A slip system-based kinematic hardening model application to in situ neutron diffraction of cyclic deformation of austenitic stainless steel

International Journal of Fatigue 36 (2012) 181–193

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

A slip system-based kinematic hardening model application to in situ neutron diffraction of cyclic deformation of austenitic stainless steel J.A. Wollmershauser a,⇑, B. Clausen b, S.R. Agnew a a b

Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904-4745, USA Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

a r t i c l e

i n f o

Article history: Received 19 November 2010 Received in revised form 13 July 2011 Accepted 15 July 2011 Available online 27 July 2011 Keywords: Bauschinger Neutron diffraction Kinematic Hardening Backstress

a b s t r a c t Accurate prediction of the Bauschinger effect is considered a litmus test for the validity of strengthening theories. The effect is known to arise from ‘backstresses’ having intergranular and intragranular sources. Polycrystal plasticity models inherently capture intergranular effects but typically neglect intragranular (dislocation-based) sources of backstress. The negative impact of this omission is made apparent by comparisons of model predictions with in situ neutron diffraction measurements of the hystereses and internal stresses within a sample subjected to fully-reversed tension–compression cyclic deformation. An elasto-plastic self-consistent (EPSC) model is modified to include a Voce-type non-linear kinematic hardening rule, similar to the phenomenological Armstrong–Frederick–Chaboche model, but implemented at the slip system level. This additional physically-based hardening evolution enables the polycrystal model to account for hardening due to reversible, geometrically necessary dislocation structures, such as pileups, as well as the more isotropic hardening effect due to forest dislocations. The model accurately predicts the macroscopic hysteresis loops and internal strains observed during the aforementioned in situ low cycle fatigue tests. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The Bauschinger effect, first documented in the 1880s [1,2], describes a material response common to most crystalline metals. In the simplest case, the Bauschinger effect comprises a reduced compressive (or tensile) yield strength and a more extended elastoplastic transition, if a material has first been strained plastically in the tensile (or compressive) direction. This simple description may be generalized to include the tendency of some metals to develop anisotropy during proportional and non-proportional loading. This type of flow anisotropy is important when modeling any metal forming processes which impose non-proportional strain paths, particularly for the prediction of spring-back [3]. The significance of the Bauschinger effect in cyclic loading is also well documented. Studies show that the shape of the hystereses loops [4] and even fatigue life predictions [5] are strongly influenced by the effect. The early literature provides two overarching mechanistic concepts to account for a generalized Bauschinger effect in metals [6]. Pre-dislocation theory explanations [7] identified internal stresses and macroscopic residual stresses developed due to inhomogeneous deformation of individual grains (i.e., type II intergranular ⇑ Corresponding author. E-mail address: [email protected] (J.A. Wollmershauser). 0142-1123/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2011.07.008

strains). The intergranular stresses built up between grains having different crystallographic orientations (or phases with differing strengths and/or stiffnesses) ‘assist’ reverse deformation during strain path changes. It was later observed that single crystals also demonstrate a Bauschinger effect [8,9]. Without backstresses which develop from the interaction between ‘hard’ and ‘soft’ grains, Bauschinger effects in single crystals must be explained in terms of intragranular mechanisms. As a result, dislocation-based theories were proposed (e.g., [10,11]). The Bauschinger effect was, therefore, suggested to originate from lowered resistance to dislocation motion in the reverse direction than the previous forward direction. However, it is pointed out that the two sources of backstress do not contradict one another and it has been acknowledged that both intergranular and intragranular effects give rise to the backstresses that generate the Bauschinger effect within polycrystals [6,12]. Recent theories more explicitly address the heterogeneous nature of the backstress and decompose contributions to the backstress into a multi-scale and multi-phenomena problem that is the result of a hierarchical behavior of a range of dislocation events, such as trapping and escape [13–16]. In these theories, intergranular backstresses are thought to arise from grain orientation dependent sources (as described above). Intragranular backstresses are theorized to arise as a result of differential resistance to slip in the forward and reverse directions caused by pile ups

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(or other geometrically necessary dislocation configurations) and cell structures. Using a composite model that describes dislocation cell structures as areas of hard (wall) and soft (interior) regions, Mughrabi [13] and related works [17,18] have well captured the Bauschinger effect when specifically addressing the influence of a type of heterogeneous dislocation structure on the intragranular backstress associated with high levels of deformation and/or fatigue (excluding the initial stages of fatigue). However, other models, such as the phenomenological, take a more simplified approach that is not assigned to such specific dislocation structures and show good correlation with experiments over a range of accumulated strain (i.e., dislocation configurations). The phenomenological models that have been developed to quantitatively predict Bauschinger effects frequently utilize nonlinear kinematic hardening rules [19]. Initially proposed by Armstrong and Frederick (A–F) [20], and developed for visco-plasticity and multi-axial states of stress by Chaboche [21], non-linear hardening models relate macroscopic stress to macroscopic strain. The uniqueness of non-linear kinematic hardening models was first demonstrated by McDowell [14], where it was shown that these formulations provide better accounting for both the hardening modulus and direction of yield surface translation during nonproportional cyclic hardening. These phenomenological modeling approaches offer computational efficiency and ease of implementation within commercial finite element codes [22,23]. The downside is that they apply best to a specific initial material state and to the specific loading conditions used in the model fitting process. Additionally, such models typically do not directly account for the mechanistic sources of backstress or even distinguish between intergranular and intragranular contributions. Polycrystal plasticity models provide a simple approach that allows consideration of sources of backstress. Though the various polycrystal plasticity models make use of a variety of homogenization schemes to relate the constitutive response of individual grains to the behavior of the polycrystalline aggregate, these relations can inherently account for intergranular backstresses. Additionally, these models typically account for both anisotropic elasticity and plasticity (or viscoplasticity) where the behavior of the grains is based upon the fundamental crystallographic mechanisms of plastic deformation (such as dislocation slip and mechanical twinning). As a consequence, these models are not constrained to an initial material state (such as a particular crystallographic texture) or to a specific deformation path (since they readily incorporate the evolution of crystallographic texture) and are much more predictive and robust than macroscopic plasticity [24]. Tóth et al. [25] predicted a macroscopic Bauschinger effect when modeling tension–compression cycles for an elasto-viscoplastic Taylor–Lin polycrystal, even for aggregates of fcc crystals, which are only mildly plastically anisotropic. However, the particular homogenization scheme employed will dictate the level of intergranular backstress which develops, and, therefore, can impact the prediction of the Bauschinger effect. Two commonly employed homogenization schemes, the Taylor–Lin and the Sachs models, bound the types of enforced interaction. The Taylor–Lin model assumes equal strain in all of the grains which is equal to that of the aggregate strain. The Sachs model assumes equal stress in all of the grains which is equal to that of the aggregate stress. As has been shown by Barton et al. [26], the Taylor–Lin assumptions leads to large intergranular strains while the Sachs model develops none. Various Finite Element (FE), relaxed constraints (RC), and self-consistent (SC) modeling schemes provide intermediate interaction assumptions. FE enforces capability between the grains while maintaining a weak sense of equilibrium, and SC models allow the stress and strain in individual grains to vary from one grain to another, while requiring that equilibrium and compatibility are enforced between

ellipsoidal idealizations of the grains and a surrounding aggregate which has the average properties of all the grains in the aggregate. Though both involve assumptions, FE models are typically thought to be more ‘realistic’, because specific grain-to-grain relationships (i.e., nearest neighbors) can be defined and because intragranular stress–strain variations can be explicitly incorporated, while SC models typically assume a uniform strain within individual grains (it is noted that more modern incarnations of SC models account for intragranular variations). However, FE models require long computation times, often employing parallel computing, to model truly realistic scenarios. Additionally, when specifically accounting for nearest neighbor effects in a particular RC model dubbed ALAMEL [27] Van Houtte et al. showed that, though they are important for capturing texture associated with large deformations, nearest neighbor interactions do not dominate the anisotropic effects associated with strain path changes [28]. Intragranular strain variations are, perhaps, a more significant issue which is intimately related to the phenomenon in question and will be discussed later in the paper. The elasto-plastic self-consistent (EPSC) model [29] is among the most widely applied crystal plasticity models, for the interpretation of in situ neutron diffraction data [29–34], a method to experimentally probe grain-level deformation behavior within bulk samples through measurements of internal strains. Notably, the experimental technique has recently been employed to investigate fatigue [35,36]. The high penetration depth of neutrons in most materials makes it possible to probe the internal structure and phase distribution within bulk samples, including conventional mechanical test sample geometries. With the use of timeof-flight detectors at a spallation neutron source, a complete diffraction pattern can be obtained quickly while the sample is under load. Changes in peak location are associated with internal elastic strain development, which has distinct evolutions during elastic and plastic loading. An advantage of coupling elasto-plastic self-consistent (EPSC) modeling [29] and in situ neutron diffraction is that the model and neutron experiment probe the polycrystal in similar ways allowing successful interpretation of the diffraction results. Neutron diffraction polls a number of grains with a common orientation in a variety of grain neighborhoods, yielding an ‘‘averaged’’ response, and the EPSC model allows determination of the average response of a similar subset of grains embedded within an ‘‘averaged’’ environment. However, as pointed out by Lorentzen et al. [35] and shown by Mulay et al. [37], the inherent intergranular stresses of the unmodified EPSC model fail to capture subtleties of the internal strain evolution in cyclic deformation experiments. The primary reason for the poor prediction during subsequent strain path directions is the models lack of incorporation of intragranular effects. The main issue when attempting to account for intragranular effects during forward-reverse loading is the treatment of slip when the sense changes direction. Lorentzen et al. [35] redefined the slip system hardening of an EPSC model by modifying the Voce hardening rule such that the critical stress to activate the ‘‘reverse’’ motion dislocations of slip systems decreased by the same amount that the stress required to continue forward motion increased. This simple ‘kinematic hardening’ approach did produce an increased Bauschinger effect, but the model predictions still deviated significantly from the experimental results in subsequent loadings. Similarly, Turkmen et al. [38] modified the slip system hardening of an elasto-viscoplastic model under the Taylor–Lin homogenization scheme. Though the purpose of this work was to capture pseudosaturation effects of fatigue, it was noted that the implemented critical strain increment to the Voce hardening rule could not capture the features of the elasto-plastic transition (i.e., the Bauschinger effect). Importantly, Cailletuad [39] employed a simple kinematic (translational) and an isotropic (expansion)

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hardening term into a visco-plastic self-consistent (VPCS) model and showed that such an approach provides very good agreement with the stress–strain response at small and large strain fatigue and good predictive capabilities during more complex loading paths. However, a need for a better description of microscale hardening and a better description of intergranular hardening which includes a variable accommodation factor (such as that utilized in EPSC), is suggested in that prior work. The present work builds upon these prior efforts by employing a simple non-linear kinematic hardening law (similar to that of an A–F approach) applicable for small strains to the slip system hardening within the EPSC model and compares the predictions of internal elastic lattice strains. The small strain model is first described and then parameterized by fitting to experimental stress–strain data of an austenitic stainless steel, 317L. Predictions of the internal strain developments are then compared with in situ neutron diffraction studies of a small number of fatigue cycles. Three modeling scenarios are explored and compared against the experimental data; (1) the original EPSC model employing typical ‘‘isotropic’’ (forest dislocation) hardening, (2) a case employing only kinematic hardening, and (3) a final case employing both kinematic and ‘‘isotropic hardening’’. Lastly, the final case (kinematic + ‘‘isotropic’’ hardening), with an unmodified parameter set, is compared to the experimental data of Lorentzen et al. [35] for the same 317L alloy. 2. Model description The elasto-plastic self-consistent (EPSC) code of Turner and Tomé [29] serves as a framework for the presently developed hardening model. The EPSC model is viewed as a good platform for the present formulation because of the aforementioned inherent incorporation of intergranular type stresses [25,35,37], the ability to be validated experimentally with diffraction data [29,40], and the physics-based approach to plastic strain accommodation (i.e., explicitly accounting for dislocation slip-induced shear on slip planes and along Burgers vector directions). However, it is suggested that other crystal plasticity-based approaches, such as those implemented within FE codes [41,42] represent approaches in which the concepts could be implemented and further explored. Within the EPSC model, the polycrystal is treated as a collection of single crystals with discrete orientations that reflect the initial texture of the material. Each grain is considered as an elasto-plastically anisotropic ellipsoidal inclusion embedded within a homogenous effective medium (HEM). The response and properties of the HEM are calculated as a weighted average of all of single crystals. The equilibrium and compatibility between the inclusions and HEM are rigorously enforced using a modified Eshelby [43] inclusion formalism. As applied to the isothermal case, and valid for infinitesimal deformations and rotations, the grain-level, small strain constitutive equation relates the stress rate, r_ , in a crystal, c, to the elastic strain rate,

r_ c ¼ C : e_ c 

X

! ms c_ s

ð1Þ

s

by taking the difference between the total strain rate in the crystal, P e_ c , and the plastic strain rate, s ms c_ s . Plastic strain is accomplished by distinct shear rates, c_ , on slip systems s and resolved by the Schmid tensor m, which is symmetric part of the dyadic product of the unit vectors in the slip and slip plane normal directions. C is the normal fourth rank elastic stiffness tensor of the crystal and ‘‘:’’ denotes the scalar product. The form of the single crystal yield criterion follows the work of Goh et al. [44], and is similar to an approach employed by Xu and

Jiang [45]. The yield criterion for each slip system now includes an explicit backstress term, ssbs :

ssfor ¼ ms : rc  ssbs

ð2Þ

ssfor is typically the only threshold stress utilized in the EPSC model [46,47]. It is denoted the forest strength because it is the evolution of the forest dislocation density which is primarily responsible for the evolution of this strength value, even though the initial value accounts for effects of the intrinsic lattice resistance, solid solution alloying, particle strengthening, and initial density of forest dislocations. The resolved backstress term is associated with local stresses that build up due to geometrically necessary dislocations that accumulate at hard particles and grain boundary regions [48]. Considering of the impact of discrete dislocation slip band impingement on the interface of an ellipsoid representative of a grain embedded within a homogeneous effective medium, Berbenni et al. [49] demonstrated that they induce a backstress that is not accounted for by traditional mean field theories. The present work is a mean field approximation which seeks to overcome such shortcomings. In summary, for slip to occur, we require that the resolved applied stress, reduced by the backstress (which accounts for the kinematic hardening), equals the forest strength (which may itself evolve to account for isotropic hardening effects, such as the increase in the density of forest dislocations). For a slip system to be considered active, it must initially fulfill Eq. (2) (the ‘‘potentially active’’ condition) and it must also remain active throughout a straining step as the critical stresses evolve (‘‘loading’’ condition). For infinitesimal deformation and rotations,

s_ sfor ¼ ms : r_ c  s_ sbs

ð3Þ

It is emphasized here that the backstress shear rate, s_ sbs , is always positive for active systems, though it may be negative for inactive (reverse) slip systems. The rates of change of the critical shear stresses are assumed to be related to the shear rates, generically, by Hill [50],

s_ s ¼

X

ss0 V s h c_ s0

ð4Þ

s0 s

ss0

s

where Vs is either ddsC or ddscs depending on the hardening law and h is P the latent hardening matrix. C ¼ s cs and is the total shear in the grain. We employ a non-linear evolution of the forest resistance based upon the Kocks–Mecking [51] model that accounts for dislocation accumulation and recovery. The only distinction from that original formulation is the inclusion of a stage-IV linear hardening [52] to define what is often termed an extended Voce hardening relationship.

dssfor dC

    h0;for h1;for h0;for ¼ h1;for þ h0;for  h1;for þ ðCÞ exp ðCÞ

s1;for

s1;for

ð5Þ

The extended Voce-type hardening rule is also used to describe the evolution of the backstress for active slip systems,

  dssbs h0;bs h1;bs s s ¼ h1;bs þ h0;bs  h1;bs þ ð c  c Þ 0 dc s1;bs þ xso   h0;bs  exp ðcs  cs0 Þ s1;bs þ xso

ð6Þ

where h0 and h1 are initial and asymptotic hardening rates, respectively, s1 is back-extrapolated from the terminal linear hardening region, and cs0 and xso are memory parameters [53] described below. Though not present in the derived form of the equations above, an initial critical resolved shear strength, s0, initially defines the hardening associated with the intrinsic lattice resistance, solid solution alloying, particle strengthening, and initial density of forest dislocations and, as described previously, is coupled with the forest

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hardening term (also, see Table 3). An important distinction is that the evolution of the slip system backstress depends on the shear of P that slip system, cs, whereas the total grain shear, C = scs, governs the forest stress of that slip system, see Eq. (5). This disparity is intentional, since the Voce laws (together with the latent hardening matrices) convert the shears into amounts of hardening related to distinct dislocation densities. It is acknowledged that shear is not a state variable which can be directly related to dislocation densities. That said, the forest strengthening term is most readily related to the total amount of shear in the grain, while the backstress term hardens in proportion to the amount of shear on a single slip system because it accounts for reversible geometrically necessary dislocation structures, such as pile-ups. Future implementations will employ an explicit, dislocation density-based, hardening model. However, the current model functions well for introducing the overall approach and illustrating its potential. Similar to Lorentzen et al. [35], the current formulation pairs forward and reverse slip systems in the evolution of the backstress (Eq. (6)) such that the rate of increase of the resolved backstress of the active system, s, equals the rate of decrease of the resolved backstress of the reverse system, s.

_s _s s_ s bs ¼ sbs ðif c –0Þ

ð7Þ

To accomplish this, the ‘forward’ and ‘reverse’ sense of a single slip system are paired. In short, the same dislocations will experience resistance or assistance during ‘forward’ or ‘reverse’ motion, respectively. Inactive system pairs (i.e. latent slip systems) are not hardened by the backstress term. This aspect contrasts with the evolution of the threshold stress adopted by Lorentzen et al. [35] which incorporated equal latent hardening of all systems, with the exception of the reverse sense of the active systems, which were softened. What is termed ‘‘strain memory,’’ in the kinematic hardening literature (e.g., [54]), is accounted for by the strain memory parameter, cs0 , initially defined equal to zero. When glide on a slip system reverses, this parameter is defined to be equal to the total accumulated shear on that slip system, cstot . This ‘‘remembering’’ of strain history for each slip system in a slip mode allows high initial hardening rates when slip direction has changed, since it effectively resets the Voce hardening back a point of zero strain. ‘‘Stress memory’’ is accounted for by the parameter xso , which is also initially defined as zero. If, in a given strain step (i), the direction of glide on a slip system pair changes direction, xso is defined as the positive resolved backstress, ssbs , in the previous strain step (i  1). A change in slip direction (and setting of memory parameters) does not typically occur in consecutive strain steps, since the stress must first traverse the yield surface before reverse slip is activated. However, this scenario might occur in extreme boundary condition problems. Consider the case where the backstress saturates to a constant value (i.e. h1 = 0). As the forward system hardens to saturation, the coupling causes the reverse system to soften equally. When glide is reversed in that system pair, the reverse slip direction will harden to the same level of saturation though it initially held a softened backstress. In the absence of the stress memory parameter, the reverse direction of slip would only harden to a value of 0. The stress memory parameter ensures that the both senses of a slip system backstress always have the potential to accumulate to the same finite stress level, regardless of prior loading history. In a yield surface description, the surface may harden sbs in the ‘forward’ direction and, subsequently, 2sbs in the ‘reverse’ direction. Without the incorporation of xso , the yield surface would only translate sbs in the reverse direction, placing it back at the origin. Importantly, neither of the memory parameters are ‘‘fitting parameters’’. Rather, they are automatically set during model operation according to the strain path that is executed. An evolution of these

parameters, as well as the evolution of the resolved backstress, is provided at the beginning of the results and discussion section. The self-consistent homogenization scheme requires that the elasto-plastic constitutive rule (Eq. (1)) be linearized over each strain increment, both within a grain and within the surrounding aggregate. The crystal-level linearization is expressed as follows

r_ c ¼ Lc e_ c

ð8Þ

c

where L is the fourth rank single crystal instantaneous (incremental) elasto-plastic stiffness tensor. When the conditions given by Eqs. (2) and (3) are satisfied, the shear in a grain can be related to the total shear rate (see [29]), through Eq. (8), by defining c

L ¼C:

I

X

s

m 

X

s

! ss0

s

Y m :C

ð9Þ

s0

where  the represents uncontracted tensor product, and the sums are over the active slip systems in the grain. The formulation of Xss0 (where Y = X1) is altered from previous descriptions since it depends on the evolution of the backstress. Xss0 may be derived by first combining Eqs. (1), (3), and (4) to give:

X

" ss0 V sfor hfor _ s0

s

_c

c ¼ m : c : ðe 

X

s0

# s _s

mcÞ 

s dss

X

ss0

V sbs hbs c_ s0

ð10Þ

s0

dss

where V sfor ¼ dCfor and V sbs ¼ dcbss . Rearranging and combining like terms, Eq. (10) can be rewritten as,

X ss0 X s ss0 ss0 ½V for hfor þ V sbs hbs þ ms : C : ms c_ s0 ¼ X c_ s0 ¼ ms : C : e_ c s0

ð11Þ

s0

(i.e. Xss0 is the term in square brackets.) This equation relates the shear strain increments on individual slip systems to the total strain increment of the crystal:

c_ s ¼ f s e_ c

ð12Þ

where

fs ¼

X 1 ss0 ðX Þ ms0 : C

ð13Þ

s0

and the sum is again over the active systems. In the present work, ss0 all components of the latent hardening matrix hfor are defined to be 1, indicative of the fact that all slip systems are affected equally, what is often termed ‘‘isotropic’’ slip system hardening, in contrast with more complex models that have unequal latent hardening parameters [55,56]. The backstress term describes the hardening related to geometrically necessary dislocation pile-up type configurass0 tions and, therefore, hbs is set equal to the Kronecker delta, dss0 , to induce self-hardening only. In short, it is assumed that only dislocations of the same slip system exert strong backstresses on each other. Similar to the grain level linearization above, the aggregate response is linearized as,

r_ ¼ Le_

ð14Þ

where r_ and e_ are the stress and strain rate in the aggregate, respectively, and L is the overall elasto-plastic stiffness tensor and is unknown a priori. Stress equilibrium is solved for the grain (idealized as an ellipsoidal inclusion) embedded in the HEM using the Eshelby formalism [43]. For the Eshelby formalism, it is assumed that the fields are uniform inside the inclusion and the macroscopic stress rate and strain rate are related through the interaction equation 

r_ c  r_ ¼ Lc : ðe_ c  e_ Þ

ð15Þ

where 

Lc ¼ LðSc

1

 IÞ:

ð16Þ

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I is the fourth order unity tensor and Sc is the symmetric Eshelby tensor. Combining Eqs. (8), (14), and (15), (see Hutchinson [57]) a localization tensor, Ac, is defined c

c 1

c

c

A ¼ ðL þ L Þ ðL þ LÞ:

ð17Þ

Table 1 Chemical Composition of 317L stainless steel alloy. Cr

Ni

Mo

Mn

Si

C

Fe

18.25

13.42

3.66

1.48

0.44

0.02

Balance

with the property,

e_ c ¼ Ac : e_

ð18Þ

The conditions of self-consistency require that the weighted averages, ‘‘h i’’, of stress rate and strain rate equal the macroscopic magnitudes of the stress and strain rate of the aggregate:

e_ ¼ he_ c i and

ð19Þ

r_ ¼ hr_ c i;

ð20Þ

and lead to an expression for the macroscopic elasto-plastic stiffness, 



L ¼ hðLc þ Lc Þ1 i1 hðLc þ Lc Þ1 Lc i:

ð21Þ

3. Material and experimental design To validate the model, cyclic fatigue tests of austenitic 317L stainless steel were conducted. The composition of the alloy can be found in Table 1. The sample was cut from the same sheet as that used by Clausen et al. [58], where they measured an average grain size of 28 lm. The material, initially supplied by Sandvik, Sweden in the form of a 15 mm rolled plate, contains high Ni and Mo content to ensure phase stability to high strains (unchanged magnetic permeability up to 65% cold working) while retaining a low stacking fault energy (SFE) characteristic of austenitic stainless steels (e.g., [59,60]). No additional thermo-mechanical treatments were performed to the sample. The single crystal elastic constants necessary for the EPSC model are not known for this specific alloy. However, previous studies have indicated that the measured single crystal elastic constants for the Fe–Cr–Ni alloy listed in Table 2 are appropriate [40,58,61]. Fatigue tests were conducted in a custom-built Instron load frame that sits on the Spectrometer for Materials Research at Temperature and Stress (SMARTS) beam line at Los Alamos Neutron Science Center (LANSCE) at Los Alamos National Laboratory, LANL (Los Alamos, NM). (Details of the instrument can be found in [62].) All experimental tests, including previously reported data [35] represented toward the end of the paper, were conducted at room

Table 2 Single crystal elastic constants (GPa) of Fe–Cr–Ni. C11

C12

C44

206.4

137.7

126.2

temperature. The sample geometry follows the ASTM/E 606 standard sub-size sample shown in Fig. 1, where the diameter at the fillet radius is not more than 1% larger than the diameter at the center of the gage section. An extensometer was attached to the gauge volume with rubber bands. Cyclic tension–compression tests to strains of ±0.02 were conducted at a very low nominal strain rate (on the order of 2  105) with periodic holds of approximately 10 min at constant strain or stress to collect neutron diffraction data in situ. The stress axis of the aforementioned test frame is oriented 45° relative to the incident neutron beam (see [63] for a diagram). As is common practice in in situ neutron diffraction measurements, some portions of the experiments were strain-controlled and other portions of the experiment were stress-controlled. To make sure that the yield is captured well (initial, as well as cyclic), stress-control is used during elastic loading, since the stress increases very quickly over a very small strain. Once the material has yielded, the control is switched to strain control to avoid stress plateaus (creep) that can occur during the holds for diffraction collection. Each diffraction data collection period is observable by slight load relaxations and plateaus (creep) in the stress–strain curve. Prior to the mechanical testing, the initial experimental texture was measured on the High Intensity Pressure and Preferred Orientation (HIPPO) instrument at LANSCE. As needed for the model, the measured texture was converted into 23,328 discrete orientations by dividing Euler space into 5°  5°  5° cells with varying volume weight fractions considering the symmetry operations of the cubic crystals. Using the discrete orientations to describe the initial state of the material, an iterative ‘‘inverse’’ approach was used to find the set of hardening parameters (h0, h1, s1, and s0), for each

Fig. 1. Tension–compression experimental sample geometry. Follows the ASTM/E 606 standard sub-size sample. All dimensions are in millimeters.

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hardening law, that best-fit the experimental macroscopic data (as in Figs. 3 and 5). As is common with comparison studies of neutron diffraction, the model fits were obtained to the diffraction hold points rather than the ‘‘flow curve.’’ The reason for this choice is that the EPSC model does not account for the slight viscoplastic effects associated with the load relaxations and stress plateaus during the experimental collection periods. Indeed, use of an elasto-viscoplastic model, such as the elasto-viscoplastic selfconsistent (EVPSC) model recently developed by Wang et al. [64,65], might account for such effects. However, the viscoplastic effects in the present study are small (at the highest stress level the load relaxation corresponds to <5% of the applied stress) and fitting to the flow curves or the diffraction hold points yields very similar predictions whose distinctions would not impact the results or conclusions. In the experiment, lattice strains, ehkl, are calculated by measuring the shift of each diffraction peak position at each collect 0 0 period during deformation by ehkl ¼ ðdhkl  dhkl Þ=dhkl where dhkl is 0 the measured lattice spacing and dhkl is the initial stress-free lattice spacing. The detector used for comparison is a square detector array sitting at 2h = 90° from the incident neutron beam with a width of ±13° from center to edge. The small grain size (28 lm) of the stainless steel ensures that a sufficiently large number of grains occupy the gauge volume, and therefore the irradiated volume, allowing many variously oriented grains that share a common orientation (the h k l plane normal direction) to contribute to a single diffraction peak. As described by basic diffraction theory, placement of the detector at this position causes the common orientation to be along the axial loading direction. The evolution of each individual diffraction peak describes the behavior of a large subset of commonly oriented grains deforming in a variety of grain neighborhoods and the complete diffraction pattern (all peaks) yields information characteristic of the bulk behavior. Within the EPSC model, the discrete grain orientations are polled at each deformation increment in order to identify which fulfill Bragg’s law for a given wavelength and diffraction vector that

corresponds to that of the experiment (as described above). The component of strain which is parallel to the nominal diffraction vector (i.e., pointing along the bisector of the incident beam and the detector, which is the axial direction of loading) is calculated from the current stress in the grain, the elastic constants, and conventional tensor algebra. These values of strain are then averaged over all the grains selected for a given {h k l} diffraction peak. In this light, EPSC is viewed as an ideal polycrystal model for the current study, because the simulation and experiment similarly poll the lattice strains within different orientations in the polycrystal and both present average behaviors (i.e., of grains of similar orientation immersed in various neighborhoods within an actual polycrystal). 4. Results and discussion 4.1. Modeling simple Bauschinger effects To illustrate the basic response of the model, the evolutions of the resolved backstress and memory parameters for a single grain undergoing single slip deformation during a simple forwardreverse-forward loading case are presented in Fig. 2. In this first case, only one discrete orientation (grain) is defined in the model and it can, therefore, be consider as the ‘aggregate.’ Additionally, only the forward and reverse sense of one slip system are defined (i.e., a single slip system pair). Fig. 2 shows that the resolved backstress for the ‘forward’ system, sþs bs , initially at 0, evolves first following an evolution defined by the hardening law in Eq. (6). This system direction hardens until saturation at s1,bs (i.q., ssbs ¼ 1), 1;bs

while the backstress for the ‘reverse’ system, ss bs , ‘softens’ the same amount. At a total shear strain of 0.02, the macroscopic loading direction is reversed and the ‘reverse’ system becomes active with an initially reduced resolved backstress of s1,bs. The reverse system then hardens with an initially high hardening rate until becoming saturated again at s1,bs. This high hardening rate and

Fig. 2. Evolution of the (a) resolved backstress, (b) strain memory parameter, and (c) stress memory parameter for a one grain, one slip system material that is loaded in tension, compression and tension, respectively. When a system is active, it is plotted with a solid line. When a system is inactive, it is plotted with a dashed line.

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Fig. 3. Flow curve for initial tensile loading and first subsequent tensile loading of ±2% strain fatigue tests. Solid black line corresponds to the complete experimental flow curve. Black squares are the points at which diffraction patterns were collected. Solid blue is the presently developed model with only kinematic hardening. Dashed blue line is the unmodified EPCS model with commonly employed isotropic hardening. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) s extended hardening region are a result of cs 0 and xo being defined by the material state at the point of reverse slip activation, respectively, seen in Fig. 2b and c. Again, the opposite (forward) system is softened the same amount. At 0.06 total shear strain, the loading direction is reversed again so that the material is strained in the same direction as the initial stroke. The ‘forward’ system is active again and hardens with a high hardening rate that saturates at s1,bs. The combined memory effects (resetting cs and incorporation of xso ) promote an extended elasto-plastic transition after any change in slip system directionality, which directly relate to dislocationbased mechanisms governing the Bauschinger effect, in the absence of intergranular sources of backstresses. In a polycrystal model that permits many slip systems and grain orientations, the employed hardening description uniquely allows accounting for both major sources of backstress. Fig. 3 shows the experimental results for the first tensile and compressive loading of the cyclic tests of the stainless steel sample. The complete experimental flow curve is plotted as a solid black line. The square black data points correspond to the time averaged values of stress and strain during which the neutron diffraction data were collected. It can be seen that the sample has a pronounced Bauschinger effect. The material initially yields at 292 MPa (using a 0.2% offset) in tension and hardens to 375 MPa at a strain of 0.02. In the subsequent unload and compressive reloading, the material yields at 165 MPa. In the absence of a Bauschinger effect, the material would be expected to yield during the compressive reload at 375 MPa. Also provided in the Fig. 3 is a comparison between the flow curves simulated with a single forest (isotropic) hardening law, denoted ‘Turner and Tomé’ since this is the hardening model

employed in their original EPSC publication [29], and the newly developed slip system based non-linear kinematic hardening law. Table 3 contains the hardening parameters for both models shown in Fig. 3. As reflected in the parameters of Table 3, the ‘Turner and Tomé’ model enforces no kinematic hardening and the Current Model A compared here enforces no isotropic hardening. For the Turner and Tomé model (blue dashed line), where only isotropic hardening is considered, the parameters were systematically fit to the initial tensile (forward) direction. The subsequent reloading behavior can be viewed as a prediction. As a result of the inherent intergranular backstress effects captured by the EPSC model, a slight Bauschinger effect is predicted. However, the level of intergranular backstress is inadequate to predict the level of Bauschinger effect observed experimentally. The rate of hardening is equal for all systems, including the reverse directions of the acss0 tive systems and all inactive systems (recall that h ¼ 1). Therefore, when glide reverses direction on a system pair, the elastic region (within the single crystal yield surface) will always be larger than the initial elastic region and result in a higher yield than that observed experimentally. Additionally, because the hardening law simply depends on the total accumulated shear in a grain, any subsequent loading will have a sharper elasto-plastic transition than the first transition, rather than the more protracted elasto-plastic transition observed during experimental reloads. The model which employs only the kinematic (backstress) hardening law with the parameter set ‘Current Model A’ in Table 3 is plotted with a solid blue line in Fig. 3. Because of the nature of the hardening law, the fit was able to be tuned to both the initial forward response and the reversal response. The use of kinematic hardening (even without any isotropic hardening) is able to well describe the initial forward response as well as the early yield and extended elasto-plastic transition during the reload. Slip systems that were active in the forward direction require lower applied stress to activate glide in the reverse sense. The amount of lowering depends on the amount of previous strain. Fig. 4a and b show a comparison between the experimentally measured and model predicted internal strain during the initial tensile loading and first unload and compressive reloading, respectively. The experimental measurements are shown as data points. During the initial tensile loading, all of the internal lattice strains evolve linearly (albeit, anisotropically), indicating that all grains are elastic up to a stress, r  175 MPa, at which point the diffraction peaks diverge from linearity. This nonlinearity in the internal strain development occurs at a significantly lower stress than the 0.2% offset yield identified by the macroscopic curve and signifies microyielding; the initiation of plasticity within some of the grains. Agreement between the model and experiment on the elastic behavior and onset of microyielding is a good indication that the single crystal elastic constants are adequate and initial threshold stress has been properly identified. Post yield, grains with a (2 0 0) plane normal parallel to the loading direction start accommodating a greater portion of the strain elastically while grains with a (2 2 0) diffraction vector parallel to the loading direction start accommodating less elastic strain. This indicates that the latter are yielding as they now accommodate some of their

Table 3 Hardening parameters (GPa) for fits to simple Bauschinger tests for the kinematic only (Model A) and isotropic only (Turner and Tomé) models compared in Figs. 3 and 4 and the kinematic only model used in Fig. 5. Model B is the parameter set for both complete cyclic fatigue curves. Model

Hardening term

s0

s1

h0

h1

Turner and Tomé (isotropic only)

Forest (isotropic) – Forest (isotropic) Backstress (kinematic) Forest (isotropic) Backstress (kinematic)

0.08 – 0.07 – 0.07 –

0.0175 – 0 0.027 0.015 0.027

10.0 – 0 64.0 0.065 55.0

0.5 – 0 1.8 0.045 1.3

Current Model A (kinematic only) Current Model B (kinematic and isotropic)

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Fig. 4. Evolution of the internal strain of various diffraction peaks (grain subsets) for the (a) initial tensile loading and (b) first subsequent compressive loading of the ±2% strain experiment. Data points represent experimental data. Solid lines are lattice strain predictions from the presently developed model employing only kinematic hardening (Current Model A). Dashed lines are lattice strain predictions from the unmodified EPCS model with commonly employed isotropic hardening (Turner and Tomé model).

deformation by plastic strain rather than elastic strain and that the former must, thus, carry more of the applied load. The internal strains of the (3 1 1) grains approximate linearity at all stress levels, an indication that behavior of these grains approximates that of the overall polycrystal [61,66]. The model predictions are shown as lines in Fig. 4. It is reemphasized that the modeled internal strains are not a fit to the experimental data. Both models correctly predict the evolution of the internal strain during the initial tensile loading (Fig. 4a), albeit with slight differences due to the distinctions in slip system selection. Distinctions in slip system selection are caused by differences in the way slip system hardening is implemented; i.e. the presence (isotropic) or absence (kinematic) of latent hardening. However, the two models generally predict a similar internal stress evolution in the initial tensile loading. In monotonic loading, the selected active slip systems remain active during loading and evolve similarly at these low strain levels. Therefore, latent hardening had minimal impact and gross distinctions between the two models are not apparent. It is noted that latent effects do become significant at larger strain levels (larger than those explored in this study) where the texture evolution itself can become distinct. Indeed, the small, but observable, differences between the slopes of the model predictions of the (2 0 0) reflection (in Fig. 4a) at high stresses suggests that deformations of 2% strain may approach the limit of the applicability of a single kinematic term. While there is general agreement between the model predictions during initial loading, as mirrored in the macro flow curves discrepancies between the hardening laws arise in the subsequent compressive reloading (Fig. 4b). The internal strains of the isotropic hardening law remain linear-elastic much beyond the deviation from the linear-elastic response of the experiment. The early yield and transient plastic response observed experimentally are only predicted by the kinematic hardening law. These trends are most obvious in the evolution of the (2 0 0) diffraction peak. The above results suggest that kinematic hardening (translation of the yield surface), rather than isotropic hardening (uniform increase in the size of the yield surface) dominates the hardening response of 317L austenitic stainless steel in monotonic and reversed loading, at low strains. This result echoes the recent conclusions of Choteau et al. [67], where they showed that a phenomenological kinematic model best fit the experimental data of tension– compression experiments of a 316 stainless steel (similar to the present 317L) at various amounts of prestrain up to 5% prestrain.

This is most likely due to the planar slip nature of austenitic stainless steels. With dislocation glide confined to specific planes, by the dissociation of dislocations into partial dislocations which bound low energy stacking faults, dislocation motion appears to be more reversible than it would be for wavy slip metals, which more readily form cell structures. This result conforms with studies of pure copper and Cu–Al alloys [68,69] which demonstrate that the magnitude of the Bauschinger effect, as measured by common parameters (such as Bauschinger stress, strain, and energy, see [6]), increases with decreasing stacking fault energy. Curiously, this result seems to contradict a recent work by Feaugus [70] on a 316 stainless steel. In that work, it was concluded that intergranular stresses dominate the backstress at strains below 1.5%. In fact, it was even suggested that the intragranular backstresses are zero in the low strain regime. However, Feaugus

Fig. 5. Experimental and modeled flow curves for ±2% strain cyclic fatigue tests. Black squares are points at which diffraction patterns were collected. Solid blue line is the current model employing only kinematic hardening. Red dotted line is current model employing both kinematic and isotropic hardening. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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only considers an intragranular backstress to arise from heterogeneous dislocation cell/wall structures, as described by Mughrabi [13]. However, these structures are not applicable to planar slip alloys like austenitic stainless steels at such low levels of deformation. On the other hand, Feaugus used transmission electron microscopy to show that pile-ups are the dominant dislocation structures at strains below 1.5%. The present formulation envisages pile-ups and Taylor lattices [71,72] to be the primary sources of intragranular backstress envisaged in the present formulation. A major difference between the two hardening laws investigated here, is the value of the initial hardening rate, h0 (see Table 3). Traditional dislocation forest (Taylor) hardening models give an expectation that h0 values should be G/50 or G/20 at the largest, e.g. [73]. The values for the kinematic hardening model, in particular, actually approach G (G/4 to G/3). Given the distinctions in the mechanisms responsible for hardening, this distinction in hardening rates is expected. For example, the idealized cases considered by Berbenni et al. [49] predict a backstress hardening rate G/2. 4.2. Modeling cyclic response Fig. 5 shows the experimental results of multiple ±2% strain fatigue cycles tests of the 317L austenitic stainless steel. In

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agreement with cyclic studies of 316L stainless steels [74], the 317L alloy cyclically strain hardens at a rate 30–60 MPa for each of the first few fatigue cycles with a strain amplitude of 0.02. Also shown in Fig. 5 are the model predictions of the cyclic response for the ‘Current Model A’ (employing only kinematic hardening) and ‘Current Model B’ (employing both kinematic and isotropic hardening). The parameter set for ‘Current Model B’ were found by systematically reducing the amount of kinematic hardening of ‘Current Model A’ while adding isotropic hardening until the ensuing macroscopic fatigue loops were well described. After the initial quarter cycle, Model A saturates and the following two full fatigue loops fall on top of each other (see the solid blue curve). It can be seen that the simplistic kinematic hardening description captures initial Bauschinger effects (i.e., Fig. 3). However, it cannot account for the cyclic hardening observed during repeated cycling. This limitation results from the nature of the single term kinematic hardening model employed. The red dotted line in Fig. 5 corresponds to Current Model B (see Table 3). Because of the additional hardening from the isotropic term, the kinematic parameters are slightly altered from that of the single term model. This more complete description accounts for both reversible dislocation structures and the creation of a more permanent forest dislocation structure, allowing the model to capture both the reduced yield and extended elasto-plastic

Fig. 6. Internal strain cyclic fatigue loops for the 2% macroscopic strain fatigue experiment for the (1 1 1), (2 0 0), (2 2 0), and (3 1 1) diffraction peak (i.e., grain subsets). Experimental measurements are plotted as data points. The predictions of the current model employing both kinematic and isotropic hardening are plotted as solid lines. Parameter set is found in Table 3.

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transition associated with the Bauschinger effect as well as cyclic strain hardening. Fig. 6 shows the internal strain ‘hysteresis loops’ measured using data obtained from the (1 1 1), (2 0 0), (2 2 0), and (3 1 1) diffraction peaks and compares that with the predictions of the two term kinematic/isotropic hardening model (Current Model B). Again, the internal strain prediction is good; suggesting that slip system activity and associated hardening is appropriately described. The two term model predicts the correct shape, magnitude, and width of the internal strain evolutions, including the elasto-plastic transitions. Though the predictions are good, minor deviations from the experiment (e.g., the slopes of the (2 0 0) and (3 1 1) curves) are observed. These may be due to a variety of sources, such as imprecise knowledge of the elastic constants, the fact that the curve was fit to the overall stress–strain response at diffraction collection periods without consideration for the relaxation associated with those holds, the assumptions associated with the self-consistent framework (i.e. averaging or lacking consideration of neighborhood effects), or that the additional yield surface expansion (forest) term needs more careful consideration of the details of latent hardening. As further validation of the presently developed hardening law, predictions of the two term hardening model (see Table 3, Current Model B) are compared with the smaller cyclic strain conditions originally explored experimentally by Lorentzen et al. [35].

Fig. 7. Experimental and model flow curves for the cyclic fatigue tests of Lorentzen et al. Black squares are points at which diffraction patterns were collected. Red dotted line is the two term kinematic/isotropic hardening model. Parameter set for the model is found in Table 3 as Current Model B. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Internal strain cyclic fatigue loops from Lorentzen et al. for the (1 1 1), (2 0 0), (2 2 0), and (3 1 1) diffraction peak (i.e., grain subsets). Experimental measurements are plotted as data points. Two-term kinematic/isotropic model predictions are plotted as solid lines. Parameter set for the model is found in Table 3 as Current Model B.

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Lorentzen et al. used a sample cut from the same sheet of material as that used in the present study. Therefore, it is hopeful that the same model, using the same parameters should also describe their data. Their experiment consists of tension–compression fatigue loops from +0.004 to 0.005 strain. Fig. 7 reproduces their macroscopic fatigue data (only available at the diffraction holds) and plots the results from the two term isotropic/kinematic complete hardening model. It can be seen that same model parameters describe the response of smaller strain fatigue cycles, including the re-load yield points, elasto-plastic transitions, and magnitude of cyclic hardening. Not only is the macroscopic response well capture by the same model parameters, but internal strain evolutions are also well predicted. Fig. 8 compares the internal strain predictions and the experimental measurement of Lorentzen et al. for the (1 1 1), (2 0 0), (2 2 0) and (3 1 1) diffraction peaks. The model predicts the correct shape, magnitude, and width of the internal strain evolutions, including the elastic–plastic transitions. Again, the slopes of some of the curves disagree, suggesting that some of the same sources of error stated previously may be suspect. The fact that a single parameter set (invoking both kinematic and isotropic hardening terms) captures the response for both experimental data sets, suggests that the selected hardening description is appropriate for austenitic stainless steel at small strains. Furthermore, it corroborates theories that suggest the Bauschinger effect present at small strains is largely a result of the forward and reserve motion of dislocations, though a comprehensive fatigue model is typically correlated with more details of the dislocation structure in a grain (i.e. evolution of loop patches, veins, persistent slip bands, and dislocation cells). In addition, simple adjustment of the parameters should allow applicability of the model to other materials. For example, if the current model were applied to pure copper or aluminum, it is expected that the forest term (isotropic hardening) would account for a larger proportion of the overall hardening response than it did for stainless steel, since they are wavy slip metals. Though the number of ‘fitting’ parameters for the two term model is relatively high at 7, this is not viewed as a major setback since is it on par with or lower than other models [38,67]. The use of two independent hardening terms allows decoupling of the phenomenology of some of the mechanical behaviors of austenitic stainless steel. Kinematic hardening accounts for a majority of the hardening at low strains and the features associated with small strain Bauschinger effects, while small amounts of isotropic hardening can account for features associated with cyclic strain hardening. This categorization is in line with previous work [75,76] and conclusions of Chaboche [77] that only isotropic hardening models can describe cyclic hardening. Conversely, it is pointed out by Moosbrugger [78] that cyclic strain hardening can be captured in phenomenological models for 304 stainless steel using a second kinematic hardening term. Seeking to discriminate between potential models of cyclic strain hardening is not a specific objective of the present research and it is plausible that such an addition (instead of the isotropic term) could be applied in this context to capture the cyclic hardening. However, follow on research has highlighted that latent hardening is an important ingredient in a comprehensive model and this is easier to rationalize with isotropic hardening. The simple description and implementation of the slip system level kinematic hardening rule allows it to be applied in other polycrystal plasticity models, such as FE models. A similar study performed by Turkmen et al. [38] concluded that the inclusion of a slip system-based kinematic hardening law could improve the overly sharp elasto-plastic transitions predicted by their model. They found that different polycrystal homogenization schemes (from Taylor–Lin to finite element) did not strongly influence the trends predicted by the slip system based hardening model. How-

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ever, it would be interesting to return to a comparison of various homogenization schemes with a grain-level hardening model, which does incorporate a backstress. Finally, the success of the present model is provided with a major caveat; to correctly describe a material response at large monotonic strains, more complex strain path changes, or further into fatigue life, several other details must be considered. At large strains, latent hardening, texture evolution and other kinematic details of finite deformation become extremely important. A more careful description of latent hardening, such as that employed by Holmedal et al. [79], and consideration for more complex dislocation structures, such as that done by Mughrabi [13] must be addressed before the model can be applied to cases of large strain, crossloading, or higher levels of fatigue. 5. Conclusions The hardening law in an elasto-plastic self-consistent model is modified to account for intragranular backstresses by adding an additional term incorporating non-linear kinematic hardening. Employing only kinematic hardening, the modified algorithm quantitatively models the macroscopically observed reduced yield and extended elasto-plastic transition upon the first load reversal (i.e., the Bauschinger effect) in austenitic stainless steel. This reveals that the strain hardening behavior of this planar slip metal is dominated by kinematic hardening due to reversible configurations of dislocations, rather than ‘‘isotropic’’ hardening due to forest dislocation accumulation, at least for the small strain case presently considered. As important, the modified model predicts the evolution of lattice strain as measured by in situ neutron diffraction, providing further validation that the single kinematic hardening term dominates the hardening during small strain Bauschinger tests. That said, a small amount of secondary ‘‘isotropic’’ hardening can be used to model the cyclic hardening observed during multiple load reversals. This two term hardening model quantitatively captures the evolution of stress and strain at both the macroscopic and internal lattice level during individual fatigue cycles, as well as over multiple cycles. The additional minor isotropic hardening term reveals the presence of a minimal, but required, amount of more complex, non-reversible dislocation arrangements correlated with cyclic strain hardening. Notably, the parameter set employed to describe the uniaxial in situ fatigue experiments performed in this study (±0.02 strain fatigue loops) also correctly predicts the response of a previously published experiment (+0.004 to 0.005 strain fatigue loops), providing validation that, within the framework of elasto-plastic self-consistent models and in situ neutron diffraction, the physically-based description of hardening is not simply a fit to the present data but is a more widely applicable model of small strain deformation behavior of an austenitic stainless steel. Finally, it is pointed out that the description of slip system kinematic hardening can easily be applied other models, such finite element based models. Acknowledgments The authors would like to thank Dr. Carlos Tomé for invigorating discussion regarding the validity and applicability of the model, Prof. Mark Daymond for providing the second set of fatigue data, and the attendees of the Elasto-Plastic International Meeting (EPIM) at Queen’s University, Kingston, Ontario in August 2010 for providing further insight into the roles and sources of backstresses. This work has benefited from the use of the Lujan Neutron Scattering Center at LANSCE, funded by the Department of Energy’s Office of Basic Energy Sciences. Los Alamos National Laboratory is operated by Los Alamos National Security LLC under DOE Contract

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DE-AC52-06NA25396. This research is supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.

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