View source for Slip bands in metals

{{Short description|Deformation mechanism in crystallines}}
[[File:S4 loaded slip band.gif|thumb|A slip band formed on a ferrite grain in an [[475 °C embrittlement|age hardened duplex stainless steel]]. The slip band at the centre of the image was observed at a certain load, then the load was increased with a burst of dislocations coming out of the slip band tip as a response to the load increment. This burst of dislocations and topographic change ahead of the slip band was observed across different slip bands. image length is 10 um.<ref name="Koko 118284">{{Cite journal |last1=Koko |first1=Abdalrhaman |last2=Elmukashfi |first2=Elsiddig |last3=Becker |first3=Thorsten H. |last4=Karamched |first4=Phani S. |last5=Wilkinson |first5=Angus J. |last6=Marrow |first6=T. James |date=2022-10-15 |title=In situ characterisation of the strain fields of intragranular slip bands in ferrite by high-resolution electron backscatter diffraction |journal=Acta Materialia |language=en |volume=239 |pages=118284 |doi=10.1016/j.actamat.2022.118284 |bibcode=2022AcMat.23918284K |s2cid=251783802 |issn=1359-6454|doi-access=free }}{{Creative Commons text attribution notice|cc=by4|from this source=yes}}</ref><ref name=":31">{{Cite thesis |title=In situ full-field characterisation of strain concentrations (deformation twins, slip bands and cracks) |url=https://ora.ox.ac.uk/objects/uuid:f2ba08f3-4a27-4619-92ed-bcd3834dadf0 |publisher=University of Oxford |date=2022 |degree=PhD |language=English |first=A. Mohamed |last=Koko |access-date=2023-03-02 |archive-date=2023-02-01 |archive-url=https://web.archive.org/web/20230201163423/https://ora.ox.ac.uk/objects/uuid:f2ba08f3-4a27-4619-92ed-bcd3834dadf0 |url-status=live}}{{Creative Commons text attribution notice|cc=by4|from this source=yes}}</ref>|300x300px]]'''Slip bands''' or '''stretcher-strain marks''' are localized bands of [[plastic deformation]] in metals experiencing stresses. Formation of slip bands indicates a concentrated unidirectional [[Slip (materials science)|slip]] on certain planes causing a stress concentration. Typically, slip bands induce surface steps (e.g., roughness due [[persistent slip bands]] during [[Fatigue (material)|fatigue]]) and a stress concentration which can be a crack nucleation site. Slip bands extend until impinged by a [[Grain boundary|boundary]], and the generated stress from [[dislocation]]s pile-up against that boundary will either stop or transmit the operating slip depending on its (mis)orientation.<ref>{{Citation |last1=Smallman |first1=R. E. |title=Chapter 9 - Plastic Deformation and Dislocation Behaviour |date=2014-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780080982045000092 |work=Modern Physical Metallurgy (Eighth Edition) |pages=357–414 |editor-last=Smallman |editor-first=R. E. |place=Oxford |publisher=Butterworth-Heinemann |language=en |doi=10.1016/b978-0-08-098204-5.00009-2 |isbn=978-0-08-098204-5 |access-date=2022-10-04 |last2=Ngan |first2=A. H. W. |editor2-last=Ngan |editor2-first=A. H. W. |archive-date=2022-10-04 |archive-url=https://web.archive.org/web/20221004191425/https://www.sciencedirect.com/science/article/pii/B9780080982045000092 |url-status=live }}</ref><ref>{{Cite journal |last=Sangid |first=Michael D. |date=2013-12-01 |title=The physics of fatigue crack initiation |url=https://www.sciencedirect.com/science/article/pii/S0142112312002940 |journal=International Journal of Fatigue |series=Fatigue and Microstructure: A special issue on recent advances |language=en |volume=57 |pages=58–72 |doi=10.1016/j.ijfatigue.2012.10.009 |issn=0142-1123}}</ref>

Formation of slip bands under cyclic conditions is addressed as [[Slip bands#Persistent slip bands|persistent slip bands]] (PSBs) where formation under monotonic condition is addressed as dislocation planar arrays (or simply slip-bands, see [[Slip bands#Slip bands in the absence of cyclic loading|''Slip bands in the absence of cyclic loading'']] section).<ref name="onlinelibrary.wiley.com">{{Cite journal |last1=Lukáš |first1=P. |last2=Klesnil |first2=M. |last3=Krejčí |first3=J. |date=1968 |title=Dislocations and Persistent Slip Bands in Copper Single Crystals Fatigued at Low Stress Amplitude |url=https://onlinelibrary.wiley.com/doi/10.1002/pssb.19680270212 |journal=Physica Status Solidi B |language=de |volume=27 |issue=2 |pages=545–558 |doi=10.1002/pssb.19680270212 |bibcode=1968PSSBR..27..545L |s2cid=96586802 |access-date=2022-10-03 |archive-date=2022-10-03 |archive-url=https://web.archive.org/web/20221003171410/https://onlinelibrary.wiley.com/doi/10.1002/pssb.19680270212 |url-status=live }}</ref> Slip-bands can be simply viewed as boundary sliding due to dislocation glide that lacks (the complexity of ) PSBs high plastic deformation localisation manifested by tongue- and ribbon-like extrusion. And, where [[Persistent slip bands|PSBs]] normally studied with (effective) [[Burgers vector]] aligned with the extrusion plane because a PSB extends across the grain and exacerbates during fatigue;<ref>{{Cite journal |last1=Schiller |first1=C. |last2=Walgraef |first2=D. |date=1988-03-01 |title=Numerical simulation of persistent slip band formation |url=https://dx.doi.org/10.1016/0001-6160%2888%2990089-2 |journal=Acta Metallurgica |language=en |volume=36 |issue=3 |pages=563–574 |doi=10.1016/0001-6160(88)90089-2 |issn=0001-6160 |access-date=2023-03-25 |archive-date=2023-03-25 |archive-url=https://web.archive.org/web/20230325212113/https://www.sciencedirect.com/science/article/abs/pii/0001616088900892?via%3Dihub |url-status=live }}</ref> a monotonic slip-band has a Burger’s vector for propagation and another for plane extrusions both controlled by the conditions at the tip.

== Persistent slip bands (PSBs) ==
{{confusing section|date=March 2023}}
[[File:PSBStructure.tif|thumb|upright=1.3|PSB structure (adopted from <ref>{{Cite journal |last1=Erel |first1=Can |last2=Po |first2=Giacomo |last3=Crosby |first3=Tamer |last4=Ghoniem |first4=Nasr |date=December 2017 |title=Generation and interaction mechanisms of prismatic dislocation loops in FCC metals |journal=Computational Materials Science |language=en |volume=140 |pages=32–46 |doi=10.1016/j.commatsci.2017.07.043 |doi-access=free }}</ref>)]]
Persistent slip-bands (PSBs) are associated with [[Strain (mechanics)|strain]] localisation due to [[Fatigue (material)|fatigue]] in metals and cracking on the same plane. [[Transmission electron microscopy]] (TEM) and three-dimensional discrete [[dislocation]] dynamics (DDD<ref>{{Cite web |title=grain boundray, dislocation simulation, discrete dislocation dynamics, dislocation dynamics, grain boundary, dislocation cell, cross slip, creep, deformation, strain hardening, superalloy, climb |url=https://www.dierk-raabe.com/ddd-discrete-dislocation-dynamics/ |access-date=2023-03-05 |website=www.dierk-raabe.com |language=de |archive-date=2022-12-03 |archive-url=https://web.archive.org/web/20221203155140/https://www.dierk-raabe.com/ddd-discrete-dislocation-dynamics/ |url-status=live }}</ref>) simulation were used to reveal and understand dislocations type and arrangement/[[patterns]] to relate it to the sub-surface structure. PSB – ladder structure – is formed mainly from low-density channels of mobile gliding [[Dislocation#Screw|screw dislocation]] segments and high-density walls of dipolar edge dislocation segments piled up with tangled bowing-out edge segment and different sizes of dipolar loops scattered between the walls and channels.<ref>{{Citation |last1=Differt |first1=K. |title=Dynamical model of the wall structure in persistent slip bands of fatigued metals |date=1993 |url=https://linkinghub.elsevier.com/retrieve/pii/B9781483228150500486 |work=Fundamental Aspects of Dislocation Interactions |pages=295–299 |publisher=Elsevier |language=en |doi=10.1016/b978-1-4832-2815-0.50048-6 |isbn=978-1-4832-2815-0 |access-date=2022-10-03 |last2=Essmann |first2=U. |archive-date=2018-06-30 |archive-url=https://web.archive.org/web/20180630103107/https://linkinghub.elsevier.com/retrieve/pii/B9781483228150500486 |url-status=live }}</ref><ref name=":0">{{Cite journal |last1=Verdier |first1=M |last2=Fivel |first2=M |last3=Groma |first3=I |date=1998-11-01 |title=Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications |url=https://iopscience.iop.org/article/10.1088/0965-0393/6/6/007 |journal=Modelling and Simulation in Materials Science and Engineering |volume=6 |issue=6 |pages=755–770 |doi=10.1088/0965-0393/6/6/007 |bibcode=1998MSMSE...6..755V |s2cid=250889422 |issn=0965-0393 |access-date=2022-10-03 |archive-date=2022-10-03 |archive-url=https://web.archive.org/web/20221003163859/https://iopscience.iop.org/article/10.1088/0965-0393/6/6/007 |url-status=live }}</ref>

One type of dislocation loop forms the boundary of a completely enclosed patch of slipped material on the slip plane which terminates at the free surface. Widening of the slip band: Screw [[dislocation]] can have high enough [[critical resolved shear stress|resolved shear stress]] for a glide on more than one slip plane. [[Dislocation#Cross-slip|Cross-slip]] can occur. But this leaves some segments of [[dislocation]] on the original slip plane. Dislocation can cross-slip back on to a parallel primary slip plane. where it forms a new dislocation source, and the process can repeat. These walls in PSBs are a ‘dipole dispersion’ form of stable arrangement of edge [[dislocations]] with minimal long-range [[Stress (mechanics)|stress]] field which has a minimal long-range stress field.{{clarify|date=March 2023}} This is different to slip-bands that is a planar stack of a stable array that has a strong long-range stress field.{{clarify|I cannot parse this sentence|date=March 2023}} Thus, – in the free surface – cut and open (elimination) of dislocation loops at the surface cause the irreversible/persistent surface step associated with slip-bands.<ref name=":0" /><ref>{{Cite journal |last1=Déprés |first1=C. |last2=Robertson |first2=C. F. |last3=Fivel |first3=M. C. |date=January 2006 |title=Low-strain fatigue in 316L steel surface grains: a three dimension discrete dislocation dynamics modelling of the early cycles. Part 2: Persistent slip markings and micro-crack nucleation |url=http://www.tandfonline.com/doi/abs/10.1080/14786430500341250 |journal=Philosophical Magazine |language=en |volume=86 |issue=1 |pages=79–97 |doi=10.1080/14786430500341250 |bibcode=2006PMag...86...79D |s2cid=135953582 |issn=1478-6435 |access-date=2022-10-03 |archive-date=2021-05-17 |archive-url=https://web.archive.org/web/20210517165907/https://www.tandfonline.com/doi/abs/10.1080/14786430500341250 |url-status=live }}</ref><ref>{{Cite journal |last1=Déprés |first1=C. |last2=Robertson |first2=C. F. |last3=Fivel |first3=M. C. |date=August 2004 |title=Low-strain fatigue in AISI 316L steel surface grains: a three-dimensional discrete dislocation dynamics modelling of the early cycles I. Dislocation microstructures and mechanical behaviour |url=http://www.tandfonline.com/doi/abs/10.1080/14786430410001690051 |journal=Philosophical Magazine |language=en |volume=84 |issue=22 |pages=2257–2275 |doi=10.1080/14786430410001690051 |bibcode=2004PMag...84.2257D |s2cid=137329770 |issn=1478-6435 |access-date=2022-10-03 |archive-date=2022-10-07 |archive-url=https://web.archive.org/web/20221007035523/https://www.tandfonline.com/doi/abs/10.1080/14786430410001690051 |url-status=live }}</ref>

Surface relief through extrusion occurs on the Burger's vector direction and extrusion height and PSB depth increase with PSB thickness.<ref>{{Cite journal |last1=Man |first1=J. |last2=Obrtlík |first2=K. |last3=Polák |first3=J. |date=June 2009 |title=Extrusions and intrusions in fatigued metals. Part 1. State of the art and history† |url=http://www.tandfonline.com/doi/abs/10.1080/14786430902917616 |journal=Philosophical Magazine |language=en |volume=89 |issue=16 |pages=1295–1336 |doi=10.1080/14786430902917616 |bibcode=2009PMag...89.1295M |s2cid=136919859 |issn=1478-6435 |access-date=2022-10-03 |archive-date=2022-10-06 |archive-url=https://web.archive.org/web/20221006070310/https://www.tandfonline.com/doi/abs/10.1080/14786430902917616 |url-status=live }}</ref> PSB and planar walls are parallel and perpendicularly aligned with the normal direction of the [[Critical resolved shear stress]], respectively.<ref>{{Cite journal |last1=Amodeo |first1=R. J. |last2=Ghoniem |first2=N. M. |date=1990-04-01 |title=Dislocation dynamics. II. Applications to the formation of persistent slip bands, planar arrays, and dislocation cells |url=https://link.aps.org/doi/10.1103/PhysRevB.41.6968 |journal=Physical Review B |language=en |volume=41 |issue=10 |pages=6968–6976 |doi=10.1103/PhysRevB.41.6968 |pmid=9992953 |bibcode=1990PhRvB..41.6968A |issn=0163-1829 |access-date=2022-10-03 |archive-date=2023-03-25 |archive-url=https://web.archive.org/web/20230325221757/https://journals.aps.org/prb/abstract/10.1103/PhysRevB.41.6968 |url-status=live }}</ref> And once dislocation saturate and reach its sessile configuration, cracks were observed to nucleate and propagate along PSB extrusions.<ref>{{Cite journal |last=Wood |first=W. A. |date=July 1958 |title=Formation of fatigue cracks |url=http://www.tandfonline.com/doi/abs/10.1080/14786435808237004 |journal=Philosophical Magazine |language=en |volume=3 |issue=31 |pages=692–699 |doi=10.1080/14786435808237004 |bibcode=1958PMag....3..692W |issn=0031-8086 |access-date=2022-10-03 |archive-date=2022-10-07 |archive-url=https://web.archive.org/web/20221007223415/https://www.tandfonline.com/doi/abs/10.1080/14786435808237004 |url-status=live }}</ref><ref name="Koss 1245–1252">{{Cite journal |last1=Koss |first1=D.A. |last2=Chan |first2=K.S. |date=September 1980 |title=Fracture along planar slip bands |url=https://linkinghub.elsevier.com/retrieve/pii/0001616080900802 |journal=Acta Metallurgica |language=en |volume=28 |issue=9 |pages=1245–1252 |doi=10.1016/0001-6160(80)90080-2 |access-date=2022-10-03 |archive-date=2018-06-28 |archive-url=https://web.archive.org/web/20180628001528/https://linkinghub.elsevier.com/retrieve/pii/0001616080900802 |url-status=live }}</ref><ref name="Mughrabi 1367–1379">{{Cite journal |last=Mughrabi |first=H. |date=September 1983 |title=Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals |url=https://linkinghub.elsevier.com/retrieve/pii/000161608390007X |journal=Acta Metallurgica |language=en |volume=31 |issue=9 |pages=1367–1379 |doi=10.1016/0001-6160(83)90007-X |access-date=2022-10-03 |archive-date=2022-06-16 |archive-url=https://web.archive.org/web/20220616003944/https://linkinghub.elsevier.com/retrieve/pii/000161608390007X |url-status=live }}</ref> To summarise, contrary to 2D line defects, the field at the slip-band tip is due to three-dimensional interactions where the slip band extrusion simulates a sink-like dislocation blooming along the slip band axis. The magnitude of the gradient [[Deformation (engineering)|deformation]] field ahead of the [[Slip (materials science)#slip band|slip band]] depends on the slip height and the mechanical conditions for propagation is influenced by the emitted dislocations long range field.A surface marking, or slip band, appears at the intersection of an active slip plane and the free surface of a crystal. Slip occurs in [[Dislocation avalanches|avalanches]] separated in time. Avalanches from other slip systems crossing a slip plane containing an active source led to the observed stepped surface markings, with successive avalanches from the given source displaced relative to each other.<ref name=":1">{{Cite journal |last1=Fisher |first1=John C. |last2=Hart |first2=Edward W. |last3=Pry |first3=Robert H. |date=1952-09-15 |title=Theory of Slip-Band Formation |url=https://link.aps.org/doi/10.1103/PhysRev.87.958 |journal=Physical Review |language=en |volume=87 |issue=6 |pages=958–961 |doi=10.1103/PhysRev.87.958 |bibcode=1952PhRv...87..958F |issn=0031-899X}}</ref>

Dislocations are generated on a single slip plane They point out that a dislocation segment ([[Frank–Read source]]), lying in a slip plane and pinned at both ends, is a source of an unlimited number of dislocation loops. In this way the grouping of dislocations into an avalanche of a thousand or so loops on a single slip plane can be understood.<ref>{{Cite journal |last1=Frank |first1=F. C. |last2=Read |first2=W. T. |date=1950-08-15 |title=Multiplication Processes for Slow Moving Dislocations |url=https://link.aps.org/doi/10.1103/PhysRev.79.722 |journal=Physical Review |language=en |volume=79 |issue=4 |pages=722–723 |doi=10.1103/PhysRev.79.722 |bibcode=1950PhRv...79..722F |issn=0031-899X |access-date=2022-10-03 |archive-date=2023-03-25 |archive-url=https://web.archive.org/web/20230325212130/https://journals.aps.org/pr/abstract/10.1103/PhysRev.79.722 |url-status=live }}</ref> Each dislocation loop has a stress field that opposes the applied stress in the neighbourhood of the source. When enough loops have been generated, the stress at the source will fall to a value so low that additional loops cannot form. Only after the original avalanche of loops has moved some distance away can another avalanche occur.

Generation of the first avalanche at a source is easily understood. When the stress at the source reaches r*, loops are generated, and continue to be generated until the back-stress stops the avalanche. A second avalanche will not occur immediately in polycrystals, for the loops in the first avalanche are stopped or partially stopped at grain boundaries. Only if the external stress is increased substantially will a second avalanche be formed. In this way the formation of additional avalanches with rising stress can be understood.

It remains to explain the displacement of successive avalanches by a small amount normal to the slip plane, thereby accounting for the observed fine structure of [[Slip (materials science)|slip bands]]. A displacement of this type requires that a [[Frank–Read source]] move relative to the surface where slip bands are observed.

In situ nano-compression work <ref>{{Cite journal |last1=Xie |first1=Kelvin Y. |last2=Wang |first2=Yanbo |last3=Ni |first3=Song |last4=Liao |first4=Xiaozhou |last5=Cairney |first5=Julie M. |last6=Ringer |first6=Simon P. |date=December 2011 |title=Insight into the deformation mechanisms of α-Fe at the nanoscale |url=https://linkinghub.elsevier.com/retrieve/pii/S1359646211005008 |journal=Scripta Materialia |language=en |volume=65 |issue=12 |pages=1037–1040 |doi=10.1016/j.scriptamat.2011.08.023 |access-date=2022-10-03 |archive-date=2018-06-02 |archive-url=https://web.archive.org/web/20180602150000/https://linkinghub.elsevier.com/retrieve/pii/S1359646211005008 |url-status=live }}</ref> in [[Transmission electron microscopy]] (TEM) reveals that the deformation of a-Fe at the nanoscale is an inhomogeneous process characterized by a series of short displacement bursts and intermittent large displacement bursts. The series of short bursts correspond to the collective movement of [[dislocations]] within the crystal. The large single bursts are from SBs nucleated from the specimen surface. These results suggest that the formation of SBs can be considered as a source-limited plasticity process. The initial plastic deformation is characterized by the multiplication/ movement of a few dislocations over short distances due to the availability of dislocation sources within the nano-blade. Once it has reached a stage at which the mobile dislocations along preferred slips planes have moved through the nano-blade or become entangled in sessile configurations and further dislocation movement is difficult within the crystal, plasticity is carried out by the formation of SBs, which nucleate from the surface <ref>{{Cite journal |last1=Zheng |first1=He |last2=Cao |first2=Ajing |last3=Weinberger |first3=Christopher R. |last4=Huang |first4=Jian Yu |last5=Du |first5=Kui |last6=Wang |first6=Jianbo |last7=Ma |first7=Yanyun |last8=Xia |first8=Younan |last9=Mao |first9=Scott X. |date=December 2010 |title=Discrete plasticity in sub-10-nm-sized gold crystals |journal=Nature Communications |language=en |volume=1 |issue=1 |pages=144 |doi=10.1038/ncomms1149 |issn=2041-1723 |pmc=3105591 |pmid=21266994|bibcode=2010NatCo...1..144Z }}</ref> and then propagate through the nano-blade. 
[[File:FR10G-1.GIF|thumb|upright=1.3|[[Frank–Read source]]]]
Fisher ''et al.'' <ref name=":1" /> proposed that SBs are dynamically generated from a [[Frank–Read source]] at the specimen surface and are terminated by their own stress field in single crystals. The displacement burst behaviour reported by Kiener and Minor <ref>{{Cite journal |last1=Kiener |first1=D. |last2=Minor |first2=A.M. |date=February 2011 |title=Source-controlled yield and hardening of Cu(100) studied by in situ transmission electron microscopy |url=https://linkinghub.elsevier.com/retrieve/pii/S1359645410007391 |journal=Acta Materialia |language=en |volume=59 |issue=4 |pages=1328–1337 |doi=10.1016/j.actamat.2010.10.065 |bibcode=2011AcMat..59.1328K |access-date=2022-10-03 |archive-date=2022-06-17 |archive-url=https://web.archive.org/web/20220617142517/https://linkinghub.elsevier.com/retrieve/pii/S1359645410007391 |url-status=live }}</ref> on compressing Cu single crystal nanopillars. Obviously suppressed the progress of serrated yielding (a series of short strain '''bursts''') relative to that without the spinodal nanostructure. The results revealed that during compression deformation, the spinodal nanostructure confined the movement of dislocations (leading to a significant increase in dislocation density), causing a notable strengthening effect, and also kept the [[Slip (materials science)#Slip band|slip band]] morphology planar.<ref>{{Cite journal |last1=Hsieh |first1=Yi-Chieh |last2=Zhang |first2=Ling |last3=Chung |first3=Tsai-Fu |last4=Tsai |first4=Yu-Ting |last5=Yang |first5=Jer-Ren |last6=Ohmura |first6=Takahito |last7=Suzuki |first7=Takuya |date=December 2016 |title=In-situ transmission electron microscopy investigation of the deformation behavior of spinodal nanostructured δ-ferrite in a duplex stainless steel |url=https://linkinghub.elsevier.com/retrieve/pii/S1359646216303001 |journal=Scripta Materialia |language=en |volume=125 |pages=44–48 |doi=10.1016/j.scriptamat.2016.06.047 |access-date=2022-10-03 |archive-date=2022-06-20 |archive-url=https://web.archive.org/web/20220620212238/https://linkinghub.elsevier.com/retrieve/pii/S1359646216303001 |url-status=live }}</ref>

Dislocation activity assists the growth of [[austenite]] precipitates and provide quantitative data for revealing the stress field generated by interface migration.<ref>{{Cite journal |last1=Du |first1=Juan |last2=Mompiou |first2=Frédéric |last3=Zhang |first3=Wen-Zheng |date=March 2018 |title=In-situ TEM study of dislocation emission associated with austenite growth |url=https://linkinghub.elsevier.com/retrieve/pii/S1359646217306000 |journal=Scripta Materialia |language=en |volume=145 |pages=62–66 |doi=10.1016/j.scriptamat.2017.10.014 |access-date=2022-10-03 |archive-date=2018-06-30 |archive-url=https://web.archive.org/web/20180630215631/https://linkinghub.elsevier.com/retrieve/pii/S1359646217306000 |url-status=live }}</ref> The jerky nature of the tip moving rate is probably due to the accumulation and relaxation of stress field near the tip. After leaving from the tip, the [[dislocation]] loop expands rapidly ahead of the tip thus the change in tip velocity is concomitant with dislocation emission. It indicates that the emitted dislocation is strongly repelled by the stress field present at the lath tip. When the loop meets the foil surface, it breaks into two dislocation segments that leave a visible trace, due to the presence of a thin oxide layer on the surface. The emission of a dislocation loop from the tip may also affect tip moving rate via interaction between the local dislocation loop and the possible interfacial dislocations in the semi-coherent interface surrounding the tip. consequently, the tip halted temporarily. The net [[shear stress]] acting on each dislocation results from a combination of the stress field at the lath tip (τ<sub>tip</sub>), the image stress tending to attract the dislocation loop to the surface (τ<sub>image</sub>), the line tension (τ<sub>l</sub>) and the interaction stress between dislocations (τ<sub>inter</sub>). This implies the strain field due to the transformation of [[austenite]] is large enough to cause the nucleation and emission of dislocations from an austenite lath tip.<ref name=":31" />

== Slip bands in the absence of cyclic loading ==
[[File:Slip bands.jpg|thumb|upright=1.3|Slip bands formation]]
While repeatedly reversed loading commonly leads to localisation of dislocation glide, creating linear extrusions and intrusions on a free surface, similar features can arise even if there is no load reversal.  These arise from dislocations gliding on a particular slip plane, in a particular slip direction (within a single grain), under an external load.  Steps can be created on the free surface as a consequence of the tendency for dislocations to follow one another along a glide path, of which there may be several in parallel with each other in the grain concerned.  Prior passage of dislocations apparently makes glide easier for subsequent ones, and the effect may also be associated with dislocation sources, such as a [[Frank–Read source|Frank-Read source]], acting in particular planes.
[[File:Slip band DSS.tif|thumb|Secondary electron images of [[475 °C embrittlement|age-hardened duplex stainless-steel]] observed ''in situ'' in three-point bending at applied crosshead displacements of (a) 1.2 mm and (b) 1.5 mm. Selected regions (2 and 4) are shown with higher magnification in (c) and (d). The apparent slip band height is marked as ‘''h''.’ Ferrite (<math>\alpha</math>) and austenite (<math>\gamma</math>) phases are labelled.<ref name="Koko 118284" />]]
The appearance of such bands, which are sometimes termed “persistent slip lines”, is similar to that of those arising from cyclic loading, but the resultant steps are usually more localised and have lower heights.  They also reveal the [[grain structure]]. They can often be seen on free surfaces that were polished before the deformation took place.  For example, the figure shows micrographs<ref name="PSBs in Cu">{{cite journal |last1=Campbell |first1=JE |last2=Thompson |first2=RP |last3=Dean |first3=J |last4=Clyne |first4=TW |date=2019 |title=Comparison between stress-strain plots obtained from indentation plastometry, based on residual indent profiles, and from uniaxial testing |journal=Acta Materialia |volume=168 |pages=87–99 |doi=10.1016/j.actamat.2019.02.006|bibcode=2019AcMat.168...87C }}</ref> (taken with different magnifications) of the region around an indent created in a copper sample with a spherical indenter.  The parallel lines within individual grains are each the result of several hundred dislocations of the same type reaching the free surface, creating steps with a height of the order of a few microns.  If a single slip system was operational within a grain, then there is just one set of lines, but it is common for more than one system to be activated within a grain (particularly when the strain is relatively high), leading to two or more sets of parallel lines.  Other features indicative of the details of how the plastic deformation took place, such as a region of cooperative shear caused by [[deformation twinning]], can also sometimes be seen on such surfaces.  In the optical micrograph shown, there is also evidence of grain rotations – for example, at the “rim” of the indent and in the form of depressions at [[grain boundaries]].  Such images can thus be very informative.

=== Nature of the non-cyclic slip band local field ===
[[File:Schematic of a slip band.tif|thumb|upright=1.4|Schematic of a slip band, relative to the measurement axes (𝑥<sub>1</sub>, 𝑥<sub>2</sub>, and 𝑥<sub>3</sub>), and axes related to the slip-band (𝑥, 𝑦, and 𝑧), showing the angles that describe the relationship between these axes and the traces of the slip band (𝛼, 𝜃), and the inclination angle (𝜓) of the slip trace (𝑥) and Burgers vector (𝑏) relative to the surface. ℎ is the slip band height, and 𝑞 is the slip band propagation direction assumed for [[J-integral]] calculation when using the virtual extension method.<ref>{{Cite journal |last=Parks |first=D.M. |date=December 1977 |title=The virtual crack extension method for nonlinear material behavior |url=https://linkinghub.elsevier.com/retrieve/pii/0045782577900238 |journal=Computer Methods in Applied Mechanics and Engineering |language=en |volume=12 |issue=3 |pages=353–364 |doi=10.1016/0045-7825(77)90023-8 |bibcode=1977CMAME..12..353P |access-date=2022-10-03 |archive-date=2022-06-29 |archive-url=https://web.archive.org/web/20220629171534/https://linkinghub.elsevier.com/retrieve/pii/0045782577900238 |url-status=live }}</ref> 𝑡 describes the line vector drawn here as for an edge dislocation, i.e., 𝑏⊥𝑡, and 𝑛 is the slip band plane normal.<ref name="Koko 118284" />]]
The deformation field at the slip-band is due to three-dimensional elastic and [[Deformation (physics)|plastic strains]] where the concentrated shear of the slip band tip deforms the grain in its vicinity. The elastic [[Strain (mechanics)|strains]] describe the stress concentration ahead of the [[Slip (materials science)#slip band|slip band]], which is important as it can affect the transfer of plastic deformation across grain boundaries.<ref>{{Cite journal |last1=Benjamin Britton |first1=T. |last2=Wilkinson |first2=Angus J. |date=September 2012 |title=Stress fields and geometrically necessary dislocation density distributions near the head of a blocked slip band |url=https://linkinghub.elsevier.com/retrieve/pii/S135964541200451X |journal=Acta Materialia |language=en |volume=60 |issue=16 |pages=5773–5782 |doi=10.1016/j.actamat.2012.07.004 |bibcode=2012AcMat..60.5773B |hdl=10044/1/13886 |hdl-access=free |access-date=2022-10-03 |archive-date=2023-01-01 |archive-url=https://web.archive.org/web/20230101173026/https://linkinghub.elsevier.com/retrieve/pii/S135964541200451X |url-status=live }}</ref><ref>{{Cite journal |last1=Guo |first1=Y. |last2=Britton |first2=T.B. |last3=Wilkinson |first3=A.J. |date=September 2014 |title=Slip band–grain boundary interactions in commercial-purity titanium |url=https://linkinghub.elsevier.com/retrieve/pii/S1359645414003632 |journal=Acta Materialia |language=en |volume=76 |pages=1–12 |doi=10.1016/j.actamat.2014.05.015 |bibcode=2014AcMat..76....1G |hdl=10044/1/25973 |s2cid=136904692 |hdl-access=free |access-date=2022-10-03 |archive-date=2022-06-20 |archive-url=https://web.archive.org/web/20220620042631/https://linkinghub.elsevier.com/retrieve/pii/S1359645414003632 |url-status=live }}</ref><ref>{{Cite journal |last1=Andani |first1=Mohsen Taheri |last2=Lakshmanan |first2=Aaditya |last3=Sundararaghavan |first3=Veera |last4=Allison |first4=John |last5=Misra |first5=Amit |date=November 2020 |title=Quantitative study of the effect of grain boundary parameters on the slip system level Hall-Petch slope for basal slip system in Mg-4Al |journal=Acta Materialia |language=en |volume=200 |pages=148–161 |doi=10.1016/j.actamat.2020.08.079|bibcode=2020AcMat.200..148A |s2cid=225279335 |doi-access=free }}</ref> An understanding of this is needed to support the study of yield and inter/intra-granular fracture.<ref>{{Cite journal |last1=Livingston |first1=J.D |last2=Chalmers |first2=B |date=June 1957 |title=Multiple slip in bicrystal deformation |url=https://linkinghub.elsevier.com/retrieve/pii/0001616057900445 |journal=Acta Metallurgica |language=en |volume=5 |issue=6 |pages=322–327 |doi=10.1016/0001-6160(57)90044-5 |access-date=2022-10-03 |archive-date=2022-06-17 |archive-url=https://web.archive.org/web/20220617062722/https://linkinghub.elsevier.com/retrieve/pii/0001616057900445 |url-status=live }}</ref><ref>{{Cite journal |last1=Lee |first1=T.C. |last2=Robertson |first2=I.M. |last3=Birnbaum |first3=H.K. |date=May 1989 |title=Prediction of slip transfer mechanisms across grain boundaries |url=https://linkinghub.elsevier.com/retrieve/pii/0036974889905346 |journal=Scripta Metallurgica |language=en |volume=23 |issue=5 |pages=799–803 |doi=10.1016/0036-9748(89)90534-6 |access-date=2022-10-03 |archive-date=2022-07-10 |archive-url=https://web.archive.org/web/20220710044425/https://linkinghub.elsevier.com/retrieve/pii/0036974889905346 |url-status=live }}</ref><ref>{{Cite journal |last1=Luster |first1=J. |last2=Morris |first2=M. A. |date=July 1995 |title=Compatibility of deformation in two-phase Ti-Al alloys: Dependence on microstructure and orientation relationships |url=http://link.springer.com/10.1007/BF02670762 |journal=Metallurgical and Materials Transactions A |language=en |volume=26 |issue=7 |pages=1745–1756 |doi=10.1007/BF02670762 |bibcode=1995MMTA...26.1745L |s2cid=137425735 |issn=1073-5623 |access-date=2022-10-03 |archive-date=2023-03-25 |archive-url=https://web.archive.org/web/20230325212106/https://link.springer.com/article/10.1007/BF02670762 |url-status=live }}</ref> The concentrated shear of [[Slip (materials science)#slip band|slip bands]] can also nucleate cracks in the plane of the [[Slip (materials science)#slip band|slip band]],<ref name="Koss 1245–1252" /><ref name="Mughrabi 1367–1379" /> and persistent slip bands that lead to intragranular [[Fatigue (material)|fatigue]] crack initiation and growth may also form under cyclic loading conditions.<ref name="onlinelibrary.wiley.com"/><ref>{{Citation |last1=Tu |first1=S.-T. |title=Fatigue Crack Initiation Mechanisms |date=2016 |url=https://linkinghub.elsevier.com/retrieve/pii/B9780128035818028526 |work=Reference Module in Materials Science and Materials Engineering |pages=B9780128035818028526 |publisher=Elsevier |language=en |doi=10.1016/b978-0-12-803581-8.02852-6 |isbn=978-0-12-803581-8 |access-date=2022-10-03 |last2=Zhang |first2=X.-C. |archive-date=2022-07-11 |archive-url=https://web.archive.org/web/20220711093845/https://linkinghub.elsevier.com/retrieve/pii/B9780128035818028526 |url-status=live }}</ref> To properly characterise slip bands and validate mechanistic models for their interactions with microstructure, it is crucial to quantify the local [[Deformation (engineering)|deformation]] fields associated with their propagation. However, little attention has been given to [[Slip (materials science)|slip bands]] within grains (i.e., in the absence of [[grain boundary]] interaction).

The long-range stress field (i.e., the elastic strain field) around the tip of a stress concentrator, such as a [[Slip (materials science)#slip band|slip band]], can be considered a singularity equivalent to that of a crack.<ref>{{Cite journal |last=Makin |first=M. J. |date=April 1970 |title=The mechanism of slip band growth in irradiated crystals |url=http://www.tandfonline.com/doi/abs/10.1080/14786437008238467 |journal=Philosophical Magazine |language=en |volume=21 |issue=172 |pages=815–817 |doi=10.1080/14786437008238467 |bibcode=1970PMag...21..815M |issn=0031-8086 |access-date=2022-10-03 |archive-date=2022-10-07 |archive-url=https://web.archive.org/web/20221007141627/https://www.tandfonline.com/doi/abs/10.1080/14786437008238467 |url-status=live }}</ref><ref>{{Cite journal |last=Rice |first=James R. |date=December 1987 |title=Tensile crack tip fields in elastic-ideally plastic crystals |url=https://linkinghub.elsevier.com/retrieve/pii/0167663687900305 |journal=Mechanics of Materials |language=en |volume=6 |issue=4 |pages=317–335 |doi=10.1016/0167-6636(87)90030-5 |bibcode=1987MechM...6..317R |access-date=2022-10-03 |archive-date=2022-12-05 |archive-url=https://web.archive.org/web/20221205211312/https://linkinghub.elsevier.com/retrieve/pii/0167663687900305 |url-status=live }}</ref> This singularity can be quantified using a path independent integral since it satisfies the conservation laws of elasticity. The conservation laws of elasticity related to translational, rotational, and scaling symmetries were derived initially by Knowles and Sternberg <ref>{{Cite journal |last1=Knowles |first1=J. K. |last2=Sternberg |first2=Eli |date=January 1972 |title=On a class of conservation laws in linearized and finite elastostatics |url=http://link.springer.com/10.1007/BF00250778 |journal=Archive for Rational Mechanics and Analysis |language=en |volume=44 |issue=3 |pages=187–211 |doi=10.1007/BF00250778 |bibcode=1972ArRMA..44..187K |s2cid=122386163 |issn=0003-9527 |access-date=2022-10-03 |archive-date=2023-03-25 |archive-url=https://web.archive.org/web/20230325212112/https://link.springer.com/article/10.1007/BF00250778 |url-status=live }}</ref> from the [[Noether's theorem]].<ref>{{Cite journal |last=Noether |first=Emmy |date=January 1971 |title=Invariant variation problems |url=http://www.tandfonline.com/doi/abs/10.1080/00411457108231446 |journal=Transport Theory and Statistical Physics |language=en |volume=1 |issue=3 |pages=186–207 |doi=10.1080/00411457108231446 |arxiv=physics/0503066 |bibcode=1971TTSP....1..186N |s2cid=119019843 |issn=0041-1450 |access-date=2022-10-03 |archive-date=2022-07-15 |archive-url=https://web.archive.org/web/20220715022344/https://www.tandfonline.com/doi/abs/10.1080/00411457108231446 |url-status=live }}</ref> Budiansky and [[James R. Rice|Rice]]<ref>{{Cite journal |last1=Budiansky |first1=B. |last2=Rice |first2=J. R. |date=1973-03-01 |title=Conservation Laws and Energy-Release Rates |url=https://asmedigitalcollection.asme.org/appliedmechanics/article/40/1/201/387600/Conservation-Laws-and-EnergyRelease-Rates |journal=Journal of Applied Mechanics |language=en |volume=40 |issue=1 |pages=201–203 |doi=10.1115/1.3422926 |bibcode=1973JAM....40..201B |issn=0021-8936 |access-date=2022-10-03 |archive-date=2022-10-03 |archive-url=https://web.archive.org/web/20221003171412/https://asmedigitalcollection.asme.org/appliedmechanics/article/40/1/201/387600/Conservation-Laws-and-EnergyRelease-Rates |url-status=live }}</ref> introduced the J-, M-, L-[[Invariant (mathematics)|integral]] and were the first to give them a physical interpretation as the [[Energy release rate (fracture mechanics)|strain energy-release rates]] for mechanisms such as cavity propagation, simultaneous uniform expansion, and defect rotation, respectively. When evaluated over a surface that encloses a defect, these conservation integrals represent a configurational force on the defect.<ref>{{Cite journal |date=1951-11-06 |title=The force on an elastic singularity |url=https://royalsocietypublishing.org/doi/10.1098/rsta.1951.0016 |journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |language=en |volume=244 |issue=877 |pages=87–112 |doi=10.1098/rsta.1951.0016 |bibcode=1951RSPTA.244...87E |issn=0080-4614 |last1=Eshelby |first1=J. 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A. Mathematical and Physical Sciences |language=en |volume=332 |issue=1591 |pages=527–548 |doi=10.1098/rspa.1973.0040 |bibcode=1973RSPSA.332..527P |issn=0080-4630 |last1=Palmer |first1=A. C. |last2=Rice |first2=J. R. |s2cid=7935986 |access-date=2022-10-03 |archive-date=2022-10-03 |archive-url=https://web.archive.org/web/20221003171411/https://royalsocietypublishing.org/doi/10.1098/rspa.1973.0040 |url-status=live }}</ref> and co-planar dislocation nucleation from shear loaded cracks.<ref>{{Cite journal |last=Rice |first=James R. |date=January 1992 |title=Dislocation nucleation from a crack tip: An analysis based on the Peierls concept |url=https://linkinghub.elsevier.com/retrieve/pii/S0022509605800122 |journal=Journal of the Mechanics and Physics of Solids |language=en |volume=40 |issue=2 |pages=239–271 |doi=10.1016/S0022-5096(05)80012-2 |bibcode=1992JMPSo..40..239R |access-date=2022-10-03 |archive-date=2022-07-16 |archive-url=https://web.archive.org/web/20220716065806/https://linkinghub.elsevier.com/retrieve/pii/S0022509605800122 |url-status=live }}</ref> The integrals have been applied to linear elastic and elastic-plastic materials and have been coupled with processes such as thermal <ref>{{Cite journal |last1=Kc |first1=Amit |last2=Kim |first2=Jeong-Ho |date=May 2008 |title=Interaction integrals for thermal fracture of functionally graded materials |url=https://linkinghub.elsevier.com/retrieve/pii/S0013794407003025 |journal=Engineering Fracture Mechanics |language=en |volume=75 |issue=8 |pages=2542–2565 |doi=10.1016/j.engfracmech.2007.07.011 |access-date=2022-10-03 |archive-date=2022-08-07 |archive-url=https://web.archive.org/web/20220807213003/https://linkinghub.elsevier.com/retrieve/pii/S0013794407003025 |url-status=live }}</ref> and electrochemical <ref>{{Cite journal |last1=Haftbaradaran |first1=Hamed |last2=Qu |first2=Jianmin |date=November 2014 |title=A path-independent integral for fracture of solids under combined electrochemical and mechanical loadings |url=https://linkinghub.elsevier.com/retrieve/pii/S0022509614001318 |journal=Journal of the Mechanics and Physics of Solids |language=en |volume=71 |pages=1–14 |doi=10.1016/j.jmps.2014.06.007 |bibcode=2014JMPSo..71....1H |access-date=2022-10-03 |archive-date=2023-03-25 |archive-url=https://web.archive.org/web/20230325212135/https://www.sciencedirect.com/science/article/abs/pii/S0022509614001318?via%3Dihub |url-status=live }}</ref> loading, and internal tractions.<ref>{{Cite journal |last1=Walters |first1=Matthew C. |last2=Paulino |first2=Glaucio H. |last3=Dodds |first3=Robert H. |date=July 2005 |title=Interaction integral procedures for 3-D curved cracks including surface tractions |url=https://linkinghub.elsevier.com/retrieve/pii/S0013794405000160 |journal=Engineering Fracture Mechanics |language=en |volume=72 |issue=11 |pages=1635–1663 |doi=10.1016/j.engfracmech.2005.01.002 |access-date=2022-10-03 |archive-date=2022-08-08 |archive-url=https://web.archive.org/web/20220808142545/https://linkinghub.elsevier.com/retrieve/pii/S0013794405000160 |url-status=live }}</ref> Recently, experimental fracture mechanics studies have used full-field in situ measurements of displacements <ref>{{Cite journal |last1=Becker |first1=T. H. |last2=Mostafavi |first2=M. |last3=Tait |first3=R. B. |last4=Marrow |first4=T. J. |date=October 2012 |title=An approach to calculate the J-integral by digital image correlation displacement field measurement: An Approach to Calculate the J -Integral Using Digital Image Correlation |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1460-2695.2012.01685.x |journal=Fatigue & Fracture of Engineering Materials & Structures |language=en |volume=35 |issue=10 |pages=971–984 |doi=10.1111/j.1460-2695.2012.01685.x |access-date=2022-10-03 |archive-date=2022-10-03 |archive-url=https://web.archive.org/web/20221003171410/https://onlinelibrary.wiley.com/doi/10.1111/j.1460-2695.2012.01685.x |url-status=live }}</ref><ref name=":2">{{Cite journal |last1=Koko |first1=A. |last2=Earp |first2=P. |last3=Wigger |first3=T. |last4=Tong |first4=J. |last5=Marrow |first5=T.J. |date=May 2020 |title=J-integral analysis: An EDXD and DIC comparative study for a fatigue crack |url=https://linkinghub.elsevier.com/retrieve/pii/S0142112320300050 |journal=International Journal of Fatigue |language=en |volume=134 |pages=105474 |doi=10.1016/j.ijfatigue.2020.105474 |s2cid=214391445 |access-date=2022-10-03 |archive-date=2022-06-22 |archive-url=https://web.archive.org/web/20220622030916/https://linkinghub.elsevier.com/retrieve/pii/S0142112320300050 |url-status=live }}</ref> and elastic strains <ref>{{Cite journal |last1=Barhli |first1=S.M. |last2=Saucedo-Mora |first2=L. |last3=Jordan |first3=M.S.L. |last4=Cinar |first4=A.F. |last5=Reinhard |first5=C. |last6=Mostafavi |first6=M. |last7=Marrow |first7=T.J. |date=November 2017 |title=Synchrotron X-ray characterization of crack strain fields in polygranular graphite |url=https://linkinghub.elsevier.com/retrieve/pii/S0008622317308709 |journal=Carbon |language=en |volume=124 |pages=357–371 |doi=10.1016/j.carbon.2017.08.075 |bibcode=2017Carbo.124..357B |hdl=1983/93385e0e-155e-49e6-9a3d-e4f881de013a |access-date=2022-10-03 |archive-date=2022-06-18 |archive-url=https://web.archive.org/web/20220618052747/https://linkinghub.elsevier.com/retrieve/pii/S0008622317308709 |url-status=live |hdl-access=free }}</ref><ref name=":2" /> to evaluate the local deformation field surrounding the crack tip as a [[J-integral]].

Slip bands form due to plastic deformation, and the analysis of the force on a [[dislocation]] considers the two-dimensional nature of the [[dislocation]] line defect. General definitions of the Peach–[[James Stark Koehler|Koehler]] [[Configurational mechanics|configurational force]] (𝑃<sub>𝑘𝑗</sub>) <ref>{{Cite journal |last=Lubarda |first=Vlado A. |date=2019-01-01 |title=Dislocation Burgers vector and the Peach–Koehler force: a review |journal=Journal of Materials Research and Technology |language=en |volume=8 |issue=1 |pages=1550–1565 |doi=10.1016/j.jmrt.2018.08.014 |s2cid=125242265 |issn=2238-7854|doi-access=free }}</ref> (or the elastic energy-momentum tensor <ref>{{Citation |last=Eshelby |first=J.D. |title=The Continuum Theory of Lattice Defects |date=1956 |url=https://linkinghub.elsevier.com/retrieve/pii/S0081194708601320 |series=Solid State Physics |volume=3 |pages=79–144 |publisher=Elsevier |language=en |doi=10.1016/s0081-1947(08)60132-0 |isbn=978-0-12-607703-2 |access-date=2022-10-03 |archive-date=2022-12-19 |archive-url=https://web.archive.org/web/20221219134144/https://linkinghub.elsevier.com/retrieve/pii/S0081194708601320 |url-status=live }}</ref>) on a dislocation in the arbitrary 𝑥<sub>1</sub>, 𝑥<sub>2</sub>, 𝑥<sub>3</sub> coordinate system, decompose the Burgers vector (𝑏) to orthogonal components. This leads to the generalised definition of the [[J-integral]] in equations below. For a dislocation pile-up, the [[J-integral]] is the summation of the Peach–[[James Stark Koehler|Koehler]] configurational force of the dislocations in the pile-up (including out-of-plane, 𝑏<sub>3</sub> <ref>{{Cite book |last1=Anderson |first1=PM |last2=Hirth |first2=JP |last3=Lothe |first3=J. |title=Theory of Straight Dislocations. Theory of dislocations}}</ref>).

𝐽<sub>𝑘</sub> = ∫ 𝑃<sub>𝑘𝑗</sub> 𝑛<sub>𝑗</sub> 𝑑𝑆 = ∫(𝑊<sub>𝑠</sub> 𝑛<sub>𝑘</sub>− 𝑇<sub>𝑖</sub> 𝑢<sub>𝑖,𝑘</sub>) 𝑑𝑆

𝐽<sub>𝑘</sub><sup>𝑥</sup> = 𝑅<sub>𝑘𝑗</sub> 𝐽<sub>𝑗</sub>,   𝑖,𝑗,𝑘=1,2,3

where 𝑆 is an arbitrary contour around the [[dislocation]] pile-up with unit outward normal 𝑛<sub>𝑖</sub>, 𝑊<sub>𝑠</sub> is the strain energy density, 𝑇<sub>𝑖</sub> = 𝜎<sub>𝑖𝑗</sub> 𝑛<sub>𝑗</sub> is the traction on 𝑑𝑆, 𝑢<sub>𝑖</sub> are the displacement vector components, 𝐽<sub>𝑘𝑥</sub> is 𝐽-integral evaluated along the 𝑥<sub>𝑘</sub> direction, and 𝑅<sub>𝑘𝑗</sub> is a second-order mapping tensor that maps 𝐽<sub>𝑘</sub> into 𝑥<sub>𝑘</sub> direction. This vectorial 𝐽<sub>𝑘</sub>-integral leads to numerical difficulties in the analysis since 𝐽<sub>2</sub> and, for a three-dimensional slip band or inclined crack, the 𝐽<sub>3</sub> terms cannot be neglected.<ref name="Koko 118284" />

== See also ==

* [[Crystal twinning#Deformation twinning|Deformation twinning]]
* [[Lüders band]]

==References==
{{Reflist}}

== Further reading ==
{{Commons}}

* {{Cite journal |last1=Trochidis |first1=A. |last2=E. |first2=Douka |last3=Polyzos |first3=B. |date=2000-08-11 |title=Formation and evolution of persistent slip bands in metals |url=https://www.sciencedirect.com/science/article/abs/pii/S0022509699000770 |journal=[[Journal of the Mechanics and Physics of Solids]] |volume=48 |issue=8 |pages=1761–1775 |doi=10.1016/S0022-5096(99)00077-0|bibcode=2000JMPSo..48.1761T }}
* {{Cite journal |last1=Zhang |first1=P. |last2=Zhang |first2=L. |last3=Baxevanakis |first3=K.P. |last4=Lu |first4=S. |last5=Zhao |first5=L.G. |last6=Bullough |first6=C. |date=April 2021 |title=Discrete Crystal Plasticity Modelling of Slip-Controlled Cyclic Deformation and Short Crack Growth under Low Cycle Fatigue |url=https://www.sciencedirect.com/science/article/abs/pii/S0142112320306277 |journal=[[International Journal of Fatigue]] |volume=145 |page=106095 |doi=10.1016/j.ijfatigue.2020.106095|s2cid=230576292 }}

[[Category:Materials science]]
[[Category:Materials degradation]]
[[Category:Mechanical failure modes]]
[[Category:Structural analysis]]
[[Category:Solid mechanics]]
[[Category:Crystallography]]
[[Category:Metallurgy]]