On intergranular interaction and the crystallographic mechanism of plastically deformed polycrystalline metals

Taylor principle prevailing currently has been proved to be effective only when grains or grain clusters are deformed in a rigid environment, which disagrees with the reality. Grains realize their plastic deformation in elastoplastic environment, and the equilibrium of intergranular stress and strain need to be reached simultaneously and naturally. The intergranular reaction stress (RS) during deformation can be calculated according to Hooke’s law and the yield stress should be the up-limit of the RS. The combinations of constant external stress and changing RS in RS theory induce alternate activation of deformation systems including slips and twinning with different work hardening rates, while grain orientations and inhomogeneous distribution of external stress have important influence. Some additional deformation systems near the boundaries need to be locally activated when the reduced stress up-limits are reached frequently, to balance the intergranular stress and strain incompatibilities in elastoplastic way. Based on the simpler RS theory, the simulation results of rolling texture on aluminium, low carbon steel, austenite stainless steel, titanium are very close to the reality, indicating that the intergranular elastoplastic interactions and the corresponding crystallographic behaviours of plastic deformation have been reasonably and quantitatively described, therefore, Taylor principle becomes no more necessary.


Introduction
Plastic deformation is usually the premise of recrystallization, and the formation of recrystallization texture starts from deformation texture.Therefore, the theory of plastic deformation crystallography is extremely important for understanding the formation of deformation texture and has received extensive attention [1].At present, the widely popular theories of plastic deformation crystallography and models of deformation texture formation are mostly based on the Taylor principle, e. g. the viscoplastic self-consistent (VPSC) theory [2], advanced Lamella (ALAMEL) theory [3], grain interaction (GIA) theory [4] etc., which shows some problems both in theory and practice [5].Several slip or twinning systems in deformation grains are activated simultaneously according to a power-law relationship controlled by a parameter of shear-rate sensibility exponent in all the theories above [1], which unreasonably violate the Schmidt's law.In practice, the real strain tensors of deformation grains in all kinds of metals are essentially different from that prescribed by the Taylor principle as well [5].Therefore, we would try to trace the real physical process of plastic deformation of metal polycrystals, and explore a more reasonable plastic deformation principle in theory and practice.If the grain is in a rigid environment, so that all plastic strain components in equation 1 would be elastically compressed back by 100%, thus forming a reaction stress (RS) tensor acting on the grain.At this time, the stress tensor [σij], that the deforming grain is bearing, consists of both the external stress tensor [   ] as well as the elastic RS tensor [   ] that can be calculated by Hooke's law (R=1 is the rigidity rate, E is Young's modulus and ν is Poisson's ratio) [1]: However, no matter how the RS accumulates, it must not exceed the yield level related with the yield stress σy of the polycrystal, and the up-limit of the RS tensor [   ]lim in equation 2 is expressed, in the case of rolling deformation, as (in which αij=1 are the effective coefficients of the RS) [1,5]: With the progress of the slip and accumulation of RS, the stress tensor (equation 2) borne by the slip system will continue to evolve, so that after deformation to a certain extent, another slip system may obtain the maximum critical resolved shear stress (CRSS), begins to slip instead, and penetrate the grain.The alternation of the individually activated slip systems will occur frequently during deformation, and multi-slips are realized by these alternating slips all penetrating the deformation grain.However, the observed true plastic behavior of grains is more complicated (figure 1b) [5], which includes not only penetrating slips, but also some non-penetrating slips mostly near grain boundary to coordinate the intergranular incompatibility of stress and strain (figure 1c).The nonpenetrating slips often do not show any regularity, which will lead to a randomization tendency of deformation texture.

Problems existing in the Taylor principle both in practice and theory
Figure 2a shows an example of deformation texture of pure aluminum sheet after 95% cold rolling from an initial state of roughly random orientation distribution, in which significant copper texture and brass texture are obtained simultaneously.The texture in figure 2a can be directly calculated with the help of the Taylor principle, in which combinations of five independent slip systems are activated and penetrate grains homogenously.A small enough simulation step of rolling strain ε in this paper is Δε=δb3n3=0.001according to equation 1.The simulation result (figure 2b) does not indicate any brass texture but extremely strong Taylor texture with lower Φ angle, rather than copper texture in figure 2a.The same calculations based on ALAMEL and GIA model could obtain a little brass texture, though it is too weak, but still very strong Taylor texture instead of the copper one [1,5], which indicates strong background of the Taylor principle in the two models.
If, in a rigid environment, only the slip system with the highest Schmidt factor in grains can start to slip under the combination of external stress and the reaction stress (equation 2, R=1), whereas the maximum reaction stress can always reach the yield level of the metals (equation 3, αij=1) and the small enough simulation step can also make all slip systems start to slip once its Schmidt factor becomes the highest, the simulated rolling texture of the aluminum sheet is shown in figure 2c [1,5].The texture shown in figure 2c is consistent with that of the Taylor model in figure 2b even in details, including the absence of brass texture, the extremely strong Taylor texture rather than copper one, and so on, only the orientation density increases a little owing to the limited reaction stress (equation 3).
The real metal grains realize their plastic deformation in an elastoplastic environment, and the intergranular stress and strain compatibility are reached naturally rather than rigidly.The simulation result based on equation 2 and equation 3 shows that the Taylor principle sets the grains absolutely in a rigid environment, which is inconsistent with the actual metal deformation process, so it often fails to accurately predict the texture formation during plastic deformation of metals.

The reaction stress (RS) theory
Real metal grains possess elastoplastic characteristics, and their plastic deformation takes place in the same elastoplastic environment.Therefore, for an elastic equilibrium, 50% of the plastic strain produced by the deformed grain in equation 1 needs to be absorbed by the adjacent zone in the form of elastic strain, while the other 50% will be compressed back into the deformed grain in the form of reverse elastic strain, i. e. the rigidity rate R in equation 2 should 0.5 [1,5].
In the case of rolling deformation, the external stress is usually expressed as a main stress tensor composed of a tensile stress in the rolling direction (RD, i. e. direction 1 in equation 1) and a compressive stress in the normal direction (ND, i. e. direction 3 in equation 1) of the rolling sheet.When the pass reduction increases, the main stress tensor near the sheet surface will rotate around the transverse direction about an angle θ, which gradually decreases from the surface inwards and becomes 0 at the center layer.The greater the pass reduction is, the higher the θ will be.Therefore, the rolling stress tensor in equation 2 including the RS is transformed into [1,5]: in which, R=0.5 is valid for elastoplastic matrix, b is the length of Burgers vector of a slip system, μ is its Schmidt factor under instant stress tensor [σij] (equation 4), and the average distance d between dislocations can be calculated momentarily according to the flow stress that is experiencing work hardening during deformation of the individual metals [1,5].The first term on the right side of equation 4 represents the external stress tensor, and the second term represents the RS.When θ is 0˚, the external stress becomes the normal rolling stress tensor, i. e. a tensile stress in RD and a compressive stress in ND.
The strain tensors produced respectively by the activated penetrating slips in two adjacent grains under the stress tensor shown in equation 4 are usually different, and will cause the intergranular incompatibility of strain and stress, especially near the grain boundary region, which will induce additional interaction stress.According to Schmidt's law, except for the deformation system with the highest Schmidt factor instantly under the action of stress tensor [σij], all slip and twining systems will not be activated, but they always bear a certain shear stress lower than their CRSS during deformation.However, if the shear stress borne by the un-activated slip or twining systems is superimposed with the additional stress of the interaction between grains, it may lead to the total stress reaching the CRSS, and cause some un-activated slip or twining systems to be activated in non-penetrating way (figure 1c), thus making the incompatibility between grains mostly relieved in the form of local plastic deformation, while the remaining incompatibility is coordinated in the form local elastic strain.In the process of deformation, the frequent coordination of stress and strain that occurs naturally between grains will constantly stop the rise of RS, so that it can never reach its theoretical maximum value, that is, the values of αij in equation 3 are usually far less than 1, which reduces the effect of RS on the activation selection of penetrating slip systems.The levels of the effective coefficients αij and their possible evolutions with deformation ε are closely related to the characteristics of the deformed metals, and should be determined individually in advance.
It is known, that crystallographic metals are intrinsically anisotropic and an anisotropy can also be induced by local texture, or different orientations of adjacent grains.In the case of rolling process, however, the deformed polycrystalline metals, from the statistical point of view, show often the characteristics of approximate isotropy because of the macroscopic symmetry of the samples and the symmetry of all texture components formed [1].Therefore, the metals are generally regarded as nearly elastic isotropic materials, as shown in the equation 2. On the other hand, the levels of the coefficients αij can also be adjusted according to the experimental characteristics of different metal rolling processes, in order to reflect the effects of the local anisotropy to a certain extent.
Equation 1 to 4 constitute the basic equations of the RS theory.In order to implement the RS simulation, it is necessary to first determine which slip or twinning systems will be active.When different types of systems may be alternately active, the relative CRSS between them and its possible evolution with deformation ε should also be confirmed in advance.RS theory only calculates the orientation evolution and corresponding deformation texture caused by twinning and penetrating slip.Because of the randomization effect induced by those non-penetrating slips both in deformed grain and its adjacent zone, certain random texture component vr (volume fraction of random orientations) needs to be added to the simulated texture whereas the vr level is also closely related to the characteristics of individual deformed metals [1,5].

Simulations of rolling textures of FCC, BCC and HCP metals based on the RS model
Rolling textures of aluminum, low carbon steel, austenite stainless steel, titanium, are simulated based on the RS theory, whereas the active slip or twinning systems, relative CRSS, effective coefficients αij, rotation angle θ and random texture component vr that were determined in advance are listed in table 1 [1,5].936 and 1716 homogeneously distributed initial orientations are used for the texture simulations of cubic and hexagonal metals, respectively [1]. Figure 3 to 7 give the simulated rolling textures in comparison with those of experimental observations [1,5].
RS model can reproduce rolling texture formation of FCC aluminum sheet, including all texture types and their density levels (figure 3), as well as the surface shear texture induced by obvious rotation θ (table 1) of principal stress tensor under large pass reductions (figure 4).Two types of slip systems with different work hardening effects are activated during rolling of BCC low carbon steel, in which {112}<111> slips become more and more active with the rolling strain ε (table 1), resulting in the stability of {111} fiber texture (figure5).The rolling deformation conducted by joint activation of slips and twinning in FCC austenite stainless steel is rather complicated, because not only will slip systems be work hardened more rapidly, but also RS levels (α12 and α23) will gradually decrease instead of keep constant (table 1).Nevertheless, the effect that the phenomenon of copper texture turning to Goss texture by means of twinning [1,5] can still be simulated (figure6).In general, all the simulation results agree very well with the experimental observations (figure 3 to 6).Many different types of slip systems and twin systems have been observed in HCP titanium.However, those slip and twinning systems that are rarely activated or not independent ones should be eliminated to simplify the simulation process [1].Nevertheless, there are still too many activated slip or twinning systems (table 1), which is prone to randomize or weaken the deformation texture resulting in that the vr becomes no more necessary.On the other hand, frequent twinning leads to too many twin orientation evolutions, which needs to be simulated separately from the orientation evolutions induced by slips.In order to avoid too cumbersome simulation calculation, two sets of the 1716 simulation orientations are adopted [1].One set only tracks the orientation evolutions of the continuously twinned part, and the other set tracks the orientation evolutions of the parent part of the twinning, as so, to obtain the rolling texture of titanium sheet in a simple and approximate way (table 1).The rolling texture evolution of Ti sheet has been simulated based on the RS model, and the results agree well with those of the experimental observations (figure 7), including the characteristics of very low-density levels and low Φ angle locations of the texture [1].

Summary
It is to see clearly that RS theory of the plastic deformation crystallography is extremely simple without complicated mathematical treatment, is also very intuitive and reasonable on physical background, and the corresponding simulations agree basically with the experimental observations of different metals.However, those non-penetrating slip or twinning may cause some uncertainty, so that the simulation parameters need to be adjusted for different metals with individual characteristics as well as for different processes in the actual calculations.RS theory gives good consideration to the compatibility both of intergranular stress and strain that naturally formed during elastoplastic deformation of polycrystals while Schmidt's law prevails all the time, and takes into account also many factors that have important impacts on deformation, including: coactivation of multiple deformation systems, changing work hardening rate, homogeneity of the deformation, arbitrary external stress tensor, effective extent of the RS, random effect of non-penetrating deformation, even the elastic anisotropy of the single crystal or that induced by matrix texture, etc.Of course, the RS theory still needs to be greatly improved and perfected.However, the Taylor principle could therefore be abandoned rather than trying somehow to modify it.

Figure 1 .
Figure 1.Deformation behaviours of a grain in polycrystal, a. initial behaviour, b. exp.observation in steel, c. multi penetrating and non-penetrating slips.

Table 1 .
Simulation parameters of rolling textures based on the RS theory