3.1. The Deformation Localization Pattern and Some Important Features
The plastic flow possesses two characteristics which are common to all deforming solids. First, the deformation is found to exhibit a localization behavior on the macro-scale level from the yield point to the ultimate stress. Second, the dependence of the work hardening coefficient on the deformation would occur by stages in which each hardening stage involves a certain dislocation mechanism [
1,
2]. To gain an insight into the nature of plastic deformation, the existence of an explicit connection between the two characteristics must be disclosed. Thus, the distinctive features of localization patterns remain the same, with only quantitative differences. It is noteworthy that there should be a correlation between plastic deformation non-homogeneities for the entire material volume. The localized plasticity nuclei are distributed periodically in space at a distance of
λ ≈ 10
−2 m between them. In this way, phenomena appear spontaneously at the macroscopic scale during the deformation process.
An analysis of experimental data reveals that these regularities serve to provide a unified explanation of plastic flow behavior. Four types of localized plasticity patterns have been observed experimentally for all studied materials. At each flow stage, there is a certain type of localization pattern. In particular, there is the yield plateau identified by a single mobile plastic flow nucleus, the linear work hardening stage with a set of synchronously moving nuclei, the parabolic work hardening stage with a set of equidistant stationary nuclei, and the pre-failure stage with a set of mobile nuclei that are forerunners of failure. These results together furnish a reliable proof for the one-to-one correspondence between the localization pattern and the plastic flow stages. This regularity, named the ‘correspondence rule’, is fulfilled for all the materials investigated up until now.
A key aspect of this many-faceted problem to be dealt with is the nature of localized plasticity. Hähner [
22] has proposed a very attractive idea about the wave character of plastic flow, which was close to the well-known concept on stress waves formulated by Kolsky [
23]. However, our basic viewpoint is that there are striking parallels between the localized plasticity patterns and the synergetics of the relevant dissipative structures [
24]. These structures are also known as switching autowaves, phase autowaves, and stationary dissipative structures. They have been described in detail for various chemical and biological systems [
25], and were assumed to be useful for solving the plasticity problem.
The observed pattern of localized plasticity is a projection of the autowave modes on the surface of the deforming specimen. The relationship between the localization pattern and autowaves takes its origin from a formal resemblance between these kinds of phenomena. This fact can be taken as an unequivocal physical evidence of the autowave nature of processes under consideration.
Thus, the well-known Lüders front, being observable on the yield plateau, separates the elastically and plastically deforming material volumes. As this front propagates along the sample, it leaves behind an increasing volume of the deformed material [
13]. Because of the structural changes, the deforming material volume acquires a new state, which is characterized by increased density of defects, so that deformation occurs via the dislocation mechanism. As the total deformation increases, the plastic flow exhibits an intermittent behavior on the macro-scale level. Therefore, the Lüders band motion might be regarded as “a switching autowave”.
A different scenario is realized at the linear work hardening stage where a set of mobile nuclei is observed. In this case, the nuclei move at a constant rate
Vaw =
const along the test sample conforming to the phase constancy condition (
). This pattern corresponds to a phase autowave. The parabolic stage of work hardening is characterized by the values
n = 1/2 and
Vaw = 0. The localization picture emerging at the latter stage fits the definition of a stationary dissipative structure [
24]. Finally, at the pre-failure stage (0 <
n < 1/2), the deformation pattern evolution is nearing completion. The autowave process collapse [
26], observed at this stage, is coupled with the transition to a viscous failure.
Thus, the plastic flow process occurring in the deforming medium can be regarded as the regular continuous evolution of localized autowaves. Thus, it can be claimed that the transition from one to another flow stage involves a changeover in the modes of autowaves generated by the deformation. In a word, the plastic flow stages are closely related to the respective modes of autowave processes. This enables one to infer that the localized plastic flow is the evolution of the autowave pattern. Because of a changeover in the flow stages, the autowave modes will emerge from a random strain distribution in an orderly sequence: switching autowave → phase autowave → stationary dissipative structure → autowave collapse. However, some stages of the processes may be absent.
It is well known that experimental investigations of autowaves in chemistry or biology require the use of a special-purpose reaction cell. Such cells differ widely in type and size, depending on the kind of a studied system and its chemical composition as well as the kinetics of chemical reactions involved, temperatures employed, etc. However, autowaves in deforming media are another matter. Under constant-rate loading, localized plasticity autowaves will be generated spontaneously in the stretched sample at any temperature. Hence, the deforming solids can be regarded as a universal reaction cell, which can be conveniently used for studying the generation and evolution of all kinds of autowaves. As it is established experimentally, the plastic flow exhibits a regular localization behavior, which is markedly pronounced at the linear work hardening stage. Here the localization nuclei will move in a concerted manner at a constant rate along the stretched sample, forming a phase autowave according to the mechanisms below.
First, the propagation rates of localized plasticity autowaves in the studied materials are in the range 10
−5 ≤
Vaw ≤ 10
−4 m/s. They depend solely on the non-dimensional work hardening coefficient
as follows:
Here, V0 and Ξ are the empirical constants with the dimensionality LT−1.
Second, the dispersion relation
ω(
k) for localized plasticity autowaves, where
is the frequency and
is the wave number, is defined as
in which
ω0,
k0 and
α are the constants. Using the substitutions
and
, where
and
are the dimensionless frequency and wave number, correspondingly, one can reduce Equation (4) to a canonical dimensionless form
(
Figure 3).
Third, the experiments were carried out for the polycrystalline aluminum specimens with the grain sizes in the range of 5·10
−3 ≤
δ ≤ 5 mm. The grain size dependence of the autowave length
λ(
δ) has the general form of a logistic curve [
27]
where
a1 and
a2 are the empirical coefficients,
λ0 ≈ 5 mm, and
C = 2.25. The inflection point for the plot described by Equation (5) and found from the condition
is
δ =
δ0 ≈ 0.2 mm. There are two limit cases for Equation (5), which are
at
δ <
δ0 and
at
δ >
δ0.
Based on the studies, one can conclude that the plastic flow of solids is related to macroscopic features of localized phenomena.
3.2. The Physical Basic of the Autowave Model
Now it is clear that the problem at hand is the development of the new model of plastic flow evolution during plastic deformation. This is also evident from works [
8,
9,
10,
12], revealing that the nature of the macroscopic length of localized plastic flow autowaves remains poorly understood. As mentioned above [
7], the plastic flow localization can be regarded as a spontaneous process of self-organization (‘
structure formation’) in an open system. This suggestion is supported by the features in the localized plastic flow pattern, which are identified at different work hardening stages and demonstrate the spatial distribution of deforming and undeforming layers in media. This concept has been developed for self-organization of open non-equilibrium systems [
6,
24]. A similar system can attain spatial, temporal or functional inhomogeneity without any specific action from the outside (here specific action implies any external action which causes the system to acquire a certain kind of structure). It is worth noting that the definition implies no certain underlying mechanism responsible for the self-organization process. The formation of the localization pattern can be regarded as an ordering phenomenon in the deforming medium.
The concept of self-organization is frequently and successfully used to explain the structural formation in all kinds of active media studied in physics, chemistry, materials science or biology. The deforming medium can be likened to an active medium far from thermodynamic equilibrium in which the sources of energy are distributed over the material volume.
It is also evident that plastic flow also involves self-organization phenomena. According to [
8], the generation of localized plastic flow autowaves causes a decrease in the entropy of the deforming system, which is the fundamental condition of self-organization processes. When studying the plastic flow localization in solids, one should take into account the significant changes in the acoustic characteristics during the material deformation, such as the velocity of transverse elastic waves (
Vt) [
28,
29]. Up to now, the acoustic characteristics of the deforming medium have been considered in terms of energy dissipation, especially in internal friction studies and related problems.
Another characteristic of the medium is the phonon gas viscosity [
30]. At first glance, this quantity seems to be out of place in the analysis of slow processes of the localized plasticity autowave propagation. However, it can be assumed that dislocations move inside the localized plasticity nuclei between and over the local obstacles under the control of the phonon gas viscosity
B conforming with the law
. The electron gas viscosity in metals also makes a significant contribution to the coefficient
B.
It is found that mechanical and acoustic characteristics of the deforming medium are closely related to each other. This finding is supported by the experimental evidence suggesting that the acoustic processes play an important role in the development of localized plastic flow. The available acoustic emission data imply that structural nonhomogeneities emerge in the deforming medium due to a traveling deformation front. Thus, the acoustic emission sources distributed over the material bulk are associated with the localized plastic flow nuclei occurring in the same material. To elucidate the nature of localized plasticity, a two-component model was formulated. In this case, a key role is assigned to the acoustic properties of the deforming solid so that acoustic emission pulses play the role of the information system which controls the structural transformation dynamics.
In the framework of the concept used to describe the autowave formation, the basic problem is the nature of self-organization, which manifests itself in the deforming medium as a spontaneous emergence of the pattern. To solve the problem, a self-organizing system must be separated spontaneously into dynamic and information interacting subsystems [
25].
The principles of the proposed model are as follows. In the course of plastic deformation, local stress concentrators would form and disintegrate, which are considered the dislocation pile-ups [
1]. The elementary stress relaxation is due to the breaking from a local obstacle, which involves acoustic emission [
30]. These acoustic signals can activate other stress concentrators so that the same process is repeated. Thus, acoustic emission signals propagating in the deforming medium play the role of information subsystems, whereas dislocation shears are involved in the plastic deformation proper and operate as a dynamic subsystem. The developed model is hence made up of two components, acoustic emission and dislocation mechanisms of plasticity, which have been extensively studied, though in different contexts. The generation of acoustic signals was considered in connection with the initiation of dislocation shears, while the reverse process, i.e., the initiation of shears due to acoustic pulses, has not been touched on so far.
The next step consists in assessing the validity of the proposed model. Let an acoustic signal propagate in a non-uniform dislocation substructure, which is formed by deformation, e.g., a dislocation cell with a size
0.01 mm, which is observable via transmission electron microscopy. Such a cell can be regarded through an acoustic lens with a focal length
fl defined as:
where
is the ratio of rates for the non-deformed and deformed volumes, playing the role of the acoustic refractive index. The initiation of plastic flow is due to the localization of ultrasound waves at the distance
10
−2 m from the active localized plasticity nuclei.
The delay times of thermally activated spontaneous relaxation acts [
31,
32] in the absence of acoustic pulses can be expressed as follows:
and the same under the action of inducting pulses:
Here, ωD is the Debye frequency, U0 is the potential barrier to be overcome during the relaxation event, is the activation volume of this event, b is the Burgers vector of dislocations, l is the size of the shear zone, kB is the Boltzmann constant, and T is the temperature. Based on Equations (7) and (8), the activation enthalpy of the process is set as 0.5 eV, and the acoustic pulse with the elastic strain amplitude decreases this parameter by 0.1 eV. The calculation at T = 300 K yields 5·10−5 s and 9·10−7 s These estimations provide evidence for the principal possibility of plastic flow acceleration under the action of acoustic pulses, and confirm the correctness of the model.
Therefore, Equations (7) and (8) demonstrate also the possibility of induced decay of stress concentrators. It seems that the rate of localized plasticity autowave dynamics is determined by the time of plasticity center growth from a nucleus that can be identified with the intergrowth of the Lüders band through the sample as . Moreover, it is well known that the perfect crystal lattice is a source of crystal defects responsible for the change in plastic form, which must be taken into consideration when describing self-organization processes as well. Hence, the basic premise of the paper is that the regular features of plastic flow macrolocalization are directly related to the lattice characteristics.
3.3. Fundamental Equations of the Autowave Model for Plasticity
The difference between the waves and autowaves can be shown as follows. In the simplest case, waves are described by the function
, where
ω is the frequency, and
k is the wave number. These functions are the solutions of partial hyperbolic differential equations
[
25]. The wave rate c is determined by the material constants, e.g., for the transverse elastic waves,
(here
G is the shear modulus,
ρ is the material density). The second derivative with respect to time is the sign of reversibility of processes, in particular elastic waves.
Autowaves have long been recognized as solutions of partial parabolic differential equations
[
25]. A similar equation can be obtained formally by adding a nonlinear function
to the right-hand side of the equation
, describing the transport phenomena by analogy with diffusion or heat conductivity. The coefficient
κ has the dimensionality of L2T-1 and does not refer to the material’s properties. The first derivative with respect to time,
, means that the above equation is suitable for determining the irreversible processes similar to those involved in the plastic deformation.
To offer adequate tools for describing the autowave processes, a set of two equations must be produced: one considering the rate of change for the catalytic factor and another for the damping factor. The choice of these factors is far from trivial. In case of plasticity, it is reasonable to guess that plastic deformation ε is a catalytic factor and stress σ is damping the same.
Hence, the equations for rates
and
can be derived from the general principles. The equation for
obeys the following condition of deformation flow continuity [
4]:
where
, and the value
is the transport coefficient for the autocatalytic factor; the term
is the deformation flow in the deformation gradient field. For the uniaxial elongation along the
x-axis, the autowave equation can be derived from Equation (9) as follows:
Here, is a non-linear stress-strain function, which is used to satisfy the condition .
The equation for the damping factor change rate
can be derived from the Euler equation for a liquid flow as follows [
33]:
Here, the momentum flux density tensor is
,
is the unit tensor, and
and
are the flow rate components. The stress tensor
includes viscous and elastic stresses, i.e.,
; hence,
or
The viscous stress
is due to the plastic deformation inhomogeneity; the value
is related to the variation in the elastic wave velocity in the deforming medium, i.e.,
. In turn,
is the dynamic viscosity and
is the propagation velocity of transverse elastic waves. The equation
yields
. The value
behaves as a linear function of stress, i.e.,
[
28]. Hence, the equation for stress change rate is
. Then, the autowave equation for stresses takes the form
where
and
. It is of great importance that solving a set of Equations (10) and (12) enables one to predict adequately the regularities of different plastic flow modes [
8].
The uniformity of localized plasticity phenomena for various materials implies the existence of a general law for the localized plastic flow autowaves. This section focuses on searching for a similar relationship. One can assume that there is the relationship between plastic flow macro-parameters and crystal lattice characteristics. To verify this, one can match products
and
for plastic flow and elastic deformation, respectively (
Figure 4). The quantities
and
are the interplanar spacings of the crystal lattice and the transverse ultrasound wave velocity, respectively. Both the experimentally found values (
λ and
) and textbook values (
χ and
) were used in the analysis. It was found that
The obtained relationship is valid for all the studied materials. The values of
are distributed according to the normal law [
33] relative to the average quantity
This result indicates that elastic and plastic deformation processes are closely related to each other by the plastic flow. It is noteworthy that the products
and
in Equation (13) are related to the transport coefficients
and
in Equations (10) and (12), as follows:
and
Thus, Equation (13) was called the “
elastic-plastic strain invariant”.
Further, it is of interest to discuss some remarkable properties of the invariant. For example, the product
, included in Equation (13), characterizes the stable development of the linear work hardening stage. If
, then
where
is the small displacement,
W is the interparticle potential, and
is the shear modulus [
34]. The value
is the specific acoustic resistance of the medium.
The potential
W can be expanded into a power series [
35] as follows:
where
f is the quasi-elastic coupling coefficient and
g is the anharmonicity coefficient [
34]. With the proviso that
, Equation (14) takes the form:
The invariant (13) is also suitable for describing deformation initiated by chaotically distributed dislocations. If the density of a mobile dislocation is
, then the average distance between dislocations is
. It can be assumed [
2] that
, i.e.,
. The speed of a quasi-viscous motion of dislocations is
, where
B is the coefficient of dislocation drag by the phonon and electron gases [
30]. Thus,
The values
G ≈ 40 GPa and
B ≈ 10
−4 Pa·s are conventionally employed to describe the dislocation motion. Using these values, it can be found from Equation (17) that
10
−6 m
2/s. The latter value is close to the calculated product
for materials under consideration. The above suggests that a reliable quantitative criterion has been established for analyzing the interaction between elastic and plastic deformation, which occurs on the macro- and micro-scale levels. This criterion, in its universal form, is applicable to autowaves as well as to elastic and plastic deformation during the dislocation motion. Therefore, this criterion can be considered as a useful generalization parameter of the elastic-plastic strain invariant:
In view of the above, the invariant (13) is promising for gaining an insight into the nature of localized plastic flow. All the basic regularities of localized autowaves can be deduced from Equation (12); therefore, the invariant is expected to play an important role in the development of new models of plasticity.
A very important dependence of the product on the interplanar spacing was recorded for studied materials. This dependence () falls into three sections, which correspond to the 3rd, 4th or 5th period of the periodic table. This implies a close relation between the deformation processes and the electronic structure of metals according to their position in the periodic table.
Since the invariant (13) is referred to the characteristics of simultaneously proceeding elastic and plastic processes in a deforming medium, it plays an important role of ‘master equation’ in the autowave theory of localized plastic flow under consideration. This is confirmed by the fact that the invariant (13) has a number of consequences that can be used not only to understand the nature of some important features of dynamics of localized plasticity, but also to describe them quantitatively. Certain consequences will be discussed below.
For example, one can show that the velocity of phase autowaves follows from the invariant. Indeed, the derivative of the invariant (13) with respect to the deformation
ε has the form
The value
χ is independent of the plastic strain, so that
, and
Therefore, Equation (21) can be rewritten as follows:
This result coincides with the experimentally established Equation (3), if the work hardening coefficient is expressed as the ratio of the structural parameters
[
2].
The dispersion relation for the localized plasticity autowaves can be also obtained from Equation (13) as follows:
where
. If
, then
Evidently,
and the dispersion law of autowaves acquires the quadratic form:
where
is the parameter used in Equation (5). In this case, the coefficients in Equation (25) can also be calculated by rewriting Equation (23) as
where
is the Debye temperature [
34]. Then,
. Using the reference values
χ(Fe),
χ(Al),
and
, one can find that
3.7·10
−7 m
2/s and
4.45·10
−7 m
2/s. These are consistent with the values
10
−7 m
2/s and
12.9·10
−7 m
2/s, experimentally found from the dependence
(
Figure 5).
There is a very interesting problem concerning the influence of the structure (e.g., grain size) on the length of the autowave. Supposing that it follows from the invariant (13), one obtains
If the velocities
and
depend on the grain size
δ (
Figure 5), then the differentiation of Equation (27) with respect to
δ yields:
from which it follows that
where
and
since
A solution of Equation (29) is the logistic function [
27]
Apparently, Equation (30) is equal to Equation (5) obtained experimentally for Al.
The Hall–Petch relationship [
13] is close to the dependence determined by Equation (30). It is well known that the mechanical characteristics of polycrystalline materials (yield limit and flow stress) depend on the grain size. The corresponding dependences are usually linear in the coordinates ‘property–
’. Being plotted in the coordinates ‘
’, this dependence is separated into two parts at the point
mm. Therefore, it makes sense to check the validity of Equation (13) for two ranges of grain sizes, namely
and
. The results of estimations indicate that
for both ranges.
The dimension effect of the plastic flow can also be explained in terms of Equation (13). If the autowave length is measured in samples of different length
L, then
and
For
and
, it obtains
and
, that is,
, as established previously in the experiments [
8].
It is worth mentioning that the autowave equation of deformation can be derived from the invariant Equation (13) as follows:
where
is the plastic deformation. It is possible to introduce the differential operator
and to act on both sides of Equation (33). Thus,
The ultrasound velocity depends weakly on the strain, so that
in Equation (34). Since
, then
This relationship is equivalent to Equation (10) for the strain rate, if .
A key question is the relationship between the autowave approach and the dislocation theory, which is used usually [
1,
2] to explain the nature of plastic deformation and work hardening in real crystals. The dislocation models are mostly based on the Taylor–Orovan equation [
2]
bringing the plastic strain rate
into correspondence with the mobile dislocation density
and their rate
. A comparison of Equations (34) and (36) shows that the term
in Equation (34) is analogous to
in Equation (36). Indeed, it can be assumed that
and
. If
then for the dislocation chaos
, where
l is the free path of dislocations. In this case
By setting
, one can write
, so that:
Evidently, Equation (38) differs from the Taylor–Orovan Equation (36) by the term which is responsible for the macroscopic strain redistribution over the volume at a certain distance from the existing fronts. This means that the Taylor–Orovan equation is a special case of Equation (38), which includes the diffusion-like component of the strain flow along with the hydrodynamic component .
Hence, it follows that the autowave plastic flow model is reduced to the dislocation model the same in the limit case. It is obvious that the application of the Taylor–Orovan Equation (36) at small dislocation densities (small strain) allows one to obtain reliably correct results. However, for large strains corresponding to the high defect densities and the nonlinear dislocation properties, it is more expedient to use the universal autowave Equation (38).
It is noteworthy that the invariant enables one to explain the origin of localized plasticity autowaves. Indeed, according to the Taylor–Orovan Equation (36), the condition
set to a testing machine is fulfilled by
. This condition can be violated under strain hardening, as well as at the decrease in the density of mobile dislocations with increasing strain or the decrease of dislocation velocity when the effective stress decreases from
σ to
[
2] (here
is the total dislocation density). In this case, the condition
is satisfied only if the diffusion-like strain mechanism described by the term
in Equation (7) is incorporated and the relaxation act of the localized plastic flow nucleus at the distance
from the initial nucleus is induced by one of the above-considered mechanisms. This assumption can be used to explain the generation of the macroscopically localized plastic flow autowave.
Finally, Equation (13) enables one to calculate the work hardening coefficient. For this aim, using the autowave velocity from Equation (22) and the dispersion component from Equation (25), the following expression can be written:
According to Equation (39),
, which is close to the experimental values at this stage in single and polycrystalline metals and alloys [
2].
Thus, the consequences from Equation (13) explain many regularities of plastic flow. For this reason, the invariant can be thought as the master equation for the autowave theory of plasticity.