Chapter 6:Multiscale Modeling and Simulation of Deformation 245 conservation of the dislocation flux and the continuity of the dislocation curves [2]. As for the cases where the simulation box represents the complete specimen with finite domain and arbitrary loading conditions,the above boundary conditions are no longer valid;and a special treatment for the finite domain is needed.This treatment is implemented within the framework of the multiscale model and its discussion is presented in Sect.6.2.3 Evaluation of the macroscopic plastic strain In metals,macroscopic deformation is the result of slip on different slip systems.The area swept by a gliding dislocation represents the area of the newly slipped region due to this motion.In the framework of DD,the increment of the plastic strain can be explicitly calculated from the area swept by the dislocation segments using this relation [39] (6.11) 台2V where Ns is the total number of dislocation segments,Is is the segment length,v is the segment glide velocity,bs is the segment Burgers vector,ns is the normal to the slip plane of the segment,and I is the volume of the RVE. 6.2.3 Multiscale DD Model The coupling of continuum mechanics and DD calculations provides the physical link between the meso-and the microscales.At the continuum level,the typical laws governing an elastic continuum are implemented along with Hooke's law for the elastic regime,as usual.No constitutive law for the plastic behavior of the material is prescribed.Instead,the continuum level plastic strain is explicitly calculated from the actual motion of the underlying dislocations and homogenized at each material point. Another quantity that is explicitly calculated in DD and passed to the continuum scale is the internal stress from dislocations (and any other defects exhibiting long-range,self-induced stress fields).In this manner, the continuum level back-stress concept and its direct effect on hardening are naturally incorporated.Furthermore,this framework allows the rigorous treatment of boundary conditions for free surfaces and interfaces separating
conservation of the dislocation flux and the continuity of the dislocation curves [2]. As for the cases where the simulation box represents the complete specimen with finite domain and arbitrary loading conditions, the above boundary conditions are no longer valid; and a special treatment for the finite domain is needed. This treatment is implemented within the framework of the multiscale model and its discussion is presented in Sect. 6.2.3. Evaluation of the macroscopic plastic strain In metals, macroscopic deformation is the result of slip on different slip systems. The area swept by a gliding dislocation represents the area of the newly slipped region due to this motion. In the framework of DD, the increment of the plastic strain can be explicitly calculated from the area swept by the dislocation segments using this relation [39] s p s s s ss s 1 ( ), 2 N s l n bb n V υ ε = = ⊗+⊗ ∑ (6.11) where Ns is the total number of dislocation segments, ls is the segment length, υs is the segment glide velocity, bs is the segment Burgers vector, ns is the normal to the slip plane of the segment, and V is the volume of the RVE. 6.2.3 Multiscale DD Model The coupling of continuum mechanics and DD calculations provides the physical link between the meso- and the microscales. At the continuum level, the typical laws governing an elastic continuum are implemented along with Hooke’s law for the elastic regime, as usual. No constitutive law for the plastic behavior of the material is prescribed. Instead, the continuum level plastic strain is explicitly calculated from the actual motion of the underlying dislocations and homogenized at each material point. Another quantity that is explicitly calculated in DD and passed to the continuum scale is the internal stress from dislocations (and any other defects exhibiting long-range, self-induced stress fields). In this manner, the continuum level back-stress concept and its direct effect on hardening are naturally incorporated. Furthermore, this framework allows the rigorous treatment of boundary conditions for free surfaces and interfaces separating Chapter 6: Multiscale Modeling and Simulation of Deformation 245
246 F.Akasheh and H.M.Zbib heterogeneous media through the concept of image stresses and eigen- stresses,respectively,as will be demonstrated below.This framework also facilitates the application of general loading conditions in DD simulations. Treatment of finite domains The stress fields employed in the DD calculations are those for a disloca- tion in infinite homogeneous media.In the case of finite domains,the stress fields are truncated at the boundaries and,thus,the dislocation can experience a force depending on its position relative to the free surfaces. The stress field calculations in this situation can be handled through the concept of superposition [5,34,39].The elastic fields for the finite domain problem can be found by summing the elastic fields from two solutions: that for the dislocations as if they existed in an infinite medium,and the solution to a complementary problem where the domain is finite and tractions equal but opposite to those caused by the infinite stress fields at the finite domain boundary (Fig.6.3) u=u+u, 8D=80+8', (6.12) GD=aD+o, where the superscript Doo indicates a defect field quantity as if the defect existed in an infinite homogeneous medium,while the superscript co Solution for dislocation Complementary field in infinite and problem solution: homogenous medium u,8,o Fig.6.3.Superposition principle application for the rigorous treatment of finite boundaries
heterogeneous media through the concept of image stresses and eigenstresses, respectively, as will be demonstrated below. This framework also facilitates the application of general loading conditions in DD simulations. Treatment of finite domains The stress fields employed in the DD calculations are those for a dislocation in infinite homogeneous media. In the case of finite domains, the stress fields are truncated at the boundaries and, thus, the dislocation can experience a force depending on its position relative to the free surfaces. The stress field calculations in this situation can be handled through the concept of superposition [5, 34, 39]. The elastic fields for the finite domain problem can be found by summing the elastic fields from two solutions: that for the dislocations as if they existed in an infinite medium, and the solution to a complementary problem where the domain is finite and tractions equal but opposite to those caused by the infinite stress fields at the finite domain boundary (Fig. 6.3) Fig. 6.3. Superposition principle application for the rigorous treatment of finite boundaries DD * DD * DD * , , , uu u ε ε ε σ σ σ ∞ ∞ ∞ = + = + = + (6.12) where the superscript D∞ indicates a defect field quantity as if the defect existed in an infinite homogeneous medium, while the superscript * = + - t Complementary problem solution: u*, ε*, σ* Solution for dislocation field in infinite and homogenous medium = + - t Complementary problem solution: u*, ε*, σ* Solution for dislocation field in infinite and homogenous medium 246 F. Akasheh and H.M. Zbib