Atomistic investigation into the mechanical properties of the ferrite-cementite interface: The Bagaryatskii orientation

Accepted Manuscript Atomistic investigation into the mechanical properties of the ferrite-cementite interface: The Bagaryatskii orientation Matthew Guziewski, Shawn P. Coleman, Christopher R. Weinberger PII:

S1359-6454(17)30937-0

DOI:

10.1016/j.actamat.2017.10.070

Reference:

AM 14169

To appear in:

Acta Materialia

Received Date: 19 September 2017 Revised Date:

26 October 2017

Accepted Date: 30 October 2017

Please cite this article as: M. Guziewski, S.P. Coleman, C.R. Weinberger, Atomistic investigation into the mechanical properties of the ferrite-cementite interface: The Bagaryatskii orientation, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.10.070. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Atomistic Investigation into the Mechanical Properties of the Ferrite-Cementite Interface: The Bagaryatskii Orientation Matthew Guziewskia,∗, Shawn P. Colemanb , Christopher R. Weinbergera,c a

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Department of Mechanical Engineering, Colorado State University, Fort Collins, Colorado 80524, USA b U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, USA c School of Advanced Material Discovery, Colorado State University, Fort Collins, Colorado 80524, USA

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Abstract

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Atomistic modeling is used to investigate the mechanical response to compressive and tensile straining of the Bagaryatskii orientation relationship between ferrite and cementite within pearlite. A range of interlamellar spacings and ferrite to cementite ratios are considered and values for important mechanical properties, including elastic modulus, yield stress, flow stress, and ductility, are determined. These values are fit to simple elasto-plastic models, thus allowing for the easy interpolation of states not simulated. Transverse loading can be described using simple 1-D composite theory, while longitudinal loading requires the consideration of the strain compatibility of the interface. The mechanical properties are shown to be largely dependent on the volume ratios of the cementite and ferrite, with the interlamellar spacing having an increasing role as it reaches smaller values. Additionally, the effect of the interface is discussed and characterized, including its role as a nucleation site for dislocations in both the ferrite and cementite.

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Keywords: nanostructure; plastic deformation; Fe-C alloys; molecular dynamics; 1. Introduction

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Although carbon steels are among the most commonly used materials in modern society and have been studied extensively for decades, there are still many outstanding questions regarding their microstructure and the associated mechanical properties. Pearlite is one of the most common of these steel microstructures, forming from the eutectic transformation of austenite. Alternating layers of ferrite and cementite form colonies in a lamella structure [1], with all colonies sharing the same orientation [2]. There has been significant work done towards understanding the macroscopic mechanical properties of pearlite, particularly in relation to the lamella thickness and austenite colony size. The strength of pearlite has been √ shown to follow a Hall-Petch type relation to interlamellar spacing (S) [3–5], σy = σ0 + k S, while the ductility is affected by both the spacing and the preceding austenite colony [6]. Upon the onset of plastic deformation, a yield point phenomenon can be observed in the stress-strain response [7–9]. This phenomenon is a result of carbon interstitials in the steel pinning dislocations until a critical stress is reached. These relations all highlight the role ∗

Corresponding author Email address: [email protected] (Matthew Guziewski)

Preprint submitted to Elsevier

November 1, 2017

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that atomic scale behavior and structure can play in macroscopic response, and while there are clearly important relations between the atomic scale physics and mechanical response in pearlite, there has been very little exploration of this topic. Interfaces in particular are known to influence the material properties of laminates [10] and multilayer systems [11]. In pearlite, Modi et al. [12] found that below a critical interlammelar spacing (≈ 600 nm) the ultimate tensile strength and ductility remain constant, suggesting this may be due to the residual stress of the ferrite-cementite interface. Both the character of the interface, as well as the interface density, or interlammelar spacing, have been shown to affect the mechanical response in other heterogeneous systems. In the Cu-Nb system, the interfacial structure has been shown to influence the formation of twins [13], while the number of interfaces per volume was found to have a profound effect on thermal stability [14]. Experimental observations of these properties can often be difficult though for reasons of both production and measurement. Experimental limitations make atomistics an ideal method for evaluating these systems, as both creation and deformation of these bilayers is a relatively straightforward endeavor. As such, atomistics have been used extensively to study the mechanical response of interfaces [15]. For homophase interfaces, grain boundaries and lamella thickness have been observed to influence the strength of the material in FCC [16], BCC [17], and HCP [18] crystals. Interfaces have also been found to serve as dislocation sources [19, 20] in metals. In studies exploring dynamic conditions, such as copper grain boundaries under shock, it has been shown that coherent grain boundaries will nucleate voids while incoherent grain boundaries will not [21] and that the interface structure can influence the flow stress [22, 23]. There has also been work done on the shear response of heterogeneous interfaces, including Cu-Nb, examining the anisotropy of the shear resistance [24], and Cu-Ta [25], relating the yield stress to the interfacial energy. These studies all highlight the importance of considering the energetics and interface structure when examining mechanical properties. As such, when considering the response of an interface, orientation relationship (OR) and interfacial chemistry will almost certainly have an effect on the observed results. One common interfacial structure that the OR and interfacial chemistry have been observed to influence is misfit dislocations, which have been shown to form in the ferritecementite interface by both the authors [26] and others [27]. As the name suggests, these dislocations form from the mismatch of the two component lattices. The spacing of these dislocations can be described by the Frank-Bilby equation [28], while the structure and energetics of the dislocations have been shown to be influenced by interfacial chemistry [26]. These interfacial dislocations have been shown to be nucleation sites for dislocations emitted into the constituent components forming the interfaces [20, 29–31]. The emitted dislocations allow for plastic flow through the composite and have been observed in pearlite [32]. Ceramic-metal composites are already an active area of study both experimentally [33] and atomistically [34], and as such there has already been the development of 3D elasto-plastic models for these systems [35]. Characterization and quantification of the elastic and plastic response in the cementite-ferrite ceramic-metal interface could therefore allow for its implementation into more robust mesoscale models in the future. In this work, atomistics are utilized to analyze the tensile and compressive mechanical response of the Bagaryatskii FeC-Fe interface in pearlite, with future works considering other pearlitic ORs and interfacial chemistries. Stress-strain relations are generated for both 2

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transverse and longitudinal loadings for a variety of volume ratios and interlamellar spacings, and values for elastic moduli, yield stress, flow stress, and ductility are quantified and fit to elasto-plastic models. The elastic response of both the transverse and longitudinal loading can be described using simple composite theory with an additional size effect term. Under the iso-strain, transverse loading, the ferrite and cementite are observed to behave largely independent of each other, with the overall mechanical response being described through the weighted volume average of the stress state of each of the components. For the iso-stress, longitudinal loading, the strain compatibility of the interface after the ferrite yields proves to be a dominant factor, influencing the yield stress, flow stress, and ductility of the system. The magnitude of this effect is dependent on the volume ratios of the cementite and ferrite, with increasing ferrite values resulting in higher yield stress and ductility and lower flow stress. For all loading conditions, ferrite is observed to plastically deform first, emitting dislocations of the a2 h111i{110} type. The system then plastically flows until dislocations are emitted into the cementite matrix, with slip planes varying with loading conditions. The development of these models to describe the mechanical response allows for a simple way to predict mechanical response with varying amounts of ferrite and cementite, and will allow for the comparison of the various ORs in the future.

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Figure 1: Overview of the Bagaryatskii OR. The cementite and ferrite are found to have the following symmetries: [100]θ ||[1¯ 10]α , [010]θ ||[111]α , and (001)θ ||(11¯2)α . Previous work by the authors also revealed the presence of interfacial dislocations [26].

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2. Bagaryatskii Orientation Relationship Among the most commonly observed orientation relationships for ferrite and cementite is the Bagaryatskii OR [36], in which the ferrite and cementite lamella are related as follows (Fig. 1): [100]θ ||[1¯10]α [010]θ ||[111]α (001)θ ||(11¯2)α

with θ denoting cementite and α for ferrite and using the convention a < b < c for the crystallographic directions of cementite. There are several other ORs commonly reported in literature, including the Isaichev [2] and the Pitsch-Petch [37]. These different ORs will produce different effective elastic constants, interfacial structures and interfacial energies. In 3

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Figure 2: Energy map showing the difference of atomic energy in the interface relative to the bulk, highlighting the formation of two sets of dislocations in the interface, one with line direction [100]θ ||[1¯ 10]α and another with line direction [010]θ ||[111]α . Reproduced with permission from [26].

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Figure 3: Potential interfacial chemistries within the cementite unit cell for the Bagaryatskii OR. The dotted lines represent where the cementite structures forms the interface with ferrite. The FeC-Fe chemistry was found to have the lowest interfacial energy and as such is the focus of this work. Reproduced with permission from [26].

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addition to the obvious impact that varying the elastic constants will have on mechanical response, the interfacial structure and energy previously were shown to affect mechanical response [24]. It is therefore important to consider each OR individually and not attempt to create one simple generalized model for pearlite. Previous work by the authors [26] used atomistics to examine the energetics and structure of the interface that forms in the Bagaryatskii OR. It was found that an array of dislocations forms at the interface with line directions in the [100]θ ||[1¯10]α and [010]θ ||[111]α directions (Fig. 2). Following a continuum formulation based on O-Lattice theory and anisotropic elastic theory developed by Vattr´e and Demkowicz [38], these Burgers vectors were found to be pure edge in character. The Burgers vectors of these dislocations were determined to remain in the interface plane with magnitudes of 4.26 ˚ A and 2.51 ˚ A in the [010]θ ||[111]α ¯ and [100]θ ||[110]α directions, respectively. Additionally, it was shown that the chemistry of the interface, particularly that of the cementite, played a significant role, with the FeC-Fe interfacial chemistry (Fig. 3) found to be the most energetically favorable after being tested by multiple interatomic potentials. For the interatomic potential used in this work, the interfacial energy was found to be 0.52 J/m2 . As the interfacial chemistry with the lowest energy will be the most likely to form, the focus of this work will be on the FeC-Fe interface. 3. Methodology

3.1. Interatomic potential In atomistics, a simulation is only as accurate as the interatomic potential that is used. As such, extensive testing was undertaken in the authors’ previous work on the interfacial energy and structure of the Bagaryatskii OR to find a suitable potential. Lattice constants, elastic constants, and cohesive energy for nine known iron-carbides were calculated and compared with existing values in literature. Potentials that varied significantly from reported values were removed from consideration. Additionally, an evolutionary algorithm, USPEX [39], was utilized to determine if the potentials predicted any stable iron-carbide structures that have 4

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not been observed experimentally. Known iron-carbides, along with 50 random structures were initially considered, and after 45 generations a convex hull was constructed for each potential. For the iron-carbon system, the convex hull should form a straight line between ferrite and graphite, and all potentials that did not reproduce this were eliminated. Six potentials were originally considered and two were deemed suitable for these simulations, a Tersoff potential developed by Henriksson [40] and a Modified Embedded Atom Method (MEAM) developed by Liyaange [41]. The downselection process for the potentials is discussed in depth in the Appendix of [26]. The Henriksson Tersoff potential was used in this work for reasons of computational efficiency. The Tersoff style potential is less computationally expensive than the MEAM and the box size required to minimize the in-plane strains, a function of the lattice constants of the potentials, was smaller for the Tersoff. Tables 1 and 2 show the Tersoff potential predicted values for lattice and elastic constants for ferrite and cementite.

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Table 1: Comparison of ferrite predicted lattice constant (˚ A) and elastic constants (GPa) with experimental [42] values.

a C11 C12 C44 Tersoff 2.86 225 142 128 Experiment 2.87 242 147 112

Table 2: Comparison of cementite predicted lattice constant (˚ A) and elastic constants (GPa) with density functional theory [43] (DFT) and experimental [44] values.

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a b c C11 C22 C33 C12 C23 C13 C44 Tersoff 4.47 5.07 6.45 332 356 373 183 133 174 68 DFT 4.51 5.08 6.73 322 388 345 164 156 162 15 (Experiment) (4.52) (5.09) (6.74)

C55 C66 118 128 134 134

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3.2. Simulation Parameters The interface was created using LAMMPS [45] in the Bagaryatskii OR with an FeC-Fe interfacial chemistry. The in-plane dimension were made to be large enough (85 ˚ A x 203 ˚ A) to reduce strain below 0.25%, with the dimensions determined by the lattice constants of the constituent phases. Various lamella thicknesses were considered for both the cementite and the ferrite, however, the resultant strain normal to the interface plane is zero regardless of the thicknesses. The simulation cell was made to be periodic in all directions, thus reproducing the lamellar structure of pearlite. These unrelaxed structures then underwent a molecular statics energy minimization at 0 K and zero pressure using the conjugate gradient method and a Parrinello-Rahmen barostat. This minimization produces the interfacial dislocations that were observed in the authors previous work and ensures an initial zero net stress state for the system. The system was then equilibrated at 5 K and zero pressure for 20 ps using a Nos´e-Hoover thermostat and Parrinello-Rahmen barostat in order to approximate an isothermal-isobaric ensemble. Simulations were performed at this low temperature in order to minimize thermal noise when attempting to characterize the deformation mechanisms. 5

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This is a common approach for this type of simulation [46, 47]. After equilibration, the simulations are then strained at a rate of 109 s−1 , or 0.0001% per timestep, using nonequilibrium molecular dynamics with a modified Nos´e-Hoover thermostat. This approach adjusts the atomic velocities to account for the changing shape and size of the simulation cell. This modification uses the SLLOD algorithm [48] and has been shown to be produce the correct velocity gradients and proper values for work performed by stress [49]. Simulations were then strained to 40% with the box shape continuously updated every timestep.

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3.3. Deformation Characterization The bulk stress state of the system was determined using the standard virial formulation, which has been shown to converge to the Cauchy stress [50]. It is additionally possible to to determine the individual stress states of both the ferrite and cementite by summing the per-atom virial [51] of the atoms of the desired phase and dividing by the volume the phase occupies. The stress of a constituent phase is then given by:

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Np N 1X 1 X (np) (np) (n) (n) ri F j ] [m(n) (vi − v¯i )(vj − v¯j ) + σij = − V n=1 2 p=1

(1)

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where V is the volume of the phase, n is an atom in the phase being considered, p is any atom in the simulation, v is atomic velocity, v¯ is the mean velocity of the deforming box, r(np) is the displacement vector between atoms n and p and F (np) is the force between the two. Since the box is being deformed, it is necessary to determine this volume of the phase at every time step, and this is done using Voronoi tesselation [52] on all the atoms in the phase and summing the calculated volumes. This formulation converges to the standard virial when all atoms are considered. These component stress-strain relations were observed to be iso-stress when normal to the interface and iso-strain in the in-plane directions, which is consistent with composite theory. As the formation of dislocations was observed in both the ferrite and cementite, it was also necessary to use dislocation characterization techniques to visualize them. Due to the common and well studied BCC character of the ferrite, these dislocations were analyzed using the dislocation extraction algorithm, or DXA [53], that has been implemented into the OVITO [54] visualization program. In this approach, common neighbor analysis [55] (CNA) is used to determine atoms in the bulk crystallographic structure. The CNA algorithm takes all neighbors within a cutoff radius of an atom and determines three values for each pair: the number of common neighbors of the two atoms, the number of pairs within these common neighbors, and the longest chain of atoms formed. These values are then matched to the known, theoretical values of common crystal structures (FCC, BCC, HCP). A Burgers circuit is then generated on the atoms not in the selected crystallographic structure (BCC for ferrite), and a Burgers vector is determined. Additionally, as DXA only requires the current state of the simulation, it is able to account for a significant amount of deformation in the crystal. However, because the cementite crystal structure has not been implemented into DXA, an alternate approach was required. Atomic slip vector analysis [56] is used here, with the slip vectors (s) defined by: sα = −

n 1 X (xαβ − X αβ ) ns α6=β

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(2)

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where ns is the number of slipped vectors and xαβ and X αβ are the spatial vectors between atoms α and β in the current and reference states, respectively. These slip vectors were visualized using the Atomviewer [57] program. Unlike DXA, the calculation of slip vectors requires both the current state and a reference state. As the nucleation of dislocations in cementite was seen to occur in the vicinity of 15% strain, there was significant deformation of the crystal. As such, there was ambiguity in the determination of exact Burgers vectors. Nevertheless, slip vector analysis was very useful in the determination of the slip planes in cementite. 4. Results

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Strain controlled tensile and compressive deformation of the simulation cells was undertaken in the [100]θ ||[1¯10]α , [010]θ ||[111]α , and [001]θ ||[11¯2]α crystallographic directions. For the sake clarity and brevity, going forward these directions will be referred to using the deformation direction of the cementite. While much of the focus here is on the approximate 7:1 ratio of ferrite to cementite that occurs during the eutectic transformation from austenite [58], additional ratios, as well as various lamella thicknesses at these ratios, are also considered. It has been proposed for other ferrite-cementite systems that reducing the 7:1 ratio and increasing the volume fraction of cementite can reduce the magnitude of the yield point drop [59]. Using advanced processing techniques, such as nitrocarburizing [60], it is already possible to increase the cementite volume fraction, making the quantification of the mechanical response over a range of values all the more important. In addition to deforming the pearlitic microstructure, simulations using the same procedure discussed in Section 3.2 were also performed for both the ferrite and cementite individually. This allows for the relation of the component properties to those of the system. Elastic moduli in the relevant crystallographic directions can be found in Table 3. Table 3: Elastic Moduli (GPa) for ferrite and cementite in the relevant crystallographic directions.

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[100]θ ||[1¯10]α Cementite 149.3 Ferrite 260.1

[010]θ ||[111]α 166.2 300.9

(001)θ ||(11¯2)α 201.0 306.7

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Figure 4 shows the mechanical response for the 7:1 ferrite to cementite ratio with a cementite lamella thickness of 4 nm. It can be observed for all simulations that three main events can be sequentially observed: 1) elastic deformation, 2) ferrite plasticity, and 3) cementite plasticity. The specifics of these deformation mechanisms will be discussed in the subsequent sections. 4.1. Transverse Loading 4.1.1. Elastic Deformation Due to the lamellar structure of cementite, it is possible to make a first order approximation of its behavior using simple composite theory. For a transverse (in-plane) straining, an iso-strain state exists in the direction of loading. In such a strain state, the effective elastic

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Figure 4: Stress-strain response of the entire pearlite structure in compression (blue) and tension (red) for deformation in the a)[100]θ ||[1¯ 10]α direction, b) [010]θ ||[111]α direction, c) (001)θ ||(11¯2)α direction for a 7:1 ferrite to cementite ratio with cementite lamella thickness of 4 nm.

moduli of the system can be related to the component moduli simply using volume ratios: Eθ V θ Eα V α + V V

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Eeff =

This formulation suggests that at a given ferrite to cementite ratio, the effective elastic moduli will remain constant regardless of lamella thickness. It can be observed from Figure 5 that this is not the case for the simulations performed, the effective moduli is seen to decrease with decreasing lamella thickness. This suggests there is a size effect to the effective elastic moduli. It has been commonly reported in literature that at the nanoscale the stress state is often influenced by the size of the crystal [61, 62], with the common approximation:

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σ = σ0 +

C t

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where σ0 represents the elastic-plastic stress state and the second term the size effects. C is a material constant and t is the thickness. A similar approximation is made for the elastic modulus. Accounting for the both the ferrite and cementite having common interfacial dimensions, the elastic relations can be expressed as:

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Eef f = Eθ r + Eα (1 − r) +

C t

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where r is the volume ratio of the cementite and t is the thickness of the bi-layer. The effective moduli were determined from simulations for ferrite to cementite ratios ranging from 1:1 to 7:1 and cementite lamella thicknesses ranging from 2 nm to 8 nm. Using the moduli for ferrite and cementite in the appropriate direction found in Table 3, C was calculated by minimizing the resultant error between calculated and predicted values. Excellent agreement was found between the analytical model formulation and the simulations, as the values predicted by Equation 3 were all within 10% of the values found in atomistics. The C constants were determined to be 2316 GPa·˚ A in the [100]θ direction and 909 GPa·˚ A for the [010]θ . These values reflect the contribution of the interface, and the stress field it produces, to the effective modulus of the system with the positive values suggesting stiffening. This results in a much more robust method to determining the effective elastic modulus than attempting to calculate an interfacial stiffness and interfacial thickness. 8

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Figure 5: Variation of effective elastic moduli for the entire pearlite structure at a 1:1 ferrite to cementite ratio under [001]θ tensile straining. Lamella thicknesses of 2 nm (blue), 4 nm (red), 6 nm (green) and 8 nm (purple). Variation of the effective moduli implies a size effect for the elastic modulus.

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4.1.2. Ferrite Plasticity The ferrite was found to plastically deform first in all simulations, regardless of ferrite to cementite ratio or lamella thickness. Upon reaching yield, a2 h111i{110}α dislocations loops nucleate at the interface (Fig. 6), specifically from from the interfacial dislocations into the ferrite matrix. This is a result of the increased stress state present around the interfacial dislocations, which allows for the necessary resolved shear stress to be reached at a significantly lower applied stress than in bulk ferrite. The dislocation loops expand until reaching the opposite interface, at which point they become straight screw dislocations (Fig. 6). This behavior has also been observed in other bi-layer systems [63–65]. After the nucleation events, the stress drops to the flow stress (Table 4) as the initial nucleation stress is higher than that required for continued plastic flow. The resultant flow stress does vary with loading, but it is independent of the ferrite lamella thickness (Fig. 7) and is consistent with the flow stress from bulk ferrite simulations under the same loading conditions. It is worth noting that while the flow stress of the ferrite does not vary with the lamellar thickness, the stress state of the whole system will still be a volume average of the stresses in the ferrite and cementite. Thus the reduction in the bulk stress of the system from the nucleation of dislocations in the ferrite will increase with an increasing ferrite to cementite ratio. Table 4: Yield and flow stresses (GPa) for ferrite and cementite and ductility between ferrite and cementite plasticity under various transverse loading states.

Yield Flow Yield Cementite Flow Accommodated Strain Ferrite

[100]θ Compression Tension 20.6 27.3 13 10 32.8 25.2 15 17 1.2% 5.1%

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[010]θ Compression Tension 45.4 24.8 13 12 37.9 18.6 20 15 8.0% 5.0%

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Figure 6: Formation of a2 h111i{110}α type dislocations in the ferrite matrix, visualized both by removing all BCC atoms as determined by CNA as well through the use of DXA. Both dislocation loops and straight screw dislocations can be observed within the ferrite.

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4.1.3. Cementite Plasticity Since transverse loading results in an iso-strain state, the ferrite and cementite generally behave independently of each other. After the ferrite yields, it continues to plastically flow while cementite continues to elastically load prior to yield. Similar to the ferrite, this occurs at significantly lower stress than bulk cementite. The amount of plastic strain that occurs between these plasticity events varies with loading (Table 4), as the stress state required for dislocation nucleation will be different in each loading direction. The amount of accumulated plastic strain does not vary with lamella thickness or ratio, since in an iso-strain state the stress in the individual lamella is independent of these factors. Like the ferrite, once yield in the cementite occurs, dislocations will again nucleate from the interface, however, these dislocations do not solely nucleate from the interfacial dislocations. The dislocations instead form upon favorable slip planes within the cementite. This variation in nucleation sites is likely due to the change in the interfacial dislocation structure after the ferrite plasticity events; there is no longer a pristine interface and the interfacial dislocations will have been disrupted or destroyed. Due to the more complex crystal structure of cementite, the slip planes vary with loading state. For straining in the [100]θ direction, dislocations form on the {111} family of planes, while for straining in the [010]θ direction the dislocations for on the {110}. Similar to the ferrite, dislocations form until the plastic flow stress of cementite is reached (Table 4) and the specific flow stress is independent of lamella thickness (Fig. 8). These flow stresses are higher than those of bulk cementite, suggesting the interface plays a role in limiting plastic flow. After reaching the plastic flow stress, the entire system will then continue to deform plastically. 4.2. Longitudinal Loading Whereas in the transverse loading state the ferrite and cementite behave almost independent of each other, the longitudinal loading state provides a much more coupled response. As such, the mechanical response will be presented as simply elastic deformation and plasticity, as opposed to discussing the individual behavior of both the ferrite and the cementite.

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Figure 8: Stress-strain response for the cementite component of the pearlite with a 1:1 ferrite to cementite ratio under [100]θ tensile loading for lamella thicknesses of 6 nm (blue), 4 nm (red) and 2 nm (green) for ferrite to cementite ratios of 7:1, 2:1, and 1:1, respectively. All lamella thicknesses are observed to converge to the same flow stress regardless of ferrite:cementite ratios or lamella size.

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Figure 7: Stress-strain response for the ferrite component of the pearlite under [100]θ tensile loading for lamella thicknesses of 14 nm (blue), 8 nm (red) and 4 nm (green) with a constant associated ferrite thickness (2 nm). All lamella thicknesses are observed to converge to the same flow stress for both different ferrite:cementite ratios and lamella size.

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4.2.1. Elastic Deformation Similar to the approach used in the transverse elastic deformation, simple composite theory is again considered. For longitudinal loading, the effective compliance of the system is the sum of the volume ratios of the component compliances:

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1 Vθ Vα = + Eef f V Eθ V Eα

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Combining this equation with a size effect term yields the expression: r Eθ

1 C 1−r + t + Eα

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Eef f =

(4)

Fitting simulation results to this equation yields a C value of -338 GPa·˚ A, smaller in absolute terms than those from the transverse simulations (Table 5). This suggests the interface plays a smaller role in the elastic response, while the negative value suggests softening from the interface. While Equation 4 fully describes the elastic response in compression, there is an additional event that occurs during the longitudinal tensile simulations. From Figure 9 it can be observed that there is a stress drop at approximately 5% strain for simulations of all ratios. The magnitude of this stress drop decreases, however, as the ferrite to cementite ratio is increased. Simulations were performed in which the strains were reduced back to zero after this stress drop, revealing that this is a plastic event. Closer inspection of the atomic displacements reveals atomic movement along the core of the [010]θ ||[111]α (line direction) dislocations. This suggests plastic flow is occurring within the interface. It is worth noting 11

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that for the 7:1 ferrite to cementite ratio the resultant stress drop is very small, and would likely be imperceivable during macroscopic analysis. Table 5: Values of C (GPa·˚ A) from Equations 3 and 4 for the various loading states.

[010]θ 909

[001]θ -338

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[100]θ 2316

Figure 10: Flow stress vs volume ratio plot for [001]θ tension (red) and compression (blue) and the associated fit to Equation 5). The nearly identical curves suggest that the response is independent of direction of straining and solely a function of volume ratio. Stress oscillates around the flow stess value, shown by the error bars.

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Figure 9: Tensile, longitudinal straining of pearlite with cementite lamellar thickness of 4 nm. Ferrite to cementite ratios of 7:1 (purple), 3:1 (green), 2:1 (red), and 1:1 (blue) all reveal a stress drop during straining, with the magnitude of the drop decreasing at higher ferrite to cementite ratios. The 1:1 ratio is also unloaded, revealing that this is a plasticity event resulting from plastic flow in the interfacial dislocations.

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4.2.2. Plasticity Much like the transverse loading states, plasticity in longitudinal loading begins with the nucleation of a2 h111i{110} dislocations from the interfacial dislocations. However, due to the iso-stress state of the system, the mechanical response is quite different. A simple one dimensional model would predict that the system would both yield and plastically flow at the ferrite flow stress, as all further straining would be accommodated by the ferrite. The flow stress of the system would therefore remain constant regardless of the ferrite to cementite ratio. The results of the tensile longitudinal loading, however, show that the resultant flow stress varies with the ratio (Fig. 10), with larger ratios observed to have a lower flow stress. This ratio dependent response can be explained by considering the compatibility within the interfacial plane. Once the ferrite begins to yield, the higher stiffness of the cementite acts a limiting factor to how much the ferrite can contract, and thus limits how much of the strain it will carry. This relation can be shown to follow: σflow = σ1 + K1 12

Vθ Vα

(5)

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σy = σ 0 − K 0

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where Vθ and Vα are the volumes of the cementite and ferrite, respectively, and σ1 and K1 are material and orientation dependent constants. Simulation results are fit to this equation (Fig. 10), and the resultant constants can be found in Table 6. The value for σ1 should be near the flow stress of bulk ferrite since as Vα approaches infinity, σflow = σ1 . These values are observed to be fairly close with σ1 = 8.58 GPa and the flow stress of bulk ferrite being 9.22 GPa. The constraint of the cementite also affects the yield stress, providing additional stress to that of the applied stress, σy = σapp + σconst , and resulting in a similar relation for yield stress: Vθ Vα

(6)

Vθ Vα

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p = 0 − K2

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with σ0 and K0 being material and orientation dependent constants. The ductility of the system is a function of the yield stress, the flow stress, and the effective elastic modulus. As these are all volume ratio dependent quantities, it too can be expressed in a similar form: (7)

The ductility reported here is evaluated as the strain if the simulation were elastically unloaded just prior to cementite plasticity. Equations 6 and 7 are fit to simulation results (Figs. 11 and 12) and the resultant constants are shown in Table 6. Table 6: Constants for Equations 5, 6, and 7 using least squares fit to simulation data

K0,t 2.29 GPa

K0,c 3.32 GPa

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σ0,c 28.06 GPa

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σ0,t 19.15 GPa

σ1 8.58 GPa

K1 7.57 GPa

0 K2 19.15% 2.29%

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After plasticity begins in the ferrite, the system deforms in an elasto-plastic state at the flow stress until the cementite yields. There are two potential processes by which this could occur, either the plasticity mechanisms in the ferrite will be exhausted and the system will then continue deform elastically until the cementite yields, or the cementite will yield before the plasticity events are exhausted. The exhaustion of plasticity mechanisms in the ferrite would have a size dependence as, assuming a constant maximum allowable dislocation density, larger ferrite lamella would be able to accommodate more strain. The latter would be independent of size, as the stress state in the cementite is a function of only the ferrite to cementite ratio. Simulations reveal that there is no size dependence to the ductility of the system, results that are consistent with the findings of Modi et al. [12]. Simulation results also show that the cementite yield stress has the opposite trend of that of the ferrite; increasing the ferrite volume ratio results in lower yield. This is still consistent with Equation 6, as the constraints would produce opposite stress states in the ferrite and cementite. Similar to the transverse loading, dislocations in the cementite nucleate along favorable slip systems, in this case the (011)θ plane, instead of from the interfacial dislocations. The system quickly fails upon formation of the dislocations, which is unsurprising as cementite is a ceramic. The mechanical response of the longitudinal compression simulations is very similar to that of the longitudinal tension. As the volume ratio of ferrite is increased, the compressive yield stress increases and the flow stress decreases. While it may seem counterintuitive that 13

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Figure 11: Predicted yield stress in tension (red) and compression (blue) under [001]θ loading for various volume ratios and associated fit of Equation 6. Higher compressive yield strength can be observed due to the compressive strength of cementite, a ceramic. Error in values is smaller than data markers.

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Figure 12: Ductility between ferrite and cementite yield in [001]θ tensile straining as a function of volume ratio and associated fit to Equation 7. Error bars represent strain from ferrite yield to failure.

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the same responses occur in both tension and compression, this is consistent with the interface compatibility. As the sign of the applied stress is changed, so will the sign of the constraint stresses. As a result, in absolute terms, the interfacial constraints will affect the mechanical response the same in both tension and compression. This is further demonstrated by nearly identical flow stress prediction curves (Fig. 10). Since for both tensile and compressive loading the ferrite forms a2 h111i{110} dislocations, σ1 would be expected to be the same, as these should flow at the same stress. Similarly, K1 is a function of the elastic response in the interface plane; which should be roughly the same in both tension and compression. There are variations between the yield response in compression and tension, however. While the compression yield strength values still fit well to Equation 6 (Fig. 11), there are different values for both σ0 and K0 (Table 6). The variation in σ0 is to be expected, materials commonly yield at different stress in tension and compression, and the higher compressive value is to be expected due to the higher compressive strength of ceramics. The reason for the variation in K0 is slightly more complex. While the distribution of strain between ferrite and cementite does affect the trend of yield stress, the exact magnitude of this contribution, which is expressed as the K0 value, will be a function of the stress state caused by the interfacial constraint and how it relates to the required dislocation nucleation stress state. These required nucleation stress states are different in tension and compression, hence the variation in value. Another crucial difference between the compressive and tensile simulations is that in compression, the cementite yields almost immediately after the ferrite. Therefore, there is very little plastic strain between the two events and the system simply plastically flows at the larger value of the individual component flow stresses. As the flow stresses found in compression are nearly identical to the ferrite values in tension, this is likely the ferrite flow stress. This is consistent with the bulk flow stresses where ferrite has a flow stress 9.87 GPa vs. 5.20 GPa in the cementite.

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5. Summary

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Atomistic simulations are used to characterize and quantify the mechanical response of the Bagaryatskii OR in pearlite. Simulations reveal that the interface and volume ratios of ferrite and cementite play a significant role in this response. For both transverse and longitudinal loading, the effective elastic modulus is observed to be related to the volume ratio with an additional size dependent term, which essentially accounts for the density of interfaces within the system. Yield stresses in transverse loading are observed to have a small size effect, with these effects becoming negligible at sizes greater than approximately 10 nm. While Modi et al. found yield stresses to be independent of size below ≈ 600 nm, the smallest interlammelar spacing observed in their work are more than an order of magnitude larger than the largest in this work. As such, these observations are in an entirely different regime. Flow stresses within both the ferrite and cementite are observed to be constant regardless of size, with the ferrite flow stress consistent with bulk values, while the cementite flow stress is noticeably higher than the bulk values. The strains at which plasticity begins in both the ferrite and cementite also remain constant, meaning that the ductility of the system is the same regardless of size or volume ratio, consistent with the results of Modi et al. Using the tabulated values for elastic modulus, yield stress, flow stress, and ductility, it is now possible to interpolate mechanical response in the transverse direction over a large range of volume ratios. The mechanical response in the longitudinal direction was found to be slightly more complicated. As ferrite was observed to plastically deform first, the compatibility of the interface was found to play a major role in the yield stress, flow stress, and ductility of the system. Relations were developed for these values for both tensile and compressive loading, again allowing for the interpolation of these values over a range of volume ratios. For specific volume ratios, strength and ductility were found to be size independent, again consistent with the findings of Modi et al. Ferrite was observed to play the dominant role in plastic flow. In tension, the system flows at the constrained ferrite value and then rapidly fails once cementite begins to nucleate dislocations. In compression, the system simply flows at the constrained ferrite flow stress. Both of these observations are consistent with metalceramic laminates, which are known to have high strength in compression but are brittle in tension. The development of these models will allow for the potential tuning of size and volume ratio to desired mechanical properties. One example would be the ability to balance strength, which increases with ferrite volume ratio, with a more continuous plastic response, which occurs as the ferrite volume ratio decreases. Additionally, the values calculated here will allow for easy comparison to other ORs and interfacial chemistries for pearlite, further aiding the potential tuning of material properties. 6. Acknowledgments This research was supported though a grant from the Petroleum Research Fund, PRF #54697-DNI10. This work utilized the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder and Colorado State University. The RMACC Summit supercomputer is a joint effort of the University of Colorado Boulder and Colorado State University. 15

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