On the Origin of Delayed Elasticity in Inorganic Glasses
DOI:
https://doi.org/10.52825/glass-europe.v4i.3253Keywords:
Delayed Elasticity, Anelasticity, Glasses, Viscoelasticity, Shear TransformationAbstract
Delayed elasticity is a universal feature of glasses, manifesting as a recoverable deformation accompanying viscous flow across the glass-transition domain. While heuristic models have been proposed to explain its origin, the underlying mechanisms remain poorly understood. Here, the origin of delayed elasticity is investigated using a mesoscopic shear transformation (ST) model within the linear viscoelastic regime. Glasses undergo continuous, thermally activated STs that generate long-range stress fields, leading to a randomly fluctuating internal stress. In the linear viscoelastic regime, applied macroscopic stresses are much smaller than the characteristic stress fluctuations induced by interactions between shear transformation zones (STZs). The model reveals that delayed elasticity emerges from the collective behavior of STZs rather than from their individual mechanical history which cannot be preserved under such fluctuating internal stresses. When STZs have no intrinsic distribution of energy barriers, localization into shear bands mays occur, leading to faster plastic strain accumulation in certain regions and the generation of a certain counteracting "back stress." However, more realistically, when an intrinsic distribution of energy barriers exists, localization is suppressed, and delayed elasticity arises instead from the cooperative dynamics of STZs. The back stress, which drives delayed elasticity recovery, originates from these collective interactions, through long-range stress fields, rather than from individual mechanical history/memory. Thus, the memory behind delayed elasticity, often considered local, is fundamentally collective.
Downloads
References
[1] U. Fotheringham, "Viscosity of glass and glass-forming melts", in Springer handbook of glass, Springer, pp. 79–112.
[2] L. Boltzmann, "On the theory of the elastic aftereffect", Poggendorff’s Ann. Phys. Chem, vol. 7, pp. 624–645.
[3] R. Simha, "On Relaxation Effects in Amorphous Media", J. Appl. Phys., vol. 13, no. 3, pp. 201-207. DOI: 10.1063/1.1714856.
[4] A. S. Argon, "Mechanisms of inelastic deformation in metallic glasses", J. Phys. Chem. Solids, vol. 43, no. 10, pp. 945–961. DOI: 10.1016/0022-3697(82)90111-1.
[5] A. O. Rankine, "The behaviour of over-strained materials", Science Progress in the Twentieth Century (1906-1916), vol. 1, no. 3, pp. 465–482. [Online]. Available: https://www.jstor.org/stable/43769106.
[6] W. Weber, "Ueber die Elasticität der Seidenfäden", Ann. Phys. (Berl.), vol. 110, no. 2, pp. 247–257. ISSN: 1521-3889. DOI: 10.1002/andp.18351100204.
[7] W. Weber, "Ueber die elasticitäet fester körper", Ann. Phys. (Berl.), vol. 130, no. 9, pp. 1–18. DOI: 10.1002/andp.18411300902.
[8] H. Cavendish, "XXI. Experiments to determine the density of the earth", Philos. Trans. R. Soc. Lond., vol. 88, pp. 469-526, see p.478. DOI: 10.1098/rstl.1798.0022.
[9] C. W. F. Everitt, "Gravitation, relativity and precise experimentation", in Proc. of the First Marcel Grossmann Meeting on General Relativity, pp. 545–615. [Online]. Available: https://einstein.stanford.edu/content/sci_papers/papers/Everitt_Gravitation-Precise-Expt-1977.pdf.
[10] F. Kohlrausch, "Ueber die elastische Nachwirkung bei der Torsion", Annalen der physik, vol. 195, no. 7, pp. 337–368. DOI: 10.1002/andp.18631950702.
[11] L. Boltzmann, "Zur Theorie der elastischen Nachwirkung , Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, 70 (2): 275–306", German. DOI: 10.1017/CBO9781139381420.031.
[12] O. E. Meyer, "Theorie der elastischen Nachwirkung", Annalen der Physik, vol. 227, no. 1, pp. 108–119. DOI: 10.1002/ANDP.18742270106.
[13] E. Wiechert, "Gesetze der elastischen Nachwirkung für constante Temperatur", Ann. Phys. (Berl.), vol. 286, no. 11, pp. 546–570. DOI: 10.1002/andp.18932861110.
[14] M. Dörries, "Prior History and Aftereffects: Hysteresis and "Nachwirkung" in 19th-Century Physics", Historical studies in the physical and biological sciences, vol. 22, no. 1, pp. 25–55. DOI: 10.2307/27757672.
[15] M. G. Ianniello, "Elastic Nachwirkung, Brownian motion, and the tide against determinism: 1835-1920", Historical studies in the physical and biological sciences, vol. 24, no. 1, pp. 41–100. DOI: 10.2307/27757712.
[16] B. B. DeWitt, "On the Transparency of the Ether", Nebraska University Studies, vol. I, no. I, pp. 1-17. [Online]. Available: https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1002&context=univstudiespapers.
[17] H. Lamb, "On a Certain Theory of Elastic After-Strain", Nature, vol. 41, no. 1064, pp. 463–463. DOI: 10.1038/041463b0.
[18] H. Markovitz, "Boltzmann and the Beginnings of Linear Viscoelasticity", T. Soc. Rheol., vol. 21, no. 3, pp. 381-398. DOI: 10.1122/1.549444.
[19] K. E. Guthe, "Some Peculiarities in the Elastic Properties of Certain Substances", in Proceedings of the Iowa Academy of Science, vol. 15, pp. 147–156. [Online]. Available: https://scholarworks.uni.edu/pias/vol15/iss1/24.
[20] E. C. Bingham, "Plasticity and elasticity", J. Franklin Inst. B, vol. 197, no. 1, pp. 99-115. ISSN: 0016-0032. DOI: https://doi.org/10.1016/S0016-0032(24)90500-X.
[21] K. H. Borchard, "Elastic After-Effect in Glass", Sprechsaal, vol. 67, no. 20, p. 297.
[22] W. Thomson, Elasticity and heat, Adam and Charles Black.
[23] W. T. B. Kelvin, Mathematical and Physical Papers, vol. 3, At the University Press.
[24] A. Love, A treatise on the mathematical theory of elasicity, vol. Vol.1, Cambrigde: at the university press. [Online]. Available: https://hal.science/hal-01307751v1/file/LoveVol1.pdf.
[25] E. Bergler, "Contributions to the theory of glass formation and the glassy state", J. Am. Ceram. Soc., vol. 15, no. 12, pp. 647–678. DOI: 10.1111/j.1151-2916.1932.tb13902.x.
[26] M. F. Sayre, "Elastic After-Effect in Metals", J. Rheol., vol. 3, no. 2, pp. 206–211. ISSN: 0097-0360. DOI: 10.1122/1.2116452.
[27] F. W. Preston, "The time factor in the testing of glassware", J. Am. Ceram. Soc., vol. 18, no. 1-12, pp. 220–224. DOI: 10.1111/j.1151-2916.1935.tb19384.x.
[28] N. W. Taylor, "Mechanism of fracture of glass and similar brittle solids", J. Appl. Phys., vol. 18, no. 11, pp. 943–955. DOI: 10.1063/1.1697579.
[29] D. Griggs, "Creep of rocks", The Journal of Geology, vol. 47, no. 3, pp. 225–251. DOI: 10.1086/624775.
[30] A. A. Michelson, "The Laws of Elastico-Viscous Flow", Proc. Natl. Acad. Sci., vol. 3, no. 5, pp. 319–323. ISSN: 1091-6490. DOI: 10.1073/pnas.3.5.319.
[31] C. W. Washburne, "Some Wrong Words", The Journal of Geology, vol. 51, no. 7, pp. 495–497. DOI: 10.1086/625173.
[32] G. Wiedemann, "I. On torsion", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 9, no. 53, pp. 1-15. DOI: 10.1080/14786448008626791.
[33] C. Barus, Maxwell's theory of the viscosity of solids and certain features of its physical verification. DOI: 10.2475/ajs.s3-36.213.178.
[34] W. J. M. Rankine, A Manual of applied Mechanics: With numerous Diagrams, Ch. Griffin.
[35] M. H. Tresca, "On further applications of the flow of solids", Proceedings of the Institution of Mechanical Engineers, vol. 29, no. 1, pp. 301–345. DOI: 10.1016/0016-0032(78)90311-3.
[36] G. J. Bair, "II, THE CORRELATION OF PHYSICAL PROPERTIES WITH ATOMIC ARRANGEMENT", J. Am. Ceram. Soc., vol. 19, no. 1–12, pp. 347–358. ISSN: 1551-2916. DOI: 10.1111/j.1151-2916.1936.tb19849.x.
[37] N. W. Taylor, and P. S. Dear, "Elastic and viscous properties of several soda-silica glasses in the annealing range of temperature", J. Am. Ceram. Soc., vol. 20, no. 1-12, pp. 296–304. DOI: 10.1111/j.1151-2916.1937.tb19906.x.
[38] N. W. Taylor, "Anomalous Flow in Glasses", J. Phys. Chem., vol. 47, no. 3, pp. 235–253. ISSN: 1541-5740. DOI: 10.1021/j150426a004.
[39] G. O. Jones, "Viscosity and related properties in glass", Rep. Prog. Phys., vol. 12, no. 1, p. 133. DOI: 10.1088/0034-4885/12/1/307.
[40] A. S. Argon, "Delayed elasticity in inorganic glasses", J. Appl. Phys., vol. 39, no. 9, pp. 4080–4086. DOI: 10.1063/1.1656927.
[41] C. M. Zener, and S. Siegel, "Elasticity and Anelasticity of Metals", J. Phys. Chem., vol. 53, no. 9, pp. 1468–1468. DOI: 10.1021/j150474a017.
[42] C. Zener, and J. H. Hollomon, "Problems in Non-Elastic Deformation of Metals", J. Appl. Phys., vol. 17, no. 2, pp. 69–82. ISSN: 1089-7550. DOI: 10.1063/1.1707696.
[43] J. V. Fitzgerald, "Anelasticity of glass: I, Introduction", J. Am. Ceram. Soc., vol. 34, no. 10, pp. 314–319. DOI: 10.1111/j.1151-2916.1951.tb13475.x.
[44] J. V. Fitzgerald, "Anelasticity of Glass: II, Internal Friction and Sodium Ion Diffusion in Tank Plate Glass, A Typical Soda-Lime-Silica Glass", J. Am. Ceram. Soc., vol. 34, no. 11, pp. 339–342. DOI: 10.1111/j.1151-2916.1951.tb13481.x.
[45] J. V. Fitzgerald, "Anelasticity of Glass: III, Effect of Heat-Treatment on the Internal Friction of Tank Plate Glass", J. Am. Ceram. Soc., vol. 34, no. 12, pp. 388–390. DOI: 10.1111/j.1151-2916.1951.tb13020.x.
[46] J. V. Fitzgerald, "Anelasticity of Glass: IV, Correlation of Electrical Strain with Mechanical Strain in Glass", J. Am. Ceram. Soc., vol. 34, no. 12, pp. 390–391. DOI: 10.1111/j.1151-2916.1951.tb13021.x.
[47] L. C. Hoffman, "Chemical Composition and the Anelasticity of Glass", Ph.D. dissertation.
[48] A. S. Argon, "Investigations of the Strength and Anelasticity of Glass", Ph.D. dissertation. [Online]. Available: https://dspace.mit.edu/handle/1721.1/57750.
[49] A. S. Argon, and L. T. Shi, "Development of visco-plastic deformation in metallic glasses", Acta Metall., vol. 31, no. 4, pp. 499–507. DOI: 10.1016/0001-6160(83)90038-X.
[50] M. Goldstein, "Viscous liquids and the glass transition: A potential energy barrier picture", J. Chem. Phys., vol. 51, pp. 3728-3739. DOI: 10.1063/1.1672587.
[51] S. Parke, "Anelasticity and viscoelasticity in glass", British Journal of Applied Physics, vol. 14, no. 5, pp. 243–248. DOI: 10.1088/0508-3443/14/5/306.
[52] E. C. Bingham, Fluidity and plasticity, McGraw-Hill.
[53] R. Hill, The mathematical theory of plasticity, vol. 11, Oxford university press.
[54] A. Nicolas, E. E. Ferrero, K. Martens, and J. Barrat, "Deformation and flow of amorphous solids: Insights from elastoplastic models", Rev. Mod. Phys., vol. 90, no. 4, p. 045006. DOI: 10.1103/RevModPhys.90.045006.
[55] D. Rodney, A. Tanguy, and D. Vandembroucq, "Modeling the mechanics of amorphous solids at different length scale and time scale", Model. Simul. Mater. Sci. Eng., vol. 19, no. 8, p. 083001. DOI: 10.1088/0965-0393/19/8/083001.
[56] K. E. Jensen, D. A. Weitz, and F. Spaepen, "Local shear transformations in deformed and quiescent hard-sphere colloidal glasses", Phys. Rev. E, vol. 90, no. 4, p. 042305. DOI: 10.1103/PhysRevE.90.042305.
[57] A. Tanguy, "Elasto-plastic behavior of amorphous materials: a brief review", C. R. Phys., vol. 22, no. S3, pp. 1–17. DOI: 10.5802/crphys.49.
[58] T. Rouxel, H. Ji, V. Keryvin, T. Hammouda, and S. Yoshida, "Poisson's ratio and the glass network topology-relevance to high pressure densification and indentation behavior", Adv. Mater. Res., vol. 39, pp. 137–146. DOI: 10.4028/www.scientific.net/AMR.39-40.137.
[59] T. Rouxel, H. Ji, T. Hammouda, and A. Moréac, "Poisson’s ratio and the densification of glass under high pressure", Physical Review Letters, vol. 100, no. 22, p. 225501. DOI: 10.1103/PhysRevLett.100.225501.
[60] K. W. Peter, "Densification and flow phenomena of glass in indentation experiments", J. Non-Cryst. Solids, vol. 5, no. 2, pp. 103–115. DOI: 10.1016/0022-3093(70)90188-2.
[61] E. Barthel, V. Keryvin, G. Rosales-Sosa, and G. Kermouche, "Indentation cracking in silicate glasses is directed by shear flow, not by densification", Acta Mater., vol. 194, pp. 473-481, 2020. ISSN: 1359-6454. DOI: https://doi.org/10.1016/j.actamat.2020.05.011.
[62] B. Mantisi, A. Tanguy, G. Kermouche, and E. Barthel, "Atomistic response of a model silica glass under shear and pressure", Eur. Phys. J. B, vol. 85, no. 9, p. 304, 2012. DOI: 10.1140/epjb/e2012-30317-6.
[63] J. T. Hagan, "Shear deformation under pyramidal indentations in soda-lime glass", J. Mater. Sci., vol. 15, no. 6, pp. 1417–1424, 1980. DOI: 10.1007/BF00752121.
[64] T. Rouxel, and J. Sangleboeuf, "The brittle to ductile transition in a soda - lime - silica glass", J. Non-Cryst., vol. 271, no. 3, pp. 224–235. DOI: 10.1016/S0022-3093(00)00109-5.
[65] E. R. Homer, and C. A. Schuh, "Mesoscale modeling of amorphous metals by shear transformation zone dynamics", Acta Mater., vol. 57, no. 9, pp. 2823–2833. DOI: 10.1016/j.actamat.2009.02.035.
[66] W. A. Zdaniewski, G. E. Rindone, and D. E. Day, "The internal friction of glasses", J. Mater. Sci., vol. 14, no. 4, pp. 763–775. DOI: 10.1007/bf00550707.
[67] S. Reinsch, R. Müller, J. Deubener, and H. Behrens, "Internal friction of hydrated soda-lime-silicate glasses", J. Chem. Phys., vol. 139, no. 17, p. 174506. DOI: 10.1063/1.4828740.
[68] S. V. Borovinskii, and O. V. Mazurin, "On the concept of volume viscosity", in Progress and Trends in Rheology II: Proceedings of the Second Conference of European Rheologists, Prague, June 17–20, 1986, pp. 79–81. DOI: 10.1007/978-3-642-49337-9_16.
[69] J. J. Mills, and J. L. Sievert, "Stress Relaxation Modulus of a Commercial Glass", J. Am. Ceram. Soc., vol. 56, no. 10, pp. 501-505. DOI: 10.1111/j.1151-2916.1973.tb12397.x.
[70] M. J. Crochet, J. De Bast, P. Gilard, and G. Tackels, "Experimental study of stress relaxation during annealing", J. Non-Cryst. Solids, vol. 14, no. 1, pp. 242–254. DOI: 10.1016/0022-3093(74)90033-7.
[71] M. S. Rekhson, and S. M. Rekhson, "Shear, Uniaxial, and Biaxial Stress Relaxation Functions", J. Am. Ceram. Soc., vol. 69, no. 9, pp. 704-708. DOI: 10.1111/j.1151-2916.1986.tb07475.x.
[72] S. M. Rekhson, "CHAPTER 1 - Viscoelasticity of Glass", in Viscosity and Relaxation, D. Uhlmann, and N. Kreidl, Eds., vol. 3, Elsevier, pp. 1-117. ISSN: 0927-4472. DOI: 10.1016/B978-0-12-706703-2.50004-0.
[73] L. Duffrène, R. Gy, H. Burlet, and R. Piques, "Multiaxial linear viscoelastic behavior of a soda - lime - silica glass based on a generalized Maxwell model", J. Rheol., vol. 41, no. 5, pp. 1021-1038. DOI: 10.1122/1.550824.
[74] F. T. Trouton, "On the coefficient of viscous traction and its relation to that of viscosity", Proc. R. Soc. London A., vol. 77, no. 519, pp. 426–440. DOI: 10.1098/rspa.1906.0038.
[75] L. Li, E. R. Homer, and C. A. Schuh, "Shear transformation zone dynamics model for metallic glasses incorporating free volume as a state variable", Acta Mater., vol. 61, no. 9, pp. 3347-3359. DOI: 10.1016/j.actamat.2013.02.024.
[76] P. Debye, "Einige resultate einer kinetischen theorie der isolatoren", Physik Z., vol. 13, p. 97.
[77] J. C. Maxwell, "On the Dynamical Theory of Gases", Philos. Trans. R. Soc. London, vol. 157, pp. 49–88. [Online]. Available: http://www.jstor.org/stable/108968.
[78] Y. Gueguen, V. Keryvin, T. Rouxel et al., "A relationship between non-exponential stress relaxation and delayed elasticity in the viscoelastic process in amorphous solids: Illustration on a chalcogenide glass", Mech. Mater., vol. 85, pp. 47–56. DOI: 10.1016/j.mechmat.2015.02.013.
[79] R. Becker, "Elastische nachwirkung und plastizität", Zeitschrift für Physik, vol. 33, no. 1, pp. 185–213.
[80] R. C. Welch, J. R. Smith, M. Potuzak et al., "Dynamics of Glass Relaxation at Room Temperature", Phys. Rev. Lett., vol. 110, p. 265901. DOI: 10.1103/PhysRevLett.110.265901.
[81] R. Böhmer, K. L. Ngai, C. A. Angell, and D. J. Plazek, "Nonexponential relaxations in strong and fragile glass formers", J. Chem. Phys., vol. 99, no. 5, pp. 4201-4209. DOI: 10.1063/1.466117.
[82] T. Mura, "General theory of eigenstrains", Micromechanics of defects in solids, pp. 1–73. DOI: 10.1007/978-94-009-3489-4_1.
[83] J. D. Eshelby, "The determination of the elastic field of an ellipsoidal inclusion, and related problems", Proceedings of the royal society of London. Series A. Mathematical and physical sciences, vol. 241, no. 1226, pp. 376–396. DOI: 10.1098/rspa.1957.0133.
[84] J. Chattoraj, and A. Lemaitre, "Elastic signature of flow events in supercooled liquids under shear", Phys. Rev. Lett., vol. 111, no. 6, p. 066001. DOI: 10.1103/PhysRevLett.111.066001.
[85] E. Orowan, "Creep in metallic and non-metallic materials.", Proc. First U.S. Natn. Cong. Appl. Mech., pp. 453-47, New York: ASME.
[86] O. V. Mazurin, "Glass relaxation", J. Non-Cryst. Solids, vol. 87, no. 3, pp. 392-407. DOI: 10.1016/S0022-3093(86)80013-8.
[87] V. V. Bulatov, and A. S. Argon, "A stochastic model for continuum elasto-plastic behavior. I. Numerical approach and strain localization", Modelling and Simulation in Materials Science and Engineering, vol. 2, no. 2, p. 167. DOI: 10.1088/0965-0393/2/2/001.
[88] E. R. Homer, D. Rodney, and C. A. Schuh, "Kinetic Monte Carlo study of activated states and correlated shear-transformation-zone activity during the deformation of an amorphous metal", Phys. Rev. B, vol. 81, p. 064204. DOI: 10.1103/PhysRevB.81.064204.
[89] F. V. Loock, L. Brassart, and T. Pardoen, "Implementation and calibration of a mesoscale model for amorphous plasticity based on shear transformation dynamics", Int. J. Plasticity, vol. 145, p. 103079. DOI: 10.1016/j.ijplas.2021.103079.
[90] V. Keryvin, "Indentation as a probe for pressure sensitivity of metallic glasses", J. Phys. Condens. Matter, vol. 20, no. 11, p. 114119. DOI: 10.1088/0953-8984/20/11/114119.
[91] A. S. Argon, "Plastic deformation in metallic glasses", Acta Metall., vol. 27, no. 1, pp. 47–58. DOI: 10.1016/0001-6160(79)90055-5.
[92] D. F. Castellanos, and M. Zaiser, "Avalanche Behavior in Creep Failure of Disordered Materials", Phys. Rev. Lett., vol. 121, p. 125501. DOI: 10.1103/PhysRevLett.121.125501.
[93] D. F. Castellanos, and M. Zaiser, "Statistical dynamics of early creep stages in disordered materials", Eur. Phys. J. B, vol. 92, pp. 1–12. DOI: 10.1140/epjb/e2019-100124-0.
[94] Q. Hao, S. Liang, G. Xing et al., "Decoupling elasticity, anelasticity, and plasticity in static and dynamic stress relaxation of metallic glasses", Sci. China Technol. Sci., vol. 68, no. 9, p. 1920203. DOI: 10.1007/s11431-025-2970-4.
[95] A. S. Argon, and H. Y. Kuo, "Free energy spectra for inelastic deformation of five metallic glass alloys", J. Non-Cryst. Solids, vol. 37, no. 2, pp. 241–266. DOI: 10.1016/0022-3093(80)90155-6.
[96] P. Schall, D. A. Weitz, and F. Spaepen, "Structural Rearrangements That Govern Flow in Colloidal Glasses", Science, vol. 318, no. 5858, pp. 1895-1899. DOI: 10.1126/science.1149308.
[97] J. S. Langer, "Shear-transformation-zone theory of plastic deformation near the glass transition", Phys. Rev. E, vol. 77, no. 2, p. 021502. DOI: 10.1103/physreve.77.021502.
[98] J. H. Simmons, R. Ochoa, K. D. Simmons, and J. J. Mills, "Non-Newtonian viscous flow in soda-lime-silica glass at forming and annealing temperatures", J. Non-Cryst., vol. 105, no. 3, pp. 313–322. DOI: 10.1016/0022-3093(88)90325-0.
[99] C. A. Schuh, T. C. Hufnagel, and U. Ramamurty, "Mechanical behavior of amorphous alloys", Acta Mater., vol. 55, no. 12, pp. 4067–4109. DOI: 10.1016/j.actamat.2007.01.052.
[100] F. Puosi, J. Rottler, and J. Barrat, "Time-dependent elastic response to a local shear transformation in amorphous solids", Phys. Rev. E, vol. 89, p. 042302. DOI: 10.1103/PhysRevE.89.042302.
[101] A. Nicolas, F. Puosi, H. Mizuno, and J. Barrat, "Elastic consequences of a single plastic event: Towards a realistic account of structural disorder and shear wave propagation in models of flowing amorphous solids", J. Mech. Phys. Solids, vol. 78, pp. 333-351. DOI: 10.1016/j.jmps.2015.02.017.
[102] CEA, "Cast3M,".
[103] D. M. Barnett, and W. Cai, "Properties of the Eshelby tensor and existence of the equivalent ellipsoidal inclusion solution", J. Mech. Phys. Solids, vol. 121, pp. 71-80. DOI: 10.1016/j.jmps.2018.07.019.
[104] J. Frenkel, "Über die Wärmebewegung in festen und flüssigen Körpern", Zeitschrift für Physik, vol. 35, no. 8-9, pp. 652–669.
[105] S. Sandfeld, Z. Budrikis, S. Zapperi, and D. F. Castellanos, "Avalanches, loading and finite size effects in 2D amorphous plasticity: results from a finite element model", J. Stat. Mech: Theory Exp., vol. 2015, no. 2, p. P02011. DOI: 10.1088/1742-5468/2015/02/P02011.
[106] Z. Zhen, D. Jun, and M. Evan, "Shear transformations in metallic glasses without excessive and predefinable defects", Proc. Natl. Acad. Sci., vol. 119, no. 48, p. e2213941119. DOI: 10.1073/pnas.2213941119.
[107] R. S. Lakes, Viscoelastic Solids, CRC press.
[108] T. Meidav, "Viscoelastic properties of the standard linear solid", Geophys. Prospect, vol. 12, no. 1, pp. 80–99. DOI: 10.1111/j.1365-2478.1964.tb01891.x.
[109] T. Egami, "Atomic level stresses", Prog. Mater. Sci., vol. 56, no. 6, pp. 637-653. DOI: 10.1016/j.pmatsci.2011.01.004.
[110] C. Bernard, G. Delaizir, J. Sangleboeuf et al., "Room temperature viscosity and delayed elasticity in infrared glass fiber", J. Eur. Ceram. Soc., vol. 27, no. 10, pp. 3253-3259. DOI: 10.1016/j.jeurceramsoc.2006.12.001.
[111] X. Jin, D. Lyu, X. Zhang, Q. Zhou, Q. Wang, and L. M. Keer, "Explicit analytical solutions for a complete set of the Eshelby tensors of an ellipsoidal inclusion", J. Appl. Mech., vol. 83, no. 12, p. 121010. DOI: 10.1115/1.4034705.
[112] L. Duffrène, R. Gy, H. Burlet et al., "Generalized Maxwell model for the viscoelastic behavior of a soda - lime - silica glass under low frequency shear loading", Rheol.Acta, vol. 36, pp. 173–186. DOI: 10.1007/s003970050034.
[113] T. Rouxel, "Elastic properties and short-to medium-range order in glasses", J. Am. Ceram. Soc., vol. 90, no. 10, pp. 3019–3039. DOI: 10.1111/j.1551-2916.2007.01945.x.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Yann Gueguen
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2026-03-23
Published 2026-04-02