Revisiting “Steady-State” Monotonic and Cyclic Deformation: Emphasizing the Quasi-Stationary State of Deformation

The analysis of microstructural changes in mechanical steady state will be based on a wide range of published earlier experimental data, most of which have not been analyzed comprehensively in similar fashion before. Essentially, the data refer to experimental documentation of mechanical properties, accompanied by measurements of electrical resistivity, magnetic properties, X-ray diffraction, and direct observations by TEM (transmission electron microscopy). These data provide information on

As will be discussed in Section IV–B, characteristic features of quasi-stationary deformation are that, after mechanical steady state has been attained, a small but non-negligible persistent increase of the dislocation density ρ is noted, stemming, for example, from a continuing decrease of the spacings s (mesh size) in the dislocation networks, and also from increases of the density of geometrically necessary dislocations (GNDs) in the dislocation cell walls/subboundaries, as evidenced from increasing misorientations across the cell walls/subboundaries, compare References 8, 9. These findings show that the dislocation density continues to increase in steady state and thus contradicts the current general opinion that the dislocation density is constant in steady state. Very recently, and in a somewhat different context, Huang arrived at the same conclusion.[16]

Another important feature of deformation-induced dislocation distributions is that dislocations are not distributed randomly but tend to cluster, forming dislocation wall and cell structures. It is important to clarify whether and how this phenomenon of the so-called dislocation patterning, compare Kubin,[17] affects the constitutive equations of plasticity.

The microstructural data will be related to the (shear) flow stress τ, as described by the Taylor flow-stress law[18]:where G is the shear modulus, b the modulus of the Burgers vector, and α a geometrical factor that takes into account the arrangement of the dislocations. Thus, the α-factor should, for example, provide evidence of dislocation patterning. Usually, a value of α ≈ 0.35 to 0.4 is used, as estimated for dislocation intersection, i.e., forest cutting, first by Saada[19] and later by Schoeck and Frydman.[20] Variations of the α-factor and changes of the dislocation density during the so-called steady-state deformation are normally not considered.[21‐23] In fact, one finds in the literature statements like “paradoxically, realistic strain hardening properties in uniaxial deformation are obtained without accounting for dislocation patterning”[21] or that there exists “a relative insensitivity of the dislocation strengthening relation to the arrangement of the microstructure.”[23] However, in the present work, evidence will be provided, showing that as the dislocation density continues to increase mildly during mechanical steady-state deformation, the α-factor varies and can in fact decrease by about 20 pct. This conclusion was first drawn qualitatively about 45 years ago from a study on cyclically deformed α-iron single crystals in cyclic saturation[24] and has been substantiated since, compare References 8, 9, 25. In the next Section, it will be shown in a simple dislocation model that the α-factor is expected to decrease as the dislocation distribution becomes more heterogeneous in the process of clustering of dislocations and/or the development of a cell structure.

Examples of dislocation microstructures in different deformed metals and alloys are shown in Figure 4 for unidirectional deformation[26,27] (Figures 4(a) and (b)), cyclic deformation[9,28,29] (Figures 4(c) and (d)), and high-temperature creep[30,31] (Figures 4(e) and (f)). It is obvious that these dislocation microstructures are rather complex and differ considerably. Nonetheless, they have one thing in common. In all cases, the distribution of the dislocations is rather heterogeneous, with areas of low and high local dislocation densities, respectively.

It is desirable to describe at least semi-quantitatively the properties and the flow stresses of deformed materials containing different heterogeneous dislocation distributions. In the simplest possible procedure, the flow stress for a homogeneous dislocation distribution with a total mean dislocation density ρ is described in terms of the Taylor flow stress \( \tau_{\text{Taylor}} \) and denoted as τ_hom with an arrangement factor \( \alpha_{ \hom } \):

Usually, the value of \( \alpha_{ \hom } \) is considered to lie in the range \( \alpha_{ \hom } \) ≈ 0.35 to 0.4 which is characteristic of dislocation cutting, as stated earlier.

In the case of a heterogeneous dislocation distribution, the simplest approach is to describe the flow stress \( \tau_{{{\text{het}} }} \)in terms of the two-component composite model[25,32,33] which considers only dislocation cell wall regions of high local dislocation density ρ_w with a volume fraction f_w and cell interior regions of lower local dislocation density ρ_c with a volume fraction f_c. For simplicity, the α-factor is taken to be the same in the cell walls and in the cell interiors. Then, the flow stress \( \tau_{{{\text{het}} }} \) can be written in terms of a rule of mixtures with the weighted contributions of the dislocation cell walls and cell interiors:

When the terms are lumped together, the flow stress \( \tau_{{{\text{het}} }} \) can be expressed aswith an arrangement factor \( \alpha_{het } \). As shown earlier,[33] it is easy to show that, in general, \( \alpha_{\text{het}} \) < \( \alpha_{ \hom } \), and that the arrangement factors \( \alpha_{ \hom } \) and \( \alpha_{\text{het}} \) are related in a simple one-dimensional model in good approximation as

In this model, one finds \( \alpha_{\text{het}} \) < \( \alpha_{ \hom } \) in all cases, except in the case of a homogeneous distribution, characterized by f_c = f_w, where one finds \( \alpha_{\text{het}} \) = \( \alpha_{ \hom } \), as expected. In this limit, there is no distinction between cell walls and cell interiors. Thus, in the model described earlier, the terms f_c and f_w are interchangeable.[9,33] With increasing heterogeneity, the volume fraction f_c of cell interiors decreases with increasing deformation to typical values in the range of, say 0.3 to 0.1. Inserting these values into Eq. [6], one obtains, with \( \alpha_{ \hom } \) ≈ 0.4, values of

0.24 < \( \alpha_{\text{het}} \) < 0.36 in the range of 0.1 < f_c < 0.3.

It follows from the cross check that, at the same time, f_w increases, in agreement with the experimental finding that the volume fraction of the walls increases with increasing deformation.[27] It is pointed out that, in the original work[33] and in an earlier publication,[9] the range of values of the volume fractions were formulated in terms of 0.1 < f_w < 0.3 instead of 0.1 < f_c < 0.3, and the same basic conclusions regarding \( \alpha_{\text{het}} \) were drawn. However, strictly speaking, this formulation would have implied incorrectly that the volume fraction of walls would decrease with increasing deformation, in disagreement with experiment.[27]

In summary, it is predicted that the arrangement factor \( \alpha_{\text{het}} \) is expected to decrease with increasing heterogeneity of the dislocation distribution in the course of increasing deformation, from approximately 0.36 to about 0.24, i.e., by about 30 pct. In the following sections, based on the assessment of substantial experimental evidence. this rather large effect will be substantiated and verified.

Basinski modified the Taylor flow-stress equation, based on the argument that, when a dislocation bows out between two obstacles, the spacing between the obstacles, i.e., 1/√ρ, should appear in the flow-stress equation as an outer cut-off radius.[34] Thus, Basinski proposed the following corrected flow-stress law:

When the logarithmic term is lumped into the α-factor, the effect of the Basinski correction corresponds qualitatively also to a decrease of the α-factor in the Taylor flow-stress law, as in the previously discussed case of increasing heterogeneity. However, as elaborated by Sauzay and Kubin,[23] the Basinski correction corresponds (only) to a modest decrease of the α-factor, namely 11 pct per decade in dislocation density. Thus, for example, assuming a value α = 0.35 for a dislocation density of 10¹³ m⁻², a value α = 0.31 would follow, if the dislocation density increased by one order of magnitude to 10¹⁴ m⁻². In other words, a tenfold increase in dislocation density would cause the α-factor to decrease by less than only 12 pct.

By comparison with the effect of heterogeneity in the composite model, an important conclusion follows: Even without any increase of dislocation density (!), the effect of increasing heterogeneity during deformation on the α-factor is significantly larger than the effect of the Basinski correction, and can amount up to about 20 to 30 pct. To the author’s knowledge, this rather strong effect of heterogeneity on the “effective” value of the arrangement factor α (for a given constant dislocation density!) has generally not been considered in the past or in more recent comprehensive discrete dislocation dynamics modeling (DDD) simulations, compare, for example, the work of Kubin and co-workers.[21,22] The present statements will be substantiated amply in the following by a large body of experimental evidence.

Under the so-called steady-state conditions, it is assumed that the flow stress τ remains constant, i.e.,

This implies that the dislocation density ρ does not change, i.e., that the dislocation production and annihilation rates are equal and compensate each other. At the same time, the α-factor is considered to remain constant during deformation. In the present work, these assumptions are questioned in the light of experimental observations. Further details follow in the next section by considering the assumptions that are made in order to derive a constant steady-state creep rate.

As stated earlier and as is well known, the condition for steady-state creep is strain rate \( \dot{\varepsilon } \) = const. In the context of the present work, it is important to discuss critically the underlying assumptions that lead to this definition of steady-state creep. For this purpose, the derivation of the well-known recovery creep model[35] is recalled.

Quite generally, the shear flow stress can be expressed according to Seeger[36] as follows:

where \( \tau_{G } \) represents the so-called athermal stress that depends on temperature only through the temperature dependence of the shear modulus G and is given by the Taylor flow-stress law, as introduced in Eq. [1]:

On the other hand, \( \tau^{* } \left( {\dot{\varepsilon },T} \right) \) is the so-called effective or thermal stress that contains the dependence on temperature T and strain rate \( \dot{\varepsilon } \). Following Bailey and Orowan, see Cottrell,[35] the differential change of the (axial) flow stress σ can be written in terms of a strain-dependent hardening term h and a time-dependent recovery term r as follows:

Steady state is attained, when the dislocation production rate \( \dot{\rho }^{ + } \) is exactly balanced by the dislocation annihilation rate \( \dot{\rho }^{ - } \), i.e.,

Then, dσ = 0, and one obtains from Eqs. [10] and [11] the well-known following result for the “steady-state” creep rate:

In the following, one major point will be to show that the assumption made in Eq. [12] and the assumption made earlier, namely that the dislocation densities (and hence the flow stress) remain constant, Eq. [7], are not exactly fulfilled in the course of deformation. In fact, it will be shown that during the quasi-stationary deformation of an initially undeformed material, after a mechanical steady state has previously been attained, a persisting increase of the dislocation density is found, implying that in violation of Eq. [12].

It will also be shown that, at the same time, the arrangement factor α decreases non-negligibly during quasi-stationary deformation. These two effects are largely self-compensating, thus rendering the overall effect on the flow stress marginal. A more detailed discussion will be given in Sections VI–C and VII–C.

In many metals and alloys, dislocation cell/subgrain structures form in high-temperature creep. Figure 5 shows two examples of dislocation subboundaries of a creep-deformed Al-11 wt pctZn alloy.[31] In general, such subboundaries consist of networks composed of dislocations of two or more interacting slip systems. The spacings s in the subboundaries can be determined from such micrographs. Here, it is of interest to note that, even during the so-called steady-state creep, after the strain rate has become constant, these cell structures continue to undergo appreciable changes. According to our earlier definitions, compare Section III, this case of mechanical steady-state creep is simply a preliminary stage in the process of quasi-stationary creep deformation. In the subsequent assessment of the microstructural changes which persist in this stage of creep deformation, the following features of the dislocation pattern should be documented carefully: changes of the cell/subgrain size d, changes of the dislocation spacings s (mesh size), changes of the dislocation density ρ, and changes of the lattice misorientations θ across the cell boundaries.

In Figures 6(a) and (b), the variations of the subgrain size d and the mesh size s in creep-deformed Al-11 wt pctZn alloy[31] are shown. Even after the so-called steady state has been attained, defined by constancy of the subgrain size d beyond a strain of about 0.24 (marked by the red bold broken vertical line in the figure), the dislocation spacings s (mesh size) in the networks continue to decrease. In the same work, an even stronger change after entering the steady-state regime is found for the misorientations θ across the subgrain boundaries, as shown in Figure 7.

In line with this work, Blum,[10] in a study on creep-deformed Al-5 at. pctZn, also noted that the dislocation spacings s continued to decrease after the subgrain size d had become constant, as shown in Figure 8. In addition, Blum was able to show that the α-factor decreased systematically by about 25 pct with increasing strain ε, namely from 0.38 to 0.27, as the strain increased from 0.02 to 0.32, compare Table I. This behavior is in satisfactory semi-quantitative agreement with the prediction derived in Section V–B.

Kassner et al.[37,38] found very similar results, as those shown in Figure 6 for creep-deformed Al-11 wt pctZn alloy, in a study of creep-deformed 304 stainless steel. These data are shown in Figure 9. In another study, Kassner and McMahon showed that, in creep-deformed aluminum, some low-angle boundaries transform into high-angle boundaries[39] with larger misorientations with increasing strain, compare Figure 10. In summary, it is evident that in the so-called steady-state creep, after the subgrain size has become constant, the misorientations continue to increase, while at the same time the dislocation spacings s decrease. The decrease of s at constant cell size d implies that the overall dislocation density ρ has increased. Since misorientations are generally accommodated by geometrically necessary dislocations (GNDs), it is suggested that GNDs contribute appreciably to the increase of the dislocation density. In related work, Kassner and Perez-Prado[37] substantiated the noted increase of misorientation angle and the continuing decrease of the dislocation spacings by considering other published work on different materials.

In summary, after attainment of mechanical steady-state creep deformation, defined by the constancy of the cell/subgrain size, the following microstructural changes continue persistently:

Finally, it should be noted that the microstructural changes observed during quasi-stationary high-temperature creep do not come to a halt, because typical creep strains are rather small and do not exceed some 10 pct. Thus, a final constant steady-state microstructure is usually not reached. In contrast, to be shown subsequently, in cyclically deformed materials, where large cumulative plastic strains of some 100 pct are reached, the microstructure ultimately attains a true microstructural steady state.

The preceding discussion has shown that non-negligible microstructural changes occur at constant flow stress in quasi-stationary creep. In order to understand this behavior, the following flow-stress equation in the form introduced by Ashby,[40] with a modification by the author,[41] is discussed:

In this equation, ρ_s and ρ_G denote the densities of statistically stored and geometrically necessary dislocations, respectively. The factor β takes into consideration that only a fraction β < 1 of the geometrically necessary dislocations that act as forest dislocations enhances the flow stress.[41,42] Based on the preceding discussion of the observed microstructural changes, it follows that the term \( \rho_{\text{s}} + \beta \rho_{\text{G}} \) increases during deformation. However, since the α-factor decreases at the same time, these two effects are almost self-compensating. Altogether, the increase of \( \rho_{\text{s}} + \beta \rho_{\text{G}} \) thus has little effect on the flow stress.

In cyclic deformation, dislocations glide to and fro. At very small strain amplitudes and hence very small dislocation glide paths, it becomes improbable that two dislocations of opposite sign will annihilate mutually when they meet. For a particular type of dislocations, arranged in groups of n dislocations, the condition that dislocations will not annihilate can be formulated for a plastic shear strain amplitude γ_pl and an annihilation distance y in terms of a parameter β as follows[42]:

This implies that as long as β <1, the same dislocations can travel to and fro reversibly without any microstructural changes, whereas for β ≥1, annihilations will occur and slip becomes irreversible. In this latter case, steady state will be maintained through a dynamic equilibrium between generation and annihilation of dislocations. Under conditions of partly irreversible slip, a cyclic slip irreversibility p can be defined as the ratio between the irreversible part γ_pl,irr of the shear strain amplitude and the shear strain amplitude γ_pl as follows:

Thus, slip will be reversible for p = 0 and increasingly irreversible for p > 0. In the first (trivial) case, steady state is maintained without any microstructural changes, whereas in the second case, steady state obtains through a dynamic equilibrium between generation and annihilation of dislocations.

Figure 11 shows a sequence of TEM micrographs obtained on copper single crystals after cyclic deformation to quite different numbers of cycles.[43] The specimens are viewed in \( (1\bar{2}1) \) sections along the line direction of the primary edge dislocations and perpendicular to the primary Burgers vector b. The sequence shows, from left to right, the well-known “ladder” structure of persistent slip bands (PSBs) with primary edge dislocation walls in the early stage (number of cycles N = 5000), and the variations observed at a later stage (N = 1.34 × 10⁵ cycles) and at a much later stage (N = 3.6 × 10⁶ cycles). The very high number of cycles in the latter case could only be obtained by performing the cyclic deformation in vacuum, whereby fatigue failure is delayed by about a factor 10.[43] In this sequence of micrographs, there is a very slight increase of the cyclic stress amplitude, resulting from a weak “secondary” cyclic hardening.[44] The main purpose is to illustrate the nature of the very slow specific microstructural changes that occur during prolonged cycling at almost constant stress amplitude. In early saturation (number of cycles N = 5000), no dislocations are located along the interface between the PSB ladder structure and the adjacent so-called matrix (vein) structure, compare References 43, 45. Then, after N = 1.34 × 10⁵ cycles, the PSB–matrix interface has become wiggly and is decorated with secondary dislocations,[43] and a kind of cell structure has developed inside the PSBs. The varying background contrast indicates noticeable misorientations with respect to the matrix structure. Finally, after even larger numbers of cycles (N = 3.6 × 10⁶ cycles), the regular dislocation walls have been completely replaced by closed cell structures, elongated along the direction of the Burgers vector, with marked misorientations. In the present context, it is important to note that the microstructural changes observed, namely an increase of (secondary) dislocation density, including GNDs (inferred from the misorientations) and the actual misorientations develop very slowly at almost constant stress amplitude.

These findings will be addressed in more detail in the subsequent Figures 12, 13, 14, and 15, which illustrate different types of evolutions of the dislocation microstructure in real so-called steady-state cyclic deformation. Referring to the work of Wilkens et al.,[46] Figure 12(a) shows on the left two cyclic hardening curves (cyclic peak shear stress τ vs number of cycles N). Referring to the curve for the larger plastic shear strain amplitude γ_pl = 4.5 × 10⁻³ (saturation is indicated by a vertical arrow after about N = 200 cycles), Figure 12(b) shows the development of misorientations, as measured by the broadening of X-ray rocking curves for reflections of type {200} as a function of number of cycles. It is important to note that saturation of the broadening is observed about a factor of 10 later than mechanical cyclic saturation, namely after about 2000 cycles. Since the misorientations are essentially accommodated by GNDs, these results imply that the density of the GNDs continues to increase well beyond mechanical saturation at constant stress.

Figure 13 from the work of Polák[47] shows the development of the torsional shear stress in copper polycrystals fatigued in torsion at 77 K and the corresponding changes of the electrical resisitivity Δρ_el as a function of number of cycles. The electrical resisitivity change Δρ_el is directly proportional to the total defect density, i.e., essentially to the total density of deformation-induced dislocations and point defects. Polák showed by annealing experiments that about 50 pct of the electrical resistivity changes are due to the dislocations. Referring only to the curves for the largest amplitude in Figure 13, the following is noted. Whereas saturation of the torsional stress occurs already after about 100 cycles, the change in electrical resistivity saturates only after about 1000 cycles, i.e., a factor of 10 later. It follows that the density of defects continues to increase long after mechanical saturation has been attained.

In ferromagnetic materials like α-iron and nickel, measurements of the coercive force H_c can be used advantageously to study the evolution of defect density.[24,48] Since the coercive force H_c is directly proportional to √ρ,[24] it is directly proportional to the flow stress τ. The cyclic saturation behavior of both α-iron single crystals[24] and nickel single crystals[48] has been studied by magnetic measurements of the coercive force. In Figures 14(a) and (b), results obtained on cyclically deformed nickel single crystals are shown. It is evident that the coercive force H_c continues to increase beyond almost 2000 cycles, whereas the shear stress amplitude τ has saturated already after about 400 cycles. Quite similar results were obtained in the studies on α-iron single crystals.[9,24] In both cyclically deformed α-iron single crystals and nickel single crystals, a truly microstructural steady state, characterized by attainment of a constant value of the coercive force H_c, comparable to a saturation of the electrical resisitivity changes in Figure 13, was not attained. This is attributed to the fact that the number of cycles was too small. In all probability, the coercive force would have saturated, had the number of cycles been sufficiently large.

To summarize, the conclusions drawn from Figure 14, together with the similar conclusions drawn from Figure 13, and the observations that follow from Figure 12, provide proof that, in the so-called steady-state cyclic saturation, the density of the defects, essentially dislocations, continue to increase and that GNDs constitute a substantial part of the increasing dislocation density. As will be discussed in more detail in Section VII–D, it follows from the results of Polák’s work on torsionally deformed copper[47] and from the cyclic deformation studies on ferromagnetic α-iron single crystals[24] and nickel single crystals[48] that the arrangement factor α must have decreased systematically, as the dislocation density continued to increase after mechanical steady state had been attained.

Finally, Figures 15(a) and (b) show TEM micrographs of the dislocation distribution in cyclically deformed α-iron single crystals[28] in early saturation and in deep saturation, respectively. These micrographs document that, during continued cycling deeper in saturation, the cell boundaries become sharper, and the overall dislocation density and the misorientations (which are indicative of GNDs) increase. This finding matches the conclusions drawn earlier from Figures 12, 13, and 14 and confirms that, at more or less constant stress, non-negligible microstructural changes occur and persist. The explanation follows along similar lines as explained earlier in Section VI–C in the case of high-temperature creep.

Referring again to the flow-stress equation, Eq. [1]:the simultaneous constancy of both τ and \( \rho \), can only be explained as follows:

Since the first case is trivial, the second case, i.e., a decrease of the arrangement factor α as a result of a change of the dislocation arrangement, is more probable and considered realistic and will be considered in more detail subsequently.

The decrease of the arrangement factor α during mechanical saturation can be estimated as follows. In the case of Polák’s measurements of the electrical resisitivity of copper deformed in torsion at 77K,[47] the electrical resistivity continued to increase after the shear stress had already become constant (mechanical saturation) by about 55 pct until it reached a constant value (microstructural saturation). Since this increase is proportional to the dislocation density ρ, it follows that α must have decreased by a factor 1/√1.55, i.e., from the initial value α ≈ 0.4 to α ≈ 0.32.

In a similar fashion, the decrease of α can be estimated also, based on the change of the coercive force H_c in the cases of cyclically deformed nickel and α-iron single crystals. In this case, the coercive force H_c increased by ≈ 50 pct in cyclic “mechanical” saturation, hence, α must have decreased accordingly. Since H_c is proportional to √ρ, α must have decreased by a factor of 1/1.5, i.e., from α ≈ 0.4 to α ≈ 0.27. Within the limits of this crude estimate, this result is in satisfactory agreement with the values found above, based on electrical resisitivity measurements.

These estimates are of necessity only approximate. Nonetheless, it can be concluded that the arrangement factor α decreases non-negligibly by 20 pct or more with increasing deformation (and increasing heterogeneity) in cyclic saturation in the so-called steady state. The qualitative agreement of this behavior with the predictions of the composite model (Section V–B) is gratifying. Moreover, it should be noted that the present results for cyclic deformation are in line with the findings in the case of high-temperature creep, Table I. It is concluded that the heterogeneity of the dislocation distribution has an appreciable effect on the flow stress and that this has hitherto been overlooked.

The preceding analysis and discussion of available published work on both high-temperature creep and also cyclic deformation in saturation have provided clear evidence for persisting ongoing microstructural changes in the so-called steady-state deformation. Thus, correctly speaking, what is commonly called steady-state deformation should in fact be referred to as quasi-stationary deformation. In general, this comprises the initial attainment of a mechanical steady state and, finally, also a microstructural steady state with a “constant” microstructure, characterized by a dynamic equilibrium between the production and annihilation of microstructural defects.

It follows from the preceding survey of experimental details and the discussion that a correct description of the transition to a steady-state deformation must address the following aspects:

The present analysis has provided evidence of details of the microstructural evolution that occur during quasi-stationary deformation. These details have been overlooked and/or ignored in earlier work. Thus, in a more rigorous approach, the following changes and details in the evolution of the dislocation microstructure at constant stress (!) deserve more attention and should be considered explicitly, where necessary:

Current flow-stress laws, in which it is assumed that the arrangement factor α is constant, are inadequate to describe satisfactorily all details of the microstructural evolution and the stress-strain behavior. As discussed earlier, the simple Taylor flow-stress law could be retained in a formal sense, if one used adjusted “effective” values of the α-factor.[9] However, it would be much more satisfactory, if refined flow-stress laws and microstructure-based constitutive equations were developed which consider explicitly the interaction of different slip systems and take into account the effects of heterogeneity (patterning).

Considering the impressive progress made in dislocation modeling, analytically or by 3D Discrete Dislocation Dynamics (DDD)[21,22,57‐60] some years ago and also very recently,[61,62] it is hoped that eventually it will be possible to model in some detail the specific microstructural features of plastic deformation which are revealed in experiment, including the evolution of the α-factor, during monotonic and/or cyclic hardening or softening, and also in quasi-stationary deformation. Then, with this capability, dislocation modeling could become a predictive tool rather than only a technique that strives in many cases just to confirm and/or reproduce what is known from experiment.