Chapter 6:Multiscale Modeling and Simulation of Deformation in Nanoscale Metallic Multilayered Composites F.Akasheh and H.M.Zbib School of Mechanical and Materials Engineering Washington State University,Pullman,WA,USA 6.1 Introduction Nanoscale metallic multilayered (NMM)composites represent an important class of advanced engineering materials which have a great promise for high performance that can be tailored for different applications.Tradi- tionally,NMM composites are made of bimetallic systems produced by vapor or electrodeposition.Careful experiments by several groups have clearly demonstrated that such materials exhibit a combination of several superior mechanical properties:ultrahigh strength reaching 1/3 to 1/2 of the theoretical strength of any of the constituent materials [28],high ductility [25],morphological stability under high temperatures and after large deformation [22],enhanced fatigue resistance of an order of magnitude higher than the values typically reported for the bulk form [35],and improved irradiation damage resistance [17,27],again,as compared to the bulk.However,the basic understanding of the behavior of those materials is not yet at a level that allows them to be harnessed and designed for engineering applications. The problem lies in the complexity and multiplicity of factors that govern their behavior.Although the concept of creating a stronger metal from two weaker ones by combining them in laminates has been proposed and understood by Koehler in 1970 [20],the nanometer scale introduces a new domain of complexity.At this length scale,the discrete nature of dislocations and their interactions becomes increasingly significant in dictating the response.Depending on the lattice structure and lattice para- meters mismatch of the two materials,the layers can be under very high
Chapter 6: Multiscale Modeling and Simulation of Deformation in Nanoscale Metallic Multilayered Composites F. Akasheh and H.M. Zbib School of Mechanical and Materials Engineering Washington State University, Pullman, WA, USA 6.1 Introduction Nanoscale metallic multilayered (NMM) composites represent an important class of advanced engineering materials which have a great promise for high performance that can be tailored for different applications. Traditionally, NMM composites are made of bimetallic systems produced by vapor or electrodeposition. Careful experiments by several groups have clearly demonstrated that such materials exhibit a combination of several superior mechanical properties: ultrahigh strength reaching 1/3 to 1/2 of the theoretical strength of any of the constituent materials [28], high ductility [25], morphological stability under high temperatures and after large deformation [22], enhanced fatigue resistance of an order of magnitude higher than the values typically reported for the bulk form [35], and improved irradiation damage resistance [17, 27], again, as compared to the bulk. However, the basic understanding of the behavior of those materials is not yet at a level that allows them to be harnessed and designed for engineering applications. The problem lies in the complexity and multiplicity of factors that govern their behavior. Although the concept of creating a stronger metal from two weaker ones by combining them in laminates has been proposed and understood by Koehler in 1970 [20], the nanometer scale introduces a new domain of complexity. At this length scale, the discrete nature of dislocations and their interactions becomes increasingly significant in dictating the response. Depending on the lattice structure and lattice parameters mismatch of the two materials, the layers can be under very high
236 F.Akasheh and H.M.Zbib stress states;and interfaces may contain misfit dislocation structures.The miscibility of the materials and the chemical potential strongly affect the nature of interfaces and,hence,their interaction with dislocations.The fact that interfaces form an unusually high-volume fraction of the material makes them a major factor in governing the behavior.The combined complexity and interactions among all of the above-mentioned factors explains the deficiency in the theoretical understanding of the response of NMM composites. The strong dependence of the mechanical behavior of NMM composites on unit dislocation processes and interfaces poses a challenge to modeling and simulating their behavior.Classical plasticity does not consider the physical mechanism underlying the deformation of the modeled continuum and fails to predict the dependence of the response of metallic structures on their size.Although classical crystal plasticity provides the correct physical framework for modeling dislocation-dependent plasticity,it fails to predict size effect and related phenomena because it does not accom- modate geometrically necessary dislocations associated with gradients in plastic deformation.If any,it would be strain gradient plasticity theories that could provide the suitable framework for modeling NMM composites, although this remains a challenging problem and is far from being resolved at the present state of the field. Multiscale modeling is one of the most promising modeling paradigms which appeared in the last decade for modeling macroscopic phenomena whose roots lie at a finer scale.The approach is based on the appropriate coupling of two models for each of the scales involved.In the case of NMM composites,such coupling involves the continuum mesoscale and dislocation microscale models,although a further coupling to the atomic scale is possible but practically very complex.Three-dimensional dislocation dynamics (DD)analysis is one of the most recent and powerful tool to model the behavior of metallic materials at the microscale in a more physical manner than existing plasticity models [8,21,33,40,411.Since its development in the early 1990s,DD analysis has made significant advancement and proved useful in addressing several problems of interest in materials science and engineering.When coupled with the continuum level finite element (FE)analysis,the result is a multiscale model of elastoviscoplasticity which explicitly incorporates the physics of disloca- tion motion and interactions among themselves and with external loads, surfaces,and interfaces [37,38].Such a model provides a very useful tool perfectly suited to studying the behavior of micro-and nanosized metallic structures.The mechanical behavior of NMM composites is clearly one example of those problems
stress states; and interfaces may contain misfit dislocation structures. The miscibility of the materials and the chemical potential strongly affect the nature of interfaces and, hence, their interaction with dislocations. The fact that interfaces form an unusually high-volume fraction of the material makes them a major factor in governing the behavior. The combined complexity and interactions among all of the above-mentioned factors explains the deficiency in the theoretical understanding of the response of NMM composites. The strong dependence of the mechanical behavior of NMM composites on unit dislocation processes and interfaces poses a challenge to modeling and simulating their behavior. Classical plasticity does not consider the physical mechanism underlying the deformation of the modeled continuum and fails to predict the dependence of the response of metallic structures on their size. Although classical crystal plasticity provides the correct physical framework for modeling dislocation-dependent plasticity, it fails to predict size effect and related phenomena because it does not accommodate geometrically necessary dislocations associated with gradients in plastic deformation. If any, it would be strain gradient plasticity theories that could provide the suitable framework for modeling NMM composites, although this remains a challenging problem and is far from being resolved at the present state of the field. Multiscale modeling is one of the most promising modeling paradigms which appeared in the last decade for modeling macroscopic phenomena whose roots lie at a finer scale. The approach is based on the appropriate coupling of two models for each of the scales involved. In the case of NMM composites, such coupling involves the continuum mesoscale and dislocation microscale models, although a further coupling to the atomic scale is possible but practically very complex. Three-dimensional dislocation dynamics (DD) analysis is one of the most recent and powerful tool to model the behavior of metallic materials at the microscale in a more physical manner than existing plasticity models [8, 21, 33, 40, 41]. Since its development in the early 1990s, DD analysis has made significant advancement and proved useful in addressing several problems of interest in materials science and engineering. When coupled with the continuum level finite element (FE) analysis, the result is a multiscale model of elastoviscoplasticity which explicitly incorporates the physics of dislocation motion and interactions among themselves and with external loads, surfaces, and interfaces [37, 38]. Such a model provides a very useful tool perfectly suited to studying the behavior of micro- and nanosized metallic structures. The mechanical behavior of NMM composites is clearly one example of those problems. 236 F. Akasheh and H.M. Zbib
Chapter 6:Multiscale Modeling and Simulation of Deformation 237 Section 6.2 explores the subject of modeling and simulation of NMM composites using multiscale modeling.The basics of dislocation-based metal plasticity and its mathematical modeling through DD analysis are reviewed.Multiscale coupling of continuum mechanics and dislocation dynamics are then presented.Background on the mechanical behavior of NMM composites is presented in Sect.6.3.Finally,the benefits of multiscale and other modeling tools for NMM composites are demonstrated using different examples. 6.2 Multiscale Modeling of Elastoviscoplasticity Decades of research,since the existence of dislocations in crystal was first theorized,have established that metal plasticity is governed by the response of crystal defects,mainly dislocations,to external and internal loading.Macroscopically observed deformation of metals is the cumulative result of the motion of a very large number of dislocations.Although the theory of dislocations provides a complete description of the stress,strain, and displacement fields of a dislocation as well as of their motion under the effect of forces acting on them,the extension of this theoretical under- standing to provide accurate physics-based prediction of the mechanical behavior of metals is practically impossible. A typical density of dislocation in a moderately worked metal amounts to 10 x 102m2.A cubic millimeter of such metal contains about 1,000 m of curved dislocation lines.The huge computational demand in calculating the dynamics of such densities of dislocations,further complicated by the fact that dislocations have long-range interactions and can react with each other upon colliding to form intricate configurations with possibly new characteristics,is beyond the existing and near future computational capacities. On the other hand,alternative continuum level modeling,although computationally feasible,remains phenomenological in nature.Even in the case of strain gradient plasticity and geometrically necessary dislocation- based theories,success of one theory in capturing certain aspects of size effects has been problem dependant;and it remains that no general framework is agreed upon.The status quo is mainly due to the complexity and multiplicity of dislocation interactions leading to size effects.For example,it is well known that a dislocation has a distortion field associated with it,which results in a long-range stress field that decays inversely proportional to the distance from the dislocation core.As the dimensions of the specimen become smaller,the interactions between
Section 6.2 explores the subject of modeling and simulation of NMM composites using multiscale modeling. The basics of dislocation-based metal plasticity and its mathematical modeling through DD analysis are reviewed. Multiscale coupling of continuum mechanics and dislocation dynamics are then presented. Background on the mechanical behavior of NMM composites is presented in Sect. 6.3. Finally, the benefits of multiscale and other modeling tools for NMM composites are demonstrated using different examples. 6.2 Multiscale Modeling of Elastoviscoplasticity Decades of research, since the existence of dislocations in crystal was first theorized, have established that metal plasticity is governed by the response of crystal defects, mainly dislocations, to external and internal loading. Macroscopically observed deformation of metals is the cumulative result of the motion of a very large number of dislocations. Although the theory of dislocations provides a complete description of the stress, strain, and displacement fields of a dislocation as well as of their motion under the effect of forces acting on them, the extension of this theoretical understanding to provide accurate physics-based prediction of the mechanical behavior of metals is practically impossible. A typical density of dislocation in a moderately worked metal amounts to 10 × 1012 m−2 . A cubic millimeter of such metal contains about 1,000 m of curved dislocation lines. The huge computational demand in calculating the dynamics of such densities of dislocations, further complicated by the fact that dislocations have long-range interactions and can react with each other upon colliding to form intricate configurations with possibly new characteristics, is beyond the existing and near future computational capacities. On the other hand, alternative continuum level modeling, although computationally feasible, remains phenomenological in nature. Even in the case of strain gradient plasticity and geometrically necessary dislocationbased theories, success of one theory in capturing certain aspects of size effects has been problem dependant; and it remains that no general framework is agreed upon. The status quo is mainly due to the complexity and multiplicity of dislocation interactions leading to size effects. For example, it is well known that a dislocation has a distortion field associated with it, which results in a long-range stress field that decays inversely proportional to the distance from the dislocation core. As the dimensions of the specimen become smaller, the interactions between Chapter 6: Multiscale Modeling and Simulation of Deformation 237
238 F.Akasheh and H.M.Zbib these stress fields become increasingly significant,making the nonlocal effects increasingly pronounced.Furthermore,when the dimensions of a specimen become comparable to the range of the defect structure stress field,size effect arises due to the interaction of this field with the free surfaces (image stresses). The Hall-Petch effect,which implies that strength is inversely pro- portional to the square root of a characteristic microstructural length scale, e.g.,the grain size in microsized grains or the individual layer thickness in microscale multilayered structures,can be directly attributed to dislocation pileups at grain boundaries or layer interfaces,respectively.The stress needed to activate dislocation sources also depends on the grain size and their location within the grain,which reflects as a size effect in the early stages of deformation.Another size effect originates from low-energy dislocation structures,like cell structure or dislocation walls,which tend to form by dislocation patterning and reorganization.Capturing all this complexity is a formidable task for any phenomenology-based theory Plasticity in metals is an example of a problem that is multiscale in nature:The macroscopically observed behavior has its origin in the complex physics occurring at the microscale.A multiscale model for plasticity would implement a continuum level framework which avoids phenomenology by explicitly incorporating the physics of plasticity at the microscale through the DD analysis.The link between the two models is two-way:the DD model calculates and passes the plastic strain and the internal stress field due to dislocations at each material point (after proper homogenization),while the continuum model accounts for boundary con- ditions and internal surfaces and interfaces through the solution of an auxiliary boundary value problem and the superposition concept as detailed below. In Sect.6.2.1,we provide a brief background on dislocations in metals The theoretical aspects of DD and their implementation in DD simulations are presented in Sect.6.2.2.Then the multiscale dislocation dynamics plasticity model is presented in Sect.6.2.3. 6.2.1 Basics of Dislocations in Metals Dislocations are linear defects in crystals identified by their Burgers vector and line sense.Depending on the crystal structure,a dislocation can have one out of a finite set of Burgers vectors and can glide on one of a finite set of crystallographic planes.For example,in face-centered cubic (FCC) metals,there are six possible Burgers vectors,all of a/2(011)-type,a being the lattice parameter,and four {111)slip planes.A combination of a
these stress fields become increasingly significant, making the nonlocal effects increasingly pronounced. Furthermore, when the dimensions of a specimen become comparable to the range of the defect structure stress field, size effect arises due to the interaction of this field with the free surfaces (image stresses). The Hall–Petch effect, which implies that strength is inversely proportional to the square root of a characteristic microstructural length scale, e.g., the grain size in microsized grains or the individual layer thickness in microscale multilayered structures, can be directly attributed to dislocation pileups at grain boundaries or layer interfaces, respectively. The stress needed to activate dislocation sources also depends on the grain size and their location within the grain, which reflects as a size effect in the early stages of deformation. Another size effect originates from low-energy complexity is a formidable task for any phenomenology-based theory. Plasticity in metals is an example of a problem that is multiscale in nature: The macroscopically observed behavior has its origin in the complex physics occurring at the microscale. A multiscale model for plasticity would implement a continuum level framework which avoids phenomenology by explicitly incorporating the physics of plasticity at the microscale through the DD analysis. The link between the two models is two-way: the DD model calculates and passes the plastic strain and the internal stress field due to dislocations at each material point (after proper homogenization), while the continuum model accounts for boundary conditions and internal surfaces and interfaces through the solution of an auxiliary boundary value problem and the superposition concept as detailed below. In Sect. 6.2.1, we provide a brief background on dislocations in metals. The theoretical aspects of DD and their implementation in DD simulations are presented in Sect. 6.2.2. Then the multiscale dislocation dynamics plasticity model is presented in Sect. 6.2.3. 6.2.1 Basics of Dislocations in Metals Dislocations are linear defects in crystals identified by their Burgers vector and line sense. Depending on the crystal structure, a dislocation can have one out of a finite set of Burgers vectors and can glide on one of a finite set of crystallographic planes. For example, in face-centered cubic (FCC) metals, there are six possible Burgers vectors, all of a / 2 011 〈 〉 -type, a being the lattice parameter, and four {111} slip planes. A combination of a F. Akasheh and H.M. Zbib to form by dislocation patterning and reorganization. Capturing all this dislocation structures, like cell structure or dislocation walls, which tend 238
Chapter 6:Multiscale Modeling and Simulation of Deformation 239 Burgers vector and a slip plane defines the slip system of a dislocation. The Burgers vector defines the direction of slip of the material,while the slip plane defines the plane on which the slip motion occurs.On its plane, the dislocation can have an arbitrary line sense,which can change as the dislocation glides.Although the Burgers vector is a characteristic of a dislocation,its slip plane is not because a dislocation can change its glide plane,a process known as cross-slip.Dislocations glide under the effect of shear stress resolved in the slip plane along the slip direction(direction of Burgers vector).Notice the difference between slip direction,which pertains to the direction of motion of the atoms,and the dislocation line motion.The macroscopically observed plastic deformation of a metallic continuum structure is the result of the irreversible glide motion of a large number of dislocations on multiple slip systems each with its own spatial orientation.The macroscopic plastic strain tensor is thus expressed by the following relation,which reflects the tensorial addition of several multiple contributions to slip each in a certain direction on a parti- cular ( EP=∑产(S⑧i叭)m (6.1) where is the plastic strain increment,Bis the slip system index,is the increment of slip on slip system B is the unit slip direction,and )is the slip plane normal. Gliding dislocations can also collide with each other resulting in special types of interactions(short-range interactions)which are very complicated in nature and depend strongly on the interacting dislocations'slip systems, line senses,and approach trajectory.The main interactions include annihilation,jog formation,junction formation,and dipole formation. Furthermore,dislocations can also be trapped,ceasing to move either due to short-range interactions that leave them locked or due to long-range effects like pileups against obstacles or simply due to the occurrence of regions in the material where the stress field is not high enough to drive dislocations. 6.2.2 DD Simulations The idea behind conducting DD simulations is to explicitly model the behavior of a dislocation population under applied load taking into consi- deration all the topological and kinematical characteristics of dislocations
Burgers vector and a slip plane defines the slip system of a dislocation. The Burgers vector defines the direction of slip of the material, while the slip plane defines the plane on which the slip motion occurs. On its plane, the dislocation can have an arbitrary line sense, which can change as the dislocation glides. Although the Burgers vector is a characteristic of a dislocation, its slip plane is not because a dislocation can change its glide plane, a process known as cross-slip. Dislocations glide under the effect of shear stress resolved in the slip plane along the slip direction (direction of Burgers vector). Notice the difference between slip direction, which pertains to the direction of motion of the atoms, and the dislocation line motion. The macroscopically observed plastic deformation of a metallic continuum structure is the result of the irreversible glide motion of a large number of dislocations on multiple slip systems each with its own spatial orientation. The macroscopic plastic strain tensor p ε is thus expressed by the following relation, which reflects the tensorial addition of several multiple contributions to slip each in a certain direction ( ) sˆ β on a particular ( ) nˆ β p () () () sym ( ), s n ˆ ˆ ββ β β ε γ = ⊗ ∑ (6.1) where p ε is the plastic strain increment, β is the slip system index, ( ) β γ is the increment of slip on slip system β, ( ) sˆ β is the unit slip direction, and ( ) nˆ β is the slip plane normal. Gliding dislocations can also collide with each other resulting in special types of interactions (short-range interactions) which are very complicated in nature and depend strongly on the interacting dislocations’ slip systems, line senses, and approach trajectory. The main interactions include annihilation, jog formation, junction formation, and dipole formation. Furthermore, dislocations can also be trapped, ceasing to move either due to short-range interactions that leave them locked or due to long-range effects like pileups against obstacles or simply due to the occurrence of regions in the material where the stress field is not high enough to drive dislocations. 6.2.2 DD Simulations The idea behind conducting DD simulations is to explicitly model the behavior of a dislocation population under applied load taking into consideration all the topological and kinematical characteristics of dislocations Chapter 6: Multiscale Modeling and Simulation of Deformation 239
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