Atomistic investigation into interfacial effects on the plastic response and deformation mechanisms of the pearlitic microstructure

Acta Materialia 180 (2019) 287e300

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Atomistic investigation into interfacial effects on the plastic response and deformation mechanisms of the pearlitic microstructure Matthew Guziewski a, b, *, Shawn P. Coleman b, Christopher R. Weinberger a, c a

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

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 March 2019 Received in revised form 5 September 2019 Accepted 8 September 2019 Available online 13 September 2019

Atomistic modeling is used to investigate the mechanical response and deformation mechanisms at 5 K temperature within the commonly reported orientation relationships between ferrite and cementite within pearlite: Bagaryatskii, Pitsch-Petch, Isaichev, and their associated near orientations. For each orientation, compressive and tensile simulations were performed in the transverse and longitudinal directions for a range of ferrite to cementite volume ratios. Important mechanical properties such as peak stress, flow stress, and the activated slip systems in both lamella are reported. Significant variation in mechanical response is found between the various orientation relationships. In the transverse direction, the responses are well described by composite theory; longitudinal loading requires further consideration of the strain compatibility of the interface. Plasticity within the ferrite is found to initiate from the interface and is well described by Schmid factors; slip and failure in the cementite are affected by slip transfer mechanisms across the interface between the lamella. Simulation results are used to create a simple model for predicting deformation behavior in pearlite, allowing for greater understanding of the plasticity and failure mechanisms within the various reported orientations, and raising the possibility of the targeted creation of specific microstructures based on the intended mechanical loading. Published by Elsevier Ltd on behalf of Acta Materialia Inc.

Keywords: Nanostructure Plastic deformation Fe-C alloys Molecular dynamics

1. Introduction Steel is among the most commonly used materials in modern society, owing to its availability, high strength, and low cost. As such, its microstructures have been extensively studied for decades. These various microstructures influence an array of mechanical properties [1], and thus understanding their behavior is fundamental to understanding the response of steel. Within carbon steels, pearlite is among the most commonly observed microstructures, consisting of alternating layers of ferrite (a-Fe) and cementite (Fe3C). Much of the existing work on pearlite has focused at the macroscopic level, such as understanding how various processing routes affects the resulting pearlite colony [2,3]. Size effects have also been studied and found to influence the mechanical response, with the interlamellar spacing affecting the strength of pearlite in a Hall-Petch type relation [4e6], and ductility being

* Corresponding author. U.S. Army Research Laboratory, Aberdeen Proving Ground, MD, 21005, USA. E-mail address: [email protected] (M. Guziewski). https://doi.org/10.1016/j.actamat.2019.09.013 1359-6454/Published by Elsevier Ltd on behalf of Acta Materialia Inc.

influenced by both size of the initial austenite colony, as well as the interlamellar spacing [7]. In addition to these size effects, the residual stress state present at the interface between the ferrite and cementite has also been found to influence properties such as ultimate tensile strength and ductility [8]. This interface, and its associated microscale characteristics, is therefore very important to understanding the mechanical response of pearlite, yet it has been significantly less studied. Within a pearlite colony, the ferrite and cementite lamella have been found to maintain a consistent orientation relationship (OR) relative to each other [9]. However, this OR is not the same for all pearlite colonies, with multiple variations being observed. Three ORs in particular have been experimentally reported most often: the Bagaryatskii OR [10]:


½100q jj 110 a ½010q jj½111a

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  ð001Þq jj 112 a the Isaichev OR [11]:

½010q jj½111a 
½101q jj 011 a     101 q jj 211 a and the Pitsch-Petch OR [12]:


 ½100q 2:6+ from 131 a
 ½010q 2:6+ from 113 a ð001Þq jjð521Þa

with a denoting ferrite and q, cementite, and the habit plane in each given by the planar symmetry. The convention a  b  c has been used here, and throughout this work, to describe the indices of the cementite. There is no consensus however, as to which of these ORs is the most common or favorable, as a search of the literature reveals studies where each is found to form [13e18]. Additionally, Zhang el al [19] postulated that the Bagaryatskii and Pitsch-Petch ORs are actually incorrect. Two “near” ORs were instead proposed, varying by a small angle rotation of the habit plane from their parent ORs: Near Pitsch-Petch


½010q jj 131 a ð103Þq jjð110Þa Near Bagaryatskii

½010q jj½111a   ð103Þq jj 101 a

For these ORs the habit plane of the cementite is assumed to be the same as the parent OR, (001)q, while the ferrite lattice is rotated to create the symmetries shown above. Previous atomistic studies by the authors [20,21] and others [22] have shown that careful examination of each OR is necessary, as they contain unique interfacial dislocation sets. Interfacial structures are known to influence the material properties of laminates [23,24] and multilayer systems [25], and pearlite is no exception. Pearlite's high strength for example, has often been attributed to the cementite interface acting as a barrier to dislocation motion [26,27]. It has also been proposed that this interface acts as a nucleation site for dislocations [28,29], which has been observed computationally by the authors [30]. Due to experimental limitations in creating pearlite in specific ORs, as well as the difficulty in characterizing the microscale

response, atomistic simulations are a prime candidate for studying these systems. It is fairly straightforward to create and deform bicrystals that mirror the structure of pearlite, and atomistics a common tool for studying the mechanical response of interfaces [31e35]. This includes work by the authors characterizing the mechanical response of one of the ORs of pearlite, the Bagaryatskii [30]. However, many of the Bagaryatskii mechanical responses were influenced by factors that vary with different ORs. This includes the nucleation of dislocations into the ferrite from the interfacial misfit dislocations, which have different character in each OR, as well as how the alignment between the cementite and activated ferritic slip system influences the deformation response in the cementite. It is therefore necessary to expand the mechanical study to include the other ORs in order to understand the effect of the interface on the mechanical response of pearlitic microstructure as a whole. In this work, molecular dynamics simulations are used to analyze the tensile and compressive mechanical response of the five most commonly reported ORs within pearlite. For loadings both normal and parallel to the interface, stress-strain relations are generated and the resultant slip systems characterized. While the elastic behavior can be well described by simple composite theory, the plastic response requires the consideration of interface structure, Schmid factors, and slip transfer concepts. For all loading conditions, ferrite is observed to plastically deform first, generally through the nucleation of dislocations of the 2a < 111 > f110g type. Cementite deformation occurs on the f100g, f001g, f110g, f101g and f011g families of slip planes, with the specific plane determined by its alignment with the existing slip system in ferrite. While the net response in transverse loading can be described through the volume ratio of the two lamella, in the longitudinal direction interface compatibility must also be considered. Trends observed through the analysis of these various ORs are presented, connected to several metrics, and simple relations to predict deformation are developed, allowing for an increased understanding of plasticity and failure in pearlite. 2. Methodology 2.1. Atomistic models Atomistic simulations for the five most commonly reported pearlitic interface ORs were modeled using the Tersoff potential by Henrikssonn [36] within LAMMPS [37]. Dimensions within the interface plane were chosen to create periodic boundaries for both the ferrite and the cementite, and made to be sufficiently large as to reduce the initial straining of the lattices below 0.5%. In the longitudinal direction, periodic boundary conditions were also create in order to simulate the lamella structure of pearlite. The cemenite lamella was made to be approximately 4 nm for all simulations, with the size of the ferrite made to be seven times this for all transverse loadings, and made to match the volume ratio under consideration for longitudinal loading. The specifics of the simulation construction and choice of interatomic potential are described in the authors previous work [21] and highlighted in the supplementary material. For each OR, an array of interfacial dislocations were found to form at the interface, arising from the misfit of the two lattices. The spacing and direction of these dislocations was found to be well defined by O-lattice theory, a variation of the Frank-Bilby equation [38]. The Burgers vectors were found to either be of the 12h111〉a character, or lie in a shared high symmetry direction of the two lattices in the given OR. Table 1 shows the determined Burgers vector, along with the specific crystallographic orientation of the simulation, for each OR. The character of these interfacial dislocations is of particular relevance to this work as they

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Table 1 Specific crystallographic orientation of the various ORs evaluated in this work, as well as the Burgers vectors of the interfacial dislocations that form as a result of lattice mismatch. x

Orientation y

z

Interfacial Burgers Vectors

Bagaryatskii

½100q jj½110a

½010q jj½111a

½001q jj½112a

Near Bagaryatskii

½100q jj½1 12 11a

½010q jj½111a

½001q jj½23 10 13

Pitsch-Petch

½100q jj½10 31 12a

½010q jj½11 10 35a

½001q jj½521a

Near Pitsch-Petch

½100q jj½12 39 17a

½010q jj½113a

½001q jj½134 53 27

Isaichev

½101q jj½110a

½010q jj½111a

½101q jj½112a


 1 ½100q jj 110 a ; ½111a 2 * 1 1
111 a ; ½111a 2 2  1
1 ½010q jj 113 a ; ½100q jj ½131a 2 2 * 1
1 111 a ; ½010q jj ½113a 2 2
 1 110 a ; ½111a 2

*Actual Burgers vector is the projection of this Burgers vector onto the interface plane.

are a common nucleation site for dislocations into the lamella [28,29] and will likely have the highest local stress state on the interface [39]. To deform each OR, the interface models were equilibrated at
-Hoover thermostat and 5 K and zero pressure, using the Nose Parrinello-Rahmen barostat to approximate an isothermal-isobaric ensemble. The temperature of 5 K was chosen to minimize thermal noise, making characterization of deformation mechanisms easier, which is a common approach for this type of study [40,41]. After this equilibration, the simulation was strained at a temperature of 5 K and a rate of 109 s1 to 25% strain using non-equilibrium mo
-Hoover thermostat that adlecular dynamics and modified Nose justs atomic velocities with the changing shape and size of the box. This utilizes the SSLOD algorithm [42], which produces the proper velocity gradients and work performed by stress values for the strained boxes [43]. In order to allow for the activation of different slip systems, for each OR compression and tensile simulations were performed deforming the box uniaxially in two transverse directions parallel to the interface plane (x and y directions in Table 1), as well as longitudinally in the direction perpendicular to the interface plane (z direction in Table 1). This uniaxial deformation was made to be unconstrained, with no displacement boundary conditions placed upon the dimensions perpendicular to the direction of loading. To better quantify the influence of the interface on the mechanical response of the pearlite systems, the ferrite and cementite lamellar regions were modeled separately as single phase slabs for each transverse loading direction. This provides for an approximation of the mechanical response for each individual lamella, which can be used in conjunction with simple 1-D composite theory to predict quantities such as elastic modulus, peak stress, and flow stress. Any significant variation from these are attributed to interfacial effects, thus giving greater insight into the system. These individual lamella regions were modeled as 40 Å thick slabs by orienting either the ferrite or cementite region as they exist in each pearlite OR. A vacuum region of 20 Å is introduced to create free surfaces in the direction normal to corresponding OR interface. Each simulation is deformed in the same manner as the pearlite simulations described above, uniaxially and unconstrained at 5 K
-Hoover thermostat. using a modified Nose In order to determine the net response of each simulation, the standard virial formulation was used, an approach which converges to the Cauchy stress [44]. A similar approach also allows for the determination of the individual stress states in each lamella of the pearlite though the summing of the per-atom virial [45]:

sij ¼ 

Np  N    1X 1X ðnÞ ðnÞ ðnpÞ ðnpÞ mðnÞ vi  vi vj  vj þ ri F j V n¼1 2 p¼1

(1)

where 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 ðnpÞ is the displacement vector between atoms n and p and F ðnpÞ is the force between the two. Due to the changing dimensions of the simulation cell as it is being deformed, V is determined by summing the atomic volume of each atom as determined by Voronoi tessellation [46]. All values are updated at each timestep. Two approaches were required to define and visualize the dislocations that were found to form during the simulations. For ferrite, the dislocation extraction algorithm (DXA) [47] within the OVITO [48] visualization program was used. The more complex cementite crystal structure, however, is not among the structures that DXA recognizes, and so an alternate approach was used. Atomic slip vector analysis [49] within the Atomviewer [50] program was employed to analyze the cementite dislocations. In this formulation, the slip vectors (s) are defined by:

sa ¼ 

n   1 X xab  X ab ns

(2)

asb

where ns is the number of slipped vectors and xab and X ab are the spatial vectors between atoms a and b in the current and reference states, respectively. However as the cementite dislocations did not nucleate until higher strain states, the deformation of the crystal did introduce some error into the determination of the exact Burgers vectors. The slip vector analysis did find the general directionality of the Burgers vector, as well as the activated slip plane, which in conjunction with experimentally observed and theoretically proposed slip systems was sufficient to infer the exact Burgers vector. Additionally, an atomic strain metric was used to aid in the visualization of slip planes. Here the atomic deformation gradient tensor (F) from the undeformed box is used to determine the Green-Lagrangian strain tensor (E) using the expression E ¼ 1 ðFFT  IÞ, where I is the identity tensor. This strain tensor can 2 then be broken into deviatoric and hydrostatic scalar values for each atom. While these specific values will not be reported here, the regions of higher deviatoric strain values were found to correspond to the dislocations predicted by DXA and slip vector analysis, and thus proved useful in visualization of the plasticity mechanisms.

2.2. Slip transfer metrics As a ceramic, cementite is a brittle material and therefore its plastic deformation mechanism are often overlooked. However, dislocations have been observed in cementite prior to its failure [51]. Within pearlite, the plastic deformation of cementite has been shown to be strongly related to the plastic deformation of ferrite in

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cold drawn pearlitic steel wires [52], where cementite slip and failure was found to occur on planes parallel to the slip planes in ferrite. This is consistent with the concept of slip transfer across the interface [53], a possibility that was proposed within pearlite from a purely crystallographic perspective by Karkina et al. [54]. Three simple criteria for slip transfer have been previously suggested by Lee et al. [55e57]: 1. The resolved shear stress of the emitted dislocation should be high. 2. The magnitude of the residual Burgers vector left at the interface should be small. 3. The angle between the lines of intersection between each slip plane and the interface should be small. These criteria can be quantified to allow for straightforward analysis of slip transfer [58]. The first criteria can be fully described by the Schmid factor of the dislocation that forms in the cementite.

m ¼ cos f cos l

(3)

where f and l are the angles between the direction of loading with the slip plane and Burgers vector, respectively. The Luster-Morris parameter [59], m0 ¼ cos j cos k, is a commonly used metric for slip transfer, where j is the angle between the slip plane normals and k is the angle between the Burgers vectors of the two slip systems (Fig. 1). The k term in this expression partially describes the second criteria, allowing the creation of a second metric to evaluate slip transfer.

b0 ¼ cos k

(4)

It is worth nothing that this fails to account any potential differences in magnitude between the Burgers vectors in the two lamella. A third metric, simply taken to be the cosine of the angle between the lines of intersection of the slip planes and the interface (q) covers the third criteria for slip transfer.

p0 ¼ cos q

(5)

One additional metric will also be defined here that is similar to the three slip transfer metrics. As there are interfacial dislocations in the pearlite system prior to deformation, it is useful to define a value that shows the alignment between dislocations and the activated slip system in the ferrite. This value, q0 , is taken to be the cosine of the angle between the line of intersection of the ferrite slip system with the interface and the line direction of the closest aligned interfacial dislocations (j).

q0 ¼ cos j

(6)

These metrics are all functions of geometry, thus allowing them to be determined for any set of slip systems in the ferrite and cementite as long as the OR and direction of loading are known. 2.3. Composite theory The initial study by the authors [30] on the mechanical response of only the Bagayatskii OR showed that due to the lamellar structure of pearlite, the system was farily well described by composite theory. This was particularly true for the elastic response, as the elastic moduli were found to match extremely well with the observed atomistic results. For an iso-strain state, which exists in the transverse (in-plane) straining of this system, the effective elastic moduli of the system can be related to the elastic moduli of the individual lamella using volume ratios:

E V Ea Va Eeff ¼ q q þ V V

where Vq and Va are the volumes of the individual lamella and V is the total volume. This formulation suggests that at a given ferrite to cementite ratio, the effective elastic moduli will remain constant regardless of lamella thickness. This was true for the Bagayatskii OR simulations, with the exception of interlamellar spacings below 10 nm. Size effects on the stress state at these length scales are common [60,61], but since even very fine pearlite is observed to have interlamellar spacings of more than 5 times this value [62], Equation (7) provides an apt description of the elastic behavior. For the iso-stress, longitudinal loading, composite theory can also be used, with the effective compliance of the system being the sum of the volume ratios of the component compliances:

1 V Va ¼ q þ Eeff VEq VEa

Vq Va

sflow ¼ s1 þ K1 Fig. 1. Example of slip transfer metrics used to calculate and where q is the angle between the lines of intersection of the slip systems with the interface and k is the angle between the Burgers vectors of the two slip systems. p0 ,

(8)

The plastic response of the pearlite interface under transverse loading was also found to be well defined by composite theory in the authors initial work [30]. The peak and flow stresses of each lamella were found to be independent of size and volume ratio, and the net response of the system was simply the volume average of the stress states of the two lamella. In the longitudinal loading of the system, there is a much more coupled plastic response for the two lamella. This arises from the necessity of considering interface compatibility, in addition to basic composite theory, as the deformation occurs. Interface compatibility is the constraint that the two lamella must remain ideally bonded, and thus have identical interface dimensions, and has been shown experimentally to be important in other brittle-ductile composites [63]. Due to the similar rates at which the ferrite and cementite expand or contract during their elastic response, this is constraint is generally only relevant once the ferrite begins to yield. It is at this point that the cementite acts as a constraint, thus increasing the stress state of the system. The following expressions for peak and flow stresses, the derivation of which can be found in the supplementary material, are found to arise from this constraint:

sy ¼ s0 þ K0

b0

(7)

(9) Vq Va

(10)

This is consistent with observations from the authors’ past work, which found that during longitudinal loading of the Bagaryatskii

M. Guziewski et al. / Acta Materialia 180 (2019) 287e300

OR, peak stress and flow stress only change with volume ratio of the lamella, not size. The s and K terms here will be dependent on both material and orientation. Thus in order to examine their validity, the ferrite to cementite ratio was varied from 1:1 to 7:1 for all ORs. While pearlite that is formed from austenite during the eutectoid transformation always has an approximate 7:1 ratio [64], advanced processing techniques such as nitrocarburizing [65] have made it is possible to increase the cementite volume fraction. Additionally, it has been proposed for other ferrite-cementite systems that reducing this ratio and increasing the volume fraction of cementite can influence mechanical properties [66], highlighting the importance of this analysis. It is important to note that the s and K values presented here are solely meant to show that Equations (9) and (10) are valid fits for the mechanical response of the pristine samples being considered here. These values do not account for all the potential factors such as impurities, non-planar interfaces, and the different strain rates found in experimental values, but nonetheless can provide some insight into the manner in which pearlite deforms and are useful for comparing behavior between the ORs reported here. 3. Results In order to study the mechanical response and deformation mechanisms of the pearlite microstructure, atomistic models of the most common orientations were constructed. As it is not possible to examine all loading states for each OR, three orthogonal directions of loading (two in transverse directions and one in the longitudinal) were chosen to study a diversity of stress states in the two lamella, with both tensile and compressive loadings considered. For all cases, the order of deformation events were the same: elastic deformation of both lamella, plasticity in the ferrite, plasticity in the cementite, failure. As such, simulation results will be presented in this order, along with relevant trends that can be discerned about each by comparing the response of the various ORs. 3.1. Individual ferrite and cementite lamella Before considering the response of pearlite, it is important to consider how individual lamella of ferrite or cementite would respond to the loading states being considered. Thus values for the elastic modulus, peak stress, and flow stress were determined for the transverse loadings of the individual lamella in single phase simulations that matched the straining direction that occurred within the various OR simulations. These values, along with the activated slip systems, can be found in the Supplemental section of this work. The slip systems that activated during the plastic response of the individual ferrite and cementite lamella were found to be well defined by the Schmid factor. The ferrite slabs were found to plastically deform along the two commonly reported BCC slip systems, 2ah111if110g and 2ah111if112g. Plasticity in the cementite slabs was found to occur within the f100g, f001g, f110g, f101g and f011g families of planes, which matches experimental observations [67]. 3.2. Elastic response While the focus of this work is on the plastic response of pearlite, it is important to briefly note how the system deforms elastically before these plastic events occur. In the early stages of deformation, both crystals deform elastically, and the moduli of both the ferrite and cementite within the pearlite were roughly equivalent to the associated moduli of the individual lamella loaded in the corresponding direction. These values were found to be

291

related to the net pearlite response by composite theory (Eqs. (7) and (8)), suggesting the influence of the interface during this phase of the deformation is minimal. As ferrite and cementite have moduli on the same order for the high symmetry loading directions considered in this work, there was no significant development of internal stress patterns in the simulations performed. However, this may not be the case for all loading directions, as cementite is known to have significant elastic anistropy [68], with a C55 value which is very low. 3.3. Ferrite plasticity After the elastic phase of deformation, the ferrite yields first, with dislocations forming at the interface. In general, the activated slip systems within the single phase simulations were consistent with those that activated within the pearlitic microstructure simulations. However, there were several deviations between these simulations that are attributed the microstructure, specifically interfacial effects. For both brevity and clarity, in the following discussion of the response of each OR, the loading directions will be described simply by the crystallographic direction in the cementite. The associated direction in the ferrite can be found in Table 1. 3.4. Bagaryatskii OR For all transverse loading conditions considered in the Bagaryatskii OR, the 12h111if110ga family of slip systems is activated in compression, while the 12h111if112ga is activated in tension (Table 2). This is despite the 12h111if112ga slip systems having the higher Schmid factor for both loading directions. The tensioncompression asymmetry in the family of the activated slip system is consistent with recent work performed by Shimokawa et al. on the Bagaryatskii OR [69], in which this same behavior was observed in the ½100q loading, and found to be a result of the compressive state lowering the critical resolved shear stress. This behavior arises from the emission of dislocation into the ferrite from the interfacial dislocations. If the Burgers vector of the activated slip system is assumed to be equal and opposite in compression and tension, the resultant interfacial Burgers vector after emission can be given by the following: In both cases, the magnitude of the resultant interfacial Burgers vector is smaller in the compressive state, making emission of this type of dislocation more favorable. This, in conjunction with the f110ga slip plane family requiring a lower resolved shear stress than the f112ga , shown both experimentally [70] and in the single phase simulations performed for this work, is enough to cause the f110ga to activate in compression despite its lower Schmid factor. The activation of different slip systems at an interface for tensile and compressive loadings has been observed in other lamellar structures [71] as well. Longitudinal loading of the Bagaryatskii OR again finds that different slip systems activate in ferrite under tension and compression. Under tensile longitudinal loading, the 12 ½111ð101Þa slip system, which has the highest Schmid factor (0.408) among common slip planes in this loading, activates first. However, in compressive longitudinal loading, the 12 ½111ð112Þa slip system activates despite a lower Schmid factor (0.314). This tensioncompression asymmetry has a similar impetus as that which was observed in the transverse loading, except in this case it is the dislocation emitted in tension which would decrease the resultant magnitude of the interfacial dislocation's Burgers vector. It is also worth noting that the activated 12 ½111ð112Þa slip system's line of intersection with the interface aligns perfectly with one of the existing interfacial dislocations (q0 ¼1). This likely serves as a

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Table 2 RSS Slip systems and the resolved shear stresses (GPa) for peak stress (tRSS y ) and plastic flow (tf ) in ferrite and cementite for various transverse loading states of the Bagaryatskii OR. Values in parenthesis represent the Schmid factor for the slip system. ½100q loading :

½010q loading :




 1
   1
1
110 a ± 111 a ¼ 331 a ðTÞ; 111 a ðCÞ 2 2 2

 1 1
½111a ± 111 a ¼ ½011a ðTÞ; ½100a ðCÞ 2 2

½100q

½010q

Tension

Compression

Tension

Compression

Ferrite

½111ð112Þ(0.471)

½111ð101Þ(0.408)

½111ð112Þ(0.314)

½111ð101Þ(0.272)

Cementite

tRSS y

b0 p0 Ferrite

½111ð110Þ(0.359) 0.90 0.99 9.71

½111ð110Þ(0.359) 0.90 0.96 7.50

½111ð011Þ(0.427) 0.83 1.00 11.10

½111ð110Þ(0.359) 0.90 0.96 7.87

tRSS f

Cementite Ferrite

9.00 3.87

10.54 4.80

10.88 2.80

13.33 3.32

Cementite

7.54

5.71

3.63

9.13

Slip System Slip Transfer

Fig. 2. Comparison of stress-strain curves for the Bagaryatskii and Near Bagaryatskii ORs for a 7:1 ferrite to cementite ratio with cementite lamella thickness of 4 nm. Response is seen to be similar in compressive, transverse loading due to the activation of the same slip systems in both ORs, while other loadings differ due to the different slip systems being activated.

nucleation site, decreasing the required resolved shear stress, and allows for this system to activate despite its lower Schmid factor. 3.4.1. Near Bagaryatskii OR The near ORs provide the opportunity to evaluate how small changes in orientation affect the mechanical response of the system. Fig. 2 compares stress-strain curves for Bagaryatskii and the Near Bagaratskii simulations containing a 7:1 ferrite to cementite volume ratio. The curves show that there are differences in the mechanical response for several of the loading states, despite the ORs only varying by a few degrees. These difference are minor in transverse compression, the Near Bagaryatskii is observed to have nearly identical elastic moduli and similar peak stresses as its parent OR. When the resolved shear stresses are considered

(Table 3), the differences between the two ORs is reduced further. These minor deviations can be attributed to the different dislocation structures at the interface, which will result in different localized stress states. Once the ferrite begins to nucleate dislocations, the activated slip systems during transverse compressive loading of Near Bagaryatskii are the same as those activated in the Bagaryatskii, resulting in a very similar stress-strain response. Differences in mechanical response for the Near Bagaryatskii and its parent OR are more pronounced for transverse tensile loadings (Fig. 2). The elastic responses are roughly the same, which is expected due to the similar orientations. However, for both the ½100q and ½010q loadings there are significant variations with regards to the ferrite plasticity. This deviation can be explained through the activation of different slip system families in the two

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Table 3 RSS Slip systems and the resolved shear stresses (GPa) for peak stress (tRSS y ) and plastic flow (tf ) in ferrite and cementite for various transverse loading states of the Near Bagaryatskii OR. Values in parenthesis represent the Schmid factor for the slip system. ½100q

½010q

Tension

Tension

Tension

Compression

Ferrite

½111ð110Þ(0.439)

½111ð101Þ(0.405)

½111ð101Þ(0.272)

½111ð101Þ(0.272)

Cementite

tRSS y

b0 p0 Ferrite

½111ð101Þ(0.393) 0.92 1.00 6.64

½111ð110Þ(0.359) 0.86 0.96 8.04

½111ð110Þ(0.359) 0.86 0.96 14.22

½111ð110Þ(0.359) 0.86 0.96 9.74

tRSS f

Cementite Ferrite

5.84 4.40

9.40 4.80

14.31 3.73

11.25 3.24

Cementite

4.86

6.18

9.17

4.82

Slip System Slip Transfer

ORs. In the Bagaryatskii, f112ga type slip planes were activated, while in the Near Bagaryatskii they are of the f110ga type. For the ½100q tensile loading of the Near Bagaryatskii, slip occurrs on the 1 ½111ð110Þ slip system. Due to the rotation of the ferrite relative a 2 to loading in this OR, the Schmid factor of this slip system is 7.6% larger than the same slip system in the Bagaryatskii. Additionally, the line of intersection of the activated ferrite slip system in the Near Bagaryatskii aligns perfectly with an interfacial dislocation (q0 ¼1), further reducing the necessary resolved shear stress and resulting in the nucleation of this slip system instead of the 1 ½111ð112Þ which occurred in the Bagaryatskii. The larger Schmid a 2 factor indicates that the Near Bagaryatskii would yield earlier, which is shown to be true in Fig. 2. For the ½010q tensile loading, plasticity in the Near Bagaryatskii occurs on the same 12 ½111ð101Þa slip system that was observed in compression. This differs from its parent OR's behavior, which activated the higher Schmid factor 1 ½111ð112Þ system under tension. Here the rotation of ferrite a 2 relative to the Bagaryatskii OR alters the Bagaryatskii's activated slip system's line of intersection from perfectly aligned with an interfacial dislocation (q0 ¼1) as it was in the parent OR, to varying from the line directions of the Near Bagaryatskii, which has a different interfacial dislocation structure (Table 1), by angles of 37 and 89 (q0 values of 0.79 and 0.02, respectively). This creates a much less favorable nucleation site for the 12 ½111ð112Þa , and results in the activation of a different slip system. There are also significant differences between the Bagaryatskii and Near Bagaryatskii during longitudinal tensile loading. This variation appears to be a result of different interfacial disclocation character/structure between the two ORs. In the Near Bagayatskii, one of the interfacial dislocations, the 12 ½111a , has a large out of interface plane component (Table 1). As the preferred state of these dislocations is within the interface plane, this dislocation is less stable than the interfacial dislocations within the Bagaryatskii, a state which is exasperated by the tensile state normal to the interface. As result, a different slip system is activated than in the Bagaryatskii, with a 12 ½111a dislocation being emitted into the ferrite in order to reduce its energy. This results in the following transformation if the interfacial dislocation:

 
 1
1
111 a  111 a ¼ 011 a 2 2 The resultant Burgers vectors lies much closer to the interface plane, and is identical in character to the Burgers vector that was observed in the Bagaryatskii. The ð211Þa slip plane that contains the emitted dislocation has a line of intersection with the interface that closely corresponds to the line direction of this dislocation, differing by less than 3 (q0 ¼0.998). This slip system, shown in Fig. 3a, has an extremely high Schmid factor of 0.487. During longitudinal compressive loading, the instability of the out of interface

plane dislocation of the Near Bagaryatskii OR appears to be somewhat muted, and as a result, the activated slip system differs from that which occurs in tension. Slip occurs on the ½111ð110Þa system with a Schmid factor of 0.439. This Schmid factor is larger than the that of the Bagaryatskii, and as such yield occurs at a lower stress. 3.4.2. Pitsch-Petch and Near Pitsch-Petch ORs While there was significant deviation between the Near Bagaryatskii and its parent OR, the other near OR, the Near Pitsch-Petch has a nearly identical response to that of the Pitsch-Petch (Fig. 4). This similarity is a result of identical slip systems being activated in both ORs for all loading states. As the peak stress values are found to vary by less than 5%, all values presented in this section will be with regards to the Pitsch-Petch OR (Table 4). However, the Schmid factors for both ORs (Table 5) will be presented, and there will be additional discussion on why the Near Pitsch-Petch behaves so similarly to its parent OR for all loading states, while the Bagaryatskii and Near Bagaryatskii behave differently in certain cases. In all transverse loadings of the Pitsch-Petch, dislocations are emitted into the ferrite (Fig. 5a) in the slip system with the highest Schmid factor, which is consistent with the simulations performed on the individual lamella. In compression, this occurs through the emission of dislocations having 12 ½111a Burgers vectors. These dislocations are emitted from the same interfacial dislocations in both ½100q and ½010q loading due to the nearly identical relative orientation of the interfacial dislocation (approximately 45 ) to both loading directions. This emission lowers the energy of a nonideal 12 ½113a dislocation to a more favorable ½011a :

  1
1
113 a  111 a ¼ ½011a 2 2 For transverse, tensile loading, plasticity in ferrite initiates from simultaneous emission of dislocations (Fig. 5b) from the same interfacial dislocations as in compression, with both loading directions resulting in the activation of the ½111ð011Þ slip system, with this emission reducing the 12 ½113a interfacial dislocation to ½001a . The exact transformation is found to be:

  1
1
113 a  111 a ¼ ½001a 2 2 For tensile loading in the longitudinal direction, dislocations are of the 12 ½111ð211Þa character (Schmid factor 0.346) which has a line of intersection that closely corresponds to the line direction of the interfacial dislocation. This dislocation emission reaction gives a ½101a resultant interfacial Burgers vector, again reducing the magnitude of the interfacial dislocation:

  1
1
113 a  111 a ¼ ½101a 2 2

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Fig. 3. a) Dislocation emission originating at the interfacial dislocations of the Near Bagaryatskii OR b) Formation of a void when dislocation reaches the opposite interface. Dislocations are highlighted using the atomic shear strain.

Fig. 4. Comparison of stress-strain curves for the Pitsch-Petch and Near Pitsch-Petch ORs for a 7:1 ferrite to cementite ratio with cementite lamella thickness of 4 nm. The stressstrain response is seen to be similar in all loadings due to the activation of the same slip systems in both ORs.

Table 4 RSS Slip systems and the resolved shear stresses (GPa) for peak stress (tRSS y ) and plastic flow (tf ) in ferrite and cementite for various transverse loading states of the Pitsch-Petch OR. Values in parenthesis represent the Schmid factor for the slip system. ½100q

½010q

Tension

Compression

Tension

Compression

Ferrite

½111ð011Þ(0.422)

½111ð110Þ(0.458)

½111ð011Þ(0.395)

½111ð011Þ(0.457)

Cementite

tRSS y

b0 p0 Ferrite

½111ð110Þ(0.359) 0.84 0.97 5.72

½111ð110Þ(0.359) 0.99 0.96 11.05

½111ð110Þ(0.359) 0.84 0.97 6.20

½111ð011Þ(0.427) 0.99 0.91 9.88

tRSS f

Cementite Ferrite

8.17 2.34

9.04 2.87

7.15 3.82

9.02 2.80

Cementite

5.78

7.38

5.77

7.81

Slip System Slip Transfer

The near identical response of the Pitsch-Petch and Near PitschPetch appears to primarily arise from the specific rotation of the ferrite lattice that is the difference between the two ORs. This rotation is only 2.6 about an axis that is normal to interface plane, resulting in very similar Schmid factors, a well as the creation of interfacial dislocations that are much more stable than those

observed in the Near Bagaryataskii. As a result, the same slip systems are activated for all loading states of Near Pitsch-Petch and its parent OR. It should be noted though that the common response of the Pitsch-Petch and Near Pitsch-Petch likely would not occur for all loading states. Due to the loading directions chosen both having an approximate 45 angle to one of the dislocation sets that is

M. Guziewski et al. / Acta Materialia 180 (2019) 287e300 Table 5 Slip transfer metrics for the Near Pitsch-Petch OR. For all loading states, the same slip systems are activated in the Near Pitsch-Petch and the Pitsch-Petch ORs (Table 4). ½100q

ma b0 p0

½010q

½001q

Tension

Compression

Tension

Compression

Tension

Compression

0.397 0.83 0.96

0.470 0.99 0.97

0.371 0.83 0.96

0.445 0.99 0.92

0.350 e e

0.434 0.93 0.99

295

different orientation in these two ORs, and thus the cementite component of these interfacial dislocations is different. This, along with different interfacial dislocation spacing and a different internal stress state due to different elastic behavior, would account for the observed variation in peak stresses between the Bagaryatskii and Isaichev ORs. This does however highlight the effect that the both the interface and the cementite lamella has on the ferrite response. 3.4.4. Important factors in ferrite dislocation formation Three notable factors in the nucleation of ferrite dislocations from the interface can be discerned after comparing the response of all ORs. These are:

Fig. 5. Plasticity mechanisms in the ferrite lamella of the Pitsch-Petch OR during transverse loading showing a) the formation of dislocation loop from the interface under compressive loading, and b) the simultaneous emission of dislocations from the interfacial dislocations.

present in both the Pitsch-Petch ORs, this shared dislocation set was primary source of plasticity for all the considered loadings of both ORs, and thus the similar response. It is likely that specific loading states may have plasticity that instead begins on the nonshared dislocation set in the two ORs, in which case it would be expected that the behavior would diverge to some degree. Considering the character of these interfacial dislocations, a h113〉a in the Pitsch-Petch, and an out of plane component to the Burgers vector in the Near Pitsch-Petch, that the emission of a dislocation to reduce the energy of the interfacial dislocations would likely be the mechanism for plasticity. 3.4.3. Isaichev OR Within the Isaichev OR, the orientation of the ferrite relative to the loading is identical to that of the Bagaryatskii OR, and the interfacial dislocations have identical Burgers vectors within the ferrite lattice; thus the same slip systems are observed to nucleate in the ferrite for the two ORs (Table 6) for all loadings. There are noticeable differences between the peak stresses of the two ORs however. This is likely due to differing local stress states at the interface for the two ORs. While the character of the interfacial dislocations within the ferrite is the same, the cementite has a

1. Schmid factor of the slip system, not surprisingly, appears to be the more important factor in the nucleation of ferrite dislocations, as in most cases it is the slip system with the highest Schmid factor that is activated. However there are several cases in which this is not true, suggesting that the next two factors also play a role. 2. Reduction in the magnitude of non-ideal interfacial dislocations. In most of the cases in which the activated slip system did not have the highest Schmid factor, the dislocation was emitted from a less than favorable interfacial dislocation, such as a 1 ½113 or one with an out of plane component. As this emission a 2 reduces the energy of the interfacial dislocation, the energetic cost to nucleate it is lower than it would otherwise be, thus allowing it to be activated before higher Schmid factor dislocations. 3. The slip system aligns well with an interfacial dislocation. This appears to be the least important factor, as there are many cases of aligned slip systems not activating. Nonetheless, as there are multiple instances of this being true when lower Schmid factor slip systems are activated, it does likely play a role in making the nucleation of the dislocation easier. These factors allow for a prediction of the slip system that would be activated in ferrite. Since the ferrite slip systems are the major factor in determining which cementite slip system will be activate, which will be shown in the next section, these insights can potentially be very useful in understanding how pearlite deforms and fails. 3.5. Cementite plasticity and failure While it might be expected that the ductile ferrite would continue to deform until all slip systems were exhausted, this is not the case as the still elastically deforming cementite limits the rate at which the ferrite can deform. This creates an internal stress state

Table 6 RSS Slip systems and the resolved shear stresses (GPa) for peak stress (tRSS y ) and plastic flow (tf ) in ferrite and cementite for various transverse loading states of the Isaichev OR. Values in parenthesis represent the Schmid factor for the slip system. ½010q

½101q Tension

Compression

Tension

Compression

½111ð101Þ(0.408) ½100ð001Þ(0.468)

½111ð112Þ(0.314)

½111ð101Þ(0.272)

Cementite

½111ð112Þ(0.471) ½001ð100Þ(0.468)

tRSS y

b0 p0 Ferrite

0.94 0.77 9.71

0.85 0.52 7.34

½111ð011Þ(0.427) 0.92 0.84 7.82

½111ð011Þ(0.427) 0.86 1.00 12.49

tRSS f

Cementite Ferrite

8.15 4.02

10.33 5.29

8.51 3.07

14.73 3.82

Cementite

7.37

7.58

4.56

9.73

Slip System Slip Transfer

Ferrite

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that causes the nucleation of dislocations into the cementite at the interface within the pearlite. Whereas ferrite dislocations were found nucleate from the interfacial dislocations, the creation of the ferrite dislocations destroys the pristine nature of the interface, and thus the interfacial dislocations were found to no longer be the preferred nucleation point within the cementite. Instead, it is the ferrite dislocations themselves that are nucleation sites for cementite dislocations, and therefore the slip metric values discussed in Section 2.2 become very important. All plasticity in the cementite was found to occur within slip systems with some combination high Schmid factors (m) and good alignment between the Burgers vectors (b0 ) and line of intersections (p0 ) of the ferrite and cementite slip system. The Supplemental material lists the cementite slip system with the highest value for each of these metrics, as we well as the slip system that was actually activated. The relative importance of each slip transfer metric will be discussed in greater detail in Section 3.5.5 after the results of the various ORs are presented. Due to the periodic nature of the simulation cells, it is not possible to observe the failure mechanisms for many of the loadings considered. However in longitudinal tensile loading, it is possible to observe this failure, generally through delamination. Additionally, the volume dependence of the peak and flow stresses during longitudinal loading, which is discussed in length in Ref. [30], will be shown, and the fitting parameters from Equations (9) and (10) given. 3.5.1. Bagaryatskii OR For transverse loadings of the Bagaryatskii OR, dislocations in cementite were found to have the 12h111if110gq or 12h111if011gq character (Table 2). However, the specific slip system that was activated did not always occur in the system with the largest Schmid factor, highlighting the fact that there are other relevant factors with regards to cementite plasticity. The activated cementite slip systems did however have either the highest b0 or p0 slip transfer metric values in the given loading of all considered slip systems, suggesting that the alignment of the slip system with the ferrite slip system is certainly a factor, and likely an important one. For example, in ½010q compressive loading, the ½111ð110Þq slip system with a Schmid factor of 0.359 is activated over the ½111ð011Þq with a Schmid factor of 0.427 due to a larger p0 value (0.96 vs 0.85). In longitudinal loading, the cementite was found to slip within the ½111ð011Þq system (Schmid factor of 0.427) under compression and the ½111ð101Þq system (Schmid factor of 0.393) under tension, where failure occurred shortly after cementite plasticity due to delamination. Both these cementite slip systems had the highest b0 and p0 values relative to activated ferrite system, which were 0.90 and 1.00, respectively in tension and 0.83 and 1.00, respectively under compression. The peak and flow stresses under longitudinal loading of the Bagaryatskii OR were found to vary with the volume ratio of ferrite to cementite (Fig. 7a) with fitting parameters given in Table 7. 3.5.2. Near Bagaryatskii OR In examining the transverse response of the Near Bagaryatskii, it was observed that during ferrite plasticity the compressive

response mirrored that of the Bagaryatskii due to the activation of the same slip systems in the two ORs. This trend continues during cementite plasticity, with the same cementite slip systems being activated. Not surprisingly, values for m, b0 , and p0 (Table 3) are also found to be similar between the two ORs, with the activated cementite slip system again having high values of all three metrics. The same is found to be the case for the tensile transverse loading where the activated slip system in the cementite is a 12h111if110gq or 12h111if011gq character slip system that has large m, b0 , and p0 values. Perhaps the most glaring difference between the Bagaryatskii and Near Bagaryatskii is the lack of a ductile phase during under longitudinal tensile loading. In this loading there is actually no cementite plasticity, as once the emitted ferrite dislocation completely traverses the ferrite lammella, the cementite interface prevents further slip causing an increased localized stress state that creates a void and begins delamination of the two lamella (Fig. 3b). The creation of a void over further plasticity appears to be related to the stability of the interface between the ferrite and cementite, and will be discussed in greater detail in the following section. During longitudinal compression, the stress state appears to prevent the formation of a void and thus cementite plasticity is again present, occurring within the ½111ð011Þq slip system, Schmid factor 0.426, which has b0 and p0 values of 0.92 and 0.86, respectively. Fig. 7 shows that there is again a volume ratio dependence, and fitting parameters for both the tensile and compressive longitudinal loading can be found in Table 7. 3.5.3. Pitsch-Petch and Near Pitsch-Petch ORs The Pitsch-Petch and the Near Pitsch-Petch were seen to have nearly identical stress-strain response, with the same slip system being activated in the ferrite, and the same is true in cementite (Table 4). The rotation of the Near Pitsch-Petch does cause a slight change in the b0 and p0 values, as seen in Table 5. For all transverse loadings, it is again observed that the activated cementite slip has high m, b0 , and p0 values and, including the largest value of one of at least one of the metrics, and is of the 12h111if110gq or 12h111if011gq character. The importance of the p0 value is once again observed, in this case during ½100q tensile loading where between two slip systems with identical b0 values, it was the slip system with the higher p0 value and not Schmid factor that was activated. Much like the Near Bagaryatskii, the Pitsch-Petch and Near Pitsch-Petch are found to have no cementite plasticity during tensile longitudinal loading, and fail due to delamination. It is notable that there is significantly more ductility before failure in the Pitsch-Petch and Near Pitsch-Petch ORs than in the Near Bagaryatskii. This is likely due to the interface dislocation structures of the ORs, and their ablility to absorb the emitted dislocations. The authors’ past work found the Near Bagaryatskii to have an interfacial dislocation density that was more than three times that of the Pitsch-Petch (Table 8). As a result, the Pitsch-Petch ORs can absorb significantly more dislocations before pile-up occurs and void formation begins. As the Pitsch-Petch has a lower dislocation density than the Near Pitsch-Petch, this would suggest that delamination should occur first in the Near Pitsch-Petch. Inspection of Fig. 4 shows that this is indeed the case.

Table 7 Calculated constants (in GPa) for the volume ratio dependent flow and yield stresses (Eqs. (9) and (10)) of the all ORs during longitudinal loading. Values were determined using a least squares fit of simulation data.

Bagaryatskii Near Bagaryatskii Pitsch-Petch/Near Pitsch-Petch Isaichev

s0;t

s0;c

K0;t

K0;c

s1;t

s1;c

K1;t

K1;c

19.15 19.74 14.13 18.50

28.06 16.41 21.58 14.52

2.29 1.47 2.62 2.71

3.32 6.56 1.66 5.32

8.58 e 10.14 7.30

8.58 11.35 7.43 12.58

7.57 e 2.75 4.06

7.57 5.64 5.54 2.53

M. Guziewski et al. / Acta Materialia 180 (2019) 287e300

297

Table 8 Interfacial dislocation density (  10-3 Å-2) and inelastic strain after yield during longitudinal, tensile loading for each of the pearlite ORs.

Dislocation Density Inelastic Strain

Isaichev

Bagaryatskii

Pitsch-Petch

Near Pitsch-Petch

Near Bagaryatskii

0.3 > 20%

0.9 14%

1.1 11%

2.2 9%

4.2 4%

It is also again found that in both tension and compression the response is function of volume ratio (Fig. 8) and well described by Equations 9 and 10. 3.5.4. Isaichev OR Unlike the Bagaryatskii, the Pitsch-Petch, and their associated near ORs in which the ð001Þq is the habit plane in the cementite, the Isaichev OR's habit plane is the ð101Þq . This provides opportunity to evaluate the response of other slip planes within cementite as many of the f110gq , f101gq , and f011gq type slip planes observed in the other ORs have zero Schmid factors. The ferrite plastic response of the Isaichev was found to match that of the Bagaryatskii since this lamella had identical orientation in the two ORs, however in the cementite there was varation betweem the two. Plasticity continues to follow observed trends in ½010q loading, with slip occurring in slip systems with high m, b0 , and p0 values, however the activated slip systems for ½101q loading appears to be entirely determined by the Schmid factor, as the activated slip systems in this loading have b0 and p0 values as low as 0.85 and 0.52, respectively. This change is likely due to slip systems activated, the h001if100gq and h100if001gq , which had Schmid factors of zero in the other ORs for the loading considered. These slip systems have been observed experimentally [67], and have been found to have among the lowest stacking fault energies of the cementite slip systems [72]. The low stacking fault energy makes these slip systems much easier to form, and therefore slip transfer mechanisms are not required to nucleate this type of dislication. The Isaichev is also unique in that cementite plasticity was not observed within the Isaichev OR for either tensile or compressive loading (Fig. 6). Additional simulations were run to a greater strain state, and failure had still not occurred at 40% strain. As failure occurred significantly earlier in all other ORs, this suggests qualitatively that there is something unique about the Isaichev. Returning to the density of interfacial dislocations, from Table 8 it can be observed that the Isaichev has a much lower dislocation density than all other ORs. Thus following previously observed trends, it is likely that this interface is able to accommodate more dislocations and thus allowing higher ductility in the ferrite. Additionally, atomistic simulations found the Isaichev to have the lowest interfacial energy of all ORs, further supporting the idea that this interface is particularly stable. It therefore seems likely that this stable

interface allows the system to deform primarlily in the ferrite without any of the inelastic behavior in the cementite that precedes failure. The cementite still acts as a constraint on the ferrite, as the peak and flow stresses of the Isaichev are found to have a volume dependence (Fig. 9). 3.5.5. Relative importance of slip transfer metrics As the slip metrics presented in Section 2.2 are all functions of the orientation of the cementite relative to the activated ferrite slip system and the loading direction, it becomes possible to consider these values for slip systems in the cementite that were not activated. Within the supplemental section of this work, the slip systems with the highest m, b0 , and p0 , as well as any other relevant slip systems, for each loading state is presented. From this data several trends emerge which potentially give insight into which slip systems will activate or not activate within the cementite. The clearest trends were as follows: 1. Schmid factor is not the only factor in determining the slip system that activates. For nearly every loading there were multiple slip systems with larger Schmid factors, but lower b0 and p0 values, than the system that was activated. This highlights that it is not just the resolved shear stress that is determining the plastic behavior in the cementite and the importance of considering other factors such as slip transfer. 2. If a slip system has the largest m, b0 , and p0 value of all potential slip systems, it will be the system that activates. This further shows the validity of applying slip transfer metrics to this system, as in all cases where a single slip system had the largest value of all three metrics, it was the one that was activated. 3. The activated slip system generally had the largest b0 value of all potential slip systems. 4. The p0 value in general appears to be slightly more important than the Schmid factor. The last two trends were in fact found to be true for all considered loading states outside of the ½101q loading, which was the only loading where h001if100gq and h100if001gq type slip systems had non-zero Schmid factors. This further highlights what was mentioned in the discussion of the Isaichev results, that the b0 and p0 metrics reflect cementite slip systems that have favorable nucleation sites. For slip

Fig. 6. Stress-strain response of the Isaichev in compression (blue) and tension (red) for deformation in the a) ½101q direction, b) ½010q direction, c) ½101q direction for a 7:1 ferrite to cementite ratio with cementite lamella thickness of 4 nm. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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Fig. 7. Yield and flow stress as a function of volume ratio for longitudinal loading of the a) Bagaryatskii and b) Near Bagaryatskii ORs.

Fig. 8. a) Yield and b) flow stress as a function of volume ratio for compressive and tensile longitudinal loading of the Pitsch-Petch OR.

These trends actually lead to a relatively simple expression for predicting which slip system in the cementite will be activated for a given ferrite slip system when no h001if100gq or h100if001gq type are available, mþb0 þp0 . Outside of the ½101q loading, the slip system with largest value from this expression was in fact the activated slip system in the cementite. The relevance of these metrics are also consistent with experimental work by Zhang et al. [52], which found the slip and failure planes of cementite within pearlite to be align well with the slip systems within the ferrite. As these slip systems often precede failure in the cementite, this is potentially valuable information to understand the brittle response of the cementite and possibly other ceramics that have interfaces with metals.

4. Summary

Fig. 9. Yield and flow stress as a function of volume ratio for longitudinal loading of the Isaichev OR.

systems with higher stacking fault energy, this is the most important factor, while for those with lower stacking fault energy, the h001if100gq and h100if001gq , it is the resolved shear stress that is the dominant factor.

This work surveys the deformation mechanisms in astransformed pearlite at a temperature of 5 K through comparison of the response of multiple ORs. Several notable trends emerged, which can lend better understanding to this system. These include: 1. Plastic deformation occurs first in the ferrite. This generally favoring the 2ah111if110g or 2ah111if112g slip system with the largest Schmid factor. 2. Interfacial dislocation structure does sometimes play a role in the activated ferrite slip systems, as the emission of dislocations

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into the ferrite can reduce the energy of the interfacial dislocations and result in slip systems with lower Schmid factors being activated. 3. The deformation mechanisms in the cementite is related to the response in the ferrite, and is well described through slip transfer metrics, and the expression mþb0 þp0 , when h001if100gq or h100if001gq type slip systems are unavailable. When h001if100gq or h100if001gq type are available, it is the slip system with the largest Schmid factor. 4. The interfacial dislocation density can affect failure mechanisms, as there appears to be an inverse relation between this value and the ductility of the system in longitudinal loading. One important aspect of these observations is that many trends can be extended by simple consideration of the interface geometry. Interfacial dislocation structure can be determined through consideration of the ferrite and cementite lattices at the interface, which is a function of the OR. Similarly, the Schmid factors and other slip transfer metrics, are determined by loading direction and orientation of the crystals. This allows for a procedure to infer possible deformation mechanisms in pearlite for alternative loadings and ORs prior to atomistic or experimental observation. For ferrite plasticity, determination of the 2ah111if110g and a 2h111if112g slip systems with the highest Schmid factors for the direction of loading would likely be sufficient for a reasonable guess as to what slip system will activate without the need to perform simulations. Considering the interfacial dislocation structure, would allow for further refinement. Then for the given ferrite slip system, slip transfer metric could be calculated for the given loading and OR. The cementite slip system with the largest mþb0 þp0 would then likely be the one that is activated. It is worth noting again that these rules are for pristine samples and that specific slip systems activated in the simulations performed here may not occur in all systems, as things such as defects and alloying content will certainly influence the manner in which the system deforms. Additionally, as the performed simulations only consider perfect crystals, properties such as yield stress, flow stress, and ductility are much higher than observed values experimentally. However, the trends observed here are extremely relevant, and in conjunction with empirical data, could allow for simple models to be constructed for plasticity in the pearlitic microstructure. Acknowledgment Much of this research was supported though a grant from the Petroleum Research Fund, PRF #54697-DNI10. Portions of this research were also sponsored by the Army Research Laboratory and accomplished under Cooperative Agreement Number W911NF-162-0008. This work was supported in part by computer time from the DOD HPCMP at the ARL DOD Supercomputing Resource Center (DSRC), the Navy DSRC, the US Army Corps of Engineers Research and Development Center DSRC, and the US Air Force Research Laboratory DSRC. This work also 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. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.actamat.2019.09.013. References [1] S.R. Phillpot, D. Wolf, S. Yip, Effects of atomic-level disorder at solid interfaces,

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