- Email: [email protected]
Deformation patterns in cross-sections of twisted bamboo-structured Au microwires
Acta Materialia 97 (2015) 216–222
Contents lists available at ScienceDirect
Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Deformation patterns in cross-sections of twisted bamboo-structured Au microwires M. Ziemann a, Y. Chen a, O. Kraft a, E. Bayerschen b, S. Wulfinghoff c, C. Kirchlechner d,e, N. Tamura f, T. Böhlke b, M. Walter a, P.A. Gruber a,⇑ a
Karlsruhe Institute of Technology, Institute for Applied Materials, Engelbert-Arnold-Strasse 4, D-76131 Karlsruhe, Germany Karlsruhe Institute of Technology, Institute of Engineering Mechanics, Kaiserstrasse 10, D-76131 Karlsruhe, Germany RWTH Aachen University, Institute of Applied Mechanics, Mies-van-der-Rohe-Str. 1, D-52074 Aachen, Germany d Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Strasse 1, D-40237 Düsseldorf, Germany e Department Materials Physics, Montanuniversitaet Leoben, A-8700 Leoben, Austria f Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA b c
a r t i c l e
i n f o
Article history: Received 23 January 2015 Revised 22 May 2015 Accepted 5 June 2015
Keywords: Microtorsion Strain gradients Laue microdiffraction Deformation patterns Crystal plasticity Equivalent plastic strain
a b s t r a c t In order to investigate an almost pure extrinsic size effect we propose an experimental approach to investigate the deformation structure within single crystalline cross-sections of twisted bamboo-structured Au microwires. The cross-sections of individual h1 0 0i oriented grains of 25 lm thick Au microwires have been characterized by Laue microdiffraction. The diffraction data were used to calculate the misorientation of each data point with respect to the neutral fiber in the center of the cross-section as well as the kernel average misorientation to map the global and local deformation structure as function of the imposed maximum plastic shear strain. The study is accompanied by crystal plasticity simulations which yield the equivalent plastic strain distributions in the cross-section of the wire. The global deformation structures are directly related to the activated slip systems, resulting from the real orientations of the investigated grains. When averaging the degree of deformation along ring segments, an almost continuous but non-linear increase of misorientation from the center toward the surface is observed, reflecting the overall strain gradient imposed by torsion. For the local deformation structure, pronounced and graded deformation traces are observed which often pass over the neutral fiber of the twisted wire and which are obviously reflecting domains of high geometrically necessary dislocations content. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Multi-axial loading conditions are ubiquitous in the common use of structural materials. It is known that strain gradients, due to multi-axial loading conditions, have a significant influence on the hardening behavior of metallic materials in small dimensions. In several investigations it was found that the hardening scales inversely with the length in which plastic deformation occurs and thus is leading to increasing flow stresses with decreasing component size when strain gradients are present. This was demonstrated originally by Fleck et al. [1] for torsion of microwires. Consequently, it has been argued that the impact of strain gradients on the mechanical behavior has to be taken into account by mathematical descriptions of the deformation behavior of metals in small dimensions [2–8]. The basis of common strain gradient theories is the assumption that the strain distribution along the ⇑ Corresponding author. http://dx.doi.org/10.1016/j.actamat.2015.06.012 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
strain gradient is linear and that hardening arises from the accumulation of both, randomly stored – so called statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) [1,2,9]. SSDs exist and interact randomly and are associated with uniform deformation. When local strain gradients are present, GNDs are additionally stored in order to accommodate the lattice curvature. In this context, the size effect in hardening results from the fact that, for a given maximum strain, the strain gradient and thus the density of GNDs must be higher for smaller sample sizes. In order to investigate this so called extrinsic size effect and to create well defined stress and strain states with and without strain gradients, several setups for tension and torsion experiments on metallic microwires have been established [10–14]. Given by limitations of current manufacturing techniques, long metallic wires are naturally polycrystalline. To account for this, special emphasis has been given to the ratio of grain size and wire diameter and thereby it is proposed that the observed size effects are related to both, the relevant microstructural length scale (grain size,
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
dislocation density) and the sample size [14–17]. However, the interplay of the different length scales, particularly in combination with imposed strain gradients in torsion, is not known as their individual contribution to the overall size effects could not be quantified up to now [17]. Consequently, it is indeed questionable if the assumptions of the classical strain gradient theories with respect to strain distribution and accumulation of GNDs may hold for polycrystalline microstructures and if it is reasonable to apply them to describe torsion experiments of polycrystalline wires. In order to investigate the specific effect of a strain gradient, the interplay between grain size and strain gradient would have to be eliminated. For torsion experiments, where the radial strain distribution is assumed to be linear, this may be achieved by creating at least a single crystalline cross-section in the wire. Therefore, we propose an experimental approach to investigate the deformation structure within single crystalline cross-sections of bamboostructured Au microwires. The cross-sections of individual h1 0 0i oriented grains of twisted Au microwires are characterized by Laue microdiffraction to map the crystal misorientation along the strain gradient from the outer circumference of the wire to its center. Wires with increasing maximum plastic shear strain cr = R, pl, from:
cr ¼
u l
r
0 ðcenterÞ 6 r 6 R ðsurfaceÞ;
217
Fig. 1. Torsional response of Au microwires loaded up to defined maximum plastic shear strains.
ð1Þ
where u is the rotation angle and l the gauge length, are investigated to determine how the plastic deformation proceeds radially. The experiments are accompanied by crystal plasticity (CP) simulations to determine the equivalent plastic strain within the wires. Both, simulations and experiments, show that the local deformation structure within the cross-section of the wire is governed by the crystal orientation of the individual grain and its glide systems. The global deformation indeed shows a gradually increasing deformation from the center toward the surface, and thus reflects the overall strain gradient. 2. Experimental methods 2.1. Sample preparation As base material, polycrystalline Au wires made from high purity Au (99,99%, supplied by Heraeus, Germany) with a diameter D of 25 lm were used. To obtain a so called bamboo microstructure, the Au wires were annealed for 168 h at 800 °C using a high vacuum glass tube furnace. The torsion tests were performed using a custom-built torsion test setup. A detailed description of the sample mounting, experimental setup and data evaluation can be found in Ref. [10]. Au wires with a gauge length of 50 mm were deformed to different maximum plastic shear strains cr = R, pl of 0.4%, 0.6%, 1.0% and 1.6% with a constant rotation velocity of 0.086 rad/s for all tests (see Fig. 1). After twisting the wires, they were cut centric related to the gauge length and both pieces were glued with silver paint on a sample stub for preparation of defined cross-sections. The microstructure of the wires was characterized using a Dual Beam SEM-FIB microscope (FEI Nova NanoLab 200) additionally equipped with an electron-backscatter-diffraction (EBSD) system from Oxford Instruments. The scanning electron microscope (SEM) mode was used for imaging, the focused ion beam (FIB) for preparation of the individual cross-sections and the EBSD system for determining the crystal orientation of the grains. After annealing, the wire consists of a chain of mostly elongated cylindrical grains (Fig. 2(a)). The crystallographic orientation along the wires main axis z is predominantly [0 0 1] and [1 1 1], with the [0 0 1]-oriented grains being observed most frequently. Due to the majority of [0 0 1] oriented grains and their generally high
Fig. 2. (a) SEM micrograph showing the bamboo microstructure of an annealed Au wire with a diameter of 25 lm. (b) Selection of an [0 0 1] oriented grain based on EBSD and (c) ready-prepared cross-section of the selected grain.
aspect ratio, these grains were chosen systematically and individual cross-sections of wires with different cr = R, pl were cut by FIB (Fig. 2(b) and (c)). [0 0 1]-oriented grains with a length of at least 30 lm were chosen and the cross-sections were prepared in the center between two grain boundaries.
2.2. Laue microdiffraction The Laue microdiffraction experiments have been conducted at the X-ray microdiffraction beamline 12.3.2 of the Advanced Lights Source (ALS), Berkeley, USA [18]. Fig. 3 shows a schematic of the experimental setup and exemplary data of the experimental procedure. The sample stub was mounted on the stage of the 5-axis diffractometer of the beamline. The wires were scanned with a polychromatic X-ray microbeam (6–22 keV, 0.8 0.9 lm2 FWHM) in reflection geometry with a sample tilt of 45°. A large area detector (169 170 mm2, Pilatus 1M, Dectris) was positioned at a 2h angle of 90° and a sample to detector distance of about 100 mm. In addition, a laser distance measurement system, an optical microscope and a fluorescence detector system were available to
218
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
Fig. 3. (a) Schematic of the experimental setup at the microdiffraction beamline of the Advanced Light Source (ALS, Berkeley). (b) Fluorescence scan of the wire together with an exemplary Laue pattern of an individual pixel on the wire surface and a schematic illustrating the determination of the Kernel Average Misorientation angle (KAM). In this schematic, aci denote the angles between the orientation matrices Rc and Ri of the center of the kernel and the individual neighboring pixels, respectively. The fluorescence scan was used to locate the surface of the wire and Laue patterns were taken with a step size of 0.8 lm across the wire. The graded intensity in the fluorescence scan results from the inclination of the fluorescence detector with respect to the wire cross-section.
align the sample in the microbeam. Based on a fast fluorescence scan with a step size of 1 lm the final scanning area around the cross-section of individual Au microwires was defined. Subsequently, Laue diffraction patterns were collected with a step size of 0.8 lm (44 44 patterns). The individual Laue diffraction patterns were indexed using XMAS (X-ray Microdiffraction Analysis Software) [19] to determine the crystal orientation for each position of the raster scan in form of the full orientation matrix Ri. The orientation matrices were used to characterize the deformation structure within the single crystalline cross-section of the Au wires. In order to determine the local misorientation distribution, the Kernel Average Misorientation angle (KAM) between all individual data points and their eight nearest neighbors was calculated (see Fig. 3(b)). To characterize the global deformation behavior along the cross-section of the wires, the misorientation of the individual data points with respect to the center of the cross-section has been analyzed. Therefore, the center of the cross-sections have been determined by analyzing the local distribution of the integral intensity of all Laue spots within the individual Laue diffraction patterns. This integral intensity of single Laue diffraction patterns was summated along defined circles and by maximizing this total intensity along the circles, the center within the individual cross-section could be determined. Subsequently the rotation matrix Ric between the orientation matrices Rc and Ri of the center of the cross-section and the individual data points, respectively, was determined. Based on the rotation axis and rotation angle of Ric the misorientation vector mic was obtained. For the analysis of the misorientation within the wire cross-section only the geometric mean of the x- and y-component (in-plane components) of the misorientation vector was used. Finally the misorientation with respect to center of the cross-section was averaged over 15 ring segments starting from the center and plotted as function of the radial distance from the center.
strain in the energetic framework and the additional grain boundary yield criterion are neglected. Consequently, in contrast to Wulfinghoff et al. [21] no internal length scale is introduced in the framework. This is motivated by the focus on bamboo-like microwires with single crystalline cross-section. The goal from the modeling perspective is to investigate if the equivalent plastic strain within individual cross-sections can be used for comparison to the experimental quantities determined by Laue microdiffraction. However, one has to take into account, that the diffraction data only reflect elastic lattice distortions as result of the applied graded plastic deformation. A small strain framework is utilized and an equivalent plastic strain is introduced (see Table 1). Contrary to Wulfinghoff et al. [21] and for simplicity of reading all equations are presented here solely in terms of the equivalent plastic strain ceq. Thus, the numerically necessary additional micromorphic field variable f ceq is not shown in the theoretical summary. The stored energy contains Wh, an isotropic hardening contribution of Voce-type, besides the classic elastic Hooke-type energy We. In the field equations (derivable from the principle of virtual power) the Cauchy stress r, the scalar resolved shear stress sa on the individual octahedral slip systems a and the dissipative stresses sda are considered. The flow rule of overstress-type considers the hardening in form of the additional microstress b, the drag stress sD, and the reference shear rate c_ 0 as well as the strain rate sensitivity exponent p. In addition, for the isotropic Voce-hardening relation, the material parameters sC0 (initial yield stress), sC1 (saturation yield stress) and H (initial hardening modulus) need to be specified. The Finite Element implementation of the theory at hand is a special case of the more general implementation discussed in [21] and can be derived with the simplifications mentioned above. The boundary conditions are chosen such that a microwire is loaded with Dirichlet (displacement) boundary conditions. In the direction of the central specimen axis the displacement is unrestricted by the boundary conditions except at one node of the bottom surface of the cylinder. The spatial position of this node of the microwire is only fixed in x-direction. On the opposing top and bottom surface of the cylinder the in-plane displacements are then prescribed. Consequently, any possible rigid translation of the specimen in space is removed. Since only small deformation behavior is computationally investigated, the occurring warping deformations are negligibly small. Therefore, no artificial stress
Table 1 Theoretical model adapted from Wulfinghoff et al. [21] after neglecting of gradient contributions and grain boundary contributions. Additive decomposition Plastic strain
ee ¼ e ep
Equivalent plastic strain
ceq ð^kÞ ¼
Stored energy
W ¼ W e ðe; ep Þ þ W h ðceq Þ
Elastic energy
3. Crystal plasticity modeling Crystal plasticity models are intended to take into account dislocation activity and accumulation. A strain gradient crystal plasticity model incorporating an equivalent plastic strain accounting for intrinsic and extrinsic size effects and dislocation-induced plastic deformation on the continuum scale has been introduced in [20,21]. This has also been applied to microtorsion experiments on coarse grained Au wires [22]. However, in the present work on bamboo-structured microwires this model is used in a simplified version. The main difference of the theoretical framework at hand is that all gradient contributions of the equivalent plastic
Isotropic Voce hardening Linear momentum balance Slip system shear stress Traction BC Dirichlet BC Dissipation Flow rule & dissipative stress
ep ¼
X
k symðda na Þ a a Z X
a t
k_ a dt ¼
X
k ; k_ a P 0; a a
a ¼ 1 . . . 24
1 ðe ep Þ C½e ep 2 ! C 2 Hc 1 C W h ¼ s1 sC0 ceq þ s1 sC0 exp C eq C H s1 s0
We ¼
divðrÞ ¼ 0 8x 2 B
sa ðsda þ bÞ ¼ 0 8x 2 B; sa ¼ r symðda na Þ rn ¼ t on @Bt on @Bu u¼u X D¼ sd k_ a a a * +p sd sC0 k_ a ¼ c_ 0 a ;
sD
sda ¼ sa b; b ¼ @ ceq W h
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
concentrations are observed in the investigated torsion loading upon fixing the spatial position of one bottom surface node in x-direction. At the beginning of the simulation, the equivalent plastic strain is assumed to vanish throughout the model, i.e. the crystal is undeformed, completely. Although this assumption is not expected to be valid in general, e.g. in the case of predeformed microwires, it seems reasonable to use it in the context of investigating the overall strain gradient imposed on annealed microwires by CP simulations. The microwire with a diameter of 25 lm and a length of 31.25 lm is loaded up to 0.02 relative maximum shear of the top and bottom surfaces, and unloaded until the torque Mt, normalized by the wire radius R approximately vanishes, i.e. MT/R3 6 0.1 MPa. For the anisotropic elastic constants of gold values from the literature were chosen (C1111 = 186 GPa, C1122 = 157 GPa, C1212 = 42 GPa, cf. Ref. [23]). Since the determination of the material parameters related to initial yield strength and hardening would require extended experiments with completely single crystalline microwires seeing beyond the scope of this work, the same material parameters as in [21] are chosen here. Although the mentioned work considered copper and not gold microwires, still an fcc material with comparable anisotropy is used. The simulations were carried out with a mesh of 160,992 degrees of freedom. Convergence of the results is emphasized by the fact that an increase to 415,744 degrees of freedom leads to a decrease in the maximum relative error between the normalized-torque shear curves of less than 1%. 4. Results and discussion As already mentioned above, the global deformation field within a cross-section can be determined by calculating the misorientation of every data point with respect to the center element. Fig. 4 shows the resulting maps for wires twisted to a maximum plastic shear strain cr = R, pl = 0.4%, 0.6%, 1.0% and 1.6%, respectively. As the different deformed grains reveal a wide spread in misorientation, the scale is normalized by the related maximum misorientation. When comparing the maps it is seen that some of the cross-sections exhibit a graded rudimental symmetrical strain field, whereas the others show pronounced areas with stronger
219
orientation gradients, more or less opposing each other. To understand this non-uniform deformation behavior in detail, the following important feature has to be considered: If a h1 0 0i oriented single crystalline cylinder made of an fcc metal is ideally aligned concerning its rotation axis, exactly four independent slip systems will be simultaneously activated under torque with the onset of yielding. As a result, in contrast to the common idea of a graded annular deformation structure, an indeed symmetrical, but non-homogeneously graded strain field is generated within a cross-section as shown in Fig. 5. Here, CP simulations are used to depict the evolution of the spatial distribution of the equivalent plastic strain ceq, pl in dependence of the applied twist. Comparable to the torsion tests, the calculations were performed such that after a defined plastic deformation, the structures were unloaded again. The pronounced quadruple symmetry, however, disappears increasingly at certain misalignments. Fig. 6 shows this behavior exemplified by means of CP simulations for misalignments of 8° related to the y-axis (a) and for misalignments of 8° related to both the x-axis and the y-axis (b). The value of 8° was chosen based on the fact that the average tilt between the [0 0 1] orientation of the investigated grains and their individual rotation axis was found to be about 4° (determined by Laue experiments). Since the line of symmetry of the single grains may also reveal a small tilt related to the rotation axis of the entire wire z (compare Fig. 2, front part of the wire) an additional misalignment of 4° is assumed. The tilt between a wires rotation axis and the experimental rotation axis Z is 1°. Taking the results of the CP simulations into account, it becomes conceivable that the [0 0 1] direction of the chosen grains did not match exactly with the experimental rotation axis Z. It is obvious that some grains exhibited a slight but significant tilt mainly related to the x- or y-axis, whereas the other grains exhibited a slight but significant spatial tilt. Although the maps from Fig. 4 reveal local orientation gradients within the cross-sections, when averaging the misorientation for different annuluses between center and surface, an almost continuously graded deformation behavior is found for every sample as shown in Fig. 7(a). Although grains extracted from the same wire show big differences in overall misorientation, the general trend
Fig. 4. Normalized misorientation maps from cross-sections of different twisted bamboo-structured Au microwires with a diameter of 25 lm. The corresponding grains have orientations close to the [0 0 1]-orientation with respect to the experimental rotation axis Z. In this analysis, the rotation of every data point is referred to the center element and the misorientation is normalized to the maximum misorientation that occurred in the particular cross-section. The arrows show the in-plane h1 0 0i orientations.
220
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
Fig. 5. CP simulation: Evolution of the spatial distribution of the equivalent plastic strain with increasing twist within an ideally aligned h1 0 0i oriented single crystalline cylinder of Au.
Fig. 6. CP simulation: Evolution of the spatial distribution of the equivalent plastic strain with increasing twist within a misaligned h1 0 0i oriented single crystalline cylinder of Au – (a): misalignment of 8° related to the y-axis, (b): misalignment of 8° related to both the x- and y-axis.
of increasing misorientation with increasing twist is clearly visible when additionally averaging the results of grains extracted from the same wire (Fig. 7(b)).
Based on different orientations of the single grains, a non-homogeneous deformation behavior along a wires main axis during loading was expected naturally. However, that even similar
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
221
Fig. 7. Averaged misorientation vs distance to the center, calculated for the cross-sections shown in Fig. 4 – (a): individual grains and (b): averaged for both grains extracted from the same wire.
Fig. 8. Averaged equivalent plastic shear strain vs distance to center, calculated for the cross-sections shown in Figs. 5 and 6, respectively.
oriented grains exhibit such big differences in deformation is somehow surprising, particularly since according to CP simulations slight misalignments are only causing marginal differences in the equivalent plastic strain distribution, as illustrated in Fig. 8. When averaging ceq, pl along different ring segments for the cross-sections shown in Figs. 5 and 6, respectively, it is seen that the integral degree of deformation is decreasing moderately with increasing misalignment. It is feasible that the resulting deformation within a single grain is significantly affected by the deformation behavior of the adjacent grains, which is again depending on their orientation. This feature has to be taken into account in future investigations. In contrast to the curves depicted in Fig. 8, the curves depicted in Fig. 7 show an indeed graded, but not linearly increasing deformation from the center to the surface. Toward the outer part of the wire, the increase in lattice rotation is clearly less pronounced. Since in a single crystalline material from high purity elastically distortions are mainly resulting from both the interplay between dislocations and dislocation multiplication, it is argued that a significant number of dislocations is annihilating at the free surface during plastic deformation.
Fig. 9. Normalized KAM maps from cross-sections of different twisted bamboo-structured Au microwires with a diameter of 25 lm – the arrows show the in-plane h1 0 0i orientations.
222
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
Besides the identification of the global deformation behavior, the microdiffraction data were also used to determine the local deformation behavior within the twisted grains. The maps in Fig. 9 depict the KAM of every data point related to its encircling neighbors. Here, with exception of one grain from a wire twisted to cr = R, pl = 0.4% (almost no deformation), deformation traces are visible. A comparison of these extraordinary structures with the related global deformation fields (Fig. 4) shows that the traces are mainly correlating with the border areas of stronger deformed regions within a cross-section and the transition regions of small equivalent plastic strain (compare Figs. 5 and 6). Most of the traces reveal a pronounced graded shape and some additionally pass over the neutral fiber of the twisted grains. Based on classical ideas about deformation of metals [2] one may assume that the traces mainly consist of GNDs, usually required to accommodate stronger lattice curvatures. Although the misorientation data do not allow to distinguish between pure elastic distortions and distortions caused by dislocations, the rather strong misorientation gradients within the individual traces strongly support the assumption that GNDs are accumulating in this regions and that dislocation substructures have formed within the cross-section of the twisted wire. In order to verify this interpretation and to identify the role of dislocation arrangement discrete dislocation dynamics (DDD) simulations are currently underway. When viewing the results of the present investigations it is clearly seen, that Laue microdiffraction, supported by CP simulations, is generally an excellent tool to determine both the global and local deformation behavior within twisted microspecimen – at least for (pseudo) single crystalline metals. Based on both well selected samples and statistical analyses, the method can be used to study general aspects of deformation in torsion beyond yielding, as well as to clarify specific aspects, for example concerning the physical background of already observed size effects [1,12,17]. 5. Summary and conclusion In order to study the radial evolution of irreversible deformation within twisted thin bamboo-structured Au wires, Laue microdiffraction experiments were performed. For that purpose, wires with a diameter of 25 lm were twisted to different degrees of maximum plastic strain. Subsequently, EBSD investigations were carried out on central pieces of the samples to determine the crystal orientation of individual grains with respect to the longitudinal axis z of the wires. Based on the EBSD measurements, sufficiently long, almost [0 0 1] oriented grains were centrically cut perpendicular to the wires main axis by means of the FIB technique. The such prepared cross-sections were then investigated using Laue microdiffraction. Supported by CP simulations it is found, that the determined global deformation structures are directly related to the activated slip systems, resulting from the real orientations of the investigated grains concerning their rotation axis. Furthermore, when averaging the degree of deformation for different annuluses, an almost continuous increase of strain from the center toward the surface is observed for every investigated sample. An important finding is that the increase in deformation is less pronounced toward the surface. It is argued that during twisting a significant number of dislocations annihilates at the free surface. This is assumed to be also valid for coarse grained microwires and thus, should be considered in modeling. In addition to the global deformation behavior, the local deformation behavior was investigated by means of KAM calculations. Here, pronounced deformation traces within the cross-sections are observed. When comparing these traces with the maps showing the global
deformation behavior it is found that they are mainly located at border areas of stronger deformed regions. It is assumed that the strong misorientation gradient within the traces is correlated to the formation of dislocation substructures within the twisted wire. Acknowledgments The authors acknowledge the support rendered by the German Research Foundation (DFG) under Grants BO1466/5-1 and GR 3677/2-1. The funded projects ‘‘Dislocation based Gradient Plasticity Theory’’ and ‘‘Experimental Characterization of Micro Plasticity and Dislocation Microstructure’’ are part of the DFG Research Group 1650 ‘‘Dislocation based Plasticity’’. References [1] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain Gradient Plasticity – Theory and Experiment, Acta Metall. Mater. 42 (2) (1994) 475–487. [2] M.F. Ashby, The deformation of plastically non-homogeneous materials, Philos. Mag. 21 (170) (1970) 399–424. [3] R. Deborst, H.B. Muhlhaus, Gradient-dependent plasticity – formulation and algorithmic aspects, Int. J. Numer. Methods Eng. 35 (3) (1992) 521–539. [4] H. Gao, Y. Huang, W.D. Nix, J.W. Hutchinson, Mechanism-based strain gradient plasticity – I. Theory, J. Mech. Phys. Solids 47 (6) (1999) 1239–1263. [5] W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solids 46 (3) (1998) 411–425. [6] N.A. Fleck, J.W. Hutchinson, A phenomenological theory for strain gradient effects in plasticity, J. Mech. Phys. Solids 41 (12) (1993) 1825–1857. [7] N.A. Fleck, J.W. Hutchinson, Strain gradient plasticity, in: Advances in Applied Mechanics, Elsevier, 1997, pp. 295–361. [8] D.J. Dunstan, B. Ehrler, R. Bossis, S. Joly, K.M.Y. Ping, A.J. Bushby, Elastic limit and strain hardening of thin wires in torsion, Phys. Rev. Lett. 103 (15) (2009) 155501. [9] J.F. Nye, Some geometrical relations in dislocated crystals, Acta Metall. 1 (2) (1953) 153–162. [10] M. Walter, O. Kraft, A new method to measure torsion moments on smallscaled specimens, Rev. Sci. Instrum. 82 (3) (2011) 035109. [11] D.J. Dunstan, J.U. Gallé, B. Ehrler, N.J. Schmitt, T.T. Zhu, X.D. Hou, K.M.Y. P’ng, G. Gannaway, A.J. Bushby, Micromechanical testing with microstrain resolution, Rev. Sci. Instrum. 82 (9) (2011) 093906. [12] D. Liu, Y. He, X. Tang, H. Ding, P. Hu, P. Cao, Size effects in the torsion of microscale copper wires: experiment and analysis, Scr. Mater. 66 (6) (2012) 406–409. [13] W.-Y. Lu, B. Song, Quasi-static torsion characterization of micro-diameter copper wires, Exp. Mech. 51 (5) (2011) 729–737. [14] B. Yang, C. Motz, M. Rester, G. Dehm, Yield stress influenced by the ratio of wire diameter to grain size – a competition between the effects of specimen microstructure and dimension in micro-sized polycrystalline copper wires, Philos. Mag. 92 (2012) 3243–3256. [15] X.X. Chen, A.H.W. Ngan, Specimen size and grain size effects on tensile strength of Ag microwires, Scr. Mater. 64 (2011) 717–720. [16] D.J. Dunstan, A.J. Bushby, Grain size dependence of the strength of metals: the Hall–Petch effect does not scale as the inverse square root of grain size, Int. J. Plast. 53 (2014) 56–65. [17] Y. Chen, O. Kraft, M. Walter, Size effects in thin coarse-grained gold microwires under tensile and torsional loading, Acta Mater. 87 (2015) 78–85. [18] M. Kunz, N. Tamura, K. Chen, A.A. MacDowell, R.S. Celestre, M.M. Church, S. Fakra, E.E. Domning, J.M. Glossinger, J.L. Kirschman, G.Y. Morrison, D.W. Plate, B.V. Smith, T. Warwick, V.V. Yashchuk, H.A. Padmore, E. Ustundag, A dedicated superbend X-ray microdiffraction beamline for materials, geo-, and environmental sciences at the advanced light source, Rev. Sci. Instrum. 80 (3) (2009) 035108. [19] N. Tamura, A.A. MacDowell, R. Spolenak, B.C. Valek, J.C. Bravman, W.L. Brown, R.S. Celestre, H.A. Padmore, B.W. Batterman, J.R. Patel, Scanning X-ray microdiffraction with submicrometer white beam for strain/stress and orientation mapping in thin films, J. Synchrotron Radiat. 10 (2) (2003) 137– 143. [20] S. Wulfinghoff, T. Böhlke, Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics, Proc. R. Soc. A 468 (2012) 2682– 2703. [21] S. Wulfinghoff, E. Bayerschen, T. Böhlke, A gradient plasticity grain boundary yield theory, Int. J. Plast. 51 (2013) 33–46. [22] E. Bayerschen, S. Wulfinghoff, T. Böhlke, Application of strain gradient plasticity to micro-torsion experiments, Proc. Appl. Math. Mech. 14 (2014) 313–314. [23] J. Rösler, H. Harders, M. Bäker, Elastisches Verhalten, in: Mechanisches Verhalten der Werkstoffe, Teubner, 2006, pp. 31–61.
Contents lists available at ScienceDirect
Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Deformation patterns in cross-sections of twisted bamboo-structured Au microwires M. Ziemann a, Y. Chen a, O. Kraft a, E. Bayerschen b, S. Wulfinghoff c, C. Kirchlechner d,e, N. Tamura f, T. Böhlke b, M. Walter a, P.A. Gruber a,⇑ a
Karlsruhe Institute of Technology, Institute for Applied Materials, Engelbert-Arnold-Strasse 4, D-76131 Karlsruhe, Germany Karlsruhe Institute of Technology, Institute of Engineering Mechanics, Kaiserstrasse 10, D-76131 Karlsruhe, Germany RWTH Aachen University, Institute of Applied Mechanics, Mies-van-der-Rohe-Str. 1, D-52074 Aachen, Germany d Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Strasse 1, D-40237 Düsseldorf, Germany e Department Materials Physics, Montanuniversitaet Leoben, A-8700 Leoben, Austria f Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA b c
a r t i c l e
i n f o
Article history: Received 23 January 2015 Revised 22 May 2015 Accepted 5 June 2015
Keywords: Microtorsion Strain gradients Laue microdiffraction Deformation patterns Crystal plasticity Equivalent plastic strain
a b s t r a c t In order to investigate an almost pure extrinsic size effect we propose an experimental approach to investigate the deformation structure within single crystalline cross-sections of twisted bamboo-structured Au microwires. The cross-sections of individual h1 0 0i oriented grains of 25 lm thick Au microwires have been characterized by Laue microdiffraction. The diffraction data were used to calculate the misorientation of each data point with respect to the neutral fiber in the center of the cross-section as well as the kernel average misorientation to map the global and local deformation structure as function of the imposed maximum plastic shear strain. The study is accompanied by crystal plasticity simulations which yield the equivalent plastic strain distributions in the cross-section of the wire. The global deformation structures are directly related to the activated slip systems, resulting from the real orientations of the investigated grains. When averaging the degree of deformation along ring segments, an almost continuous but non-linear increase of misorientation from the center toward the surface is observed, reflecting the overall strain gradient imposed by torsion. For the local deformation structure, pronounced and graded deformation traces are observed which often pass over the neutral fiber of the twisted wire and which are obviously reflecting domains of high geometrically necessary dislocations content. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Multi-axial loading conditions are ubiquitous in the common use of structural materials. It is known that strain gradients, due to multi-axial loading conditions, have a significant influence on the hardening behavior of metallic materials in small dimensions. In several investigations it was found that the hardening scales inversely with the length in which plastic deformation occurs and thus is leading to increasing flow stresses with decreasing component size when strain gradients are present. This was demonstrated originally by Fleck et al. [1] for torsion of microwires. Consequently, it has been argued that the impact of strain gradients on the mechanical behavior has to be taken into account by mathematical descriptions of the deformation behavior of metals in small dimensions [2–8]. The basis of common strain gradient theories is the assumption that the strain distribution along the ⇑ Corresponding author. http://dx.doi.org/10.1016/j.actamat.2015.06.012 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
strain gradient is linear and that hardening arises from the accumulation of both, randomly stored – so called statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) [1,2,9]. SSDs exist and interact randomly and are associated with uniform deformation. When local strain gradients are present, GNDs are additionally stored in order to accommodate the lattice curvature. In this context, the size effect in hardening results from the fact that, for a given maximum strain, the strain gradient and thus the density of GNDs must be higher for smaller sample sizes. In order to investigate this so called extrinsic size effect and to create well defined stress and strain states with and without strain gradients, several setups for tension and torsion experiments on metallic microwires have been established [10–14]. Given by limitations of current manufacturing techniques, long metallic wires are naturally polycrystalline. To account for this, special emphasis has been given to the ratio of grain size and wire diameter and thereby it is proposed that the observed size effects are related to both, the relevant microstructural length scale (grain size,
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
dislocation density) and the sample size [14–17]. However, the interplay of the different length scales, particularly in combination with imposed strain gradients in torsion, is not known as their individual contribution to the overall size effects could not be quantified up to now [17]. Consequently, it is indeed questionable if the assumptions of the classical strain gradient theories with respect to strain distribution and accumulation of GNDs may hold for polycrystalline microstructures and if it is reasonable to apply them to describe torsion experiments of polycrystalline wires. In order to investigate the specific effect of a strain gradient, the interplay between grain size and strain gradient would have to be eliminated. For torsion experiments, where the radial strain distribution is assumed to be linear, this may be achieved by creating at least a single crystalline cross-section in the wire. Therefore, we propose an experimental approach to investigate the deformation structure within single crystalline cross-sections of bamboostructured Au microwires. The cross-sections of individual h1 0 0i oriented grains of twisted Au microwires are characterized by Laue microdiffraction to map the crystal misorientation along the strain gradient from the outer circumference of the wire to its center. Wires with increasing maximum plastic shear strain cr = R, pl, from:
cr ¼
u l
r
0 ðcenterÞ 6 r 6 R ðsurfaceÞ;
217
Fig. 1. Torsional response of Au microwires loaded up to defined maximum plastic shear strains.
ð1Þ
where u is the rotation angle and l the gauge length, are investigated to determine how the plastic deformation proceeds radially. The experiments are accompanied by crystal plasticity (CP) simulations to determine the equivalent plastic strain within the wires. Both, simulations and experiments, show that the local deformation structure within the cross-section of the wire is governed by the crystal orientation of the individual grain and its glide systems. The global deformation indeed shows a gradually increasing deformation from the center toward the surface, and thus reflects the overall strain gradient. 2. Experimental methods 2.1. Sample preparation As base material, polycrystalline Au wires made from high purity Au (99,99%, supplied by Heraeus, Germany) with a diameter D of 25 lm were used. To obtain a so called bamboo microstructure, the Au wires were annealed for 168 h at 800 °C using a high vacuum glass tube furnace. The torsion tests were performed using a custom-built torsion test setup. A detailed description of the sample mounting, experimental setup and data evaluation can be found in Ref. [10]. Au wires with a gauge length of 50 mm were deformed to different maximum plastic shear strains cr = R, pl of 0.4%, 0.6%, 1.0% and 1.6% with a constant rotation velocity of 0.086 rad/s for all tests (see Fig. 1). After twisting the wires, they were cut centric related to the gauge length and both pieces were glued with silver paint on a sample stub for preparation of defined cross-sections. The microstructure of the wires was characterized using a Dual Beam SEM-FIB microscope (FEI Nova NanoLab 200) additionally equipped with an electron-backscatter-diffraction (EBSD) system from Oxford Instruments. The scanning electron microscope (SEM) mode was used for imaging, the focused ion beam (FIB) for preparation of the individual cross-sections and the EBSD system for determining the crystal orientation of the grains. After annealing, the wire consists of a chain of mostly elongated cylindrical grains (Fig. 2(a)). The crystallographic orientation along the wires main axis z is predominantly [0 0 1] and [1 1 1], with the [0 0 1]-oriented grains being observed most frequently. Due to the majority of [0 0 1] oriented grains and their generally high
Fig. 2. (a) SEM micrograph showing the bamboo microstructure of an annealed Au wire with a diameter of 25 lm. (b) Selection of an [0 0 1] oriented grain based on EBSD and (c) ready-prepared cross-section of the selected grain.
aspect ratio, these grains were chosen systematically and individual cross-sections of wires with different cr = R, pl were cut by FIB (Fig. 2(b) and (c)). [0 0 1]-oriented grains with a length of at least 30 lm were chosen and the cross-sections were prepared in the center between two grain boundaries.
2.2. Laue microdiffraction The Laue microdiffraction experiments have been conducted at the X-ray microdiffraction beamline 12.3.2 of the Advanced Lights Source (ALS), Berkeley, USA [18]. Fig. 3 shows a schematic of the experimental setup and exemplary data of the experimental procedure. The sample stub was mounted on the stage of the 5-axis diffractometer of the beamline. The wires were scanned with a polychromatic X-ray microbeam (6–22 keV, 0.8 0.9 lm2 FWHM) in reflection geometry with a sample tilt of 45°. A large area detector (169 170 mm2, Pilatus 1M, Dectris) was positioned at a 2h angle of 90° and a sample to detector distance of about 100 mm. In addition, a laser distance measurement system, an optical microscope and a fluorescence detector system were available to
218
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
Fig. 3. (a) Schematic of the experimental setup at the microdiffraction beamline of the Advanced Light Source (ALS, Berkeley). (b) Fluorescence scan of the wire together with an exemplary Laue pattern of an individual pixel on the wire surface and a schematic illustrating the determination of the Kernel Average Misorientation angle (KAM). In this schematic, aci denote the angles between the orientation matrices Rc and Ri of the center of the kernel and the individual neighboring pixels, respectively. The fluorescence scan was used to locate the surface of the wire and Laue patterns were taken with a step size of 0.8 lm across the wire. The graded intensity in the fluorescence scan results from the inclination of the fluorescence detector with respect to the wire cross-section.
align the sample in the microbeam. Based on a fast fluorescence scan with a step size of 1 lm the final scanning area around the cross-section of individual Au microwires was defined. Subsequently, Laue diffraction patterns were collected with a step size of 0.8 lm (44 44 patterns). The individual Laue diffraction patterns were indexed using XMAS (X-ray Microdiffraction Analysis Software) [19] to determine the crystal orientation for each position of the raster scan in form of the full orientation matrix Ri. The orientation matrices were used to characterize the deformation structure within the single crystalline cross-section of the Au wires. In order to determine the local misorientation distribution, the Kernel Average Misorientation angle (KAM) between all individual data points and their eight nearest neighbors was calculated (see Fig. 3(b)). To characterize the global deformation behavior along the cross-section of the wires, the misorientation of the individual data points with respect to the center of the cross-section has been analyzed. Therefore, the center of the cross-sections have been determined by analyzing the local distribution of the integral intensity of all Laue spots within the individual Laue diffraction patterns. This integral intensity of single Laue diffraction patterns was summated along defined circles and by maximizing this total intensity along the circles, the center within the individual cross-section could be determined. Subsequently the rotation matrix Ric between the orientation matrices Rc and Ri of the center of the cross-section and the individual data points, respectively, was determined. Based on the rotation axis and rotation angle of Ric the misorientation vector mic was obtained. For the analysis of the misorientation within the wire cross-section only the geometric mean of the x- and y-component (in-plane components) of the misorientation vector was used. Finally the misorientation with respect to center of the cross-section was averaged over 15 ring segments starting from the center and plotted as function of the radial distance from the center.
strain in the energetic framework and the additional grain boundary yield criterion are neglected. Consequently, in contrast to Wulfinghoff et al. [21] no internal length scale is introduced in the framework. This is motivated by the focus on bamboo-like microwires with single crystalline cross-section. The goal from the modeling perspective is to investigate if the equivalent plastic strain within individual cross-sections can be used for comparison to the experimental quantities determined by Laue microdiffraction. However, one has to take into account, that the diffraction data only reflect elastic lattice distortions as result of the applied graded plastic deformation. A small strain framework is utilized and an equivalent plastic strain is introduced (see Table 1). Contrary to Wulfinghoff et al. [21] and for simplicity of reading all equations are presented here solely in terms of the equivalent plastic strain ceq. Thus, the numerically necessary additional micromorphic field variable f ceq is not shown in the theoretical summary. The stored energy contains Wh, an isotropic hardening contribution of Voce-type, besides the classic elastic Hooke-type energy We. In the field equations (derivable from the principle of virtual power) the Cauchy stress r, the scalar resolved shear stress sa on the individual octahedral slip systems a and the dissipative stresses sda are considered. The flow rule of overstress-type considers the hardening in form of the additional microstress b, the drag stress sD, and the reference shear rate c_ 0 as well as the strain rate sensitivity exponent p. In addition, for the isotropic Voce-hardening relation, the material parameters sC0 (initial yield stress), sC1 (saturation yield stress) and H (initial hardening modulus) need to be specified. The Finite Element implementation of the theory at hand is a special case of the more general implementation discussed in [21] and can be derived with the simplifications mentioned above. The boundary conditions are chosen such that a microwire is loaded with Dirichlet (displacement) boundary conditions. In the direction of the central specimen axis the displacement is unrestricted by the boundary conditions except at one node of the bottom surface of the cylinder. The spatial position of this node of the microwire is only fixed in x-direction. On the opposing top and bottom surface of the cylinder the in-plane displacements are then prescribed. Consequently, any possible rigid translation of the specimen in space is removed. Since only small deformation behavior is computationally investigated, the occurring warping deformations are negligibly small. Therefore, no artificial stress
Table 1 Theoretical model adapted from Wulfinghoff et al. [21] after neglecting of gradient contributions and grain boundary contributions. Additive decomposition Plastic strain
ee ¼ e ep
Equivalent plastic strain
ceq ð^kÞ ¼
Stored energy
W ¼ W e ðe; ep Þ þ W h ðceq Þ
Elastic energy
3. Crystal plasticity modeling Crystal plasticity models are intended to take into account dislocation activity and accumulation. A strain gradient crystal plasticity model incorporating an equivalent plastic strain accounting for intrinsic and extrinsic size effects and dislocation-induced plastic deformation on the continuum scale has been introduced in [20,21]. This has also been applied to microtorsion experiments on coarse grained Au wires [22]. However, in the present work on bamboo-structured microwires this model is used in a simplified version. The main difference of the theoretical framework at hand is that all gradient contributions of the equivalent plastic
Isotropic Voce hardening Linear momentum balance Slip system shear stress Traction BC Dirichlet BC Dissipation Flow rule & dissipative stress
ep ¼
X
k symðda na Þ a a Z X
a t
k_ a dt ¼
X
k ; k_ a P 0; a a
a ¼ 1 . . . 24
1 ðe ep Þ C½e ep 2 ! C 2 Hc 1 C W h ¼ s1 sC0 ceq þ s1 sC0 exp C eq C H s1 s0
We ¼
divðrÞ ¼ 0 8x 2 B
sa ðsda þ bÞ ¼ 0 8x 2 B; sa ¼ r symðda na Þ rn ¼ t on @Bt on @Bu u¼u X D¼ sd k_ a a a * +p sd sC0 k_ a ¼ c_ 0 a ;
sD
sda ¼ sa b; b ¼ @ ceq W h
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
concentrations are observed in the investigated torsion loading upon fixing the spatial position of one bottom surface node in x-direction. At the beginning of the simulation, the equivalent plastic strain is assumed to vanish throughout the model, i.e. the crystal is undeformed, completely. Although this assumption is not expected to be valid in general, e.g. in the case of predeformed microwires, it seems reasonable to use it in the context of investigating the overall strain gradient imposed on annealed microwires by CP simulations. The microwire with a diameter of 25 lm and a length of 31.25 lm is loaded up to 0.02 relative maximum shear of the top and bottom surfaces, and unloaded until the torque Mt, normalized by the wire radius R approximately vanishes, i.e. MT/R3 6 0.1 MPa. For the anisotropic elastic constants of gold values from the literature were chosen (C1111 = 186 GPa, C1122 = 157 GPa, C1212 = 42 GPa, cf. Ref. [23]). Since the determination of the material parameters related to initial yield strength and hardening would require extended experiments with completely single crystalline microwires seeing beyond the scope of this work, the same material parameters as in [21] are chosen here. Although the mentioned work considered copper and not gold microwires, still an fcc material with comparable anisotropy is used. The simulations were carried out with a mesh of 160,992 degrees of freedom. Convergence of the results is emphasized by the fact that an increase to 415,744 degrees of freedom leads to a decrease in the maximum relative error between the normalized-torque shear curves of less than 1%. 4. Results and discussion As already mentioned above, the global deformation field within a cross-section can be determined by calculating the misorientation of every data point with respect to the center element. Fig. 4 shows the resulting maps for wires twisted to a maximum plastic shear strain cr = R, pl = 0.4%, 0.6%, 1.0% and 1.6%, respectively. As the different deformed grains reveal a wide spread in misorientation, the scale is normalized by the related maximum misorientation. When comparing the maps it is seen that some of the cross-sections exhibit a graded rudimental symmetrical strain field, whereas the others show pronounced areas with stronger
219
orientation gradients, more or less opposing each other. To understand this non-uniform deformation behavior in detail, the following important feature has to be considered: If a h1 0 0i oriented single crystalline cylinder made of an fcc metal is ideally aligned concerning its rotation axis, exactly four independent slip systems will be simultaneously activated under torque with the onset of yielding. As a result, in contrast to the common idea of a graded annular deformation structure, an indeed symmetrical, but non-homogeneously graded strain field is generated within a cross-section as shown in Fig. 5. Here, CP simulations are used to depict the evolution of the spatial distribution of the equivalent plastic strain ceq, pl in dependence of the applied twist. Comparable to the torsion tests, the calculations were performed such that after a defined plastic deformation, the structures were unloaded again. The pronounced quadruple symmetry, however, disappears increasingly at certain misalignments. Fig. 6 shows this behavior exemplified by means of CP simulations for misalignments of 8° related to the y-axis (a) and for misalignments of 8° related to both the x-axis and the y-axis (b). The value of 8° was chosen based on the fact that the average tilt between the [0 0 1] orientation of the investigated grains and their individual rotation axis was found to be about 4° (determined by Laue experiments). Since the line of symmetry of the single grains may also reveal a small tilt related to the rotation axis of the entire wire z (compare Fig. 2, front part of the wire) an additional misalignment of 4° is assumed. The tilt between a wires rotation axis and the experimental rotation axis Z is 1°. Taking the results of the CP simulations into account, it becomes conceivable that the [0 0 1] direction of the chosen grains did not match exactly with the experimental rotation axis Z. It is obvious that some grains exhibited a slight but significant tilt mainly related to the x- or y-axis, whereas the other grains exhibited a slight but significant spatial tilt. Although the maps from Fig. 4 reveal local orientation gradients within the cross-sections, when averaging the misorientation for different annuluses between center and surface, an almost continuously graded deformation behavior is found for every sample as shown in Fig. 7(a). Although grains extracted from the same wire show big differences in overall misorientation, the general trend
Fig. 4. Normalized misorientation maps from cross-sections of different twisted bamboo-structured Au microwires with a diameter of 25 lm. The corresponding grains have orientations close to the [0 0 1]-orientation with respect to the experimental rotation axis Z. In this analysis, the rotation of every data point is referred to the center element and the misorientation is normalized to the maximum misorientation that occurred in the particular cross-section. The arrows show the in-plane h1 0 0i orientations.
220
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
Fig. 5. CP simulation: Evolution of the spatial distribution of the equivalent plastic strain with increasing twist within an ideally aligned h1 0 0i oriented single crystalline cylinder of Au.
Fig. 6. CP simulation: Evolution of the spatial distribution of the equivalent plastic strain with increasing twist within a misaligned h1 0 0i oriented single crystalline cylinder of Au – (a): misalignment of 8° related to the y-axis, (b): misalignment of 8° related to both the x- and y-axis.
of increasing misorientation with increasing twist is clearly visible when additionally averaging the results of grains extracted from the same wire (Fig. 7(b)).
Based on different orientations of the single grains, a non-homogeneous deformation behavior along a wires main axis during loading was expected naturally. However, that even similar
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
221
Fig. 7. Averaged misorientation vs distance to the center, calculated for the cross-sections shown in Fig. 4 – (a): individual grains and (b): averaged for both grains extracted from the same wire.
Fig. 8. Averaged equivalent plastic shear strain vs distance to center, calculated for the cross-sections shown in Figs. 5 and 6, respectively.
oriented grains exhibit such big differences in deformation is somehow surprising, particularly since according to CP simulations slight misalignments are only causing marginal differences in the equivalent plastic strain distribution, as illustrated in Fig. 8. When averaging ceq, pl along different ring segments for the cross-sections shown in Figs. 5 and 6, respectively, it is seen that the integral degree of deformation is decreasing moderately with increasing misalignment. It is feasible that the resulting deformation within a single grain is significantly affected by the deformation behavior of the adjacent grains, which is again depending on their orientation. This feature has to be taken into account in future investigations. In contrast to the curves depicted in Fig. 8, the curves depicted in Fig. 7 show an indeed graded, but not linearly increasing deformation from the center to the surface. Toward the outer part of the wire, the increase in lattice rotation is clearly less pronounced. Since in a single crystalline material from high purity elastically distortions are mainly resulting from both the interplay between dislocations and dislocation multiplication, it is argued that a significant number of dislocations is annihilating at the free surface during plastic deformation.
Fig. 9. Normalized KAM maps from cross-sections of different twisted bamboo-structured Au microwires with a diameter of 25 lm – the arrows show the in-plane h1 0 0i orientations.
222
M. Ziemann et al. / Acta Materialia 97 (2015) 216–222
Besides the identification of the global deformation behavior, the microdiffraction data were also used to determine the local deformation behavior within the twisted grains. The maps in Fig. 9 depict the KAM of every data point related to its encircling neighbors. Here, with exception of one grain from a wire twisted to cr = R, pl = 0.4% (almost no deformation), deformation traces are visible. A comparison of these extraordinary structures with the related global deformation fields (Fig. 4) shows that the traces are mainly correlating with the border areas of stronger deformed regions within a cross-section and the transition regions of small equivalent plastic strain (compare Figs. 5 and 6). Most of the traces reveal a pronounced graded shape and some additionally pass over the neutral fiber of the twisted grains. Based on classical ideas about deformation of metals [2] one may assume that the traces mainly consist of GNDs, usually required to accommodate stronger lattice curvatures. Although the misorientation data do not allow to distinguish between pure elastic distortions and distortions caused by dislocations, the rather strong misorientation gradients within the individual traces strongly support the assumption that GNDs are accumulating in this regions and that dislocation substructures have formed within the cross-section of the twisted wire. In order to verify this interpretation and to identify the role of dislocation arrangement discrete dislocation dynamics (DDD) simulations are currently underway. When viewing the results of the present investigations it is clearly seen, that Laue microdiffraction, supported by CP simulations, is generally an excellent tool to determine both the global and local deformation behavior within twisted microspecimen – at least for (pseudo) single crystalline metals. Based on both well selected samples and statistical analyses, the method can be used to study general aspects of deformation in torsion beyond yielding, as well as to clarify specific aspects, for example concerning the physical background of already observed size effects [1,12,17]. 5. Summary and conclusion In order to study the radial evolution of irreversible deformation within twisted thin bamboo-structured Au wires, Laue microdiffraction experiments were performed. For that purpose, wires with a diameter of 25 lm were twisted to different degrees of maximum plastic strain. Subsequently, EBSD investigations were carried out on central pieces of the samples to determine the crystal orientation of individual grains with respect to the longitudinal axis z of the wires. Based on the EBSD measurements, sufficiently long, almost [0 0 1] oriented grains were centrically cut perpendicular to the wires main axis by means of the FIB technique. The such prepared cross-sections were then investigated using Laue microdiffraction. Supported by CP simulations it is found, that the determined global deformation structures are directly related to the activated slip systems, resulting from the real orientations of the investigated grains concerning their rotation axis. Furthermore, when averaging the degree of deformation for different annuluses, an almost continuous increase of strain from the center toward the surface is observed for every investigated sample. An important finding is that the increase in deformation is less pronounced toward the surface. It is argued that during twisting a significant number of dislocations annihilates at the free surface. This is assumed to be also valid for coarse grained microwires and thus, should be considered in modeling. In addition to the global deformation behavior, the local deformation behavior was investigated by means of KAM calculations. Here, pronounced deformation traces within the cross-sections are observed. When comparing these traces with the maps showing the global
deformation behavior it is found that they are mainly located at border areas of stronger deformed regions. It is assumed that the strong misorientation gradient within the traces is correlated to the formation of dislocation substructures within the twisted wire. Acknowledgments The authors acknowledge the support rendered by the German Research Foundation (DFG) under Grants BO1466/5-1 and GR 3677/2-1. The funded projects ‘‘Dislocation based Gradient Plasticity Theory’’ and ‘‘Experimental Characterization of Micro Plasticity and Dislocation Microstructure’’ are part of the DFG Research Group 1650 ‘‘Dislocation based Plasticity’’. References [1] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain Gradient Plasticity – Theory and Experiment, Acta Metall. Mater. 42 (2) (1994) 475–487. [2] M.F. Ashby, The deformation of plastically non-homogeneous materials, Philos. Mag. 21 (170) (1970) 399–424. [3] R. Deborst, H.B. Muhlhaus, Gradient-dependent plasticity – formulation and algorithmic aspects, Int. J. Numer. Methods Eng. 35 (3) (1992) 521–539. [4] H. Gao, Y. Huang, W.D. Nix, J.W. Hutchinson, Mechanism-based strain gradient plasticity – I. Theory, J. Mech. Phys. Solids 47 (6) (1999) 1239–1263. [5] W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solids 46 (3) (1998) 411–425. [6] N.A. Fleck, J.W. Hutchinson, A phenomenological theory for strain gradient effects in plasticity, J. Mech. Phys. Solids 41 (12) (1993) 1825–1857. [7] N.A. Fleck, J.W. Hutchinson, Strain gradient plasticity, in: Advances in Applied Mechanics, Elsevier, 1997, pp. 295–361. [8] D.J. Dunstan, B. Ehrler, R. Bossis, S. Joly, K.M.Y. Ping, A.J. Bushby, Elastic limit and strain hardening of thin wires in torsion, Phys. Rev. Lett. 103 (15) (2009) 155501. [9] J.F. Nye, Some geometrical relations in dislocated crystals, Acta Metall. 1 (2) (1953) 153–162. [10] M. Walter, O. Kraft, A new method to measure torsion moments on smallscaled specimens, Rev. Sci. Instrum. 82 (3) (2011) 035109. [11] D.J. Dunstan, J.U. Gallé, B. Ehrler, N.J. Schmitt, T.T. Zhu, X.D. Hou, K.M.Y. P’ng, G. Gannaway, A.J. Bushby, Micromechanical testing with microstrain resolution, Rev. Sci. Instrum. 82 (9) (2011) 093906. [12] D. Liu, Y. He, X. Tang, H. Ding, P. Hu, P. Cao, Size effects in the torsion of microscale copper wires: experiment and analysis, Scr. Mater. 66 (6) (2012) 406–409. [13] W.-Y. Lu, B. Song, Quasi-static torsion characterization of micro-diameter copper wires, Exp. Mech. 51 (5) (2011) 729–737. [14] B. Yang, C. Motz, M. Rester, G. Dehm, Yield stress influenced by the ratio of wire diameter to grain size – a competition between the effects of specimen microstructure and dimension in micro-sized polycrystalline copper wires, Philos. Mag. 92 (2012) 3243–3256. [15] X.X. Chen, A.H.W. Ngan, Specimen size and grain size effects on tensile strength of Ag microwires, Scr. Mater. 64 (2011) 717–720. [16] D.J. Dunstan, A.J. Bushby, Grain size dependence of the strength of metals: the Hall–Petch effect does not scale as the inverse square root of grain size, Int. J. Plast. 53 (2014) 56–65. [17] Y. Chen, O. Kraft, M. Walter, Size effects in thin coarse-grained gold microwires under tensile and torsional loading, Acta Mater. 87 (2015) 78–85. [18] M. Kunz, N. Tamura, K. Chen, A.A. MacDowell, R.S. Celestre, M.M. Church, S. Fakra, E.E. Domning, J.M. Glossinger, J.L. Kirschman, G.Y. Morrison, D.W. Plate, B.V. Smith, T. Warwick, V.V. Yashchuk, H.A. Padmore, E. Ustundag, A dedicated superbend X-ray microdiffraction beamline for materials, geo-, and environmental sciences at the advanced light source, Rev. Sci. Instrum. 80 (3) (2009) 035108. [19] N. Tamura, A.A. MacDowell, R. Spolenak, B.C. Valek, J.C. Bravman, W.L. Brown, R.S. Celestre, H.A. Padmore, B.W. Batterman, J.R. Patel, Scanning X-ray microdiffraction with submicrometer white beam for strain/stress and orientation mapping in thin films, J. Synchrotron Radiat. 10 (2) (2003) 137– 143. [20] S. Wulfinghoff, T. Böhlke, Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics, Proc. R. Soc. A 468 (2012) 2682– 2703. [21] S. Wulfinghoff, E. Bayerschen, T. Böhlke, A gradient plasticity grain boundary yield theory, Int. J. Plast. 51 (2013) 33–46. [22] E. Bayerschen, S. Wulfinghoff, T. Böhlke, Application of strain gradient plasticity to micro-torsion experiments, Proc. Appl. Math. Mech. 14 (2014) 313–314. [23] J. Rösler, H. Harders, M. Bäker, Elastisches Verhalten, in: Mechanisches Verhalten der Werkstoffe, Teubner, 2006, pp. 31–61.