⎤ ⎡ ⎤ Ω Ω Ω ⎧ ⎞ ⎛ ⎡ ⎤ =− + Ω − − ⎨ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎩ ⎣ ⎦ ⎣ ⎦ ⎫ ⎞ ⎛ ⎡ ⎤ Ω Ω Ω ⎪ ⎡ ⎤ − − + Ω + − − ⎟ ⎜ ⎬ ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ − − ⎣ ⎦ ⎪ ⎣ ⎦ ⎝ ⎠ ⎭ ,a τ τ > (5b) and 1/4 (2) 2 2 2 2 2 2 1 ( ) ( ) ( ) ( , ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) exp , , 2 4 2 b b b c b k b T x x c x k x x x x x erfc ρ τ τ ρ τ τ τ τ τ τ π ⎡ ⎤ =− ⎢ ⎥ ⎣ ⎦ ⎧ ⎫ ⎡ ⎤ Ω Ω Ω ⎞ ⎛ ⎡ ⎤ × + Ω − − ≤ ⎨ ⎬ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ ⎩ ⎭ (6a) ( ) 1/4 2 (2) 2 2 2 2 2 2 2 2 2 2 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( , ) ( ) exp ( ) ( ) ( ) 2 4 2 1 ( ) ( ) ( ) ( ) exp 2 4 2 b b b b b b c b k b x x x T x x erfc x c x k x x x x x erfc ρ τ τ τ τ ρ τ τ π τ τ τ τ τ τ τ τ π ⎡ ⎤ ⎡ ⎤ Ω Ω Ω ⎧ ⎞ ⎛ ⎡ ⎤ =− + Ω − − ⎨ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎩ ⎣ ⎦ ⎣ ⎦ ⎫ ⎞ ⎛ ⎡ ⎤ Ω Ω Ω ⎡ ⎤ − − + Ω + − − ⎟ ⎜ ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ − − ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ , ,b τ τ ⎪ ⎬ > ⎪⎭ (6b) respectively, and τ , τa and τb are the nondimensional times defined by 2 2 2 0 0 0 / , / , / a a b b t b t b t b τ κ τ κ τ κ = = = , (7) in which (0) 0 κ κ = and ( ) /( ) x k cρ κ = is the thermal diffusivity. In Eqs. (5) and (6), erfc( ) is the complementary error function and Ω1(x) and Ω2(x) are defined by 0 0 1 2 0 1 1 ( ) ', ( ) ' ( ') ( ') x b x x dx x dx b x b x κ κ κ κ Ω = Ω = ∫ ∫ . (8) Figure 1. An FGM plate containing an array of periodic edge cracks with alternating lengths subjected to a thermal shock The above temperature filed induces thermal stresses in both longitudinal (perpendicular to the crack direction) and transverse (perpendicular to the x-y plane) directions in the FGM plate. The
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