Yield strength of a material pre-processed by simple shear

Modern techniques of severe plastic deformation used as a means for grain refinement in metallic materials rely on simple shear as the main deformation mode. Prediction of the mechanical properties of the processed materials under tensile loading is a formidable task as commonly no universal, strain...

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Datum:	2015
Hauptverfasser:	Chen, C., Beygelzimer, Y., Toth, L.S., Estrin, Y., Kulagin, R.
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Sprache:	English
Veröffentlicht:	Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України 2015
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spelling	irk-123456789-1074062016-10-20T03:02:38Z Yield strength of a material pre-processed by simple shear Chen, C. Beygelzimer, Y. Toth, L.S. Estrin, Y. Kulagin, R. Modern techniques of severe plastic deformation used as a means for grain refinement in metallic materials rely on simple shear as the main deformation mode. Prediction of the mechanical properties of the processed materials under tensile loading is a formidable task as commonly no universal, strain path independent constitutive laws hold. In this paper we derive an analytical relation that makes it possible to predict the mechanical response to uniaxial tensile loading for a material that has been pre-processed by simple shear and presents a linear strain gradient in it. A facile recipe for mechanical tests on solid bars required for this prediction to be made is proposed. As a trial, it has been exercised for the case of commercial purity copper rods. The results of the derivation of the true stress-strain curve for large tensile deformation of copper are presented. The method proposed is recommended for design with metallic materials that underwent preprocessing by simple shear. Для прогнозу властивостей субмікрокристалічних металів, отриманих методами інтенсивних пластичних деформацій, необхідно знати напруження при одновісному пластичному розтягуванні після обробки великим простим зсувом (деформація зсуву більше 10). Механічні властивості матеріалів при таких непропорційних шляхах деформування вивчають в основному за допомогою трубчастих зразків. Через втрату стійкості трубок при крученні такі експерименти можливі лише для малих пружньопластичних деформацій, що не перевищують кількох відсотків. У статті запропоновано й обґрунтовано метод визначення напруги течії матеріалу, попередньо обробленого великим простим зсувом. Метод заснований на двох стандартних випробуваннях: крученні з вільними торцями й одновісному розтягуванні. Отримано співвідношення, що дозволяє по напрузі течії неоднорідного зразка, попередньо підданого крученню, знайти напругу пластичного розтягування його поверхневого шару з певною деформацією простого зсуву. Шляхом простих аналітичних оцінок показано, що пружні залишкові напруги першого роду, які виникають після розвантаження зразка, попередньо підданого крученню, практично не впливають на межу тікучості при розтягуванні. Для прогноза свойств субмикрокристаллических металлов, полученных методами интенсивных пластических деформаций, необходимо знать напряжение их течения при одноосном растяжении после обработки простым сдвигом большой величины (деформация сдвига более 10). Механические свойства материалов при таких непропорциональных путях деформирования изучают в основном с помощью трубчатых образцов. Из-за потери устойчивости трубок при кручении такие эксперименты возможны лишь для малых упругопластических деформаций, не превышающих нескольких процентов. В статье предложен и обоснован метод определения напряжения течения материала, предварительно обработанного большим простым сдвигом. Метод основан на двух стандартных испытаниях: кручении со свободными торцами и одноосном растяжении. Получено соотношение, позволяющее по напряжению течения неоднородного образца, предварительно подвергнутого кручению, найти напряжение пластического растяжения его поверхностного слоя с определенной деформацией простого сдвига. Путем простых аналитических оценок показано, что упругие остаточные напряжения первого рода, возникающие после разгрузки образца, предварительно подвергнутого кручению, практически не влияют на предел текучести при растяжении. 2015 Article Yield strength of a material pre-processed by simple shear / C. Chen, Y. Beygelzimer, L.S. Toth, Y. Estrin, R. Kulagin // Физика и техника высоких давлений. — 2015. — Т. 25, № 3-4. — С. 133-140. — Бібліогр.: 16 назв. — англ. ос. 0868-5924 PACS: 62.20.–x http://dspace.nbuv.gov.ua/handle/123456789/107406 en Физика и техника высоких давлений Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Modern techniques of severe plastic deformation used as a means for grain refinement in metallic materials rely on simple shear as the main deformation mode. Prediction of the mechanical properties of the processed materials under tensile loading is a formidable task as commonly no universal, strain path independent constitutive laws hold. In this paper we derive an analytical relation that makes it possible to predict the mechanical response to uniaxial tensile loading for a material that has been pre-processed by simple shear and presents a linear strain gradient in it. A facile recipe for mechanical tests on solid bars required for this prediction to be made is proposed. As a trial, it has been exercised for the case of commercial purity copper rods. The results of the derivation of the true stress-strain curve for large tensile deformation of copper are presented. The method proposed is recommended for design with metallic materials that underwent preprocessing by simple shear.
format	Article
author	Chen, C. Beygelzimer, Y. Toth, L.S. Estrin, Y. Kulagin, R.
spellingShingle	Chen, C. Beygelzimer, Y. Toth, L.S. Estrin, Y. Kulagin, R. Yield strength of a material pre-processed by simple shear Физика и техника высоких давлений
author_facet	Chen, C. Beygelzimer, Y. Toth, L.S. Estrin, Y. Kulagin, R.
author_sort	Chen, C.
title	Yield strength of a material pre-processed by simple shear
title_short	Yield strength of a material pre-processed by simple shear
title_full	Yield strength of a material pre-processed by simple shear
title_fullStr	Yield strength of a material pre-processed by simple shear
title_full_unstemmed	Yield strength of a material pre-processed by simple shear
title_sort	yield strength of a material pre-processed by simple shear
publisher	Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/107406
citation_txt	Yield strength of a material pre-processed by simple shear / C. Chen, Y. Beygelzimer, L.S. Toth, Y. Estrin, R. Kulagin // Физика и техника высоких давлений. — 2015. — Т. 25, № 3-4. — С. 133-140. — Бібліогр.: 16 назв. — англ. ос.
series	Физика и техника высоких давлений
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first_indexed	2025-07-07T19:54:59Z
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fulltext	Физика и техника высоких давлений 2015, том 25, № 3–4 © Cai Chen, Yan Beygelzimer, Laszlo S. Toth, Yuri Estrin, Roman Kulagin, 2015 PACS: 62.20.–x Cai Chen1,2, Yan Beygelzimer1,3, Laszlo S. Toth1,2, Yuri Estrin4,5, Roman Kulagin6 YIELD STRENGTH OF A MATERIAL PRE-PROCESSED BY SIMPLE SHEAR 1Laboratory of Excellence on Design of Alloy Metals for low-mаss Structures (DAMAS), Université de Lorraine, Metz, France 2Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux (LEM3), UMR 7239, CNRS / Université de Lorraine, F-57045 Metz, France 3Donetsk Institute for Physics and Engineering named after O.O. Galkin, National Academy of Sciences of Ukraine 4Centre for Advanced Hybrid Materials, Department of Materials Engineering Monash University, Clayton VIC 3800, Australia 5Laboratory of Hybrid Nanostructured Materials, NITU MISIS, Moscow, Russia 6Institute of Nanotechnology (INT), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, Eggenstein-Leopoldshafen 76344, Germany Received October 20, 2015 Modern techniques of severe plastic deformation used as a means for grain refinement in metallic materials rely on simple shear as the main deformation mode. Prediction of the mechanical properties of the processed materials under tensile loading is a formidable task as commonly no universal, strain path independent constitutive laws hold. In this paper we derive an analytical relation that makes it possible to predict the mechanical response to uniaxial tensile loading for a material that has been pre-processed by simple shear and presents a linear strain gradient in it. A facile recipe for mechanical tests on solid bars required for this prediction to be made is proposed. As a trial, it has been ex- ercised for the case of commercial purity copper rods. The results of the derivation of the true stress-strain curve for large tensile deformation of copper are presented. The method proposed is recommended for design with metallic materials that underwent pre- processing by simple shear. Keywords: strain hardening, torsion, tension, strain gradient, strain path change, copper Для прогнозу властивостей субмікрокристалічних металів, отриманих методами інтенсивних пластичних деформацій, необхідно знати напруження при од- новісному пластичному розтягуванні після обробки великим простим зсувом (де- формація зсуву більше 10). Механічні властивості матеріалів при таких непро- порційних шляхах деформування вивчають в основному за допомогою трубчастих Физика и техника высоких давлений 2015, том 25, № 3–4 134 зразків. Через втрату стійкості трубок при крученні такі експерименти можливі лише для малих пружньопластичних деформацій, що не перевищують кількох відсотків. У статті запропоновано й обґрунтовано метод визначення напруги течії матеріалу, попередньо обробленого великим простим зсувом. Метод засно- ваний на двох стандартних випробуваннях: крученні з вільними торцями й од- новісному розтягуванні. Отримано співвідношення, що дозволяє по напрузі течії неоднорідного зразка, попередньо підданого крученню, знайти напругу пластичного розтягування його поверхневого шару з певною деформацією простого зсуву. Шля- хом простих аналітичних оцінок показано, що пружні залишкові напруги першого роду, які виникають після розвантаження зразка, попередньо підданого крученню, практично не впливають на межу тікучості при розтягуванні. Ключові слова: деформаційне зміцнення, кручення, розтягування, градієнт дефор- мації, зміна шляху деформування, мідь 1. Introduction Processing of metals by simple shear is a good way to improve their mechani- cal characteristics. This comes to bearing especially in the production of ultrafine grained materials by severe plastic deformation (SPD). SPD techniques, such as high-pressure torsion, high-pressure tube twisting, equal-channel angular pressing, twist extrusion, and shear extrusion have one thing in common – they all are based on simple shear [1,2] and almost all of them present a strain gradient in them. Products fabricated by these techniques are often designed for structures that operate under extremely large tensile loads. Design with such materials there- fore requires a reliable tool for predicting the flow stress for metals, which under- went processing by gradient simple shear of a given magnitude. The large strain behavior of metals is usually studied in torsion of cylindrical bars because very large strains can be readily achieved in torsion. Indeed, during tensile testing – which is the most commonly employed characterization technique – the uniform deformation is limited because of early necking. However, the mechanism of strain hardening is quite special for torsion because of the small number of the operating slip systems that lead to smaller equivalent stresses for torsion compared to tension or compression [3,4]. This presents difficulties in the characterization of the material behavior at large strains. The problem of construction of stress-strain curves for torsion of solid bars has been resolved by Fields and Backofen [5] who established a formula for obtaining the flow stress at the outer radius of the twisted bar. However, no such formula is available for tension of a sample with a strain gradient, particularly when a bar is tested in ten- sion after being pre-twisted in torsion. This problem is resolved in the present paper. The importance of such testing is that stress-strain curves can be obtained for tension for very large strains, up to the same strains as in torsion, by tension of pre-twisted bars. For this purpose only thin-walled tubes were pre-twisted so far where the strain gradient can be neglected [6–12]. However, the maximum plastic strain is very limited in torsion of such tubes, which is not the case for the torsion of solid bars. In the following we first present the theoretical basis and then show that the role of the residual stresses, which are inherent in gradient structures, can be ne- Физика и техника высоких давлений 2015, том 25, № 3–4 135 glected at large strains. Finally, the new technique is applied for the large strain torsion of copper bars. 2. Theoretical basis We consider a solid cylindrical bar twisted in large strain torsion. It can be shown with using the equilibrium equation that for large uniform torsion, the local plastic shear strain γr is proportional to the local radius r [13]: r R r R γ = γ . (1) Here γR is the shear strain at the outer radius R of the sample. When the twisted bar is subjected to tensile testing, the local tensile flow stress depends on the ten- sile true strain ε and on the local shear pre-strain γr; σ(r) = σ(ε,γr). The force re- quired for plastic stretching of a rod previously subjected by torsion is given by ( ) ( ) 0 , 2 , d R R rF r rε γ = π σ ε γ∫ . (2) This integral can be developed as follows: ( ) ( )2 0 0 2, 2 , d , d RR R R r r r R r SF r r R γ γ⎛ ⎞ε γ = π σ ε = σ ε γ γ γ⎜ ⎟ ⎝ ⎠ γ∫ ∫ , (3) where S is the cross-sectional area of the bar. We introduce the quantity ( ), Rσ ε γ , which is the apparent tensile flow stress of the bar: ( ) ( ) ( )2 0 , 2, , d R R R r r r R F S γε γ σ ε γ = = σ ε γ γ γ γ ∫ . (4) After differentiation with respect to γR, the following expression is obtained: ( ) ( ) ( ), 2 , ,R R R R R ∂σ ε γ = σ ε γ −σ ε γ⎡ ⎤⎣ ⎦∂γ γ . (5) Hence, it follows: ( ) ( ) ( ), , , 2 RR R R R ∂σ ε γγ σ ε γ = σ ε γ + ∂γ . (6) This formula allows one to find the stress-strain curve ( ), Rσ = σ ε γ using the ex- perimentally measured curve for the apparent stress ( ), Rσ ε γ and its derivative with respect to γR. 3. The role of residual stresses Residual stresses arise after unloading a plastically twisted solid bar sample [13]. Assuming that the entire cross section of the sample was in plastic state un- Физика и техника высоких давлений 2015, том 25, № 3–4 136 der torsion, the local residual shear stress ( )res , Rrτ γ after a shear strain γR reached at the outer radius of the bar is given by the following equation [14]: ( ) ( )res 4, , ( ) 3R R R rr r R τ γ = τ γ − τ γ . (7) Here ( ), Rrτ γ is the local shear flow stress before unloading and τ is the mean shear stress across the bar, ( ) 2 3 0 3( ) , d R R Rr r r R τ γ = τ γ∫ . (8) The effect of the residual stresses during the subsequent tensile testing is a reduc- tion of the yield stress of the rod in tension, as they are present in the yield condi- tion. It can be shown, however, that the residual stresses can be ignored when the technique presented above is applied. This is due to a rapid relaxation of the re- sidual stresses at the beginning of plastic deformation. As will be shown below, the strain required for this relaxation is very small. The residual elastic strain γres associated to the residual elastic stress τres is res res G τ γ = , (9) where G is the elastic shear modulus. According to the associated flow rule, the following relation between the components of the strain increments and the acting stresses is valid: res3d d VMeτ γ = σ , (10) where eVM is the von Mises equivalent strain. It follows then from Eqs. (9) and (10) that res res 3d d VM G eγ = −γ σ . (11) Here the negative sign takes into account that γres decreases in absolute value, so that the sign of dγres is opposite to the sign of γres. By integrating Eq. (11) at a constant stress σ, we obtain res 3~ exp VM G e⎛ ⎞γ −⎜ ⎟σ⎝ ⎠ . (12) Therefore, the characteristic equivalent plastic strain eVM required for the relaxa- tion of residual stresses for torsion-tension can be estimated as 3VMe G σ = . (13) Using characteristic stress values that correspond to the tensile test of copper after torsion, the following estimate is obtained: eVM ~ 10–3. This estimate shows that already after a very small tensile strain, the effect of the residual stresses can be Физика и техника высоких давлений 2015, том 25, № 3–4 137 neglected. The physical meaning of the above analysis is that the stress state moves along the yield surface very rapidly from a combined tension-torsion into a pure tension state during the initial stage of the tensile test. 4. Experimental results Experiments were carried out on commercially pure copper samples at room temperature. The initial microstructure of the material can be characterized by an average grain size of about 30 μm with a weak crystallographic texture. The di- mensions of the deforming part of the samples were as follows: 7 mm in diameter and 40 mm in gauge length. The torsion testing was done in a free-end torsion machine to different rotation angles at a constant angular speed of 0.2 rad/s. The selected values of the rotation angle (in radian) were: 11.43, 22.85, 34.29, 45.71, 57.14, and 68.57. They correspond to a shear strain of 1, 2, 3, 4, 5, and 6, at the outer radius of the sample, respectively. These values were converted into the von Mises equivalent strain using the formula 3VMe = γ . The small lengthen- ing of the bar during the free end torsion testing (less than 2%) was neglected in the analysis of the experimental data. The shear flow stress acting at the outer ra- dius of the bar was calculated by the Fields and Backofen formula [5] and con- verted into equivalent von Mises stress using the formula 3σ = τ . The obtained stress-strain curve for torsion curve is displayed in Fig. 1 (curve 2). The tensile tests were done in a 10 ton Zwick machine at a strain rate of 0.05 mm/s. The results are shown in Fig. 2 for the twisted samples and in Fig. 1 for the non- twisted ones for larger strain (curve 1). After torsion rupture took place under tension already after about 2–4% plastic strain. With the available specimens, six points on the average stress-strain curve were obtained (labeled 3 in Fig. 1). They were taken from the tensile curves at 1% strain. A continuous curve was fitted to these points to calculate the derivative in Eq. (6), and the resulting large strain tensile test curve base on Eq. (6) was plotted as curve 4. It can be seen that the initial part of this curve matches the continuous tensile curve well. At large deformations, from about a strain of 1.5, the curve levels off at a constant stress level of about 415 MPa. Fig. 1. Stress-strain curves obtained for pure copper solid bars in tension (curve 1), in torsion (curve 2) and in tension after tor- sion with different magnitude of the twist (curve 3). Curve 4 was constructed using Eq. (6) Физика и техника высоких давлений 2015, том 25, № 3–4 138 The stress-strain curve 4 in Fig. 1 obtained for tensile testing after torsion was constructed for the material behavior at the outer radius of the twisted bar, but it is also valid for the inner points in the bulk of the bar. 5. Discussion As can be seen from Fig. 1, the monotonic torsion and tensile stress-strain curves do not coincide, despite the use of the equivalent stress and strain quanti- ties as a common platform: the torsion curve lies below the tensile one. This effect known for a long time and was examined in the past [3,4]. The main reason for it was mentioned in Section 1, viz. the scarcity of slip systems in torsion compared to tension. One particularity of the present results is that for low tensile strains the tensile flow stress after torsion agrees well with the flow stress in monotonic tension. In- deed, lower stresses are expected for a strain path change because the micro- structure that develops in the first path is not stable for the new path, thus many dislocations that were immobile in the first path can glide in the second one. However, it has been shown in Ref. [4] that the dislocation density is higher in torsion compared to tension, which can compensate for this effect. Another particularity of the results in Fig. 1 is that the tensile flow stress is constant after torsion at large strains, starting from about 1.5 strain. This effect was observed for the first time because the present technique is the first one to provide access to tensile flow stress after large strain torsion. Its origin might be rooted in the fragmentation of the grains which is occurs under severe plastic de- formation [3,15]. Further studies are needed to identify the exact reasons for this material behavior. An interesting observation can be drawn from the stress-strain curves in Fig. 1, namely, it is apparent that the equivalent flow stress is higher in tension than in torsion. This effect is due to the anisotropy that develops within the bar; a non- isotropic orientation distribution of grain orientations, i.e. a shear texture, appears [3,16]. Therefore, the tensile yield strength can be significantly enhanced through Fig. 2. Stress-strain curves in tension obtained without pre-torsion (0) and after different amounts of shear in torsion γ: 1 – 1, 2 – 2, 3 – 3, 4 – 4, 5 – 5, 6 – 6 Физика и техника высоких давлений 2015, том 25, № 3–4 139 pre-processing the material by simple shear. It then follows that less energy is re- quired for strengthening the material by simple shear compared to tensile straining. 6. Conclusions In this work we presented a new theoretical derivation for obtaining tensile stress-strain curves for the bars pre-strained by torsion. Experiments on Cu pro- viding exemplary data for the proposed derivation were conducted. The following main conclusions can be drawn: 1. In spite of the strain gradient inherent in a torsion-deformed bar, it is possi- ble to obtain tensile stress-strain curves for it. The procedure includes a series of torsion tests of bars to different amounts of strains, followed by deforming them in uniaxial extension. 2. Application of the proposed algorithm to large strain torsion of Cu yielded a tensile stress-strain curve which saturates quite early, from an equivalent von Mises strain of about 1. This work was funded by the French State program «Investment in the future» operated by the National Research Agency (ANR) ANR-11-LABX-0008-01, La- bEx DAMAS. Cai Chen acknowledges the doctorate scholarship of the China Scholarship Council No. 20120807 0035. Yuri Estrin acknowledges funding sup- port from the Ministry of Education and Science of the Russian Federation (Grant # 14.A12.31.0001) 1. R.Z. Valiev, T.G. Langdon, Adv. Eng. Mater. 12, 677 (2010). 2. Y. Estrin, A.V. Vinogradov, Acta Mater. 61, 782 (2013). 3. C. Tome, G.R. Canova, U.F. Kocks, N. Christodoulou, J.J. Jonas, Acta Metall. 32, 1637 (1984). 4. T. Ungar, L.S. Toth, J. Illy, I. Kovacs, Acta Metall. 34, 1257 (1986). 5. D.S. Fields, Jr, W.A. Backofen, Proc. Am. Soc. Test. Mater. 57, 1259 (1957). 6. A.A. Ilyushin, V.S. Lensky, Strength of Materials, Pergamon Press, Oxford (1967). 7. J. Bell, A.S. Khan, Int. J. Solids Structures 16, 683 (1980). 8. P. Delobelle, P. Robinet, P. Geyer, P. Bouffioux, J. Nuclear Mater. 238, 135 (1996). 9. H. Takahashi, K. Fujiwara, T. Nakagawa, Int. J. Plasticity 14, 489 (1998). 10. A.S. Khan, R. Liang, Int. J. Plasticity 16, 1443 (2000). 11. J. Faleskog, I. Barsoum, Int. J. Solids and Structures 50, 4241 (2013). 12. R. Mehrabi, M.T. Andani, M. Kadkhodaei, M. Elahinia, Experimental Mechanics 55, 1151 (2015). 13. L.M. Kachanov, Foundations of the Theory of Plasticity, Chapter 4. Torsion, North- Holland Publishing Co., Amsterdam (1971), pp. 123–144. 14. J.J. Jonas, F. Montheillet, S. Shrivastava, Scripta Metall. 19, 235 (1985). 15. R.Z. Valiev, Y. Estrin, Z. Horita, T.G. Langdon, M.J. Zehetbauer, Y.T. Zhu, JOM 58, 33 (2006). 16. F. Montheillet, M. Cohen, J.J. Jonas, Acta Metall. 32, 2077 (1984). Физика и техника высоких давлений 2015, том 25, № 3–4 140 Кай Чен, Ян Бейгельзимер, Ласло С. Тот, Ю. Эстрин, Роман Кулагин ПРЕДЕЛ ПРОЧНОСТИ МАТЕРИАЛОВ, ОБРАБОТАННЫХ ПРИ ПОМОЩИ ПРОСТОГО СДВИГА Для прогноза свойств субмикрокристаллических металлов, полученных методами интенсивных пластических деформаций, необходимо знать напряжение их течения при одноосном растяжении после обработки простым сдвигом большой величины (деформация сдвига более 10). Механические свойства материалов при таких не- пропорциональных путях деформирования изучают в основном с помощью трубча- тых образцов. Из-за потери устойчивости трубок при кручении такие эксперименты возможны лишь для малых упругопластических деформаций, не превышающих нескольких процентов. В статье предложен и обоснован метод определения напря- жения течения материала, предварительно обработанного большим простым сдви- гом. Метод основан на двух стандартных испытаниях: кручении со свободными торцами и одноосном растяжении. Получено соотношение, позволяющее по на- пряжению течения неоднородного образца, предварительно подвергнутого круче- нию, найти напряжение пластического растяжения его поверхностного слоя с опре- деленной деформацией простого сдвига. Путем простых аналитических оценок по- казано, что упругие остаточные напряжения первого рода, возникающие после раз- грузки образца, предварительно подвергнутого кручению, практически не влияют на предел текучести при растяжении. Ключевые слова: деформационное упрочнение, кручение, растяжение, градиент деформации, изменение пути деформирования, медь

Yield strength of a material pre-processed by simple shear

Institution

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