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Phase space of solids under deformation
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Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
Phase space of solids under deformation Yu.V. Grinyaev, S.G. Psakhie and N.V. Chertova* Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia This paper analyzes sequences of various strain-induced defects. The defect is considered as an internal stress source, and hence the deformation is described by a phase curve in the 2D stress-strain space. Notice that the defect as a local discontinuity has eigenenergy and this, if taken into account, increases the phase space dimension because of the parameters to be introduced to characterize defect structures formed under deformation. Analysis of the potential energy of a deformed system makes it possible to elucidate the origin of various defects and to introduce the notion of strain levels in the phase space. Thus, we can distinguish strain levels for which the nature and the number of parameters of the system remain unchanged under deformation and associate the transition of the deformed system from one strain level to another with qualitative and quantitative changes. In this context, the deformation from elasticity to fracture is described by the phase space curve and the phase space dimension is determined, in addition to stresses and strains, by the parameters of defect structures formed under deformation. A standard stress-strain curve is the projection of the phase space curve on the stress-strain plane. Keywords: phase space, strain levels, defects, gauge theory
1. Introduction Research on the formation of various structural defects, from point to three-dimensional, is important for understanding the mechanisms by which plastic deformation of materials originates and evolves. The system under study is a macroscopic dynamic system with many elements whose stochastic behavior transforms into deterministic behavior of the system as a whole. Let us consider the evolution of a mechanically loaded solid. The deformation of the solid is treated as a macrophenomenon in a complex system containing many structural elements that interact in their relative displacements (translations and rotations). In other words, the deformed solid is a nonequilibrium system and its deformation is described in the context of local thermodynamics, i.e., each structural element is in equilibrium, while the system as a whole is not. For simplicity, the deformed solid is a homogeneous isotropic single-phase medium. Mechanics of solids treats of mechanical processes in which the spacing between material points of the medium is changed. The change in the spacing and its attendant forces are determined through mechanical state parameters, namely * Corresponding author Dr Nadezhda V. Chertova, e-mail: [email protected] Copyright © 2008 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2008.11.003
through stress and strain tensors. For elastic deformation, these parameters unambiguously characterize the mechanical state of a deformed solid. For inelastic deformation, there is no one-to-one correspondence between the mechanical parameters, since to one value of stresses may correspond an infinitely many values of strains depending on loading history. This fact is due to local structural changes responsible for plastic deformation and irreversible total strains under unloading. More than 20 years ago, Panin [1, 2] put forward a concept in which defect nucleation is preceded by so-called highly excited states of a crystal. Thus, describing the deformation of solids requires a macroscopic theory that takes into account the physical processes associated with the defect nucleation and dynamics. In classical mechanics of solids, irreversible deformation is described by formulating different nonlinear material relations with the use of only mechanical stress-strain parameters. It is also assumed that nonlinear relations, in a way, will take into account structural changes occurring under plastic deformation. However, the structural changes must inevitably lead to a change in the material relations. Hence, the description of deformation by the mechanical stressstrain parameters alone is incapable of clarifying the physical mechanisms of plasticity and is inadequate beyond
Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
the elasticity limit where structural states need to be taken into account. Classical theory of defects treats of the physical mechanisms responsible for plastic behavior of materials. The basic problem in the theory is to determine the stress and strain fields of a single defect or simple groups of defects, the forced interaction of defects with each other and with fields of applied loads [38]. The defect is considered only as an internal stress source, and not as a local discontinuity (a structural element). For successive description of deformation, let us take into account that a defect (the defect core) has eigenenergy. In addition to the mechanical state parameters, this description should involve structural state parameters characterizing the evolution of the corresponding defect structures. Hence, the phase space dimension for describing the deformation should be determined by both the mechanical stress-strain parameters and the structural state parameters of different defects. In this case, each specific set of the parameters in the phase space can be assigned to a certain strain level. Let us consider what each strain level determines exactly in physical terms and how one strain level differs from another [9]. For identification, each strain level is described in phase space terms. The description involves the type and the number of dynamic variables, and the type and the number of states corresponding to this level. The evolution and behavior of a deformed system in the phase space are governed by the potential energy of the system. The system transforms from state to state, violating the relations between the dynamic variables and synthesizing new ones with the nature of the variables remained the same. In other words, the system evolves at a certain strain level, turning from one steady state to another through successive bifurcations for no change of dynamic variables. This means that the system is kept at a certain strain level due to the self-identity of the variables and main state parameters. If at a certain strain level the bifurcation responsible for a change of variables becomes inevitable, the system passes to another strain level, otherwise it inevitably fails. If this passage occurs, the system is said to evolve crosswise the level.
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not sufficiently advanced, and primarily because of the high time (1014 s) and space (109 m) resolutions required for experimental studies of the defect nucleation at the atomic level. In this situation, an efficient way of studying the generation mechanisms and the dynamics of structural defects is computer simulation of mechanical loading based on molecular dynamics. Using direct molecular-dynamic simulation, the authors of [15] were first to demonstrate the feasibility of protodefect generation by thermofluctuation mechanisms as the realization of precursor (highly excited) states under dynamic load. In the work, the researchers investigated the initiation of plastic deformation in a copper crystallite under high-speed loading. It was shown that as the threshold strain (potential energy) is reached in the simulated crystallite at final temperatures, jump-like production of protodefects (structural changes) begins (Fig. 1, curve 3). The protodefect generation decreases the potential energy of the crystallite (Fig. 1, curve 2). The decrease in potential energy as against the value corresponding to the strain under geometrical changes (Fig. 1, curve 1) is due to the eigenstress field of structural defects that compensates the external stress field. As well as producing the stress field in the deformed system, the protodefects have eigenenergy. Increasing the fraction of protodefects causes an increase in their eigenenergy (about 0.036 eV per defect). Finally, the situation may arise in which the total eigenenergy of protodefects becomes greater than the elastic energy of the system and thus responsible for the behavior of the crystallite. Thus, the generation of the new structure (protodefects characterized by their density) in the crystallite can be considered as the transition of the deformed system in the phase space from the elastic strain level to the next strain level level of protodefects (levels 1 and 2, respectively, in Fig. 2(b). As the protodefect eigenenergy reaches a certain threshold, the further increase in protodefect fraction leads to
2. Strain levels 2.1. Strain level of protodefects Research on the formation of various structural defects is particularly urgent for describing the behavior of materials under dynamic loading, since the deformation in this case gives rise to so-called dynamic defects [1013]. One of the fundamental problems in solid state physics and materials science is indubitably the study of possible thermofluctuation mechanisms of local structural distortions and transformations at the atomic level that can be referred to socalled precursor states [14] or highly excited states introduced by V.E. Panin [1, 2]. Unfortunately, this concept is
Fig. 1. Potential energy per atom, ÏÅ, displaced upward by 0.05 eV for 0 K (1) and at 50 m/s for 300 K (2), and the strain dependence of the protodefect density n (3)
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à
b
Fig. 2. Schematic strain dependence of the potential energies (à): Ï el is the elastic energy of a deformed solid, Ï Å is the protodefect eigenenergy, Ï D is the eigenenergy of translational defects, ÏT is the eigenenergy of rotational defects, and strain levels in the phase space at the corresponding threshold energies (b): elastic level arising at the onset of loading (1), level of protodefects (2), level of translational defects (3), level of rotational defects (4), and level of local discontinuities (5)
fracture of the system if it is kept at the strain level of protodefects. A decrease in protodefect eigenenergy is possible where the protodefects form a new defect type with eigenenergy lower than the total protodefect energy. The possibility of the formation of translational defects (dislocations) from a protodefect system is demonstrated in [15]. 2.2. Strain level of translational defects The strain-induced behavior of a material with defects can be described using a gauge theory formalism that makes possible models of mechanical and structural states of the deformed system [1619]. In this approach, an elastic Lagrangian invariant to global symmetry groups (translation and rotation groups) is employed. The symmetry groups are gauge groups if the initial Lagrangian is invariant under their uniform action. In going to the plastic range, the deformed material is broken down into small elastically deformed structural elements whose displacements and rotations relative to each other correspond to the motion of translational and rotational defects. In the gauge approach, the process leads to localization of the gauge groups, i.e., elements of the groups become functions of coordinates and time. It should be emphasized that the material in this case retains its continuity and the total strain satisfies the compatibility condition. The localization of the translation and rotation groups affects the invariance of the initial elastic Lagrangian. The gauge invariance restoration based on minimum replacement and minimum coupling makes possible a new Lagrangian invariant under inhomogeneous action of a gauge group. For construction
of the new Lagrangian, we should replace the ordinary differentiation with a covariant one and write additional Lagrangians dependent on gauge field derivatives in the form: L L0 LD LT. (1) Here L0 is the elastic Lagrangian modified in view of translational and rotational defects, and LD and LT are the Lagrangians of specified defect fields [1619]. Consider possible strain levels in a deformed solid with defects through analyzing the potential energy of gauge model (1). In the elastic range, the deformation is described by a phase space curve with coordinates V (stress) and H (strain). Taking into account Hookes law, the density of the potential (elastic) energy of the deformed solid can be expressed as:
Ïel | C (Vext )2, (2) ext where Ñ is the compliance modulus, V is the external elastic stress in a material. The expression for the density of the potential energy (further referred to as the potential energy) is approximate, because there is no need for exact expressions in our further analysis. If the material is deformed only elastically, the potential energy increases in proportion to the squared stress, finally resulting in brittle fracture. Any evolving system tends to minimize the potential energy and searches for ways, if any, to accomplish this. In a plastic material, the passage to the inelastic range gives rise to translational defects whose elastic fields partially compensate the external stress fields. In this case, the potential energy in the modified Lagrangian L0 of Eq. (1) is expressed as follows: Ïel0 | C (Vext Vint,á )2,
(3)
int,á
is the elastic fields of translational defects in a where V material. Thus, effective stress fields equal to the difference between the external stresses and the internal defect stresses are produced in the material and the elastic energy becomes lower than the elastic response. The potential energy | C (Vint ) 2 is normally assigned to defects, though clearly the stresses in a material govern the potential energy of the medium rather than that of the defects, no matter what their source. Defects have potential eigenenergy or the core energy, which rarely, if ever, is taken into account in analyzing plastic deformation. This is because the potential core energy of a single defect in an unloaded material is about an order of magnitude lower than the elastic energy of the medium. The situation reverses if we consider elastoplastic deformation. The energy density of translational defects entered in the Lagrangian LD is expressed as:
ÏD
S1D 2,
(4)
where S1 is the linear energy density of defects and D is the tensor density of translational defects.
Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
Let us compare the expressions for potential energy (3) and defect eigenenergy (4). In a plastically deformed solid, the potential energy decreases with increasing defect density for as long as the effective stresses (Vext Vint,D ) are sufficient for defect motion, i.e., for plastic deformation. At the same time, defect eigenenergy (4) increases with an increase in defect density. At a certain point in time, the energies become comparable, and eigenenergy (4) may even far exceed elastic energy (3). In this case, the energy of the defect structure rather than the elastic energy of the medium controls the behavior of the deformed system. Hence, the inelastic behavior of a deformed solid with translational defects should be described, along with mechanical parameters (stresses and strains), by parameters characterizing the defect structure, e.g., by the density tensor and the flux density tensor of translational defects (dislocations), and this increases the phase space dimension of the deformed system. The foregoing can be summarized as follows. The state of a deformed system in the elastic range is described by two mechanical parameters, namely by stresses and strains. As the parameters change, the system passes from one elastic state to another, being kept at this elastic (or name it mechanical) strain level (Fig. 2(b), level 1). The system evolves at the elastic level until its shear stability is lost giving rise to local structural transformations translational defects. From the gauge theoretical standpoint, this corresponds to translation group localization. The formation of translational defects brings the system to another strain level the level of translational defects with its own parameters. In this case, the system is considered to pass from the mechanical level to the level of translational defects (Fig. 2(b), level 3). Thereafter, the system may evolve at the level of translational defects, passing from one translational defect structure to another. The strain level of translational defects can be related to the class of nonmisoriented defect structures discussed in detail in [20]. In the above work, it is shown that the dislocation structures follow one after another in a strictly specified sequence through successive bifurcations for no change of the nature of the state parameter (scalar dislocation density). Each structure type exists in a specific range of defect density invariant for different materials and deformation patterns. This makes it possible to develop a deformation theory applicable to most of materials, surely, without regard for features typical of a particular medium. 2.3. Strain level of rotational defects Increasing the density of translational defects increases their eigenenergy, and at a certain threshold, the deformed system is again to decide whether to fail or to pass to a new strain level the level of rotational defects. The formation of rotational defects, which differ from translation defects, suggests that the system passes to the new strain level. When
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passed to the level of rotational defects, the material is fragmented and the density of translational defects during the process decreases, on average, to values typical of annealed crystals [21]. Hence, the eigenenergy of translational defects decreases giving way to that of rotational defects ÏT S2T2 (Fig. 2, à), where S 2 is the linear energy of rotational defects and T is their density, and so the stress fields of rotational defects rather than those of translational defects will contribute to elastic energy (3). The transitions from one strain level to another corresponding to phase transformations are subject to experimental observation. Experiments show that the transitions from nonmisoriented to misoriented structures are more pronounced as against those between nonmisoriented structures of one strain level [20]. As the eigenenergy of rotational defects reach its threshold, local discontinuities appear in the material, i.e., the deformed system passes to the strain level of local discontinuities (Fig. 2(b), level 5). 3. Conclusion The foregoing analysis suggests that in the simplest case of a homogenous isotropic single-phase material, we can distinguish the following strain levels: 1. The elastic level is a strain level for which the phase space parameters are V and H. At this level, no structural changes are found in the system. 2. The level of protodefects is a strain level at which local structural changes (protodefects) are produced in the system and the phase space dimension is thus increased. In this case, an additional phase space parameter is the protodefect density n. 3. The level of translational defects is a strain level at which local structural changes (translational defects) are produced in the system, i.e., the translation symmetry is locally broken. In this case, an additional phase space parameter is the scalar or tensor density of translational defects 4. The level of rotational defects is a strain level at which local structural changes (rotational defects) are produced in the system, i.e., the rotation symmetry is locally broken. At this level, an additional phase space parameter is the density of rotational defects. Note that at all the above strain levels, the material retains its continuity. In the case where translational and rotational defects are unable to provide continuity of the material, the system passes to the next hierarchical level. 5. The level of local discontinuity is a strain level at which local discontinuities appear in the system. At this level, the rotational fracture mechanism is realized resulting in a decrease in the eigenenergy of rotational defects ÏT S2T2. 6. The level of global discontinuity is a strain level at which major cracks appear in the system and the specimen
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is broken into parts. This happens where the capabilities of the above strain levels are exhausted or they are not realized, the material thus fails and its mechanical parameters V and H lose their physical meaning. It should be noted that for brittle materials, there exist only strain levels 1, 5 and 6. For high-strength materials, where translational slip is precluded, strain levels 1, 4, 5 and 6 dominate. For plastic materials, all strain levels from 1 to 6 are realized under deformation. If fragmentation in a material is hindered, the deformed system passes from level 3 to level 5 following translational fracture mechanisms. If levels 3 and 4 are operative to the point of level 5, the system is deformed by mixed translational-rotational fracture mechanisms. The elastic level (level 1) is fundamental, since it survives to the point of fracture. Strain levels 1 and 6 bound the domain of deformation description. From this standpoint, strain levels 25 can be interpreted as mesolevels (intermediate levels) in the phase space. The evolution of a deformed system is determined by the phase space parameters of the highest level to which the system was brought under loading. The phase space parameters of the lower levels can either be involved in the description or not, depending on a particular system and loading pattern. The previous lower strain level is an accommodation level for the next higher level. The more exhaustive the capabilities of the lower levels, the lower the deformation capacity of a material. In the limiting case, where the capabilities of mesolevels 24 are completely exhausted, the material behavior is ideally brittle. Thus, the life of a deformed solid is determined by its mesolevels in the phase space. The work was partially supported by the Russian Foundation for Basic Research (project code 06-08-96917-r_ofi.) References [1] V.E. Panin, V.E. Egorushkin, Yu.A. Khon and T.F. Elsukova, Atomvacancy states in crystals, Russ. Phys. J., 25, No. 12 (1982) 1073.
[2] V.E. Panin, Yu.V. Grinyaev, V.E. Egorushkin et al., Spectrum of excited states and vortex mechanical field in a crystal under deformation, Russ. Phys. J., 30, No. 1 (1987) 24. [3] E. Kröner, Allegemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Rational Mech. Anal., 4 (1960) 273. [4] J.B. Eshelby, The continuum theory of lattice defects, Solid State Physics, No. 3 (1956) 79. [5] R. de Wit, Theory of diclinations, J. Res. Nat. Bureau Stand. 77A, 49 (1973) 359, 607. [6] A.M. Kosevich, Dislocations in Elasticity Theory, Naukova Dumka, Kiev, 1978 (in Russian). [7] J.P. Hirth and J. Lothe, Theory of Dislocations, Wiley, New York, 1982. [8] J. Friedel, Dislocations, Pergamon Press, Oxford, 1964. [9] J.S. Nicolis, Dynamics of Hierarchical Systems: An Evolutionary Approach, Springer-Verlag, Berlin, 1986. [10] A.I. Dmitriev and S.G. Psakhie, Molecular-dynamic study of nanofragmentation during relaxation in after-loaded solid, Techn. Phys. Letters, 30, No. 8 (2004) 677. [11] A.I. Dmitriev, Molecular dynamics study into the features of coordinated collective motion of atoms in the loaded material near the free surface, Phys. Mesomech., 8, No. 34 (2005) 71. [12] S.G. Psakhie, T.Yu. Uvarov and K.P. Zolnikov, On the new mechanism of defect generation at the interface. Molecular-dynamic simulation, Phys. Mesomech., 3, No. 3 (2000) 67. [13] S.G. Psakhe and K.P. Zolnikov, Possibility of a vortex mechanism of displacement of the grain boundaries under high-rate shear loading, Comb., Explos., and Shock Waves, 34, No. 3 (1998) 366. [14] V.A. Polukhin, Nanostructure and Precursor State Modeling, UrB RAS, Ekaterinburg, 2004 (in Russian). [15] S.G. Psakhie, K.P. Zolnikov, D.S. Kryzhevich and A.G. Lipnitskii, On structural defect generation induced by thermal fluctuations in materials with a perfect lattice under dynamic loading, Phys. Letters A, 349 (2006) 509. [16] Yu.V. Grinyaev and N.V. Chertova, The gauge theory of plastic deformation in continuum mechanics, Russ. Phys. J., 33, No. 2 (1990) 123. [17] Yu.V. Grinyaev and N.V. Chertova, Field theory of defects. Part I, Phys. Mesomech., 3, No. 5 (2000) 17. [18] Yu.V. Grinyaev and N.V. Chertova, Field theory of defects, Part II, Phys. Mesomech., 9, No. 12 (2006) 31. [19] N.V. Chertova and Yu.V. Grinyaev, Description of heterogeneous solids in the framework of the gauge approach, Phys. Mesomech., 9, No. 56 (2006) 69. [20] N.A. Koneva and É.V. Kozlov, Physical nature of stages in plastic deformation, Russ. Phys. J., 33, No. 2 (1990) 165. [21] V.V. Rybin, Structural kinetic aspects of the physics of evolution of plastic deformation, Russ. Phys. J., 34, No. 3 (1991) 186.
Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
Phase space of solids under deformation Yu.V. Grinyaev, S.G. Psakhie and N.V. Chertova* Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia This paper analyzes sequences of various strain-induced defects. The defect is considered as an internal stress source, and hence the deformation is described by a phase curve in the 2D stress-strain space. Notice that the defect as a local discontinuity has eigenenergy and this, if taken into account, increases the phase space dimension because of the parameters to be introduced to characterize defect structures formed under deformation. Analysis of the potential energy of a deformed system makes it possible to elucidate the origin of various defects and to introduce the notion of strain levels in the phase space. Thus, we can distinguish strain levels for which the nature and the number of parameters of the system remain unchanged under deformation and associate the transition of the deformed system from one strain level to another with qualitative and quantitative changes. In this context, the deformation from elasticity to fracture is described by the phase space curve and the phase space dimension is determined, in addition to stresses and strains, by the parameters of defect structures formed under deformation. A standard stress-strain curve is the projection of the phase space curve on the stress-strain plane. Keywords: phase space, strain levels, defects, gauge theory
1. Introduction Research on the formation of various structural defects, from point to three-dimensional, is important for understanding the mechanisms by which plastic deformation of materials originates and evolves. The system under study is a macroscopic dynamic system with many elements whose stochastic behavior transforms into deterministic behavior of the system as a whole. Let us consider the evolution of a mechanically loaded solid. The deformation of the solid is treated as a macrophenomenon in a complex system containing many structural elements that interact in their relative displacements (translations and rotations). In other words, the deformed solid is a nonequilibrium system and its deformation is described in the context of local thermodynamics, i.e., each structural element is in equilibrium, while the system as a whole is not. For simplicity, the deformed solid is a homogeneous isotropic single-phase medium. Mechanics of solids treats of mechanical processes in which the spacing between material points of the medium is changed. The change in the spacing and its attendant forces are determined through mechanical state parameters, namely * Corresponding author Dr Nadezhda V. Chertova, e-mail: [email protected] Copyright © 2008 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2008.11.003
through stress and strain tensors. For elastic deformation, these parameters unambiguously characterize the mechanical state of a deformed solid. For inelastic deformation, there is no one-to-one correspondence between the mechanical parameters, since to one value of stresses may correspond an infinitely many values of strains depending on loading history. This fact is due to local structural changes responsible for plastic deformation and irreversible total strains under unloading. More than 20 years ago, Panin [1, 2] put forward a concept in which defect nucleation is preceded by so-called highly excited states of a crystal. Thus, describing the deformation of solids requires a macroscopic theory that takes into account the physical processes associated with the defect nucleation and dynamics. In classical mechanics of solids, irreversible deformation is described by formulating different nonlinear material relations with the use of only mechanical stress-strain parameters. It is also assumed that nonlinear relations, in a way, will take into account structural changes occurring under plastic deformation. However, the structural changes must inevitably lead to a change in the material relations. Hence, the description of deformation by the mechanical stressstrain parameters alone is incapable of clarifying the physical mechanisms of plasticity and is inadequate beyond
Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
the elasticity limit where structural states need to be taken into account. Classical theory of defects treats of the physical mechanisms responsible for plastic behavior of materials. The basic problem in the theory is to determine the stress and strain fields of a single defect or simple groups of defects, the forced interaction of defects with each other and with fields of applied loads [38]. The defect is considered only as an internal stress source, and not as a local discontinuity (a structural element). For successive description of deformation, let us take into account that a defect (the defect core) has eigenenergy. In addition to the mechanical state parameters, this description should involve structural state parameters characterizing the evolution of the corresponding defect structures. Hence, the phase space dimension for describing the deformation should be determined by both the mechanical stress-strain parameters and the structural state parameters of different defects. In this case, each specific set of the parameters in the phase space can be assigned to a certain strain level. Let us consider what each strain level determines exactly in physical terms and how one strain level differs from another [9]. For identification, each strain level is described in phase space terms. The description involves the type and the number of dynamic variables, and the type and the number of states corresponding to this level. The evolution and behavior of a deformed system in the phase space are governed by the potential energy of the system. The system transforms from state to state, violating the relations between the dynamic variables and synthesizing new ones with the nature of the variables remained the same. In other words, the system evolves at a certain strain level, turning from one steady state to another through successive bifurcations for no change of dynamic variables. This means that the system is kept at a certain strain level due to the self-identity of the variables and main state parameters. If at a certain strain level the bifurcation responsible for a change of variables becomes inevitable, the system passes to another strain level, otherwise it inevitably fails. If this passage occurs, the system is said to evolve crosswise the level.
229
not sufficiently advanced, and primarily because of the high time (1014 s) and space (109 m) resolutions required for experimental studies of the defect nucleation at the atomic level. In this situation, an efficient way of studying the generation mechanisms and the dynamics of structural defects is computer simulation of mechanical loading based on molecular dynamics. Using direct molecular-dynamic simulation, the authors of [15] were first to demonstrate the feasibility of protodefect generation by thermofluctuation mechanisms as the realization of precursor (highly excited) states under dynamic load. In the work, the researchers investigated the initiation of plastic deformation in a copper crystallite under high-speed loading. It was shown that as the threshold strain (potential energy) is reached in the simulated crystallite at final temperatures, jump-like production of protodefects (structural changes) begins (Fig. 1, curve 3). The protodefect generation decreases the potential energy of the crystallite (Fig. 1, curve 2). The decrease in potential energy as against the value corresponding to the strain under geometrical changes (Fig. 1, curve 1) is due to the eigenstress field of structural defects that compensates the external stress field. As well as producing the stress field in the deformed system, the protodefects have eigenenergy. Increasing the fraction of protodefects causes an increase in their eigenenergy (about 0.036 eV per defect). Finally, the situation may arise in which the total eigenenergy of protodefects becomes greater than the elastic energy of the system and thus responsible for the behavior of the crystallite. Thus, the generation of the new structure (protodefects characterized by their density) in the crystallite can be considered as the transition of the deformed system in the phase space from the elastic strain level to the next strain level level of protodefects (levels 1 and 2, respectively, in Fig. 2(b). As the protodefect eigenenergy reaches a certain threshold, the further increase in protodefect fraction leads to
2. Strain levels 2.1. Strain level of protodefects Research on the formation of various structural defects is particularly urgent for describing the behavior of materials under dynamic loading, since the deformation in this case gives rise to so-called dynamic defects [1013]. One of the fundamental problems in solid state physics and materials science is indubitably the study of possible thermofluctuation mechanisms of local structural distortions and transformations at the atomic level that can be referred to socalled precursor states [14] or highly excited states introduced by V.E. Panin [1, 2]. Unfortunately, this concept is
Fig. 1. Potential energy per atom, ÏÅ, displaced upward by 0.05 eV for 0 K (1) and at 50 m/s for 300 K (2), and the strain dependence of the protodefect density n (3)
230
Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
à
b
Fig. 2. Schematic strain dependence of the potential energies (à): Ï el is the elastic energy of a deformed solid, Ï Å is the protodefect eigenenergy, Ï D is the eigenenergy of translational defects, ÏT is the eigenenergy of rotational defects, and strain levels in the phase space at the corresponding threshold energies (b): elastic level arising at the onset of loading (1), level of protodefects (2), level of translational defects (3), level of rotational defects (4), and level of local discontinuities (5)
fracture of the system if it is kept at the strain level of protodefects. A decrease in protodefect eigenenergy is possible where the protodefects form a new defect type with eigenenergy lower than the total protodefect energy. The possibility of the formation of translational defects (dislocations) from a protodefect system is demonstrated in [15]. 2.2. Strain level of translational defects The strain-induced behavior of a material with defects can be described using a gauge theory formalism that makes possible models of mechanical and structural states of the deformed system [1619]. In this approach, an elastic Lagrangian invariant to global symmetry groups (translation and rotation groups) is employed. The symmetry groups are gauge groups if the initial Lagrangian is invariant under their uniform action. In going to the plastic range, the deformed material is broken down into small elastically deformed structural elements whose displacements and rotations relative to each other correspond to the motion of translational and rotational defects. In the gauge approach, the process leads to localization of the gauge groups, i.e., elements of the groups become functions of coordinates and time. It should be emphasized that the material in this case retains its continuity and the total strain satisfies the compatibility condition. The localization of the translation and rotation groups affects the invariance of the initial elastic Lagrangian. The gauge invariance restoration based on minimum replacement and minimum coupling makes possible a new Lagrangian invariant under inhomogeneous action of a gauge group. For construction
of the new Lagrangian, we should replace the ordinary differentiation with a covariant one and write additional Lagrangians dependent on gauge field derivatives in the form: L L0 LD LT. (1) Here L0 is the elastic Lagrangian modified in view of translational and rotational defects, and LD and LT are the Lagrangians of specified defect fields [1619]. Consider possible strain levels in a deformed solid with defects through analyzing the potential energy of gauge model (1). In the elastic range, the deformation is described by a phase space curve with coordinates V (stress) and H (strain). Taking into account Hookes law, the density of the potential (elastic) energy of the deformed solid can be expressed as:
Ïel | C (Vext )2, (2) ext where Ñ is the compliance modulus, V is the external elastic stress in a material. The expression for the density of the potential energy (further referred to as the potential energy) is approximate, because there is no need for exact expressions in our further analysis. If the material is deformed only elastically, the potential energy increases in proportion to the squared stress, finally resulting in brittle fracture. Any evolving system tends to minimize the potential energy and searches for ways, if any, to accomplish this. In a plastic material, the passage to the inelastic range gives rise to translational defects whose elastic fields partially compensate the external stress fields. In this case, the potential energy in the modified Lagrangian L0 of Eq. (1) is expressed as follows: Ïel0 | C (Vext Vint,á )2,
(3)
int,á
is the elastic fields of translational defects in a where V material. Thus, effective stress fields equal to the difference between the external stresses and the internal defect stresses are produced in the material and the elastic energy becomes lower than the elastic response. The potential energy | C (Vint ) 2 is normally assigned to defects, though clearly the stresses in a material govern the potential energy of the medium rather than that of the defects, no matter what their source. Defects have potential eigenenergy or the core energy, which rarely, if ever, is taken into account in analyzing plastic deformation. This is because the potential core energy of a single defect in an unloaded material is about an order of magnitude lower than the elastic energy of the medium. The situation reverses if we consider elastoplastic deformation. The energy density of translational defects entered in the Lagrangian LD is expressed as:
ÏD
S1D 2,
(4)
where S1 is the linear energy density of defects and D is the tensor density of translational defects.
Yu. V. Grinyaev, S.G. Psakhie and N.V. Chertova / Physical Mesomechanics 11 5–6 (2008) 228–232
Let us compare the expressions for potential energy (3) and defect eigenenergy (4). In a plastically deformed solid, the potential energy decreases with increasing defect density for as long as the effective stresses (Vext Vint,D ) are sufficient for defect motion, i.e., for plastic deformation. At the same time, defect eigenenergy (4) increases with an increase in defect density. At a certain point in time, the energies become comparable, and eigenenergy (4) may even far exceed elastic energy (3). In this case, the energy of the defect structure rather than the elastic energy of the medium controls the behavior of the deformed system. Hence, the inelastic behavior of a deformed solid with translational defects should be described, along with mechanical parameters (stresses and strains), by parameters characterizing the defect structure, e.g., by the density tensor and the flux density tensor of translational defects (dislocations), and this increases the phase space dimension of the deformed system. The foregoing can be summarized as follows. The state of a deformed system in the elastic range is described by two mechanical parameters, namely by stresses and strains. As the parameters change, the system passes from one elastic state to another, being kept at this elastic (or name it mechanical) strain level (Fig. 2(b), level 1). The system evolves at the elastic level until its shear stability is lost giving rise to local structural transformations translational defects. From the gauge theoretical standpoint, this corresponds to translation group localization. The formation of translational defects brings the system to another strain level the level of translational defects with its own parameters. In this case, the system is considered to pass from the mechanical level to the level of translational defects (Fig. 2(b), level 3). Thereafter, the system may evolve at the level of translational defects, passing from one translational defect structure to another. The strain level of translational defects can be related to the class of nonmisoriented defect structures discussed in detail in [20]. In the above work, it is shown that the dislocation structures follow one after another in a strictly specified sequence through successive bifurcations for no change of the nature of the state parameter (scalar dislocation density). Each structure type exists in a specific range of defect density invariant for different materials and deformation patterns. This makes it possible to develop a deformation theory applicable to most of materials, surely, without regard for features typical of a particular medium. 2.3. Strain level of rotational defects Increasing the density of translational defects increases their eigenenergy, and at a certain threshold, the deformed system is again to decide whether to fail or to pass to a new strain level the level of rotational defects. The formation of rotational defects, which differ from translation defects, suggests that the system passes to the new strain level. When
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passed to the level of rotational defects, the material is fragmented and the density of translational defects during the process decreases, on average, to values typical of annealed crystals [21]. Hence, the eigenenergy of translational defects decreases giving way to that of rotational defects ÏT S2T2 (Fig. 2, à), where S 2 is the linear energy of rotational defects and T is their density, and so the stress fields of rotational defects rather than those of translational defects will contribute to elastic energy (3). The transitions from one strain level to another corresponding to phase transformations are subject to experimental observation. Experiments show that the transitions from nonmisoriented to misoriented structures are more pronounced as against those between nonmisoriented structures of one strain level [20]. As the eigenenergy of rotational defects reach its threshold, local discontinuities appear in the material, i.e., the deformed system passes to the strain level of local discontinuities (Fig. 2(b), level 5). 3. Conclusion The foregoing analysis suggests that in the simplest case of a homogenous isotropic single-phase material, we can distinguish the following strain levels: 1. The elastic level is a strain level for which the phase space parameters are V and H. At this level, no structural changes are found in the system. 2. The level of protodefects is a strain level at which local structural changes (protodefects) are produced in the system and the phase space dimension is thus increased. In this case, an additional phase space parameter is the protodefect density n. 3. The level of translational defects is a strain level at which local structural changes (translational defects) are produced in the system, i.e., the translation symmetry is locally broken. In this case, an additional phase space parameter is the scalar or tensor density of translational defects 4. The level of rotational defects is a strain level at which local structural changes (rotational defects) are produced in the system, i.e., the rotation symmetry is locally broken. At this level, an additional phase space parameter is the density of rotational defects. Note that at all the above strain levels, the material retains its continuity. In the case where translational and rotational defects are unable to provide continuity of the material, the system passes to the next hierarchical level. 5. The level of local discontinuity is a strain level at which local discontinuities appear in the system. At this level, the rotational fracture mechanism is realized resulting in a decrease in the eigenenergy of rotational defects ÏT S2T2. 6. The level of global discontinuity is a strain level at which major cracks appear in the system and the specimen
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is broken into parts. This happens where the capabilities of the above strain levels are exhausted or they are not realized, the material thus fails and its mechanical parameters V and H lose their physical meaning. It should be noted that for brittle materials, there exist only strain levels 1, 5 and 6. For high-strength materials, where translational slip is precluded, strain levels 1, 4, 5 and 6 dominate. For plastic materials, all strain levels from 1 to 6 are realized under deformation. If fragmentation in a material is hindered, the deformed system passes from level 3 to level 5 following translational fracture mechanisms. If levels 3 and 4 are operative to the point of level 5, the system is deformed by mixed translational-rotational fracture mechanisms. The elastic level (level 1) is fundamental, since it survives to the point of fracture. Strain levels 1 and 6 bound the domain of deformation description. From this standpoint, strain levels 25 can be interpreted as mesolevels (intermediate levels) in the phase space. The evolution of a deformed system is determined by the phase space parameters of the highest level to which the system was brought under loading. The phase space parameters of the lower levels can either be involved in the description or not, depending on a particular system and loading pattern. The previous lower strain level is an accommodation level for the next higher level. The more exhaustive the capabilities of the lower levels, the lower the deformation capacity of a material. In the limiting case, where the capabilities of mesolevels 24 are completely exhausted, the material behavior is ideally brittle. Thus, the life of a deformed solid is determined by its mesolevels in the phase space. The work was partially supported by the Russian Foundation for Basic Research (project code 06-08-96917-r_ofi.) References [1] V.E. Panin, V.E. Egorushkin, Yu.A. Khon and T.F. Elsukova, Atomvacancy states in crystals, Russ. Phys. J., 25, No. 12 (1982) 1073.
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