Influence of M 23C 6 Carbides on the Heterogeneous Strain Development in Annealed 420 Stainless Steel

Understanding the local strain enhancement resulting from different microstructure features in metal alloys is crucial in many engineering processes as it plays an important role in the work hardening and in other processes such as recrystallization and damage. Isolating the contribution of precipitates to the development of heterogeneous strain can be challenging due to the presence of grain boundaries or other microstructure features that might cause ambiguous interpretation. In this work a statistical analysis of local strains measured by electron back scatter diffraction and crystal plasticity based simulations are combined to determine the effect of M<sub>23</sub> C<sub>6</sub> carbides on the deformation of an annealed AISI 420 steel. Results suggest that carbides provide a more effective hardening at low plastic strain by a predominant long-range interaction mechanism than that of a pure ferritic microstructure. Carbides influence local strain by elastic incompatibilities with the ferritic matrix and also the spatial interactions between ferrite grains. The development of strain observed near ferrite grain boundaries is enhanced by the presence of carbides. However, this effect is mitigated at regions with high density of carbides and ferrite grain boundaries. Generation of artificial microstructures with controlled distribution of precipitates emerges as a powerful tool for the understanding of heterogeneous strains development.

representative volume element; finite element crystal plasticity Abstract: Understanding the local strain enhancement resulting from different microstructure features in metal alloys is crucial in many engineering processes as it plays an important role in the work hardening and in other processes such as recrystallization and damage. Isolating the contribution of precipitates to the development of heterogeneous strain can be challenging due to the presence of grain boundaries or other microstructure features that might cause ambiguous interpretation. In this work a statistical analysis of local strains measured by electron back scatter diffraction and crystal plasticity based simulations are combined to determine the effect of M23C6 carbides on the deformation of an annealed AISI 420 steel. Results suggest that carbides provide a more effective hardening at low plastic strain by a predominant long-range interaction mechanism than that of a pure ferritic microstructure. Carbides influence local strain by elastic incompatibilities with the ferritic matrix and also the spatial interactions between ferrite grains. The development of strain observed near ferrite grain boundaries is enhanced by the presence of carbides. However, this effect is mitigated at regions with high density of carbides and ferrite grain boundaries. Generation of artificial microstructures with controlled distribution of precipitates emerges as a powerful tool for the understanding of heterogeneous strains development .  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65

Introduction
It is well known that strain development in metallic alloys is critically affected by the microstructural characteristics such as grain size of matrix phases as well as size, density and nature of existing precipitates. These characteristics influence the dislocation motion in the structure and play a fundamental role in the mechanical behaviour of the metallic alloys.
Understanding the evolution of complicated dislocation structures in metals and their effect on the hardening behaviour of the materials during deformation is a major issue in materials science. Dislocations are commonly categorized into redundant and non-redundant dislocations, respectively called Statistically Stored Dislocations (SSDs) and Geometrically Necessary Dislocations (GNDs). GNDs share similar Burgers vector and they allow the accommodation of lattice curvature due to non-homogeneous deformation. The exact manner in which GNDs contribute to the strengthening of materials is not completely understood.
Existing GNDs locally interact with moving dislocations by forming jogs that provide macroscopic isotropic hardening under strain development [1,2]. Pile-ups of GNDs also lead to the development of long-range back stresses, which result in kinematic hardening [3]. The relative significance of each mechanism varies with the overall imposed strain and size of microstructure elements related to material strengthening [2].
The literature relating the microstructure to the properties via the development of GNDs in metal alloys is abundant [4][5][6][7][8][9]. GNDs typically accumulate at the grain boundaries due strain incompatibilities of grains with different orientation or constituents with dissimilar properties such as hard precipitates in a soft metal matrix. The relation between GNDs and grain size has both theoretical and experimental validations and it can be linked to the well-known Hall-Petch effect [9,10], considering that at small grain size, the grain boundary layer in which GNDs typically accumulate encompasses a relatively large volume fraction of the material [6]. The relation between hard precipitates and GNDs in metal alloys is still unclear.
It was extensively reported that large-size carbides and other brittle particles have a notably adverse effect on the low-temperature toughness because they improve the formation and propagation of micro-cracks [11,12]. De Cock et al. [13] proposed that coarse cementite carbides develop a deformation zone with a high dislocation density, which promotes the formation of a recovered and equiaxed ferritic matrix. Vivas et al. [14] suggested a similar mechanism induced by coarse M 23 C 6 carbides to explain the formation of fine equiaxed ferrite grains in ferritic/martensitic chromium steel during creep tests. These regions with high population of large M 23 C 6 carbides and small size ferrite grains with low angle boundaries tend to develop microcavities, worsening the material performance to creep. It is, therefore, necessary to study and understand the local strain enhancement and lattice distortion resulting from different microstructure features, i.e. grain boundaries, precipitates or a combination of both, because they not only play an important role in the work hardening of the material but also in other processes such as recrystallization and damage inheritance and fracture.
In this work, we study the effect of large M 23 C 6 carbides in the local strain development and strain hardening of an annealed AISI420 stainless steel. To isolate the effect of carbides from ferrite grain boundaries, two approaches are followed. As a first approach, real material is subjected to interrupted tensile tests at different strain levels and the development of local strain arising from different microstructure features is characterized by electron backscatter diffraction (EBSD). The second approach uses a digital recreation of the microstructure with different 3D representative volume elements (RVE) and crystal-plasticity based simulation of the strain development in the microstructure under uniaxial tensile deformation using DAMASK software [15].

Microscopic modelling based on crystal plasticity
The strain and stress development in different digital recreations of the microstructure was simulated combining a crystal plasticity model and the spectral solver based on FFT (Fast Fourier Transform) provided by DAMASK software. Here, only the constitutive equations for the elastic and plastic deformation are broadly presented. For a complete description of simulation procedure, the reader is referred to Refs. [15,16]. The intricate stress interactions between the grains of a polycrystalline material are modelled numerically using the spectral 4 element (SE) method. Each grain is represented by one or more finite elements, and the polycrystal is subjected to boundary conditions that simulate the deformation under specific constraints. The single crystal plasticity model is combined into the SE framework to define the constitutive relation at each integration point of the element. The deformation in the continuum theory of crystal plasticity is described as a multiplicative decomposition into elastic, F e , and plastic, F p , parts of the deformation gradient F, where the elastic part accounts for lattice distortion and rotation, and plastic distortion arises due to slip: The elastic stress is expressed in form of the 2 nd Piola-Kirchhoff stress S, and depends only on the elastic strain expressed as the Green-Lagrange strain tensor E and the material specific stiffness C, according to For cubic crystals in this study, the elastic stiffness matrix is composed of three independent terms, C 11 , C 12 and C 44 . It is worth to note that reversible dislocation glide, i.e. dislocation anelasticity, is not considered in the model. The evolution of plastic strain is given by: where L p is the plastic velocity gradient. A widely adopted phenomenological description for the hardening is used in the present work, which is based only on slip of multiple slip systems β i . The evolution of critical shear stress, , i.e. the hardening, of individual slip systems in a single crystal is given by: The instantaneous slip-system hardening moduli h βη , in general, depend on the history of slip and provides information about additional hardening caused by interactions of fixed slip systems β and active slip systems η. h βη is determined by 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   5 The parameters h 0 , and τ sat are respectively the reference hardening, the critical slip resistance and the saturation shear stress, and depend on the crystal structure and the slip system. The parameter a, typically a ≥ 1, has not a direct physical meaning, but has a direct influence on the development of hardening. The latent hardening parameter, q βη , defines the interaction between system β and η and is set to 1, if β and η are coplanar, otherwise q βη = 1.4.

⁄
The shear strain rate of the system η is restricted by it resolved shear stress, , and : where n is related to the strain rate sensitivity of slip and is the reference shear rate, being both material-specific variables. The shear rates of all slip systems can be then used to determine the plastic velocity gradient: where N denotes the number of slip systems (N = 12 for iron {110} bcc based on [17,18] and 12 for M 23 C 6 carbide {111} fcc ), m the normalized slip direction and n the unit normal of the slip plane.

Experimental procedure
The AISI 420 steel used in this study contains 0.32 wt.% C, 0.2 wt.% Si, 0.3 wt.% Mn and 13.7 wt.% Cr and it was received in the form of fully annealed sheets of 0.45 mm thickness.
Sub-size tensile test specimens following the ASTM E8/E8M−13a standard [19] and miniature tensile test specimens, with dimensions shown in Figure 1, were machined with the long axis (gauge section) oriented along the sheet rolling direction. Sub-size specimens were tested in an Instron 5500R electromechanical tensile test machine, with load cell of 50 kN, at room temperature and in extension control mode. A clip-on extensometer with knife-edges, a gauge length of 7.8 mm and a maximum extension of ±2.5 mm was used to record the elongation during the tensile test. Three strain rates, 0.01 s -1 , 0.001 s -1 and 0.0001 s -1 , were applied to evaluate the strain rate sensitivity.
Miniature specimens were subjected to interrupted tensile tests to approximately 0.5 mm, 10 mm and 15 mm of cross-head displacement in a Deben micromechanical tester. The plastic strain achieved in each deformation step has been derived by the change in length between the 6 centre of two indents placed along the gauge and separated by 1 mm distance, as shown in Figure 1b. These indents were made by a position-controlled Vickers hardness tester applying a load of 1 N. Extra indents were made to delimit a square area of approximately 60 x 50 μm 2 upon which EBSD analysis was conducted in the unstrained condition and after each deformation step. Previous to the interrupted test and after making the indents, the top surface of the tensile specimen was prepared following the same procedure as for microstructure characterization by EBSD. This procedure consists of grinding and polishing the specimens with a final polishing step to 0.03 μm colloidal silica solution during 60 min to mitigate the plastic strains introduced at the surface during the process.
A FEI Quanta 450 scanning electron microscope equipped with a Field Emission Gun (FEG-SEM) and EDAX-TSL, OIM Data Collection software were used to obtain EBSD patterns.
The set-up conditions are detailed as: acceleration voltage of 20 kV, spot size #5 corresponding to beam current of 2.4 nA, working distance of 16 mm, tilt angle of 70°, and step size of 50 nm in a hexagonal scan grid. Plasma cleaning step was carried out before each EBSD test to make sure that no changes in indexation was caused by pollution/damage of the sample due to an earlier scan. Post-processing and analysis of the orientation data was performed with TSL OIM® Analyses 6.0 software. A grain confidence index (CI) standardization was applied to the raw data, with a minimum tolerance angle and grain size of 5° and 6 pixels respectively. It was considered that grains are formed by multiple pixel rows.
Thereafter, neighbour-orientation correlation with a tolerance angle of 5° and a minimum confidence index of 0.1 was implemented.

Characterization of the initial microstructure and RVE Generation
In order to recreate simulation RVEs as closely as possible representing the real material, the initial microstructure of the steel was extensively studied and quantified using EBSD and XRD. The microstructure of annealed AISI 420 consists of ferrite with various precipitate particles, predominantly M 23 C 6 carbides and MX carbonitrides [21]. Figure 2a shows a secondary electron image of the microstructure before deformation in which ferrite grains smaller than 10 μm are highly populated with precipitates. Large round precipitates are identified as M 23 C 6 Fe-Cr carbides by energy dispersive X-ray spectroscopy (EDS). This was confirmed by X-ray diffraction (XRD) analysis. M 23 C 6 diffraction peaks are clearly resolved in the diffractogram of Figure 2b. M 23 C 6 carbides mainly precipitate along prior austenite grain boundaries and boundaries with a large misorientation angle [22][23][24]. M 23 C 6 carbides tend to coarsen easily because the solubility of iron and chromium, the major constituents in this carbide, is high. Homogeneously distributed nanometre size precipitates, likely MX nitrides, can be also distinguished within ferrite grains. The MX carbonitrides typically precipitate finely and densely in the matrix bcc phase, mainly along dislocations, and do not grow significantly at high temperatures [25]. Although not confirmed by EDS, their presence in the microstructure is detected by XRD (Figure 2b).
The overlapped phase and image quality EBSD maps are shown in Figure 3. White lines delimit ferrite grain boundaries with misorientation angles larger than 10 °. Image quality map reveals some ferrite grains sharing boundaries with angles smaller than 10 °. Ferrite exhibits a broad grain size distribution, with clusters of small equiaxed grains surrounding regions of elongated larger grains which might be a reminiscence of a recrystallized rolled microstructure. M 23 C 6 carbides, identified as an FCC phase in EBSD, are stochastically distributed along the ferrite matrix. Dark spots, with low image quality, can be appreciated in the image quality map indexed as BCC phase. These spots might be emerging carbides situated in the range of the depth penetration of the electron beam or carbides that were wiped out during the sample preparation process. To account for these carbides in the quantification process, points with image quality lower than a threshold based upon bimodal distribution are filtered and added to the phase quantification. A carbide fraction of 0.032 is measured by EBSD, which is a small fraction compared to the 0.11±0.01 estimated by XRD. MX precipitates have FCC lattice and might be confounded with M 23 C 6 carbides. However, the estimated size by SEM analysis falls below the 50 nm EBSD step size, and presumably 8 cannot be resolved by this technique. A threshold of 6 kernels (for a size larger than 100 nm) is adopted for grain identification to avoid accounting these precipitates as M 23 C 6 carbides.
The size distributions of ferrite grains and M 23 C 6 carbides are shown in Figure 4 and the relevant statistics are collected in Table 1. These values were obtained from several EBSD maps from surfaces perpendicular to the normal and transverse direction. More than 1000 grains were included in the analysis. The results were fitted to a lognormal distribution with characteristic μ and σ 2 parameters, which represent, respectively, the mean and variance of the The ferrite inverse pole figure map in Figure 3b indicates a preferential orientation of {111} bcc parallel to the normal direction, i.e. the γ fibre, which is typical for rolled steels [26]. Texture was translated in terms of an orientation distribution function (ODF), which along with the microstructure statistics was used to generate a representative volume element of AISI 420 steel microstructure by StatsGenerator filter of DREAM.3D software [27,28]. The algorithm first creates a collection of idealized ellipsoidal grains having a distribution of size, shape and shape orientation equivalent to those observed in the experimental microstructure. Secondly, the generated grains are placed inside a representative volume. A number of constraints are used to determine the arrangement and spatial location of the grains inside this volume.

Calibration of materials parameters
The calibration of the crystal plasticity constitutive parameters for ferrite, τC,0, τsat, h0, a and n, was performed based on strain-stress data from tensile test (see Figure 6) and the implementation of a modified Nedler-Mead (NM) simplex algorithm following a similar procedure as described in [29]. NM simplex algorithm [30], outstands for its simplicity and easy implementation. Its deterministic character and independence on gradient information makes it suitable for the relatively low dimensional optimization inverse problem. The algorithm iteratively adjusts the parameters by performing crystal plasticity simulations of the uniaxial tensile deformation of RVE and comparing the resulting strain-stress data to the experimental reference. When the deviation, evaluated in the present work by the error sum of squares, meets a given tolerance, the algorithm finalises. The fitting process was performed for three strain rates to best optimise the n parameter. The bounds of calibrated parameters were defined based on typical values for ferrite presented in Table 2. There is a lack of dedicated data of ferrite parameters in stainless steel and in general for steel systems.
Therefore, the bounds were largely expanded, maintaining the limits of typical values in metals. In the crystal plasticity model, plastic deformation initiates only by slip and is regulated for a single crystal by τ C,0 . It can be considered that the calibrated value of τ C,0 will effectively account for other strengthening mechanisms, e.g. presence of grain boundaries, dislocations, precipitates, etc., which also affect the material yielding. Therefore, in the polycrystal and for uniaxial tension, estimation of τ C,0 was made by connecting this parameter with the yield stress (σ y ) and the arithmetic mean of the Taylor factor, M, by: The experimental σ y for 0.0001 s -1 strain rate, determined by the 0.2% offset method, resulted to be 222 MPa. The measured M for the ferrite in present steel is 2.9, which leads to a value of 76.5 MPa for τ C,0 , applying Eq.(9).
Finding experimental measurements of the mechanical properties of single crystal of M 23 C 6 carbide is more challenging and thus, a rather qualitative choice of material parameters has been made. Ab-initio calculation of elastic properties of different M 23 C 6 carbides can be found in Liu et al. [31]. M 23 C 6 carbides could be considered as rigid particles compared to the ferrite matrix. There is evidence that M 7 C 3 can yield to plastic deformation at moderate temperatures, where dislocation gliding and deformation twins mechanisms operate [32], 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 10 however no information was found on its yield stress. Inoue et al. [33] determined the hardness of (Fe-Cr) 23 C 6 depending on several alloying elements and concluded that Mn, present in AISI 420 composition, has a minor effect. Values of 1100 HV0.3 were reported, which can be translated to 10.8 GPa [19]. Assuming that the yield strength is one third of the hardness Vickers value [34] and considering M = 3, τ C,0 results in 1200 MPa. The rest of the parameters were selected in order to emulate a particle exhibiting high hardening. Simulations revealed that despite the stress being partitioned to carbides, it did not develop to values high enough to initiate plastic deformation. Hence, the hardening related parameters, except τ C,0 , can be considered anecdotic for M 23 C 6 in the frame of the present study.   [7], which were optimised for ferrite in a dual phase steel. τ C,0 and τ sat , also adopted for ferrite in a dual phase steel, are higher in Ref. [35], which indicates that any comparison should be done with care. Anyhow, the low τ C,0 and τ sat values in Ref. [36] are consistent with the fact that they are for plain iron ferrite. It is worth to note that the calibrated τ C,0 for the present steel is close to the value estimated by Eq. (9). The reference hardening, h 0 , is significantly higher in the present work compared to all consulted references, whereas a takes an analogous value to most of them. The strain rate sensitivity obtained for the studied range of strain rate is considerably lower than for ferrite in other studies as it is deduced from the high n value.

Local strain development by EBSD analysis
The local evolution of strain during tensile deformation of AISI420 microstructure was evaluated by the analysis of the Kernel Average Misorientation (KAM) parameter from EBSD scans as credited in a number of studies [37,38]. KAM accounts for the local average crystal misorientation Δ<θ> around the distance Δ x from a measurement point [39], which can be connected to the lattice curvature tensor (κ) by the Nye tensor associated to GND density (α). In a simplified one-dimensional (scalar) representation and assuming only parallel edge dislocation of the same sign, this tensor relation is expressed as [40] :  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 11 ⁄ ⁄ (10) where dθ/dx can be approximated as Δ<θ> / Δ x and b is the modulus of the Burgers vector, which is (a/2)<111> for bcc lattice. The lattice parameter, a, of the ferrite phase in AISI20 steel is 2.8723±0.0001 Å, which was calculated from XRD peaks and the Nelson-Ridley method [41]. c is a constant that depends on the geometry of the boundaries, having values of 2 and 4 for pure tilt and pure twist boundaries, respectively. In reference [38], it is demonstrated that using Nye's tensor, α = 3, which represent a mixed-type boundaries and is selected for the present study. More sophisticated methods considering extended components of the Nye tensor [42,43] and cross-correlation EBSD techniques for the estimation of GND density can be found in literature e.g. [5,9,44]. Although present results could potentially differ from these studies, it is likely that they would follow the same trends.
Strain heterogeneities were revealed by KAM maps at all deformation levels, as representatively shown in Figure 7a for the 0.066 plastic strain condition. High KAM values are observed near grain boundaries (delimited by white lines) and particularly close to carbides (black ellipsoids), while low KAM values are rather more spread throughout the material. KAM is very sensitive to the EBSD step size [42,44], and the overestimation due to the measurement noise is drastically increased when decreasing the step size. The method proposed by Kamaya [45], illustrated in Figure 8a, is used to estimate the measurement error. In absence of measurement noise, the extrapolated <θ(x)> values to x = 0 should tend to zero.
In Figure 8a, it can be observed that <θ(x = 0)> ranges between 0.4 ° and 0.6 ° at different strains, which can be considered as an estimate of the measurement noise. Assuming that under very small (shear) strains dγ = tg(dθ) ≈ dθ and the estimated measurement noise, the shear plastic strain detection limit is in the range of 0.007 -0.01, which falls below the chosen macroscopic plastic strains. Hence, it is considered that measurement noise is not significantly Electronic copy available at: https://ssrn.com/abstract=3558254 12 affecting evaluation of local strains by analysing KAM maps except for the unstrained condition. Figure 8b shows an ascending linear relation between the plastic strain and the average KAM values (or equivalent GND density by applying Eq. (10)), which up to the strain level analysed here (0.139) well reproduces the Ashby model [40] and is consistent with the observations reported by other studies on ferritic steels [42,43]. Standard deviation in the distributions of KAM was used to generate the error bars. The increase of standard deviation with increasing plastic strain is explained by the strain heterogeneities present in the specimens. This is evidenced by the evolution of KAM distribution with plastic strain in Figure 9a, which appears highly skewed with a pronounced tail to the high KAM side of the peak. It is worth noting that the KAM distribution (including measurement noise) in the unstrained condition already exhibits significantly high values compared to the strained specimens. In the unstrained specimen, high KAM values are mainly concentrated at M 23 C 6 /ferrite interfaces (see Figure 10), which may be explained by thermal strains arising from differences in the thermal expansion coefficients during cooling [5] or from lattice mismatch [46]. The average KAM value for the carbide phase is higher than for ferrite in the unstrained condition. This  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  14 on the evolution of KAM with plastic strain compared to others is difficult to assess by only looking at the plots in Figure 11 because of the combined effects of N G and N C . The following multivariate linear regression model, which includes the macroscopic plastic strain ε P , is fitted: with K / i = 0, 1, 2, 3, 4 are the model coefficients and A the parcel area. K is the intercept and represents the average KAM in the case of unstrained condition for the ferrite phase without carbides or grain boundary contribution. The interaction of NC and NG is introduced by the last term. Table 3 collects the resulting coefficients after applying multilinear regression analysis based on model of Eq. (11), along with the standard error and significance parameters.
From the linear regression model it can be observed that K 0 is on the order of the measurement noise, which means that mean KAM should tend to 0 in absence of grain boundaries and carbides, which is what is physically expected. The high values of K 1 indicate that KAM is primarily influenced by strain, and the value of 2.6° matches well with the slope of linear relation of KAM with plastic strain presented in Figure 11. The increase of KAM with the number of carbides (i.e. carbide density) is more effective than with the number of grains (i.e. grain boundary density), as resulting from comparing K 2 and K 3 , but this is only evident in the 50x50 grid. The positive effect of these variables was expected from the analysis of box and whisker plots. However, the negative sign associated with the interaction parameter K 4 means that the regions with high density of grain boundaries and carbides will counteract the development of KAM by the positive effect of K 2 and K 3 .
Former analyses disregard the influence of the position of carbides in the microstructure on the intensity of KAM. Grid parcels in the EBSD map were classified according to the position of carbides by a categorical variable, CP, which is coded as described in Table 4. Figure 12 shows  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   15 cementite. Their results showed that when the cementite diameter is at micron scale, the thickness of GND layer is independent of carbide size. Ma et al. [48] argued, however, that when cementite diameter is at submicron size, as it is the case of the present study, a ratio should be maintained between the thickness of GND layer and the diameter of the particle.
Average size of intragrain M 23 C 6 carbides is smaller than that of carbides at grain boundaries, which might explain the lower values of average KAM. Nevertheless, average KAM in parcels containing carbides at grain boundaries is typically higher than that in parcels in which only grain boundaries were recorded. From Figure 12, it is also observed that the average KAM of different CP at high plastic strains tends to equal with NG and NC.

Isolation of carbide effect by model microstructures
The effect of carbides on the mechanical response is evaluated by a new representative volume element, RVE FERR. , sharing the microstructure topology and texture with RVE CARB. , but in which material properties for the M 23 C 6 carbide are substituted for those of ferrite phase. Any difference in the crystal plasticity modelling between RVE FERR and RVE CARB. will consequently be attributed only to an incompatibility arising from a mismatch in the strength between two phases. Figure 13 shows the strain and stress curves after tensile deformation of RVE CARB. and RVE FERR. with a strain rate of 0.0001 s -1 . Differences in the yielding of RVE CARB. , RVE FERR.
are insignificant. However, the initial hardening rate of RVE CARB. is slightly higher than that of RVE FERR. , which results in a lower ultimate tensile strength in RVE FERR. Figure 14a shows the distribution of equivalent Von Mises strain for the ferrite phase in RVE CARB. and RVE FERR. at macroscopic plastic strains similar to those of the experimental tests. For the initial unstrained condition (not shown in Figure 14) the strain distribution is represented by a Dirac delta function. A broadening of strain distribution is observed in both RVEs with increasing strain. Broadening is more pronounced in RVE CARB. at all strain levels, indicating a more heterogeneous strain, by development of regions with high strain levels and others in which strain development is hindered compared with fully ferritic microstructure. It is not possible to establish a direct correspondence between experimental and simulated results, considering that the former only accounts for plastic deformation leading to GNDs (but not due to the motion and generation of SSDs), and the latter includes in the statistics the deformation at every point in the RVE regardless the operating mechanism. Nevertheless, it can be concluded that the simulations are consistent with the experimental measurements.
Electronic copy available at: https://ssrn.com/abstract=3558254 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   16 The ferrite strain distribution function resulting from deformation of RVE CARB. , h . , can be decomposed in two parts: 1) f, which accounts for texture and strain incompatibilities between ferrite grains affecting the deformation of ferrite, i.e. the strain distribution function resulting from RVE FERR. and 2) g, which considers the effects of carbides, including the influence of carbides on the ferrite-ferrite grain interactions. The following procedure was followed in order to obtain g . . First, since h = f * g, where * stands for convolution, g was obtained from h of RVE CARB. by deconvolution of f. The shapes of f and g can be considered representative of each individual effect to the global strain heterogeneities captured by the shape of h. However, the summation of the f and g areas does not correspond to the area of h.
This fact was already predicted upon analysing the strain distribution curves in Figure 14a. develop higher strain levels regardless the presence or absence of carbides, which indicates that carbides not only influence local strain directly by elastic incompatibilities with the ferritic matrix, but also the spatial interactions between ferrite grains.

Assessment of micromechanical model
The

⁄ 2 (14)
where is the elastic anisotropy factor, is the elastic modulus of the crystallographic plane <hkl> and are the elastic compliances. The equation relating the elastic compliances with the elastic stiffness matrix components can be found in, e.g., Knowles et al. [49]. A high population of grains oriented with <111> normal to the load direction in the RVE, with E 111 = 306 GPa, explains the high Young's modulus obtained in the simulated tensile curves. Figure   5 shows that as the number of ferrite grains decreases in RVE* CARB. , the grain orientation with a random texture RVE leads to E = 216 MPa, which is still higher than that for the alloy, but credits the choice of C ij .
Another issue arises during the transition from elastic to plastic deformation. The instantaneous work hardening, ⁄ , is plotted as a function of stress in Figure 17 for a better assessment of elastic to plastic transition according to Arechabaleta et al. [50]. In the experimental plot, an abrupt monotonic decrease of Θ below the material yield stress indicates an anelastic regime, consequence of dislocation bow-out by Orowan mechanism, but with limited dislocation glide. This trend is more progressive in AISI 420 steel than in pure iron and low alloy ferritic steel [50], which points to an effect of the carbides or the reduced ferrite grain size. Initial instantaneous modulus of around 275 GPa is attributed to inaccuracies in the strain measurement at low deformation. This value rapidly evolves to a more reasonable value of 200 GPa. Above 220 MPa stress, a change in trend is observed and Θ slightly increases to subsequently smoothly decrease to Θ = 0 GPa when maximum uniform elongation is achieved. Cheng et al. [51] observed this behaviour for overaged aluminium alloys as the precipitates reached an average size at which dislocations cannot longer shear it, which it is assumed the case for large M 23 C 6 carbides in AISI 420.
The modelled curves do not accurately capture the real material hardening behaviour until approximately 275 MPa, well into the plastic regime. However, they do recreate an abrupt decrease in Θ after a prolonged plateau maintaining the initial instantaneous modulus. This decrease cannot be attributed to a reversible anelastic behaviour, since this is not included in the modelling, but to a progressive activation of glide in differently oriented grains. Following a generalization of Schmid's law, any slip system will be active if and only if the corresponding shear stress and stress rate on that system reach critical values [52,53]. The question that arises is which stress value should be considered for the initiation of yielding. If the point in which first slip systems start to glide is selected, there are no significant differences between RVE CARB. and RVE FERR. . This would indicate a negligible influence of the carbide phase at low strains, which contribution to the deformation of the artificial 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 19 microstructures is only expected to be a local increase in strain due to elastic incompatibilities. However, differences in the stress values increase as the strain increases for the artificial microstructures, reaching a maximum of 18.6 MPa near maximum uniform elongation. This makes σ y , defined as the 0.2%-offset yield stress, slightly higher in RVE CARB.
(208 MPa) compared to RVE FERR. (203 MPa). However, it cannot be explained directly by classical strengthening mechanisms in the synthetic microstructures.
Phenomenological constitutive formulation suffers from the drawback that the material state is only described in terms of the critical resolved shear stress, , and not in terms of lattice defects population. Nevertheless, if a carful interpretation is made, useful information can be extracted from the application of this model. Based on the hardening law adopted in this work, Acharya and Bassani [54] formulated that h βη depends both on the slips and their gradients via the incompatible lattice deformations, i.e., Nye's GND density evolution with strain, : where e ijkl denotes the alternating vector. Under multi-slip deformation of ferrite in this study, a distinction should be made for short-and long-range character of the interaction between a mobile dislocation and the GND content. Short-range interactions may be fully accounted for in the hardening rate matrix, h βη . The contribution to stress of long-range interactions due to the presence of GNDs can be considered to arise solely from the presence of an incompatibility. Incompatibility will develop in ferrite-ferrite and ferrite-M 23 C 6 boundaries as a direct consequence of the individual orientations of grains, combined with the anisotropy of the elastic stiffness tensor. Under these assumptions, and considering that the ferrite parameters are the same, the differences observed between RVE CARB. and RVE FERR. would be due to the contribution of M 23 C 6 carbides to hardening due to both long-range and short-range interactions.
The results should be also interpreted considering the limitations in the model when tackling spatial grain interactions, which affect the manner in which grains deform and rotate influencing the development of texture. An advantage of crystal plasticity finite element predictions is that they can reproduce experimental findings well. However, they are can be stated that observations made in RVE CARB. and RVE FERR. would be reasonably well reproduced in equivalent real microstructures.

Effect of carbides on plastic flow
The work hardening of AISI 420 annealed microstructure, magnitude 330 MPa, is higher than that for other annealed ferritic low carbon and stainless steels, which are in the range between 100 MPa to 200 MPa [58]. This fact is explained by high density of large M 23 C 6 and small MX precipitates. The effect of small ferrite grains is also significant, following from simulations with modelled full ferritic microstructure resulting in only less than 20 MPa of work hardening difference with respect to modelled microstructure containing large M 23 C 6 carbides. However, the calibration of parameters for ferrite also accounted for the effect of small MX precipitates, which were not included in the simulated microstructures. Therefore, the effect of small ferrite grain size should be corroborated in other carbide free ferritic microstructures.
Large M 23 C 6 carbides would also result in high initial hardening rates, which are typically observed for metal alloys with a low volume fraction of non-shearable precipitates [51]. This  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   21 has been attributed to two mechanisms: 1) the storage of geometrically necessary dislocations, 2) the storage of elastic energy of the second phase. The calibrated h 0 hardening parameter is higher compared to other ferritic steels [7,35,36]. High h 0 is typical for martensitic steels with high dislocation density and small grain size [7,59] Figure 17), or equivalent, up to approximately 0.05 strain. At higher plastic strains the hardening behaviour of both RVEs is equivalent. This suggests that the main contribution of large M 23 C 6 carbides to hardening occurs at low plastic strain. Fleck et al. [2] argued that the dominant source of hardening at levels of plastic strain below a few percent is by the generation of long-range elastic stresses generated by a gradient of the strain α. At higher levels of plastic strain, the long-range elastic stresses will be relaxed by cross slip and the principal source of hardening becomes isotropic hardening associated with short-range dislocation interaction. In the hardening model, h βη and interdependently evolve with macroscopic plastic strain according to Eq.(5), Eq.(6) and Eq. (7) which complicates discerning the character of carbides interaction. However, the saturation stress is the same in RVE CARB. and RVE FERR since this represents the physical limit to the density of dislocations that might be stored in the ferrite crystal. Therefore, the difference of 20 MPa in the ultimate tensile strength between RVE CARB. and RVE FERR arises only due to differences in hardening below 0.05 strain, and can be assumed a consequence of long-range interactions.

Figure 13
True strain true stress curves obtained from experimental (Exp.) and modelled (RVE CARB. and RVE FERR. ) tensile tests.