Equivalent Strain in Simple Shear Deformations

Transcription

1 Equivalent Strain in Simple Shear Deformation Yan Beygelzimer Donetk Intitute of Phyic and Engineering The National Academy of Science of Ukraine Abtract We how that imple hear and pure hear form two group of tranformation with different propertie. The equivalent train i viewed a an external control parameter of the deformation proce at low homologou temperature. The von Mie train atifie group-theoretic propertie of both group, upporting it ue for meauring the equivalent train. The Hencky train, on the other hand, doe not atify the imple hear group propertie, implying that it i not appropriate for meauring the equivalent train in imple hear. The paper alo propoe a hypothei explaining the abence of metal hardening in large imple hear deformation. Thi hypothei explain why excluding rotation a precribed by the finite train theory i not valid.. Introduction Fueled by advance in evere platic deformation [], there ha been much recent dicuion about computing the equivalent train e in imple hear. Since the increment of equivalent train can be defined via the increment of platic work, the author of [ - 5] argue for uing the von Mie train e M =, () where i the hear train. Onaka [6,7], on the other hand, argue that one mut ue the Hencky relation e + H = ln + () 4 The jutification given by Onaka i the need to exclude rotation from the deformation-gradient tenor in large imple hear deformation. There i another line of reaoning that led ome author (ee, e.g., [8]) to formula (). Starting with Ludwik [9], it wa widely believed that the ame value of e guarantee eentially the ame level of hardening, regardle of the deformation mode. Seen in thi light, the etimate () i unreaonably high if one compare the hardening achieved by imple hear and that achieved by other deformation mode (e.g., elongation). In [0], the equivalent train i defined a a meaure of hardening. Thi definition gave an expreion for the equivalent train in imple hear predicting that e aturate at large value of. Thi aturation i in agreement with an empirically oberved fact that, in large deformation caued by high preure torion, material reach a tationary tructure without hardening [ - ]. When viewed a a meaure of hardening, the equivalent train become a phyical (intead of

2 a geometric) parameter that depend on the material propertie. Here we take a ynergetic approach [4], viewing the equivalent train a an external control parameter of the deformation proce. Uing group-theoretic propertie of geometric tranformation [5] and the principle of additivity, we how that the equivalent train of imple hear mut be linear in. Thi approach give evidence in favor of uing the von Mie train for computing the equivalent train in imple hear. The paper alo propoe a hypothei explaining the abence of metal hardening in large deformation caued by imple hear. Thi hypothei explain why excluding rotation a precribed by the finite train theory i not valid.. The von Mie train and group theoretic propertie of imple and pure hear We view the equivalent train a an external calar control parameter determining the deformation proce at low homologou temperature. We will how that, for imple hear, uch a control parameter mut be a linear function in hear train. For thi, conider imple hear a a geometric tranformation defined by relation x = SX, () where X and x are the initial and the final coordinate of a point, repectively; and S = 0 0 (4) 0 0 i the tranformation operator. It i eaily verified that the et of all imple hear tranformation along a fixed hear direction atifie the four axiom of group theory [5]. Indeed, conider two conecutive tranformation (4) with hear train and repectively: It i clear that S = S 0 0 = 0 0 ( ) and S 0 0 = S( ) = (5) S S = = = S( + ) (6) Thu a compoition of two imple hear tranformation along the ame direction i a imple hear tranformation along that direction, and the et defined by (4) atifie the axiom of cloure. S S S = S S ), Similarly, it i eay to ee that (4) atifie the aociativity axiom, ( ( ) ( ) S contain the identity element I = S( 0), and that for every tranformation ( ) invere S ( ) = S( ) uch that S ( ) S( ) = I. S it contain the

3 We will aociate each tranformation S ( ) with the value of it characteritic control parameter, the equivalent train e = e ( ), where the ubcript tand for imple hear. Conider two conecutive imple hear tranformation S ( ) and S ( ) with correponding equivalent train e = ( ) и e = ( ). (7) e e It follow from (6) that the total hear i characterized by the equivalent train ( + ) e = e. (8) Σ On the other hand, the equivalent train i additive [6]; thi mean that Plugging in expreion (7) and (8), we get e + Σ = e e (9) ( ) e ( ) = e ( + ) e (0) + A olution to thi functional equation i a linear function (ee, e.g., [7]) e = C, () for ome contant C. To match the von Mie value at low equivalent train [6], e ( ) = e M = yielding C =. The derivation of relation () doe not require to be mall, and o e = i valid for any value of, agreeing with equation (). Thi linear dependence of the equivalent train from follow from group-theoretic propertie of imple hear and the additivity of the equivalent train. A imilar argument give a logarithmic dependence in the cae of contant-volume pure hear. Indeed, the tranformation operator in thi cae ha the form λ 0 ( ) P λ = 0 λ 0, () 0 0 where λ i the relative elongation. One can eaily verify that the et of pure hear tranformation form a group whoe element atify the relation P λ P λ = P λ () ( ) ( ) ( ) λ Unlike the analogou relation for imple hear (6), it i nonlinear in the ene that the parameter of the total tranformation i the product intead of the um of the two component. It i thi difference in group-theoretic propertie of imple and pure hear that lead to a different dependence of the equivalent train from in thee procee. It follow from relation () and the additivity property that the equivalent train e p (where p refer to pure hear ) atifie equation

4 ( λ ) e ( λ ) e ( λ ) e + = (4) p p p λ intead of equation (0) a in the cae of imple hear. A olution of thi functional equation i the logarithm function (ee, e.g., [7]), e p λ = K ln, (5) ( ) λ where K i a contant. Thi dependence of the equivalent train on the relative elongation correpond to an expreion for the von Mie train, which in the cae of pure hear i determined by the following formula [6]: e M ( λ) = ln λ (6) In thi cae, the von Mie train again follow from group-theoretic propertie and the additivity principle.. Dicuion The previou ection argued for uing the von Mie train in both pure and imple hear, without relying on any model of the material and without making any aumption on the hear train. Thu the von Mie train can be ued a a control parameter not only in the cae of iotropic bodie where de M define the incremental work, but alo in other etting for example, in aniotropic bodie [6] or in the cae of train gradient platicity [8]. Two quetion arie in light of Section and the dicuion [-8]:. Why doe train hardening in imple hear differ ubtantially from that in pure hear, for the ame value of the von Mie train, when the von Mie train become ufficiently large?. Why doe excluding rotation, a dictated by the finite train theory, lead to a wrong reult? To anwer thee quetion, we will firt try to anwer the contrapoitive of quetion : A polycrytal pecimen repreent a ytem with multitudinou degree of freedom. So why doe a ingle calar control parameter determine hardening and the average grain ize (at relatively mall equivalent train and low homologou temperature)? Thi i urpriing, like other example of imple behavior in complex ytem, uch a turbulent flow in liquid. Kolmogorov [9] hypotheized the tructural elf-imilarity of turbulent flow and it defining caling law. Barenblatt [0] give other example of imple behavior of complex ytem related to their elf-imilarity. We believe that the anwer to the contrapoitive lie preciely in the elfimilarity of metal tructure evolution during deformation. Such elf-imilarity ha been oberved experimentally for ufficiently large train (ee, for example, []). It i alo the elf-imilarity that i reponible for the power low σ = f e (7) ( ) of the equivalent tre σ from the equivalent train e, repeatedly confirmed in experiment

5 (ee, e.g., [6]). In other word, the tre-train curve i an expreion of caling behavior, which, according to [0], i common to all elf-imilar procee. We will how that f () e i an power low during the elf-imilar tage of microtructure evolution. Conider three conecutive tate of the ytem, pecified by ( ) ;σ e, ( e ), and ( ) ;σ e ;σ. Since the evolution of metal tructure i elf-imilar, the etimate (7) i invariant with repect to the choice of unit for meauring e. Let u firt chooe e a the unit meaure. Then f () e mut have the following form: e () f e = σ ϕ, (8) e where ϕ () =. For tate ( ) ;σ e and ( ) e ;σ, according to (8), we get σ e = ϕ, σ e (9) σ e = ϕ, σ e (0) If we chooe e a our unit meaure, a imilar argument will lead to the following relation: σ e = ϕ. σ e () Multiplying (9) and () and comparing with (0), we get e e e ϕ = ϕ ϕ e e e () Thi implie that ϕ ( x) atifie the following equation ϕ( x x ) = ϕ( x ) ϕ( ), () x A olution to uch an equation i an power low (ee, for example, [7]): n ( x ) = x ϕ, (4) where n i a parameter. Thu n σ = Ae, (5) where A i a parameter. Grain refinement can be viewed a recurive grain ubdiviion []. Kolmogorov [] invetigated a fairly general model of recurive particle ubdiviion, howing that elf-imilar tructure emerge at a certain tage of thi proce if the ubdiviion mechanim i contant and cale-invariant. Other reearch [, 4-6] howed the univerality and cale-invariance of grain refinement during platic deformation. Thi lead to the concluion that a long a the

6 deformation mechanim i unchanged, metal tructure will evolve in a elf-imilar manner, reulting in a univeral power low tre-train curve. Thu a deviation from thi curve mut be related to a change in the deformation mechanim. Paper [0,7] make a cae for a hypothei that a certain percolation mechanim et in during imple hear. Thi mechanim explain the lack of deformation hardening (quetion above) and the formation of a tationary microtructure oberved experimentally [-], a well a a number of other phenomena in large imple hear deformation [8]. The anwer to quetion, we believe, i related to the ame percolation mechanim. According to [7], thi mechanim ha a relaxation nature, periodically relieving internal tre via mall rotation of grain cluter. Thi proce allow the material to repair it tructure. Thi way, the deformed material guarantee a tationary character of tranformation (4) expreed uing property (6). Large imple hear deformation are realized a a um of mall independent tep, explaining the pecial character of rotation in thi etting. 4. Concluion Simple and pure hear deformation form two group of geometric tranformation with different propertie. The von Mie train atifie group-theoretic propertie of both imple and pure hear. The Hencky train, on the other hand, doe not atify the propertie of the imple hear group. Thi bring evidence that the von Mie train i the correct meaure of the equivalent train in imple hear. Since the von Mie train i jutified uing group-theoretic argument, without replying on any model of the material, thi point to the applicability of the von Mie train a a control parameter not only in iotropic bodie but in other etting a well. Reference [] R. Z. Valiev, I. Sabirov, A. P. Zhilyaev and T.G. Langdon, JOM: The Journal of the Mineral, Metal, and Material Society, Vol. 64, No. 0 (0) p [] S. C. Shrivatava, J. J. Jona and G. Canova, Journal of the Mechanic and Phyic of Solid, Vol. 0, Iue (98) p [] J. J. Jona, C. Ghoh and S. Shrivatava, Material Tranaction, Vol. 5, No. 9 (0) p [4] S. Shrivatava, C. Ghoh and J. J. Jona, Philoophical Magazine, Vol. 9, No. 7 (0) p [5] J. Jona, C. Ghoh, V. Baabe and S.Shrivatava, Philoophical Magazine, Vol.9,No 6 8 (0)p. 8. [6] S. Onaka, Philoophical Magazine Letter, Vol. 90, No. 9 (00) p. 6. [7] S. Onaka, Philoophical Magazine, Vol. 9, Iue 8 (0), p [8] I. Saunder and J. Nutting, Met. Sci., Vol. 8 (984) p [9] P. Ludwik, Element der technologihen Mechanik, Berlin, 909. [0] Y. Beygelzimer, R. Z. Valiev and V. Varyukhin, Material Science Forum, Vol (0) pp

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