The Features of Localized Plasticity Autowaves in Solids

The localized plastic deformation and the law-like regularities underlying its development in solids are considered. The characteristic features of localized plasticity are analyzed for a wide range of materials. Thus a correlation is established between the products of scales and of process rates obtained for the elastic and plastic deformation. It is a favorable ground for hypothesizing causal links between the elastic and plastic deformation by introducing an elastic-plastic invariant, which is the master equation of the autowave plasticity model being developed. Localized plasticity phenomena are proposed to be addressed in the frame of autowave and quasi-particle approach.


Introduction
During recent decades we have focused on the macroscale development of plastic flow. The results are presented in papers [1][2][3] and in the monograph 4 . We have gradually come to accept that the plastic flow behavior has a characteristic attribute: the deforming medium would spontaneously separate into actively deforming layers alternating with inactive non-deformed layers (Fig. 1). As a result, an intricate arrangement of localized plas ticity patterns would emerge and vary in space and with time. The patterns have been correlated with the flow stages on the stress-strain curves, σ(ε), plotted for the test samples. The available experimental evidence suggests that the patterns in question have space scales ~10 -2 m and characteristic times ~10 2 …10 3 s. The latter two values are virtually unaffected by the kind of material and only slight ly so, by the loading conditions.
The stratification of the plastically deforming medium (localization of deformation) is equivalent to the structure formation in the same; hence, this is actually related to the medium's self-orga niza tion 5,6 . Our understanding of this fact offers a clearer view of the nature of plasticity. Thus the author is responsible for much of the work on the patterns in question and for identifying them as 'localized plastic flow autowaves' 7 . The autowave generation is known to involve a decrease in the entropy of the deforming medium 8 , which counts in favor of the self-organization concept. The autowave length, λ, and the time period, T, have been determined experimentally for similar processes; the values obtained are generally in the range 0.5·10 -2 ≤λ≤2·10 -2 m and 10 2 ≤T≤10 3 s. At about the same time, the theory of solitary plastic waves was put forward for an explanation of plastic flow 9 .
Here reference should be made to the conceptual representation of other workers 11,12 , concerning the nature of periodic processes involved in the plastic deformation. Today the autowave model is substantiated by abundant theoretical and experimental evidence [12][13][14][15][16][17] . It is therefore claimed herein that the auto wave approach has presently received support among scientists.

Methods and Materials
The experimental studies of autowave processes of plastic flow were done with the aid of double-exposure speckle photogra phy (Fig. 2 a); due to the stepwise stressing, the method was adapted to high plastic deformation 4 . The application of the method enables reconstruction of the displacement vector field for the flat sample, i.e.|r(x,y)|>>χ (here χ is the inter pla nar spacing). Using specially developed software, plastic distortion tensor components were calculated for different times and different points on the sample surface (Fig. 2 b). To obtain a generalizable set of data, the experiments were done for a variety of materials having dissimilar nature, which also differed in structure as well as in physical and mechanical properties. The range of studied materials included pure metals and alloys, alkali halide crystals and some rocks 3,4 . The metals and alloys had FCC, BCC, HCP or tetragonal lattice; these were in single-crystal or polycrystalline state, polycrystalline metals and alloys differing in grain size.
We will discuss herein the general regularities of localized plasticity observed for all studied materials.
Autowave processes are likely to originate in the so-called active medium with energy sources distributed throughout its volume 6,18,19 . The elastic stress fields in the vicinity of stress concentrators play the role of energy sources; hence, the deforming medium meets the condition for autowave generation. An analysis of plastic flow should be based on space-time nonlinear kinetics relations derived for strains and stresses suitable for descriptions of response to loading of a nonlinear deforming medium. The derivation of such relations is considered herein. Moreover, a new universal approach to the phenomenon of solids plasticity is introduced in the form of logical implication. The aim of the given paper is an attempt at revealing the major regularities of the plastic flow by generalization of the diverse database, which would enable elaboration of a persuasive theory of this multivariate phenomenon.

The autowave nature of plastic deformation and the elastic-plastic invariant
To gain an insight into the nature of localized plastic deformation, the regular features of its development have to be assessed on the base of qualitative data. The regularities exhibited by the localized plastic flow behavior have been determined with the maximal accuracy for the easy glide and linear work hardening stages. In these cases, the localization patterns comprise a set of localized plasticity nuclei having space and time periods, i.e. autowave length, λ, and characteristic time, T, respectively (Fig. 2 C); hence, the autowave propagation rate, 10 -5 ≤V aw =dλ/dT≤ 10 -4 m/s. Next Fig. 3 a-c shows the experimentally obtained data characterizing the localized plastic flow autowaves.

Distinctive features of localized plasticity autowaves
It is found that the autowave rate depends on a dimensionless characteristic of the plastic flow process, i.e. θ=E -1 ·dσ/dε (here E is the elastic modulus). This dependence has the form where V 0 and Ξ are constants, which differ for the easy glide and linear work hardening stages. (Fig. 3 a). It should be emphasized that the autowaves observed for the latter two stages are described by the following quadratic dispersion relation (Fig. 3 b), where ω=2π/T is frequency; k=2π/λ is wave number and ω 0 , k 0 and α are constants. The mi nus and plus signs correspond to the easy glide and linear work hardening stages, respectively. By substituting ω=ω 0 ·ῶ and , it is easily reduced to the form (here ῶ and k u are dimensionless frequency and wave number, respectively). The grain size dependence of autowave length, λ(δ), is obtained experimentally for the linear work hardening stage in aluminum (Fig. 3 where λ 0 , a 1 and a 2 are empirical constants and C is an integration constant. Function (3) is the solution of the . Experimental data analysis suggests that the autowave length depends weakly on the grain size. Thus an increase in the value δ from 10 -5 to 10 -2 m would cause the value λ to increase from ~6•10 -3 m to ~1.6•10 -2 m.

Introduction of elastic-plastic invariant
At first glance it would seem that Eqs. (1), (2) and (3) are not interrelated; however, we are going to take a good look at these relationships to find out what relation binds them. With this aim in view, quantitative data processing of deformation patterns was performed for studied materials. It has been found that the auto wave characteristics, λ and V aw , taken together with the spacing, X, and the transverse elastic wave rate, V t , make up the following ratio (see Table 1) The calculated data suggest that the average quantity

. Equation (4) is called 'elastic-plastic invariant'
by virtue of the fact that it relates the characteristics of elastic waves, X and V t , to the characteristics of localized plastic flow autowaves, λ and V aw . The elastic waves are involved in the elastic stress redistribution by the deformation, while the localized plasticity autowaves are in point of fact deformation pattern evolution. Hence, Eq. (4) relates the above two types of processes. This reasoning may play a strategic role in envisaging new conceptual representations of the nature of localized plastic deformation. As is shown in Fig. 4, invariant (4) possesses universality: it is valid for the autowaves pro pa gating at the easy glide and linear stages of work hardening as well as for the motion of individual dislocations 20 .
where the value ζ 1 =(ω D X)ρ=V t ρ is the medium's specific acoustic resistance 22 .

Consequences of the elastic-plastic invariant
Eqs. (1) -(3) can be derived from Eq. (4), which suggests that the elastic-plastic invariant plays an important role in the development of autowave concept of localized plasticity.
First, Eq. (4) is differentiated with respect to deformation, ε, as By writing the above expression with respect to V aw , we obtain The value X is unaffected by the deformation; hence, which coincides with Eq. (1). The coincidence is due to the fact that dV aw /dλ<0; besides, the work hardening coefficient, θ, is given by the ratio of two structural parameters having the dimension of length, i.e. λ and X<<λ [23][24][25] ; hence, θ~X/λ. Let Eq. (4) be written as where ΘẐ=XV t . In view of V aw =dλ/dT=dω/dk, we obtain dω=(Θ/2π)·k·dk.. Thus we obtain Hence, dispersion law of quadratic form is given as By taking into account the dependence of rates V t and V aw on the grain size, δ 5 , differentiation of Eq. (13) is performed with respect to δ as Transformation of Eq. (14) yields the following differential equation which can also be written as Consider also the other consequences of the invariant. Let Eq. (4) be written as where the plastic deformation ε≈λ/X>>1. By applying the operator ∂/∂t=D ε ·∂ 2 /∂x 2 to the right and left parts of Eq. (17), consequently, we obtain The ultrasound propagation rate depends weakly on the plastic deformation, ε 26 ; hence, V t ≈ const. In this case, we obtain which is equivalent to the following differential equation for the deformation rate , , where the coefficient D ε has the dimension L 2 •T -1 . Equation (20) has the form of the reaction-dif fu si on equation for a concentration, which is obtained when the nonlinear function, φ(ζ), is entered into the right part of Fick's second law for diffusion 27 . Equation (20) falls into the category of nonlinear relations employed for describing autowave processes, which occur in different kinds of open systems, provided that adequate variables are chosen for solving the problem 28,29 . In what follows, Eq. (20) is discussed in detail.

Plastic deformation viewed as autowave generation process
By elaborating the autowave concept, we must consider the dependence of autowave patterns on the work hardening law acting at a given plastic flow stage. To make out individual flow stages on the curve σ(ε), the Lüdwick equation 30 where K is a hardening modulus; σ 0 =const and 0≤n≤1 is a hardening exponent which varies discretely. To verify whether Eq. (21) is operative, the Kocks-Mecking method 31 was used for recognizing plastic flow stage. The results obtained in both cases were found to fit neatly. The in dividual work hardening stages can be distinguished on the flow curve subject to the condition that n n n nK A total of four flow stages have been distinguished, i.e. easy glide/yield plateau, linear work hardening stage, parabolic work hardening stage and pre-failure stage. Likewise, the possible macro-locali zation patterns are limited in number: a total of four types of localized plasticity patterns are found to emerge in the deforming sample. Table 2 presents the localization patterns matched against the autowave modes.
Evidently, a one-to-one correspondence exists between the autowave modes and the respective plastic flow stages (Table 2 and Fig. 5). In view of the above, the sample tested in constant-rate tension can be regarded as a universal generator of autowaves 32 that requires no maintenance of temperature or reagent concentration as is the case with, e.g. chemical reactors 6,19 .
The autowave plastic deformation exhibits regular features, which are consequences of Eq. (4); all the empirical coefficients in Eqs. (1) -(3) have been assigned a physical meaning. In the frame of this concept the micro-scale level is related to the macro-scale features of localized plastic deformation by Eq. (4), which should be regarded as 'master equation' of the plasticity theory being developed.

Nonlinear equations for localized plastic flow
In general, the processes involved in self-organization are conventionally regarded as an interplay of an activator and a damper 6,28 . An understanding of how the both factors relate to the localized plastic flow could illuminate the roles they play. It is therefore assumed that the plastic deformation, ε, is an activator and the elastic stresses, σ, a damper.
It would be reasonable to describe the kinetics of the activator using Eq. (20) , which was inferred as a consequence of invariant (4). In turn, the stress relaxation kinetics can be described using the Euler equation for viscous liquid flux 33 , which has the form is momentum flux density tensor; δ ik , unit tensor; p, pressure; ν i and ν k , flux velocity components. The stress tensor σ ik =pδ ik +σ vis is the sum of elastic and viscous stresses, respectively, i.e. σ el =-pδ ik and σ vis . In the case of plastically deforming medium, σ=σ el +σ vis , i.e.
(here M is elastic modulus for the sys tem 'sample -testing machine'; ρ m is mobile dislocation density and V disl is dislocation motion rate).
Due to the inhomogeneous inter nal elastic strain field, viscous stresses, σ vis , will form in the deforming medium. The viscous stresses are re la ted to the variations in the elastic wave rate which are linear with respect to stresses, i.e. V t =V t0 +κσ 26 . Here V t0 is the transverse elastic wave rate in the absence of stres ses and κ=const. Assuming that σ vis =B∂V t /∂x (here B is medium's dynamic viscosity), we can write ∂σ vis /∂t=V t ∂/∂x (B∂V t /∂x)= BV t ∂ 2 V t /∂x 2 .  We conclude this Section by saying that the above analysis provides a unified explanation for the autowave nature of plasticity; hence, substantial revision of traditional notions in this field is required. Now that the work hardening process is regarded as evolution of the autowave modes, deformation kinetics analysis must involve different principles so as to formulate a new viewpoint of the nature of multi-stage plastic flow.
Hence, the relaxation rate is given for viscous stres ses as ∂σ vis /∂t = BV t ∂ 2 V t /∂x 2 = BκV t ∂ 2 σ/∂x 2 . Apparently, where D σ =BκV t is a transport coefficient having the dimension from Eqs. (20) and (23), correspondingly. These have to do with the steady moti on of deformation fronts along the sample, with the local stress concentrators occurring on the fronts being activated one by one. The diffusion components are given by the terms D ε ∂ 2 ε/∂x 2 and D σ ∂ 2 σ/∂x 2 from Eqs. (20) and (23). These are responsible for the deformation initiated in material volumes at macroscopic distance ~λ from the active deformation front.
Thus, the resultant system of equations

The two-component model. The autowave version
Note that the coefficients D ε and D σ from the system of Eqs. (24a) and (24b) have dimensions L 2 •T -1 , which coincide with those of the products λ•V aw and X•V t from invariant (4). Hence, The above suggests that invariant (4) is equivalent to the ratio D ε /D σ =2/3<1, i.e. D σ ≡ D damp > D ε ≡ D activ . The condition D damp >D activ is a prerequisite for auto wave ge ne ration in the active medium 6 . The autowave structure formation should be regarded as a basic attribute of the self-organizing active medium 6 . In a general case, the likelihood that self-organization processes will be initiated in the active medium depends on whether the medium itself is capable of separating spontaneously into information and dynamic subsystems 35 . The main features of plastic flow might be explained by assuming that the information subsystem is related to acoustic emission pulses generated by dislocation shears and the dynamic subsystem, to shears proper 36 . The evolution of two subsystems is described by Eqs. (20) and (23) .
Based on the assumptions above, a two-component model of plastic flow is proposed which operates in accordance with the scheme presented in Fig. 6. Due to the stress concentrator decay (1), stress relaxation will occur which causes generation of acoustic emission pulses. The stress relaxation results in the liberation of energy which is absorbed by the remaining stress concentrators which act as energy-sink (2). This phenomenon is known as acoustic-plas tic effect 37 . As new stress relaxation acts are initiated, shear processes will continue to occur in the dynamic subsystem, generating thereby a series of acoustic pulses. Thus the basic idea of the model being developed is that the acoustic emission and the acousticplastic effect are in no way interdependent. This issue has been neglected thus far. In support of this interpretation one can argue that the elastic-plastic invariant is given by Eq. (5), which contains the term ξ 1 =V t ρ for medium's acoustic resistance. To verify the model's validity, the expectation times, τ* and τ**, are estimated for the thermally activated relaxation acts 38 . In the event that the external stress alone is operating, T Y (26) in the event that both the external stress and the acoustic pulse are operating, In Eqs. (26) and (27) the activation enthalpy, U 0 -γ σ ≈ 0.5 eV 38 . Due the action of acoustic pulse having amplitude, ε ac ≈ 2•10 -6 , reduction in the activation enthalpy is ΔU ac ≈ γε ac E ≈ 0.1 eV and k B T≈ 1/40 eV. Under the above conditions, τ* ≈5•10 -5 s and τ** ≈9•10 -7 s << τ*. The estima te, rough as it is, evidently speaks for the proposed model. It is thus proved that the events occurring in the acoustic (information) and dislocation (dynamic) subsystems are interrelated.
The effect of transverse ultrasound wave splitting in the field of elastic stresses can be used to estimate the autowave length 39 . Assume that a transverse ultrasound pulse is emitted by an elementary shear. The maximal power in the acoustic emission spectrum corresponds to the frequency, ω a ≈ 10 6 Hz 37 . Due to the pulse splitting occurring in an elastically stressed area, two orthogonal polarized waves will form; these have lengths ς 1 =ν 1 /ω a and ς 2 =ν 2 /ω a and propagation rates ν 1 ≠ν 2 . A difference in the wavelengths is given in 39 An estimate of the same values can be obtained by assuming that a difference in principal normal stresses, σ 2 -σ 1 ≈ 10 8 Pa; material density, ρ≈ 5•10 3 kg/m 3 and sound rate, V t ≈3•10 3 m/s. Using this line of reasoning, we obtain that δς≈ 10 -4 m. This rough estimate suggests that the maximal energy of the elastic wave is accumulated at distance ~ς 2 /δς≈ 10 -2 m ~λ from the pul se origin. This is a plausible explanation for the fact that the new localized plasticity front emerges at distance ~λ from the existing deformation front.

The two-component model. The quasi-particle version
The above numerical analysis demonstrates that the products λ·V aw ·ρ·X 3 obtained for fourteen metals are close to the quantum Planck constant h = 6.626•10 -34 J•s 40 . On the strength of da ta presented in Table 3, we write where n 1 The calculation of the t t -criterion demonstrates that the values 〈h〉 and h really belong to the sampling from one and the same general population with the probability higher than 95%. Thus, Eq. (29) actually determines the Planck constant. This leads us to believe that quantum mechanics prin ciples are also suitable for investigations performed in the frame of plastic deformation physics. This finding led us to believe that a hypothetical a quasiparticle might be introduced for addressing the localized plasticity autowave. This procedure is conventionally applied in solids physics 42 . By omitting the index 'i' from Eq. (29), we obtain which is easily identified with the well-known de Broglie equation, which gives the effective mass of a quasiparticle moving with velocity V aw as m ef ≈ ρX 343 . Thus, a quasi-particle is postulated which corresponds to the localized plasticity autowave; its effective mass is given as m a-l =h/λV aw . The hypothetical quazi-particle has been called 'autolocalizon' 44,45 . Using Eq. (32), the average mass of the autolocalizon was calculated for fourteen me tals; the value obtained, 〈m a-l 〉= 1.7±0.2 a.m.u. An attempt of similar kind is worthy of notice. Thus the authors in 46 introduced a quasi-particle, so-called 'crackon', which was identified with the tip of growing brittle crack. It was also attempted to introduce so-called 'frustron', which might be appropriate for descriptions of the initial stage of lattice defect generation 47 . A striking analogy can be drawn between the auto localizons and the 'rotons'. The latter quasi-particles were introduced in the theory of liquid He 4 super fluidity 48 . The auto localizon and the ro ton have a dispersion law of quadratic form; the effective mass of roton, m rot ≈0.64 a.m.u. 48 , is close to that of auto localizon. In view of the above, the given approach could be elaborated in more detail.
Let Eq. (29) be written as where m ph is taken to be phonon mass. Now invariant (4) may be written as (36) which reduces to the balance of masses as 2m a-l =3m ph , i.e. three phonons would produce a pair of autolocalizons. This suggestion undoubtedly requires verification. It might be well to point out that si milar complex processes occurring in the phonon gas were described earlier 49,50 .
In the frame of quasi-particle approach the deforming medium might be viewed as a mixture of phonons and autolocalizons. It is thus suggested that the random walk of the Brownian particle is 48 where the time, τ=2π/ω≈10 3 s; the dynamic viscosity of the phonon gas, B≈5•10 -4 Pa•s (see above) and k B T 1/40 eV for T=300 K. Hence, the effective size of autolocalizon, r 10 a l 3 10 .
. . | X -m and the quasi-particle walk, S≈10 -2 m≈λ, which agrees with both the experimental value and that calculated from Eq. (28) . r - (38) we obtain S 2 /τ ≈ D ε ≈ 1.3•10 -7 m 2 /s. The above suggests that the processes involved in the plastic flow can be addressed effectively in the frame of quasi-particle and auto wave approaches.

The autowave deformation and the Taylor-Orowan equation
The problem this approach seeks to resolve is this: little is known about its relation to the dislocation theory which serves as the basis for the overwhelming majority of traditional models in plasticity physics [23][24][25]51,52 . Therefore, it is absolutely necessary to relate of the autowave equations derived herein to the dislocation mechanisms of plasticity. Note that the idea about quantization of dislocation deformation is in no way objectionable, since the Burgers vector, b=a 1 +a 2 +a 3 , is usually considered as 'a quantum of shear deformation' and its components, a i , are topological quantum numbers 50,53 . Now consider the function (20). For a homogeneous distribution of dislocations with average spacing d, ε'≈d -1 • (b/d)≈b·d -2 ≈bρ m (here b/d is shear per dislocation and d -2 ≈ρ m is mobile dislocation density). Given D ε ≈L•V disl (here L≈αx is dislocation path and V disl =const, dislocation rate), we obtain D' ε =αV disl . Hence, Eq. (20) reduces to the equality (39) where the first term from the right side coincides with the Taylor-Orowan equation for dislocation deformation, i.e.
fo =bρ m V disl , which might be considered now as a special case of Eq. (39). However, Eq. (39) is applicable to more general cases, e.g. media having high dislocation density, provided the diffusion-like term D ε ε" is added to the right side of the Taylor-Orowan equation. To clarify this purely formal procedure, we shall give some explanation of the condition to be met. Given fo =const by constant-rate tensile loading, the required level of dislocation flux, ρ m V disl , has to be maintained. If the condition is not met, owing to, e.g. a decrease in mobile dislocation density, the medium will initiate deformation processes in front of the plasticity nucleus remote from the nuclei already in existence -otherwise sample fracture will occur. This representation provides a new way of tackling nonlinear problems connected with the nature and evolution of dislocation substructures 24,25,52 .

Localized plasticity autowaves and work hardening coefficient
A number of well-established models link work hardening processes to the long-or short-range interaction of dislocations 51,52 . In what follows, the autowave mechanism has to be related to the work hardening phenomena in terms of plasticity physics. Assume that the work hardening coefficient is given as θ≈W/Q 23 (here W≈Eb 2 ρs is the energy stored by plastic defor ma tion; ρ s , immobile dislocation density; Q≈σbL d ρ m , energy dissipated by mobile dislocati ons having density, ρm, and path, L d ). Now it can be written . (40) where ε e =σ/E; L d = Λ•(ε-ε*) -1 ; Λ depends on the kind of material 24 аnd ε* is the strain for the onset of linear work hardening stage. It follows from Eqs. (1) and (40) With growing density of immobile defects, ρ s , the value V aw will grow less. With increasing amo unt of energy dissipated to he at, the deforming medium would warm up, which greatly increases the likelihood of thermally activated plastic deformation and of autowave rate growth. Now let dV aw ∼L d ; hence, we obtain 24,25 / , zb 3 / . i (42) where z is the number of dislocations in a planar pileup. It follows from the above that dV aw ∼L d ∼Λ∼θ -2 . The data obtained for a range of materials suggest that ε-ε*≈const 24 ; however, the value θ is found to vary for different materials, i.e. dθ ≠ 0. Then we can write Integration of Eq. (43) yields V aw ∼θ -1 . Finally, a special case of Eq. (1) is written as where θ*=dσ/dε is a dimension characteristic of the plastic flow process (see above). The values calculated for alloyed single γ-Fe crystals are as follows: . Thus the work hardening process is addressed above in the frame of conventional dislocation model by assuming that a changeover in the work hardening stages is due to variation in the distribution of stress concentrators 51 . Hence, the same factor might be responsible for the generation of new autowave modes in the active medium.

Conclusions
Solids plasticity is addressed above in the frame of autowave concept. The given approach might fall far short of a completely theoretical prediction. In point of fact, it might be regarded as an attempt at envision of the complexities of the plastic flow process, which would enable formulation of contemporary views of this phenomenon. Hopefully, more rigorous techniques would be developed for tackling the problem of plasticity. Thus far we have the pleasure of bringing to your notice the following conclusions.
1. The localization behavior of plastic deformation has been studied for all the plastic flow stages. Localized plasticity patterns are found to emerge in the deforming medium. A total of four pat tern kinds have been recognized in studied materials. It is found that the kind of pattern observed for the given flow stage strictly corresponds to the work hardening mechanism involved in the deformation. 2. The kinetics of plastic deformation and stresses is described using autowave equations derived on the base of continuum media mechanics. 3. On the base of experimental data elastic-plastic deformation invariant is obtained, which is a master equation for plasticity mechanics. In the frame of autowave model the invariant serves to relate the characteristics of localized plasticity autowaves to those of elastic waves. The distinctive features of localized plasticity autowaves are described by the consequences from the invariant.

A two-component model of localized plasticity
is proposed which is based on the assumption that shear processes interact with acoustic pulses. The autowave model is complemented by the quasi-particle model; the both versions are of equivalent status. 5. A relationships are established between the autowave representations and certain conventional dislocation models, which precludes any controversy between the proposed auto wave approach and the existing work hardening theories.

Acknowledgements
The work was performed within the Program of Fundamen-tal Research of State Academies of Sciences for the period of 2013-2020 and was supported by Tomsk State University in the framework of the Competitiveness Improvement Program.