13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Finite Element modeling 2.1. Fatigue crack growth law Paris and co-workers [4] have developed estimation for crack growth life for internal initiation, termed fish-eye using the Paris-Hertzberg crack growth rate law [8]. In the present work we use this law to describe the growth of the fatigue crack ௗ ௗ ே ൌܾ ቀ∆ ா√ ቁଷ(1) where b is the Burger's vector modulus, E the elastic modulus and ∆ ܭ the effective stress intensity factor range. 2.2. Thermal dissipation model We consider a problem of a fatigue crack in an elastic-plastic material. The temperature field in the cylinder is evaluated by solving the heat transfer equation ܥ ߩ డ் డ௧ ൌλ∆ܶ ܹߚ ሶ (2) where is the mass density, C the specific heat, the heat conduction coefficient, and ܹ ሶ the plastic dissipation. The multiplicative factor is the so-called Taylor-Quinney factor, which takes into account the energy storage. 2.3 Computation of the temperature field during the fish eye crack growth The temperature field is computed in two steps, in a first step we perform an elastic plastic finite element analysis to compute the plastic energy dissipation as a function of the crack growth, and in a second step we perform a thermal finite element analysis to compute the evolution of the temperature field. 2.3.1 Computation of the plastic energy dissipation The plastic energy dissipation is obtained by 3D elastic plastic finite element analysis. It was shown that in VHCF regime, the crack growth is not a significant portion of life in VHCF fatigue with fish-eye failure [9]. However, the number of cycles involved is still important ( 5 10 cycles) and a direct nonlinear computation will lead to prohibitive computational time. In this respect we choose to compute the energy dissipation per cycle / irr dW dN during a single load cycle on a stationary crack for different radius ia ( 0 1 in n fi a a a a a ), as proposed by Klingbeil [10]. The mean energy dissipation per cycle is computed ܹ ሶ ൌΩ ௗௐೝೝ ௗே ൌ ߳݀:ߪ ௬ (3) where is the frequency of the loading. To simulate the plastic fatigue regime, and not a monotonic loading, two cycles are computed for each radius ia and only the last cycle is used to
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