CC BY
45
УДК 539.4, 53.09
Collective properties of defects, multiscale plasticity, and shock induced phenomena in solids
O.B. Naimark1, Yu.V. Bayandin1, and M.A. Zocher2
1 Institute of Continuous Media Mechanics UrB RAS, Perm, 614013, Russia
2 Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
A statistically based approach is developed for the construction of constitutive equations that provides linkages between defect-induced mechanisms of structural relaxation, thermally activated plastic flow, and material response to extreme loading conditions. The collective properties of defects have been studied to establish the interaction of multiscale defect dynamics and plastic flow, and to explain the mechanisms leading to the universal self-similar structure of shock wave fronts. An explanation for structural universality of the steady-state plastic shock front (the four power law) and the self-similarity of shock wave profiles under reloading (unloading) is proposed. Structural characterization under transition from thermally activated dislocation glide to nonlinear dislocation drag effects is developed in terms of scaling invariants (effective temperatures) related to mesodefect induced morphology formed during the different stages of plastic deformation.
Keywords: structural relaxation, collective modes of defects, multiscale plasticity, shock waves
Коллективные свойства дефектов и явления многоуровневой пластичности в твердых телах при ударно-волновом нагружении
О.Б. Наймарк1, Ю.В. Баяндин1, M.A. Zocher2
1 Институт механики сплошных сред УрО РАН, Пермь, 614013, Россия
2 Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
Определяющие соотношения, основанные на стататистическом описании коллективного поведения ансамблей дефектов, применяются для изучения связи механизмов структурной релаксации, обусловленных дефектами, термоактивационными механизмами пластического течения и поведением материалов при экстремальных режимах нагружения. Предложено объяснение степенной универсальности устойчивых пластических волновых фронтов и автомодельности ударно-волновых фронтов при догрузке и разгрузке. Изучены структурные особенности перехода от термоактивационного дислокационного скольжения к нелинейным закономерностям атермической пластичности на основе вычисления масштабных инвариантов (эффективных температур), характеризующих морфологию деформационных структур, на различных стадиях деформации.
Ключевые слова: структурная релаксация, коллективные моды дефектов, многоуровневая пластичность, ударные волны
1. Introduction
Experimental studies pertaining to the response of solids over a wide range of loading rate serve to shed light upon some of the fundamental processes at work in material plasticity and failure. In particular, these studies reveal linkages between material behavior and the evolution of ensembles of mesoscopic defects (typical among which are dislocation substructures, microcracks, and microshears). The impact of this is particularly pronounced whenever the loading is dynamic (especially in the case of shock wave loading), for in this case characteristic times for the evolu-
tion of defect ensembles may be of the same order as the overall loading duration. As a consequence, the widely-held assumption that phenomenology can be applied linking internal state variables to global stress-strain variables (adiabatic limit) and the evolution of damage breaks down. In this case it is paramount that the behavior of defect ensembles and the multiscale nature of defect interaction be specifically accounted for. This leads to a multiscale approach wherein the role of the defect ensemble on the local scale relates to a localized change in symmetry of the displacement field, and the role of defect ensembles on the
© Наймарк О.Б., Баяндин Ю.В., Zocher M.A., 2017
global scale relates to the generation of new collective modes. These modes could drive system behavior as it relates to relaxation (plasticity) and the evolution of damage. On the local scale the change in symmetry follows from the dislocation nature of defects (formally from the gauge field theory) and must be reflected in the corresponding internal state variables (the order parameters). On the global scale the change in symmetry follows from the group properties of evolution equations for the order parameters with specific nonlinearity, which leads to spatial-temporal localization in defect distribution and the potential generation of new structures that are important in relaxation phenomena and the evolution of damage. These new structures may be associated with new and important characteristic scales of length and time.
In contrast to the traditional view of continuum plasticity, recently conducted research (both experimental and theoretical) has shown crystal (and polycrystal) plasticity to be characterized by large spatial-temporal fluctuations possessing scale-invariant characteristics [1, 2]. It is observed that plastic flow proceeds in bursts of intermittent deformation having power-law size distributions. Deformation patterns are characterized by long-range correlations, self-similarity and self-affine surface morphology (deformation induced roughness). Nonlinear dynamics of the yield stress resulting in a serrated stress-strain curve (conventionally known as the PLC effect or jerky flow) is also observed for many metals and alloys. According to the traditional view of classical continuum plasticity, wherein the flow of material is homogeneous, spatial-temporal fluctuations occur only in the advent of instabilities brought about by anomalous strain (or strain rate) softening. These instabilities develop in the form of localized plasticity, often possessing band structure with a length scale ~10-100 ^m. Concomitant with the development if this localized flow is a decrease in stress on the stress-strain curve. An alternative mechanism that may be responsible for the development of spatial-temporal fluctuation is the development of shear bands and subsequent pronounced correlations between these bands. Studies of the development of localized shear bands in crystals using high-speed framing cameras that enable an analysis of band statistics on the crystal surface have shown varying dynamics relating to shear band physics. As a consequence varying dynamics in yield stress fluctuations is observed as the transition from stochastic dynamics of amplitude pulsation to the power law with the increase of the strain rate.
Note that the features of stochastic behavior related to the stress fluctuations that have been described here are characteristic of self-organized criticality (statistics 1/f) when the events on all temporal and spatial scales exhibit a power-spectrum and the system behavior is not characterized by a finite correlation dimension. It is possible to conclude that the dynamics of jerky flow imply a transition from the dynamics of chaos, to the statistics 1/f under in-
creasing strain rate. This transition is usually observed, when the relaxation time of plastic flow approaches the characteristic loading time that is typical for dynamic and shock wave loading.
In classical continuum plasticity, plastic deformation of a crystalline solid is viewed as smooth and quasi-laminar flow. The crystalline solid is considered to be a homogeneous continuum, where plastic flow can proceeds in a spatially homogeneous manner in the absence of any plastic instabilities (such as occur with strain softening). In this paradigm, fluctuations are averaged (smoothed) out on any length scale above that of a representative volume element, which is small in comparison to the dimensions of the body undergoing deformation. In this case, as already alluded to, spatial-temporal nonhomogeneities occur only in the case of plastic instabilities (such as occur with strain softening). Instabilities of this type give rise to spatial-temporal nonho-mogeneities with the dynamics of running solitary waves or stochastic deformation modes.
The classical paradigm of stable homogeneous plastic flow is of course a gross simplification that does not account for the incremental flow that is associated with the mechanism of momentum transfer resultant from dislocation motion. Not surprisingly, the classical view has been subject to challenges, both from an experimental and from a theoretical point of view. A qualitative new picture of plastic flow has arisen. Instead of a chorus of uncorrelated dislocation motions resulting in homogeneous flow, numerous localized bursts of deformation occur under the conditions of pronounced long-range spatial-temporal correlations.
The nucleation and evolution of these regions of localized flow are associated with plastic instabilities possessing qualitatively new features. Instead of coordinated spatial-temporal oscillations, the stochastic bursts of localized plasticity are observed on scales that are spatially and temporally free. This means that the development of plastic deformation has a nature similar to certain criticality phenomena (e.g., phase transitions), and, as a consequence, pronounced self-similarity features of plastic flow may be established.
Self-similarity is an important feature of many natural phenomena and is characteristic of strongly correlated many body systems wherein fluctuations over all scales (from microscopic length a to a divergent correlation length £) lead to the appearance of "anomalous dimensions" and fractal properties. Keep in mind that the divergence of scales in the real systems is limited by the macroscopic length of the system L that determines the range of scales between a and L, over which the anomalous behavior can occur.
Statistical theory related to the evolution of mesoscopic defects (e.g., microcracks, microshears) has enabled us to establish a new type of criticality phenomenon in solids with defects—structural-scaling transitions. Concomitantly, to develop an associated thermodynamics for solids with defects and propose a phenomenology based on a gene-
0.8 Time, js
1.2
£ 0.8 1
£ 0.4-
O
m
0.0
fT—~
f,- i ! 11 11 ii _
0.06 0.08 0.10 0.12 0.14 0.16 Scaled time, jus/mm
Fig. 1. Experimental data for reshocking and unloading of Al in real variables (a) and scaled variables (b) [5]
ralization of the Ginzburg-Landau theory leading to the development of wide range constitutive equations [1, 3, 4]. The key results of statistical theory and statistically based phenomenology are: (1) the establishment of two order parameters responsible for structure evolution—the defect density tensor (deformation induced by defects) and the structural scaling parameter, which reflects the scaling transition in the course of nucleation and growth of defects, and (2) the generation of characteristic collective modes of defects responsible for relaxation (shear transformation zones) and failure (damage transformation zone). These modes have the nature of solitary wave and blow-up dis-sipative structures providing the mechanisms for plastic relaxation and damage-failure transition that can be excited in the resonance regime of dynamically loaded and shocked materials. Dynamic shock wave experiments combined with structural studies have led us to an understanding of linkages between the evolution of these modes and material response over a wide range of strain rates and loading intensities. Moreover, they have led to our proposal of an interpretation of the universality of the steady state shock wave front as a consequence of the dynamic balance between nonlinear stress-strain (more specifically, nonlinearity of free energy release) and dissipation properties of the system. Established by Swegle and Grady [5, 6], the four
power law for plastic strain rate and stress amplitude (the so-called, structured plastic wave front) and the elastic-like anomaly of wave profiles [7] under reloading and unloading, support the existence of a self-similar nature relating the mechanisms governing relaxation to the consequence of the collective behavior in mesodefect ensembles. Experimental and theoretical investigations of shock wave structure (Fig. 1) are of great importance since there are many open questions related to the nature of relaxation under shock wave loading, the physics taking place in the shock front, and the thermodynamics of shock wave phenomena [8]. With the advent of VISAR diagnostics, shock wave profiles can now be easily captured in high resolution. Its use has revealed a number of general characteristics pertaining to shock wave physics: for example, the sharp rise of strain rate (Fig. 2), the four power universality of the dependence of strain rate versus stress amplitude [5, 6], the elastic-like anomaly of wave profiles under reloading and unloading, and a drop in yield stress under the impact of high shock stress [8].
2. Mesodefect properties
The response of a material to the imposition of dynamic loading depends upon its current microstructural state (grain
9.00 GPa (4) (a); stress amplitude versus strain rate (b)
size and orientation distribution, dislocation density, dislocation network structure, etc). Based upon empirical evidence, it is well-known that dislocation density increases with plastic deformation and that the morphology of dislocation substructures evolve as well. These phenomena occur under a variety of loading scenarios, including fatigue, creep, dynamic and shock wave loading. Moreover, it has been observed that dislocation substructure evolution in not random, but tends to follow a consistent pattern with a limited number of substructure morphologies. Temperature and dislocation interaction act as the primary drivers of substructure evolution. The transition from one type of dislocation substructure to another often produces sharp changes in the mechanical properties of both metals and alloys. Changing dislocation substructure is a fundamental mechanism for dislocation friction (viscoplastic material responses) and deformation hardening.
Each type of dislocation substructure is associated with a particular range of dislocation density, and it is important to note that these ranges are common for a large classes of materials. The reason for such commonality relates to the inherent property of dislocation ensembles as essentially nonequilibrium systems possessing self-similarity features in the sense of characteristic nonlinear responses. It is observed experimentally that the forces that lead to a change in dislocation structure exhibit a rather low level of sensitivity to the externally applied load, but a rather high level of sensitivity to localized stresses induced by dislocation interaction. It is the collective properties of dislocation ensembles that exert the greatest influence in dislocation substructure transition and dislocation substructure formation.
The driving force behind the construction of dislocation ensembles is nature's tendency to seek a minimization of total energy, which is at least partially satisfied through the creation of dislocation substructures [9, 10]. The energy of a dislocation substructure includes two primary parts: the energy related to the existence of the substructure, and the energy due to dislocation interaction. Construction of the dislocation substructure leads to a change in both of these parts. Since the system seeks to minimize its total energy, the energy of a newly formed dislocation substructure will be less than the energy of the preceding substructure any time substructure change takes place. This leads to a self-organizing tendency for the formation of substructures in the dislocation ensemble. The existence of critical dislocation densities corresponding to the creation of specific dislocation substructures was discussed in [10]. Typical values of critical dislocation density are presented in Table 1 for the copper alloys. These data tend to support an assumption that the primary independent variable for substructure morphology is the dislocation density. Dislocation substructure, in turn, influences mechanical response, deformation, hardening, and linkages between various defect ensembles. This serves to emphasize a view
Table 1
Types of transition in dislocation substructures [10]
Type of transition <pc >-10-1°,cm -2
Chaotic substructure ^ tangles 0.2
Tangles ^ orientated cells 0.2-0.5
Orientated cells ^ disorientated cells 0.4-0.6
Pile-ups ^ homogeneous net substructure 0.2-0.3
Disorientated cells ^ net substructure strips 0.5-2.0
Disorientated cells ^ strips 3.0
Strips ^ substructure with continuous disorientation 3.0-4.0
of the significance of collective effects in the dislocation ensemble and more pronounced sensitivities of structural transformation to the current value of the dislocation density, as opposed to the current external load.
In the normal course of events, local fluctuation of dislocation density precipitates a transformation from one substructure to another. An increase in fluctuations leads to a change in the distribution function for dislocation density caused by second mode generation. Such regularities in the evolution of defect substructures lead one to consider these substructures as independent subsystems of the material during deformation. In this case the list of governing state variables, like temperature and stress, can be extended through the addition of order parameters related to the dislocation substructures.
Mesodefects (like microcracks and microshears) may be considered paramount among defect structures when it comes to plastic flow and damage [11]. Other defects, such as point defects, dislocations, and dislocation pile-ups, produce smaller perturbations in local elastic fields and possess less energy in comparison to microcracks and micro-shears. Moreover, the nucleation and growth of these defects (that are closer to the macroscopic level) are resultant from the previous rearrangement of dislocation substructures and any entropy producing events. The density of these defects may reach 1012-1014 cm , but keep in mind that each mesoscopic defect consists of a dislocation ensemble and exhibits the properties of that ensemble.
Studies pertaining to the distribution of microcracks and microshears resultant from diverse loading conditions, have revealed important features of statistical self-similarity and a universality in the distribution function in self-similar (scaled) coordinates related to the size and concentration of these mesodefects (Fig. 3 [12]). This statistical self-similarity reflects invariance in the form of the distribution function for mesodefects of different structural levels. This fact has important implications for the development of a statistical multiscale (multifield) theory for the evolution of defect ensembles.
Fig. 3. Distribution of microcracks and microshears in dimension and dimensionless coordinates: n is the microcrack (microshears) concentration, l is characteristic size; nsc, lsc are scaling parameters [12]
3. Modeling of continuum with mesodefects
Constitutive relations that are dependent in general on a set of internal state variables, are typically constructed using a sophisticated homogenization procedure and relationships between internal state variables and microstructure evolution. Real solids are complex in structure, including a hierarchy of different scale levels and, under the influence of load, may undergo changes on all structural levels. In modeling material behavior, these changes have been understood to include plastic deformation and myriad damage processes, including nucleation, evolution, and the interaction of defects on a given structural level as well as the interaction of defects between structural levels. Heretofore, no unified multifield (or multiscale) theory of solid behavior has been developed that can accurately describe the variety, complexity, and interaction mechanisms, of processes taking place across all levels of structure. Thus, in order to construct a model capable of accounting for all of these features, processes, and mechanisms that influence material behavior, prime attention must be given to choices of the physical level of microstructure, and consequently, to the type of defects.
3.1. Phase field modeling
Phase field modeling was originally developed as a mathematical formulation applicable to the modeling of interfacial problems. It has mainly been applied to solidification dynamics, including viscous fingering. Attempts have been made to apply this methodology to the dynamics of fracture. The general idea in phase field modeling is not to track the location of interphases or defects explicitly, but
to introduce a so-called phase field. The method substitutes boundary conditions at an interface with a partial differential equation for the evolution of an auxiliary field (the phase field), which takes on the role of an order parameter. This phase field takes distinct values in each of the phases with a smooth change occurring between these values in the zone around the interface, which is diffuse over a finite width. A discrete location of the interface may be defined as the collection of all points where the phase field takes a certain value. A phase field model is usually constructed in such a way that in the limit of an infinitesimal interface width (the so-called sharp interface limit) the correct interfacial dynamics are recovered. This approach enables solution of the problem by integrating a set of partial differential equations for the whole system, thus avoiding an explicit treatment of the boundary conditions at the interface. Phase field models were first introduced in [13-15] and have experienced growing interest in several areas.
In the simplest case of two phase material, which can consist for example of a hard and a solid phase, the value ^ = 1 is assigned to the hard and ^ = 0 to the soft phase. If the transition between the phases is now smoothed out on a small numerical length scale £ (instead a sharp transition from ^ = 0 to ^ = 1), the motion of interphases can be mapped to a partial differential equation for the plastic field of the entire domain. In particular, the boundary conditions at interphases are automatically satisfied.
As an example in the application of formalism of phase field modeling, we consider the conventional analysis of damage evolution, where the "order parameter" ^ is identified with the damage parameter. The presentation of elas-
(1)
tic energy is typically presented as:
/el = M)ej +Mf) 4/2,
where is the shear modulus and A(^) are the Lame coefficients, ejj is the elastic strain, and ej = etj -ellSjj3 is the deviator of elastic strain, /s(^) = 3y^ (V^)2/2 with interface width ^ and the surface energy y. A double wall potential is used for the "damage energy part":
/dw = 6y^2 (1 -^)VI.
An expression for total potential—nonequilibrium free energy U = J dV (/el + / + /dw), leads to dissipative phase field dynamics with a kinetic coefficient D having dimension [D] = m2s-1:
di=D SU
dt ~ 3y^ S^ '
This well-known formalism of phase field modeling reduces to classical results [16] in the case of near equilibrium thermodynamic systems but there are serious limitations in the case of "mesoscopic systems", that are out of equilibrium due to the impact of mesoscopic defects. This limitation is a consequence of the "gauge" symmetry of damage parameters related to defects, the corresponding part of defect induced energy (free energy release), "thermalization" conditions for a system with mesoscopic defects, and definition of the "phase field dynamics" due to spinodal decomposition in appropriate metastability areas of a nonequilibrium potential.
3.2. Microscopic and macroscopic variables for mesosodefect ensemble
Structural parameters associated with mesodefects have been introduced as the derivative of the dislocation density tensor [1, 17]. These defects are described by symmetric tensors of the form
sik = SViVk (2) in the case of mesodefects with a normal opening mode (similar to microcracks) and
Sik = V2 s (Vi 4 + h Vk) (3)
for slip modes (microshear). Here v is a unit vector normal to the base of a microcrack or slip plane of a microscopic shear, l is a unit vector in the direction of shear, s is the volume of a microcrack or the shear intensity for a microscopic shear.
Changes in the diffeomorphic structure of the displacement field due to these defects has important consequences with respect to changes in the symmetry of the system "solid with defects". This symmetry aspect can be used to good advantage in the modeling of arbitrary defects both in crystalline and amorphous materials without any assumption concerning the dislocation nature of the defects that derived originally from the properties of crystalline materials. Averaging the microscopic tensor sik yields the macroscopic tensor of the microcrack or microshear density:
Pik = n<sik X (4)
Corresponding to the deformation that is induced by the defects, n is the defect concentration.
4. Statistical model of continuum with mesodefects
4.1. Effective field method
The effective field method is frequently used to refer to the case wherein an auxiliary field (real or virtual) is introduced into a theoretical model in order to construct a simplified way of taking into account the effect of complicated factors, such as interparticle interactions, which are either too difficult to evaluate rigorously, or are insufficiently well-understood in detail. The effective field method was proposed by Leontovich [18, 19] in the fields of statistical physics and thermodynamics. In accordance with this method, given an arbitrary nonequilibrium state of any thermally uniform system that is characterized by definite values of internal parameters, the transition into an equilibrium state with the same values of those internal parameters may be accomplished by introducing an additional force field. By definition, the entropy of this nonequilibrium state is equal to the entropy of the equilibrium state (due to the presence of the additional force field) characterized by the same values of the considered material parameters.
The microscopic kinetics for the parameter sik is determined by the Langevin equation sik = Kik (slm) - Fik, where Kik = BE/dsik, E is the energy of the defect, and Fik is a random part of the force field that satisfies the relations <Fik(t))= 0 and <Fik(0Fik(t))= QS(t -1'). The parameter Q characterizes the mean value of the energy relief of the initial material structure.
A statistical model of the defect ensemble was developed in terms of the solution to the Fokker-Plank equation in [1]
9 ~ 3 KkW + 1Q^W. (5)
-W = -
dt dsik
2 dsikdsik
According to the statistical self-similarity hypothesis, the defect distribution function can be represented in the form W = Z_1 exp(-E/Q), where Z is the normalization constant. It follows from (5) that the statistical properties of the defect ensemble can be described after a determination of the defect energy E and the "thermalization" factor Q.
In the term of microscopic and macroscopic variables, and according to the presentation of these mesodefects in terms of dislocation substructure, the energy of these defects can be written in the following form [11]: E = E0 -- Hiksik + asik, where the quadratic term represents the "self-energy" of defects and the term Hiksik describes the interaction of these defects with the external stress oik and with the ensemble of the defects in the effective field
approximation: Hik = Oik + XPik = Oik + Xn(Sik). Here a and X are material constants. An averaging procedure gives the self-consistency equation for the determination of the defect density tensor
Pik = nIsikW(s, v> l)dsik-For the dimensionless variables
pik = Vn 4alQPik , 4 = 4alQ sik, &ik = Gk/yfQv ,
the self-consistency equation takes the form:
P ik = J sikZ -
//
exp
W
1 ,
a ik + g P ik
s ik sik
d ss
ik'
(6)
which includes the dimensionless material parameter 8 = = a/(An). Employing dimensional analysis we estimate that a ~ G/V0, X ~ G, n ~ R-3. Here G is the elastic modulus, V0 ~ r03 is the mean volume of the defect nuclei, and R is the distance between defects. Finally we obtain for 8 the value 8 ~ (Rr0)3 that is in correspondence with the hypothesis concerning statistical self-similarity of the defect distribution on different structural levels.
The solution to the self-consistency equation (6) has been found for the case of uniaxial tension and simple shear (Fig. 4). The existence of characteristic nonlinear behavior of the defect ensemble in the corresponding ranges of 8 (8 > 8* - 1.3,8c < 8 < 8* 8 < 8c - 1) was established, where 8c and 8* are bifurcation points. It has been shown [1, 17] that the above ranges of 8 are characteristic for quasi-brittle (8< 8c -1), ductile (8c < 8 < 8*), and nanocrystal-line (8>8* - 1.3) responses of materials. It is evident from this solution that the behavior of the defect ensemble in the corresponding ranges of 8 is qualitatively different due to differing nonlinearity of the free energy of a solid with defects. The characteristic form of free energy in the range 8<8c -1 is shown in Fig. 5. Transformation from stable material response for fine grain materials to metastable behavior for ductile materials with intermediate grain size occurs when the value of 8 = 8* -1.3. In this case interaction between the orientation modes of defects has a more pronounced character. This means that the metastability exhibits orientation ordering in the defect ensemble. The kinetics of orientation transition and the growth of defect density (and, as a consequence, a decrease in 8) results in specific spinodal decomposition of metastability, the generation of collective modes of defects, and a mechanism of momentum transfer that is conventionally known as plastic flow.
P2
Pc Pi
• 8C < 8 < 8* ^
» / 8<5C\ \
—\ 1 % * 1 \ x / 5 >8» J/
1 t l I l ] / —*
The dependence of our statistical integral on the unit structural parameter 8 and the corresponding non-linearity types (Fig. 4) reflect, most probably, on universal properties of media undergoing a local change of symmetry due to the generation of defects. Nonlinearity and related group properties of the kinetic equations for pik define the types of collective modes that can impact the dynamics of an entire system.
Taking into account the physical meaning of the structural-scaling parameter, a natural generalization for the distribution function can be introduced to assume independent statistics for 8 as the variation of structural scales R and
r0 in the initial state of the system:
N( E) = J d8f (8)- 1 ' 1 '
Z (8)
exp
i E
(7)
where f(8) is the distribution function for the initial "sensitivity" of the system in terms of 8, and E = 8(oiksik - s2k) + +piksik. Averaging in this case involves an integration over all order parameters of the mesoscopic system
pik = I f(8) I SikZ_lexP(( °ik + V8 pik )s ik -Sl )dsik d8.
Generalization of the Boltzmann-Gibbs statistics (nonextensive statistics, superstatistics [20, 21]) is based on a similar assumption. It follows from (7) that the structural-scaling parameter 8 plays the role of the thermalization factor for solids with mesoscopic defects.
4.2. "Effective temperature " of nonequilibrium state of solid with defects
An intensively studied problem is how to determine the "thermalization" condition of plastically deformed solids taking into account the multiscale nature of defect induced mechanisms of structural relaxation providing the plastic flow. Solids undergoing plastic flow represent a characteristic example of nonequilibrium systems with "slow dynamics" [22]. These systems are far from equilibrium and therefore conventional thermodynamic concepts, including a conventional definition of temperature, cannot be applied. This represents a fundamental problem for plasticity considering the Orowan law which explains the motion of
Fig. 4. Characteristic responses of materials on defect growth [1]
Fig. 5. Free energy dependence on stress and defect density for 8<8C =1 [1]
Fig. 6. Schematic stress-strain diagram of aluminum monocrystal
mesodefects in terms of a thermally activated pass through a multiwall potential relief induced by obstacles with varying levels of energy. With regard to slow evolution of the thermodynamic state, a possible solution to this problem is associated with a generalization of the fluctuation-dissipation theorem for nonequilibrium systems.
It is shown in [23] that the determination of nonequilibrium free energy in the "effective field" approximation enables one to link the mean square fluctuations of internal variables with susceptibility of the nonequilibrium system on the variation of these variables and the "effective temperature" in the nonequilibrium state: T
(S - s) =
d2F ds2
Metals undergoing quasi-static plastic deformation are a prime example of nonequilibrium systems with slow dynamics. Stress-strain diagrams of ductile materials exhibit weak rate sensitivity over a wide range of strain rates and present relatively flat curves in the plastic region. These features of the nonequilibrium states governed by the dynamics of dislocation substructures reflect the nature of "slow" relaxation related to the metastability of the thermodynamic potential.
The determination of an "effective" temperature in terms of thermodynamic characteristics of the current state of a plastically deforming material was performed for a mono-crystaline aluminum specimen under uniaxial plastic strain (Fig. 6). The spatial distribution of plastic shear fluctua-
tions was studied by measuring the surface roughness with a high-resolution interferometer New View 5000 (the vertical resolution is 0.1 nm and the horizontal up to 0.5 mm). Figure 7 represents a typical profile of the specimen surface relief.
The following representation of structural sensitivity was employed X = (d2 F/ de2 ) 1 with generalizations of the fluctuation-dissipation theorem [22]: X = 1/ T AC. Here the correlation function increment AC represents the difference of correlation functions for plastic strain fluctuations in two points that are close to one another on the stress-strain curve, Fig. 6 [23].
Plastic strain fluctuations were estimated by the roughness height of the free surface (Fig. 7) of the deformed specimen according to the following formula: Ae = Ah(l)/1, where l is the chosen scale of structural resolution of the interferometer New View 5000.
The results of our calculations are presented in Fig. 8 where the T(l) dependence and can be used to distinguish the range of scales of dislocation substructures with the dynamics that is governed by temperatures and thermal activation kinetics of plastic deformation (temperature that is close to the temperature of experiment ~15°C) and the "effective temperatures" that are independent on the scales l up to the integral scale of the plastically deformed area.
Another important result is the revelation of the structural scale of transition (ls ~104-103 cm) from dislocation carrier dynamics governed by ordinary temperature versus the effective temperature of the "slow dynamics" of dislocation substructures responsible for the athermal mechanisms of plastic flow.
5. Collective properties of ensembles of defects. Structural-scaling transitions
5.1. Phenomenology of solid with defects. Free energy
Adopting the statistical approach one may propose the existence of a mesoscopic nonequilibrium potential, which describes the evolution of mesodefects in a scenario that is related to the nonlinearity types. The curves presented in Fig. 4 correspond to the solution of the equation 9F/ dp = 0, where F is a mesoscopic potential (nonequilibrium free
113144.5 K\xm nm
0 100 200 300 400 x, \im
Fig. 7. 3D surface roughness (a) and typical 1D roughness profile (b)
Fig. 8. "Effective temperature" for characteristic strains: 4.0(1), 4.4 (2), 5.1 (3), 5.8 (4), 6.3% (5)
energy). In the case of simple shear (p = pxz, a = axz), the "minimal expansion" for F takes the form of a 6th power polynomial and its form is similar to the Ginzburg-Landau expansion [1, 19]
F = 1/2 A(8,8*)p2 -1/4Bp4 + 16C(8,8c)p6 -- Dap + x(Vlp)2. (8)
The gradient term in (8) describes the nonlocality effect under mesodefect interaction; A, B, C, D and % are parameters that characterize nonlinearity properties of the solid with mesodefects.
Kinetics of the order parameters pik and 8 follows from the following evolution inequality AF/At = dF/dp p + + dFId88 < 0 as given by the motion equations (the Ginz-burg-Landau approximation)
^ = -rp(A(8,5.)p-Bp3 + C(5,5C)p5 -at
- Da-V i (xV lp )),
at 1
1 dA
(9) (10)
__2-1 dC 6
v 2 98 P 6 98 Py where Tp h r8 are kinetic coefficients. It follows from the solution of Eq. (6) that the transitions that occur at the bifurcation points 8c and 8* (Fig. 4) result in sharp changes
to the distribution function and the formation of collective modes of defects.
5.2. Collective properties of defect ensembles
The type of transitions that are associated with the critical points fall under the bifurcation type category— with the group properties of Eqs. (9) and (10) extant for different ranges of the structural-scaling parameter 8 (8>8*, 8c <8<8*, 8< 8c). For 8>8* the eigenforms express spatially-periodic modes S1 (Fig. 9) on the scale of A with weak anisotropy (orientation) determined mainly by the force field a. For 8^8* the solution of Eq. (9) undergoes qualitative changes due to the divergence of the inner scale A: A - - ln(8 - 8*). The periodic solution transforms into the "breathers" for 8^8* (in the domain 8 > 8*) and the autosolitary waves p(Z) = p(x - Vt) in the orientation metastability region (S2, Fig. 9, a). The wave amplitude p, wave front velocity V, and the width of the wave front Ls are determined by the parameters of nonequi-librium transition
p = 1/2(pa - pm)[1 -th(ZLs-1)], (11)
Ls = 4/(pa - pm) (2%/A)1/2.
The velocity of wave fronts is expressed as: V =
= XA(pa -pm Vrp' where (pa - pm) is the Jump in the value ofp in the metastability region. It follows from the statistical model that the aforementioned transition separates the periodic mesodefect distribution modes and the autosolitary modes. These autosolitary waves are a sign of strain localization zones within the front, where the orientation transition is realized. A transition through the bifurcation point 8c is accompanied by the appearance of spatial-temporal structures of a qualitatively new type (Fig. 9, a, dashed lines) characterized by explosive accumulation of defects as t ^ tf in the spectrum of spatial scales (denoted as "blow-up" dissipative structures S3, Fig. 9, b) [24]. It can be shown that for 8< 8C and p > pC , solution of Eq. (9) results in a developed stage of the kinetics of p that in the limit of characteristic times t ^ tf can be described by a self-similar solution of the following form:
Pi JX > a s2 7y Oi^X'.
r\
\ st
an
laa
WV
) m
Fig. 9. Collective modes of defects
p(x, t) = |(t)f (Z), Z=*M0, (12)
|(t)~(t - tf)<p(t)~(t - tf)d, where m and d are parameters related to the type of nonli-nearity resultant from Eq. (6) for 8<SC and p > pC, and tf is the characteristic temporal scale of self-similar solution (12).
The specific form of the function f (Z) can be determined by solving the corresponding eigenvalue problem. Variable Lf, the so-called fundamental length [25], has the character of a spatial periodicity in the solution (12). This self-similar solution describes the kinetics of the collective mode of defects in the "blow-up" regime:p(x, t) — — <» for t — tf on the spectrum of spatial scales LH = kLf, k = 1, 2, ..., K. In this case the complex "blow-up" structures appear on the scales LH = kLf (Fig. 9, b) when the distance between simple structures, denoted LC is close to Lf.
6. Constitutive model of solids with defects
6.1. Defect induced mechanisms of momentum transfer
The physical mechanism for the transfer of momentum taking place during plastic flow revolves around multiscale motion of deformation carriers (mesodefects at different structural levels). This statement expresses the principal difference between the irreversible deformation that is caused by dislocation structure rearrangement and the more conventionally understood mechanism of momentum diffusion that occurs with the flow of viscous liquids.
Considerable effort has been expended during last few decades in an attempt to gain a better understanding of the mechanisms of plastic flow and the development of phe-nomenological plasticity that is capable of covering a wide range of stress and strain rate intensities. Despite this, attempts to incorporate structural aspects into the formulation of the plasticity constitutive equations have not succeeded in explaining most of the principal questions related to the specific nature of the deformation carriers that are responsible for plastic flow. Still unresolved are the regularities of strain localization that occur with shear banding, the linkage of structure evolution with deformation hardening, and an explanation of the structure of the plastic wave front in shocked materials.
Using a methodology employing a statistical description of mesodefect evolution and the incorporation of nonequilibrium free energy in materials with mesodefects undergoing structural-scaling transitions, field equations were obtained [1, 4, 26] that link the kinetics of structural-scaling transitions with the relaxation properties of these material systems. The kinematics of a specific volume of medium containing mesodefects was represented in the following form (for the shear conditions) exz = eV + pxz, where exz = dux/ dz, e^z is the kinematic strain rate characterizing the relative motions of subdomains that are
formed at different scales due to structure rearrangement (formation of dislocation substructures); and pxz is the strain rate related to the mesodefects generated within the specified volume. Dissipative function D of this medium (the part related to drop in temperature) reads:
n a v AF
D = ®xzexz~--Px
APx
-dF 5,0, as ,
where the symbol a(-)/apxz represents the variational derivative. The condition that the dissipative function be positively definite leads to a system of equations for the tensor variables (using the Onsager principle) and structural scaling parameter:
axz=rAz + r2 Pxz> (13)
aF
APx
5 = -rs—, 5 as,
• = -r2 exz+r3pxz'
F
(14)
(15)
where r1; r 2, r3 and rS are kinetic coefficients, which depend, in general, on pxz (the invariants ofpik). Equations (13)—(15) provide two mechanisms of momentum transfer: the viscous flow related to the term r1evxz and the mechanism of "structural relaxation" related to the me-sodefect kinetics as given by Eq. (13). The term r2evz reflects the crosscorrelation effects of the interaction of two mechanisms. Linkage of the 8-kinetics in the range SC < S < 8», with momentum transfer provide an alternate scenario for the pxz kinetics that depends on the stress rate in the metastability domain and can be close to the Arrhenius sliding kinetics for low strain rate (e< 105 s-1) deformation and by the autosolitary wave collective modes localized on the corresponding spatial scales at higher strain rates (e> 105 s-1).
6.2. Resonance excitation of collective modes of defects
Linking structural scaling transitions in mesodefect ensembles with the excitation of collective modes of meso-defects provides an explanation of the viscosity anomaly that was observed first in the Sakharov experiment [27] focused on the mechanisms of viscosity in shocked condensed matter (in liquid and solid state). Experimentally estimated characteristic relaxation times in these experiments revealed a universal asymptotic of viscosity v —^ 104 Pz for strain rates e —105 s-1 at the shock wave front with initially induced periodic disturbances. Using the definition of viscous strain rate evxz = exz - pxz that follows from (9), Eq. (10) can be reformulated in the following way:
a xz =
r -
(2 )
(16)
V x J J
In cases wherein the kinematics of the medium is subject to an autosolitary wave that is excited in resonance on the shock wave front, exz = pxz and (16) has the form
Gxz = r2exz. This means that the influence of dynamic viscosity r disappears. Subjection of the effective viscosity to the self-similar solution (8) on the shock wave front was also manifested in the experiment as the fourth order universality of strain rate 8 on the stress amplitude Gamp: 8 - AGamp. Numerical studies relating to the structure of shock wave fronts [28-30] based on Eqs. (13)—( 15) have also supported linkage of the aforementioned fourth power law with the kinetics of structural-scaling transitions in the S-range S* < S < SC as will be discussed in Sect. 7.
Structural-scaling transitions and the formation of a cascade of autosolitary waves in the range S* < S < SC can provide new structural scales, the current rescaling of S, and, as a consequence, the current sensitivity of material to the generation of defects and structural relaxation.
Fig. 10. Structural-scaling transitions in mesodefect ensemble
7. Kinetics of structural-scaling transitions and mechanical responses of shocked materials
Nonlinear material response with increasing strain rates are manifested in solids through changes to the yield stress and structural manifestations in the wave front that are caused by the mechanisms of hardening and structural relaxation. Attempts to link the mechanisms of defect-induced structural relaxation and the nature of the shock wave front were undertaken in [28-35]. Taking a statistical approach enabled us to establish self-similar features of multiscale defect evolution in the form of collective modes providing for sharp changes in the mechanisms of structural relaxation leading to "splitting" of the shock front into elastic precursor and plastic waves due to a strong rearrangement of mesodefect structure in the course of transition in the metastability domain (Fig. 10).
7.1. Structure of shock wave fronts 7.1.1. The Hugoniot elastic limit
A two-wave shock structure is generally observed in the moderate range of pressure. The first wave, the so-called elastic precursor, propagates with the sound velocity, and the second (plastic) wave travels somewhat slower. The two-wave structure is generally unstable, yet each wave can be considered stable after some distance from the loading surface, at least in the sense of shape stability for constant propagation velocity of corresponding parts.
Still today a complete understanding of the physics behind shock wave structure, including the Hugoniot phenomenon, the range of the Hugoniot variation with pressure (or strain rate) increase, and the mechanism of elastic precursor decay, remain open questions. Transition in the metastability region (Fig. 10) is characterized by a finite "jump" of the order parameter Ap (defect-induced deformation) and a finite width of the metastability domain (the range of stress of the transition Ag) and is accompanied by a sharp increase in compliance (a sharp drop in the effec-
tive modulus). Assuming a jump of Apc_c ("c-c" is the line of "equilibrium transition") during equilibrium transition in the metastability region leads to the following change in effective compliance:
At___L,
Geff Geff Gc_c
where G'Gff, Gff are effective moduli corresponding to the values ofp before and after transition, and Gc_c is the stress of the "equilibrium" transition.
Sharp decay (on a time scale corresponding to the shock wave rise) of the effective modulus leads to splitting of the wave front. This evolution in the wave front is a consequence of the genesis of defects with pronounced orientation modes (orientation transition), providing transition to the plastic wave front. This similarity in the structure evolution under defect induced structural transition and phase transition was noted by Duvall [36]. The existence of stress range in the metastability domain for this transition Ag establishes the amplitudes of the elastic limit (the Hugoniot elastic limit ghel).
The value of the characteristic time of transition in the metastability domain is given by the well-known formula tm ~ t0 exp(AF/(kT)), where AF is the free energy barrier, that is the difference in free energy values corresponding to upper and low "thermodynamic" branches, and t0 is a constant. It follows from this formula the time of transition is determined by the barrier value AF with a magnitude that depends upon the level of the stress of transition in the metastability domain. The inverse value of the strain rate Tload ~ 8lo1ad is the characteristic time of loading and the comparison of this time with the time of transition Tm determines the value of the barrier AF and the corresponding stress of the transition g hel. Experiments show that the maximum values of the Hugoniot attain for strain rates 8load ~108-109 s-1, which corresponds to the limit "depth" of the metastability domain on the stress scale and a limit of zero for the free energy barrier AF.
7.1.2. Relaxation of elastic precursor
The physics of elastic precursor decay and the stability of the plastic wave front have been the subject of discussion under study of dynamic strength and the evolution of the propagating wave front ever since the papers by Duvall [36], Gilman [37] and Barker [38] that are devoted to the study of viscoplasticity under shock compression. It is our belief that understanding these linked phenomena and their relation to shock wave splitting will require that one take into consideration the structural mechanisms that provide the threshold character of stress relaxation at the onset of the elastic precursor and the "soft" relaxation of elastic precursor propagation and the role of structure evolution, which provides steady-state stability of the propagating plastic front.
The mechanism of early threshold relaxation has been considered as a transition in microshear ensembles that leads to jump-like deformation induced by this structural transition. Later, the mechanism of structural relaxation is linked more closely with structural transitions involving more "rough" dislocation substructures that can be described in term of the kinetics of structural parameter 8 using (15). The kinetics of 8 results in a decrease of the structural scaling parameter with increasing p, which leads to a continuous orientation transition in dislocation substructures with their consequent "roughening". For instance, the points along the trajectory DF (Fig. 10) belong to consequent metastable dependencies for the current 8 having their own values of maximum ^HEL, representing the current relaxation limits for the Hugoniot. Thus, the elastic precursor decay could be linked with scaling transitions and, as a consequence, with a shifting of the metastability domain and threshold maximum value into a region of lower stress.
7.1.3. Stability of plastic wave front
Experimental studies pertaining to the effects of relaxation on the plastic wave front were carried out in [27] in the course of an indirect measurement of viscosity. Employing the Doppler interferometry technique, the stability of a steady-state plastic wave front has been established [5, 9]. A unique feature of wave profiles is the universality of the steady-state plastic front: the steady-state profile propagates without a change of shape as a consequence of stable balance between the competitive processes of the nonlinear dependence between stress and strain and the dissipa-tive properties of the material.
To follow the results in Sect. 5, the similarity of plastic wave front for different amplitudes of shock pulse appears to be a consequence of the subjugation of the mechanisms of structural relaxation to the dynamics of autosolitary collective modes in the ensemble of microshears. An effective change of scaling characteristics in term of decreasing 8 under the influence of growth in microshear density (DHF-trajectory, Fig. 10) and provides for a stable autosolitary
response of increasing stress in the shock wave front. The rate of transition p between thermodynamic branches in metastability regions reaches a maximum at the boundary of the metastability region (point b, Fig. 10). This explains the universality of strain rate dependence on stress amplitude p ~ AixJmp that was established for a large class of materials over the range of strain rate e> 105 s1. This relation follows from the self-similar solution (8) when the "driving force" of the transition performs as the difference in the free energy values along the trajectory dG and the levels of energy of the metastability thresholds (the analog of point b) for the curves corresponding to the current values of 8.
This result reflects the importance of collective effects (orientation transitions in defect ensembles), which provide this universality of material responses in the steady-state shock wave and the specific viscosity law for solids. At the same time this four power law universality suggests the following assumption; that the complex processes of plastic flow caused by structure evolution may become simpler under shock wave loading. The study of these general mechanisms is important for the understanding of the role of collective effects in defect ensembles responsible for the plastic deformation of solids.
Thus, the high level of stability that is exhibited by the propagating plastic front can be linked with the "slow" dynamics of orientation transitions that are controlled by the 8-kinetics and having the autosolitary dynamic nature given by the solution (11) of the Eqs. (9) and (10). Universality of the strain rate dependence on stress amplitude ep = Aa4Jmp is therefore a consequence of the special type of the free energy nonlinearity of the medium with defects that describe structural-scaling transitions in mesodefect ensembles and the existence of the self-similar solution— the autosolitary waves, which subjugate the ability of the material to relax and provide for the universality (self-similarity) of the plastic wave front.
7.2. Universality ofplastic wave front and shock-induced structural scaling in copper
With the aim of developing an experimental methodology for shock wave loading, diagnostic accumulation, validation of the model and determination of material parameters, shock loading of plate copper specimens have been achieved using a high explosive plane wave generator. The dynamic response of the specimen was monitored using a VISAR velocity interferometer. The free surface velocity profile is presented in Fig. 11. The portions corresponding to the elastic precursor, plastic wave front, rarefaction wave and periodic pulsations of velocity due to a spall plane are recognized in the recorded signal.
The results of a study of target morphology along a cross section in the direction of wave propagation is shown in Fig. 12 as the digital 3D-profiles measured by New View
Fig. 11. The free surface velocity profile for shocked copper
5000 (magnification x1300). The cross section of the target was polished and etched with the goal of revealing defect-induced morphology. The scanning relief shows the presence of ensembles of localized shear bands with an orientation angle close to 30° with respect to the direction of wave propagation. A regular change of width and spacing between bands is observed that could be interpreted as spatial scaling in the shear band arrangement. A study of this spatial scaling could lead to the establishment the linkages between self-similar solutions and symmetry properties of the system under the generation of localized strain modes.
The results of correlation analysis of New View roughness data z(x) are presented in Fig. 13 in the terms of the correlation function
C(r) = ((z(x + r)_z(x))2)X2 -rH
in log-log format (log C(r) ~ Hlogr) for quasi-static (curves 3-5), dynamic (curve 2) and shock wave loading (curve 1). The last dependence (curve 1) is practically linear with a constant value of the scaling exponent (Hurst exponent), reflecting a very pronounced long-range correlation of mesodefects. This result was expected from theoretical prediction and points to linkages between the dynamic self-similarity (the four power law) and spatial scaling, which reflects the subjugation of the ability to relax at the shock wave front to the set of autosolitary modes of defects.
7.3. Modeling of shock wave fronts
7.3.1. Numerical simulation of plane shock-wave propagation
Using dimensionless variables X = g/G and a scaled representation of the free energy ¥ = F/Fm, a system of equations of motion that link the mechanisms of structural relaxation and the kinetics of stress can be stated as follows [28]:
= 36 p0 9t 95' 9t
=-r
dV
' 98 '
dP--r
9t" dp ^2'
(17)
Equations (17) represent the momentum conservation law and kinetic equations for structural-scaling parameter S and mesodefect density tensor p, respectively. Plastic strain rate ep, deviatoric X' and isotropic X0 parts of stress X are specified as:
ep -P/XT " ^ = 2_ep,
dp 9t 3 9t 9t
^-K^, Z-Z0 + X', dT dT ' 0 '
(18)
and closure of the initial boundary problem is achieved through the initial and boundary conditions X|ç=0 =2r(T), x|ç=0- 0, XU- 0,
dp 95
- 0, ^P
5-0 d5
- 0 Pi-0- 0'
5-1
8,-0-60, £P -0- 0 P|,-0-P0
(19)
(20)
-, r
TT
■l1 a
Tp Ta
■XT
pa
Tl Tp
Tl Tpa
a — 2
T p Ta — XT pa
K
pa
rcK
PC{
t-—, 5-X, x:
Ti h
r'T| - C
TpTa XTpa
h
(21)
PC
2
t
8
Fig. 12. Optical photograph of shear bands of defect-induced roughness in the cross section of target (after polishing and etching) in the direction of wave propagation
Fig. 13. Correlation function of defect-induced roughness for shock wave loading (1), dynamic (2) and quasi-static curves (3-5)
1.4 T
Fig. 14. Results of numerical simulations: stress profile (a), free surface velocity (b)
Here v is velocity, 8 = dv/dx is strain rate, ep is plastic strain rate, h is plate thickness, Cl is longitudinal sound speed, p is density, a is stress, K is bulk elastic modulus, 2 0 and are isotropic and deviatoric parts of the stress, respectively. The parameter X = (P0Cl2/Fm)-1 characterizes the scaled elastic modulus caused by defects. Dimen-sionless parameters r*, rp, ra, rpa and rc relate to the kinetics of structural scaling transitions x§, orientation transitions Tp, stress relaxation kinetics Ta, the cross-correlation of relaxation mechanisms between Tpa and processes related to the nonlocality effects Tc.
Material parameters for Armco iron were estimated to be: r* = 0.0035, Tp = 6.8, ra = 3.4, rpa = 3.36, Tc =
= 0.0087, k = 4.5, Cl = 1.0, F = 6.5
Ta= 3.4, r pa = 3.36,
: 2890 m/s, p0 = 7800 kg/m3, x =
1010 Pa.
7.3.2. Universality of structured steady-state plastic front
Equations (14)-(18) have been solved numerically for the previously discussed plate impact test with free boundary conditions on the right and the specified stress amplitude £ amp on the left surface. Usual nominal values for
amp
other variables were specified. Results of numerical simulation for a single shock in the Armco iron plate are presented in Fig. 14 as plots of stress profile and free surface velocity. These results illustrate clear allocation of elastic precursor and pronounced plastic wave front profile. These data correspond well to experimental data obtained in tests involving Armco iron. The self-similarity of shock wave
profiles was confirmed in the course of numerical simulation as the universality of the particle velocity versus time plot in some dimensionless (self-similar) coordinates. The results of simulation for the series of stress amplitudes are shown in Fig. 15.
Dependencies of plastic strain rate versus stress amplitude reflect different scenarios for relaxation dependent upon the ratio of stress rise time and the kinetics of structural-scaling transition in the metastability domain. For example, thermally assisted Arrhenius kinetics (branch I, Fig. 15, a) corresponds to low strain rate (ep < 105 s1) and the subjection of relaxation kinetics to the dynamics of mesodefect collective modes in the form of autosolitary waves for the range of strain rates (105 < ep < 108 s1). The last scenario leads (after some transient branch II) to the self-similar response with the four power law relationship that is in correspondence with the data of Swegle and Grady for 105 < ep < 108 s1 (Fig. 15, a, branch III) and an asymptotic regime of the "overdriven shock" [39] for high strain rates ep < 108 s1 (Fig. 15, b, branch III).
By definition, an overdriven shock wave is one in which the plastic wave has overrun the elastic precursor to produce a front steeper than that attainable by adiabatic elastic compression. As was shown in [40], the mean flow stress has a power-law dependence on the plastic strain rate (instead the Arrhenius law) for the strong shock path near 1012 s1. The power-law representation has also been found in [40] to fit molecular dynamics shock-wave simulations.
500
'amp
"amp
Fig. 15. Strain rate versus stress amplitude at plastic wave front: r8 « 1 (a), 3.5 -10 8 (b)
Fig. 16. Numerical results of plane shock wave propagation for reshock conditions
7.3.3. Self-similarity of shock wave front under reloading (unloading)
Experimental studies have revealed self-similarity of wave fronts in shocked aluminum over a wide range of strain rates and load history, including reloading and unloading [7]. Reloading demonstrated that the material exhibits anomalous elastic-like response, which differs from the initial elastic material response. This quasi-elastic deformation observed during reshock implies that either the material exhibits time-dependent yield behavior for shock stress exceeding 10 GPa, or that the average shear stress state induced by the initial shock loading differs from the critical shear stress. The observation of quasi-elastic response from a preshocked state raises several questions about the dynamic flow process.
In our studies numerical simulations have been used to describe the material response to reshock. Plate impact was modeled with boundary conditions involving a double step on the face side of plate with free surface condition on back surface. Results of simulation are shown in Fig. 16 for different times and reveal the pronounced elastic precursor in the first wave and a second allocation of elastic precursor in the reshock wave. This effect follows the metastability form of nonequilibrium free energy and the linkage of relaxation ability of shocked materials with multiscale relaxation related to structural internal variables: mesodefect density tensor and structural scaling parameter. For the first
shock impulse there is the usual separation of the shock wave front into plastic and elastic fronts according to the kinetics of previously discussed structural variables in the metastability domain. The second "elastic" precursor appears due to a relaxation of the deviatoric part of stress that occurs in the Hugoniot state after the first shock. This mechanism follows from the constitutive equations, where deformation is split due to the mechanisms of defect-induced structural relaxation on the defect-induced strain and the "viscous" plastic flow. After stress relaxation the system could have an elastic-like response (Fig. 16).
Results of numerical simulations for reshock loading with different initial conditions (stress amplitude and target thickness) are presented in Fig. 17 in both variables carrying dimension and in scaled variables, where Vsc is the scale of mass velocity (surface velocity amplitude for first shock) and tsc is characteristic time scale (tsc = h/Cx, h is the thickness, C1 is the sound velocity).
The simulation of the shock wave profiles showed a collapse of the pull-back velocity for different stress amplitudes. The self-similarity of the plastic shock-wave fronts in solids (including reshock regimes) is a consequence of the existence of two independent mechanisms defining structural relaxation. These mechanisms are related to the structural-scaling transition in terms of two independent variables (defect density tensor and structural-scaling parameter). The kinetics of these parameters at the steady-state plastic wave front is realized due to the generation of autosolitary strain modes that provides the self-similar scenario of relaxation on the large range of structural scales. This result supports the assumption concerning the linkage of specific (defect-induced) nonlinearity of material behavior and relaxation (dissipative) properties in shocked materials in some characteristic range of load intensity (or strain rate).
8. Comparative analysis of multiscale model and MTS-PTW model
A plastic constitutive relation depends in general on a set of internal variables that represent the microstructural state. One approach to continuum plasticity modeling dis-
Fig. 17. Free surface velocity versus time in dimension (a) and scaled self-similar variables (b)
cussed above employs internal state variables and is constructed using a sophisticated homogenization procedure. This approach has been taken by Follansbee and Kocks [41] who developed a constitutive model employing the mechanical threshold stress (MTS), flow stress at 0 K, as an internal state variable. This model accounts for thermally activated dislocation motion as a mechanism of plastic flow and is therefore limited to strain rates on the order of 104-105 s-1. Applications of the MTS model have indicated large strain or strain rate history effects. Measurements involving homogeneous stress-strain states are usually limited to strain rates <103 s-1. The availability of mechanical threshold stress data measured in copper at strain rates as high as 104 s-1 enables an extension of this model into the regime where strain rate sensitivity (determined at constant strain) is known to increase dramatically. This suggest the feasibility of a single model based only on thermal activation controlled glide that describes results over a wide range of strain rates (10-4 s-1 <e < 104 s-1).
In the MTS model the mechanical threshold stress is separated into two components: 6 = 6 a + 6,, where the component 6a characterizes the rate independent interactions of dislocations with long-range barriers such as grain boundaries, whereas the component 6, characterizes the rate dependent interactions with short-range obstacles. At finite temperature, thermal activation can lead to a lowering of the second component of the flow stress, while leaving the first unchanged. The relation between the flow stress and the mechanical threshold stress becomes
6 = 6a +6, = 6a + s(e, T)6t, (22)
where the strain rate and temperature dependencies are included in the factor s.
With Eq. (22) as the foundation for a model, there are two separate issues to deal with: first, the form of the factor s in Eq. (22) must be specified, and second, evolution of the mechanical threshold stress must be described. The factor s in Eq. (22) specifies the ratio between the applied stress and the mechanical threshold stress. The value of s is defined by the glide kinetics. This factor is s < 1 for thermally activated controlled glide because the contribution of thermal activation energy reduces the stress required to force a dislocation pass an obstacle. If deformation is instead in the dislocation drag controlled regime, then an applied stress greater than the mechanical threshold stress is required for continued deformation. Results pertaining to copper suggest that that at strain rates e < 104 s-1 the rate controlling mechanism is thermal activation and the contribution of dislocation drag is negligible. In the thermally activated glide regime the interaction kinetics for short range obstacles are described by an Arrhenius expression of the form
where AG is the Gibbs free energy, e0 is a constant, and k is the Boltzmann constant.
The strain hardening rate 0 = d6/d£ is used to characterize the differential variation of the structure parameter with strain. The fundamental relation between the competing processes is written as 9 = 90 - 9X (T, e, 6), where 60 is the hardening due to the dislocation accumulation and 6T is the dynamic recovery rate.
It is emphasize that the particular evolution law chosen to fit the results is merely a convenient mathematical form which happens to describe the data. The major conclusion of this work is that there is strong evidence that the dislocation accumulation rate, or stage II hardening rate, begins to increase dramatically with a strain rates exceeding ~103 s-1. This observation explains the origin of "the increased strain rate sensitivity" of the flow stress at constant strain found in copper and other f.c.c. metals at these strain rates.
One interesting implication of this finding is that at strain rates exceeding ~105 s-1 strain hardening will quickly saturate and the stress-strain behavior will approach perfect plasticity. The reason for this is that the strain rate sensitivity of the athermal hardening rate exceeds the strain rate sensitivity of the saturation stress at high strain rates.
At strain rates up to at least 104 s-1 the strain rate controlling mechanism is the thermally activated interaction of dislocations with obstacles, usually other dislocations [39]. Applied stress fields results in dislocation motion and dislocation intersection. For sufficiently low applied stresses the intersecting dislocations do not immediately pass through one another because of short distance repulsion (energy barrier), but thermal fluctuations can eventually drive the more mobile dislocation through its partner. The thermally assisted transition rate is given by transition state theory. The dislocation transition rate, and therefore the plastic strain rate, is of the Arrhenius form [40]:
e = e oexp(-ao(6)/(^T)), (23)
where the activation energy A®(6), a decreasing function of the applied stress, is the difference in the biased crystal potential between the saddle point and the initial minimum, and e o is proportional to the dislocation vibration frequency in the direction of the saddle point. Most models of plastic deformation are based on a linear representation of the activation energy a®(6) = a®(0) - va6, where va is activation volume. When the stress increases, the Arrhenius form for the strain rate becomes less accurate. The Arrhenius equation shows that the stress is a function of the variable kT ln(e0 /e). This suggests an analogous combination kT ln(y4/e) as adopted in the model of Preston, Tonks and Wallace (the PTW model) which is discussed below (here k and Y are dimensioness material constants).
The work hardening saturation stress and yield stress in the thermal activation regime are given by [39]:
= s0 - (S0 - s^)erf[kT ln(y4/e)], ,= J0 - (J - J-)erf [kTln(Y4/e)].
(24)
Each of these equations can be written in the Arrhenius form (23) with a double wall potential expressed in terms of "thresholds" AO(a)« erf-1[(c0 -a)/(c0 - c^)], where c = s or y. The material constants s0 and s^ are the values that as takes at zero temperature and very high temperature, respectively; j0 and y^ have analogous interpretations. Applications of mentioned for the strain rates up to 1012 s-1 have been discussed by Preston et al. [39] in the framework of the PTW-model.
Statistically based models of solids with mesodefects lead to an interpretation of the double wall potential AO(a) related to nonlinearity of the metastability renormalized by the kinetics of structural scaling parameter 8. The parameters of the PTW-model s0 and s^ (y0 and y^) are similar to aJ and aHEL, respectively (Fig. 10). The saturation stresses discussed above are critical points that characterizes the current state of the material in terms of the structural scaling parameter 8 since this parameter defines types of ¿"-curves (Fig. 10), and therefore different values of aJ and aHEL in the course of structural-scaling transitions. Equations (24) can be interpreted as material evolution by the curve DHF (Fig. 10), controlled by evolution of collective modes of defect ensembles according to the kinetics in term of the structural-scaling parameter 8. As discussed in [3], drag controlled dislocation glide is described by the kinetics of structural scaling parameter 8. The current value of 8, as given in Eq. (15) can be expressed as
d8 - dF (p, 8)
dt t8 d8 and provides the current sensitivity of the material to dislocation glide, which is described by the kinetic equation for p in the "saturated" regime (the path DF, Fig. 10):
T8=r8 A, F = F/A,
dp dt
(
dF (p, a, 8) tp dP
Tp =rp A.
(25)
This type of dislocation glide can be realized only for strain rates such that (e)-1 = Tload > T8. In the "overdriven shock regime" Tload << t8 and, consequently, 8 = 0, and the "thermodynamic" path is along branches (similar to Dd) with current sensitivity h = (da/dp)-1. In this case the branches are steeper than Ob and the wave front overruns the elastic precursor.
This analysis allows us to conclude that an alternate scenario of plastic wave evolution is a consequence of the ratio of characteristic times T.,
Tload, Tp
T8 and t,r'
~ td exp(af/(kT)), where ttr is the transition time in the metastability range. Kinetic equation (25) produces different values of ttr for different e ~ (Tload)-1 —different "penetration depth" as the scenario of the spinodal decomposi-
tion in the metastability area. For high strain rate related to the "overdriven shock regime" tload << ttr in the point b (path bd) and any thermally activated glide cannot be realized.
9. Summary
The responses of metals subjected to high stress states and very high strain rates (>105 s-1) is an area of active research since a fundamental understanding of the dynamic responses are critical to a number of technical applications. The kinetics of viscoplastic flow and damage-failure transition are controlled by different multiscale mechanisms in corresponding ranges of strain rates. The work presented here is intended to address the viscoplastic deformation of metals in shock waves and construct wide-range constitutive models in the range of strain rates 103-109 s-1.
A physically based model is desirable because the macroscopic properties and responses of metals are intimately related to the underlying microstructure. The aforementioned models are based on glide kinetics that describes dislocations overcoming obstacles with the assistance of thermal fluctuations (i.e., thermally-activated dislocation depinning). This treatment is valid for applied stresses below the mechanical threshold, which translates to rates of deformation below 105 s-1. Above the mechanical threshold, thermally-activated dislocation glide is no longer the mechanism that controls viscoplastic flow. Constitutive models of metals have been constructed [42] specifically for the shock loading regime and it was assumed that all microstructural processes have saturated above a critical strain rate (~105 s-1) and the effective plastic strain was used as a state variable that is somewhat unsatisfactory [40]. The PTW model [39] addresses the rate-dependent flow stress of metals over a wide range of strain rates (1031012 s-1) by merging a model of thermally-activated dislocation glide at lower deformation rates with a power-law relation that describes the flow response in very strong (overdriven) shock waves and specific thermodynamics of the material state [43]. The power-law form for overdriven shocks (strain rates in the range of 107-1010 s-1) is suggested by the thermodynamic calculations in [3] and the assumption about thermally-activated dislocation glide that could be extrapolated into this regime.
Comparing the MTS-PTW models, to our mesoscopic model, which enables us to link the mechanism of structural relaxation, multiscale defect kinetics under the spinodal decomposition of metastability of a nonequilibrium potential with generation of collective modes of defects. It allows one to propose an explanation in terms of a thermo-dynamic metastability potential that accounts in a fundamental way for phenomena such as yield stress hardening under thermo-activation flow, "the singularity" of transition to the self-similar ("structured steady-state") and "over-
driven" shock wave fronts; phenomena that are treated in a more phenomenological way in the MTS and PTW models.
It has been shown, that the transition from thermo-acti-vation flow to the structured steady-state plastic front and "overdriven" shock wave fronts is a consequence of the qualitative change of the thermodynamics of the system "solid with defects" due to the generation of collective modes of defects that play the role of new "thermodynamic" coordinates and provide new "thermalization" conditions for nonequilibrium systems. The power universality of the structured steady-state plastic fronts in the corresponding range of strain rates (>105 s-1) is the result of subjugation of the mechanisms of structural relaxation to the dynamics of collective (autosolitary) modes of defects, that have the nature of a self-similar solution for the variable related to defect density.
Our statistically based approach and mesoscopic model enable us to formulate a statistically based thermodynamic potential and constitutive equations that are universally applicable over a wide range of strain rates. Moreover, this new model requires far fewer parameters in comparison with the MTS and PTW models (7 instead of 15). Experimental validation of wide-range constitutive model has been achieved using the data of experiments on dynamic and shock wave loading (in the recovery conditions) and the following study of scaling regularities of defect induced structure evolution (using New View and atomic force microscopy data) in corresponding ranges of strain rates.
Research was supported by the Program of Fundamental Research of UrB RAS (project 15-10-1-18).
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Поступила в редакцию 06.02.2017 г.
Сведения об авторах
Наймарк Олег Борисович, д.ф.-м.н., зав. лаб. ИМСС УрО РАН, [email protected] Баяндин Юрий Витальевич, к.ф.-м.н., нс ИМСС УрО РАН, [email protected] Zocher Marvin A., Doct. Sci., Los Alamos National Laboratory, USA, [email protected]
