13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- tan m mL H β β = (13) 2.2. Thermal Stress in UHTC strip under cold shock After the temperature is obtained, the transient thermal stress in a fully free strip can be obtained from the classical beam analysis [21]. For plane-stress ( ) , yy x t σ has the form ( ) ( ) ( )( ) 0 , ( , ) ( , ) ( , ) yy x t E T x t Ax B Txt Txt T σ α = + − − ⎡ ⎤ ⎣ ⎦ (14) where E and α are the temperature-dependent elastic modulus and thermal expansion coefficient, respectively. Values for constants A and B are determined from: ( ) ( ) 2 2 0 0 , 0 and , 0 a a yy yy x t dx x t xdx σ σ = = ∫ ∫ (15) 3. Thermal stress intensity factor and thermal shock cracking The transient thermal stress intensity factor (TSIF) for an edge crack of quasi-static fracture problem can be obtained by using a non-dimensional weight function and the thermal stress. ( ) ( ) 0 I , 2 c yy x t dx c K t cF c a σ π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∫ (16) where c is the crack length, and F is the weight function[22], KI(t) represents the TSIF at t. It is obvious that the TSIF is proportional to thermal shock temperature difference ∆T, crack propagation will not occur if the thermal shock has no reached a critical value ∆Tc, at which the peak value of transient TSIF reaches the fracture toughness KIC. At ∆T=∆Tc, the peak value KI(t) reaches KIC and crack propagation is initiated. Therefore, when ∆T≥∆Tc, as the thermal shock proceeding, the crack propagation will occur when TSIF KI(t) > KIC and results in severe damage in UHTC plate; And it will be arrested after extending when KI(t) > KIC again, thus the crack will reach a final length cf after thermal shock determined by Eq. (17). ( ) ( ) f 0 f I f IC c f , , and 0 2 c yy x t dx c K t c F K T T t c a σ π ⎛ ⎞ = = Δ >Δ > ⎜ ⎟ ⎝ ⎠ ∫ (17) Therefore, the thermal shock residual strength can be determined by the final crack length as shown in Eq. (18). IC R f f K c c F a σ π = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (18) 4. Results and discussion Refractory diborides of zirconium (ZrB2) based ceramics are the most used UHTCs. Take it as an example, the temperature-dependent material properties of the diborides of zirconium based UHTCs are shown Table 1 [8, 16, 23]. And the thickness of UHTC strip 2a = 5mm. Fig. 2 shows temperature has significant effect on crack resistance [5]. It shows that a higher temperature leads to a higher crack resistance. The highest crack resistances are about 5.5MPa·m1/2 and 6MPa·m1/2 for 20oC and 600oC respectively. Due to lack of temperature-dependent crack resistance data, while determining the thermal shock residual strength with temperature-dependent material properties, we consider that it is still the same as the crack resistance at 600oC when
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