Slip bands in metals

A slip band formed on a ferrite grain in an 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.[1][2]

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 on certain planes causing a stress concentration. Typically, slip bands induce surface steps (e.g., roughness due persistent slip bands during fatigue) and a stress concentration which can be a crack nucleation site. Slip bands extend until impinged by a boundary, and the generated stress from dislocations pile-up against that boundary will either stop or transmit the operating slip depending on its (mis)orientation.[3][4]

Formation of slip bands under cyclic conditions is addressed as persistent slip bands (PSBs) where formation under monotonic condition is addressed as dislocation planar arrays (or simply slip-bands, see Slip bands in the absence of cyclic loading section).[5] 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 PSBs normally studied with (effective) Burgers vector aligned with the extrusion plane because a PSB extends across the grain and exacerbates during fatigue;[6] 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)

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PSB structure (adopted from [7])

Persistent slip-bands (PSBs) are associated with strain localisation due to fatigue in metals and cracking on the same plane. Transmission electron microscopy (TEM) and three-dimensional discrete dislocation dynamics (DDD[8]) 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 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.[9][10]

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 resolved shear stress for a glide on more than one slip plane. 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 field which has a minimal long-range stress field.[clarification needed] This is different to slip-bands that is a planar stack of a stable array that has a strong long-range stress field.[clarification needed] 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.[10][11][12]

Surface relief through extrusion occurs on the Burger's vector direction and extrusion height and PSB depth increase with PSB thickness.[13] PSB and planar walls are parallel and perpendicularly aligned with the normal direction of the Critical resolved shear stress, respectively.[14] And once dislocation saturate and reach its sessile configuration, cracks were observed to nucleate and propagate along PSB extrusions.[15][16][17] 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 field ahead of the 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 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.[18]

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.[19] 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 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 [20] 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 [21] and then propagate through the nano-blade.

Frank–Read source

Fisher et al. [18] 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 [22] 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 band morphology planar.[23]

Dislocation activity assists the growth of austenite precipitates and provide quantitative data for revealing the stress field generated by interface migration.[24] 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 (τtip), the image stress tending to attract the dislocation loop to the surface (τimage), the line tension (τl) and the interaction stress between dislocations (τinter). 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.[2]

Slip bands in the absence of cyclic loading

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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, acting in particular planes.

Secondary electron images of 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 () and austenite () phases are labelled.[1]

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[25] (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

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Schematic of a slip band, relative to the measurement axes (𝑥1, 𝑥2, and 𝑥3), 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.[26] 𝑡 describes the line vector drawn here as for an edge dislocation, i.e., 𝑏⊥𝑡, and 𝑛 is the slip band plane normal.[1]

The deformation field at the slip-band is due to three-dimensional elastic and plastic strains where the concentrated shear of the slip band tip deforms the grain in its vicinity. The elastic strains describe the stress concentration ahead of the slip band, which is important as it can affect the transfer of plastic deformation across grain boundaries.[27][28][29] An understanding of this is needed to support the study of yield and inter/intra-granular fracture.[30][31][32] The concentrated shear of slip bands can also nucleate cracks in the plane of the slip band,[16][17] and persistent slip bands that lead to intragranular fatigue crack initiation and growth may also form under cyclic loading conditions.[5][33] To properly characterise slip bands and validate mechanistic models for their interactions with microstructure, it is crucial to quantify the local deformation fields associated with their propagation. However, little attention has been given to 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 band, can be considered a singularity equivalent to that of a crack.[34][35] 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 [36] from the Noether's theorem.[37] Budiansky and Rice[38] introduced the J-, M-, L-integral and were the first to give them a physical interpretation as the 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.[39] That work paved the way for the field of Configurational mechanics of materials, with the path-independent J-integral now widely used to analyse the configurational forces in problems as diverse as dislocation dynamics,[40][41] misfitting inclusions,[42] propagation of cracks,[43] shear deformation of clays,[44] and co-planar dislocation nucleation from shear loaded cracks.[45] The integrals have been applied to linear elastic and elastic-plastic materials and have been coupled with processes such as thermal [46] and electrochemical [47] loading, and internal tractions.[48] Recently, experimental fracture mechanics studies have used full-field in situ measurements of displacements [49][50] and elastic strains [51][50] 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–Koehler configurational force (𝑃𝑘𝑗) [52] (or the elastic energy-momentum tensor [53]) on a dislocation in the arbitrary 𝑥1, 𝑥2, 𝑥3 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–Koehler configurational force of the dislocations in the pile-up (including out-of-plane, 𝑏3 [54]).

𝐽𝑘 = ∫ 𝑃𝑘𝑗 𝑛𝑗 𝑑𝑆 = ∫(𝑊𝑠 𝑛𝑘− 𝑇𝑖 𝑢𝑖,𝑘) 𝑑𝑆

𝐽𝑘𝑥 = 𝑅𝑘𝑗 𝐽𝑗, 𝑖,𝑗,𝑘=1,2,3

where 𝑆 is an arbitrary contour around the dislocation pile-up with unit outward normal 𝑛𝑖, 𝑊𝑠 is the strain energy density, 𝑇𝑖 = 𝜎𝑖𝑗 𝑛𝑗 is the traction on 𝑑𝑆, 𝑢𝑖 are the displacement vector components, 𝐽𝑘𝑥 is 𝐽-integral evaluated along the 𝑥𝑘 direction, and 𝑅𝑘𝑗 is a second-order mapping tensor that maps 𝐽𝑘 into 𝑥𝑘 direction. This vectorial 𝐽𝑘-integral leads to numerical difficulties in the analysis since 𝐽2 and, for a three-dimensional slip band or inclined crack, the 𝐽3 terms cannot be neglected.[1]

See also

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References

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  1. ^ a b c d Koko, Abdalrhaman; Elmukashfi, Elsiddig; Becker, Thorsten H.; Karamched, Phani S.; Wilkinson, Angus J.; Marrow, T. James (2022-10-15). "In situ characterisation of the strain fields of intragranular slip bands in ferrite by high-resolution electron backscatter diffraction". Acta Materialia. 239: 118284. Bibcode:2022AcMat.23918284K. doi:10.1016/j.actamat.2022.118284. ISSN 1359-6454. S2CID 251783802. This article incorporates text from this source, which is available under the CC BY 4.0 license.
  2. ^ a b Koko, A. Mohamed (2022). In situ full-field characterisation of strain concentrations (deformation twins, slip bands and cracks) (PhD thesis). University of Oxford. Archived from the original on 2023-02-01. Retrieved 2023-03-02. This article incorporates text from this source, which is available under the CC BY 4.0 license.
  3. ^ Smallman, R. E.; Ngan, A. H. W. (2014-01-01), Smallman, R. E.; Ngan, A. H. W. (eds.), "Chapter 9 - Plastic Deformation and Dislocation Behaviour", Modern Physical Metallurgy (Eighth Edition), Oxford: Butterworth-Heinemann, pp. 357–414, doi:10.1016/b978-0-08-098204-5.00009-2, ISBN 978-0-08-098204-5, archived from the original on 2022-10-04, retrieved 2022-10-04
  4. ^ Sangid, Michael D. (2013-12-01). "The physics of fatigue crack initiation". International Journal of Fatigue. Fatigue and Microstructure: A special issue on recent advances. 57: 58–72. doi:10.1016/j.ijfatigue.2012.10.009. ISSN 0142-1123.
  5. ^ a b Lukáš, P.; Klesnil, M.; Krejčí, J. (1968). "Dislocations and Persistent Slip Bands in Copper Single Crystals Fatigued at Low Stress Amplitude". Physica Status Solidi B (in German). 27 (2): 545–558. Bibcode:1968PSSBR..27..545L. doi:10.1002/pssb.19680270212. S2CID 96586802. Archived from the original on 2022-10-03. Retrieved 2022-10-03.
  6. ^ Schiller, C.; Walgraef, D. (1988-03-01). "Numerical simulation of persistent slip band formation". Acta Metallurgica. 36 (3): 563–574. doi:10.1016/0001-6160(88)90089-2. ISSN 0001-6160. Archived from the original on 2023-03-25. Retrieved 2023-03-25.
  7. ^ Erel, Can; Po, Giacomo; Crosby, Tamer; Ghoniem, Nasr (December 2017). "Generation and interaction mechanisms of prismatic dislocation loops in FCC metals". Computational Materials Science. 140: 32–46. doi:10.1016/j.commatsci.2017.07.043.
  8. ^ "grain boundray, dislocation simulation, discrete dislocation dynamics, dislocation dynamics, grain boundary, dislocation cell, cross slip, creep, deformation, strain hardening, superalloy, climb". www.dierk-raabe.com (in German). Archived from the original on 2022-12-03. Retrieved 2023-03-05.
  9. ^ Differt, K.; Essmann, U. (1993), "Dynamical model of the wall structure in persistent slip bands of fatigued metals", Fundamental Aspects of Dislocation Interactions, Elsevier, pp. 295–299, doi:10.1016/b978-1-4832-2815-0.50048-6, ISBN 978-1-4832-2815-0, archived from the original on 2018-06-30, retrieved 2022-10-03
  10. ^ a b Verdier, M; Fivel, M; Groma, I (1998-11-01). "Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications". Modelling and Simulation in Materials Science and Engineering. 6 (6): 755–770. Bibcode:1998MSMSE...6..755V. doi:10.1088/0965-0393/6/6/007. ISSN 0965-0393. S2CID 250889422. Archived from the original on 2022-10-03. Retrieved 2022-10-03.
  11. ^ Déprés, C.; Robertson, C. F.; Fivel, M. C. (January 2006). "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". Philosophical Magazine. 86 (1): 79–97. Bibcode:2006PMag...86...79D. doi:10.1080/14786430500341250. ISSN 1478-6435. S2CID 135953582. Archived from the original on 2021-05-17. Retrieved 2022-10-03.
  12. ^ Déprés, C.; Robertson, C. F.; Fivel, M. C. (August 2004). "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". Philosophical Magazine. 84 (22): 2257–2275. Bibcode:2004PMag...84.2257D. doi:10.1080/14786430410001690051. ISSN 1478-6435. S2CID 137329770. Archived from the original on 2022-10-07. Retrieved 2022-10-03.
  13. ^ Man, J.; Obrtlík, K.; Polák, J. (June 2009). "Extrusions and intrusions in fatigued metals. Part 1. State of the art and history†". Philosophical Magazine. 89 (16): 1295–1336. Bibcode:2009PMag...89.1295M. doi:10.1080/14786430902917616. ISSN 1478-6435. S2CID 136919859. Archived from the original on 2022-10-06. Retrieved 2022-10-03.
  14. ^ Amodeo, R. J.; Ghoniem, N. M. (1990-04-01). "Dislocation dynamics. II. Applications to the formation of persistent slip bands, planar arrays, and dislocation cells". Physical Review B. 41 (10): 6968–6976. Bibcode:1990PhRvB..41.6968A. doi:10.1103/PhysRevB.41.6968. ISSN 0163-1829. PMID 9992953. Archived from the original on 2023-03-25. Retrieved 2022-10-03.
  15. ^ Wood, W. A. (July 1958). "Formation of fatigue cracks". Philosophical Magazine. 3 (31): 692–699. Bibcode:1958PMag....3..692W. doi:10.1080/14786435808237004. ISSN 0031-8086. Archived from the original on 2022-10-07. Retrieved 2022-10-03.
  16. ^ a b Koss, D.A.; Chan, K.S. (September 1980). "Fracture along planar slip bands". Acta Metallurgica. 28 (9): 1245–1252. doi:10.1016/0001-6160(80)90080-2. Archived from the original on 2018-06-28. Retrieved 2022-10-03.
  17. ^ a b Mughrabi, H. (September 1983). "Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals". Acta Metallurgica. 31 (9): 1367–1379. doi:10.1016/0001-6160(83)90007-X. Archived from the original on 2022-06-16. Retrieved 2022-10-03.
  18. ^ a b Fisher, John C.; Hart, Edward W.; Pry, Robert H. (1952-09-15). "Theory of Slip-Band Formation". Physical Review. 87 (6): 958–961. Bibcode:1952PhRv...87..958F. doi:10.1103/PhysRev.87.958. ISSN 0031-899X.
  19. ^ Frank, F. C.; Read, W. T. (1950-08-15). "Multiplication Processes for Slow Moving Dislocations". Physical Review. 79 (4): 722–723. Bibcode:1950PhRv...79..722F. doi:10.1103/PhysRev.79.722. ISSN 0031-899X. Archived from the original on 2023-03-25. Retrieved 2022-10-03.
  20. ^ Xie, Kelvin Y.; Wang, Yanbo; Ni, Song; Liao, Xiaozhou; Cairney, Julie M.; Ringer, Simon P. (December 2011). "Insight into the deformation mechanisms of α-Fe at the nanoscale". Scripta Materialia. 65 (12): 1037–1040. doi:10.1016/j.scriptamat.2011.08.023. Archived from the original on 2018-06-02. Retrieved 2022-10-03.
  21. ^ Zheng, He; Cao, Ajing; Weinberger, Christopher R.; Huang, Jian Yu; Du, Kui; Wang, Jianbo; Ma, Yanyun; Xia, Younan; Mao, Scott X. (December 2010). "Discrete plasticity in sub-10-nm-sized gold crystals". Nature Communications. 1 (1): 144. Bibcode:2010NatCo...1..144Z. doi:10.1038/ncomms1149. ISSN 2041-1723. PMC 3105591. PMID 21266994.
  22. ^ Kiener, D.; Minor, A.M. (February 2011). "Source-controlled yield and hardening of Cu(100) studied by in situ transmission electron microscopy". Acta Materialia. 59 (4): 1328–1337. Bibcode:2011AcMat..59.1328K. doi:10.1016/j.actamat.2010.10.065. Archived from the original on 2022-06-17. Retrieved 2022-10-03.
  23. ^ Hsieh, Yi-Chieh; Zhang, Ling; Chung, Tsai-Fu; Tsai, Yu-Ting; Yang, Jer-Ren; Ohmura, Takahito; Suzuki, Takuya (December 2016). "In-situ transmission electron microscopy investigation of the deformation behavior of spinodal nanostructured δ-ferrite in a duplex stainless steel". Scripta Materialia. 125: 44–48. doi:10.1016/j.scriptamat.2016.06.047. Archived from the original on 2022-06-20. Retrieved 2022-10-03.
  24. ^ Du, Juan; Mompiou, Frédéric; Zhang, Wen-Zheng (March 2018). "In-situ TEM study of dislocation emission associated with austenite growth". Scripta Materialia. 145: 62–66. doi:10.1016/j.scriptamat.2017.10.014. Archived from the original on 2018-06-30. Retrieved 2022-10-03.
  25. ^ Campbell, JE; Thompson, RP; Dean, J; Clyne, TW (2019). "Comparison between stress-strain plots obtained from indentation plastometry, based on residual indent profiles, and from uniaxial testing". Acta Materialia. 168: 87–99. Bibcode:2019AcMat.168...87C. doi:10.1016/j.actamat.2019.02.006.
  26. ^ Parks, D.M. (December 1977). "The virtual crack extension method for nonlinear material behavior". Computer Methods in Applied Mechanics and Engineering. 12 (3): 353–364. Bibcode:1977CMAME..12..353P. doi:10.1016/0045-7825(77)90023-8. Archived from the original on 2022-06-29. Retrieved 2022-10-03.
  27. ^ Benjamin Britton, T.; Wilkinson, Angus J. (September 2012). "Stress fields and geometrically necessary dislocation density distributions near the head of a blocked slip band". Acta Materialia. 60 (16): 5773–5782. Bibcode:2012AcMat..60.5773B. doi:10.1016/j.actamat.2012.07.004. hdl:10044/1/13886. Archived from the original on 2023-01-01. Retrieved 2022-10-03.
  28. ^ Guo, Y.; Britton, T.B.; Wilkinson, A.J. (September 2014). "Slip band–grain boundary interactions in commercial-purity titanium". Acta Materialia. 76: 1–12. Bibcode:2014AcMat..76....1G. doi:10.1016/j.actamat.2014.05.015. hdl:10044/1/25973. S2CID 136904692. Archived from the original on 2022-06-20. Retrieved 2022-10-03.
  29. ^ Andani, Mohsen Taheri; Lakshmanan, Aaditya; Sundararaghavan, Veera; Allison, John; Misra, Amit (November 2020). "Quantitative study of the effect of grain boundary parameters on the slip system level Hall-Petch slope for basal slip system in Mg-4Al". Acta Materialia. 200: 148–161. Bibcode:2020AcMat.200..148A. doi:10.1016/j.actamat.2020.08.079. S2CID 225279335.
  30. ^ Livingston, J.D; Chalmers, B (June 1957). "Multiple slip in bicrystal deformation". Acta Metallurgica. 5 (6): 322–327. doi:10.1016/0001-6160(57)90044-5. Archived from the original on 2022-06-17. Retrieved 2022-10-03.
  31. ^ Lee, T.C.; Robertson, I.M.; Birnbaum, H.K. (May 1989). "Prediction of slip transfer mechanisms across grain boundaries". Scripta Metallurgica. 23 (5): 799–803. doi:10.1016/0036-9748(89)90534-6. Archived from the original on 2022-07-10. Retrieved 2022-10-03.
  32. ^ Luster, J.; Morris, M. A. (July 1995). "Compatibility of deformation in two-phase Ti-Al alloys: Dependence on microstructure and orientation relationships". Metallurgical and Materials Transactions A. 26 (7): 1745–1756. Bibcode:1995MMTA...26.1745L. doi:10.1007/BF02670762. ISSN 1073-5623. S2CID 137425735. Archived from the original on 2023-03-25. Retrieved 2022-10-03.
  33. ^ Tu, S.-T.; Zhang, X.-C. (2016), "Fatigue Crack Initiation Mechanisms", Reference Module in Materials Science and Materials Engineering, Elsevier, pp. B9780128035818028526, doi:10.1016/b978-0-12-803581-8.02852-6, ISBN 978-0-12-803581-8, archived from the original on 2022-07-11, retrieved 2022-10-03
  34. ^ Makin, M. J. (April 1970). "The mechanism of slip band growth in irradiated crystals". Philosophical Magazine. 21 (172): 815–817. Bibcode:1970PMag...21..815M. doi:10.1080/14786437008238467. ISSN 0031-8086. Archived from the original on 2022-10-07. Retrieved 2022-10-03.
  35. ^ Rice, James R. (December 1987). "Tensile crack tip fields in elastic-ideally plastic crystals". Mechanics of Materials. 6 (4): 317–335. Bibcode:1987MechM...6..317R. doi:10.1016/0167-6636(87)90030-5. Archived from the original on 2022-12-05. Retrieved 2022-10-03.
  36. ^ Knowles, J. K.; Sternberg, Eli (January 1972). "On a class of conservation laws in linearized and finite elastostatics". Archive for Rational Mechanics and Analysis. 44 (3): 187–211. Bibcode:1972ArRMA..44..187K. doi:10.1007/BF00250778. ISSN 0003-9527. S2CID 122386163. Archived from the original on 2023-03-25. Retrieved 2022-10-03.
  37. ^ Noether, Emmy (January 1971). "Invariant variation problems". Transport Theory and Statistical Physics. 1 (3): 186–207. arXiv:physics/0503066. Bibcode:1971TTSP....1..186N. doi:10.1080/00411457108231446. ISSN 0041-1450. S2CID 119019843. Archived from the original on 2022-07-15. Retrieved 2022-10-03.
  38. ^ Budiansky, B.; Rice, J. R. (1973-03-01). "Conservation Laws and Energy-Release Rates". Journal of Applied Mechanics. 40 (1): 201–203. Bibcode:1973JAM....40..201B. doi:10.1115/1.3422926. ISSN 0021-8936. Archived from the original on 2022-10-03. Retrieved 2022-10-03.
  39. ^ Eshelby, J. D. (1951-11-06). "The force on an elastic singularity". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 244 (877): 87–112. Bibcode:1951RSPTA.244...87E. doi:10.1098/rsta.1951.0016. ISSN 0080-4614. S2CID 14703976. Archived from the original on 2022-10-03. Retrieved 2022-10-03.
  40. ^ Agiasofitou, Eleni; Lazar, Markus (May 2017). "Micromechanics of dislocations in solids: J -, M -, and L -integrals and their fundamental relations". International Journal of Engineering Science. 114: 16–40. arXiv:1702.00363. doi:10.1016/j.ijengsci.2017.02.001. S2CID 119531197. Archived from the original on 2022-08-15. Retrieved 2022-10-03.
  41. ^ Kim, Hokun; Kim, Soon; Kim, Sung Youb (March 2021). "Lattice-based J integral for a steadily moving dislocation". International Journal of Plasticity. 138: 102949. doi:10.1016/j.ijplas.2021.102949. S2CID 233799154. Archived from the original on 2023-03-25. Retrieved 2022-10-03.
  42. ^ Markenscoff, Xanthippi; Ni, Luqun (January 2010). "The energy-release rate and "self-force" of dynamically expanding spherical and plane inclusion boundaries with dilatational eigenstrain". Journal of the Mechanics and Physics of Solids. 58 (1): 1–11. Bibcode:2010JMPSo..58....1M. doi:10.1016/j.jmps.2009.10.001. Archived from the original on 2022-02-21. Retrieved 2022-10-03.
  43. ^ Rice, Jr; Drugan, WJ; Sham, T-L (1980-01-01), Paris, Pc (ed.), "Elastic-Plastic Analysis of Growing Cracks", Fracture Mechanics, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959: ASTM International, pp. 189–189–33, doi:10.1520/stp36972s, ISBN 978-0-8031-0363-4, archived from the original on 2018-06-02, retrieved 2022-10-03{{citation}}: CS1 maint: location (link)
  44. ^ Palmer, A. C.; Rice, J. R. (1973-04-03). "The growth of slip surfaces in the progressive failure of over-consolidated clay". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 332 (1591): 527–548. Bibcode:1973RSPSA.332..527P. doi:10.1098/rspa.1973.0040. ISSN 0080-4630. S2CID 7935986. Archived from the original on 2022-10-03. Retrieved 2022-10-03.
  45. ^ Rice, James R. (January 1992). "Dislocation nucleation from a crack tip: An analysis based on the Peierls concept". Journal of the Mechanics and Physics of Solids. 40 (2): 239–271. Bibcode:1992JMPSo..40..239R. doi:10.1016/S0022-5096(05)80012-2. Archived from the original on 2022-07-16. Retrieved 2022-10-03.
  46. ^ Kc, Amit; Kim, Jeong-Ho (May 2008). "Interaction integrals for thermal fracture of functionally graded materials". Engineering Fracture Mechanics. 75 (8): 2542–2565. doi:10.1016/j.engfracmech.2007.07.011. Archived from the original on 2022-08-07. Retrieved 2022-10-03.
  47. ^ Haftbaradaran, Hamed; Qu, Jianmin (November 2014). "A path-independent integral for fracture of solids under combined electrochemical and mechanical loadings". Journal of the Mechanics and Physics of Solids. 71: 1–14. Bibcode:2014JMPSo..71....1H. doi:10.1016/j.jmps.2014.06.007. Archived from the original on 2023-03-25. Retrieved 2022-10-03.
  48. ^ Walters, Matthew C.; Paulino, Glaucio H.; Dodds, Robert H. (July 2005). "Interaction integral procedures for 3-D curved cracks including surface tractions". Engineering Fracture Mechanics. 72 (11): 1635–1663. doi:10.1016/j.engfracmech.2005.01.002. Archived from the original on 2022-08-08. Retrieved 2022-10-03.
  49. ^ Becker, T. H.; Mostafavi, M.; Tait, R. B.; Marrow, T. J. (October 2012). "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". Fatigue & Fracture of Engineering Materials & Structures. 35 (10): 971–984. doi:10.1111/j.1460-2695.2012.01685.x. Archived from the original on 2022-10-03. Retrieved 2022-10-03.
  50. ^ a b Koko, A.; Earp, P.; Wigger, T.; Tong, J.; Marrow, T.J. (May 2020). "J-integral analysis: An EDXD and DIC comparative study for a fatigue crack". International Journal of Fatigue. 134: 105474. doi:10.1016/j.ijfatigue.2020.105474. S2CID 214391445. Archived from the original on 2022-06-22. Retrieved 2022-10-03.
  51. ^ Barhli, S.M.; Saucedo-Mora, L.; Jordan, M.S.L.; Cinar, A.F.; Reinhard, C.; Mostafavi, M.; Marrow, T.J. (November 2017). "Synchrotron X-ray characterization of crack strain fields in polygranular graphite". Carbon. 124: 357–371. Bibcode:2017Carbo.124..357B. doi:10.1016/j.carbon.2017.08.075. hdl:1983/93385e0e-155e-49e6-9a3d-e4f881de013a. Archived from the original on 2022-06-18. Retrieved 2022-10-03.
  52. ^ Lubarda, Vlado A. (2019-01-01). "Dislocation Burgers vector and the Peach–Koehler force: a review". Journal of Materials Research and Technology. 8 (1): 1550–1565. doi:10.1016/j.jmrt.2018.08.014. ISSN 2238-7854. S2CID 125242265.
  53. ^ Eshelby, J.D. (1956), The Continuum Theory of Lattice Defects, Solid State Physics, vol. 3, Elsevier, pp. 79–144, doi:10.1016/s0081-1947(08)60132-0, ISBN 978-0-12-607703-2, archived from the original on 2022-12-19, retrieved 2022-10-03
  54. ^ Anderson, PM; Hirth, JP; Lothe, J. Theory of Straight Dislocations. Theory of dislocations.

Further reading

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