13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- The objective of this work is to investigate the effects of cooling rate on the TSIFs for an array of parallel edge cracks with alternating lengths in a functionally graded ceramic plate. A linear ramp function is used to describe the rate of boundary temperature variations at the surfaces of the FGM plate. The remainder of the paper is organized as follows. Section 2 reviews a closed-form, asymptotic solution of temperature field for short times and the thermal stresses in the periodically cracked FGM plate. Section 3 describes an integral equation method to obtain the TSIFs at the tips of long and short cracks. Section 4 presents numerical results of TSIFs for an Al2O3/Si3N4 FGM. Section 5 provides concluding remarks. 2. Temperature and Thermal Stress Fields This section reviews the temperature and thermal stress solutions for a long FGM plate with an array of parallel edge cracks with alternating lengths as shown in Fig. 1, where a1 is the length of the long cracks, a2 is the length of the short cracks, h is the crack spacing, and b is the plate thickness. The thermal properties of the FGM plate are arbitrarily graded in the thickness direction (x-direction). Initially the temperature of the plate is a constant T0 which can be assumed to be zero without loss of generality. The temperature then gradually changes to -Ta and -Tb at the surfaces x = 0 and x = b of the plate, respectively. We use a linear ramp function to describe the variations of the boundary temperatures. The initial and boundary conditions for the heat conduction problem are thus ( ,0) 0, 0 T x x b = ≤ ≤ , (1) ( ) / , 0 (0, ) , a a a a a T t t t t T t T t t − ≤ ≤ ⎧ = ⎨ − > ⎩ , (2a) ( ) / , 0 ( , ) , b b b b b T t t t t T b t T t t − ≤ ≤ ⎧ = ⎨ − > ⎩ , (2b) where T = T(x, t) is the temperature, t is time, and ta and tb are two temporal parameters describing the rates of temperature variation (cooling/heating rates) at the plate surfaces. The one-dimensional heat conduction in the plate is governed by the following basic equation ( ) ( ) ( ) T T k x x c x x x t ρ ∂ ∂ ∂ ⎡ ⎤ = ⎢ ⎥ ∂ ∂ ∂ ⎣ ⎦ , (3) where k(x) is the thermal conductivity, ρ (x) the mass density, and c(x) the specific heat. Jin [12] obtained a closed form solution of the temperature field for short times in the FGM plate with continuous and piecewise differentiable properties as follows (1) (2) ( , ) ( , ) ( , ) b a a T x T T x T x T T τ τ τ ⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠ , (4) where ( , ) (1) τ T x and ( , ) (2) τ T x are given by 1/4 (1) 0 0 0 2 2 1 1 1 1 1 ( , ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) exp , , 2 4 2 a a c k 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 − − ≤ ⎨ ⎬ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ ⎩ ⎭ (5a)
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