Relationship of Internal Stress Fields with Self-Organization Processes in Hadfield Steel under Tensile Load

Abstract

The effect of tensile strains on the microstructure of Hadfield steel was studied by transmission electron microscopy (TEM). Stages of the obtained stress–strain curves were observed, and correlated well with the evolution of the dislocation substructure. Based on an analysis of TEM images, quantitative parameters were determined, such as the material volume fractions, in which slip and twinning occurred, as well as twinning, which developed in one, two and three systems. Some transformation mechanisms were reported that caused great hardening of Hadfield steel. In particular, a complex defect substructure formed in a self-organized manner due to the formation of cells, the dislocations retarded by their walls, as well as the deceleration of dislocations on twins and, vice versa, of twins on dislocations. These factors affected both the average and excess local density of dislocations. Additionally, they resulted in elastic stress fields, which manifested themselves in the curvature–torsion gradient of the crystal lattice. A high level of stresses caused by solid-solution strengthening prevented the relaxation of elastic ones, contributing to the strain hardening of the Hadfield steel.

1. Introduction

Hadfield steel possesses a good combination of strength, ductility and tribological properties [1,2,3]. Hence, it is widely applied as a structural material in a variety of industries for the manufacture of armor products, hammers of crushing and grinding machines, teeth of excavator buckets, railroad turnouts, etc. Hadfield steel belongs to the austenitic class, characterized by the dominance of one slip system in any orientation within a wide strain range under tensile load [4]. In such cases, microstructural transformations are the subject of a significant number of studies using various methods, including infrared thermography (IRT) [5,6], in situ digital image correlation (DIC) [5,6,7], synchrotron X-ray [8], neutron [9] and X-ray diffraction (XRD) [10,11], electron backscatter diffraction (EBSD) [12,13], as well as transmission electron microscopy (TEM) [14,15,16]. Some mechanisms of its plastic deformation have been described as well [17,18].

Many authors have discussed the reasons for enhancing strength of the Hadfield steel. For example, an increase in hardness from 200 up to 520–600 HB as a result of impact loads has been associated with martensitic transformations [3,19]. At the same time, only 0.5–1.5% of martensite has formed of the 110G13L grade at high strains [20], which cannot be a determining factor in enhancing strength, since limiting dislocation density is achieved and deformation twins are formed [19,20]. Another approach to hardening such medium-manganese steels is connected to the formation of the ε phase as a result of shear processes without manganese redistribution via diffusion [21]. An important feature of this process is the formation of new martensite regions, which is not only due to the growth of previously formed martensite lamellae [22,23]. A number of authors have shown that the enhanced hardness of such steels after plastic deformation is mainly caused by the increased density of dislocations, as well as the concentration of twins and stacking faults (SFs) [24,25,26,27]. At the initial deformation stage, twins are located in one shear system, the number of which increases then. The formed martensite region sizes are of the nanoscale and undergo transformation into α-martensite. Its amount depends on the orientation of austenite grains relative to the load direction [25].

Despite the large number of papers that have been devoted to the study of the medium-manganese austenitic grades, there is no generally accepted theory of the hardening of both Hadfield steel and manganese austenite so far. In this regard, an advanced synergistic approach should be noted, which has been actively developed for more than two decades. It is based on the fact that a deformable body is a highly nonequilibrium multilevel system capable of internal self-organization [28,29]. Some authors have reported quite interesting results in the study of different classes of materials and at various scales. For example, a periodic change in the amplitude of increments of local ε_xx elongations is associated with the mutually self-consistent development of centers of the local ε_xx elongations, ε_xy shears and ω_z rotations during the deformation of single-crystal Hadfield steel [30].

Based on the analysis of the published data, some specific patterns can be noted that significantly affect both the strength and ductility of Hadfield steel [31]: (1) low-stacking fault energy; (2) variations in the sliding mechanisms for partial and/or full dislocations, as well as deformation twinning with rising strains; (3) changes in the slip mechanisms during the formation of deformation-induced ε-martensite; (4) the presence of several types of the deformation-induced martensitic transformations.

As shown in [28,32], the formation of ordered spatial structures at different scales is observed when an external field is applied to a strongly nonequilibrium open (poly- or single-crystal) system. This process is controlled by the internal characteristics of the crystal and is aimed at dissipating energy input from an environment in order to preserve crystal integrity for as long as possible. Consequently, the formation and successive change in the type of dislocation substructures should be considered a self-organizing process at the microlevel. It should be noted that self-organizing processes are those in which more complex and perfect structures arise [33]. They are possible in systems (1) that are thermodynamically open, (2) in which the emergence and amplification of fluctuations is energetically favorable, (3) that are characterized by positive feedback, (4) that have a sufficient amount of interaction between external factors that must be accumulated and enhanced by themselves and (5) that possess a sufficient number of interacting elements [31,33]. Hence, all these conditions are met by the deformation processes occurring in Hadfield steel. Therefore, it is of interest to relate changes in its microstructure during deformation with such self-organization processes.

The aim of the present study was to reveal a relationship between self-organization processes in a defect substructure in the deformed Hadfield steel and the evolution of both dislocations and twins.

2. Materials and Methods

The study was performed using plates of Hadfield steel (Fe-13%Mn-(1.0–1.3)%C (wt.%) of the 110G13 grade according the Russian classification), quenched from a temperature of 1050–1100 °C. The plate dimensions were 140 × 15 × 1.5 mm.

The plates were deformed with an ‘Instron’ uniaxial tension machine at a strain rate of 10^–3 s^–1. Based on the obtained data, both true stresses and true strains were assessed using the σ_true = (1 + ε)σ and ε_true = ln(1 + ε) ratios, where σ is engineering stress and ε is engineering strain [34]. The θ = dσ_true/dε_true strain hardening coefficient was determined as a result of differentiation of the stress–strain curve in the ‘σ_true–ε_true’ coordinates.

For TEM examinations with an ‘EM-125′ microscope (Sumy Plant of electron microscopes, Sumy, Ukraine), foils were fabricated from the deformed Hadfield steel plates via the electric spark method and subsequent electropolishing. The dislocation substructure was investigated by analyzing the obtained TEM images, for which quantitative description was carried out by measuring the following parameters: both scalar and excess dislocation densities, the density of deformation twins, and the sizes of dislocation cells in the substructure.

Strains were calculated using the following formula:

$$\epsilon \hspace{0.33em}=\hspace{0.33em}ln(1+\frac{\mathsf{\Delta }l}{l_0}).$$

(1)

where $\mathsf{\Delta }l$ is the sample length change and $l_0$ is the initial sample length.

The samples were strained up to 2, 5, 10, 14, 20, 28, and 36%. Before that, a reference row of diamond pyramid imprints had been preliminarily applied on the polished sample surface with a step of 50 μm. At each strain value, a change in the distance between the reference points was assessed.

The scalar dislocation density was determined via the secant method with a correction for the invisibility of dislocations [35]. A rectangular grid was used as a test line. Then, the scalar density of dislocations was calculated using TEM images using the following formula:

$$\rho =\frac{M}{t}\left(\frac{n_1}{l_1}+\frac{n_2}{l_2}\right)\left[{\text{cm}}^{-2}\right],$$

(2)

where M is the micrograph magnification, and n₁ and n₂ are the number of intersections of the l₁ horizontal and l₂ vertical lines by dislocations, respectively. At high strains, the density of dislocations was great, so the TEM images were used with a direct magnification in the microscope column of at least 40,000.

The ρ_± excess dislocation density was measured locally via the misorientation gradient [35]:

$${\rho }_{\pm }=\frac{1}{b}\cdot \frac{\partial \varphi }{\partial l}\left[{\text{cm}}^{-2}\right],$$

(3)

where b is the Burgers vector of dislocations and $\frac{\partial \varphi }{\partial l}$ is the foil curvature gradient or the χ lattice curvature, which is explained below. The χ values were determined by shifting the $\partial l\hspace{0.33em}=\hspace{0.33em}\Delta l$ extinction contour at the $\partial \varphi \hspace{0.33em}=\hspace{0.33em}\Delta \varphi$ controlled foil tilt angle in the microscope column using a goniometer. In this case, it was required that the effective reflection vector was perpendicular to the goniometer tilt axis. Otherwise, a recalculation was needed, because the effective reflection plane no longer contained the goniometer inclination axis. The foil cross section, on which the measurement was carried out, had to not contain interface boundaries or misorientations in the contour movement path, i.e., the bending of the foil had to be continuous. In terms of misorientations in the FCC alloys based on nickel, copper, and iron, the contour width was ~1° [35]. This value in combination with the contour width enabled the determination of the misorientation gradient.

The density of deformation microtwins was determined via the secant method [36] (a rectangular grid was used as a test line):

$${\rho }_{tw}=M\left(\frac{n_1}{l_1}+\frac{n_2}{l_2}\right)\left[{\text{cm}}^{-1}\right].$$

(4)

where M is the micrograph magnification and n₁ and n₂ are the number of intersections of the l₁ horizontal and l₂ vertical lines by twins, respectively.

The scalar density of dislocations was assessed from the TEM images [35]. According to the theory of dislocations, they were divided into both ρ₊, positively charged, and ρ₋, negatively charged. The scalar dislocation density was calculated using the following expression:

ρ = ρ₊ + ρ₋.

(5)

The excess dislocation density was calculated as follows:

ρ_± = ρ₊ − ρ₋.

(6)

According to [35,36], it was possible to determine the χ lattice curvature, which was related to the ρ_± excess dislocation density, as in the following:

$$\chi =\frac{\partial \varphi }{\partial l}=\frac{1}{r}=b{\rho }_{\pm }.$$

(7)

where b is the Burgers vector, φ is the angle of inclination of the crystallographic plane, and l is the distance on the plane. The $\frac{\partial \varphi }{\partial l}$ value could be determined either from the S width of the bending extinction contour, or from its displacement when the foil was tilted using a goniometer [35,36]. In this study, the first option was applied.

3. Results

In the Hadfield steel, deformation typically led to the formation of microtwins [27,37], while areas of their initiation and deceleration were the sources of internal stresses. This phenomenon was also confirmed in the present study. In particular, Figure 1 presents an instance of bending near a twin deceleration region at a tensile strain of 13%. According to Figure 1a, twinning in a grain developed in both ($1\overline{1}1$)[$\overline{1}12$] and ($\overline{1}11$)[$1\overline{1}2$] systems along the maximum and minimum loaded planes (indicated by solid lines in Figure 1a; the Schmid factor values are also presented). At the intersection of these systems of microtwins, deceleration occurred. As a result, two parallel A and B contours were observed on the TEM image. When the goniometer inclination angle (φ) changed from 0 up to 10.5 degrees, the A [002]_γ and B [$00\overline{2}$]_γ bending contours moved towards each other (Figure 1a,e,f). This meant that the crystal was bent around an axis parallel to the bending contours in the twin deceleration region.

Figure 2 shows another example of complex bending near the twin deceleration region in the Hadfield steel at a tensile strain of 25%. In this case, the result of deceleration was a complex closed loop, in which twinning had developed along two maximally loaded planes. Twins of the (111)[$\overline{2}11$] system had been decelerated by the ($\overline{1}\overline{1}1$)[112] ones. Respectively, a complex pattern of elastic–plastic bending of the crystal lattice arose.

With the development of strains, the curvature–torsion gradient increased in the crystal lattice along the plate. This was reflected in the changing width of the extinction contours along their axis and their curvature. As a result, the contours were bent and the angle of their contact with the deformation twin boundary changed. Preliminary estimated values of the changes in the amplitudes of the curvature–torsion of the crystal lattice, both on average over the plate volume and in regions containing deformation microtwins, depending on the plastic strains, are shown in Figure 3. Based on their analysis, it could be stated that the regions of localization of the largest values of the curvature–torsion amplitude of the crystal lattice were the zones of the nucleation of deformation microtwins. With the rising plastic strain, curves 1 and 2 gradually converge in Figure 3. As noted above, twinning was random through the plate volume in the strain range from 0 up to 5%, so the average χ₁ value over the plate volume was low. At ε_true > 5%, twinning developed intensively and the χ₁ value increased sharply.

As it was found in [27] based on the results of TEM investigations, the fraction of the Hadfield steel volume, in which intense twinning occurred, increased with rising tensile strains. In this case, twinning developed most intensively in the strain (ε) interval of 5–20%. In this study, the TEM images in Figure 1 and Figure 2 showed that mutually intersecting systems of microtwins could be present in one grain. An important feature of the material response to rising tensile strain was the enhancement in the proportion of the material which included two or even three systems of twins, as well as the tendency to form packets from microtwins. At the same time, the number of microtwins per package increased with the rising tensile strain (from 3–4 at ε~10% up to 6–8 at ε~20%). Finally, almost the entire plate volume was covered by twinning at ε~36%. As a result, both amplitudes of the curvature–torsion of the crystal lattice, namely the χ₁ averaged over the plate volume and the χ₂ value in the deformation microtwin regions, actually coincided; i.e., χ₁~χ₂ according to Figure 3.

4. Discussion

4.1. The ‘σ–ε’ and ‘θ–ε’ Dependences

It was found for all metals, alloys and steels, that the ‘σ–ε’ dependences in the plastic strain region possessed stages differing in the strain hardening intensity. For single crystals and single-phase materials, such staging was discovered in early studies (see reviews [37,38]). Then, it was confirmed for polycrystals of pure metals with different syngonies [39,40,41], as well as for solid solutions and intermetallic compounds of FCC metals [42,43,44]. Data on the stages of plastic stress–strain curves for various metallic materials were summarized in [45,46,47,48].

An analysis of a large amount of data showed that the ‘σ–ε’ dependences were multi-stage in general. The following stages were distinguished one after another. Firstly, the π transitional one was accentuated, which followed the yield point and demonstrated either an increase or a decrease in the θ = dσ/dε strain-hardening coefficient. After π, stage II was typically observed, characterized by an almost constant coefficient (θ). A further increase in external stresses resulted in the parabolic hardening stage, stage III, with a decreased coefficient (θ). Then, the transition from stage III to IV was reflected by a very low and constant coefficient (θ). Up until fracture, further dynamics of the changing coefficient (θ) could be characterized by either decreases or increases [19,20,47,48]. For the 110G13 Hadfield steel, both the stress–strain curve and the strain hardening coefficient (θ) versus the plastic strain (ε) dependence during a tensile test were given in [27] with marked the stages described above (Figure 4).

As can be seen in Figure 4, stage II could be divided into two substages with different hardening factors: ${\theta }_{I{I}_1}$ = 3 × 10³ MPa and ${\theta }_{I{I}_2}$ = 2.1 × 10³ MPa. A turning-point of the ‘σ–ε’ curve corresponded to a strain of 5%, coinciding with the beginning of the twinning process and decreasing the strain hardening coefficient (θ) (discussed in detail below). In many papers [1,20,24,49,50,51,52,53,54,55,56,57,58,59,60,61,62], the high hardenability of the Hadfield steel was highlighted when studying the mechanical properties of both poly [1,20,55,56,57,58,59,60,61] and single crystals [1,20,24,59,60,61,62]. At the same time, high values of the strain hardening coefficient (θ) and the linear nature of the ‘σ–ε’ dependence of the 110G13 Hadfield steel were associated with both sliding deformation and mechanical twinning.

Since it was known that the dislocation substructure type determined the staging of the flow curves and the strain hardening coefficient (θ) [37,39,40,63,64], their changes had to be expected for austenitic steels in general and the Hadfield grades in particular, as a result of the dislocation substructure transformation.

On the other hand, SFs were observed simultaneously with flat dislocation clusters when studying the dislocation substructure of polycrystal austenitic stainless steels [56]. Additionally, the important role of SFs of strengthening elements of the dislocation substructure was shown. Therefore, an increase in the strain hardening coefficient (θ) and the formation of a cellular dislocation substructure could be expected with the development of dislocations and SFs in the austenitic steels under tensile loads. In this case, if microtwins were formed, the density of which increasing with rising strains, then they could result in additional strengthening effects, enhancing the strain hardening coefficient (θ).

Rising strains in austenitic steels were accompanied by enhancing stresses and changes in the deformation mechanism from slip to twinning, which could become the main one. In this case, the number of active twinning systems could determine staging of the flow curves and the strain hardening coefficient (θ). It is known [42,43] that if twinning develops due to a deformation of the slip mechanism, then competition between cross slip and twinning could be observed. After slip, the development of twinning in one system has to suppress the cross-slip processes and causes an increase in the plasticity of crystals, i.e., the twinning-induced plasticity (TWIP) effect has to be observed. At low plastic strains (i.e., ε < 5%), a low strain hardening coefficient (θ) could be expected. The development of twinning in several systems had to result in a high θ value close to that in martensitic transformations. In [27], the influence of the dislocation substructure type, both slip and twinning deformation mechanisms, as well as the number of existing slip and twinning systems on the staging of the flow curve and the strain hardening coefficient (θ) of Hadfield steel was investigated in detail.

It is known that steel hardening results in the accumulation of a scalar density of dislocations organized into a substructure, which contributes to the formation of stress fields. An uncharged dislocation ensemble (i.e., one without excess dislocations) gives shear-stress fields, caused by the dislocation substructure, according to the following formula [65]:

$$\sigma =\alpha Gb\sqrt{\rho },$$

(8)

where α varies within 0.05–1.00, depending on the dislocation ensemble type [64] (α = 0.625 for a charged dislocation ensemble [66]), G is the shear modulus, b is the Burgers vector, and ρ is the scalar density of dislocations.

In the case of a charged dislocation ensemble, when the excess dislocation density corresponded to the relation,

ρ_± = ρ₊ − ρ₋ ≠ 0,

(9)

moment stresses were formed, and the following expression could be applied:

$${\sigma }_{\partial }={\alpha }_{c}Gb\sqrt{{\rho }_{\pm }},$$

(10)

where α_c = 1 is the Strunin coefficient [67].

The dislocation substructure and the scalar dislocation density were characterized by the excess dislocation density. The latter also contributed to the formation of internal stress fields (moment stresses), which were identified via the presence of extinction contours in the steel. The point is that the bending extinction contour was caused by the diffraction contrast observed on the TEM image of the deformed crystal structure. This was the locus of points where a given family of atomic planes remained parallel to themselves and, therefore, were in the same reflective position. This was the reason for using TEM technique to measure internal stress fields, since only their presence led to the foil bending or the corresponding curvature of the crystal lattice, if the foil retained the plate shape [68]. The procedure for measuring the magnitude of internal stresses and the excess dislocation density was reduced to determining the χ = Δϕ/Δl curvature gradient of the foil or the crystal lattice (the details are discussed in the next subsection). Then, the excess density of dislocations (ρ_±) was measured locally along the misorientation gradient [64,69]:

$${\rho }_{\pm }=\frac{1}{b}\times \frac{\Delta \varphi }{\Delta l}$$

(11)

where b is the Burgers vector.

4.2. Mechanisms Providing the High Hardening of the Hadfield Steel

As was shown above, the Hadfield steel possessed a high strain hardening coefficient (θ) and was characterized by the atypical ‘σ–ε’ curve (Figure 3). The key point that distinguished the Hadfield steel from other materials was the linear long-term hardening stage, stage II. Figure 5 schematically shows the main hardening stages for the FCC metallic materials [37], so the greatest hardening was typically observed only at stage II and could be either short or long, depending on the plastic strain value. It was short (a few percent) as a rule, but was prolonged by up to 20% for ordered alloys with high energy values for antiphase boundaries [37]. Respectively, such materials were considered unique ones. The obtained stress–strain curve (Figure 3) indicated that the Hadfield steel had the same unique properties as the ordered FCC alloys. Firstly, it possessed the long stage, stage II, reaching ~20%. Secondly, the strain hardening coefficient (θ) was also high; it was ~3000 MPa before twinning and ~2100 MPa after that. The third feature of the Hadfield steel was the formation of the dislocation cellular substructure, which provided the θ strain hardening coefficient (θ) at stage II [37,40]. A feature of the cellular substructure was strong shear inhibition, since gliding dislocations could cross no more than 5–6 cell walls during their advance, after which the shear decayed [37,40]. For instance, Figure 6 schematically demonstrates the strong shear inhibition in a cellular substructure.

The fourth feature of the Hadfield steel was the simultaneous microtwinning with the dislocation slip. As is known, microtwins are a strong obstacle to the dislocation glide, and the shear could only cross several twins located parallel to each other. Figure 7 schematically shows the dislocation shear deceleration in a package of microtwins.

The fifth feature was the fact that microtwins contributed to great internal stresses, the estimated calculations of which were given in [27].

The sixth feature was related to the SF energy (SFE) value. It is known that gliding dislocations in a FCC lattice are split into partial and interconnected SFs (Figure 8) [70]. Depending on the SFE value, the presence of SFs between partial dislocations determined the splitting distance (r) of a full dislocation. This distance played an important role in plastic deformation. If the r value was negligible, dislocations easily carried out cross-slip or climbed and left their slip planes. In this case, dislocations could bypass obstacles during their sliding. If the r value was great, then the deformation process was followed only by twinning and dislocation slip played a lesser role. An intermediate case was also possible at an average r value. This took place at SFE levels of within 25–50 erg/cm². With such splitting, a cellular substructure was formed, the wall of which were quite stable and difficult to overcome by sliding dislocations. At the same time, the twinning mechanism could be activated at these SFE values and could make a noticeable contribution to strain hardening (if it was not the main one).

The above-mentioned main features differentiated the Hadfield steel from other FCC materials. However, they were associated only with dislocation glide and twinning, forming a contribution to the so-called substructural strengthening (σ_substrO). At the same time, σ_s-s solid-solution hardening (σ_s-s) also had a significant effect on the mechanical properties of the Hadfield steel. It included a deceleration of interstitial atoms and substitution of those dissolved in the solid solution, as well as of the ones of interstitial elements precipitated on dislocations, sub-boundaries and grain boundaries. Due to the presence of microcarbides in the Hadfield steel, albeit in small amounts [71,72], dispersion strengthening (σ_disp) had to be considered as well.

Thus, according to the above data, flow stresses were contributed to in the Hadfield steel by three factors. However, this analysis of the main strengthening mechanisms was not complete, since the contribution of σ_is internal stresses (σ_is) was not taken into account. Therefore, it required additional determination and evaluation.

In view of the above, the flow stress (σ) of the Hadfield steel could be expressed, taking into account all four factors:

σ = σ_substr + σ_s-s + σ_disp + σ_is.

(12)

Contributions of each component of Equation (12) to the flow (σ) stress and strengthening of the Hadfield steel are presented below.

4.3. Features of Strain Hardening and Self-Organization Processes in the Hadfield Steel during Plastic Deformation

According to Equation (12), the flow stress (σ) of the Hadfield steel included four components, and the values of each were important for predicting its deformation behavior. The solid-solution hardening (σ_s-s) did not depend on the ε value, i.e.,

σ_s-s(ε) = const.

(13)

Figure 9 shows the flow stress curve (σ) and the solid-solution hardening (σ_s-s) contribution equal to 110–120 MPa. The difference between these two values (σ′ = σ–σ_s-s) was strain hardening. σ_s-s solid-solution hardening (σ_s-s) level could be determined in different ways, some of which are discussed below.

The first method was based on the following algorithm. Contributions of (σ_elast)elastic stress fields (σ_elast)and the substructural hardening (σ_substr) could be subtracted from σ stress value (σ), i.e.,

σ_s-s = σ − σ_substr + σ_elast.

(14)

At a strain of 5%, the σ_s-s value was 110 MPa. It should be noted that the σ_disp contribution that could give carbides (see Formula (12)) was not considered in Equation (14), since the microstructural studies showed their low concentration in the Hadfield steel. Therefore, carbides could not have a noticeable effect on the defective substructure and the dispersion hardening (σ_disp) contribution could not be taken into account, respectively.

The second method was based on drawing the σ = f(ρ^1/2) dependence in accordance with the well-known expression for the substructural (dislocation) contribution [65]:

$${\sigma }_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{t}\mathrm{r}}\hspace{0.33em}=\hspace{0.33em}m\alpha Gb\sqrt{\rho }.$$

(15)

In a defect-free material (at the ρ dislocation density is → 0), and the σ_substr contribution is → 0 as well. Therefore, a σ value of 120 MPa was obtained via the σ_substr = f(ρ^1/2) experimental dependence after extrapolating ρ → 0 (Figure 10), which corresponded to the solid-solution hardening (σ_s-s) contribution. Thus, two close σ_s-s values of 110 and 120 MPa were estimated, and the corresponding data are also shown in Figure 9 (curve 2). Hence, it was obvious that σ′ strain hardening (σ′) of the Hadfield steel was determined by the remaining components,

σ′ = σ − σ_s-s,

(16)

which are shown in Figure 9 (curve 3).

Thereby, the strain hardening (σ′) of the Hadfield steel was determined via the substructural contribution of dislocations and twins:

σ′ = σ_disl + σ_tw.

(17)

Both of these contributions (σ_disl and σ_tw) formed elastic stress fields. In addition, the σ_disl~$\sqrt{\rho }$ and σ_disl~1/D dependences had to be considered, where D is the size of dislocation cells. The D values were determined via the method of random secants [36], both by measuring them in different foil planes and by averaging all available planes. Consequently, it was most expedient to divide the σ′ value.

It should be noted that elastic stress fields were formed by both dislocations and twins, but it was reasonable to separate their contributions. The reason was the fact that dislocation-induced stresses included two components: contact and barrier deceleration. The contact one, arising from the intersection of dislocations and the formation of dislocation reactions [73,74,75,76], was proportional to $\sqrt{\rho }$:

$${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}\hspace{0.33em}=\hspace{0.33em}m{\alpha }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}\mu b\sqrt{\rho }.$$

(18)

Dislocations also contributed to barrier deceleration [42]:

σ_disl = σ_contact + σ′_barrier,

(19)

where σ′_barrier is deceleration by the walls of dislocation cells, and σ′_barrier~1/D [38,74,77,78], i.e.,

$${{\sigma }^{\prime }}_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}}=\hspace{0.33em}{\alpha }_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}}\mu b{D}^{-1},$$

(20)

where ${\alpha }_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}}$ = const~2–4 [79].

Figure 11 shows that the dependence σ = f(1/D) characterized the cellular substructure by its linearity. At 1/D = 0, the intersection point was the sum of both σ_s-s and σ_contact contributions, i.e., the dislocations that were not organized into the walls of dislocation cells.

The cellular substructure formed in the Hadfield steel obeyed the basic equations of such a typical case [80,81], namely

D = Cρ^−1/2.

(21)

This was the well-known Holt relation [82] that was determined via self-organization of a dislocation ensemble, which is confirmed in Figure 12. Since the Holt relation reflected high work-hardening degrees [81], it was the formation of the cellular substructure that primarily determined the high hardenability of the Hadfield steel.

Packets of twins also affected the σ′_barrier contribution. The data reported in [27] clearly describe the formation of twin packs found on the TEM images obtained in this study (Figure 1 and Figure 2). Entanglement of dislocations was observed at the twin–matrix boundary, reflecting the strong deceleration of dislocations. At the same time, dislocation-free boundaries of twins existed simultaneously, the quantity of which was much greater [27]. Thus, barrier deceleration meant that the flow stress (σ) was directly proportional to the density of twins, ρ_tw. This fact reflected well a dependence shown in Figure 13.

Elastic fields could be calculated in several ways. Firstly, they could be calculated from the total scalar density of dislocations [66,76,83]:

$${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}}\hspace{0.33em}=\hspace{0.33em}m{\alpha }_{\mathrm{d}\mathrm{i}\mathrm{s}}\mu b\sqrt{\rho },$$

(22)

where α_dis is a parameter characterizing the elastic interaction of dislocations.

Secondly, they could be calculated from the density of excess or geometrically necessary dislocations [65,66,67,84,85]:

$${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}}\hspace{0.33em}=\hspace{0.33em}m{\alpha }_{\mathrm{d}\mathrm{i}\mathrm{s}}^{\prime }\mu b\sqrt{{\rho }_{\pm }},$$

(23)

which is shown in Figure 14.

The third mechanism, linking elastic stress fields with the defect substructure, was the σ_disl~ρ_tw proportionality. This fact is well reflected in Figure 15, demonstrating a very important feature of the Hadfield steel’s work hardening, which has not been highlighted by anyone so far. Almost all parameters of its defect substructure changed proportionally to each other. For example, both ρ_tw and ρ_± values were affected by the elastic stress fields (Figure 16).

The scalar density (ρ) of dislocations and the density of twins (ρ_tw) were also linearly related (Figure 17). The data presented in Figure 10, Figure 11, Figure 15, Figure 16 and Figure 17 warranted the conclusion that self-organization processes took place in the Hadfield steel, contributing to the hardening mechanisms. This phenomenon was confirmed by the obtained results.

Self-organization in the defect subsystem occurred for a number of reasons. First, dislocations that formed the cellular substructure and were subsequently retarded by the walls of cells. Second, twins decelerated dislocations and affected the dislocation density. The strain inhomogeneity due to twinning contributed to the excess dislocations and curvature–torsion of the crystal lattice. Third, dislocations were decelerated on twins and, vice versa, twins were decelerated on dislocations. Fourth, the entire set of defects formed elastic stress fields.

The above Figure 14 shows both σ_sustr and σ_tw dependences along with the σ_dis one, i.e., hardening associated with the dislocation substructure and microtwins. Due to the high initial scalar density of dislocations, substructural hardening made a stable significant contribution (Figure 14, curve 2), which was already great at the yield point. The contribution of twins to strain hardening was negligible at first, but sharply increased at strains of 15–20% (Figure 14, curve 3). Thereby, as can be seen in Figure 14, three constituents affected the work hardening (σ′):

σ′ = σ_subst + σ_tw + σ_disl.

(24)

Respectively, both twins and the dislocation substructure contributed to the σ_disl value (Figure 15 and Figure 16), although the latter well shielded elastic stress fields. Finally, the strain hardening coefficient of the Hadfield steel could be written as follows:

$${\theta }_{II}\hspace{0.33em}=\hspace{0.33em}\frac{d\sigma }{d\epsilon }\hspace{0.33em}=\hspace{0.33em}\frac{d{\sigma }_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{t}}}{d\epsilon }\hspace{0.33em}+\hspace{0.33em}\frac{d{\sigma }_{\mathrm{t}\mathrm{w}}}{d\epsilon }\hspace{0.33em}+\hspace{0.33em}\frac{d{\sigma }_{dis}}{d\epsilon }.$$

(25)

It can be concluded that the evolution of the defect substructure, the increase in the scalar density of dislocations and the density of twins contributed to work the hardening of the Hadfield steel, which is in full accordance with the up-to-date theory of dislocations and twins. This was the first important result of the study. The second one was that the complex defect substructure of the Hadfield steel formed in a self-organized manner. The conclusion is that solid-solution hardening did not directly contribute to work hardening. However, it was the high σ_s-s value that ensured deceleration of dislocations and twins, preventing the relaxation of elastic stresses, and thus determining the strain hardening of the Hadfield steel.

5. Conclusions

The obtained results warrant the following conclusions.

The TEM examination of the microstructure in the deformed Hadfield steel revealed a complex defect substructure, formed in a self-organized manner due to the formation of cells, dislocations retarded by their walls, as well as the deceleration of dislocations on twins and, vice versa, the deceleration of twins on dislocations. These factors affected both the average and excess local density of dislocations. Additionally, they resulted in elastic stress fields, which manifested themselves in the curvature–torsion of the crystal lattice. A high level of stresses caused by solid-solution strengthening prevented the relaxation of elastic ones, contributing to strain hardening of the Hadfield steel.

Based on the analysis of the stress–strain curves of the Hadfield steel, their staging was determined. It was shown that the high value of the strain hardening coefficient at stage II of linear strain hardening (about 20%) was caused by the formation of the dislocation cellular substructure. The beginning of the intensive microtwinning process contributed to the change in the strain hardening coefficient. Hence, it is necessary to consider the contribution of solid-solution strengthening in the strain hardening level when analyzing the deformation behavior of Hadfield steel.