Nanophase metallic materials show a maximum in strength as grain size decreases to the nano scale, indicating a break down of the Hall-Petch relation. Grain boundary sliding, as a possible accommodation mechanisms, is often the picture that explain computer simulations results and real experiments. In a recent paper, Bringa et al. Science 309, 1838 (2005), we report on the observation of an ultra-hard behavior in nanophase Cu under shock loading, explained in terms of a reduction of grain boundary sliding under the influence of the shock pressure. In this work we perform a detailed study of the effects of hydrostatic pressure on nanophase Cu plasticity and find that it can be understood in terms of pressure dependent grain boundary sliding controlled by a Mohr-Coulomb law

Bringa, E M

Caro, A

Leveugle, E

UNT Digital Library

UCRL-JRNL-225797Do grain boundaries innanophase metals slide?E. M. Bringa, E. Leveugle, A. CaroNovember 2, 2006Applied Physics LettersDisclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.  Do grain boundaries in nanophase metals slide?E. M. Bringa,1, ∗ E. Leveugle,2 and A. Caro11Chemistry and Materials Science Directorate,Lawrence Livermore National Laboratory, Livermore, CA 945502Engineering and Materials Science,University of Virginia, Charlottesville, VA 22903, USA(Dated: February 7, 2006)AbstractNanophase metallic materials show a maximum in strength as grain size decreases to the nanoscale, indicating a break down of the Hall-Petch relation. Grain boundary sliding, as a possibleaccommodation mechanisms, is often the picture that explain computer simulations results andreal experiments. In a recent paper, Bringa et al. Science 309, 1838 (2005), we report on theobservation of an ultra-hard behavior in nanophase Cu under shock loading, explained in terms ofa reduction of grain boundary sliding under the influence of the shock pressure. In this work weperform a detailed study of the effects of hydrostatic pressure on nanophase Cu plasticity and findthat it can be understood in terms of pressure dependent grain boundary sliding controlled by aMohr-Coulomb law.∗Electronic address: ebringa@llnl.gov1The Hall-Petch relation predicts that as grain size decreases the hardness of polycrys-talline materials increases as the inverse square root of the grain size. But the possibilityof taking full advantage of this feature by designing hard materials by grain size refinementseems to be limited to grain sizes above a critical value (10-30 nm for Cu or Ni); below it,the strength ceases to increase, or eventually decreases, with decreasing grain size. Grainboundary (GB) sliding or other types of grain boundary accommodation mechanisms arethought to become dominant in this regime, that has been widely investigated both viasimulation techniques and modeling [1–10].Inhibiting softening by GB plasticity is then a key issue to obtain harder materials.Also for practical applications of these materials the stabilization of GB on aging is ofimportance. Both goals, namely GB plasticity and GB stability, can probably be achievedby the addition of carefully chosen alloying elements. However there seems to exist anothermean to avoid sliding, although in a very peculiar set up, as we recently proved in moleculardynamics simulations and experiments of shock loading nanophase Cu [11]. The highpressure developed on the wake of a shock wave appears to efficiently prevent GB sliding. Apreliminary explanation based on the assumption that GB sliding is similar to mechanicalfriction between two rough surfaces in contact, justified us to use a simple Mohr-Coulomblaw [12], for the pressure dependence of the friction coefficient to satisfactorily interpretthe results of our simulations. The ability to lock the boundaries in this way would extendthe region of normal Hall-Petch behavior to smaller grain sizes and for some particularapplications where pressure is relevant, like damage of spacecrafts by dust particles or targetsfor inertial confinement fusion, may provide a way to design materials with even greaterhardness.Computer simulations of plasticity at the nanoscale provides a plausible picture of theprocesses in terms of a crossover between dislocation - dominated regime for large grainsize, and grain boundary sliding for small grain size. Qualitative and simple quantitativeinterpretation of grain boundary sliding as linear viscous flow was given in Ref. [13].Systematic simulation studies covering a large grain size domain have recently been pub-lished [3], confirming the end of the validity of the Hall-Petch relation at about 15 nm innc-Cu. It is then quite accepted now that the increase in hardening by reducing the grainsize has a limit when the competing softening mechanisms of GB sliding comes into play.Even if that kind of computer simulations are done at extremely high deformation rates,2the results belong to the category of homogeneous strain, in the sense that there no stressgradients developed in the sample. Moreover, samples are uniaxially strained while thetransverse cross section is allowed to relax. In shocks, where we found the ultra-high strength[11], the situation is quite different. Shock waves develop a huge pressure behind the frontsince they travel faster than the speed of sound and lateral relaxation cannot therefore occur.Volume is reduced by a significant amount and plastic deformation is the way the materialfollows to relax stress from uniaxial towards hydrostatic.In our work on shock loading of nc-Cu we interpreted the pressure-induced hardeningvia two mechanisms, one is what we are discussing here, i. e. GB sliding inhibition, andthe other is the dislocation mobility sensitivity to pressure. The latter is a well knowmechanisms developed to interpret the results of shock loading in single or coarse grainpolycrystals [11, 14]. Using a very simple picture for the competition and overlap betweenboth mechanism at the crossover scale, we got the hardness behavior represented in Fig 1in Ref. [11].In this work we address a simple question in a clear simulation set up: is Gb slidingaffected by pressure as it is implicit in the elementary idea of friction? To answer thisquestion we use computer simulations to study the effect of hydrostatic pressure on thegrain boundary motion under homogeneous stress at two (both high) strain rates, namely3.0 108 and 1.0 109 1/sec. We note that these strain rates are similar to those in shocks.A similar question was recently addressed in [15] by simulations at ultra fine grain sizesin different states of uniaxial and biaxial loading. An asymmetry in tension-compression, aswell as a failure of standard yield criteria lead also the authors to propose a Mohr-Coulomblaw to interpret the results, in a similar way as we report here, although they did not considerhydrostatic load and their stresses where one order of magnitude smaller that ours.σzz = σ0zz + αP (1)In these simulations, with uniaxial stress along z in addition to hydrostatic load, wemonitor the zz component of the stress tensor minus the hydrostatic pressure, i. e. σzz −P ,that gives us a quantitative measurement of how easy is for the material to plasticallyflow under pressure. To extract conclusions regardless of the strain rate dependence of theresponse of the material, which induces an additional variable that affects the response,we measure in particular the stress necessary to induce some fixedvalue (0.5, 1 or 2%) of3FIG. 1: Hydrostatic pressure vs volume for nanophase Cu (squares); also shown is a quadratic fitand its derivative, the bulk modulus B.plasticstrain versus pressure. Because we use these fixed values of plastic strain, we needthe elastic response of the material under pressure to subtract the elastic contribution todeformation. This elastic response is sample-dependent as the small number of grains ina simulation sample precludes the use of the statistical treatment of elastic response of ananisotropic medium.The conditions used in these simulations, in particular the small grain size (5 nm), aresuch that no dislocation activity is present, i. e. all plasticity is related to GB’s activity, aswe carefully checked.We first then study the elastic response of a cubic nanophase sample constructed usingVoronoy cells and containing half a millon atoms distributed in 72 randomly oriented grainswith average grain size of 5 nm. The EAM potential used in these simulations is taken from[16]. Figure 1 shows the P-V relation and its derivative, the bulk modulus. It is evident thatdue to the large deformation range explored and the highly defected structure of the sample,the response in not linear; we fit a quadratic polynomial to this curve and extract a bulkmodulus that depends linearly on pressure. For comparison purposes, the bulk modulus ofa perfect crystal for this potential is 181 GPa [16]. For the type of loading used in thedeformation runs, the Young modulus Y is used for the elastic part, which shows a similarbehavior.Stress-strain results for ²˙ = 3.0 108/sec and several hydrostatic pressures are shown in4FIG. 2: Stress-strain curves for different hydrostatic pressures at ²˙ = 3.0108/sec. Straight lines arethe elastic responses at each pressure. Values in parenthesis indicate pressure in GPa.Figure 2, together with the measured elastic response to uniaxial load (straight lines withslope = Y(P). For negative pressures (tension) grater than 5 GPa the sample breaks. Bysubtracting the elastic to the total strain, we get the plastic contribution to it, and by lookingat the stress corresponding to different pre-set values of plastic deformation, namely 0.5, 1,and 2%, we obtain the shear stress versus pressure reported in Figure 3 and represented byEq.(1).It is important to realize that our definition of shear stress is peculiar. The σ − ² plotsat these strain rates do not show well defined yield and flow stresses. They are very strainrate sensitive and may, at these strain rates, show a maximum. Our definition of strengthas the stress needed to reach a given percentage of plastic deformation is a sensible way toavoid this complexity in the analysis of the response curve.It is remarkable in this figure the strong linear correlation between the points and thesimilarity of the slopes, α’s, at different plastic deformation. In fact α(0.5%) = 0.03,α(1.0%) = 0.036, α(1.5%) = 0.04. Similar results were obtained for the other set of runs at²˙ = 1.0 109/sec.The hypothesis is then clearly confirmed: if one considers GB sliding to occur againstsome ”friction” force, this force would increase directly proportional to the pressure, makingthe barrier for GB sliding higher with pressure. This mechanism suppresses the sliding forthe stress levels studied under shock conditions [11].5FIG. 3: Stress necessary to produce a given amount of plastic deformation versus hydrostaticpressure, as obtained form Fig (2). A remarkable linear correlation, at all 3 levels of plastic strainused for the definition of strength, supports the interpretation of GB plasticity in terms of aMohr-Coulomb law.The high values of hydrostatic pressure used in this work, within the range of pressuresinduced by shock waves, introduces a significant tension - compression asymmetry, as shownin the lower curve of Fig 3, that leads to failure for negative pressures grater than 5 GPa.It is therefore not possible to explore the validity of the Mohr-Coulomb law in tension, incontrast to the situation studied in [15] using biaxial stresses ten times smaller than ours.In summary, in this work we prove that the relation between the stress needed to deformplastically a nanophase sample and hydrostatic pressure is remarkable linear, like in a simplefriction mechanisms, and can therefore be captured with a Mohr-Coulomb law developedfor granular materials. We provide the slope in the linear relation and find it to be quiteinsensitive to the total plastic deformation or the strain rate used to calculate it.The implications of this findings support our previous results on ultra-high hardness undershock conditions. In computer simulations of shocks in nc Cu, where the Poisson volumerelaxation effect is restrained from occurring, we found that the flow stress reaches ultra-high values not seen before for this model material, in striking contrast with the resultsreported so far for simulations and experiments under quasi-static, uniform stress, wherevolume relaxation is naturally allowed [11]. The softening of the flow stress of nc metalsbelow ∼ 15 nm grain size does not happen, or is significantly reduced, under shock loading6because the transition from dislocation mediated plasticity at large grain sizes, to GB slidingat small grain sizes is suppressed. Suppression arises because the barriers for GB slidingincrease with pressure, as proved in this work. The quantitative evaluation of the pressuresensitivity reported here is of direct use in continuum models of plasticity [17].Comment on the barriers for dislocationsAcknowledgmentsThis work was performed under the auspices of the U.S. Department of Energy by theUniversity of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.[1] V. Yamakov, D. Wolf, S. R. Phillpot, and H. Gleiter, Nature Mater. 1, 1 (2002).[2] J. Schiots, F. D. D. Tolla, and K. W. Jacobsen, Nature 391, 561 (1998).[3] J. Schiots and K. W. Jacobsen, Science 301, 1357 (2003).[4] H. V. Swygenhoven, Science 296, 66 (2002).[5] R. Siegl, M. Yan, and V. Vitek, Modelling Simul. Mater. Sci. Eng. 5, 105 (1997).[6] V. Yamakov, D. Wolf, S. R. Phillpot, A. K. Mukherjee, and H. Gleiter, Nature Mater. 3, 43(2004).[7] H. Hahn, P. Mondal, and K. A. Padmanabhan, Nanostruct Mater. 9, 603 (1997).[8] H. Conrad, Mater Sci Eng A 341, 216 (2003).[9] A. C. Lund, T. G. Niegh, and C. A. Schuh, Phys. Rev. B 69, 012101 (2004).[10] H. V. Swygenhoven and A. Caro, Appl. Phys. Lett. 71, 1652 (1997).[11] E. M. Bringa, A. Caro, Y. M. Wang, M. Victoria, J. M. McNaney, B. A. Remington, R. F.Smith, B. R. Torralva, and H. V. Swygenhoven, Science 309, 1838 (2005).[12] C. A. Schuh and A. C. Lund, Nature Mater. 2, 449 (2003).[13] H. V. Swygenhoven, M. Spaczer, A. Caro, and D. Farkas, Phys. Rev. B 60, 22 (1999).[14] D. J. Steinberg, S. G. Cochran, and M. W. Guinan, J. Appl. Phys. 51, 1498 (1980).[15] A. C. Lund and C. A. Schuh, Acta Mater. 53, 3193 (2005).7[16] Y. Mishin, D. Farkas, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 59, 3393(1999).[17] B. Jiang and G. J. Weng, J. Mech. Phys. Solids 52, 1125 (2004).8

Do grain boundaries in nanophase metals slide?

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Do grain boundaries in nanophase metals slide?

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