13th International Conference on Fracture June 16–21, 2013, Beijing, China -4in hardness due to the second-step-aging is retarded in SA-P specimen aged at 473 K, and the saturation of hardness does not appear during the time investigated. The same trend can be seen in SA-U0 specimens aged at 473 K and 673 K, while the increase in hardness occurs earlier than SA-P specimens. The increase in hardness, ∆HV, during the second-step-aging is considered to arise from the formation of atmosphere of solute atoms or precipitates due to this aging, as will be mentioned later. It is well known that the evolution of precipitates frequently obeys the Johnson-Mehl-Avrami equation in which the volume fraction of precipitates can be written as [11] f =fo {1−exp(−kt n)}, (1) where fo is the saturation value of volume fraction, t is time, and k and n are constants. It is believed that the hardening during the second-step-aging till the peak-hardening is caused by the shear-cutting of the newly formed precipitates by mobile dislocations. Thus the increase in hardness may be given by ∆HV = Ar 1/2f1/2, where A is a constant and r is the radius of particles, according to the theory of precipitation hardening [12]. As will be discussed later, the coefficients of lattice diffusion of substitutional solute atoms are too small to contribute to the precipitation at 473 K. Instead their diffusion along dislocations (pipe diffusion) is considered to play a major role in the precipitation at such low temperatures. In this case the particle size is given by [13] r =B{Dd/(RT)} 1/5t1/5, (2) where B is a constant, Dd is the coefficient of pipe diffusion, R is the gas constant and T is the absolute temperature. Taking into account the incubation time ti, which appears as the retardation of the hardness increase at 473K, we can obtain the following formula for the increase in hardness. ∆HV = β(t −ti) 1/10{1 −exp[−k(t −t i) n]}1/2, (3) where β= ABfo{Dd/(RT)} 1/10. This equation reproduces the measured values of ∆HV fairly well in each specimen and for each second-step-aging condition, as is shown by solid curves in Fig. 3. Table 3 shows the values of parameters in Eq. (3). The values of βand n at 473 K are nearly equal in SA-U0 and SA-P specimens, which suggests that the precipitation proceeds with the same Fig. 3. Change of hardness with time Fig. 4. Change of evaluated yield during second-step-aging. strength with aging time. Table 3. The values of parameters in equation (4) and the activation energy of diffusion. Specimen T(K) β(GPa/ks 1/10) n k (/ksn) t i (ks) Qd (kJ/mol) 300G (SA-U0) 473 0.281 2.35 4.63x 10-4 ~0 113 673 0.636 1.44 0.0531 0.02 350G (SA-P) 473 0.283 2.34 4.37x 10-7 6.47 82.9 673 0.511 0.54 0.0377 0.01 6.0 6.5 7.0 7.5 8.0 0.1 1 10 100 1000 473 K 673 K 473 K 673 K HV (GPa) Aging time t (ks) 350G (SA-P) 300G (SA-U0) -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 0.0010 0.0015 0.0020 0.0025 350G (SA-P) 300G (SA-U0) ln( β 10T) 1/T (1/K)
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