240 F.Akasheh and H.M.Zbib and their long-and short-range interactions as described above.Short- range interactions due to dislocation collision are accounted for through a set of physics-based rules learned from either atomic scale simulations or careful experimental observations.In short,DD analysis is the numerical implementation of the theory of dislocations to analyze the dynamics of a dislocation system in materials. Generally,the simulation box in DD represents a representative volume element(RVE)of a larger specimen,although in some cases freestanding microsized components can make the simulation box.Unless a certain initial dislocation structure is desired,the simulation starts with a randomly generated dislocation structure.Dislocations are modeled as general curved lines in three-dimensional space made of an otherwise elastic medium characterized by its shear modulus,Poisson's ratio,and mass density.Dislocation lines are discritized into small segments,each asso- ciated with a dislocation node [39].The nodes are the points at which forces on a dislocation from all dislocations in the system and from external loads are calculated.The governing equation for dislocation motion is then used to estimate the velocity,and hence the displacement, of each node in response to the net applied force.The node positions are updated accordingly,generating the new dislocation configuration and the process is repeated. In this scheme,the analysis of the dynamics of continuous line objects reduces to those of a finite number of nodes.Typical to numerical algorithms,the mesh size(here the length of a segment)can be refined to obtain the desired accuracy in representing the topology of curved dislocation lines and their dynamics.The above sequence of calculations is repeated as time marches in appropriately chosen time steps,to the desired point of evolution of the dislocation system or the overall stress or strain levels.The details of the approach outlined above will be explored in the following section. Dislocation equation of motion The theory of dislocations provides the following governing equation for the motion of a straight dislocation segment s [11,14,18]: mδ，+v=P M (6.2) Typical of a Newtonian-type equation of motion,it expresses the relation between the velocity of "an object"and the dislocation segment of effective mass ms,moving in a viscous medium with a drag coefficient of
and their long- and short-range interactions as described above. Shortrange interactions due to dislocation collision are accounted for through a set of physics-based rules learned from either atomic scale simulations or careful experimental observations. In short, DD analysis is the numerical implementation of the theory of dislocations to analyze the dynamics of a dislocation system in materials. Generally, the simulation box in DD represents a representative volume element (RVE) of a larger specimen, although in some cases freestanding microsized components can make the simulation box. Unless a certain initial dislocation structure is desired, the simulation starts with a randomly generated dislocation structure. Dislocations are modeled as general curved lines in three-dimensional space made of an otherwise elastic medium characterized by its shear modulus, Poisson’s ratio, and mass density. Dislocation lines are discritized into small segments, each associated with a dislocation node [39]. The nodes are the points at which forces on a dislocation from all dislocations in the system and from external loads are calculated. The governing equation for dislocation motion is then used to estimate the velocity, and hence the displacement, of each node in response to the net applied force. The node positions are updated accordingly, generating the new dislocation configuration and the process is repeated. In this scheme, the analysis of the dynamics of continuous line objects reduces to those of a finite number of nodes. Typical to numerical algorithms, the mesh size (here the length of a segment) can be refined to obtain the desired accuracy in representing the topology of curved dislocation lines and their dynamics. The above sequence of calculations is repeated as time marches in appropriately chosen time steps, to the desired point of evolution of the dislocation system or the overall stress or strain levels. The details of the approach outlined above will be explored in the following section. Dislocation equation of motion The theory of dislocations provides the following governing equation for the motion of a straight dislocation segment s [11, 14, 18]: s s s s 1 m F . M υ υ + = (6.2) Typical of a Newtonian-type equation of motion, it expresses the relation between the velocity of “an object” and the dislocation segment of effective mass ms, moving in a viscous medium with a drag coefficient of 240 F. Akasheh and H.M. Zbib
Chapter 6:Multiscale Modeling and Simulation of Deformation 241 1/Ms under the effect of a net force Fs.The effective mass per unit dislocation length m has been given for the edge and screw components of a dislocation as follows [14]: 元"C-16m-40+8+14y+50y-2r+6 (6.3a) and W () (6.3b) with r=v1-(v/C)2 and =1-(v/C)2.C and Ci are the trans- verse and longitudinal sound speeds in the elastic medium,v is the dislocation speed,and Wo is the line energy of a dislocation per unit length given as W=(Gb2/4)In(R/)[13].In the later expression,G is the shear modulus,b is the magnitude of the Burgers vector,and R and ro are the external and internal cutoff radii,respectively.Ms is the dislocation mobility and it is typically a function of temperature and pressure.The net force Fs acting on a dislocation line can have several contributions to it depending on the problem.In general, F=Fpcicrls+Fdislocaion+F Fexcmal +Foale +Fmage+Fomotic +Fthcmal (6.4) where FPciers is the force from lattice friction opposing the motion of a dislocation,Fselr is the force from the two neighboring dislocation segments directly connected to the segment under consideration,Fdislocation is the net force from all other dislocation segments in the simulation domain,Fextemal is the force due to externally applied loads,Fbsae is the interaction force between a dislocation and the stress field of an obstacle, Fimage is the force experienced by a dislocation due to its presence near free surfaces or interfaces separating phases of different elastic properties, Fosmotie is the driving force in climb,and Fthermal is the force on the dislocation from thermal noise.In general,the force due to a general stress field o is given by F=1,ob×5, (6.5) where /s is the segment length and o is the stress field "felt"by the dislocation segment,while bs and are the Burgers vector and the line sense,respectively,of the dislocation segment.For example,in the case of
1/Ms under the effect of a net force Fs. The effective mass per unit dislocation length m has been given for the edge and screw components of a dislocation as follows [14]: ( ) 2 o 13 1 35 edge 4 l l l 16 40 8 14 50 22 6 W C m γ γ γ γγ γ γ υ −− − − − = − − + ++ − + (6.3a) and o 1 3 screw 2 ( ) W m γ γ υ − − = −+ (6.3b) with 2 γ υ = −1(/ ) C and 2 l l γ υ = −1(/ ) C . C and Cl are the transverse and longitudinal sound speeds in the elastic medium, υ is the dislocation speed, and Wo is the line energy of a dislocation per unit length given as 2 o o W Gb R r = ( / 4 )ln( / ) π [13]. In the later expression, G is the shear modulus, b is the magnitude of the Burgers vector, and R and ro are the external and internal cutoff radii, respectively. Ms is the dislocation mobility and it is typically a function of temperature and pressure. The net force Fs acting on a dislocation line can have several contributions to it depending on the problem. In general, s Peierls dislocation self external obstacle image osmotic thermal , FF F F F F FF F = + ++ + ++ + (6.4) where FPeierls is the force from lattice friction opposing the motion of a dislocation, Fself is the force from the two neighboring dislocation segments directly connected to the segment under consideration, Fdislocation is the net force from all other dislocation segments in the simulation domain, Fexternal is the force due to externally applied loads, Fobstacle is the interaction force between a dislocation and the stress field of an obstacle, Fimage is the force experienced by a dislocation due to its presence near free surfaces or interfaces separating phases of different elastic properties, Fosmotic is the driving force in climb, and Fthermal is the force on the dislocation from thermal noise. In general, the force due to a general stress field σ is given by ss ss Fl b = σ ⋅ ×ξ , (6.5) where ls is the segment length and σ is the stress field “felt” by the dislocation segment, while bs and ξs are the Burgers vector and the line sense, respectively, of the dislocation segment. For example, in the case of Chapter 6: Multiscale Modeling and Simulation of Deformation 241
242 F.Akasheh and H.M.Zbib externally applied loads,the relevant stress field is o,the net stress from all external loads along segment s and its force contribution will be F The details of the calculation of Fistion and Fe are not trivial and will be further detailed below. (a -loop b loop a (b) Fig.6.1.(a)Integration of the stress field at a point p due to a dislocation loop and (b)the corresponding integration in the framework of DD by the linear element approximation Evaluation of F As mentioned above,this force contribution comes from all of the dis- location segments in the system except for those two connected to the dislocation node under consideration.Dislocation theory provides the stress field of an arbitrary dislocation loop C at an arbitrary point p defined by the position vector r through the following expression [13](see Fig.6.1a)
externally applied loads, the relevant stress field is a σ , the net stress from all external loads along segment s and its force contribution will be a F lb external s s = ⋅× σ ξ . The details of the calculation of Fdislocation and Fself are not trivial and will be further detailed below. Fig. 6.1. (a) Integration of the stress field at a point p due to a dislocation loop and (b) the corresponding integration in the framework of DD by the linear element approximation Evaluation of Fdislocation As mentioned above, this force contribution comes from all of the dislocation segments in the system except for those two connected to the dislocation node under consideration. Dislocation theory provides the stress field of an arbitrary dislocation loop C at an arbitrary point p defined by the position vector r through the following expression [13] (see Fig. 6.1a). O x p r r’ y z i i+1 j (a) (b) loop a loop b O x p r r’ y z i i+1 j (a) (b) loop a loop b 242 F. Akasheh and H.M. Zbib
Chapter 6:Multiscale Modeling and Simulation of Deformation 243 0=- R (6.6) G6V.(bxdr)(V@V-IV2)R. 4π(1-y)yc where R is position vector of p relative to the dislocation segment position r'and I=e,⑧e,+e2⑧e2+e⑧e,is the unit dyadic.In the numerical implementation,dislocation curves are discretized into linear segments; and the above integrals over closed loops become sums over linear segments of length /s;and the contribution from all segments is summed up to find the stress field at any desired point p G (6.7) Furthermore,the integration over the segment length can be evaluated algebraically using the linear element approximation found in [3,13]. According to this approach,the stress field at point p from a dislocation segment bound by nodes i and i+1 can be evaluated as [39](see Fig.6.1b) Gaa(p)=- (6.8) Evaluation of F When applied to calculate the stress field at dislocation node j which belongs to the same dislocation segment whose stress contribution is being considered,the above procedure does not work due to the singular nature of the stress field at the dislocation core.To overcome this obstacle,a regularization scheme developed in [41]is implemented.Consider the dislocation bend consisting of a semi-infinite line and segment (j+1),as shown in Fig.6.2a.The glide force per unit length acting on a point on segment (,j+1)at a distance is explicitly given for the case where the adjacent segment is semi-infinite in length as [13] (6.9)
2 1 1 ( ) d d( ) 8 R4 R ( d )( )R, 4 (1 ) C C C G G b l lb G bl I σ αβ π π π ν ′′ ′ ′ ′ ′ = − ×∇ ⊗ + ⊗ ×∇ − ∇ ⋅ × ∇⊗∇− ∇ − ∫ ∫ ∫ v v v (6.6) where R is position vector of p relative to the dislocation segment position r′ and 1 12 23 3 Ie ee e e e =⊗+⊗+⊗ is the unit dyadic. In the numerical implementation, dislocation curves are discretized into linear segments; and the above integrals over closed loops become sums over linear segments of length ls; and the contribution from all segments is summed up to find the stress field at any desired point p s 2 1 2 1 1 ( ) d d( ) 8 R4 R . ( d )( )R 4 (1 ) N s s s s G G b l lb G bl I αβ π π σ π ν ′′ ′ ′ − = ′ ′ ⎧ ⎫ − ×∇ ⊗ + ⊗ ×∇ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ − ∇ ⋅ × ∇⊗∇− ∇ ⎩ ⎭ − ∫ ∫ ∑ ∫ (6.7) Furthermore, the integration over the segment length can be evaluated algebraically using the linear element approximation found in [3, 13]. According to this approach, the stress field at point p from a dislocation segment bound by nodes i and i + 1 can be evaluated as [39] (see Fig. 6.1b) 1 () . i i p σ αβ α σ σ β αβ + = − (6.8) Evaluation of Fself When applied to calculate the stress field at dislocation node j which belongs to the same dislocation segment whose stress contribution is being considered, the above procedure does not work due to the singular nature of the stress field at the dislocation core. To overcome this obstacle, a regularization scheme developed in [41] is implemented. Consider the dislocation bend consisting of a semi-infinite line and segment (j, j + 1), as shown in Fig. 6.2a. The glide force per unit length acting on a point on segment (j, j + 1) at a distance λ is explicitly given for the case where the adjacent segment is semi-infinite in length as [13] g g ( , ). 4 F G f b L θ πλ = (6.9) Chapter 6: Multiscale Modeling and Simulation of Deformation 243
244 F.Akasheh and H.M.Zbib This expression can be used to find the average force per unit length on segment (,j+1)by integrating it over the length of the segment yielding )a) (6.10) where B is an adjustable parameter that compensates for the energy contained in the dislocation core.Equation(6.10)is an equivalent expres- sion to an alternative expression where an adjustable core cutoff radius ro is used.To adapt the above solution to the case of a finite segment (j-1, D),the superposition principle is used and the net glide component of the force on segment (j,j+1)due to segment (j-1,)can be found by subtracting,from (6.10),the interaction force between additional semi- infinite segment and (,j+1)calculated using the standard procedure (Fig.6.2b). i+l (a j+1 (b) Fig.6.2.Calculation of the Peach-Koehler force on a dislocation segment due to its direct neighboring segment Treatment of boundary conditions Typically,the simulation box used in DD analyses is an RVE representative of an infinite medium.To account for this model,special boundary condi- tions are needed.Two types of boundary conditions are applied in DD(1) reflection boundary conditions,which ensure the continuity of dislocation curves [41]and(2)periodic boundary conditions,which ensure both the
This expression can be used to find the average force per unit length on segment (j, j + 1) by integrating it over the length of the segment yielding g g avg ( , ) ln , 4 F G L f b LL b θ β π ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (6.10) where β is an adjustable parameter that compensates for the energy o is used. To adapt the above solution to the case of a finite segment (j − 1, j), the superposition principle is used and the net glide component of the force on segment (j, j + 1) due to segment (j − 1, j) can be found by subtracting, from (6.10), the interaction force between additional semiinfinite segment and (j, j + 1) calculated using the standard procedure (Fig. 6.2b). Fig. 6.2. Calculation of the Peach–Koehler force on a dislocation segment due to its direct neighboring segment Treatment of boundary conditions Typically, the simulation box used in DD analyses is an RVE representative of an infinite medium. To account for this model, special boundary conditions are needed. Two types of boundary conditions are applied in DD (1) reflection boundary conditions, which ensure the continuity of dislocation curves [41] and (2) periodic boundary conditions, which ensure both the = - j j+1 j-1 j j+1 θ L λ (a) (b) = - j j+1 j-1 j j+1 θ L λ (a) (b) F. Akasheh and H.M. Zbib contained in the dislocation core. Equation (6.10) is an equivalent expression to an alternative expression where an adjustable core cutoff radius r 244