13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- evaluate the plastic dissipation. Such an approach neglects the contribution of the actual crack extension during any given load cycle. However, for Paris-Regime crack growth, both the plastic work and the surface energy contributions associated with the actual crack extension in any given cycle are negligible compared to the total plastic dissipation. In VHCF regime the imposed load is small compared to the yield strength and the contribution of the plastic wake will be neglected in our work. Finally an elastic- perfectly plastic constitutive model is considered in this work. 2.3.1 Computation of the temperature field In a second step, the plastic dissipation is used as a moving thermal source in a transient heat conduction problem. The thermal problem is solved with an implicit integration scheme. To define the time discretization ( 0 1 n t t t ), we consider the following simplifications in the history of the crack propagation: the crack is supposed to remain circular during the crack growth process, the crack closure effect is neglected and we use the analytical formula for a circular crack in an infinite media 2 eff K K a (4) The loading is given by the energy dissipation per cycle computed in Equation (3). The meshes used are the meshes used during the mechanical computation to avoid errors due the data transfer on / ( ) irr i dW dN a between different meshes. 3. Numerical results The specimen is modeled by a cylinder C which has a circular cross-section of radius Rc and a height 2L. A small circular crack perpendicular to the cylinder axis lies in the center of the cylinder, as shown in Fig. 1. The radius of the crack is denoted by a(t) and its eccentricity from the center of C is denoted by e. C is submitted to a cyclic loading: During each cycle the stress is varied linearly from an initial minimum value ߪ ത to a maximum value ߪ ത௫ and back to the initial value ߪ ഥ. An important parameter used to characterize the cyclic load is the so-called load ratio, defined as ܴൌ ߪ ത ߪ ത௫ ⁄ . Because of symmetry conditions only one quarter of the cylinder is modeled. Figure 1. Specimen modeling The tested material is a high-strength steel SAE 5120. Fatigue test are performed at ultrasonic fatigue frequency 20kHz with a stress ratio of 0.1 R and stress amplitude 400Mpa , using compressed air of 20oC to cool the specimen. The material is approximated as elastic-perfectly plastic with thermomechanical properties 200 E GPa (elastic modulus),
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