MODELING THE MECHANICAL BEHAVIOR OF NANOSCALE METALLIC MULTILAYERS (NMM) WITH COHERENT AND INCOHERENT INTERFACES

Transcription

1 MODELING THE MECHANICAL BEHAVIOR OF NANOSCALE METALLIC MULTILAYERS (NMM) WITH COHERENT AND INCOHERENT INTERFACES By CORY TYSON OVERMAN A dissertation/thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING WASHINGTON STATE UNIVERSITY School of Mechanical and Materials Engineering AUGUST 2009

2 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of CORY TYSON OVERMAN find it satisfactory and recommend that it be accepted. Hussein M. Zbib, Ph. D., Chair David F. Bahr, Ph. D. Balasingam Muhunthan, Ph. D. ii

3 ACKNOLEDGMENTS Firstly, I would like to acknowledge the support and expertise of Dr. Hussein Zbib throughout my time at Washington State University. His attitude and love for the research he does is one thing that I would like to take away from my experience at WSU. I would also like to thank Dr. David Bahr for his expertise and support throughout this project. I also owe many thanks to Dr. Firas Akasheh for his support and knowledge of MDDP. Without Firas work this project would not have come to completion. And last, but not least, I would like to thank Sreekanth Akarapu, and Dr. Mastorakos for the many intellectual conversations, and friendships that were formed during this experience. All of your generosities have helped in more ways than you could imagine. iii

4 MODELING THE MECHANICAL BEHAVIOR OF NANOSCALE METALLIC MULTILAYERS (NMM) WITH COHERENT AND INCOHERENT INTERFACES Abstract by Cory Tyson Overman Washington State University August 2009 Chair: Hussein M. Zbib In a previously published work (Akasheh et al., JAP, 101, , 2007), we investigated the effect on the channeling strength of a single interfacial dislocation intersecting the path of a layer-confined glide dislocation in the Cu/Ni cube-on-cube system. In spite of the enhanced strength predictions of the model compared to the isolated glide model, it was not able to capture the experimentally observed dependence of the strength of NMM composites on their individual layer thickness in the thickness range of a few nanometers. In this work, we use dislocation dynamics (DD) analysis to examine the influence of networks of interfacial dislocations whose nature and distribution are commensurate with the level of relaxation and loading of the structure. Misfit and pre-deposited interfacial dislocation arrays, as well as realistic combinations of both, are studied iv

5 and the dependence of strength on layer thickness is reported, along with observed dislocation-dislocation reactions. To extend the bilayer Cu/Ni simulations, we will also use Multiscale Discrete Dislocation Plasticity (MDDP) to model Ni/Cu/Nb tri-layer composites, and have implemented an interface model to account for the attractive nature observed in incoherent interfaces. Random distributions of threading dislocations on symmetric slip systems are placed in the copper and nickel layers, and strain rate loading is applied uni-axially. This study focuses on the effects of coherent and incoherent interfaces in large-scale simulations, and the resulting layer thickness dependence on strength. The bi-layer data has shown that the implementation of penetrable interfaces in DD captures the strength dependence at layer thicknesses on the order of 1-10 nm, an effect which has been previously observed experimentally, but had not yet been captured via DD simulations. The results also show previously unobserved reactions that occur as a result of interface crossing. The tri-layer data shows also shows a strength dependence on layer thickness, and additionally shows the importance of interfacial design in terms of modeling the phenomenological evolution of plasticity in nanoscale materials. DD and MDDP analyses not only capture the effect of long-range stresses on dislocation motion, but also the effect of short-range interactions and reactions which has proven to be crucial in understanding the strength and microstructural evolution observed in real systems. v

6 TABLE OF CONTENTS Page ACKNOLEDGMENTS...iii ABSTRACT...iv LIST OF FIGURES...ix LIST OF TABLES... xiv CHAPTER ONE: INTRODUCTION Background Motivation Size Effects in Metallic Polycrystalline Solids Nanoscale Metallic Multilayer (NMM) Systems Size Effects in NMM Systems Crystalline Interfaces Interfaces in NMM Composites Coherent Interfaces Incoherent Interfaces Objectives and Approach CHAPTER TWO: MULTILAYER MDDP Multiscale Dislocation Dynamics Plasticity Dislocation Dynamics Continuum Mechanics Coupling (MDDP) Layer Discretization Homogeneous Finite Domain Problem vi

7 2.4 Heterogeneous Finite Domain Problem Dislocations in one region of a bi-material medium Dislocations in both domains of a bi-material medium Dislocations in Multilayer materials FEA Implementation Implementation of Coherent Interfaces in DD Implementation of Incoherent Interfaces CHAPTER THREE: NMM COMPOSITE SETUP AND RESULTS Previous work and motivation in NMM composites Bilayer Bi-layer problem setup Bi-Layer Results Case Case Case Case Case Tri-Layer Setup Tri-Layer Results Case 1 Macroscopic Behavior Case 1 Microscopic Behavior Case 2 Macroscopic Behavior vii

8 CHAPTER FOUR: CONCLUSIONS viii

9 LIST OF FIGURES Figure 1.1: Bright field TEM micrograph of a typical Cu/Nb NMM composite....4 Figure 1.2: Flow strength of varying film thicknesses. Shows the large deviation from the Hall-Petch behavior....5 Figure 1.3: Hardness for varying film thicknesses. Shows that the Hall-Petch relation breaks down for both FCC/FCC and FCC/BCC systems...6 Figure 1.4: Dislocation source of size effects in multilayered systems...8 Figure 1.5: Schematic illustration of coherency stress and misfit formation 25. Below the critical thickness the minimum energy configuration is at infinite dislocation spacing. Above the critical thickness the minimum energy configuration occurs at some equilibrium spacing of misfit dislocations. E e is the elastic strain energy, E T is the sum of the elastic strain energy and the misfit dislocation energy, S is the misfit dislocation spacing, b is the misfit dislocation burgers vector,! is the misfit strain, and a x is the lattice spacing of the associated film Figure 1.6: Peach-Kohler force on a screw dislocation parallel to the interface as a function of distance from the interface. The dashed line represents the equivalent Peach-Kohler force from a free surface. Note that at L/b D < 5 the attractive force from the interface is stronger than the attractive force from a free surface Figure 2.1: Schematic illustration of the motion of an edge dislocation through a crystal Figure 2.2: Schematic illustration of the geometrical characterization of an edge dislocation segment. " is the line sense, and b is the burgers vector ix

10 Figure 2.3: Schematic representation of dislocation discretization in Dislocation Dynamics. The current node is labeled i, the forward neighbor is labeled i+1 and the backward neighbor is labeled i Figure 2.4: Illustration of the velocity and glide-force vectors for an arbitrary dislocation loop Figure 2.5: Flow chart for the numerical reaction formation in dislocation dynamics Figure 2.6: Numerical efficiency of representing a junction formation as the children segments. Nodes labeled F are free nodes, and nodes labeled J are junction nodes.25 Figure 2.7: Multilayer capabilities with multiple initial dislocation configurations Figure 2.8: Schematic of finite boundary solution via superposition Figure 2.9: Schematic of finite boundary solution via superposition, with region two being stiffer than region one Figure 2.10: Dislocations residing in both regions of a bi-material medium Figure 2.11: Schematic illustration of the multilayer extension with arbitrary loading, displacement constraints, and dislocation distribution Figure 2.12: Coherency stress model using a hyperbolic cosine kernel Figure 2.13: Comparison between hyperbolic model and constant stress model Figure 2.14: MD results used to determine the properties for the elastic hardening-plastic constitutive model used in MDDP, shown as dashed lines Figure 2.15: Location of interface dislocations due to yielding (# 0 ) of the incoherent interface Figure 2.16: Constitutive response of the incoherent interface Figure 3.1: Orientation visualized using the Thompson tetrahedron x

11 Figure 3.2: Coherent interface geometry Figure 3.3: Interface strength evaluation Figure 3.4: Bi-layer general problem setup Figure 3.5: Misfit dislocation configurations Figure 3.6: Quasi-Static loading Figure 3.7: Threading strength dependence on layer thickness Figure 3.8: Threading behavior of case one, at a layer thickness of 30b Figure 3.9: Comparison between vertically aligned and low energy configurations at 20b layer thickness Figure 3.10: FEA of the stress fields associated with vertically aligned and low energy configurations of misfit dislocations Figure 3.11: Interface penetration of low energy configuration at 30b layer thickness Figure 3.12: Typical confined layer threading with misfit dislocations Figure 3.13: Pre-deposited threaders moving out of the interface at a layer thickness of 20b Figure 3.14: Threading behavior at 30b layer thickness with pre-deposited threaders (case 3) Figure 3.15: Junction and jog formation between pre-deposited threaders and the threading dislocation Figure 3.16: Time lapse of a junction formation Figure 3.17: Jog formation and annihilation between threading dislocation and predeposited threader xi

12 Figure 3.18: Time-lapse of the threading dislocation passing through the interface at a layer thickness of 20b (case 5) Figure 3.19: Mosaic of snap-shots during the formation of a cross-slip node at a layer thickness of 30b (case 5) Figure 3.20: Typical trilayer setup Figure 3.21: Initial distribution of threading dislocations in tri-layer setup Figure 3.22: Stress - Strain curve for varying layer thicknesses Figure 3.23: Strength dependence on layer thickness Figure 3.24: Yield strain dependence on layer thickness Figure 3.25: Dislocation density evolution with respect to applied stress (Case 1) Figure 3.26: Typical confined layer threading at large layer thicknesses Figure 3.27: Formation of a super-threader by single-layer threader interaction Figure 3.28: Formation of a super-threader by super-threader and single layer threader intersections. The color key is given in figure Figure 3.29: Junction formation between super-threader and non-activated single layer threader Figure 3.30: Formation of super threader on the (-1-11) slip plane Figure 3.31: Stress-strain curves for niobium thickness effect. Nickel and copper thicknesses are held constant at 48b. The niobium thickness ratio is calculated as t Nb /t Ni or t Nb /t Cu. The parenthesized value in the thickness ratio legend is the actual thickness of the niobium layer Figure 3.32: Yield strength dependence on niobium thickness ratio Figure 3.33: Relative hardening after full onset of plasticity xii

13 Figure 3.34: Dislocation density evolution with respect to stress (Case 2) Figure 4.1: Implementation of the conservative interfacial dislocation segment in DD. 105 xiii

14 LIST OF TABLES Table 3.1: Crystallographic orientations in the bi-layer setup Table 3.2: Summary of material parameters in the bi-layer simulations Table 3.3: Misfit dislocation spacing for various layer thicknesses Table 3.4: Defect combinations for bi-layer cases Table 3.5: Crystallographic orientations in the tri-layer setup Table 3.5: Crystallographic slip-systems used in the tri-layer system Table 3.6: List of thicknesses used for the tri-layer cases Table 3.7: Material properties used in tri-layer simulations xiv

15 Dedication This dissertation/thesis is dedicated to my wife and family who have provided their benevolent emotional and financial support throughout my time in Pullman. xv

16 CHAPTER ONE: INTRODUCTION 1.1 Background Motivation In the design of mechanical systems is it often crucial to know the strengths and failure modes of the system in consideration. The strength of a system is governed by both the geometrical constraints and material properties of the components that the system is comprised of. To maximize the reliability and efficiency of mechanical systems it is fundamental to understand and predict the phenomenological evolution of the materials in both the elastic and plastic deformation regime. Historically models of plasticity were the result of mathematically describing a continuum response via macroscopic observation. Such models include the Von-Mises and Tresca yield criterions, continuum elastic-plastic models, Hall-Petch yield relation, etc. The parameters internal to the aforementioned models can be calibrated to match the experimentally observed macroscopic response to external loading conditions. These models have proven to work well for bulk material response, but like many parametric models, they are only valid in a well-defined region. At smaller length scales, such as micrometer or nanometer lengths, the limitations associated with these models is apparent, as they deviate from the experimental observations. For example, Misra et. al. has shown that the yield strength of Cu/Nb NMM composites follows the Hall-Petch relation at thicknesses above that of 75nm 1. Below 75nm the yield strength deviates from the Hall-Petch prediction. To accurately predict the plastic response of materials such as thinfilms, knowledge of the microstructural evolution of the system needs to be 1

17 accounted for. For this reason it is appropriate to obtain a model, which encompasses not only the macroscopic response of the material, but also the microscopic evolution of the substructure, or microstructure. Such models have been considered using strain gradient continuum approaches 2,3,4,5,6,7, multi-scale dislocation dynamics plasticity 8, and molecular dynamics simulations Size Effects in Metallic Polycrystalline Solids In bulk polycrystalline metallic materials the dependence of yield strength on grain size has been recognized since the early 1950 s 10,11,12. In Hall s famous paper titled The Deformation and Ageing of Mild Steel: III Discussion and Results it was shown through experimentation and calculations by Eshelby, Frank, and Nabbaro that the yield strength of mild steels is heavily dependent upon the average grain diameter. This observation gave rise to the famous Hall-Petch relation shown in equation 1.1. " ys = " i + k y d #1 2 (1.1) # ys is the yield strength of the polycrystalline material, # i is a lattice friction term, k y is a hardening parameter, and d is the average grain diameter. The dislocation analogy often used to formulate this relation considers a single frank reed source inside a grain of a poly crystalline material. The material is loaded until the resolved stress inside the grain is large enough to activate the frank reed source (# i ). It is assumed that the other grains are at orientations that require higher applied stresses to obtain the same resolve shear stress. The activation of the source produces loops, which extend out to the grain boundary. The grain boundary acts as an energy barrier to inhibit the motion of the dislocation loops, therefore causing pileups at the grain boundary. As the loops pile up a large stress concentration is formed at the leading dislocation. Equation 1.2 shows that the force per 2

18 unit length on the leading dislocation in the pileup is proportional to the number (N) of dislocations involved in the pileup 13. F L = N"b (1.2) where F/L is the force per unit length of dislocation, N is the number of dislocations involved in the pile-up, # is the applied stress, and b is the burgers vector. After additional stress is applied to the specimen and the stress concentration from the pile up reaches a critical value, the dislocations either pass through the grain boundary or nucleate new dislocations in the adjacent grain. At this point the specimen is said to have yielded. From this analogy one can show that as the grain size decreases, a lower number of dislocation loops are allowed to pile up, which lowers the stress concentration at the leading dislocation loop and requires a higher externally applied stress to yield the material 13. Size effects have historically been observed in polycrystalline materials. At large grain sizes, generally greater than tens of nanometers, the yield stress as a function of grain size follows the well known Hall-Petch formulation 14,15,16. For example in the case of polycrystalline nickel, Erb showed that electrodeposited bulk nickel specimens followed the Hall-Petch behavior down to 30nm grain diameter. At grain sizes smaller than 30nm the yield strength showed an extreme deviation from the Hall-Petch relation 17. This deviation alludes to the fact that dislocation pile-up mechanisms are not responsible for the majority of plastic strain in nanocrystalline materials, and additional modes of deformation must exist. Other models have been proposed some of which include alternative dislocation models, grain-boundary shearing models 18, and two-phase-based models 19. 3

19 1.1.3 Nanoscale Metallic Multilayer (NMM) Systems Nanoscale metallic multilayer systems generally fall under the category of thin film materials, where the aspect ratio of the film is nominally 1000[Ref. Nix]. These films consist of a periodic arrangement of repeating units, where a unit can be any number of material types. For example, a repeating unit may consist of Cu/Ni, Cu/Nb, Ni/Nb, Cu/Ni/Nb, etc. The thickness of the individual films may range from hundreds of microns to a couple of nanometers. Figure 1.1 shows a typical multilayered structure 1. Figure 1.1: Bright field TEM micrograph of a typical Cu/Nb NMM composite. In a metallic multilayered film, the layers can be any material that is capable of being deposited onto a substrate via any thin-film deposition technique. Some of these techniques include chemical vapor deposition (CVD), and physical vapor deposition (PVD). A commonly used technique under the PVD category is sputtering of thin films. Using sputtering techniques allows precise control of the layer thickness, which can vary from layer to layer. This is very desirable from a design perspective because the versatility in manufacturing allows for large flexibilities in designs. 4

20 1.1.4 Size Effects in NMM Systems Recently the size effect observed in multilayer systems has drawn much attention. A size effect has been experimentally observed in both FCC/FCC composites and FCC/BCC composites. At large layer thicknesses, typically greater than 100nm, the yield strength of the multilayer system can be predicted using the familiar Hall-Petch relation. At layer thicknesses less than that of 100nm the multilayer system shows a significant deviation from the Hall-Petch relation. Figure 1.2 shows the deviation of flow strength as a function of layer thickness for a Cu/Nb multilayer film 1. The solid black line represents the Hall-Petch formulation. Figure 1.2: Flow strength of varying film thicknesses. Shows the large deviation from the Hall-Petch behavior. Figure 1.3 shows the deviation from the Hall-Petch relation for both FCC/FCC and FCC/BCC multilayer systems 1. 5

21 Figure 1.3: Hardness for varying film thicknesses. Shows that the Hall-Petch relation breaks down for both FCC/FCC and FCC/BCC systems. Just as in polycrystalline size effects, many similar theories have been presented for the size effect in multilayered systems. From a dislocation perspective, the size effect in multilayered systems can be explained nicely using a theory presented by Misra, Hirth, and Hoagland 1. Refer to figure 1.4 during this discussion of the size effect in multilayered systems. At large length scales, generally greater than 100nm, the layers are large enough to allow for dislocation pile-ups. As seen in figure 1.4, the third regime shows how the Hall-Petch relation can be used to describe the variation in yield strength with thickness. With this formulation the yield stress (#) in the film is a function of h, layer thickness, and is given by equation 1.3. " # 1 h (1.3) As the length scale is decreased, we enter regime two. In this regime the layer thickness is on the order of a few nanometers to tens of nanometers. There is no longer enough 6

22 room for dislocation loop formation and pile-ups to occur, this is where the Hall-Petch formulation begins to breakdown. At these layer thicknesses threading dislocations are thought to be the primary deformation mechanism 20,21. Threading dislocations are glide dislocations that are formed during the layer growth. Typically the threaders originate at surface defects in the substrate and extend from the substrate to the next layer 22. In multilayer systems the threaders in each layer are inherited from the previous layer during growth. The motion of the threaders within each layer gives rise to a behavior called confined layer slip. The restrictions imposed by the interfaces and smaller layer thicknesses limits the motion of the threaders to two dimensions, which brings about a logarithmic strength (#) dependence on layer thickness (h) shown in equation 1.4. " # ln ( h b) h (1.4) where b is the burgers vector. In regime one, the layer thickness is anywhere from one to five nanometers. In this regime the threading dislocations have a very small free length, and the volume fraction of the interfaces is significant. For example, if the interfacial region is taken to be 1.5nm and the layer thickness is 5nm, and interfaces occupy 31% of the volume of the thin film. Due to the small threader free length, the confined layer slip model over predicts the yield strength of the material, and it is thought that interface/dislocation interactions and interfacial shearing become important. 7

23 Figure 1.4: Dislocation source of size effects in multilayered systems Crystalline Interfaces Interfaces in NMM Composites The films that will be dealt with in this thesis will be comprised of copper and nickel, or copper, nickel, and niobium. The combinatorial difference between FCC/FCC structures and FCC/BCC structures has been shown to cause dramatic differences in the plastic response of the two multilayered structures. As discussed in section the interfaces occupy a significant fraction of the total volume of the multilayer at small layer thicknesses. For this reason it is important to take into account the interfacial properties when modeling NMM composites. 8

24 The metallic multilayer systems show two prominent interfaces. The properties of the interfaces are derived from the crystallographic structures of the adjoined layers, and in the case of FCC/FCC (copper/nickel) multilayer structures, a coherent interface is formed. The coherency across this interface is due to the similarities between the crystals that make up the interfacial region. Coherent interfaces show very large strengths, which are associated with the lattice mismatch between the two layers. In the case of FCC/BCC (copper/niobium or nickel/niobium) multilayer structures, a incoherent interface is formed. The incoherency is caused from the dramatic difference in crystallographic structure between the adjoined layers. This type of interface tends to be very weak in shear, and attracts dislocations toward the interfacial region Coherent Interfaces As stated previously, the coherent interface arises when the adjoining layers share crystallographic properties. The high strengths of these interfaces comes about from the small lattice mismatch between the two crystals. This mismatch causes a biaxial or coherency stress/strain in the film, and tends to inhibit the motion of dislocations across the interface. While the interface tends to restrict easy glide across the interface, it still allows the passing of dislocations, and is therefore also considered transparent 24. The misfit strain in an elastic film can be given by equation 1.5, where a f is the lattice parameter of the film, and a s is the lattice parameter of the substrate. " e = #a a $ a f % a s a f $ a s % a f a s (1.5) Another interesting property associated with these interfaces is the formation of misfit dislocations. At small layer thicknesses the coherency strain caused by the interface can 9

25 be completely absorbed by the layers, but as the layer thickness increases, it becomes energetically favorable for the crystal to form misfit dislocations 25. The formation of these dislocations leads to relaxation of the coherency stress in the film. The thickness at which the misfit dislocations are formed is called the critical thickness, and is associated with the modulus mismatch, lattice mismatch, and dislocation spacing. The strain in the film with the creation of misfit dislocations can be approximated by equation 1.6, where b is the burgers vector of the misfit dislocation, and S is the misfit dislocation spacing 25. " e = " # b S $ % a f # a ( s ' & a f ) * # % b ( ' & S * (1.6) ) The critical thickness can be evaluated by comparing the strain energy associated with a dislocation free film to that of a relaxed film with an array of equilibrated dislocations at the interface 26,27,28. The energy of an elastic film subjected to a biaxial misfit strain of! is given by equation 1.7, where E e is the elastic strain energy, E is Young s modulus, h is the layer thickness, $ is poison s ratio, and! is the biaxial misfit strain 25. E e = Eh"2 1#$ ( ) (1.7) At thicknesses larger than the critical thickness, the spacing of misfit dislocations is governed by the modulus of the films and the lattice mismatch. The energy of the film with misfit dislocation can be approximated using equation 1.8, where µ is the shear modulus, and % is a numerical constant on the order of unity 25. ( )2 E T = Eh " # b S 1#$ ( ) + µb 2 2% 1#$ ( ) '&h* ln ), (1.8) ( b + 10

26 Figure 1.5 demonstrates the creation of coherency stresses and misfit dislocations caused by a coherent interface. In this thesis two face center cubic structures; copper and nickel will form the coherent interface. Figure 1.5: Schematic illustration of coherency stress and misfit formation 25. Below the critical thickness the minimum energy configuration is at infinite dislocation spacing. Above the critical thickness the minimum energy configuration occurs at some equilibrium spacing of misfit dislocations. E e is the elastic strain energy, E T is the sum of the elastic strain energy and the misfit dislocation energy, S is the misfit dislocation spacing, b is the misfit dislocation burgers vector,! is the misfit strain, and a x is the lattice spacing of the associated film. 11

27 1.2.3 Incoherent Interfaces As stated previously, the incoherent interface arises when the adjoining layers have dramatically different crystallographic structures. In this thesis the incoherent interface will consist of copper and niobium, or nickel and niobium. Because the crystals are completely different there is no alignment of atomic planes or directions. This discrepancy between crystals eliminates nearly all of the coherency stress as was seen in coherent interfaces. The incoherency tends to make these interfaces weak in shear. The low shear strength of the interface gives rise to an attraction of dislocations that are close to the interface. As the dislocation approaches the interface, the applied stress from the dislocation increases until it reaches the yield point of the interface. After yielding, the interface can t support the shear stress from the dislocation, which pulls the system out of equilibrium and gives rise to an attractive force between the dislocation and the interface. The attractive nature means that the incoherent interface will become a sink to dislocations, and in general won t allow dislocations to cross the interface. This property gives rise to the term opaque interface. Figure 1.6 shows the attractive nature of an incoherent interface on a screw dislocation that is parallel to the interface

28 Figure 1.6: Peach-Kohler force on a screw dislocation parallel to the interface as a function of distance from the interface. The dashed line represents the equivalent Peach-Kohler force from a free surface. Note that at L/b D < 5 the attractive force from the interface is stronger than the attractive force from a free surface. Molecular dynamics simulations show that after the dislocation enters the non-sharp incoherent interface, the core of the dislocation is expanded in an anisotropic fashion along the interface Error! Bookmark not defined.,29. From linear elasticity the core expansion decreases the magnitude of the self-stress field and therefore decreases the likely hood of nucleating dislocations into the neighboring layer. 13

29 Figure 1.7: Anisotropic core spreading of a screw dislocation in an incoherent interface. 1.3 Objectives and Approach All of the NMM composites that will be dealt with in this thesis are comprised of copper and nickel (bilayer), or copper, nickel, and niobium (tri-layer). The combinatorial difference between FCC/FCC structures and FCC/BCC structures has been shown to cause dramatic differences in the plastic response of the two multilayered structures. The focus of this work is to capture this phenomenological difference and microstructural evolutions using Multiscale Dislocation Dynamics Plasticity. In the bilayer (Cu/Ni) case studies the effects of interfacial defects and layer thickness on the flow strength and threading behavior of a single threading dislocation in copper are evaluated. The interfacial defects will consist of pre-deposited threading dislocations of a single character, and misfit dislocations with no coherency stress. In the tri-layer (Cu/Ni/Nb) case studies the effect of layer thickness on the macroscopic strength and threading behavior of an initial random distribution of threading dislocations in copper and nickel are evaluated. The tri-layer case will also include a shearable interface model, which has been derived from a simple shear molecular dynamics simulation. 14

30 1 Misra, A, J Hirth and R Hoagland (2005), "Length-scale-dependent deformation mechanisms in incoherent metallic multilayered composites", Acta Materialia, 53(18): E. C. Aifantis, "On the Microstructural Origin of Certain Inelastic Models," ASME J. Eng. Mat. Tech., vol. 106, pp , H. M. Zbib and E. C. Aifantis, "A Gradient-Dependent Model for Chatelier Effect," Scripta Metall, vol. 22, pp , H. M. Zbib and E. C. Aifantis, "On the Structure and Width of Shear Bands," Scripta Metall, vol. 22, pp. 703, N. A. Fleck and J. W. Hutchinson, "A reformulation of strain gradient plasticity," Journal of the Mechanics and Physics of Solids, vol. 49, pp , M. E. Gurtin, "A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations.," J. Mech. Phys. Solids, vol. 50, pp , S. D. Mesarovic, "Energy, configurational forces and characteristic lengths associated with the continuum description of geometrically necessary dislocation.," Int. J. Plasticity., vol. 21, pp , Zbib H., Diaz de la Rubia T., 2002, IJP, 18, Schiotz J., Jacobsen K. W., A Maximum Strength in Nanocrystalline Copper, Science, Hall E.O., 1951, Proc. Phys. Soc. B, 64, Petch N.J., 1953, JISI, 25, Eshelby J.D., Frank F.C., Nabarro F.R., 1951, Phil Mag, 42, Hirth J.P., Lothe J., Theory of Dislocations, 1982, pp Carlton C. E., Ferreira P. J., What is behind the Inverse Hall-Petch effect in nanocrystalline materials?, Acta Materialia, vol. 55, pp , Nieh TG, Wadsworth J. Hall Petch relation in nanocrystalline Solids. Scr Metall Mater 1991;25: Eckert J, Holzer JC, Krill CE, Johnson WL. Structural and thermodynamic properties of nanocrystalline FCC metals prepared by mechanical attrition. J Mater Res 1992;7:

31 17 Erb U., Electrodeposited Nanocrystals: Synthesis, Properties and Industrial Applications, NanoStructured Materials, Vol. 6. pp S. Takeuchi, The mechanism of the inverse Hall Petch relation in nanocrystals, Scr Mater 44 (2001), p H.W. Song, S.R. Guo and Z.Q. Hu, A coherent polycrystal model for the inverse Hall Petch relation in nanocrystalline materials, Nanostruct Mater 11 (1999), p Misra, A, J Hirth and H Kung (2002), "Single-dislocation-based strengthening mechanisms in nanoscale metallic multilayers", Philosophical Magazine A, 82(16): Matthews J.W., Blakeslee A.E., J. Cryst. Growth, 27, 118 (1974) 22 Ohring, M (2002), "The materials science of thin films: deposition and structure" 23 Misra, A, J Hirth and R Hoagland (2005), "Length-scale-dependent deformation mechanisms in incoherent metallic multilayered composites", Acta Materialia, 53(18): Hoagland, R, J Hirth and A Misra (2006), "On the role of weak interfaces in blocking slip in nanoscale layered composites", Philosophical Magazine, 86(23): Nix, W (1989), "Mechanical properties of thin films", Metallurgical and Materials Transactions A, 20(11): Matthews, J W, Blakeslee, A E, J. Cryst. Growth, 27, 118 (1974) 27 Matthews, J W, Blakeslee, A E, J. Cryst. Growth, 29, 273 (1975) 28 Matthews, J W, J. Vac. Sci. Technol., 12, 126 (1975) 29 Wang, J, R Hoagland, J Hirth and A Misra (2008), "Atomistic modeling of the interaction of glide dislocations with weak interfaces", Acta Materialia, 56(19):

32 CHAPTER TWO: MULTILAYER MDDP 2.1 Multiscale Dislocation Dynamics Plasticity As previously discussed many gradient plasticity formulations have been used to model the plastic response of metallic materials. The internal variables used to model the microstructural evolution have an abundance of assumptions and interaction conditions that need to be dealt with. An even more fundamental model called Multiscale Dislocation Dynamics Plasticity has been developed by Zbib and collogues 1 at Washington State University. In this model linear elastic continuum mechanics, is coupled with discrete dislocation dynamics to solve the multiscale hybrid elastoviscoplastic problem. Discrete dislocation dynamics (DD) makes use of the well-known dislocation theory and models the explicit three-dimensional motion of dislocations Dislocation Dynamics At room temperature and moderate length scales, plastic deformation in crystallographic materials is known to occur by the motion, nucleation, and storage of dislocations. Dislocations are one dimensional line defects that move on prescribed planes and cause slip in prescribed directions. The geometrical characterization of dislocations introduces the parameters,! and b, where! is the line sense, and b is the burgers vector. The motion of dislocations in face center cubic crystals is limited to the family of {111} slip planes, and the burgers vector can take on one of six values, (!)<110>, which is dependent upon the active slip plane. The combination of slip planes 17

33 and slip directions define what are called slip systems. In the fcc crystal there are generally 12 slip systems at room temperature. The burgers vector represents the direction and magnitude of slip caused by the motion of dislocations. See figures 2.1,2.2. Figure 2.1: Schematic illustration of the motion of an edge dislocation through a crystal. Figure 2.2: Schematic illustration of the geometrical characterization of an edge dislocation segment.! is the line sense, and b is the burgers vector. All real systems exhibit dislocations that are of mixed character and take on complex curved shapes in three-dimensional space. Adding to this complexity are the 18

34 interactions between multiple dislocations or between dislocations and other crystallographic defects. To model the kinematics of dislocations, DD employs a piecewise continuous discretization scheme in which each curved length of dislocation is broken into small straight segments. The lengths of the discretized segments are governed by the curvature of the real dislocation, where higher curvatures generate shorter segments. Figure 2.3 schematically shows how a dislocation is discretized and labeled in DD. The current node is labeled i, the forward neighbor is labeled i+1 and the backward neighbor is labeled i-1. Figure 2.3: Schematic representation of dislocation discretization in Dislocation Dynamics. The current node is labeled i, the forward neighbor is labeled i+1 and the backward neighbor is labeled i-1. 19

35 The motion of the segments is governed by a second order non-linear differential equation (2.1), where m * is the effective mass per unit length of dislocation, and M is a mobility term which is dependent upon temperature (T) and pressure (p), v i is the velocity of the i th node, and v i is the time derivative of velocity of the i th node, ie. acceleration. m i* v i + 1 M i ( T, p) v = F i i,glide"component (2.1)1 The effective mass of the dislocation is heavily dependent upon velocity, which brings about the non-linearity in the governing equation of motion. While the mobility is dependent upon thermodynamic state variables, it is also dependent upon the character of the dislocation segment, which ranges from pure screw to pure edge. F i,glide-component is the force applied to the dislocation segment. Equation 2.2 demonstrates the vast array of physical forces that can affect the net force seen by any individual segment. F i = F Peierls + F D + F Self + F External + F Obstacle + F Image + F Osmotic + F Drag + F Thermal + F Interface (2.2) F D represents the force caused by the interaction between two or more dislocations. This term is calculated as the Peach-Kohler force (equation 2.3) from the net stress field induced by all other dislocations in the simulation. F Peierls is a force that arrises from the friction within the crystalline lattice, F Self is the force that is felt from the two neighboring segments centered about the node of interest, F External is the Peach-Kohler force from the applied loads, F Obstacle is the force from other crystallographic defects such as voids or SFT s, F Image is the force from free surface effects or modulus mismatch effects, F Osmotic is the force that is attributed to non-conservative motion like dislocation climb, F Drag is a force that arises from phonon or electron effects, F Thermal is a force that arises from 20

36 thermal effects, which could come from thermal gradients or stochastic motion at high temperatures, and F Interface arises from frictional forces in coherent interfaces. N'1 1 D F D = F j, j +1 = ( & (" i,i+1 ( p) + " a ( p) ) # b j, j +1 $% j, j +1 dl + F j, j +1self L L (2.3) 1 i=1 Equation 2.3 is read as the force on the segment bounded by nodes j and j+1, is equal to the sum of the Peach-Kohler force from all other segments bounded by nodes i and i+1, plus the self-force from the two segments that bound the segment of interest (j, j+1). The singular nature of the stress field near the point of interest drives the addition of the F j,j+1self term. " a is the applied stress, and can also be the stress from other defects in the crystal such as stacking faults, substitutional atoms, and interstitial atoms. Figure 2.4: Illustration of the velocity and glide-force vectors for an arbitrary dislocation loop. 21

37 Equation 2.1 is solved using an implicit finite difference method along with a backward integration scheme. This method yields the unconditionally stable recurrence equation shown in equation 2.4, where superscript t represents the current time-step, and superscript t+#t represents the next time step. v i t +"t # 1+ 1 & % ( $ m * M ' t +"t i = v t i + )t m F t +"t * i (2.4) 1 After the calculation of the velocity, the nodes are moved over the time step $t, the increment of plastic strain is calculated, potential reactions are accounted for, and the new time step is calculated. The plastic strain is calculated from the explicit motion of each dislocation segment. Its tensorial product can be expressed as N $ A " p = i,i+1 2V n # b + b # n ' * ( i i i i) % & ( ) (2.5) i=1 where A i,i+1 is the area swept by the segment bounded by the nodes i and i+1, V is the volume of the simulation cell, n i is the slip plane unit normal vector of the i th node, b i is burgers vector of the i th node, and % is the dyadic product. The tensorial representation of the plastic spin is the main link in the coupling of Dislocation Dynamics to Continuum mechanics. Along with the plastic strain, one can obtain the plastic spin by taking the anti-symmetric part of the geometrical dyadic product. Equation 2.6 shows the tensorial product for the plastic spin. N % A " p = i,i+1 2V n # b $ b # n ( + ( i i i i) & ' ) * (2.6) i=1 22

38 The plastic spin represents the textural rotation due to the motion of dislocations. The plastic spin phenomenon is seen experimentally during rolling where the texture of the material changes throughout the process. Room temperature macroscopic plastic deformation is not only controlled by the motion of dislocations, but also by the short reactions that occur between dislocations. Reactions such as jogs, junctions, annihilations, and cross-slip mechanisms are all responsible for the macroscopic hardening or softening that occurs in metallic materials. For this reason it is important to account for the appropriate reactions in a dislocation dynamics model. Figure 2.5 shows the algorithm that is used in DD to determine if a reaction will occur. 23

39 Figure 2.5: Flow chart for the numerical reaction formation in dislocation dynamics 2. Physically speaking, the formation of a junction will create a new (child) dislocation with a burgers vector equal to the sum of the two parent burgers vectors. It is numerically efficient and convenient to represent the child junction as the two parent dislocations, but with altered properties. Figure 2.6 shows the comparison between the physical and numerical representations. The resultant burgers vector (b 3 ) is the sum of b 1 and b 2. The convenience in representing the junction with both parent segments comes about when junction zipping occurs. As a junction zips up the parent segments come together to form the junction, and when the junction unzips the parent segments simply 24

40 separate. This method reduces the numerical error and inefficiencies that would be associated with replacing the parent segments with a child segment and vice versa as the junction zipped or unzipped. Figure 2.6: Numerical efficiency of representing a junction formation as the children segments. Nodes labeled F are free nodes, and nodes labeled J are junction nodes Continuum Mechanics Coupling (MDDP) In a DD analysis the material properties, accumulated strains, and applied stresses are all assumed to be homogeneous throughout, and the simulation volume is considered a representative volume element (R.V.E.). To account for the heterogeneities observed in real systems, and finite domain problems, a finite element (FE) continuum mechanics approach is coupled with DD. To couple DD with FE, the DD simulation domain is divided into sub-cells, which are coincident with the FE mesh. In the finite element formulation the plastic strain within each sub-cell is homogenized, and the standard laws of continuum mechanics are considered. These laws include conservation of linear momentum, 25

41 div( S) + "f b = " v (2.7) and the conservation of energy, "c v T = k# 2 T + S : $ p (2.8) where S is the Cauchy stress, f b is a body force, & is the material strain, v is the material velocity, T is the temperature at a material point, ' is the density, c v is the specific heat at constant volume, and k is the thermal conductivity. The total strain is written as " = " e + " p (2.9) where & e is the elastic strain, and & p is the plastic strain. Applying Hooke s law one can obtain the stress in each element as: where C ijkl is the elastic isotropic stiffness tensor. p " ij = C ijkl (# lk $# lk ) (2.10) 2.2 Layer Discretization The original implementation of MDDP was to study single crystal materials. In single material systems the continuum properties can be considered continuous through out the entire simulation cell. The discrete nature of NMM composites drives the need to incorporate a material property discretization scheme into MDDP. Firas Akasheh has previously completed this work in which the simulation domain is broken up into discrete layers, and differing material properties can be assigned to each individual layer. This scheme allows for any number of layers with an equal number of different materials. Each layer is given it s own properties, such as, the shear modulus, poison s ratio, 26

42 density, and mobility. The initial dislocation structure is not hindered by this multilayer approach, ie. dislocations can span multiple layers, or be placed in individual layers, as shown in figure 2.7. This allows for many initial dislocation configurations that may be seen experimentally. Figure 2.7: Multilayer capabilities with multiple initial dislocation configurations. 2.3 Homogeneous Finite Domain Problem In the DD formulation, the calculation of the stress field of a dislocation is only valid in an infinite domain problem. Therefore this solution can be used only for infinite domain problems, or problems in which the boundary conditions mimic an infinite domain problem. Previously Zbib et. al. and Bultatov et. al. employed reflection, or periodic boundary conditions in which the dislocation continuity is conserved across the simulation boundary [Ref. Zbib (1998), Bulatov (2000)]. By using this boundary condition the self-stress field solution for a dislocation segment could be used without 27

43 any alteration. A problem arises when one wants to simulate dislocation evolutions with free surfaces or heterogeneous domains. In the presence of free surfaces the solution used for the infinite self-stress field must be modified to satisfy the traction free boundary condition of a free surface. This effect can be seen in the lower right hand side of figure 2.8. The stress field has a relatively long range and the non-zero parts of the field extend past the free surface boundary. The alteration to correct the infinite solution is performed using the method of superposition. In figure 2.8 the stress (S) and strain (&) fields that satisfy the internal defect stress field and the boundary conditions, are solved in two parts. The first term (S (,& ( ) is the infinite solution to the stress distribution caused by internal defects, such as dislocations or voids; and the second term (S *,& * ) is the stress field that satisfies the boundary conditions, which includes the applied tractions and the infinite solution correction for a free surface. Note that in the limit of a free surface with no applied tractions the sum of S ( and S * is equal to zero on the surface, which satisfies the boundary conditions for a free surface (traction free). S = S " + S * (2.11) 1 " = " # + " * (2.12) 1 28

44 Figure 2.8: Schematic of finite boundary solution via superposition. 2.4 Heterogeneous Finite Domain Problem In single crystal, or single material problems the homogeneous finite domain solution will suffice, but in the multilayer problem, complexities emerge when materials of differing stiffness are used. As with the homogeneous problem we will use the principle of superposition to construct solutions that satisfies the heterogeneities associated with the multilayer problem. I will start with dislocations in one region of a bi- 29

45 material medium, then address the same problem with dislocations in both mediums, and finally the extension to multilayered materials Dislocations in one region of a bi-material medium The first solution will be constructed with dislocations residing in region one, as shown in figure 2.9. Again the stress field (S) satisfying both the internal (S ( ) and external (S * ) constraints can be composed of two parts. The first term (S (1 ) is the infinite solution to the stress field of a dislocation in the entire volume with the properties of region one; the second term (S * ) is derived from the free surface effects, and as will be seen, also comes from the modulus mismatch between regions one and two. S = S "1 + S * (2.13) 1 # = # "1 + # * S * and & * are the stress and strain fields satisfying the boundary conditions: t = t a " t # on $V u = u a on part of $V (2.15) 1 For regions one and two the constitutive relation can be written as: S * = [ C 1 ]" * in V 1 (2.16) 1 S * = [ C 2 ]" * + # *21 in V 2 where C 1 and C 2 are the elastic stiffness tensors of region one and two respectively. " #21 = [ C 2 $ C 1 ]% &1 (2.17) 1 " *21 is the error associated with the infinite solution of the dislocation stress field in region two. It is convenient to note that with the condition of no applied tractions, and with region two being stiffer than region one, " *21 is positive. The positive magnitude of 30

46 " *21 implies that the infinite solution of the dislocation in region one under estimates the stress in region two. Again if region two is more compliant than region one, then " *21 would be negative, and would suggest that the infinite solution over predicts the stress in region two. Figure 2.9: Schematic of finite boundary solution via superposition, with region two being stiffer than region one. 31

47 2.4.2 Dislocations in both domains of a bi-material medium In the case of dislocations in both mediums (figure 2.10), the solution is very similar to that of a dislocation in one medium; accept that there will be two infinite solutions, and two error terms. The total stress in the entire volume can be written as: S = S 1" + S 2" + S # (2.18) 1 where S 1( is the infinite solution for a dislocation residing in region one as though the entire volume has the properties of region one. S 2( is similar to S 1(, but the dislocation is residing in region two and the entire volume is assumed to have the properties of region two. S * can be written as: S " = [ C 1 ]# " + $ "21 in V 1 (2.19) 1 S " = [ C 2 ]# " + $ "12 in V 2 where " *21 is the stress applied onto region one from a dislocation residing in region two, and " *12 is the stress applied onto region two from a dislocation residing in region one. " *21 and " *12 can be written as: " #21 = [ C 1 $ C 2 ]% 1& (2.20) 1 " #12 = [ C 2 $ C 1 ]% 2& 32

48 Figure 2.10: Dislocations residing in both regions of a bi-material medium Dislocations in Multilayer materials From the progression of sections 2.3 and , it can shown that by superposition of solutions, the previous examples can be extended to domains consisting of many materials ie. the multilayer problem. The total stress in the i th layer can be written as: N " $ S i = # S lyr + S i lyr (2.21) 1 where N is the total number of layers, and the constitutive relations for S * can be written as: S " i = [ C i ] # * $# p N ( ) + [ C lyr $ C i ] % # lyr& (2.22) 1 lyr=1 33

49 Note that the summation in 2.21 is the superposition of the infinite solutions for all dislocations in the simulation, and the summation in 2.22 is accounts for the modulus mismatch between the i th layer and all other layers in the simulation. Figure 2.11 shows the extension to multilayer materials with arbitrary loading and dislocation distributions. Figure 2.11: Schematic illustration of the multilayer extension with arbitrary loading, displacement constraints, and dislocation distribution. 2.5 FEA Implementation To couple DD with finite element methods, the stress field from DD is taken to be an internal variable, and long-range interactions are accounted for as body forces. The equations of motion can be generically written as: 34

50 m u + c u + ku = f net (2.23) where u is the displacement of a material point relative to a chosen reference frame, m is the mass, c is a damping factor, k is the stiffness, and f net is the net force from all external and internal agencies, ie. dislocations and applied loads. Using Galerkin s weighted residual method, the equations of motion can be cast into the finite element formulation: [ M] U { } + [ C] { U } + [ K] U # V [ M] = " N # V [ K] = B [ ] T N [ ]dv [ ] T [ C] B [ ]dv { } = { f net } (2.24) 1 where [M] is the mass matrix, [C] is the damping matrix, and [K] is the stiffness matrix, {U} is the nodal displacement vector, and {f net } is the net force vector. [N] is the vector of shape functions, and [B] is the gradient matrix of the shape functions. {f net } is composed of the applied force vector {f a }, the dislocation image force vector {f ( }, the plastic strain force vector {f P }, and the long-range dislocation force vector {f B }. { f net } = f a { } + { f " } + { f B } + { f P } (a.) { f a } = # t a N S { f " } = # t " S N f B # V { f P } = # C V { } = S D B [ ]ds (b.) [ ]ds (c.) [ ]dv (d.) [ ]$ P [ B]dV (e.) (2.25) 1 The long-range dislocation force vector {f B } comes about from taking the self-stress field of all dislocations in an element as an internal variable (S D ). The self-stress fields are homogenized over the element volume, and enter into the FE formulation as a body force. In simulations where the number of dislocation segments is very large, it becomes 35

51 very inefficient to calculate the Peach-Kohler force from all dislocation segments onto the segment of interest. To ease the computational load, the self-stress fields in elements far from the element of interest can be homogenized and accounted for in the body force vector {f B }. This body force accounts for the long-range interactions between dislocations in the simulation. 2.6 Implementation of Coherent Interfaces in DD As discussed previously, interfaces play an important role in the plastic deformation of NMM composites. In the case of fcc/fcc transparent interfaces there is a coherency stress that acts to impede motion across the interface. In previous work, Akasheh accounted for the coherency stress by applying a biaxial stress to the entire thickness of each layer. In doing so, he was able to model the interfaces as impenetrable, and capture the confined layer slip behavior 3,4. In realistic systems the coherency stress will decay away from the interface, and at smaller layer thicknesses or higher stresses, dislocations should cross the interface. To model the realistic decay, a hyperbolic cosine function would seem appropriate for the model kernel. Equation 2.13 demonstrates the hyperbolic cosine interface model, where ) sets the maximum amplitude of stress at the interface, * alters the rate of decay away from the interface, and + allows for non-zero residual stresses at distances far from the interface. Figure 2.12 shows the variability of the hyperbolic cosine model with zero residual stress. " coherency = # cosh($x) + % (2.13) 36

52 Figure 2.12: Coherency stress model using a hyperbolic cosine kernel. While the hyperbolic model would take into account gradients of stress away from the interface, it is somewhat computationally expensive. A more efficient method of capturing the coherency stress decay is to define a region of constant stress that extends a few atomic distances away from the theoretically sharp interface. A preliminary starting value for the stress of the interface might be the root mean square (RMS) of the entire interfacial stress function. Figure 2.13 demonstrates this model in comparison to the hyperbolic cosine interface. 37

53 Figure 2.13: Comparison between hyperbolic model and constant stress model. In large-scale systems where there may be ten or twenty thousand nodes, it becomes important to have efficient algorithms, for this reason we employee the constant stress model in our DD simulations. In DD the stress from the coherent interface arises as a frictional term (F Interface ) in equation 2.2. To give the interface a finite thickness, the frictional force is applied only to nodes that lie inside of the interfacial boundaries, which are explicitly defined in the code. Because of the finiteness of the interface, and the application of stress to nodes inside the interface, precautions need to be taken so that nodes do not jump entirely across the interface in one time step. If this were to occur, dislocations would avoid overcoming the energy barrier required to cross into the next slip system. To avoid this error the segment lengths of the dislocations were set equal to 38

54 or less than the thickness of the interface, and the maximum time step was set to allow a maximum flight distance of half the length of the shortest segment. 2.6 Implementation of Incoherent Interfaces More recently many papers have been published on incoherent interfaces in NMM composites, in particular Cu/Nb NMM composites 5,6,7,8,9. Hoagland et. al. have shown using molecular dynamics that incoherent, or opaque interfaces attract and trap dislocations in the interface. The attractive nature has been attributed to the low shear strength of the interface, and the trapping mechanism has been associated with the core spreading that occurs once a dislocation enters the interface. Both of these effects occur because of the properties of the interface. The large change in crystal systems across the interface tends to create a large interfacial region that has mixed properties. Hoagland et. al. has published results which demonstrate the weakness of an incoherent interface under simple shear 6. Using MD, a simulation was constructed to obtain the fundamental strength of incoherent interfaces by observing the shear response to a simple shear stress. This simulation consisted of copper and niobium layers of equal thickness. The shear stress was applied parallel to the interface joining copper and niobium. Figure 2.14 shows the shear stress-shear displacement relation captured in this simulation, where the relative shear displacement is measured with respect to the bottom of the simulation cell. The results show an elastic hardening-plastic behavior, visualized as dashed lines. The parameters derived from these results include the yield stress, elastic modulus, and hardening parameter. 39

55 From the previous results Hoagland has produced a dislocation model of the incoherent interface that generates an attractive force on nearby dislocations; this attractive nature acts toward the interface regardless of the character of the dislocations. In this model an infinitely long dislocation is placed L units away from the interface and is oriented parallel to the interface. As the dislocation approaches the interface due to some externally applied load, it produces an elastic shear stress on the interface. At a critical distance from the interface, the shear stress applied from the dislocation onto the interface reaches the shear yield of the interface. At this point interfacial dislocations are placed in the interface at locations where the magnitude of stress from the dislocation is equal to the yield stress of the interface. Figure 2.15 demonstrates the insertion of dislocations on the interface. As the infinite dislocation continues to approach the interface, the interface dislocations feel the applied stress and move apart to induce a plastic interfacial strain. The dislocations introduced to the interface take on properties different from both constituent materials and are restrained to move in a non-reversible fashion. For example the once the yield stress of the interface has been reached and the interface dislocations move apart to produce interfacial strain, they are not allowed the move back if the infinite dislocation for some reason moves away from the interface. This type of model produces an elastic perfectly plastic response from the interface as shown in Figure Hoagland s dislocation model assumes that the response of the interface is elastic ideal-plastic. While this may be true for small plastic strains with low densities of dislocations, it may not be true for the opposite. 40

56 In large systems, the trapping nature of the interface means that the interface accumulates large numbers of dislocations. To capture the accumulation of dislocations in the interface it would be more appropriate to use a constitutive model that includes hardening of the interface. This constitutive model is shown in figure A simple thought process can be used to imagine the hardening of the interface in large systems. At yield only a few dislocations will enter the interface, and it is likely that their separation distance is larger than the one order of magnitude of the interfacial core size. As the interface is continually deformed more dislocations enter the interface, increasing the magnitude of long-range interactions. At large plastic strains the separation distance between dislocations will be close to that of the interfacial core width. At this point additional dislocations attempting to enter the interface will see a large back stress from the interfacial dislocation structure, which attributes to hardening. In MDDP the elastic hardening-plastic constitutive model is implemented in both DD and FE portions of the code. To include the constitutive model in the FE framework an additional layer is created in the NMM composite. The elements within this layer are identified as interface elements and are given the properties of the hardening constitutive model shown in figures 2.14 and The stress on the interface elements from the dislocation structure is evaluated using equation 2.3 where the applied stress is ignored for the moment. The result from equation 2.3 gives the shear stress at the center of the element due to the self-stress field of the dislocations surrounding the element. The stress at the center of the element is assumed to be homogeneous throughout the volume, and can therefore be taken as a body force on the entire element. If the stress on the 41

57 element is greater than the yield stress from the constitutive model then the homogeneous plastic strain is calculated, and passed to the FE portion of the code as a body force in equation 2.25a. The body force is calculated as 2.25e, with the exception that the plastic strain is not from the explicit motion of dislocations, but is from the constitutive model. Figure 2.14: MD results used to determine the properties for the elastic hardening-plastic constitutive model used in MDDP, shown as dashed lines 6. 42

58 Figure 2.15: Location of interface dislocations due to yielding (" 0 ) of the incoherent interface. 43

59 Figure 2.16: Constitutive response of the incoherent interface. 1 Zbib, H and la de (2002), "A multiscale model of plasticity", International Journal of Plasticity, 18(9): Rhee, M, H Zbib, J Hirth, H Huang and la de (1998), "Models for long-/short-range interactions and cross slip in 3D dislocation simulation of BCC single crystals", Modelling and Simulation in Materials Science and Engineering, 6(4): Akasheh, F, H Zbib, J Hirth, R Hoagland and A Misra (2007), "Dislocation dynamics analysis of dislocation intersections in nanoscale metallic multilayered composites", Journal of Applied Physics, 101(8): Akasheh, F, H Zbib, J Hirth, R Hoagland and A Misra (2007), "Interactions between glide dislocations and parallel interfacial dislocations in nanoscale strained layers", Journal of Applied Physics, 102(3): Hoagland, R, R Kurtz and CH Henager (2004), "Slip resistance of interfaces and the strength of metallic multilayer composites", Scripta Materialia, 50(6): Hoagland, R, J Hirth and A Misra (2006), "On the role of weak interfaces in blocking slip in nanoscale layered composites", Philosophical Magazine, 86(23): Wang, J, R Hoagland and A Misra (2009), "Mechanics of nanoscale metallic multilayers: From atomic-scale to micro-scale", Scripta Materialia, 60(12):

60 8 Wang, J, R Hoagland, J Hirth and A Misra (2008), "Atomistic modeling of the interaction of glide dislocations with weak interfaces", Acta Materialia, 56(19): Wang, J, R Hoagland, J Hirth and A Misra (2008), "Atomistic simulations of the shear strength and sliding mechanisms of copper niobium interfaces", Acta Materialia, 56(13):

61 CHAPTER THREE: NMM COMPOSITE SETUP AND RESULTS 3.1 Previous work and motivation in NMM composites Previously Akasheh and colleagues used MDDP to model single source interactions in NMM composites. The goal of this work was to capture many of the probable reactions, and associated strengthening mechanisms that could occur between two intersecting dislocations within a single layer of a NMM material. The bilayer problem in this thesis is intended to extend the previous results to include a penetrable interface, interactions between multiple sources of interfacial defects, and show strength dependence on layer thickness. The trilayer problem was formulated to include the effects of both transparent and opaque interfaces in the evolution of large-scale systems, and observe the strength dependence on layer thickness. 3.2 Bilayer The goal of the bilayer problem is to extend the results previously captured by Akasheh to include a penetrable interface. As previously noted, Akasheh s model using biaxial coherency stress produced impenetrable interfaces. This constricted motion captured the strength behavior commonly associated with CLP models. Experimentally we know that at small layer thicknesses the strength of NMM composites decreases, and this is most likely due to dislocations crossing the interface. For this reason the bilayer problem is formulated to capture the two smaller scale strength regimes thought to exist in NMM composites ie. CLP, and interface crossing. In addition to including the transparent interface, the intersecting dislocation structures will include pre-deposited 46

62 threading dislocations, and misfit dislocations. The misfit dislocations will be placed in two orientations: a vertically opposed orientation, and a 45-degree offset, or low energy configuration Bi-layer problem setup The bi-layer DD study consists only of copper and nickel materials, which produce the transparent interfaces observed both experimentally [Ref. Misra], and in MD simulations [Ref. Hoagland]. To capture the frictional characteristics of the transparent interface we are including the constant stress interface discussed in section 2.5. The layers are arranged such that layers of nickel surround a single layer of copper. From the bottom of the simulation the layer disposition is Ni/Cu/Ni. The Kohler force effect is captured using the FE portion of the code; therefore the choice to exclusively use DD eliminates the possibility of capturing the Kohler forces generated between modulus mismatches. The intent of the bi-layer simulations is to capture reactions that occur in the copper layer, for this reason the stiffness of nickel and copper are given the same magnitude. With the previous two choices, the option of reordering the layers becomes redundant and unnecessary. The orientation of the layers is taken to be cube-on-cube, where the layers share a common edge from the family of <001> directions. For example copper can have a z-axis orientation of [001] and nickel [001], or copper [010] and nickel [001], et cetera. The chosen orientations are shown in table 3.1. These orientations were chosen to coincide with the values previously used by Akasheh, which are commonly seen in epitaxial multilayer systems. Figure 3.1 shows the orientation with respect to the common Thompson tetrahedron. 47

63 Table 3.1: Crystallographic orientations in the bi-layer setup. x-axis y-axis z-axis Copper [ 101 ] [ 101] [ 010] Nickel [ 101 ] [ 101] [ 010] Figure 3.1: Orientation visualized using the Thompson tetrahedron. Because of the copious amounts of historical data surrounding the Hall-Petch behavior, the layer thicknesses will span from seven to twenty-seven nanometers. This range of thicknesses extends across the CLP and interface penetration ranges presented by Misra. The coherent interface is 1.53nm (6b) thick, and extends equally (3b) into both layers; see figure 3.2. As discussed in section 2.6, the interface thickness governs the 48

64 maximum time step, which is proportional to the largest segment size. For this reason the largest allowable segment length is set to 5b. The interface strength was set at 0.9GPa. This value was obtained through a simple interface penetration simulation. At a layer thickness of 30b (7.7nm) the interface strength was incremented from GPa. A biaxial load was then applied to the multilayer until the threader penetrated the interface. The point at which the threading dislocation stopped penetrating the interface and threaded down the layer was considered the upper bound solution to the interface strength at 30b. At this thickness, the strength saturated at 1.0GPa. Because the layer thickness for the interface-crossing regime is around 2 nm, a value of 0.9GPa was used as the interface strength for all layer thicknesses in the bilayer simulations. Figure 3.3 shows the applied stress needed to penetrate the interface at a given interfacial strength. Table 3.2 summarizes the properties used in the bi-layer simulations. Figure 3.2: Coherent interface geometry. 49

65 Figure 3.3: Interface strength evaluation. Table 3.2: Summary of material parameters in the bi-layer simulations. Copper (Cu) Nickel (Ni) Burgers vector nm (b) (1) (1) Shear Modulus (GPa) Poison s ratio Density (kg/m 3 ) Mobility (1/Pa-s) Layer Thickness nm (b) 7-27 (26-106) 7-27 (26-106) Largest Segment Length (b)

66 Figure 3.4 shows the general setup of the bi-layer problem. The initial dislocation configurations all contain a single threading dislocation residing on the ( 111) plane with a burgers vector of [ 011 ] or [ 110]. Extending from both ends of the threading dislocation are semi-infinite defects, which resolve the erroneous end effects that would be present if the threader ended inside of the material. The semi-infinite defects are modeled as sessile segments that impose the stress field of a semi-infinite segment on the rest of the simulation domain. The threader is placed near one end of the 1000b X 250b simulation cell to allow a large amount of threading length before leaving the opposing side of the simulation. Pre-deposited threading dislocations and misfit dislocations are used as the interfacial defects. Pre-deposited dislocations are threading dislocations that have previously threaded through the layer and left straight segments along the interface. The pre-deposited dislocations lie on the ( 1 11) plane with burgers vectors [ 011 ]. The spacing of the pre-deposited dislocations is 40b, which was chosen to coincide with the equilibrium misfit spacing. Misfit dislocations arise in epitaxial systems where coherency stresses are present and layer thicknesses are larger than 5nm [Misra et al. 2002]. The misfit dislocation spacing can be calculated using equation 3.1 [Shoykhet et al. 1998], where b is the burgers vector,! m is the misfit strain, h c is the critical layer thickness, and h is the current layer thickness. Table 3.3 shows the equilibrium spacing of the misfit dislocations for the set of thicknesses used in the bilayer cases. b " = % # m 1$ h ( ' c * & h ) (3.1) 51

67 Figure 3.4: Bi-layer general problem setup. Table 3.3: Misfit dislocation spacing for various layer thicknesses. Layer Thickness (b) Misfit Spacing," (nm) Misfit Spacing, " (b)

68 From Table 3.3 one can see that the equilibrium dislocation spacing changes very little from 20b thicknesses to 100b thicknesses. For this reason, and convenience, the spacing used in the bi-layer simulations is rounded off to 40b. The misfit dislocations are placed at the center of the interface, and inhabit the (001)[100] sessile slip system. Two orientations of misfit dislocations are considered; first, pairs of misfit dislocations will be vertically opposed across the layer, and second, the low energy configuration, where the dislocations will be oriented 45-degrees from each other across the layer. Figure 3.5 schematically shows these orientation differences. The low energy (LE) configuration creates difficulties in producing the same initial structure for different layer thicknesses. Because the pre-deposited dislocations are equally spaced and vertically opposed, the spacing between LE misfits and pre-deposited dislocations changes as the thickness is increased or decreased. To avoid overlapping or unrealistically close dislocations, the starting location of the pre-deposited dislocations was altered to give the largest spacing between misfits and pre-deposited dislocations for each layer thickness. 53

69 Figure 3.5: Misfit dislocation configurations. Because the misfits are modeled as dislocation segments, and are sessile defects, the short reactions between the misfit segments and all other segments are turned off. This treatment still allows for the existence of long-range interactions between the misfits and all other segments. To be consistent with the experimental bulge testing, quasi-static biaxial loading is used with free surfaces in DD. The quasi-static loading is a technique used to find the flow stress of single source simulations in DD. To find the flow stress, the critical stress to propagate the threading dislocation must be known. To find this critical stress to propagate the threading dislocation, a small subroutine was added to the DD code. This subroutine, called ThreaderArea.F, calculates the instantaneous area that the threader has slipped through. The area is 54

70 crudely calculated assuming the leading edge of the threader as a flat propagating segment. At a chosen frequency ( nsteps) the subroutine calculates the threaded area and compares it to the previous threaded area, which was calculated nsteps earlier. If the current threaded area is less than the previous area plus some buffer area (~100) then the applied biaxial stress is increased; otherwise the stress is not changed. Because the threading dislocation tends to bow out the furthest in the center of the layer, the furthest traveled node was used to determine the threaded area. Figure 3.6 shows a typical loading curve using the quasi-static loading scheme. The long flat segment is the stress required to propagate the dislocation down the layer for this particular thickness. The gray region represents the part of the simulation where the threader left the simulation, which caused the calculated threaded area to become static and therefore the subroutine increased the applied stress. Figure 3.6: Quasi-Static loading. To test the significance of interfacial defects on the threading properties, many configurations were evaluated. Table 3.4 lists the combinations of interfacial defects that 55

71 were considered in the bilayer simulations. Case number five is identical to case three, with the exception that the sign of the burgers vector has been negated. Threader I is titled co-linear with respect to the burgers vector relation between the threader and the pre-deposited threader. Threader II is assigned a burgers vector, which is inclined by 60 degrees with respect to the pre-deposited threader. Table 3.4: Defect combinations for bi-layer cases. Case # Threader I (inclined) Threader II (co-linear) Pre-Dep. Threader Vertical Misfits ( 111) [ 011 ] ( 111) [ 110] ( 1 11) [ 110] ( 001) [ 100] Global coor. 1 X X Low Energy Misfits ( 001) [ 100] Global coor. 2 X X 3 X X X 4 X (negative) X X 5 X X X 3.4 Bi-Layer Results If predicted and modeled correctly, the inclusion of the interfacial effects in a DD model should show a strength dependence on layer thickness similar to the experimental results. Figure 3.7 shows the bi-layer strength as a function of layer thickness and interfacial dislocation distributions. The stress has been resolved onto the slip system of the threading dislocation in copper. The solid black curve shows the CLP model, and the blue line with circle points shows the experimental results. 56

72 Figure 3.7: Threading strength dependence on layer thickness. If an upper limit were set where the interface was given an infinite strength then all of the models would follow the confined layer slip behavior. Therefore the base case to make comparisons to for these simulations is the confined layer slip model. It is important to note that the CLP model is only concerned with the confinement of a threading dislocation in a layer. It does not account for interfacial effects such as obstacles or reactions that may occur in larger simulations Case 1 Referring to table 3.4, case one includes vertically opposed misfit dislocations as interfacial defects. Figure 3.7 shows that the misfits add slightly to the threading strength, but don t alter the layer dependence with respect to the CLP model. It is 57

73 interesting to note that although the layer dependence follows a CLP relationship, the threading dislocation crosses the interfaces at layer thicknesses of 20b and 30b. At thicknesses larger than 30b the threader stays confined to its initial layer. Figure 3.8 shows motion of the threading dislocation at a layer thickness of 30b. In frame one the dislocation has threaded out by about four misfit spacings (160b). The threader then starts to penetrate the interface between two misfit dislocations. After entering the next layer, the dislocation propagates completely through the interface. As the threader crosses the next layer, the frontal length of the threader is increased significantly and the threader now begins to thread down the two layers. At a length of about eleven misfit spacings the threader stops, and the stress is then increased. As a result of the increase in stress the threader begins to penetrate the bottom interface near the fixed node of the original threader. After entering the bottom layer this portion of the dislocation begins to move toward the leading edge of the threader. Now that the threader spans three layers the effective free length of the threader has been tripled and the stress required to propagate the threader is much lower. At this point the threader accelerates and moves down to the end of the simulation cell extremely fast. In frame eight the missing segments are due to the free boundaries imposed in DD. At low layer thicknesses the threading just described is characteristic in all of the simulations. The only exceptions to this are when dislocation reactions occur that inhibit free motion in the adjacent layers. 58

74 Figure 3.8: Threading behavior of case one, at a layer thickness of 30b Case 2 In case two the low energy misfit configuration is considered, where the misfit dislocations are displaced 45 degrees from each other across the layer, as shown in figure 59

75 3.5. Figure 3.7 shows that for large (>30b) layer thicknesses the strength dependence is very similar to the vertically aligned case, but at a layer thickness of 20b the threading strength is about 200MPa lower than the vertically aligned case. Figure 3.9 shows a time-lapse progression for both the vertically aligned case and low energy case at a layer thickness of 20b. In case one the threading dislocation leaves the top interface, then after the dislocation reaches the next interface in the top layer the threader penetrates the bottom interface. The low energy configuration shows a more symmetric penetration pattern, where the threader penetrates the bottom interface first, then immediately following, the top interface is penetrated and the threader enters both the top and bottom layers simultaneously. 60

76 Figure 3.9: Comparison between vertically aligned and low energy configurations at 20b layer thickness. 61

77 The stress required to penetrate the interface in the vertically aligned case is about 1.45GPa whereas the low energy configuration only requires about 1.2GPa. Figure 3.10 shows the interfacial shear stress field differences between the vertically aligned misfits and the low energy misfits. In this finite element solution the misfit dislocations were placed inside of the copper layer by one-half of a burgers vector, and the layers were 20b thick. The stress field patterns compare relatively well except for the alternating signs, and the magnitude of stress in the shear lobes between the misfits. In the vertically aligned case the stress lobes in the lower interface always have a sign opposite to the stress lobe directly above in the top interface. The low energy configuration shows a slightly different pattern where the stress lobes in the lower interface are the same sign as the opposing lobes in the top interface. The vertically aligned case also shows a maximum magnitude of stress 0.4GPa larger than the low energy configuration. This stress difference is consistent with the resultant threading strength in figure 3.7. It is also interesting to note that the shear lobes in the low energy case show a larger separation distance compared to the vertically aligned case. The combination of these differences arises solely due to the arrangement of misfit dislocations in the interface, and is the reason for the lower threading strength in figure

78 Figure 3.10: FEA of the stress fields associated with vertically aligned and low energy configurations of misfit dislocations. Figure 3.11 shows the characteristic threading motion for the low energy configuration at a layer thickness of 30b. The dislocation first threads down the copper layer then leaves the top interface. After leaving the top interface it threads out a little further and gets pinned at one of the misfit dislocations. At this point the threader leaves the bottom interface and the dislocation now threads down the entire simulation cell. 63

79 Figure 3.11: Interface penetration of low energy configuration at 30b layer thickness. As state previously with interfacial configurations that involve only misfit dislocations, the confined layer slip mechanism is observed at layer thicknesses larger than 30b. A time-lapse of this behavior is shown in figure 3.12 for a layer thickness of 100b. Figure 3.12: Typical confined layer threading with misfit dislocations. 64

80 3.4.3 Case 3 In case number three, pre-deposited threading dislocations are added to the low energy misfit configuration. The plane given to the pre-deposited threaders is oriented so that the intersections created between the pre-deposited threaders and the threading dislocation are perpendicular at the interface. Because the pre-deposited threaders are glissile dislocations, the short-range reactions are allowed to occur between the threader and pre-deposited dislocations. These short-range reactions allow the possibility of jogs, junctions, annihilations, and cross-slip to occur, which significantly changes the method by which a dislocation passes into the adjacent layer. In case three the threader and predeposited threaders have burgers vectors that are inclined to each other by 60 degrees. Figure 3.7 shows that the threading strength of this interfacial configuration closely follows the confined layer slip model. The reactions observed with this configuration include junction and jog formation. At 20b layer thickness the pre-deposited dislocations and threading dislocation show repulsive forces, and the pre-deposited dislocations are pushed out of the interface as the threading dislocation approaches them. Figure 3.13 shows this repulsive threading characteristic at a layer thickness of 20b. Once the predeposited threaders are pushed out of the interface, the trailing threader portions of the threader only have to overcome the misfit dislocations to cross the interface. 65

81 Figure 3.13: Pre-deposited threaders moving out of the interface at a layer thickness of 20b. At 30b thicknesses only pre-deposited threaders on the bottom interface leave center layer, and tend to leave with the threader as opposed to being pushed out ahead of the threader. In some instances the pre-deposited threaders stay in the interface when the threader has an approach angle opposite of the normal threading approach angle. Figure 3.14 shows this behavior. Frames 1-8 show how the pre-deposited threaders follow the threading dislocation through the next layer as opposed to the 20b case in figure

82 where the pre-deposited threaders penetrate through the entire thickness of the next layer before the threader threads through same thickness. The two lower pre-deposited threaders, A and B in frame eight, were approached by the threading dislocation with a line-sense different from the other pre-deposited threaders on the lower interface. This difference in line-sense allowed the threading dislocation to pass over threaders A and B without pulling them into the next layer. 67

83 Figure 3.14: Threading behavior at 30b layer thickness with pre-deposited threaders (case 3). Figure 3.15 shows the jog and junction formation that occurred at layer thicknesses of 40b, 50b, 75b, and 100b. Figure 3.16 shows the re-orientation and junction formation that occurs as the threader penetrates the interface and threads into the next layer. 68

84 Figure 3.15: Junction and jog formation between pre-deposited threaders and the threading dislocation. As the threading dislocation approaches the pre-deposited threader (frames 1 & 2) the pre-deposited threader re-orients itself to align with the threading dislocation (frames 3 & 4). After the re-alignment, a junction is formed between the two adjacent segments, and the as the pre-deposited threader and threading dislocation glide into the next layer, the junction zips up (frames 5,6,7). 69

85 Figure 3.16: Time lapse of a junction formation Case 4 To observe the effect of the burgers vector on the strength dependence and reaction formation, case three was modified to create case four. In case four the burgers vector of the threading dislocation was negated. Figure 3.7 shows that the negated burgers vector had almost no effect on the threading strength. At 20b layer thickness the pre-deposited threading dislocations all left the interface and penetrated into the next layer before the threading dislocation had even propagated. Because of this, the threader only interacts 70

86 with the misfit dislocations and the motion is much like the low energy misfit case. At a layer thickness of 30b many jogs were created between the threader and pre-deposited threaders. To overcome the jog, an annihilation mechanism was observed which is shown in figure 3.17 where the segments are colored by node type. In frames one and two a jog is formed in both segments. The jog acts as a pinning point for the threading dislocation, and because the threading stress is high enough, the threader begins to bow around the jog node (frame 3, 4, 5). As the threader bows around the jog node, it forms two segments of opposite line sense but with the same burgers vector. These segments are attractive and cause the an annihilation reaction to occur. The attractive nature is shown in frames four and five, and the annihilation is shown in frames five and six. Frames seven and eight show the threading dislocation passing through the next predeposited segment without a jog formation. The 40b layer thickness shows that the predominant reactions are both jogs and junctions, where the junction formation is very similar to that of the non-negated case (case 3). At layer thicknesses of 50b and higher, the threader dislocation stays within the copper layer and the pre-deposited threaders stay in the interfaces. Even though none of the dislocations leave the interface, there are a number of jog nodes are created within the interface. 71

87 Figure 3.17: Jog formation and annihilation between threading dislocation and predeposited threader. 72

88 3.4.5 Case 5 Case number five is similar to cases three and four with the exception that the burgers vector of the threading dislocation is chosen to be co-linear with the predeposited dislocations. Figure 3.7 shows that the threading strength at 20b layer thickness is much lower than the confined layer slip model would predict, and is closer to the experimental behavior observed by Misra et. al. At 20b layer thickness the long range interactions dominate, and the pre-deposited threaders leave the interface before the threading dislocation begins to move down the layer. The resulting dislocation structure in the interface after the pre-deposited threaders have left the interface is the low energy misfit dislocation structure. Therefore the threader only has to overcome the energy of the misfits to penetrate the interface. Figure 3.18 shows the progression of the threader as it passes through the interface at a layer thickness of 20b. Figure 3.18: Time-lapse of the threading dislocation passing through the interface at a layer thickness of 20b (case 5). At layer thicknesses of 30b, and 75b, the threading dislocation penetrates the interface and forms junctions and jogs with the pre-deposited threaders, but at thicknesses 73

89 of 40b, 50b, and 100b, the threader is confined to the layer throughout the simulation. At 30b, and 75b an interesting annihilation reaction occurs that creates a cross-slip node between the threader and pre-deposited threaders. Figure 3.19 shows the formation of the cross-slip node through annihilation at a layer thickness of 30b. The dislocations are colored by the slip plane they reside on, and the circle in frame one surrounds the intersection of interest. In frame two the threader has passed the pre-deposited threader and started to push it out of the interface. Frame three shows the pre-deposited threader and threader starting to re-orient themselves for annihilation. In frame four the segments are aligned and frame five shows the separation of the segments after the annihilation has occurred. The annihilation of the two segments created two cross-slip nodes that connect the threader with the pre-deposited threader. This cross-slip node is constrained to glide on the line that is created by the intersection of the two slip planes. Frames six, seven, and eight show the movement of the cross-slip nodes as the threader and pre-deposited threader continue to thread down the simulation cell. This annihilation reaction occurs repeatedly in the simulation, and one of these repeated cases is shown in frames seven, eight, and nine. 74

90 Figure 3.19: Mosaic of snap-shots during the formation of a cross-slip node at a layer thickness of 30b (case 5). 75

91 3.3 Tri-Layer Setup Both the bi-layer cases and Akasheh s previous results assume that the strengthening behavior of NMM composites can be captured using single source models. To model the macroscopic plasticity observed experimentally, one needs to extend the micro-plasticity models to include the possibility of activating multiple slip systems. A large number of activated slip systems will increase the likelihood of observing the many statistical reactions that occur in NMM composites. The tri-layer problem was formulated to capture the macroscopic plasticity, through the inclusion of both coherent and incoherent interfaces, and random distributions of threading dislocations. Figure 3.20 shows the general tri-layer configuration. The ordering of layers for the tri-layer case was chosen so that an interface was created between all of the material combinations. For example there is an interface between copper/niobum and copper/nickel, and using periodic boundary conditions a third interface is created between nickel/niobium. The ordering of the layers is as shown in figure 3.20; from the top down, niobium, copper, and nickel. 76

92 Figure 3.20: Typical trilayer setup. The Kurdjumov-Sachs orientation is used in the tri-layer system, which is the preferred growth orientation in incoherent multilayers. The Kurdjumov-Sachs orientation requires that the slip systems from each material be coincident with each other across the interface. Therefore a close packed plane from niobium must be parallel to a close packed plane from copper or nickel, and a close packed direction of niobium must be aligned with a close packed direction from copper or nickel. In the tri-layer system this is obeyed along the X and Z axes, where the interface normal is in the Z-direction. The orientation for all of the layers is shown in table 3.5. Table 3.5: Crystallographic orientations in the tri-layer setup. x-axis y-axis z-axis Niobium [111] [-1-12] [1-10] 77

93 Copper [1-10] [11-2] [111] Nickel [1-10] [11-2] [111] The initial dislocation configuration is a random distribution of threading dislocations on the family of {111} planes in copper and nickel only. The (111) plane is excluded because it is parallel to the interface, and threaders are grown on slip systems extending away from the substrate. It may be possible for glide loops or frank-reed sources to exist on this plane, but they have not been observed experimentally in NMM composites. Generating enough threaders to incorporate every possible slip system in the model creates an unrealistically high dislocation density ( /m) for a simulation cell size of 1000b X 1000b. For this reason an initial test was conducted to determine which slip systems would not be activated at relatively normal bi-axial stresses. The resulting activated slip systems exposed in this initial test run are listed in table 3.5 and are used in all of the tri-layer cases. Because the slip systems for the threaders were randomly chosen, the burgers vectors may differ between copper and nickel. This difference can be found by examining columns three and four in table 3.5; for convenience it is also noted in column five. To truly capture the macroscopic plastic deformation it would be better to have multiple threaders assigned to each slip system, but due to high densities we have limited one threader to each slip system. Figure 3.21 shows the initial distribution of threading dislocations in copper (blue) and nickel (green). The Thompson tetrahedron is also shown in the correct orientation to give an idea of the symmetry of the problem. The DD boundary conditions are considered to be periodic, which allows for large amounts of plasticity, and maintains the representative volume element concept used to formulate the finite element solution. 78

94 Table 3.5: Crystallographic slip-systems used in the tri-layer system. Slip System # Slip Plane Burgers Vector Burgers Vector Burgers Vector (pln, brg) (Copper) (Nickel) different? 1 (1,4) (1-11) [-101] [-101] Same 2 (1,5) (1-11) [0-1-1] [0-1-1] Same 3 (1,6) (1-11) [-1-10] [-1-10] Same 4 (2,2) (-111) [0-11] [01-1] Different 5 (2,3) (-111) [-10-1] [-10-1] Same 6 (2,6) (-111) [110] [-1-10] Different 7 (3,3) (-1-11) [101] [101] Same 8 (3,5) (-1-11) [0-1-1] [011] Different 79

95 Figure 3.21: Initial distribution of threading dislocations in tri-layer setup. Initially the external loading conditions were bi-axial, but due to problems with anisotropic plasticity and strain rate constraints, the loading was changed to a uni-axial condition. The loading for the results presented, is uni-axial along the y-axis. This loading condition results in a symmetric decomposition of stress onto the two slip planes that have mirror symmetries about the y-z plane. In these simulations the load is applied via strain rate conditions (1.0E6/s) in DD, but FEA is used to solve the incoherent plasticity problem, and solve the Kohler image correction. 80

96 Previously, under a strain-rate loading condition via DD, the increment of stress applied to the specimen was calculated using homogenized material properties. For example the shear modulus and poisons ratio of each layer were averaged together by their respective volume fraction, and the averaged shear modulus and poisons ratio were used to calculate the applied increment of stress. This homogenized increment of stress was then applied to every layer. While this loading condition accounted for the relaxation of stress in each layer via plasticity, it assumed that the external loading was homogeneous through out the interior of the specimen. In a physical strain-rate loading test, the stress in each layer will be proportional to the applied strain, the shear modulus, and poisons ratio specific to that layer. Therefore, each layer will carry more or less load depending on its stiffness and the amount of plasticity that has taken place in that layer. To account for the heterogeneous (multilayer) loading, a routine was developed to apply the external loading to the layers individually. An even more rigorous approach would be to apply the loading via finite element, but for the tri-layer problem the finite element solution is only for the Kohler correction. The results presented, are generated using the multilayer loading scheme, but the stress is presented as the averaged stress, n [ ( )] p "# = & f i E i "$ % "$ i (3.1) i=1 where n is the number of layers, # is the stress,! is the total strain,! p i is the plastic strain local to the i th layer, f i is the volume fraction of the i th layer, and E i is the stiffness of the i th layer. This averaged stress would be equivalent to what would be measured experimentally in a uni-axial tension test. To be consistent with the bi-layer simulations, the coherent interface formed between copper and nickel is given a thickness of 6b, and is centered on joint between the 81

97 two layers. This means that the interface extends 3b into each layer. Because interfacial defect structures are not considered in the problem, the interface strength is taken to be 2.5GPa, which is the bi-axial stress resulting from the coherency of the interface. The incoherent interface is defined to be 6b thick, and is also centered on the intersection of the two materials, in this case, copper and niobium. Since the interface between copper and niobium, or nickel and niobium is opaque, plasticity should not occur in niobium. For this reason an incoherent interface of 3b was put at the bottom of the nickel layer as shown in figure The recent literature on incoherent interfaces has only considered copper/niobium materials. Because of the lack of information on nickel/niobium interfaces the properties used for the copper/niobium interface are also used for the nickel/niobium interface. The properties obtained from MD result in a shear modulus of 18GPa, a yield stress of 550MPa, and a linear hardening coefficient of 5GPa. In the tri-layer problem we will consider two layer thickness variations. In case one we will observe the dependence of strength on the layer thickness where all of the layers share the same thickness, and case two we will observe the strength and plasticity dependence on the variation of the niobium thickness with all other layer thicknesses (Cu & Ni) held constant (48b). In case one the layer thicknesses will take on the values of; 21b, 30b, 42b, 51b, 75b, 102b. In case two, copper and nickel will have a static value of 48b, and niobium will take on the values of; 0b, 12b, 24b, 33b, 48b, 63b. Table 3.6 summarizes the layer thicknesses used in both cases. All of the thicknesses presented are measured from the center of the interfaces as shown in figure Table 3.7 shows the properties used in all of the tri-layer simulations. Table 3.6: List of thicknesses used for the tri-layer cases. 82

98 Case 1: Layer Thickness, nm (b) Case 2: Nb Thickness, Nb/Cu (Niobium Thickness (b)) 5.37 (21) 21/16 (63) 7.67 (30) 1 (48) (42) 11/16 (33) (51)! (24) (75) " (12) (102) 0 (0) Table 3.7: Material properties used in tri-layer simulations. Niobium (Nb) Copper (Cu) Nickel (Ni) Burgers vector nm (b) (1) (1) (1) Shear Modulus (GPa) Poison s ratio Density (kg/m 3 ) Mobility (1/Pa-s) Layer Thickness nm (b) 5-26 (21-102) 5-26 (21-102) 5-26 (21-102) Largest Segment Length (b) Tri-Layer Results The tri-layer results show a layer dependence on the initial yield behavior of the material, and the microstructural evolution shows a new cross-slip mechanism in which a 83

99 threading dislocation cross-slips onto the (111) plane parallel to the interface and propagates within the center of the layer. As in the bi-layer cases, interface penetration, and junction formation are also observed. The majority of macroscopic plasticity takes place through the motion of super threaders, which are threading dislocations that have penetrated the interfaces of their respective layers and propagated into the adjacent layers. Because the majority of the dislocation motion resides on symmetric slip planes, the applied stress is presented as the stress resolved onto one of the two symmetric slip planes and slip directions via the Schmidt factor; " rs = m# m = cos ( $ )cos(% ) (3.2) where $ rs is the resolved shear stress, m is the Schmidt factor, # is the applied stress, % is the angle formed between the loading direction vector and the slip plane normal, and & is the angle formed between the loading direction and the slip direction Case 1 Macroscopic Behavior Case one considers the strength dependence and microstructural evolution as the layer thicknesses are decreased equally amongst the layers. Figure 3.22 shows the stressstrain behavior observed for the selected layer thicknesses. The limiting factor controlling the final simulated strain was the number of dislocation nodes (15,000). For this reason the simulations ran out to a strain offset of about 0.2%. From figure 3.22 one can see that following the initial yield at large (75b & 102b) layer thicknesses there is a large amount of hardening which results in a secondary elastic loading phase. Layer thicknesses of 50b and 42b show an opposite effect where once yielding occurs there are less hardening effects and no secondary elastic loading occurs. At yet smaller 84

100 thicknesses there is a slight yield followed by a very sharp yield point, which will be shown to exist by way of interface penetration. Figure 3.22: Stress - Strain curve for varying layer thicknesses. From figure 3.22 one can obtain the yield strengths as a function of layer thickness. In fact figure 3.23 shows this strength dependence on layer thickness. Two yield points are listed to determine if strain hardening is also dependent upon layer thickness at small plastic strains. While strain hardening takes place, Figure 3.23 shows that strain hardening is not dependent upon layer thickness in the tri-layer case. Comparatively speaking the tri-layer model over predicts the material strength according to the idealistic CLP model. From the dislocation evolution (presented later) and this over prediction, 85

101 one would suggest that the interfacial strength is too high, or a lower initial dislocation density is needed. Figure 3.23: Strength dependence on layer thickness. Figure 3.24 is identical to figure 3.23 with the 1 st and 2 nd yield strains superimposed. From figure 3.24 one can see that the stored strain energy, or toughness does not change as a function of the tri-layer period for the 1 st yield. This is the case because the 1 st yield represents the onset of plasticity and therefore the stored strain energy is purely elastic, which is independent of layer thickness for cases where the layers share the same thickness. The 2 nd yield shows that the toughness increases only slightly at larger tri- 86

102 layer periods. This is most likely due to differences in dislocation motion at larger thicknesses. Figure 3.24: Yield strain dependence on layer thickness. Figure 3.25 shows the interrelationship between the dislocation density and resolved threading stress. Upon initial inspection one can see that the sharp yield at smaller layer thicknesses observed in figure 3.22 is due to the generation of a large number of dislocations, or the deposition of many dislocation segments along the interfaces due to threading. At larger layer thicknesses (75b & 102b) the secondary elastic loading phase 87

103 occurs where the density curve becomes horizontal, alluding to no plasticity. At most layer thicknesses the dislocation density rises sharply toward the end of the simulation, which is contrary to the well-known polycrystalline hardening models. This sharp rise at these high densities suggests that the threaders are freely moving throughout the specimen without any hindrance from previously deposited interfacial dislocation segments. This may suggest that short-range reactions are not taking place where they should be. The segments that enter the incoherent interface are immediately assigned to a sessile dislocation type, which turns off the allowance for short-range reactions within the interface. In the bi-layer cases the many of the short-range reactions between interfacial dislocations and threaders controlled the interfacial crossing behavior, which in turn affects the macroscopic behavior of the material. In a more rigorous approach it would be advantageous to include interfacial defects in a similar fashion to the bi-layer cases. 88

104 Figure 3.25: Dislocation density evolution with respect to applied stress (Case 1) Case 1 Microscopic Behavior The plastic response of the tri-layer systems is similar to that of the bi-layer system where both single layer threading and interface penetration play an important role in the dislocation evolution. At larger layer thicknesses the majority of the dislocation motion is confined to single layer threading as shown in figure

105 Figure 3.26: Typical confined layer threading at large layer thicknesses. As the blocking mechanisms active and the applied stress becomes large, the single layer threading breaks down and the dislocations begin to penetrate the coherent interfaces. A deformation mechanism that exists at all layer thicknesses is the formation of threading dislocations that span multiple layers (super-threaders). The formation of super-threaders from single layer threaders is solely due to coherent interface penetration, which is activated at the intersection of single layer threaders with pre-deposited threading segments. At all layer thicknesses the formation of super-threaders is from the interface penetration of single layer threaders originally residing in nickel. Figure 3.27 shows the formation of a super-threader from the intersection of a threading dislocation with a pre-deposited segment. Frame one shows the intersection of a threader of type (- 111)[0-11] with pre-deposited segments of type (-11-1)[-101], where the threader has approached and been blocked by the pre-deposited segments. Under additional stress the 90

106 threader proceeds to push the pre-deposited segments through the interface in a wave like manner. This wave front propagates toward the leading edge of the threader at velocities higher than the original threading speed of the single layer threader. The pre-deposited segments that penetrate the interface begin to glide across the layer, and as the wave front approaches the leading edge of the threader, the free length effectively doubles and creates a super-threader that spans both layers. This super-threader now begins to propagate down both layers at velocities much higher than that of the single layer threader. 91

107 Figure 3.27: Formation of a super-threader by single-layer threader interaction. As the super-threader threads down both layers it intersects other single layer threaders in the nickel layer and aids in creating additional super-threaders as shown in figure Frames three through eight show the wave like behavior of the penetrating segments as they proceed toward the leading edge of the single layer threader in nickel. The colorcoding used in figure 3.28 is identical to that of figure

108 Figure 3.28: Formation of a super-threader by super-threader and single layer threader intersections. The color key is given in figure

109 At layer thicknesses of 21b and 30b, the super-threader propagation is the major contributor to plastic strain, but at larger layer thicknesses the threaders stay confined to their respective layers until junctions or threader/pre-deposited threader blocking mechanisms are activated, at which point the super-threaders are formed. These other blocking mechanisms are due to the interaction between pre-deposited segments at the coherent interface and an intersecting threading dislocation. At a layer thickness of 42b an interesting hardening mechanism occurs where a super-threader is formed, but shortly after it intersects a single layer threader in the copper layer. The super-threader forms a junction with the single layer threader, but the nickel portion of the super-threader continues to propagate in nickel as a single layer threader, thus reducing the plasticity to single layer threading. Figure 3.29 shows this junction and threading behavior. Following the formation of the junction, single layer threaders dominate the plasticity until a critical stress is reached and the nickel threaders penetrate into copper. None of the simulations demonstrated an interface penetration from copper into nickel. 94

110 Figure 3.29: Junction formation between super-threader and non-activated single layer threader. At layer thicknesses of 75b, and 102b, the (-1-11) slip plane is activated and is responsible for the formation of super-threaders. At both layer thicknesses, a superthreader is first formed on the (-1-11) plane with a slip direction perpendicular to the 95

111 loading axis. The propagation of this super-threader aids in the formation of superthreaders on the two dominant symmetric slip planes (-11-1) and (1-1-1). Before the formation of the first super-threader, all of the plasticity occurs on the symmetric slip planes via single layer threading. The point at which all of the single layer threaders have been block by pre-deposited segments or formed junctions with other threaders, activates slip on the (-1-11) plane and shortly after being activated a super-threader is formed. The propagation of this super threader activates interface penetration on the two other symmetric slip systems. Figure 3.30 shows the activation and formation of a super threader on the (-1-11) slip plane. 96

112 Figure 3.30: Formation of super threader on the (-1-11) slip plane Case 2 Macroscopic Behavior Case two was formulated to determine the effects of niobium thickness on the overall macroscopic behavior of the material. If niobium is a dislocation free layer, then as copper and nickel reach full-on plasticity the niobium layer will predominantly carry the load. At this point the stress-strain curve should show a slope that is proportional to the quantity f Nb E Nb in equation 3.1. Therefore if the ratio of the niobium/copper thickness is decreased the plastic stress-strain slope should also decrease with respect to 97

113 the change in f Nb. At zero niobium thickness the simulation reduces to the bi-layer copper nickel simulation, which does not include an incoherent interface. Due to the absence of niobium, this bi-layer simulation should show a plastic slope of zero when full-on plasticity occurs in copper and nickel. Figure 3.12 shows the stress-strain curve as a function of the ratio between the niobium and copper layer thicknesses. The most noticeable effect of the thickness ratio is the inverse correlation with the material stiffness. As the niobium thickness is increased, the stiffness of the material decreases. This is due to the low modulus associated with niobium, which is about three times lower than nickel, and approximately half as large as copper (see table 3.7). The next noticeable effect is the strain at which the 1 st yield occurs in all of the layers. The strain at which the first yield occurs is identical in all of the layers with the exception of the bi-layer case. This equivalent strain event is due to the multilayer loading scheme in which copper and nickel carry the same elastic load at a given elastic strain irrespective of the niobium thickness ratio. Because the copper and nickel layer thicknesses don t change, the yield stress in these layers will occur at the same strain. The attractive nature of the incoherent interfaces increases the Peach-Kohler force on the threading dislocations, which could attribute to the lower yield strain occurring in cases where the niobium thickness is greater than zero. The deviated yield strain in the case with zero niobium thickness may be explained by the absence of the incoherent interface. 98

114 Figure 3.31: Stress-strain curves for niobium thickness effect. Nickel and copper thicknesses are held constant at 48b. The niobium thickness ratio is calculated as t Nb /t Ni or t Nb /t Cu. The parenthesized value in the thickness ratio legend is the actual thickness of the niobium layer. Figure 3.32 shows the 1 st and 2 nd yield behavior derived from figure This figure shows that hardening is present between the two yield points, but that it is not a function of layer thickness. The slight rise in yield stress at lower ratios is again attributed to the limiting bi-layer structure. 99

115 Figure 3.32: Yield strength dependence on niobium thickness ratio. Figure 3.33 shows the relationship amongst the slopes of the stress-strain curves at full-on plasticity. As previously discussed the larger niobium ratios will produce larger hardening slopes due to the influence of the volume fraction on the averaged stress. The analytical solution in figure 3.33 assumes that the entire load has been transferred to the elastic niobium layer, Slope " f Nb E Nb = t Nb E Nb ( ) t Nb + t Cu + t Ni (3.3) where t i is the thickness of the respective layer. 100

116 Figure 3.33: Relative hardening after full onset of plasticity. Figure 3.34 demonstrates the dislocation density correlation with the applied loading. At niobium thicknesses greater than zero the stress at which threading occurs is confined to a small stress region, and correlates well with the stress-strain behavior in figure The case of zero thickness shows a region where the stress required to generate additional dislocation densities is much higher than usual. This may be due to a pinning or blocking mechanism that occured due to intersections between threading dislocations, or the absence of an incoherent interface. The absence of the incoherent 101

117 interface means that additional reactions will take place at the interfaces, which would alter the strength of the composite. Figure 3.34: Dislocation density evolution with respect to stress (Case 2). 102

118 CHAPTER FOUR: CONCLUSIONS The bi-layer and tri-layer results show that DD and MDDP are very useful in capturing both the macroscopic response and microscopic evolution of NMM composite systems. The inclusion of long-range and short-range interactions has been crucial in the evolution of both bi-layer and tri-layer NMM composites. These were heavily demonstrated in the interfacial penetration of pre-deposited threaders in the bi-layer configuration. The bi-layer results demonstrate that interfacial defect configurations significantly alter the strength and behavior of the material. For example the cases in which the threading dislocation only interacted with the low energy misfit dislocations showed a drop in strength at small layer thicknesses. This drop in strength is thought to occur from the interfacial stress field that is unique to the low energy configuration of misfit dislocations. The case in which pre-deposited threaders inhabited a burgers vector colinear with the burgers vector of the threading dislocation, showed relatively large increases in strength at all layer thicknesses. This demonstrates the degree to which the interfacial slip systems alter the strength of the NMM composites. The tri-layer results show that a size effect does exist in these materials, and that a slight increase in toughness is observed at larger layer thicknesses. It is also noted that as layer thickness is increased to 75b and 102b, the deformation mechanism changes from super-threader glide to single layer glide. Even though the dislocation motion changes at larger layer thicknesses, the hardening or strengthening via junction formation are common in both. The motion of super-threaders was shown to exhibit a free glide nature, 103

119 where as the single layer threaders seem to be stopped at intersections with pre-deposited dislocation segments. The results concerning the strength dependence on niobium thickness, show that the yield stress of the specimen is controlled by the volume fraction of niobium. Because niobium is the most compliant of the three materials, an increase in volume fraction of niobium decreases the overall yield strength of the material. I has also been shown that the hardening slope of the material at full-on plasticity is proportional to the product of the niobium stiffness and volume fraction. An experimental comparison shows that the bi-layer and tri-layer strengths over predict the strength of the NMM composite. At smaller layer thicknesses this would suggest that the physical interfacial strength is much lower than predicted, and that the model should include bi-axial coherency stresses. Ideally to model the coherent interface, one would like to capture the biaxial stresses and frictional forces. In our case we are only accounting for the friction associated with dislocations crossing the interface. The inclusion of the biaxial stress near the interface would aid to drive the threading dislocations along the interface and through the layer. This additional force would tend to decrease the applied stresses and bring the results closer to the experimental observations. It would also be more realistic to leave small interfacial dislocations at the interface after a dislocation has passed from one layer to the next. The interfacial dislocation would reflect the conservation of the burgers vector across the interface due to the lattice parameter differences. Two methods could be employed to account for the conservative segment at the interface. One option would be to insert a segment in DD 104

120 with a burgers vector equal to the difference between the two burgers vectors, as shown in figure 4.1. Figure 4.1: Implementation of the conservative interfacial dislocation segment in DD. The other option would be to create a tensorial value that keeps track of the number of dislocations crossing the interface. This tensor, similar to the Nye s tensor, could then be implemented in the continuum portion of the model as a hardening parameter. The versatility and ease of implementation of MDDP make it a great tool in modeling both micro and macro plasticity. These results show that with some modifications or optimization, MDDP should capture the size effects observed experimentally. 105