13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 3. Simulation Model The plastic material behavior by localization of cyclic plastic deformation in slip bands will be considered by mechanisms which define the properties of formation, sliding, irreversibility and hardening of a slip band as follows. Formation of a slip band is assumed to occur once a critical resolved shear stress is exceeded [4,5]. The critical resolved shear stress is defined as 0 = crit τ 80MPa, which corresponds to the threshold for slip band formation observed in Ref. [6]. Motivated by the TEM micrograph in Fig. 1b the model used in this study is based on the theory of dislocation pile-ups at grain boundaries [7]. Due to the equilibrium of forces produced by external loading and repulsive forces between dislocations a characteristic dislocation distribution occurs. Taking into account the distortion of a dislocation, which is defined by the magnitude of the Burgers vector, a sliding distribution Δu(ξ) can be determined: ( ) ( ) ( ) 2 2 2 (1 ) / 2 ⋅ − Δ = − ⋅ − crit u l G ν ξ ξ τ τ , (1) where ν and G are Poisson's ratio and shear modulus in the slip plane, respectively. l is the length of slip band and ξ is the coordinate along the slip band starting at the dislocation source. τ and τcrit represent the shear stress existing in the dislocation source and the critical resolved shear stress. It is concluded that the evolution of slip bands is related to cyclic slip irreversibility [3]. To be able to consider this mechanism in the simulation model, the slip band is approximated by two closely located layers, on which dislocation motion occurs separated in tensile and compressive loading (Fig. 2a), as Tanaka and Mura suggested in their model [8]. A special procedure was adopted to accumulate the irreversible fraction of cyclic slip. Fig. 2b shows exemplarily the variation of shear stress τ as a function of time in a slip band (in both layers) provoked by external loading and shows the resulting maximum sliding Δu I in layer I and Δu II in layer II. Figure 2. (a) Approximation of slip band by two closely located layers; (b) Exemplary variation of fatigue shear stress τ as a function of time in a slip band and resulting maximum sliding Δu I in layer I and Δu II in layer II Once the critical resolved shear stress τcrit in the first tensile loading is exceeded, layer I starts to slide by conducting the pile-up sliding model. As the shear stress decreases sliding is fixed at its maximum value Δu1 I. During compressive loading layer II is activated as soon as the shear stress exceeds the critical resolved shear stress in opposite direction and concurrently in layer I, the fixed sliding Δu1 I is reduced to an irreversible fraction p·Δu1 I. It is determined by the cyclic slip irreversibility p, which is defined as the fraction of plastic shear deformation that is irreversible in a microstructural sense [3]. One full cycle later the maximum sliding Δu2 I is reduced again, but this time to an accumulated irreversible fraction denoted by the term p·(Δu1 I +Δu2 I) taking the
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