A model for creep based on microstructural length scale evolution

Materials Science and Engineering A 387–389 (2004) 576–584

A model for creep based on microstructural length scale evolution Glenn S. Daehn∗ , Holger Brehm1 , Huyong Lee, Byeong-Soo Lim2 Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA Received 9 September 2003; received in revised form 23 December 2003

Abstract This paper seeks to quantitatively link recovery and plastic deformation to develop a model for creep. A simple approach with one length scale coarsening equation is postulated in this paper to provide a descriptive and unified framework to understand recovery. We propose that recovery is a general dislocation-level coarsening process whereby the length scale, λ, is refined by dislocation generation by plastic deformation and is increased concurrently by coarsening processes. Coarsening relations generally take the form: d(λmc ) = KR(T) dt where R(T) is the rate equation for the fundamental rate controlling step in coarsening, K a free constant, dt a time increment and mc is the coarsening exponent. Arguments are presented that mc should be in the range of 3–4 for dislocation or subgrain coarsening. The coarsening equation postulated is consistent with the compared data sets in the following ways: (i) temporal evolution of one length scale λ; (ii) temperature dependence of recovery rate; (iii) adequacy of single parameter in the proper description of strength change. The coarsening equation is coupled with standard arguments for modeling plastic deformation. Combining these we can easily justify the form of the empirically derived Dorn creep equation:
n γ˙ τ =B D(T) µ where the mobility of the recovering feature, R(T) should typically scale with self diffusivity, D(T), and the value of the steady-state creep exponent, n is 2 + mc − 2c where c is a constant related to dislocation generation that should be in the range of 0–0.5. Hence, this approach predicts creep as being controlled by self diffusion and that the steady-state stress exponent should be on the order of 4–6. All the parameters in the creep model can be estimated from non-creep data. Comparison with the reported steady-state creep data of pure metals shows good agreement suggesting recovery modeled as coarsening is a fundamental element in steady-state creep. © 2004 Elsevier B.V. All rights reserved. Keywords: Creep modeling; Microstructural evolution; Recovery

There is a remarkable commonality in the way materials plastically deform. At low temperature almost all simple cubic metals have remarkable ductility and strong strain hardening. Also, if stress is normalized by elastic modulus, many materials show very similar stress–strain curves when

compared at the same homologous temperature. In 1962, Oleg Sherby published a paper [1] showing that if one plots the steady-state creep strain rate divided by diffusivity versus creep stress normalized by elastic modulus, there is a remarkable collapse of a large amount of experimental data. Twenty-one different metals or phases are represented on the diagram. Specifically:

∗ Corresponding author. Tel.: +1-614-292-6779; fax: +1-614-292-1537. E-mail address: [email protected] (G.S. Daehn). 1 Presently at Fraunhofer Gesellschaft, IWM Freiburg, Freiburg, Germany. 2 Visiting Professor from Sungkyunkwan University, Suwon, Republic of Korea.

• Scaling by diffusivity removes the temperature dependence, and data from multiple temperatures collapse to a single curve. • All the curves are relatively straight between stresses of 10−5 E and about 5 × 10−3 E, where E represents elastic modulus.

1. Goals and motivation

0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.12.082

G.S. Daehn et al. / Materials Science and Engineering A 387–389 (2004) 576–584

• All the curves roughly follow the form:  σ n ε˙ (1) ≈K D E where the K values are relatively constant and n is typically in the range of 4–6. This degree of data collapse is remarkable especially because the materials represented have a wide range of crystal structures (f.c.c., b.c.c., h.c.p.), stacking fault energies, slip systems, etc. The vast majority of the models in the literature that seek to understand the creep behavior of metals (in particular the stress exponent) are based on specific mechanistic models and detailed assumptions about how dislocations are generated, moved and are annihilated. Because a nearly common pattern is seen in many materials with many different microstructural and dislocation-level details, it seems clear that there is something going on that is more general. The purpose of this model is to understand the creep behavior of metals in a very general without reliance on specific mechanisms. In this spirit, assumptions will be kept as simple and general as possible (at great risk of over simplifying the situation). The model is intended to apply to simple, single phase, nominally pure metals at temperatures of about 0.3Tm and above (where Tm is the absolute melting temperature). With these assumptions, dislocations will tend to spend much time at relatively robust pinning points. And once they are released they will move quickly to the next set of pinning points. Barriers of small activation energy (such as Peierls barriers) will not significantly contribute to strength.

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same way with strain and time with a fixed factor between them. Under these assumptions the model that follows can be equivalently applied to dislocation strengthening or subgrain strengthening without comment as to which model for strength is ‘correct’. In either event we assume that a single parameter can be used to describe structure. That parameter could be either length scale, λ, λs , or the dislocation density (here expressed as line length per volume), ρ. From a simple dimensional argument the average slip distance can be related to the inter-dislocation spacing as g λ= √ (2) ρ The factor g will depend upon the spatial distribution of the dislocations. A lower bound on g is set by an array of dislocations that are all parallel, lying in the same direction. In this case g is unity. For a more typical nearly isotropic example, the dislocations may lie along the √ x, y and z directions in square array. Here, the g value is 3. When we use this relationship to describe yield, it will be the dislocations with the longest inter-obstacle spacing that will tend to be released, also dislocations will tend to cluster. For these two √ reasons, reasonable g values are in the range of about 3 to 20. A standard value of g = 4 will be chosen presently. With this single parameter description of structure, we shall model creep from the three elements that are undeniably present: (1) release of dislocations from obstacles; (2) refinement of the dislocation network and subgrains due to dislocation motion; (3) recovery (coarsening) of the dislocation structure. In describing these processes we will be much more guided by data than any pre-supposed theories.

2. Fundamental processes and structural description 3. Dislocation release Here, we want to use the simplest defensible description of the material microstructure. Obviously it is heterogeneous at several length scales. The relative roles of grain boundaries, subgrain boundaries and forest dislocations in material strengthening have been debated extensively. As the present focus is on behavior at relatively high homologous temperatures, and it is known that above about 0.4Tm , fine-grained materials are generally weaker than coarse grained materials. Therefore, grain size strengthening will not be considered further here. The issue with subgrains is more complicated. Subgrains are known to refine with increasing levels of stress and strain and fine subgrain sizes also correlate with higher material strength. However, it is quite difficult to separate the possible separate roles of subgrains and forest dislocations, and this topic has been discussed extensively [2]. In this treatment we assume that strength comes from dislocation density rather than from subgrain size. However, we identify two unique length scales. The first, λ, represents the average inter-dislocation spacing. The second, λs , is the average subgrain size. Typically λs is about 3–30 times greater than λ and we will assume that both will scale the

We assume dislocations are held by discrete pins and when they are released from a pinning point they glide easily over a distance on the order of λ. Implicitly assumed throughout this treatment is that the material strength is derived wholly from such pins and interactions from strengthening mechanisms such as Peierls stresses for solute drag do not contribute to strength. Thus, this model is not expected to apply at low homologous temperatures where very low energy barrier interactions contribute significantly to strength. Two types of models for dislocation release are considered in the present treatment. The Arrhenius treatment is more versatile and correct, however, it does not easily produce a clean analytical form. The athermal treatment gives very similar results and also gives a form that is easily analytically tractable. 3.1. Arrhenius treatment of dislocation release A pinned dislocation puts a force, f, on an obstacle: f = τbλ, where τ is the externally applied shear stress and b is

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the magnitude of Burgers’ vector. At each obstacle, at each local vibration, there is a probability the dislocation will break free from the obstacle and glide. That probability, Prel is given as
 −G∗ (3) Prel = exp kT where the energy barrier G∗

is given in the form commonly

used by Kocks et al. [3]:

p q f ∗ ∗ G = F 1 − kˆ

(4)

ˆ p and q are all essentially parameters that The terms F∗ , k, describe a given barrier in a single material. F∗ is in energy terms the unstressed size of the barrier that must be surmounted for dislocation release. This can be easily normalized to varied materials by expressing it as F ∗ = s1 µb3 . This has been used previously as a normalizing scheme where s1 values on the order of 0.02, 0.25 and 10 correspond to barriers such as solute interactions, dislocation junctions and precipitate interactions, respectively, for example. The dislocation obstacle can be overcome without any thermal activation if the force on the obstacle exceeds the athermal ˆ Again kˆ is normalized to the intrinsic breaking force, k. materials properties. Here, we take kˆ = s2 µb2 . With this normalizing scheme materials can be described in a rather generic way with the free parameters being s1 , s2 , p and q. Variations in these parameters can be directly correlated with variations in the micromechanics of flow. 3.2. Athermal treatment of dislocation release Eq. (4) represents a well-founded but analytically inconvenient way of describing a material’s flow stress as a function of temperature and obstacle properties. For computational convenience we will find it useful to introduce an athermal description for flow. Basically we assume that when the force on an obstacle exceeds some athermal breaking force it will be overcome. We can derive the material flow stress as sµb τf = (5) λ Here, s represents a single-parameter description of the junction strength. In fact this parameter should be temperature-dependent, but it is very weakly temperaturedependent, relative to diffusivity, for example. If dislocation density, ρ, is used as the structural metric instead of λ, the equation can be re-written as
 s √ √ τf = µb ρ = αµb ρ (6) g The latter equation is Taylor’s equation for plastic flow resistance. There is a considerable amount of data confirming the validity of this equation for f.c.c. metals at low homologous temperatures. The value of α is found to be approximately 0.3 [4]. Also, the equation quite reasonably fits b.c.c. and

other simple pure metals. In using this form, there should be a slow decrease in s with increasing temperature.

4. Dislocation accumulation Asbhy [5] produced a compelling schematic plot of dislocation density as a function of plastic shear strain. His plot was intended to schematically indicate alloy behavior, and was based on the copper single crystal data of Basinski and Basinski [4]. Since this time this plot has been schematically reproduced other places and interpreted as more general than originally intended. This plot also provides the motivation for examining if there is a general pattern for dislocation accumulation due to plastic flow. It is appealing to assume that free dislocations become immobilized after moving some distance related to the slip distance, λ, after which new dislocations must be formed to enable further deformation. In such a case we can write: k dρ (7) = = Mρ0.5 dγ λ where k or M represent material constants which must be fit to either a model of dislocation generation or experimental data. During plastic deformation there is also dynamic recovery that will reduce dislocation density, essentially while it is generated. From a practical point of view we can re-cast Eq. (7) as being the total change in dislocation density (including dynamic recovery) due to a plastic strain increment. When viewed this way, that there is considerable experimental data (which was reviewed nicely by Gilman [6]) that shows for many polycrystalline systems dislocation density accumulates linearly with strain and there is a similar pattern for many metals over a range of temperatures. A compilation of such data is shown in Fig. 1. A generic differential form that will fit both this observation as well as Eq. (7) is dρ = Mρc dγ

(8)

where M is the multiplication coefficient and c is the breeding exponent. Using this form and assuming a material has an initial dislocation density ρ0 at a strain of zero, the dislocation density as a function of strain (in the absence of any losses due to static recovery, that are discussed later) can be written as (1−c) 1/(1−c)

ρ = [Mγ(1 − c) + ρ0

]

(9)

As can be seen in Fig. 1, the data from many simple metallic systems at relatively low homologous temperatures clusters nicely, and several systems can be fit using M = 2 × 1015 m−2 and c = 0. It is impressive that the data collapses this neatly, especially since the systems studied include many very different metals examined at temperatures ranging from room temperature to creep conditions. The exponent of c = 0 will lead to parabolic hardening. We assume

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Fig. 1. Compilation of data for dislocation density as a function of imposed plastic strain for a number of annealed metallic systems. The line marked ‘assumed density’ shows the equation used in the present model. The original data for this figure comes from Refs. [7–12].

that this is the native behavior of the material. Of course Stage II hardening (which is not robustly seen in polycrystals, except at very low homologous temperatures) demands linear hardening and would require an exponent, c of 0.5. Ashby [5] has discussed how geometrically necessary dislocations will accumulate linearly with imposed strain (as the bulk of the present data indicate). In the present model, dislocation accumulation is assumed to take place in a manner that is described by Eq. (8) where the values of M and c seem to cluster around 2 × 1015 m−2 and 0, respectively, for many simple metals. We do not wish to present any theories for this; instead we will rely on the trends seen in the experimental data.

5. Recovery (substructural coarsening) Clearly at elevated temperature, dislocation density will decrease with increasing time and temperature. Also subgrains will coarsen. Plastic deformation may well have a decisive role in hastening these recovery processes. Raymond and Dorn performed an especially careful study of the effects of stress on the recovery kinetics of aluminum in creep [13]

and found that stress does not seem to change the character of the recovery, but it can change its rate. They found recovery rate increases monotonically with stress and in their study could be as much as four times as fast as stress-free recovery. Presently we assume that only static recovery is responsible for the softening of the material with increasing time at temperature, but note that applied stress may play a role in speeding the process. The materials community really does not have a singular basic approach to recovery. Instead distinct dislocation-based models that are appropriate to the given situation are usually used [14]. Here, we will assume that dislocations exist in a network structure and this network will coarsen with time in a self-similar way. There may also be a subgrain structure and this is assumed to have a size that scales proportionately with the dislocation link length, λ. With these assumptions one can treat recovery as a generalized coarsening process and leverage the advances made over the past few decades in this area [15]. Again λ represents the characteristic microstructural dimension. For simplicity and brevity we will essentially extend the derivation of grain coarsening developed by Burke and Turnbull [16]. Let us start by summarizing their derivation. They assumed that for a self-similar array of grains the

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pressure difference from one side of a boundary to another scales with σ/λ, where σ represents surface tension and λ is the grain size dimension (because this is dimensionally the equation for gas pressure in a sphere with surface tension, σ) and with regard to kinetic mechanism Burke and Turnbull assumed differential pressure drives boundary motion in at rate proportional to the boundary mobility, M. Or specifically: σ  dλ = A MP = A M (10) dt λ Upon integration one obtains the classical equation for parabolic grain growth: λ2 − λ20 = AMt

(11)

This equation will apply equally well to subgrain coarsening by boundary mobility, where the rate of boundary migration is proportional to the differential pressure on it. In the present treatment we are at least as interested in the coarsening of dislocation arrays as in grains or subgrains. For the case of coarsening a planar array of dislocations (such as those that may make up a low angle boundary), it is the force/length on the dislocation line that is the driving force for motion. Again the driving force will scale with T/λ, where T is the dislocation line tension and λ is the microstrucural length scale (i.e. proportional to the mean radius of curvature for a self-similar system). One can follow the same approach as above to again develop parabolic network coarsening. Now let us consider the dimensionality of a 3D network of lines, such as the array of dislocations and nodes described by Öström and Lagenborg [17,18]. Here, the driving force is reduction in line length per volume, instead of length per area as above. As will be shown later in a formal dimensional analysis approach [19], the driving force now scales with T/λ2 . Now if dislocations move by a viscous mechanism equation (11) will have coarsening exponent, of 3 instead of 2. To this point we have described the coarsening of three different structure types: (sub)grains, planar dislocation arrays and volumetric dislocation arrays. The former two have a coarsening exponent of 2 while the latter one has a coarsening exponent of 3. Now let us change the kinetic path and assume defect mobility controls the rate of coarsening. Now assume long-range transport will control the mobility of the defect (vacancy). Defect velocity will scale with vacancy flux that is described by Fick’s first law, where the vacancy

concentration is given by Xv . Summarizing for transport by bulk diffusion: dXv M ∝ Jvac ∝ D (12) dλ Here, we recognize that the mobility, M, term will have both temperature and length scale dependence. If we substitute this into the relations above, the coarsening exponents will all rise by 1. Hence, for coarsening of surface area per volume with bulk diffusion control, the coarsening exponent will rise from 2 to 3. This is the reason Ostwald ripening has the well-known coarsening exponent of 3 [20,21]. From this point on we will use the symbol M to designate viscous mobility of a boundary or line defect (motion by short-range diffusion across the boundary) and R(T) will be generally used for the rate controlling step in the coarsening process, including both diffusion and viscous mobility. We may take this one step further and consider pipe diffusion. Here Deff = ρDp . Since ρ scales with λ−2 , a processing coarsening by pipe-diffusion transport will have an exponent 2 higher than that through bulk diffusion. Based on all of this we can establish a general coarsening equation: d(λmc ) = KR(T) dt

(13)

The coarsening exponents take on values between 2 and 6 depending upon system morphology and mechanism. The rate controlling step R(T) can be either viscous mobility or diffusion limited. The term K must be determined from experiment or a theory based upon specific mechanistic assumptions. The possible coarsening/recovery mechanisms are summarized in Table 1. 5.1. Experimental assessment of the coarsening model There have been a number of experimental studies on the recovery of simple metals since the 1950s. Many studies have shown at homologous temperatures above 0.4 or so, the activation energies for coarsening are experimentally indistinguishable with those for diffusion (for a review see [22], also [13,23]). However, there have been relatively few studies where substructural length scales or dislocation densities have been measured directly to test the coarsening exponent or temporal form of Eq. (13). If Eq. (13) is plotted on log–log axes, it should take on a form like that as shown in Fig. 2. At very short times the initial length scale is present in the material. Also by physical constraints, this cannot be less than

Table 1 Appropriate terms for coarsening exponent, mc , and rate controlling rate, R(T), for the generalized coarsening equation d(λmc ) = KR(T) dt Rate step/geometry

Grains/subgrains

2D dislocation array

3D dislocation network

Viscous motion LR lattice diffusion LR pipe diffusion

mc = 2, R(T) = Mb (T) mc = 3, R(T) = Dl (T) mc = 5a , R(T) = Dp (T)

mc = 2, R(T) = Mg (T) mc = 3, R(T) = Dl (T) mc = 4b , R(T) = Dp (T)

mc = 3, R(T) = Mg (T) mc = 4, R(T) = Dl (T) mc = 6, R(T) = Dp (T)

a b

Assumed that the dislocation spacing will scale with (sub)grain size, both proportional to λ. Pipe diffusion increases mc by 1 vs. bulk diffusion because area fraction of dislocations scales with 1/λ.

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Fig. 2. Schematic behavior of Eq. (13) including the lower bound based on the fact that λ must be significantly larger than b and the upper bound based on the observation that even at very long annealing times dislocation density seldom drops below 1010 m−2 .

a few Burgers’ vectors. At very long times it is noted that materials will coarsen to a ‘frustration length’. Dislocation densities seldom become less than about 1010 m−2 , even after extensive annealing. Also studies of grain growth show that at very long time grain growth will stop with a grain diameter that is typically related to sample size. These considerations place the upper and lower bounds shown in Fig. 2. Between these bounds Eq. (13) is expected to be apparent. Two especially appropriate studies have been carried out that allow examination of the temporal and (in one case) thermal dependences of Eq. (13). First Oden et al. [24] used TEM to investigate the diffusional coarsening of a dislocation network developed by creep after the stress is removed. This study was carried out in a 20% Cr–35% Ni stainless steel at 700 ◦ C after creep at 127 MPa. Fig. 3 shows their data replotted in the form of Fig. 2 using g = 4 to obtain λ from dislocation density. Here, the diffusion data of

Fig. 3. Oden’s data in the form of characteristic inter-dislocation spacing as a function of the product of diffusivity and time. The line shown indicates m = 3 and a K value of 9.1 × 10−7 m.

Cermak for a similar alloy is used to normalize with time [25] and sets R(T). The specific diffusion coefficient used is 6.4 × 10−18 m2 /s. Fitting to the data yields a K value of 9.1 × 10−7 m. The recent work of Huang and Humphreys [26] allows examination of both the temporal and temperature dependence, but concentrates on subgrain growth instead of network coarsening. For a crystal deformed 70% at room temperature and recovered at varied temperatures they find coarsening exponents between 4 and 2.8 for temperatures between 250 and 400 ◦ C. Their original data is plotted in Fig. 4 in log–log form with points added at very short times to indicate the original subgrain size of the material of 0.9 ␮m. The data are also plotted with the product of time and diffusion

Fig. 4. Data of Huang and Humphreys [26] plotted on log–log axes both with time and the product of time and diffusivity. Notice that when normalized the data collapse nicely and are fit reasonably well with Eq. (13) with a value of m = 3 and K = 4 × 10−5 m, as is plotted on the graph.

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in Fig. 4 (right). The characteristic line here has an m-value near 3 and K ∼ 4 × 10−5 m, using the self diffusivity of aluminum as R(T). If subgrains and interior dislocation networks coarsen in a geometrically similar way, the coarsening constant for subgrains should be about 10 times greater than that for networks, and this is approximately seen here from these two very different data sets.

to be 0.1, and substituting in the parameters listed already and assuming a temperature of 1000 K the subgrain growth constant can be estimated as KN–H ∼ 3 × 109 . This is in the range of 200–2000 times slower than the process observed here from data and it takes on the same form with respect to temperature dependence, grain size and time.

5.2. A mechanistic comment

6. Integrating the processes

Here, we argue substructures coarsen by long-range vacancy transport. The argument for subgrain growth controlled by long-range diffusion is that in the motion of low angle grain boundaries (which are describable as arrays of dislocations), depending on their structure and curvature, some boundaries must emit vacancies to move in the driven direction, while others must absorb them. This will set up a vacancy flux that is in spirit similar to that for Nabarro–Herring diffusional creep. However, the rate of grain boundary motion at a given vacancy flux can be much greater than that in diffusional creep because now instead of atom flux having to provide all the matter to move the boundary, many fewer atoms can move the boundary by feeding the excess edge dislocations on the boundary, allowing it to move forward. It is instructive to compare our subgrain coarsening model to that for Nabarro–Herring creep, which states:
 DL σΩ ε˙ = 10 2 (14) kT d

If we use a rate-independent formulation for dislocation release, our model can be summarized as the set of equations shown in Table 2. A more detailed version of this derivation is available elsewhere [27]. To develop this model in a general way, generic values for constants are used. We note that slip Burgers’ vectors for a wide range of pure structural metals again fall in a fairly tight range. The values are almost universally between 0.25 and 0.43 nm. We will use the reference value of 0.3 nm here. Elastic moduli and diffusivity do vary widely, but these are used as normalizing parameters in the final results. With these assumptions, we can make direct predictions of the plastic behavior of simple metals. The most obvious way to do this is to tack the change in λ with strain and time due to refinement and coarsening to develop a stress strain law. The full solution to this is somewhat complex and not very illuminating, however, there are two limits of this that are quite useful. So long as the initial dislocation density is low, the material will initially strain harden (due to dislocation accumulation) with a strain hardening exponent, N (defined in σ = cεN ) of

This can be developed into a form similar to that studied here by again making some reasonable order-of-magnitude estimates. The grain size, d, is taken as the subgrain size λs . The atomic volume Ω, is taken as b3 , and the driving stress, σ, is taken as a fraction of the pressure of an isolated subgrain, of 3f(γ/λ). We can solve directly for the expected grain growth rate expected if grains were to grow by Nabarro–Herring creep:
 λ˙ DL γb3 ε˙ = = 30f 3 (15) λ kT λ Putting this into the context of Eq. (13), we can estimate the K value that would be expected if subgrains were to grow by Nabarro–Herring creep and this would place a lower bound on K. If we assume the surface tension, γ, to be 0.5 J/m2 , but the fraction, f, of this pressure driving coarsening, is taken

N=

1 2(1 − c)

(16)

Thus, if c = 0.5, linear hardening will be obtained, but a value of N = 0.5, indicating parabolic hardening will result if c = 0, as we propose it is here. This is reasonable, because parabolic hardening is often seen in the low-temperature strain hardening of annealed metals. We will show in later papers that the lower strain hardening exponents associated with higher temperature deformation can be explained based on the natural stochastic strength heterogeneity within and between grains. As deformation continues, the structure refines and this increases the rate of coarsening. A steady-state condition will be reached where the rate of dislocation accumulation and reduction balance. The steady-state characteristic spac-

Table 2 Phenomenological equations and constants in a first-order general plasticity model Phenomenon

Equation

{Range} and/or assumed value

Basis

Structure parameter (λ or ρ) Athermal yield Dislocation accumulation Recovery (structural coarsening) Other constants

Eq. Eq. Eq. Eq.

g = {1–20}, g = 4 s = {0.1–1.0}, s = 0.5 M ∼ 2 × 1015 , c = 0 k = 10−6 m, mc = 3 b = 0.3 nm

1: theoretical min.; 20: large gaps in random structure Strength compilations [1,18] Fig. 1 Figs. 3 and 4 Small range

(2) (5) (8) (13)

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ing can be determined by equating the coarsening rate and the refinement rate, both in terms of λ, and solving for the equilibrium value [27]. The steady-state characteristic spacing is
1/(2+mc −2c) 2KD(T)g2−2c (17) λss = M γ˙ With the athermal yield equation strain-rate can be written as a function of stress as
n τ γ˙ = BD(T) µ  
2+mc −2c 2Kg2−2c (bs)−(2+mc −2c) τ = (18) D(T) M µ One of the more significant findings from this investigation is that the material steady-state creep stress exponent is equal to n = (2 + mc − 2c) where we believe mc to be in the range of 3–4 for typical creep processes and c in the range of 0–0.5, with zero showing the better fit to experimental data. This immediately explains the commonly observed steady-state creep stress exponent between 4 and 6. Exponent values considerably higher than 5 can also be easily rationalized in this framework. If a microstructure is developed in such a way that it resists coarsening, or provides higher coarsening exponents (as may be the case in

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many creep-resistant materials) stress exponents over 5 are expected. This model can provide quantitative predictions of steady-state strain rates. Table 2 shows typical values for the necessary model parameters. We believe that all of these should be approximately correct for a wide range of metals as is supported by the data compilations shown here and elsewhere. Fig. 5 shows a comparison of Sherby’s [1] “all the data in the world” compilation and on top of this the prediction from this model is superposed. The constants used in Table 2 were used as inputs. In another paper [28] we have shown that so long as the activation energy for diffusion remains less than G∗ for dislocation release the scaling that is shown here is preserved very nicely. Hence, thermally activated dislocation release does not dramatically complicate the picture. The line from our prediction falls near the center of the block of data. Deviations above or below can be explained by the relative facility of coarsening processes changing with factors such as stacking fault energy and nascent dislocation and boundary structures. Also applied stress may increase the coarsening rate, which may also elevate the predicted curve.

7. Concluding remarks A simple approach with one length scale coarsening equation is postulated in this paper to provide a descriptive and unified framework to understand recovery. Also a creep model is presented based on a single length scale coarsening and refinement processes which are assumed to be independent. The coarsening equation and creep model are compared with the reported experimental data and we find a very good agreement, confirming a quantitative link between this simple approach of coarsening and the phenomenological creep behavior of pure metal. The coarsening equation postulated is consistent with the compared data sets in the following ways: (i) strain-based evolution of one length scale λ; (ii) temperature dependence of recovery rate and creep rate; (iii) quantitative consistency of the recovery kinetics with creep rates of pure metal via our model, but this is based on limited data. This model carries with it two suggestions: (1) in designing creep resistant materials it may be more effective to design for recovery resistance than for inhibition of dislocation motion, however, the same strategies will typically work in both cases; (2) a better understanding of the mechanisms and phenomenology of recovery is called for.

Acknowledgements

Fig. 5. Sherby’s 1962 compilation with the results of this model with the parameters from Table 2 superposed. The line has a slope of 5.

This work was supported by the National Science Foundation under award no. DMR-0080766 as part of the Center for Accelerated Maturation of Materials (CAMM). A von Humboldt Foundation grant largely funded the work of Hol-

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ger Brehm and SAFE, ERC of Korea funded the work of Byeong Soo Lim.

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