13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- crack loaded in fatigue in mode I. The main advantage of this problem is that it can be solved analytically. In the associated thermal problem, the thermal losses due to convection and radiation are neglected and the steady state regime is only considered. This problem is axi-symetric and the temperature variation distribution (r,t) is given by the heat equation: 2 2 δ r q (r) k t C , (2) with the Dirac function. A solution of this equation is given in [5]: at r k q r t 4 Ei 4 ( , ) 2 , (3) with u u x x u d e Ei( ) the integral exponential function and a the heat diffusivity. In order to estimate the thermal stresses the thermo-mechanical problem with the temperature variation field previously calculated needs to be solved. The behavior of the material is considered elastic and perfect plastic. It is supposed that the plastic strain occurs only in the reverse cyclic plastic zone. With alternating plasticity, the boundary condition on the reverse cyclic plastic zone radius is radial stress equal to zero. Figure 5. Decomposition of the thermomechanical problem First the thermomechanical problem can be decomposed into two problems: the first problem (purely mechanical problem) is the cracked specimen subjected only to the cyclic loading F(t)=Fm+Fa sin(2 f t) without heat source due to the crack. The stress field associated with this problem is related to a mode one stress intensity factor Kcyc(t). The second problem (purely thermal problem) is the cracked specimen subject to the line heat source q. The thermal stresses associated with this thermal loading create to a stress intensity factor named Ktemp. The thermal effect generates a compressive stress field near the crack front and thus creates a negative contribution on the stress intensity factor (Ktemp < 0) [6]. This decomposition is correct if crack closure due to the thermal effect is neglected and if the thermal effect does not affect significantly the reverse cyclic plastic zone radius. This assumption is realistic if the thermal correction remains small compared to the mechanical loading. The first pure mechanical problem is solved in a classical way and enable us to estimate the mode I stress intensity factor Kcyc(t) according to the applied force F(t). In order to solve the second problem, another decomposition is necessary. This second decomposition is illustrated in fig. 6.
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