Polycrystalline Textures Formation

Transcription

1 Polycrystallne Textures Formaton Pavel Volegov, Perm State Techncal Unversty (Perm, Russa) I. Introducton Crystallne bodes wth deal structure owng to unequal densty of atoms n varous planes and drectons of a lattce possess ansotropy of some physcal and mechancal propertes. For example, the elastc modulus, a specfc resstance, a dffuson constant have varous values for dfferent drectons n a crystal. As a rule, metals and the alloys used n engneerng, are polycrystals,.e. consst of major number of the ansotropc crystal grans (grans, subgrans, etc). In most cases n lmts of representatve macrovolume crystal grans statstcally unordered are focused one n relaton to another, therefore at a level of representatve macrovolume n all drectons of property t s possble to count dentcal then the polycrystallne body n macroscopcal sense can be counted the sotropc. However f on gauges of representatve volume n a spatal arrangement of such crystallne grans some orderlness the polycrystallne materal wll get ansotropy of propertes n some allocated drectons wll appear. In ths case usually speak about a texture as whch understand presence of the allocated (preferred) drectons n spatal orentaton of crystallne lattces of separate consttuents of a polycrystallne body. Textures are formed owng to focused acton on a body of exteror and-or nteror forces. These forces can be caused by mechancal, magnetc, electrcal or thermal actons, etc. Textures arse at varous technologcal processes: crystallzatons, plastc stran, etc. Important are strength propertes and magnetc propertes materals wth textures as ansotropy of the relevant performances of a textured materal s wdely used n engneerng. Practcally any plastc stran, except for stran under the plan of all-round squeezng, s accompaned by formaton of a crystallographc texture of ths or that type and ths or that ntensty. Fg. the Plan of crystallographc orentaton of crystal grans n a leaf materal n case of cubc (а) and rbbed (б) textures So, for example, there s a dependence between elastc modulus E and a drecton n rolled metal leafs. For many face-centered cubc lattce and body-centered lattces metals there s obvously expressed extreme E for a corner ϕ 45 (ϕ -a corner to drecton of rollng and rollng plane). However character of an extreme s varous. For body-centered lattce metals t was revealed, that strength performances ( σ В and σ Т ) are peak n drecton rollng plane, and a protracton - n drecton drecton of rollng. The known phenomenon s nterlnked to a texture press-effect, consstng that under certan condtons

2 moldngs of metal ther alloys strength propertes n a drecton of a moldng appear consderably hgher, than n a drecton of pressng of metal also at other plans of stran (for example, at a rollng). Thus, practcal value of textures s caused by ansotropy of propertes caused by them whch can be rather effectvely be used. In the gven operaton formaton of textures s consdered as a result of plastc stran. Interest to ansotropy on the part of experts n the feld of processng metals by pressure s caused, on the one hand, by that n many cases t s mportant to know how t s possble to change propertes and what of them wth the help of plastc stran. II. Defnton and the basc methods of the descrpton of textures The texture s a presence of preferred orentaton of separate crystal grans n a polycrystal. Usually due to presence of a texture n materals ansotropy of propertes s shown. Concernng polycrystallne t s model speak, that they possess a texture n that case when crystal grans are located not qute randomly. It means, that n a sample there are some drectons (or planes), along whch preferentally settle down partcular crystallographc drect (or planes) the crystal grans makng a polycrystal. The descrpton of a texture s based on defnton of orentaton of crystal grans n some system of coordnates. The select of a system of coordnates defnes an method of the descrpton of a texture. Generally the texture of the polycrystallne unt s featured by four coordnates, three of whch determne orentaton, and the fourth - probablty of ths orentaton. However the greatest dstrbuton tll the present moment three-dmensonal expedents of the descrpton of a texture (for example drect and return pole fgures), that speaks smaller methodcal dffcultes of such descrpton. The descrpton of orentaton of a separate crystal The concept of orentaton s one of basc n the analyss of a texture. To descrbe orentaton of a crystal gran, t s necessary to set ts coordnate system K cr (whch we term as ts crystallographc system of coordnates), the bound wth the chosen drectons n a crystal gran. At the same tme n an explored sample the coordnate system of K e (an exteror system of coordnates s nlet; we use further the term laboratory system of coordnates) whch axes are the reference drectons n a sample. To spot orentaton of a crystal gran s means to specfy the gyraton translatng laboratory system of coordnates K e n crystallographc system of coordnates K cr. As axes crystallographc system of coordnates K cr t s the most convenent to choose drectons, the bound wth devces of symmetry of the best order for vewed crystallne structure. For cubc crystals as axes of K cr coordnates drectons of edges of a cube [00], [00], [00] are usually used. Laboratory system of coordnates also t s defned by devces of symmetry whch are present at a sample; for example, for rolled leafs of an axs of coordnate system usually pck contermnous wth a drecton of rollng, a transverse drecton and a drecton of a normal lne to a rollng plane. The descrpton of gyraton of coordnate systems and consequently, and the descrpton of orentaton can be carred out n several ways. The basc expedents of the descrpton of orentaton are the followng: The Descrpton of orentaton by means of matrxes of gyraton The Descrpton of orentaton by a corner and an axs of rotatonal dsplacement

3 The Descrpton of orentaton wth the help of Euleran angles The Descrpton of textures by means of pole fgures The Descrpton of textures by means of a dstrbuton functon of orentatons Let's consder n more detal last two expedents of the descrpton of a texture as most frequently used on practce. The descrpton of textures by means of pole fgures The quanttatve analyss of textures s be relatve combned. It demands buld-up on the receved data (from X-ray patterns, etc.) specal «pole fgures» and ther analyss. As pole fgures (PF) of a polycrystal understand stereographc projectons of normal lnes (poles) to partcular nuclear planes {h k l } ( - the number of a gran), constructed for that volume of a polycrystal from whch the dffracton pattern s receved. At lack of a texture any orentatons of planes of dfferent crystal grans are equalty probablty. In ths case PF these planes represents the crcle of projectons unformly covered wth poles {h k l }.. Pctorally t s accepted to fgure the unform shadng of a crcle of projectons (fg. 2, a). Fg. 2 Allocaton of poles to pole fgures for varous type of textures (plan) a- absence of a texture, б-deal uncomponent rollng texture (a texture of a cube, a pole fgure {00}), в - an axal texture of a wre-drawng (the symmetry axs lays n a plane of a projecton), г - an axal texture of a wre-drawng (the axs s perpendcular planes of a projecton). At presence of a texture of a plane {h k l } dfferent crystal grans are orented n space by a partcular natural fashon, and planes of crystal grans of one textural bulders settle down so, that the relevant crystallographc parameters are focused strctly equally (f dsperson msses) or wth small разориентировкой (f dsperson of a texture s). Now the crcle of projectons wll be covered wth poles non-unformly (fg.2, б-г). Concrete character of allocaton of poles wll depend on type of a texture, ts dsperson and, certanly, from for what t s concrete planes {h k l } s constructed gven PF. The descrpton of textures by means of a dstrbuton functon of orentatons If represents dv pluralty of volumes of all parts of a sample wth orentaton g n lmts of a dfferental of orentaton dg, and V - total amount of a sample, dv f ( gdg ), V = where f(g) represents a dstrbuton functon of orentaton of volumes of a sample (further we shall term t prmely as a dstrbuton functon of orentaton - DFO). Ths functon s spotted n

4 orentaton space φ, Φ, φ 2 euleran angles where to each pont the probablty of presence at a sample of volume wth orentaton g s put n conformty. Thus, f(g) completely and unequvocally features a texture of a materal. Functon f(g) s normed so, that f ( g) dg = (snce on sense DFO there s a densty functon of probablty). Functon f(g) can be vewed as functon of many varable (or functon n orentaton space), and the sense varable depends on an expedent of the assgnment of orentaton of a separate crystal gran. III. Conceptual statement of a problem. The purposes and problems of operaton. The basc hypotheses We wll consder n ths sub tem of the purpose and a problem of the gven operaton (at the gven stage of ts development), and also we wll formulate and we gve reason for the basc hypotheses of mathematcal model. The goal of the work: Buld-up consttutve model featurng evoluton of a mcrostructure of a materal n vew of the rotaton modes of plastcty, for varous processes of metal formng. The model should also allow to feature (qualtatvely and quanttatvely) process of formaton n a sample - polycrystal of preferred orentatons of crystallne axes of grans, that s formng the texture. Thus: ) At a tentatve stage vewng polycrystals wth face-centered cubc lattce of grans s planed; 2) Vewng s necessary possble а) Volume of a polycrystal wth a tentatve casual unform dstrbuton of orentatons (absence of a texture); б) Volume of a polycrystal wth some prmary texture; 3) Among possble plans of a loadng representng nterest, such metal formng processes, as rollng and wre-drawng. By vewng expedents of the descrpton of textures the specal attenton should be gven a gang ndependent varable wth whch help orentaton of separate monocrystals and expedents of transton from the descrpton of orentaton of a monocrystal to the descrpton of a texture,.e. the descrpton of allocaton of orentatons of some volume of crystal grans s featured. The majorty of the modern models of formaton of textures n polycrystals s vewed wth processes of turns of a polycrystal grans crystallne lattces at a mcrolevel, nlettng concepts of dslocatons of orentaton dscrepancy, vewng passage of systems of slp dslocatons through boundary of grans at plastc stran, and relatng the moments wth a dslocaton densty of orentaton dscrepancy and other boundary parameters. In ths paper the descrpton of textures formaton processes on mesolevel wll be carred out wthout obvous vewng a moton of dslocatons through boundares of grans, coordnatng occurrence of the moments gvng n turns of grans, wth dscrepancy of orentatons and shears on systems of slp (dslocatons featurng a moton nsde grans) for the next grans.

5 The basc hypotheses accepted by development of model, the followng are:. The two-dmensonal contnuum n whch to each pont of a contnuum the gang of coordnates n whch except for spatal coordnates orentaton of some allocated axs of a crystallne lattce of a gran concernng some beforehand partcular axs of a laboratory system of coordnates s taken nto account s put n conformty s consdered. 2. To nput parameters of model concern ntal allocaton of grans crystallographc system of coordnates orentatons (the most smple expedent - a unform dstrbuton), the sze of a gran d, the plan of a loadng, tme (or the generalzed tme). Target parameters - termnatng allocaton of orentatons of grans, stran tensors components and stresses at a level of representatve volume. 3. Cold plastc stran of a sngle-phase polycrystallne sample wth face-centered cubc lattce a lattce s consdered. 4. At model operaton the separate gran (monocrystal) wll be represented n the shape of the exact hexagon. 5. As rotatonal dsplacements of grans we shall understand here rotatonal dsplacements of crystallne lattces of grans as the whole, wthout fragmented structures formaton. 6. Effectve mechansm of plastc stran s dslocatons slp on planes of crystallographc systems of slp (one or to several of three possble, fg. 3). 7. At the descrpton of process of plastc stran of a polycrystal we would use one of models of plastcty of the polycrystal, basng on the relevant theory of plastcty for a monocrystal. Further as such theory we use model of Ln. Fg. 3 Dslocatons systems of slp n a two-dmensonal case for face-centered cubc lattce Hypotheses, 3, 4, 5 are accepted as the frst approach of model and further wll be omtted.

6 IV. Model of Ln for polycrystals In connecton wth that for the descrpton of process of an elasto-plastc deformng of representatve volume (accordng to a hypothess 7) t s offered to use one of physcal theores of plastcty, and n the gven operaton - model of Ln, we stop on t more n detal and we consder substantve provsons and algorthms of ths model. The basc hypotheses: Model of Ln s based on the followng basc hypotheses:. Velocty of stran of the polycrystallne unt s represented by the total of elastc and plastc velocty components: e p e p e p ε = ε + ε, e = e + e, de= de + de. (.) 2. The complete strans of separate grans of polycrystal ε (n) (n=,2,,n) (as well as veloctes of stran) are equal to the complete strans of the polycrystallne unt: ε( ) = ε, e( ) = e, n n (.2) ε ( ) = ε, e ( ) = e, n. n n 3. Plastc strans are sochorc, change of volume s determned by the frst elastc stran nvarant. 4. Plastc strans are carred out by shear on crystallographc systems of slp and submt to Shmde s law; hardenng s sotropc and s defned by the total shear on all actve systems of slp. Let's consder relatons of model for any way chosen gran. At presence of one actve system of slp k shear γ (k) carred out n t (k) s bound to plastc stran e p the followng velocty relaton: p ( k) ( k) e = M γ, Σ, (.3) k (k) (k) (k) (k) (k) (k) where M = ( n0 b0 + b0 n0 )- an orentaton tensor of system of slp, n0 - a unt vector 2 (k) (k) of a normal lne of a slp plane, b = b 0 - the normalzed Burgers vector. (k) b At actvzaton of several systems of slp velocty of a devator of plastc stran s determned by expresson: K p ( k) ( k) = e M γ, (.4) k= where K s the number of the actve systems of slp. Actvzaton of system of slp k s determned by performance n t of Shmde s law (k) (k) (k) s: n b = τ, (.5) 0 0 с where s - a devator of a stress tensor. Accordng to a hypothess 4 crtcal shft stresses n each system of slp are dentcal and depend on the total shear:

7 ( k) ( k) τc = τc = f dγ. (.6) k Veloctes of elastc strans n a gran can then be spotted a relaton: K e e e- M ( k) ( k) = γ. (.7) k= Wth the help (.7), usng the velocty shape of a Hooke s law, t s easy to receve expresson for velocty of stresses: K e ( k) ( k) σ = 2Ge = 2G e - M γ (.7 ) k = Veloctes of elastc shft strans n k system of slp are equal to: K ek ( ) ( k) ( j) ( j) γ = M : e - M γ, (.8) j= and the relevant shft stresses are defned as K ek ( ) ( k) ( j) ( j) τ( ) = 2Gγ = 2 GM : γ k e M. (.9) j= The formula (.9) can be coped n the velocty shape K ek ( ) ( k) ( j) ( j) τ( ) = 2G γ = 2 GM : γ k e M. (.9 ) j= Let's consder process of a loadng n space of strans. Durng the ntal moment e 0 = 0. At ncrease e (on quantty of ntensty) n the begnnng the deformng s carred out by an elastc scheme down to achevement n some system of slp (for example, wth the number ) the shft stress equals on the module to ntal crtcal stresses τ c0 = f(0). After that moment nelastc slde on system begns at ncremental stran e > 0. Thus durng each moment of a deformng the requrement of plastc yeldng should satsfy ( ) τ = τc = f ( γ ), or () () () 2 : f GM e M γ = f ( γ ) = γ (.0) γ At performance (.0) and ncremental stran e sngle shear s prolonged untl n some other system of slp (for example, 2) shft stresses τ (2) not acheve crtcal value τ c = f( γ ). From ths tme, ncrease e causes double slde on systems and 2, thus a requrement should satsfy: () (2) τ = τ = τc = f ( γ + γ2 ) or 2 2 () () () ( 2) () () f 2 G M : γ = 2 G : γ = f ( γ + γ2 ) = (γ + γ2) e M = M e M = (γ + γ2) (.)

8 Smlarly nvolvng n slde of 3-rd, 4-th and 5-th systems of slp s consdered. Thus on each of the actve systems of slp the requrement of fludty (Shmde s law) should satsfy. Guessng, that n each slp plane (along one drect) answer the "postve" and "negatve" drecton of slde varous numbers of systems of slp, t s possble to wrte down these requrements as follows: () K K ( k) 2 GM : e M γk = f dγk, =, K, (.2) k= k= Where, γ ( k ), dγ k are non-negatve, K=,2,, 5 - number of the actve systems of slp. For transton to model of a polycrystal one of known approaches to averagng (s used as probable varants t s possble to offer orentaton averagng, or averagng on volume, or (for stresses) on a surface of representatve volume). V. Mathematcal statement of a problem In the gven secton we buld concrete mathematcal relatons on whch the algorthm of numercal calculatons of mathematcal model wll be based. For the descrpton of orentaton of crystallographc lattces of grans the dstrbuton functon of orentatons (DFO) f (Ω) s used. In a two-dmensonal case orentaton s determned by one corner - a corner between the chosen axs of a crystallographc system of coordnates e and an axs of laboratory system of coordnates e ', ths corner we desgnate φ. Then the orentaton space wll be pluralty of all values φ n lmts ϕ 0, 3 (n vew of symmetry of a lattce), then f ( Ω ) = f ( ϕ). In case of a unform 3 dstrbuton of orentatons f ( ϕ) = const =. Fg. 4 the Arrangement of axes laboratory system of coordnates and axes crystallographc system of coordnates Further t s necessary to construct the relatons, allowng to expect changes of orentaton of grans (dependng on current orentaton and from a loadng). Accordng to the basc hypotheses of model, we shall count, that the moment developng a crystallne lattce of grans, appears on grans boundary at passage of dslocatons through boundary (at plastc stran) and t s caused by varous orentaton crystallographc system of coordnates of the next grans generator boundary. Radatng from ths, one of the arguments, quantty of the moment, the off-orentatons measure the next grans r = r( ϕ, ϕ2), where ϕ, ϕ 2 - orentatons of the next

9 grans s nfluencng. We mpose on a off-orentatons measure a normalze requrement r [0,], and at absence off-orentatons ϕ = ϕ ϕ2 = 0 r = 0, and ϕ = ϕ ϕ2 = r =. The 6 last follows from propertes of symmetry for face-centered cubc lattce lattces ϕ, ϕ2 0, 3. Snce n each of grans operates on three systems of slp, and, generally speakng, for the complete descrpton of orentaton of systems of slp of the fxed gran t s possble to set orentaton of the arbtrary system of slp ϕ 0, 3 (orentatons of the others then are equal 2 ϕ ± ) then as a off-orentatons measure t s possble to pck 3 r( ϕ, ϕ ) = sn3 ϕ = sn3ϕ ϕ. (2.4) 2 2 Now the prmary goal of model s defnton moments stresses µ and buld-up of moments yeldng functons f µ ( µ ) for the arbtrary gran wth orentaton ϕ *. Let's consder (accordng to a hypothess 8), that quantty of the moment ncpent as a result of nteracton of the gven system of slp of a concrete gran wth systems of slp next, s nfluenced also wth quantty of shear on the gven system of slp as a result of plastc stran.e. as already t was spoken earler, shall relate occurrence of such moment to a moton of dslocatons n grans on the relevant systems of slp. Then we shall present the component of the moment ncpent as a result of shear on to system of slp, effectve on volume wth some fxed orentaton ϕ *, on the part of volumes wth other orentatons (n that specfc case - the next grans) as 0 3 µ * = µ *( r*, γ *) = aγ * f( ϕ) sn3( ϕ * ϕ) dϕ, (2.5) where γ * - shft as a result of stran on system of slp of the allocated volume wth orentaton ϕ * (here t s necessary to note, that accordng to model of Ln shears on systems of slp occur as γ * as ( e a result of plastc ) ( p γ *, and elastc stran γ ) *, therefore here under can be meant both, and, dependng on what stran s mplemented at present loadngs n the gven system of slp of the gven gran), a - some dmensonless parameter, whch selecton of values s, generally speakng, a separate problem, a > 0, and further we shall consder, that the parameter and s nterlnked to an elastc modulus (and defnes by elastc response of a lattce of a gran as a result of the constraned shear), - the number of system of slp n volume wth orentaton ϕ *. The formula (2.5) we shall wrte down as well n the velocty shape: 0 3 µ * = µ *( r*, γ *) = a γ * f( ϕ) sn3( ϕ * ϕ) dϕ, (2.5 ) Quantty of shear n concrete system of slp s defned by the law of plastc yeldng or from the physcal theory of plastcty, and the complete velocty moments stresses then pays off as

10 n µ * = µ *. (2.6) = Let's buld now relatons of model, usng a formalsm of the theory of plastc yeldng: We consder, that veloctes of rotatonal dsplacements ϕ for the gven gran can be presented as the total elastc (reversble) and plastc (nonreversble) components: ( e) ( ) ϕ = ϕ + ϕ p, (2.7) We consder further, that elastc veloctes makng rotatonal dsplacements are lnearly nterlnked to moments stresses veloctes: ( e) ϕ = Αµ, (2.8) Thus for a plastc component of veloctes of rotatonal dsplacements (by analogy to the theory of plastc yeldng) we can wrte down measure of yeldng: f µ µ = µ, (2.9) ( ) 0 And at f µ ( µ ) < µ 0 plastc change of an angle of rotaton does not occur. Then t s possble to wrte down velocty of plastc rotatonal dsplacement wth the help moments yeld functon: p ϕ p = λ f µ ( µ ) µ ( p) where λ - coeffcent of proportonalty. ( ) ( ), (2.0) We wll be spotted now wth moments yeld functon f µ ( µ ). By vrtue of that plastc rotatonal dsplacements of crystallne lattces of grans wll begn not at any values µ *, and at excess of the certan crtcal value by t (n analogy to the begnnng of nonreversble plastc strans) t s necessary to wrte down f ( p) ( p) µ ( µ ) ϕ = λ, (2.) µ µ = µ 0 And crtcal value µ 0 we shall calculate as follows: µ 0 = βτ0, (2. ') Where 3 µ, N m/ m 0 - crtcal value of the moment on boundary, τ, Pа 0 - a yeld strength, β - the dmensonless droppng coeffcent, As physcal backgrounds for a select of measure of actvzaton as (2. ') t s possble to state the followng: Frst, by vrtue of a hypothess about the contrbuton of boundares of grans to embodyng the rotaton mode we consder, that the mechansm of gyraton s mplemented due to plastc shears on boundares of grans; Second, crtcal stresses for shft stran on boundary are lower, than n "nteror" of a monocrystal, by vrtue of a boosted free energy of dslocatons n boundares; Let's explan measure of actvaton (2. ') on a prme example of a two-dmensonal gran of the exact shape. At monotonously ncremental stran, as a result of nteracton of the next grans on boundares, on a vewed gran operate ncremental moments stresses, wthn the framework of

11 model (and accordng to deology of Cosserat contnuum) grans carred to all volume (.e. consdered as the dstrbuted on volume). Thus resstance to rotatonal dsplacement on the part of the next grans the same as also the moment µ *, grows from zero and up to the lmtng value µ * 0 whch s defned from a requrement, that n each pont of boundary the lmtng stresses of shear βτ 0 (occurrence of droppng coeffcent β s explaned above) s acheved. Durng development of model the hypothess expressed necessty of enterng nto measure of a standard of a dverson of the shape of a gran from the deal shape (f to represent grans ellpsods as such a measure the relaton half-axles s possble to choose) also, however wthn the framework of the gven, smplfed varant of operaton of necessty of ntroducton of such coeffcent there s no vald a hypothess about the dentcal szes and the shape of grans. Introducton of coeffcent of asymmetry, most lkely, s requred at the further complcaton of model (by vewng grans of the varous sze and geometry). Let's consder a procedure of carryng out of numercal calculatons by means of mathematcal relatons of model:. We shall consder the two-dmensonal representatve volume consstng of a plenty (approxmately 000) separate grans - monocrystals, the gven the Cartesan axals (renumbered accordng to some rule: to a gran the double number, the frst - the number horzontal, second - a vertcal stratum, and a gran of even and odd stratums as on a vertcal s approprated, and across based be relatve each other on half of sze of a gran by vrtue of that grans completely area of model operaton), and to each of grans we shall put n conformty a certan orentaton ( j ϕ ),, j- the number of a current gran, ( j) ϕ [0, ) (fg. 5). 3 ( j) ( j) As orentaton ϕ we shall understand orentaton of one of systems of slp ( ϕ -a corner between a trace on a plane of model operaton of the system of slp nearest to an abscssa axs and most ths axs; thus t s not taken nto account drectons of slde n ths ( j) crystallographc system), such, that ϕ [0, ). Then, obvously, the postve drectons 3 ( j) 2 ( j) 2 of other systems of slp can be wrtten down as ϕ +, ϕ 3 3

12 Fg. 5 the Plan of numberng of grans n representatve volume, by arrows orentatons of systems of slp are desgnated 2. By vrtue of that n model for the descrpton of processes of an elasto-plastc deformng model of Ln s appled, we conduct process on strans, ncludng ther dentcal to each gran. On each step of algorthm we shall gve strans some small (but fntesmal) a ncrement e e2 e= e* =, (3.) e2 e22 Here e= e* means, that the ncrement of stran s dentcal to all grans wth the arbtrary orentaton ϕ *. 3. On ntal teraton we shall set values of crtcal shft stresses τ (0) c and the law of ( k) ( k) (0) ( k) τ * = τ * τ, γ *, where ( k γ ) * - the total shear on systems of slp n a hardenng c c ( c ) gran ϕ * on a step of a loadng k, of hardenng we shall set as γ 3 ( k) ( k) γ = * = *, - the system of slp number. The law τ * = τ + α γ *, (3.2) ( k) (0) ( k) c c E where E - an elastc modulus of a materal, α [ 0,] - droppng coeffcent. 4. Further on a settlement step k we shall calculate necessary parameters of an elasto-plastc loadng, usng for ths relaton models of Ln (secton 3.). For transton from records of expressons n veloctes to record for ncrements we shall use the plan of ntegraton of Euler. Then we shall receve the followng relatons: For the actve systems of slp Shmde s law as (.5) s carred out, and shears on the actve systems of slp are defned by a prncple of a mnmum of shear of Taylor (.3): γ * γ, (3.3)

13 =, 2 - the number of the actve system of slp. Besdes for the actve systems of slp the tangental stresses n them (for the gven gran on the gven settlement step) are equal to the relevant stresses of shear ( k) * ( k) τ * () = τc. For systems of slp on whch Shmde s law s not carred out, the ncrement of s calculated under the formula (.8): γ = - K e () ( j) M : e M γ j, (3.4) j= K - The number of the actve systems of slp, and shft stresses are equal these systems of slp, accordng to (.9) K () ( j) τ () = 2 GM : γ e M j. (3.5) j= The ncrement of a stress tensor σ *( k ) n a gran ϕ * on a settlement step k s defned accordng to a Hooke s law: K ( e) ( j) σ *( k) = 2G e( k) = 2G e M γ j. (3.5 ) j= Thus, on a settlement step k t s possble to calculate ncrements of shears on systems of slp, thus, usng the relevant values for shears from the prevous step to spot new values of shears (and the total shear): 3 3 ( k) ( k) ( k ) ( k ) γ * γ * ( γ = = * + γ *), (3.6) = = ( k) ( k) (0) ( k) and also to count crtcal values of shears τc * = τc *( τc, γ *). 5. Now, usng relatons of secton 3.2, we shall expect ncrements the moment stresses effectve on a gran as a result of nteracton of system of slp of a gran ϕ * wth systems of slp of the next grans, and the complete values of the moments (2.9, 2.0): 6 ( k) ( k) ( m) µ * = α γ * sn3( ϕ* ϕ ), (3.8) 6 m= 3 ( k) ( k) µ * = µ *, =. (3.9) ( k) ( k ) ( k) µ * = µ * + µ * 6. In conformty wth a formalsm of the theory of plastc yeldng, a ncrement of a offorentatons corner n grans we shall present as (see 2.5): ( ) ( )( ) ( )( ) ϕ k * = ϕ k e * + ϕ k p *, (3.0) where ( k)( e) ( k) ϕ * = A µ *, (3.) and f ( k)( p) ( p) ( ) * µ µ ϕ = λ. (3.2) µ µ µ = 0

14 7. Now, for fnal termnatng a settlement step, we calculate the complete values of corners offorentatons ( k) ( k ) ( k) ϕ * = ϕ * + ϕ *. (3.3) 8. Thus, n the end of each settlement step shears on systems of slp of each gran, bulders of a stress tensor of a gran, the moment, the dstrbuted on a gran and value of a corner offorentatons each gran are defned. Havng appled further procedure of averagng of stresses on orentaton volume: N ( k ) j Vj σ j V k = σ =, ( k ) where σ j - a ncrement a bulder of a tensor of strans the gran k whch have been wrtten down n laboratory system of coordnates, we shall descrbe tensely - deformed a state of representatve volume. VI. Results of operaton Let's gve values of parameters of the model used n numercal experments: N = 000, G = , ν = 0.36, τ = , b = R , = 4 * 0, α = 0.005, where G, ν - a shear modulus and Posson coeffcent, b - the module of Burgers vector, R * - the sze of a gran, N - number of grans n representatve volume, τ 0 - an ntal yeld strength of a materal, a, β, α - parameters of model. b Па Па м The data correspond to parameters for copper. We consder results of evaluatons for a case of stran of an axal compresson along an axs e of a laboratory axs of coordnates representatve volume, thus we vew strans of ntensty from 5 up 0 to 0,85 :

15 Fg. 6 Intal allocaton of orentatons (a fragment of smulated area, arrows show orentaton of grans ϕ 0, ) 3 As target parameters of model termnatng orentatons of grans of representatve volume act. To consder dynamcs of change of orentatons of grans dependng on ntensty of strans, we shall observe of change of an off-orentatons average angle grans N ϕ = ϕ N = And mean square devaton of off-orentatons corners N D( ϕ) = ( ) 2 2 ϕ ϕ N = Then n requrements of the gven numercal experment we receved: Fg. 7 Dependence of an off-orentatons average angle of grans n representatve -2 volume ϕ from ntensty of strans ε ( a = 0,5 0, = 0,7 ) β

16 Fg. 8 Dependence mean square devaton off-orentatons of grans n -2 representatve volume D( ϕ ) from ntensty of strans ε ( a = 0,5 0, β = 0,7 ) Fg. 9 Dependence of an off-orentatons average angle of grans n representatve -2 volume ϕ from ntensty of strans ε ( a = 0,2 0, = 0, 4 ) β Fg. 0 Dependence mean square devaton off-orentatons of grans n -2 representatve volume D( ϕ ) from ntensty of strans ε ( a = 0,2 0, β = 0, 4 )

17 In experments whch results are mapped on fg. 7-8 and fg. 9-0, the behavour of representatve volume surveyed at varous values of parameters of model a and β. Havng carred out the bref analyss of the submtted results, t s possble to draw a deducton, that n the gven example we have the rght to speak about texture formaton wthn the framework of representatve volume snce at major strans there are preferred orentatons, n ths case orentatons ϕ = 0. Occurrences of such preferred orentaton n ths case could be expected radatng from symmetry of a loadng (only lengthways e ) and that fact, that orentaton crystallographc system of coordnates ( j) of a gran consders orentaton of system of slp, for whch ϕ [0, ). In ths case, obvously, the 3 most favourable to grans wll be the orentaton contermnous to an axs of a loadng owng to what turns at whch there s a relaxaton moments stresses on boundares begn wth some moment and. In ths case t s such turns at whch there s, frst, a grans off-orentaton mnmzaton, and second, systems of slp symmetrze n relaton to man axes of a loadng. Fg. Dependence of ntensty of stresses σ n representatve volume -2 from ntensty of stran ε ( a = 0,5 0, β = 0,7 ), coeffcent of hardenng 0,0 α = Fg. 2 Dependence of ntensty of stresses σ n representatve volume -2 from ntensty of stran ε ( a = 0,5 0, β = 0,7 ), coeffcent of hardenng α = 0,0

18 Let's note also features of behavour of representatve volume on fg. -2. Frst, we shall note, that on dagrams the ste of only elastc stran whch yet does not gve n essental change of angles of rotaton s dstnctly traced. Then slde of dslocatons on systems slde begns, however durng even some contnuance of a loadng of plastc rotatonal dsplacements does not occur. Really, on ths "transton" ste on boundares of grans stresses, the bound wth dscrepancy of crystallographc systems of slp of neghbours collect, but owng to "small" shears these stresses are stll nsuffcent for the begnnng nonreversble plastc rotatonal dsplacements. At the further magnfcaton of strans the ncreasng number of systems of slp of varous grans, and gradually process of rotatonal dsplacements n all grans enter process of plastc stran becomes nonreversble. At a plastc relaxaton owng to rotatonal dsplacements there s a slope of stresses on a ste of plastcty of the σ ε dagram. The establshment of grans or conglomerates of grans wth whch naturally nelastc turns begn s nterestng also, on Fg. 3-6 color marks stes n whch there are such turns of crystallne lattces of grans, at varous stages of stran (values of parameters correspond to experment on fg. 9-0) Fg. 3 the General pattern of turns of grans, at stran ε = 0,05 It s possble to note, that turns of crystallne lattces of grans begn wth those grans whch ntally possessed the most "unproftable" orentatons,.e. the delta crcut of whch systems of slp has been canted concernng an axs of squeezng e on a corner of the order, thus at ntal stages 6 of a deformng t s not enough such grans, and turns n them do not gve n essental change of medal value of a corner off-orentatons and mean square devaton.

19 Fg. 4 the General pattern of turns of grans, at stran ε = 0, Fg. 5 the General pattern of turns of grans, at stran ε = 0, 25 At ncrease of strans "centres" of turns start to gve n the rotaton moton the next grans besdes enter process of turns of a gran whch orentaton essentally dffers from 6, and at major strans practcally all grans (except for possessng) start to be developed by the best orentaton plastcally (fg. 6).

20 D ϕ ( ) Fg. 6 the General pattern of turns of grans, at stran ε = 0, 45 Besdes at some values of parameters of model on a curve growth mean square devaton ε s observed. Ths phenomenon as t s supposed, s possble to explan just process of "swng" at whch grans wth whch gyraton begns, "part forcbly" the neghbours because of what n system the degree of dsorder (fg. 7) rases Fg. 7 Dependence mean square devaton off-orentatons of grans -2 representatve volume D( ϕ ) from ntensty of strans ε ( a = 0,35 0, β = 0, 4 ) Further t s necessary to work procedure of dentfcaton of model,.e. checkout of effects of model operaton on физичность and conformty to expermental data. Unfortunately, operaton on dentfcaton at the moment s not completed yet, however already now t s possble to receve some rules for rejecton of the results whch are not satsfyng expermental data on the unaxal loadng. In partcular, the lterature contans the data, that дифрактометрическими by examnatons set, that formaton of a texture s found out at a precptate already at 0-20 % (but not less), and formaton of a texture s prolonged up to degrees of stran about %. Operaton on search of such parameters of model at whch the data receved as a result of numercal experments would well be coordnated wth expermental data s at the moment carred out.

21 It s necessary to note as well that fact, that the results ncremented wth use of model, strongly dffer dependng on values of parameters of model α, β, and even at small changes of values of parameters results sometmes consderably dffer from each other, therefore at the gven stage t s mpossble to speak about stablty of model n relaton to varatons of parameters. Interestng that fact also s, that, most lkely, coeffcents α, β are not ndependent snce effects, more or less compounded wth expermental data, are ncremented at such gangs of values α, β, that product α β has the same order. Ths work s done n Perm State Techncal Unversty at Char of Mathematcal Modelng of Systems and Processes to enjoy the support of Russan Found of Fundamental Researches (РФФИ, grant )