13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- (the thermo elastic sources being not considered here regarding the test frequencies and the adiabatic character of the thermoelastic processes over a complete cycle duration (50 µs)). In the above equations, the unknown parameter is the heat transfer coefficient h. The method used to identify h is presented in the following section. 3.2. Heat loss time constant identification The heat transfer coefficient is determined for each test, from thermal field measurements when the fatigue loading is stopped and the temperature of specimen returns to thermal equilibrium. More precisely, the initial temperature was considered as the temperature when the load was stopped. During the thermal return to equilibrium, no heat source occurs. Thermal measurements θexp applied to each end of the specimen by Dirichlet method enabled us to know the boundary conditions. The unknown heat transfer coefficient h was well chosen to satisfy these conditions: (4) As a result, h was found in ranges of 30 – 100 W/m2/K. This result shows that the heat losses are caused by natural convection and also by an air flow above the specimen which aims at cooling the piezoelectric system. 4. Results for polycrystalline pure copper and discussions Fig. 3a and 3b display the average temperature over the gauge length and several thousands of cycles and the corresponding intrinsic dissipation for various maximum stress amplitudes versus the number of cycles for CuOF 99.95%. The temperature rises over cycles and never reaches a constant value. In other words, the temperature does not stabilize, showing an evolution of the heat balance and consequently of the microstructure. Despite a slight raise of the temperature at Δ/2= 44.1 MPa, the intrinsic dissipation increased very slowly with the number of cycles. It attained to 0.498 °C/s at 106 cycles and reached 0.505 °C/s at 108 cycles. It means that the heat sources were higher than the heat losses and remained active along the cycles. However, no slip bands were observed on specimen surface up to 108 cycles at this stress range. At Δ/2 = 49.6 MPa, the intrinsic dissipation increased slowly up to 107 cycles. No slip bands were either observed between 106 and 107 cycles. At 108 cycles, a clear increase of the intrinsic dissipation was recorded and slip markings were observed on the specimen surface. The intrinsic dissipation was 1.322 °C/s (Fig. 4a). At higher stress amplitude, Δ/2= 54.7 MPa, the intrinsic dissipation increases with the number of cycles more rapidly than in previous cases. First slip bands were observed at 107 cycles (Fig. 4b). They are more pronounced at 108 cycles (Fig. 4c). At Δ/2 = 69.4 MPa, the intrinsic dissipation rises very fast with the number of cycles and reaches 7.477 °C/s at 106 cycles. First slip bands were observed , ) 2 ( , ) 2 ( ( , 0) ( , 0) 0 ( ) ( , ) '( ) ² ( , ) ( ) ( , ) ( , ) exp exp 2 1 t L t L x t x t S x S x x x t x x t C k x x t t x t D − = − = = = = ∂ ∂ + ∂ ∂ − + ∂ ∂ θ θ θ θ θ θ ρ τ θ θ
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