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