The Mechanical Threshold Stress (MTS) constitutive formulation is an internal-state variable model that computes stress as the sum of the contributions from individual obstacles to dislocation motion [1] [2] [3] . These contributions are characterized by the threshold stress σ ^ that is the stress at 0 K required to promote dislocation motion past the particular obstacle. Obstacle populations include solutes, precipitates, the Peierls barrier in Body-Centered Cubic metals, and other dislocations, e.g., stored dislocations. This section provides a brief review of the theory and operative equations that comprise the MTS model.

The most general expression for the yield stress as a function of temperature and strain rate is

σ μ = σ a μ + ∑ 1 n s i ( ε ˙ , T ) σ ^ i μ o (1)

where σ_a is an athermal stress, due for instance to the contribution of the interaction of grain boundaries, σ ^ i is the threshold stress for obstacle population i, s_i is a factor between 0 and 1 that characterizes the influence of temperature and strain rate on the stress required to overcome the obstacle, μ is the shear modulus, μ_o is the shear modulus at 0 K, and n is the number of obstacle populations contributing to the stress. A general form for s_i follows from work of Kocks et al. [4] :

s i ( ε ˙ , T ) = { 1 − [ k T g o i μ b 3 ln ( ε ˙ o i ε ˙ ) ] 1 / q i } 1 / p i (2)

where T is the test temperature, ε ˙ is the test strain rate, b is the Burgers vector, k is Boltzmann’s constant, g_oi is the normalized activation energy, and ε ˙ o i , q_i, and p_i are constants. These last four variables are specific to the obstacle population (i) of interest, although as will be shown, these variables show common trends. Equation (1) and Equation (2) specifically apply to the yield stress in annealed metals. These equations are referred to as the “Yield Stress” Kinetics analysis (YSA).

In deformed metals, Equation (1) is written as

σ μ = σ a μ + ∑ 1 n s i ( ε ˙ , T ) σ ^ i μ o + s ε ( ε ˙ , T ) σ ^ ε μ o (3)

where a threshold stress term σ ^ ε arising from the contribution of mobile dislocations with stored dislocations is added (with its associated s-value). A recent paper has addressed the challenge of applying Equation (3) to a material supplied with an existing dislocation density from, for instance, a final warm working operation [5] .

Application of the MTS model to a variety of Body-Centered Cubic (BCC), Face-Centered Cubic (FCC), and Hexagonal Close Packed (HCP) metals as well as to austenitic stainless steels and superalloys has been thoroughly reviewed [2] . Section 2 through Section 5 summarize trends observed in these analyses. In particular assessed values of the normalized activation energies (Equation (2)) and mechanical threshold stresses (Equation (1)) are compared and displayed in several tables.

Equation (3) introduced an additional mechanical threshold stress σ ^ ε . The increase of σ ^ ε with strain results from the rising difficulty of moving dislocations through the array of stored dislocations. This process is referred to as “evolution”, which is used interchangeably with the term “strain hardening”. The increase of σ ^ ε with strain is described differentially, using a modified Voce equation:

d σ ^ ε d ε = θ I I ( ε ˙ ) ( 1 − σ ^ ε σ ^ ε s ( ε ˙ , T ) ) κ (4)

where θ_II is the stage II hardening rate, e.g., of a single crystal, σ ^ ε s is the saturation value of this threshold stress and κ is a constant, usually equal to one or two. Note that the saturation threshold stress has a temperature and strain-rate dependence. This is unique from that defined for the stress in Equation (1) and Equation (2). The kinetics are specified by a dynamic recovery model proposed by Kocks. [6]

ln σ ^ ε s = ln ( σ ^ ε s o ) + k T μ b 3 ( g ε s o ) ln ε ˙ ε ˙ ε s o (5)

where σ ^ ε s o is the saturation stress at 0 K, g ε s o is the applicable normalized activation energy, and ε ˙ ε s o is a constant. Equation (4) and Equation (5) are referred to as the “Evolution” Kinetics Analysis (EA).

Section 6 reviews trends in the parameters σ ^ ε s o , k, and g ε s o across several metals and alloys. These trends are displayed both graphically and in tabular form. The unique contribution of this paper is in the description of common trends in both the Yield Stress Kinetics analysis (YSA) and the Evolution Kinetics Analysis (EA) that could guide application of the MTS model to other metals and alloys. Inspection of the operative equations introduced above indicates a number of model variables. One objective of this paper is to specify ranges for many of these variables and to conclude with a listing of the independent variables in the MTS formalism (see Appendix).

One of the challenges in applying the model is the selection of the operative strengthening mechanisms, which defines the number n in Equation (1). In a pure metal, such as 0.9999 Cu, n may equal 0. That is, there is no strengthening contribution from solutes in this highly pure metal. In most FCC metals, the strengthening from solute additions, whether intentional or not, can be significant. However, the strengthening contributions of all of the impurities are difficult to assess. For most of the metals and alloys analyzed, it has been assumed that n equals 2; that is, two strengthening mechanisms are dominant. Under this assumption, Equation (1) becomes

σ μ = σ a μ + s 1 ( ε ˙ , T ) σ ^ 1 μ o + s 2 ( ε ˙ , T ) σ ^ 2 μ o . (6)

Note that nowhere has it been firmly established that the contributions from the two obstacle populations sum linearly. This question was addressed by Follansbee and Gray in the Ni-C system [7] where the fit using Equation (6), or more simply

σ = σ 1 + σ 2 (7)

was compared to that using

σ = σ 1 2 + σ 2 2 . (8)

The conclusion was that there was no improvement in the agreement with data using Equation (8) rather than Equation (7). Going forth, the linear summation of individual stress components has been assumed.

In the BCC systems, obstacle population 1 was assumed to be the Peierls barrier [8] . This is characterized by very short-range dislocation interactions with a low value of g_o1 (or g_op). The term “short-range” relates to the area swept out by the dislocation as it encounters an obstacle [4] . A second strengthening contribution was assumed to arise from dislocation interactions with impurity atoms. These were longer-range interactions with a moderate value of g_o2 (or g_oi). Some FCC and HCP systems were adequately assessed using a single strengthening contribution, presumably from solute interactions, again characterized by a moderate value of g_oi. Several others, as described in a following section, were better described using two strengthening mechanisms.

It is very difficult to ascertain which solute-dislocation interaction is responsible for the observed strengthening. In the Ni-C system [7] , the alloys were specifically supplied with three different carbon concentrations. The highest concentration was 1900 parts per million (ppm) C, which is well above the trace amounts of other elements, which are typically less than 200 ppm. The effect of C on strengthening in Fe is well known, as is the effect of O and Al on the strengthening in Ti. But in other systems, there may be numerous elements with small, but similar, concentration levels. In these metals, there may be insufficient information on strengthening due to a specific solute, and the validity of the approximation that these all can be described using a single solute strengthening mechanism is unknown.

For example, the chemical analysis of the molybdenum studied by Briggs and Campbell [9] reported 14 ppm O, 12 ppm N, 10 ppm Fe, 70 ppm Si, 100 ppm W, and trace amounts of H and Cu. The analysis published by Follansbee [8] proceeded with a 2-obstacle population model. Clearly the Peierls barrier is one of these; the other represents the contributions of dislocation interactions with the impurity elements. It is an approximation to assume these can all be lumped into a single threshold stress σ ^ i with a single value of g_oi. However, there is insufficient information to do otherwise. This would require an experimental campaign with intentionally variable solute additions. That is, molybdenum alloys with 30 ppm O, 50 ppm O, and 70 ppm O with all other elements unchanged, might yield information about the specific role of O in strengthening. This would indeed be an extensive campaign.

In this section, the procedure to extract the model parameters σ ^ i and g_oi will be reviewed. One starts with a collection of stress-strain measurements as a function of test temperature and strain rate. These measurements should be in a material that is in the annealed condition with a low starting dislocation density. The focus here is on the yield stress. Analysis of the hardening behavior generally follows. With knowledge of the Burgers vector and the temperature-dependent shear modulus [2] (pp. 136-137)¹, the yield stresses are plotted according to Equation (1) with n = 1, when a single strengthening mechanism applies, or n = 2 when two strengthening mechanisms apply. Figure 1 shows the measurements of Briggs and Campbell [9] which cover a strain rate range from 1.7 × 10⁻⁴ s⁻¹ to 100 s⁻¹ and temperatures from 77 K to 600 K. Note that the plot exhibits a distinct curvature. This suggests that a two-parameter analysis is in order. Since this is a BCC metal, the Peierls barrier serves as one of the obstacle populations. The net effect of the, albeit low, concentration of impurities (the chemical concentration detailed above translates to a purity level of 0.9996) likely contributes to the second obstacle population. The procedure requires the analyst to select values of σ ^ p , g_op, σ ^ i , and g_oi that provide a good fit with the measurements. Figure 2 shows the derived model fit along with the measurements. For this fit, of σ ^ p = 1541 MPa, g_op = 0.07, σ ^ i = 428 MPa, and g_oi = 0.27. As expected, the Peierls barrier, with a low g_o value and a high threshold stress, dominates. The strengthening contribution due to the impurities is much less and the g_o value suggests this is a longer-range dislocation-obstacle interaction. The temperature-dependence demonstrated by the Peierls barrier is quite high; in fact, its contribution goes to zero at ~500 K. (This is somewhat strain-rate dependent; at higher strain rates this strengthening contribution would persist to higher temperatures.)

Figure 3 shows the analysis of yield stress measurements in pure Zinc. Measurements reported by Risebrough in 99.999% pure material with a grain size of 20 μm [10] and measurements reported by Liu, Huang, Wu, and Zhang in material with the same purity but with a grain size of 70 μm [11] are plotted on the same coordinates used in Figure 1 and Figure 2. Because of the large grain size difference, a lower value of σ_a (5 MPa) was used for the Liu et al. analysis than for the Risebrough analysis (10 MP). In this case, the data fall nicely along a straight line, suggesting that one strengthening mechanism is operative; this strengthening is likely due to interaction of dislocations with impurity elements. The analysis yielded σ ^ i = 181 MPa, and g_oi = 0.17.

To demonstrate a case where assuming a two-obstacle model can be misleading, a fictitious alloy was created using the parameters b, σ_a, μ(T), etc. defined for Follyalloy [2] [5] . Figure 4(a) gives the yield stress versus temperature and strain rate plot for this alloy. The model shown is a two-obstacle analysis with σ ^ p = 2500 MPa, g_op = 0.19, σ ^ i = 530 MPa, and g_oi = 4.2. A red flag immediately rises with the high value of g_oi, which is an unusually high activation energy.

Figure 4(b) shows the same data set. In this case the model is a four-obstacle analysis with the model parameters shown in Table 1. Included in this table are the model parameters for the two-obstacle model. Interestingly, the fit to the data set is only slightly better in the four-obstacle analysis than in the two-obstacle analysis. The error, defined

Error = { ∑ ​ ( σ m − σ ) 2 } 1 / 2 (9)

is 0.0075 in the two-obstacle analysis and 0.0051 in the four-obstacle analysis. The major difference in the model parameters is in the value of the activation energies. This is an entirely fictitious material and data set, but it demonstrates that assuming two active strengthening mechanisms when having more than two are active can lead to unrealistic values of the model parameters, particularly the activation energies.

In the next section the model parameters assessed for a number of FCC, BCC, and HCP metals are tabulated to enable the observation of trends. In none of these cases is it assumed that more than two strengthening mechanisms are dominant. (This applies to metals in the annealed condition. When strain hardening occurs, the stored dislocation density becomes a third strengthening mechanism, as defined in Equation (3) when n = 2). While there are some “high” values of the activation energy reported, there is no case that mirrors the fictitious alloy described in this section.

In several of the systems studied, a range of compositions of one of the main solute additions has enabled an assessment of the variation of the threshold stress with composition. This was possible in Fe-C [8] , Ni-C [7] , Fe-Al [2] , Zirconium [2] , and Ti-Al [26] . In addition, experimental studies have been performed to analyze the effect of N additions in 316 L stainless steel [33] . Figure 6 shows a compilation of the results. Plotted is σ ^ i / μ o versus composition for these five alloys. Several theories and experimental studies suggest that the strength increase due to solution hardening should vary as the square-root of the composition [34] . The strengthening contributions in Figure 6 are plotted versus composition to the power of one. The limited data available support a linear model. The dashed lines in Figure 6 start close to zero at a zero concentration and show increased hardening with an increasing concentration. Carbon in Fe and Al in Ti appear to be quite effective strengtheners. Carbon in nickel and nitrogen in 316 SS are less effective strengtheners. Aluminum in Cu also is an effective strengthener. The effect of model assumptions on these observations should be considered. In Ni-C and Cu-Al, a one-obstacle model has been applied. In the other metals plotted in Figure 6, a two-obstacle model has been applied. If in fact, one or more additional solute elements contribute to strengthening (i.e., the σ ^ i term), then the effects of all of these combine to set the value of σ ^ i . If this were the case in Fe-C, then σ ^ i has been over-estimated. Evidence in opposition to this possibility is that at concentrations approaching zero, the strengthening contributions all start close to zero. If another solute were contributing to strengthening one would expect an intercept at a positive value on the ordinate. While for Fe-C, Ti-Al, and 316 SS-N, the intercepts are all positive in Figure 6, they are at relatively low values of σ ^ i / μ o , and plotting the strength contributions versus the square-root of the concentration would take the intercepts even closer to zero. Nonetheless, the importance of this model assumption needs to be

considered when evaluating solute strengthening in alloys using the MTS methodology.

Equation (4) and Equation (5) identified the governing equations for strain hardening, which is also referred to as “structural evolution”. The ensuing analysis of evolution was referred to earlier as the “Evolution” Kinetic Analysis (EA). Key to the application of these equations is the variation of σ ^ ε with strain for stress-strain curves at various temperatures and strain rates. The most rigorous way to compute the variation of σ ^ ε with strain is to estimate σ ^ ε using samples prestrained at a specified temperature and strain rate to a specified strain, and then reloaded at various temperatures and strain rates. These experiments give the variation of yield stress on these prestrained samples with temperature and strain rate. Fitting this data set to Equation (3) gives the value of σ ^ ε . Repeating this pretraining operation at the same temperature and strain rate but to different strain levels enables one to estimate the σ ^ ε versus strain curve, which can be fit to Equation (4) to give values of q_II and σ ^ ε s for that prestrain temperature and strain rate. This test sequence must be repeated at various prestrain temperatures and strain rates to give these model parameters (θ_II and σ ^ ε s ) at these temperatures and strain rates. While this rigorous test sequence necessitates a great number of stress versus strain measurements, this is precisely the approach used in copper [1] , nickel and several Ni-C alloys [7] , and Ti-6Al-4V [24] .

Figure 7(a) shows the compilation of measurements in Oxygen Free Electronic Copper [1] . The solid lines are drawn according to Equation (4) with κ = 2. As described above each value of σ ^ ε plotted in Figure 7(a) represents the analysis of yield stress measurements at various temperatures and strain rates plotted according to Equation (3). The availability of this massive data set

enabled the optimal selection of model variables in this equation [2] . The selected model variables were p_ε = 2/3, q_ε = 1, g o ε = 1.6, and ε ˙ o e = 10⁷ s⁻¹. These values were found to work equally well in nickel [7] and Ti-6Al-4V [24] . Accordingly, these values have been used for all metals and alloys; they are not treated as variables. Similarly, the measurements in Ni-C [7] led to the selection of s_i parameters in Equation (2) for the solution hardening obstacle population. In this case, as shown in Table 2, p_i = 0.5, q_i = 1.5, and ε ˙ o i = 10⁹ s⁻¹. These values have been used for many solution-hardened metals and alloys.

It is evident that as the strain rate increases the curves in Figure 7(a) trend toward higher saturation stresses— σ ^ ε s . Figure 7(b) shows the plot of saturation stress versus strain rate according to Equation (5). A value of σ ^ ε s o equal to 710 MPa forces the dashed line through the origin, as specified by Equation (5).

A less rigorous procedure for estimating the variation of σ ^ ε with strain is to solve Equation (3) for σ ^ ε directly from the stress-strain curve. The operating equation for a metal with two obstacle populations (e.g., a BCC metal with a Peierls stress and an impurity atom stress) becomes

σ ^ ε = μ o s ε ( ε ˙ , T ) [ σ μ − σ a μ − s p ( ε ˙ , T ) σ ^ p μ o − s i ( ε ˙ , T ) σ ^ i μ o ] . (10)

This is the approach taken in evaluating strain hardening in austenitic stainless steels [35] , Inconel 718 [36] , and several other of the metals and alloys described by Follansbee [2] . Figure 8 gives an example of this procedure for measurements in Inconel 718 reported by Nalawade et al. [37] at two test temperatures. Measurements at several test temperatures and strain rates enable one to evaluate Equation (5) and solve for σ ^ ε s o and g ε s o . Application of Equation (4) to the curves in Figure 7(a) and Figure 8 also gives values of the stage II hardening rate θ_II. This has been observed to have a slight strain rate dependence but no measurable temperature dependence given by

θ I I = A o + A 1 ln ε ˙ + A 2 ε ˙ (11)

where A_o, A₁, and A₂ are constants.

In the next sections, assessed values of σ ^ ε s o , g ε s o , and θ_II (actually A_o) are reviewed for several pure metals and alloys. The results are presented in tabular form. Included are references to the original data source and references to the publications that detail the data analyses.

This manuscript has outlined application of the MTS constitutive model in several FCC, BCC, and HCP metals. Section 1 and Section 2 provided an overview of the operative equations. Included in Section 2 was a discussion of how to apply Equation (1) and select a value of n for an alloy with multiple alloying additions. The rationale for linearly adding the individual strengthening contributions was also briefly considered. A critically important feature of the MTS methodology is the distinction of the kinetics affecting the yield stress (where the yield stress implies yield following any processing history) evaluated using the Yield Stress Kinetics Analysis (YSA), from the kinetics affecting strain hardening, or structure evolution, evaluated using the Evolution Kinetics Analysis (EA). A constitutive formalism that does not provide this distinction, e.g., the Johnson

Cook constitutive model [42] or the Armstrong Zerilli constitutive model [43] can replicate stress levels under constant strain rate and temperature conditions, but will be unable to accurately describe instantaneous path changes, e.g., strain rate or temperature changes. This, in turn, will affect predictions of instabilities, e.g., necking in a tensile test or shear band initiation.

Section 3 through Section 5 reviewed results of the YSA model application for several FCC, HCP, and BCC metals and alloys. For some FCC and HCP metals, the yield stress measurements can be modeled using a single obstacle population (Table 2). Application of the model to several other FCC and HCP metals necessitates a two-obstacle population model (Table 3). All of the BCC metals analyzed required a two-obstacle population model (Table 5). Common trends for the normalized activation energies across all metals and alloys were identified. Based on these observations, it is concluded that the yield stress kinetic analysis is a fairly descriptive constitutive formalism.

Section 6 reviewed experience with application with structure evolution using the EA equations. For FCC metals, there exists a clear variation of σ ^ ε s o withstacking fault energy (Table 7 and Figure 9). No trends with g ε s o were noted. For BCC metals, a weak correlation between σ ^ ε s o and the shear modulus was illustrated in Figure 10. For all metals analyzed, the stage II hardening rate (actually, A_o in Equation (11)), consistently was in the range A o / μ o = 0.037 ± 0.018. This translates to A₀ ≈ μ_o/27. Kocks and Mecking [38] report that the Stage II hardening rate is in the range θ_II ≈ μ_o/115, which is 4x less than the estimate here. This difference may reflect the common practice of evaluating Equation (4) to large-strain behavior rather than near-yield behavior.

The largest problem with the structure evolution analysis, particularly with BCC and some HCP metals, however, is in the generality of Equation (4). This equation was based on the Voce equation [44] ³. The Voce equation models the

balance between dislocation generation and recovery in strain hardening. Estrin [45] has derived the Voce equation based on dislocation density contributions. The Voce equation—or the slightly modified version with κ equal to 2 in Equation (4)—provides an adequate fit to the evolution of σ ^ ε with strain in Cu and Ni and many other metals (e.g., Figure 7(a) and Figure 8). The Voce equation breaks down, however, when strain hardening is accompanied by deformation twinning, dynamic strain aging, or stress or strain induced metallurgical transformations. The effects of deformation twinning were observed in zirconium [2] , zinc, and several other materials. Figure 11 shows that the variation of σ ^ ε with strain in 0.99999 Zn with a grain size of 70 μm [11] is not well-described using Equation (4) [2] . It is suspected that deformation twinning is effectively decreasing the grain size and leading to an increasing contribution from dislocation interactions with grain boundaries (e.g., the σ_a variable in Equation (1)) [2] . Dynamic strain aging was observed to be prevalent in niobium, titanium, austenitic stainless steels, and nickel based superalloys [46] . Signatures of the contributions of these metallurgical processes in context of the application of the MTS formalism were noted [46] .

The conclusion is that large deviations from evolution predicted by Equation (4) are possible in many metals and alloys. The Evolution Kinetics Analysis that comprises the MTS model is not as widely applicable across myriad metals and alloys as is the Yield Stress Kinetics Analysis. This conclusion may guide further research and modeling of strain hardening, particularly when dislocation storage is accompanied by deformation twinning, dynamic strain aging, or stress or strain induced phase transformations.

Much of this research was supported by Saint Vincent College. The author acknowledges the collaboration with U. F. (Fred) Kocks, who recently passed away. Fred was a colleague and friend who mentored me on topics related to deformation kinetics. The world has lost one of the top materials scientists of the 20th century.

The author declares no conflicts of interest regarding the publication of this paper.

Follansbee, P.S. (2023) A Comprehensive Review of Experience with the Application of the Mechanical Threshold Stress Model. Materials Sciences and Applications, 14, 299-323. https://doi.org/10.4236/msa.2023.145019

The number of independent variables in a constitutive equation is an important consideration. The objective is to derive equations with physical significance but with the fewest independent variables. A common objection to a proposed set of constitutive equations is that, with a great many independent variables, it is “easy” to fit the model to a given data set. This Appendix will assess this number for the MTS formalism.

For a metal that can be evaluated using a two-obstacle, σ ^ 1 and σ ^ 2 , plus an evolution obstacle, σ ^ ε , Equation (3) with n = 2 and Equation (2) for each of the (3) s_i values is the governing equation for YSA. Equation (4) and Equation (5) are the governing equations for EA. Table A1 lists the parameters in these equations. Some of the parameters are physical constants. Some are identified as “Independent Variables”. For a two-obstacle model, each threshold stress and the values of g_o1 and g_o2 are listed as independent variables. The corresponding value of g_oε is listed as a “Constrained Variable”, since as outlined in the discussion of Figure 7, this value has been taken as 1.6 for all metals and alloys.

Table 2, Table 3 and Table 5 list values of p₁, g₁, p₂, g₂, ε ˙ o 1 , and ε ˙ o 1 used in the analyses of the metals and alloys included in these tables. It is evident that common values were selected, implying that these variables were not used as

fitting parameters. Thus, these parameters in the “Constrained Variable” column. The same applies to values of ε ˙ ε s o listed for the evolution analyses for the metals listed in Table 6 and Table 8. Indeed, ε ˙ ε s o values of 10⁸ s⁻¹ and 10¹⁰ s⁻¹ are both included in Table 8. However, these values greatly exceed the test strain rates and this variable is contained with a logarithmic ratio in Equation (5), which implies this difference is not significant.

The conclusion is that for a metal that can be described using a two-obstacle model, the number of independent variables listed in Table A1 is eight (8). This paper has provided further bounds for several of these variables, including g₀₁, g₀₂, and θ_II.