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Plastic deformation: Shearing mountains atom by atom
Journal of Alloys and Compounds 577S (2013) S96–S101
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Plastic deformation: Shearing mountains atom by atom Peter Müllner Department of Materials Science and Engineering, Boise State University, Boise, ID 83725, USA
a r t i c l e
i n f o
Article history: Received 27 October 2011 Accepted 1 March 2012 Available online 23 March 2012 Keywords: Plastic deformation Disclination Twinning Hierarchical Shear band Metal Ceramic Composite Rock
a b s t r a c t Conventional wisdom established atomistic defects, dislocations, as agents of plastic deformation. On macroscopic scale, rock, wood, steel, tough ceramics, fiber reinforced composites, and silicon all deform in the same way and produce the same pattern; shear bands. The argumentation presented here, starts on the largest length scale of the problem at hand and leads through a number of hierarchical levels down to the atomistic mechanism. Shear bands develop discontinuously by the motion of a process zone. Locally, i.e. in the process zone, deformation proceeds perpendicularly to the macroscopic shear, in combination with a rotation. The microscopic shear itself may occur again in a discontinuous manner and again orthogonally to the intermediate level and so on at ever smaller scale. Material properties come into play at the highest hierarchical level, i.e. at the smallest length scale where they control the well-known micromechanisms. © 2012 Elsevier B.V. All rights reserved.
1. Introduction A large variety of materials deform in the same way. This statement may be debated when considering microscopic deformation mechanisms. Much work has been devoted to the study of deformation mechanisms at the atomic scale and this is not the topic here. Instead, we focus on the largest length scale of a given experiment yet without abandoning the discreteness of deformation. Conventional thought explores and describes mechanical properties of materials based on their structural properties and microstructure. The key length scale is the length scale of the micromechanisms, which ranges from atomic scale to the characteristic scale of the microstructure [1]. This school of thought has proven very successful. It has produced high performance materials with extraordinary properties including high strength combined with high toughness, creep resistance, light-weight combined with high strength and so on. Materials science teaches that plastic deformation proceeds through the motion of lattice dislocations, i.e. line defects with a translational displacement field. The Burgers vector and the lattice potential control the mobility of dislocations. The stress field of a dislocation decreases inversely with the distance from the dislocation line and evokes strong long-range dislocation–dislocation interaction. The mutual long-range interactions of many dislocations control the mechanical properties at large plastic strain.
E-mail address: [email protected] 0925-8388/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2012.03.002
Analytical solutions are not available and numerical calculations are cumbersome. Multi-scale modeling and discrete item dynamics have overcome to some extent these limitations (e.g. [2,3]). Yet, these methods hold to the old paradigm and take the individual dislocation as starting point and extrapolate to large scale. Here, we present an alternative approach. Instead of studying deformation on atomic scale and working our way up to larger length scales, we start at the largest length scale available and work our ways down to the microscopic scale. We do so by looking at various materials including rock, ceramics, fiber reinforced composites, wood, and metals in Section 2. We notice remarkable similarities across the length scales and suggest that these similarities bear the key to large strain deformation. We review the concept of orthogonal shear in Section 3 and outline properties of disclinations, which are useful for the study of shear bands, in Section 4. In Section 5, we discuss shear deformation as discrete and discontinuous events and give the conclusions in Section 6.
2. Shear bands across the length scales One might consider the formation of the Alps in Europe as the largest indentation experiment on Earth. Africa and Italy take on the roles of the drive and the indenter tip. Europe is the material under test and the Alps make the plastic zone. The bending and folding of geological layers and strata mark the shear bands or kink bands, which may be tens or even hundreds of meters wide. Geological kink bands and shear bands in rock (Fig. 1a) actually cover many
P. Müllner / Journal of Alloys and Compounds 577S (2013) S96–S101
S97
Fig. 1. Shear bands form over twelve orders of magnitude in all types of materials: (a) rock (Courtesy of C.J. Northrup, Department of Geosciences at Boise State University.); (b) fiber reinforced composite (reprinted from [5] “Mechanisms of kink-band formation in graphite/epoxy composites: a micromechanical experimental study,” with permission from Elsevier); (c) MAX-phase ceramics (reprinted from [6] “The MAX phases: unique new carbide and nitride materials,” with permission from Sigma Xi); (d) germanium (reprinted from [11] “On diamond-hexagonal germanium,” with permission from Cambridge).
length scales from hundreds of meters down to millimeters or even less. In wood [4] and fiber reinforced composites [5], shear bands appear with thickness in the millimeter and sub-millimeter ranges (Fig. 1b). The detailed structure of these bands is related to the composite morphology and differs from the local structure of the shear bands in rock. The macroscopic pattern, however, is strikingly similar to that of the geological shear bands. Again smaller but very similar in appearance are the shear bands which make MAX-phase ceramics ductile (Fig. 1c, [6]). Shear bands occur in many other materials including steel [7], tungsten [8], aluminum [9], silicon [10] and germanium (Fig. 1d, [11]); and their width reaches over twelve orders of magnitude from angstroms to hundreds of meters. The shear bands have a characteristic zigzag pattern in common. For wood and fiber reinforced polymers, the zigzag simply traces the deformation of the fibers. In other cases including MAX-phase ceramics and semiconductor germanium (Fig. 1c and d), the formation of this pattern is less intuitive. In contrast, the “flow of the pattern” seams to oppose the flow of the shear deformation. As we will see in Section 3, it is this “unnatural flow” which facilitates the formation of shear bands.
Collective glide of identical dislocations on parallel but separate glide planes is the simplest way to produce a simple shear by the motion of dislocations. In many cases, however, there are no glide systems at hand for this mechanism, at least not on the macroscopic length scale. A simple shear S1 (we call it primary shear with shear magnitude s1 ) can be decomposed into an orthogonal (also “secondary” or
3. Orthogonal (or secondary) shear plus rotation The macroscopic deformation produced by a shear band is a simple shear as the zigzag pattern visible in many shear bands illustrates. With “macroscopic,” we refer here to the length scale of the entire sample or the shear band thickness. The word “microscopic” is reserved to details on a scale at least one order of magnitude smaller than the shear band thickness.
Fig. 2. Decomposition of a simple shear (a and b) into (c) an orthogonal shear of identical magnitude plus (d) a rotation [7,12].
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Fig. 3. Equivalent defect configurations. (a) A wall of disclination dipoles. (b) The disclination dipoles with separation 2a1 are considered as dislocation walls. (c) The disclination dipoles with separation 2a2 are considered as dislocation walls. (d) Dislocations are partitioned between both types of dipoles. (e) The disclination dipoles 2a1 are considered each as a super dislocation. (f) The disclination dipoles 2a2 are considered each as a super dislocation. Reprinted from [17] “Deformation of hierarchically twinned martensite,” with permission from Elsevier.
“second order”) shear S2 (magnitude s2 ) plus a rotation R (rotation angle ˝, Fig. 2, [7,12]): S1 = RS2
(1)
The shear magnitudes s1 and s2 and arc ˝ are all equal. The symmetry of the stress state provides that the same shear stress which supports the shear S1 also supports the shear S2 . While S2 and R appear as separated entities in Eq. (1), they actually occur simultaneously. In situations, where no glide systems are available for S1 , there might be glide systems available for S2 . This is the case for the formation of shear bands in diamond cubic silicon and germanium (Fig. 1d, [10,11]). Here, transformation dislocations produce the secondary shear and form the diamond hexagonal structure [13]. If there is no suitable glide system available for S2 , the secondary shear may again be decomposed into a third order shear S3 and a rotation R , which is the inverse of R:
S2 = R S3 = R
−1
S3
(2)
And, thus, the primary and the third order shears are equal: S1 = RS2 = RR−1 S3 = S3
(3)
What then is the purpose of this exercise? If we were to consider homogenous deformation, the answer would be: nothing. However, we are considering heterogenous deformation via the formation of shear bands. S1 does not occur homogenously throughout the specimen but only localized in the shear band. Similarly, S2 does not occur homogenously throughout the entire shear band but only in localized volumes at a smaller length scale. S3 then does not occur homogenously throughout the region of S2 but again only localized in certain sub-regions, which we call process zones. Continuing this concept, a quasi-fractal nature of shear deformation evolves. This is not truly a fractal phenomenon because it has an upper cut-off dimension, the sample size, and a lower cutoff dimension. The lower cut-off dimension is the length scale of the microscopic mechanism. In some materials such as fiber reinforced composites and wood, there may be only two hierarchical levels, namely the primary level of the macroscopic shear band and the secondary level which is the micromechanism. In geological “experiments” manifested in deformation of the tectonic plates
and also in nano-crystalline materials, the lower cut-off dimension may be given by the size of individual grains. Here, many orders of magnitude separate the thickness of the primary shear band from the characteristic length of the micromechanism (e.g. [14]). Several hierarchical orders bridge between macroscopic shear and atomistic displacement. It is only at the smallest length scale that the nature of the material and the condition of deformation (orientation, temperature, strain rate) play a role. In metals, dislocation motion is predominant. In materials with self-accommodated martensite, deformation occurs via twinning (e.g. [15]). In geological and nano-crystalline materials, grain boundary sliding and thermally activated processes may control deformation; microcracking and delamination may occur in ceramics and composites. 4. Shear, dislocations and disclinations We make use of the disclination concept [16] in a similar way as for the description of hierarchical twinning in self-accommodated martensite [17]. A recent review on disclinations appeared in Ref. [18]. A disclination is a line defect similar to a dislocation, albeit with a rotational displacement field characterized by the Frank vector ω, with magnitude ω. In contrast, a dislocation has a translational displacement field characterized by the Burgers vector b, with magnitude b. The treatment is here limited to straight wedge disclinations and edge dislocations and the lines of all defects are parallel. For wedge disclinations, the Frank vector is parallel to and coincident with the disclination line. A single wedge disclination compares to a semi-infinite wall of infinitesimal edge dislocations. Partial disclinations are located at terminating small-angle boundaries [16] and grain boundary triple junctions [19], particularly in nano-grained materials [20,21]. Because of their large strain energy, disclinations tend to agglomerate in self-screening dipoles and quadrupoles [22]. Two wedge disclinations of equal strength |ω|, one with a positive and one with a negative sign represent a disclination dipole. There are many dislocation arrangements being equivalent to a disclination dipole. A finite dislocation wall is but one such equivalent. With equivalent we mean here that the respective defect configurations have the same long-range stress field. Close to the defect lines, i.e. within a distance from the lines equal to the spacing between defects, the
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Fig. 4. Continuous vs. discontinuous formation of a shear band. (a) Undistorted microstructure. The grid lines mark a reference frame of a suitable length scale (not necessarily the crystal lattice). (b) In an intermediate state, the distortions cause interfacial energy with enhanced chemical and/or elastic energy. (c) When the shear band is complete, the elastic component of the interfacial energy has relaxed. The interfacial energy has a minimum. (d) When the shear band forms discontinuously, the interfacial energy is minimal. The process zone (circle) carries a long-range strain field similar to that of a disclination dipole.
stress fields may vary significantly. In the extreme, a disclination dipole may be considered as a single super dislocation with Burgers vector B = 2aω, where 2a is the distance between the disclination lines. Fig. 3 depicts a wall of disclinations (a) and a number of equivalent dislocation representations (b–f). None of these representations is per se better than any other. Depending on the situation, one or the other might be more useful, though. We will see that in some situations, we need to transition from one description to another. Care must be applied regarding not to confuse different descriptions and not to double count displacements. A moving dislocation translates one half of the sample with respect to the other half by the Burgers vector. A moving disclination rotates one half of the sample with respect to the other by the amount of its strength. When a disclination dipole moves perpendicularly to the plane containing both lines, it produces a homogenous simple shear between the two planes swept by the disclination lines. Hence, a moving disclination dipole produces the displacement field of a shear band. 5. Discontinuous deformation on different hierarchical levels In 1934, Polanyi [23], Orowan [24], and Taylor [25] independently proposed that the lattice dislocation is the agent of plastic deformation. A few scientists, foremost Nabarro [26], advanced the theory of dislocations rigorously. However, the scientific community working on plasticity received the dislocation concept reluctantly until Hirsch et al. [27] and Bollmann [28], also working independently, provided experimental evidence in 1956. The advent of transmission electron microscopy introduced the “golden age” of plasticity. The dislocation is the agent of plasticity because its motion requires far less energy than the homogenous glide of the entire crystal. Homogenous glide of the entire crystal would require the simultaneous breaking of all bonds along the entire glide plane. In contrast, when a dislocation moves, bonds are broken only along the dislocation line. If a shear band formed continuously, each kink wall would evolve simultaneously across its entire length. The transition from the undistorted state (Fig. 4a) to the kinked state (Fig. 4c) then contained an intermediated transition state (Fig. 4b) of high “interfacial” energy. The interfacial energy may contain chemical and elastic components. In the analogy of rigid glide as opposed to
dislocation motion, the formation of the high-energy state (Fig. 4b) corresponds to the breaking of many bonds along the entire glide plane. If the shear band forms discontinuously by the motion of a process zone (Fig. 4d), the interfacial energy is limited to a local region, i.e. the process zone (circle in Fig. 4d). The process zone takes on the role of the dislocation in the above analogue. The process zone carries a strain field with the long-range properties of a disclination dipole. This strain energy of the disclination dipole depends logarithmically on the screening length (as does the strain energy of a dislocation), quadratically on the shear band thickness, and also quadratically on the disclination strength which equals the shear magnitude [16,18]. Thus, for thin shear bands, the strain energy is very small. In contrast, the interfacial energy depends linearly on the sample size and is independent of the shear band thickness. Hence, for thin shear bands, the interfacial energy is larger than the strain energy. For thin enough shear bands, therefore, discontinuous shear deformation is energetically favorable compared to homogenous shear deformation. “Thin enough” is so small compared to the sample size that the linear dependence of the interfacial energy overruns the logarithmic dependence of the strain energy of the disclination dipole. The deformation of self-accommodated martensite with a hierarchical twin microstructure illustrates the general principle [17]. In such microstructures, disclinations are located wherever twin boundaries of different hierarchical levels interact (Fig. 5a). At the highest hierarchical level (i.e. where twins do not contain further defects) the disclinations form disclination dipole walls. In Fig. 5b, walls of twinning dislocations represent the disclination dipoles. These twinning dislocations provide the agent for the microscopic deformation mechanism. The net defect content of the entire microstructure corresponds to a disclination dipole on the primary length scale (Fig. 5c). The three defect arrangements all have the same long-range strain field, which is most conveniently represented by the primary disclination dipole (Fig. 5c). The secondary disclination dipoles (Fig. 5a) provide the simplest model to evaluate the interfacial elastic energy component of the horizontal interface. The most detailed model with the twinning dislocations (Fig. 5b) describes the microscopic deformation mechanism. Thus, each description has its role and justification. Fig. 6 gives the interfacial energy as a function of the fraction, , of the highest order twins (or internal twins). Where is discontinuous along an interface, there is a disclination at a lower hierarchical order (large
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Fig. 5. Disclinations are present where twin boundaries interact (a). The disclinations of the internal (i.e. highest order) twins give rise to a strain energy component of the interfacial energy (Fig. 6). (b) The disclination dipoles of the internal twins can be represented by twinning dislocations. The net content of all secondary disclinations (a) or all twinning dislocations (b) corresponds to a primary disclination dipole (c). The primary disclination dipole moves laterally, when the twinning dislocations move vertically.
triangles in Fig. 5c). The interfacial energy differs on both sides of the lower order disclination. The discontinuity in Fig. 5 can move to the left or to the right by the motion of twinning dislocations along the boundaries of
Fig. 6. Interfacial strain energy as a function of density, , of internal twins. The interfacial energy has a maximum when = 0.5, which corresponds to the configuration on the right side of Fig. 5.
internal twins. These twinning dislocations move perpendicularly to the displacement of the primary disclination dipole. As the disclination dipole moves sideward, its structure and energy do not change. The only energy invested is due to the difference in interfacial energy (Fig. 6). As a result, the dipole moves under a much smaller shear stress than what would be required to increase continuously throughout the entire band [17]. We may now expand this thought to a more complex situation where deformation occurs on multiple hierarchical levels. The shear band forms by the orthogonal shear of second order twins. The shear of the second order twins is carried by third order twins, i.e. again by orthogonal shear and so on. Thus, higher order shear mechanisms (higher order twins in this case) are the agents of lower order shear (lower order twins), as illustrated in Fig. 7a. There is a process zone at each hierarchical level, which may be modeled as a disclination dipole. The self stress, micro , of the dipole enhances locally the macroscopic shear stress, macro (Fig. 7b) and triggers shear of the next higher order, i.e. deformation at lower length scale. Thus, the motion of the dipole is autocatalytic. The stress state at the largest length scale and the macroscopic properties of the shear band reach through all hierarchical levels down to the shortest length scale and control the micromechanism at atomistic length scale. It is only at the smallest length scale that properties such as number and nature of glide systems and the micromechanisms
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Fig. 7. Propagation of a process zone with higher order twins. (a) The primary structure is indicated with bold lines and large triangles. In the primary process zone (large circle), secondary shear occurs orthogonally to the primary shear. In the secondary process zones (small circles), tertiary shear occurs again on a smaller length scale, and orthogonally to the secondary shear. (b) Within the process zone, the stress field, micro , of the primary disclination dipole enhances the macroscopic stress field, macro , and triggers the operation of the secondary shear system.
(including dislocation glide, twinning, grain boundary sliding, thermally activated processes, cracking, and transformations) come into play.
Fig. 1a. Financial support from NSF under project no. DMR-1008167 is thankfully acknowledged. References
6. Conclusions Macroscopic deformation of materials occurs via the formation of shear bands. Within shear bands, shear occurs orthogonally to the macroscopic shear. On even smaller length scale, deformation can occur again orthogonally to the intermediate shear level. Thus, shear deformation proceeds on hierarchical levels where on each higher level (i.e. at smaller length scale), deformation occurs orthogonally to shear at the next lower hierarchical level (i.e. next larger length scale). At each length scale, shear is discontinuous. A process zone, which can be represented by a disclination dipole, carries shear deformation and thereby reduces the energy necessary for deformation. Throughout all length scales and hierarchical orders, the lower order deformation controls the higher order deformation. Only on the smallest length scale appear the microscopic deformation mechanisms and only at this length scale the classical, dislocation-based plasticity theory comes into play. The macroscopic loading conditions govern the deformation mode through a hierarchical mechanism of coupled orthogonal shear systems, linking large scale strains to smaller scale strains. Localized process zones, which act as agents of deformation on all hierarchical levels, and orthogonal shear systems shape mountains and displace atoms alike. Acknowledgements The author is grateful to Clyde J. Northrup of the Department of Geosciences at Boise State University for providing content of
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
E. Arzt, Acta Mater. 46 (1998) 5611. S. Brinckmann, J.Y. Kim, J.R. Greer, Phys. Rev. Lett. 100 (2008) 155502. S. Fertig, S.P. Baker, Prog. Mater. Sci. 54 (2009) 874. L. Benabou, Mech. Mater. 42 (2010) 335. S. Narayana, L.S. Schadler, Compos. Sci. Technol. 59 (1999) 2201. M.W. Barsum, T. El-Raghy, Am. Sci. 89 (2001) 334. A. Zisman, C. Teodosiu, V. Rybin, Comput. Mater. Sci. 16 (1999) 307. A. Luft, Prog. Mater. Sci. 35 (1991) 97. R. Nogueira de Coes, O.S. Hopperstad, O. Engler, O.-G. Lademo, J.D. Embury, A. Benallal, Metall. Mater. Trans. A 42 (2011) 3358–3369. P. Pirouz, R. Chaim, U. Dahmen, K.H. Westmacott, Acta Metall. Mater. 38 (1990) 313. S.Q. Xiao, P. Pirouz, J. Mater. Res. 7 (1992) 1406. V.E. Panin, Russ. Phys. J. 41 (1998) 1. P. Müllner, P. Pirouz, Mater. Sci. Eng. A 233 (1997) 139. R.Z. Valiev, I.V. Alexandrov, Y.T. Zhu, T.C. Lowe, J. Mater. Res. 17 (2002) 5. T. Saburi, C.M. Wayman, K. Takata, S. Nenno, Acta Metall. 28 (1980) 15. A.E. Romanov, V.I. Vladimirov, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, vol. 9, Elsevier, Amsterdam, 1992, p. 191. P. Müllner, A.H. King, Acta Mater. 58 (2010) 5242. A.E. Romanov, A.L. Kolesnikova, Prog. Mater. Sci. 54 (2009) 740. A. King, Interface Sci. 7 (1999) 251. P. Müllner, M. Kuschke, Scripta Mater. 36 (1997) 1451. R.D. Nyilas, M. Kobas, R. Spolenak, Acta Mater. 57 (2009) 3738. A.E. Romanov, Mater. Sci. Eng. A 164 (1993) 58. M. Polanyi, Z. Phys. 89 (1934) 660 (in German). E. Orowan, Z. Phys. 89 (1934) 634 (in German). G.I. Taylor, Proc. R. Soc. 145 (1934) 362. D. Kuhlmann-Wilsdorf, Prog. Mater. Sci. 54 (2009) 707. P.B. Hirsch, R.W. Horne, M.J. Whelan, Philos. Mag. 1 (1956) 677. W. Bollmann, Phys. Rev. 103 (1956) 1588.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Plastic deformation: Shearing mountains atom by atom Peter Müllner Department of Materials Science and Engineering, Boise State University, Boise, ID 83725, USA
a r t i c l e
i n f o
Article history: Received 27 October 2011 Accepted 1 March 2012 Available online 23 March 2012 Keywords: Plastic deformation Disclination Twinning Hierarchical Shear band Metal Ceramic Composite Rock
a b s t r a c t Conventional wisdom established atomistic defects, dislocations, as agents of plastic deformation. On macroscopic scale, rock, wood, steel, tough ceramics, fiber reinforced composites, and silicon all deform in the same way and produce the same pattern; shear bands. The argumentation presented here, starts on the largest length scale of the problem at hand and leads through a number of hierarchical levels down to the atomistic mechanism. Shear bands develop discontinuously by the motion of a process zone. Locally, i.e. in the process zone, deformation proceeds perpendicularly to the macroscopic shear, in combination with a rotation. The microscopic shear itself may occur again in a discontinuous manner and again orthogonally to the intermediate level and so on at ever smaller scale. Material properties come into play at the highest hierarchical level, i.e. at the smallest length scale where they control the well-known micromechanisms. © 2012 Elsevier B.V. All rights reserved.
1. Introduction A large variety of materials deform in the same way. This statement may be debated when considering microscopic deformation mechanisms. Much work has been devoted to the study of deformation mechanisms at the atomic scale and this is not the topic here. Instead, we focus on the largest length scale of a given experiment yet without abandoning the discreteness of deformation. Conventional thought explores and describes mechanical properties of materials based on their structural properties and microstructure. The key length scale is the length scale of the micromechanisms, which ranges from atomic scale to the characteristic scale of the microstructure [1]. This school of thought has proven very successful. It has produced high performance materials with extraordinary properties including high strength combined with high toughness, creep resistance, light-weight combined with high strength and so on. Materials science teaches that plastic deformation proceeds through the motion of lattice dislocations, i.e. line defects with a translational displacement field. The Burgers vector and the lattice potential control the mobility of dislocations. The stress field of a dislocation decreases inversely with the distance from the dislocation line and evokes strong long-range dislocation–dislocation interaction. The mutual long-range interactions of many dislocations control the mechanical properties at large plastic strain.
E-mail address: [email protected] 0925-8388/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2012.03.002
Analytical solutions are not available and numerical calculations are cumbersome. Multi-scale modeling and discrete item dynamics have overcome to some extent these limitations (e.g. [2,3]). Yet, these methods hold to the old paradigm and take the individual dislocation as starting point and extrapolate to large scale. Here, we present an alternative approach. Instead of studying deformation on atomic scale and working our way up to larger length scales, we start at the largest length scale available and work our ways down to the microscopic scale. We do so by looking at various materials including rock, ceramics, fiber reinforced composites, wood, and metals in Section 2. We notice remarkable similarities across the length scales and suggest that these similarities bear the key to large strain deformation. We review the concept of orthogonal shear in Section 3 and outline properties of disclinations, which are useful for the study of shear bands, in Section 4. In Section 5, we discuss shear deformation as discrete and discontinuous events and give the conclusions in Section 6.
2. Shear bands across the length scales One might consider the formation of the Alps in Europe as the largest indentation experiment on Earth. Africa and Italy take on the roles of the drive and the indenter tip. Europe is the material under test and the Alps make the plastic zone. The bending and folding of geological layers and strata mark the shear bands or kink bands, which may be tens or even hundreds of meters wide. Geological kink bands and shear bands in rock (Fig. 1a) actually cover many
P. Müllner / Journal of Alloys and Compounds 577S (2013) S96–S101
S97
Fig. 1. Shear bands form over twelve orders of magnitude in all types of materials: (a) rock (Courtesy of C.J. Northrup, Department of Geosciences at Boise State University.); (b) fiber reinforced composite (reprinted from [5] “Mechanisms of kink-band formation in graphite/epoxy composites: a micromechanical experimental study,” with permission from Elsevier); (c) MAX-phase ceramics (reprinted from [6] “The MAX phases: unique new carbide and nitride materials,” with permission from Sigma Xi); (d) germanium (reprinted from [11] “On diamond-hexagonal germanium,” with permission from Cambridge).
length scales from hundreds of meters down to millimeters or even less. In wood [4] and fiber reinforced composites [5], shear bands appear with thickness in the millimeter and sub-millimeter ranges (Fig. 1b). The detailed structure of these bands is related to the composite morphology and differs from the local structure of the shear bands in rock. The macroscopic pattern, however, is strikingly similar to that of the geological shear bands. Again smaller but very similar in appearance are the shear bands which make MAX-phase ceramics ductile (Fig. 1c, [6]). Shear bands occur in many other materials including steel [7], tungsten [8], aluminum [9], silicon [10] and germanium (Fig. 1d, [11]); and their width reaches over twelve orders of magnitude from angstroms to hundreds of meters. The shear bands have a characteristic zigzag pattern in common. For wood and fiber reinforced polymers, the zigzag simply traces the deformation of the fibers. In other cases including MAX-phase ceramics and semiconductor germanium (Fig. 1c and d), the formation of this pattern is less intuitive. In contrast, the “flow of the pattern” seams to oppose the flow of the shear deformation. As we will see in Section 3, it is this “unnatural flow” which facilitates the formation of shear bands.
Collective glide of identical dislocations on parallel but separate glide planes is the simplest way to produce a simple shear by the motion of dislocations. In many cases, however, there are no glide systems at hand for this mechanism, at least not on the macroscopic length scale. A simple shear S1 (we call it primary shear with shear magnitude s1 ) can be decomposed into an orthogonal (also “secondary” or
3. Orthogonal (or secondary) shear plus rotation The macroscopic deformation produced by a shear band is a simple shear as the zigzag pattern visible in many shear bands illustrates. With “macroscopic,” we refer here to the length scale of the entire sample or the shear band thickness. The word “microscopic” is reserved to details on a scale at least one order of magnitude smaller than the shear band thickness.
Fig. 2. Decomposition of a simple shear (a and b) into (c) an orthogonal shear of identical magnitude plus (d) a rotation [7,12].
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Fig. 3. Equivalent defect configurations. (a) A wall of disclination dipoles. (b) The disclination dipoles with separation 2a1 are considered as dislocation walls. (c) The disclination dipoles with separation 2a2 are considered as dislocation walls. (d) Dislocations are partitioned between both types of dipoles. (e) The disclination dipoles 2a1 are considered each as a super dislocation. (f) The disclination dipoles 2a2 are considered each as a super dislocation. Reprinted from [17] “Deformation of hierarchically twinned martensite,” with permission from Elsevier.
“second order”) shear S2 (magnitude s2 ) plus a rotation R (rotation angle ˝, Fig. 2, [7,12]): S1 = RS2
(1)
The shear magnitudes s1 and s2 and arc ˝ are all equal. The symmetry of the stress state provides that the same shear stress which supports the shear S1 also supports the shear S2 . While S2 and R appear as separated entities in Eq. (1), they actually occur simultaneously. In situations, where no glide systems are available for S1 , there might be glide systems available for S2 . This is the case for the formation of shear bands in diamond cubic silicon and germanium (Fig. 1d, [10,11]). Here, transformation dislocations produce the secondary shear and form the diamond hexagonal structure [13]. If there is no suitable glide system available for S2 , the secondary shear may again be decomposed into a third order shear S3 and a rotation R , which is the inverse of R:
S2 = R S3 = R
−1
S3
(2)
And, thus, the primary and the third order shears are equal: S1 = RS2 = RR−1 S3 = S3
(3)
What then is the purpose of this exercise? If we were to consider homogenous deformation, the answer would be: nothing. However, we are considering heterogenous deformation via the formation of shear bands. S1 does not occur homogenously throughout the specimen but only localized in the shear band. Similarly, S2 does not occur homogenously throughout the entire shear band but only in localized volumes at a smaller length scale. S3 then does not occur homogenously throughout the region of S2 but again only localized in certain sub-regions, which we call process zones. Continuing this concept, a quasi-fractal nature of shear deformation evolves. This is not truly a fractal phenomenon because it has an upper cut-off dimension, the sample size, and a lower cutoff dimension. The lower cut-off dimension is the length scale of the microscopic mechanism. In some materials such as fiber reinforced composites and wood, there may be only two hierarchical levels, namely the primary level of the macroscopic shear band and the secondary level which is the micromechanism. In geological “experiments” manifested in deformation of the tectonic plates
and also in nano-crystalline materials, the lower cut-off dimension may be given by the size of individual grains. Here, many orders of magnitude separate the thickness of the primary shear band from the characteristic length of the micromechanism (e.g. [14]). Several hierarchical orders bridge between macroscopic shear and atomistic displacement. It is only at the smallest length scale that the nature of the material and the condition of deformation (orientation, temperature, strain rate) play a role. In metals, dislocation motion is predominant. In materials with self-accommodated martensite, deformation occurs via twinning (e.g. [15]). In geological and nano-crystalline materials, grain boundary sliding and thermally activated processes may control deformation; microcracking and delamination may occur in ceramics and composites. 4. Shear, dislocations and disclinations We make use of the disclination concept [16] in a similar way as for the description of hierarchical twinning in self-accommodated martensite [17]. A recent review on disclinations appeared in Ref. [18]. A disclination is a line defect similar to a dislocation, albeit with a rotational displacement field characterized by the Frank vector ω, with magnitude ω. In contrast, a dislocation has a translational displacement field characterized by the Burgers vector b, with magnitude b. The treatment is here limited to straight wedge disclinations and edge dislocations and the lines of all defects are parallel. For wedge disclinations, the Frank vector is parallel to and coincident with the disclination line. A single wedge disclination compares to a semi-infinite wall of infinitesimal edge dislocations. Partial disclinations are located at terminating small-angle boundaries [16] and grain boundary triple junctions [19], particularly in nano-grained materials [20,21]. Because of their large strain energy, disclinations tend to agglomerate in self-screening dipoles and quadrupoles [22]. Two wedge disclinations of equal strength |ω|, one with a positive and one with a negative sign represent a disclination dipole. There are many dislocation arrangements being equivalent to a disclination dipole. A finite dislocation wall is but one such equivalent. With equivalent we mean here that the respective defect configurations have the same long-range stress field. Close to the defect lines, i.e. within a distance from the lines equal to the spacing between defects, the
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Fig. 4. Continuous vs. discontinuous formation of a shear band. (a) Undistorted microstructure. The grid lines mark a reference frame of a suitable length scale (not necessarily the crystal lattice). (b) In an intermediate state, the distortions cause interfacial energy with enhanced chemical and/or elastic energy. (c) When the shear band is complete, the elastic component of the interfacial energy has relaxed. The interfacial energy has a minimum. (d) When the shear band forms discontinuously, the interfacial energy is minimal. The process zone (circle) carries a long-range strain field similar to that of a disclination dipole.
stress fields may vary significantly. In the extreme, a disclination dipole may be considered as a single super dislocation with Burgers vector B = 2aω, where 2a is the distance between the disclination lines. Fig. 3 depicts a wall of disclinations (a) and a number of equivalent dislocation representations (b–f). None of these representations is per se better than any other. Depending on the situation, one or the other might be more useful, though. We will see that in some situations, we need to transition from one description to another. Care must be applied regarding not to confuse different descriptions and not to double count displacements. A moving dislocation translates one half of the sample with respect to the other half by the Burgers vector. A moving disclination rotates one half of the sample with respect to the other by the amount of its strength. When a disclination dipole moves perpendicularly to the plane containing both lines, it produces a homogenous simple shear between the two planes swept by the disclination lines. Hence, a moving disclination dipole produces the displacement field of a shear band. 5. Discontinuous deformation on different hierarchical levels In 1934, Polanyi [23], Orowan [24], and Taylor [25] independently proposed that the lattice dislocation is the agent of plastic deformation. A few scientists, foremost Nabarro [26], advanced the theory of dislocations rigorously. However, the scientific community working on plasticity received the dislocation concept reluctantly until Hirsch et al. [27] and Bollmann [28], also working independently, provided experimental evidence in 1956. The advent of transmission electron microscopy introduced the “golden age” of plasticity. The dislocation is the agent of plasticity because its motion requires far less energy than the homogenous glide of the entire crystal. Homogenous glide of the entire crystal would require the simultaneous breaking of all bonds along the entire glide plane. In contrast, when a dislocation moves, bonds are broken only along the dislocation line. If a shear band formed continuously, each kink wall would evolve simultaneously across its entire length. The transition from the undistorted state (Fig. 4a) to the kinked state (Fig. 4c) then contained an intermediated transition state (Fig. 4b) of high “interfacial” energy. The interfacial energy may contain chemical and elastic components. In the analogy of rigid glide as opposed to
dislocation motion, the formation of the high-energy state (Fig. 4b) corresponds to the breaking of many bonds along the entire glide plane. If the shear band forms discontinuously by the motion of a process zone (Fig. 4d), the interfacial energy is limited to a local region, i.e. the process zone (circle in Fig. 4d). The process zone takes on the role of the dislocation in the above analogue. The process zone carries a strain field with the long-range properties of a disclination dipole. This strain energy of the disclination dipole depends logarithmically on the screening length (as does the strain energy of a dislocation), quadratically on the shear band thickness, and also quadratically on the disclination strength which equals the shear magnitude [16,18]. Thus, for thin shear bands, the strain energy is very small. In contrast, the interfacial energy depends linearly on the sample size and is independent of the shear band thickness. Hence, for thin shear bands, the interfacial energy is larger than the strain energy. For thin enough shear bands, therefore, discontinuous shear deformation is energetically favorable compared to homogenous shear deformation. “Thin enough” is so small compared to the sample size that the linear dependence of the interfacial energy overruns the logarithmic dependence of the strain energy of the disclination dipole. The deformation of self-accommodated martensite with a hierarchical twin microstructure illustrates the general principle [17]. In such microstructures, disclinations are located wherever twin boundaries of different hierarchical levels interact (Fig. 5a). At the highest hierarchical level (i.e. where twins do not contain further defects) the disclinations form disclination dipole walls. In Fig. 5b, walls of twinning dislocations represent the disclination dipoles. These twinning dislocations provide the agent for the microscopic deformation mechanism. The net defect content of the entire microstructure corresponds to a disclination dipole on the primary length scale (Fig. 5c). The three defect arrangements all have the same long-range strain field, which is most conveniently represented by the primary disclination dipole (Fig. 5c). The secondary disclination dipoles (Fig. 5a) provide the simplest model to evaluate the interfacial elastic energy component of the horizontal interface. The most detailed model with the twinning dislocations (Fig. 5b) describes the microscopic deformation mechanism. Thus, each description has its role and justification. Fig. 6 gives the interfacial energy as a function of the fraction, , of the highest order twins (or internal twins). Where is discontinuous along an interface, there is a disclination at a lower hierarchical order (large
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Fig. 5. Disclinations are present where twin boundaries interact (a). The disclinations of the internal (i.e. highest order) twins give rise to a strain energy component of the interfacial energy (Fig. 6). (b) The disclination dipoles of the internal twins can be represented by twinning dislocations. The net content of all secondary disclinations (a) or all twinning dislocations (b) corresponds to a primary disclination dipole (c). The primary disclination dipole moves laterally, when the twinning dislocations move vertically.
triangles in Fig. 5c). The interfacial energy differs on both sides of the lower order disclination. The discontinuity in Fig. 5 can move to the left or to the right by the motion of twinning dislocations along the boundaries of
Fig. 6. Interfacial strain energy as a function of density, , of internal twins. The interfacial energy has a maximum when = 0.5, which corresponds to the configuration on the right side of Fig. 5.
internal twins. These twinning dislocations move perpendicularly to the displacement of the primary disclination dipole. As the disclination dipole moves sideward, its structure and energy do not change. The only energy invested is due to the difference in interfacial energy (Fig. 6). As a result, the dipole moves under a much smaller shear stress than what would be required to increase continuously throughout the entire band [17]. We may now expand this thought to a more complex situation where deformation occurs on multiple hierarchical levels. The shear band forms by the orthogonal shear of second order twins. The shear of the second order twins is carried by third order twins, i.e. again by orthogonal shear and so on. Thus, higher order shear mechanisms (higher order twins in this case) are the agents of lower order shear (lower order twins), as illustrated in Fig. 7a. There is a process zone at each hierarchical level, which may be modeled as a disclination dipole. The self stress, micro , of the dipole enhances locally the macroscopic shear stress, macro (Fig. 7b) and triggers shear of the next higher order, i.e. deformation at lower length scale. Thus, the motion of the dipole is autocatalytic. The stress state at the largest length scale and the macroscopic properties of the shear band reach through all hierarchical levels down to the shortest length scale and control the micromechanism at atomistic length scale. It is only at the smallest length scale that properties such as number and nature of glide systems and the micromechanisms
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Fig. 7. Propagation of a process zone with higher order twins. (a) The primary structure is indicated with bold lines and large triangles. In the primary process zone (large circle), secondary shear occurs orthogonally to the primary shear. In the secondary process zones (small circles), tertiary shear occurs again on a smaller length scale, and orthogonally to the secondary shear. (b) Within the process zone, the stress field, micro , of the primary disclination dipole enhances the macroscopic stress field, macro , and triggers the operation of the secondary shear system.
(including dislocation glide, twinning, grain boundary sliding, thermally activated processes, cracking, and transformations) come into play.
Fig. 1a. Financial support from NSF under project no. DMR-1008167 is thankfully acknowledged. References
6. Conclusions Macroscopic deformation of materials occurs via the formation of shear bands. Within shear bands, shear occurs orthogonally to the macroscopic shear. On even smaller length scale, deformation can occur again orthogonally to the intermediate shear level. Thus, shear deformation proceeds on hierarchical levels where on each higher level (i.e. at smaller length scale), deformation occurs orthogonally to shear at the next lower hierarchical level (i.e. next larger length scale). At each length scale, shear is discontinuous. A process zone, which can be represented by a disclination dipole, carries shear deformation and thereby reduces the energy necessary for deformation. Throughout all length scales and hierarchical orders, the lower order deformation controls the higher order deformation. Only on the smallest length scale appear the microscopic deformation mechanisms and only at this length scale the classical, dislocation-based plasticity theory comes into play. The macroscopic loading conditions govern the deformation mode through a hierarchical mechanism of coupled orthogonal shear systems, linking large scale strains to smaller scale strains. Localized process zones, which act as agents of deformation on all hierarchical levels, and orthogonal shear systems shape mountains and displace atoms alike. Acknowledgements The author is grateful to Clyde J. Northrup of the Department of Geosciences at Boise State University for providing content of
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
E. Arzt, Acta Mater. 46 (1998) 5611. S. Brinckmann, J.Y. Kim, J.R. Greer, Phys. Rev. Lett. 100 (2008) 155502. S. Fertig, S.P. Baker, Prog. Mater. Sci. 54 (2009) 874. L. Benabou, Mech. Mater. 42 (2010) 335. S. Narayana, L.S. Schadler, Compos. Sci. Technol. 59 (1999) 2201. M.W. Barsum, T. El-Raghy, Am. Sci. 89 (2001) 334. A. Zisman, C. Teodosiu, V. Rybin, Comput. Mater. Sci. 16 (1999) 307. A. Luft, Prog. Mater. Sci. 35 (1991) 97. R. Nogueira de Coes, O.S. Hopperstad, O. Engler, O.-G. Lademo, J.D. Embury, A. Benallal, Metall. Mater. Trans. A 42 (2011) 3358–3369. P. Pirouz, R. Chaim, U. Dahmen, K.H. Westmacott, Acta Metall. Mater. 38 (1990) 313. S.Q. Xiao, P. Pirouz, J. Mater. Res. 7 (1992) 1406. V.E. Panin, Russ. Phys. J. 41 (1998) 1. P. Müllner, P. Pirouz, Mater. Sci. Eng. A 233 (1997) 139. R.Z. Valiev, I.V. Alexandrov, Y.T. Zhu, T.C. Lowe, J. Mater. Res. 17 (2002) 5. T. Saburi, C.M. Wayman, K. Takata, S. Nenno, Acta Metall. 28 (1980) 15. A.E. Romanov, V.I. Vladimirov, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, vol. 9, Elsevier, Amsterdam, 1992, p. 191. P. Müllner, A.H. King, Acta Mater. 58 (2010) 5242. A.E. Romanov, A.L. Kolesnikova, Prog. Mater. Sci. 54 (2009) 740. A. King, Interface Sci. 7 (1999) 251. P. Müllner, M. Kuschke, Scripta Mater. 36 (1997) 1451. R.D. Nyilas, M. Kobas, R. Spolenak, Acta Mater. 57 (2009) 3738. A.E. Romanov, Mater. Sci. Eng. A 164 (1993) 58. M. Polanyi, Z. Phys. 89 (1934) 660 (in German). E. Orowan, Z. Phys. 89 (1934) 634 (in German). G.I. Taylor, Proc. R. Soc. 145 (1934) 362. D. Kuhlmann-Wilsdorf, Prog. Mater. Sci. 54 (2009) 707. P.B. Hirsch, R.W. Horne, M.J. Whelan, Philos. Mag. 1 (1956) 677. W. Bollmann, Phys. Rev. 103 (1956) 1588.