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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">vsgiu</journal-id><journal-title-group><journal-title xml:lang="ru">Вестник Сибирского государственного индустриального университета</journal-title><trans-title-group xml:lang="en"><trans-title>Bulletin of the Siberian State Industrial University</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2304 - 4497</issn><issn pub-type="epub">2307-1710</issn><publisher><publisher-name>Федеральное государственное бюджетное образовательное учреждение высшего образования "Сибирский государственный индустриальный университет"</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.57070/2304-4497-2023-2(44)-60-71</article-id><article-id custom-type="elpub" pub-id-type="custom">vsgiu-151</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>Раздел 2. Металлургия и материаловедение</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>Section 2. Metallurgy and Materials Science</subject></subj-group></article-categories><title-group><article-title>МЕТОДЫ МОДЕЛИРОВАНИЯ ФАЗОВЫХ ПРЕВРАЩЕНИЙ  В БЕЙНИТНЫХ СТАЛЯХ</article-title><trans-title-group xml:lang="en"><trans-title>MODELLING METHODS OF PHASE TRANSFORMATION IN BAINITIC STEEL: A REVIEW</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-8168-8809</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Филяков</surname><given-names>Артем Дмитриевич</given-names></name><name name-style="western" xml:lang="en"><surname>Filyakov</surname><given-names>Artem</given-names></name></name-alternatives><bio xml:lang="ru"><p>научный сотрудник Лаборатории электронной микроскопии и обработки изображений</p></bio><bio xml:lang="en"><p>Researcher of the Laboratory of Electron Microscopy and Image Processing</p></bio><email xlink:type="simple">filyakov.1999@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Сарычев</surname><given-names>Владимир Дмитриевич</given-names></name><name name-style="western" xml:lang="en"><surname>Sarychev</surname><given-names>Vladimir</given-names></name></name-alternatives><bio xml:lang="ru"><p> к.т.н., доцент, доцент кафедры естественнонаучных дисциплин имени профессора В.М.Финкеля</p></bio><bio xml:lang="en"><p>Cand. Sci. (Eng.), Associate Pro-fessor of the Department of Natural Sciences named after Professor V.M. Finkel</p></bio><email xlink:type="simple">sarychev_vd@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-4721-6821</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Чумачков</surname><given-names>Илья Игоревич</given-names></name><name name-style="western" xml:lang="en"><surname>Chumachkov</surname><given-names>Ilya</given-names></name></name-alternatives><bio xml:lang="ru"><p>младший научный сотрудник</p></bio><bio xml:lang="en"><p>Junior Researcher</p></bio><email xlink:type="simple">I.I.Chumachkov@yandex.ru</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Сибирский государственный индустриальный университет</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Siberian State Industrial University</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2023</year></pub-date><pub-date pub-type="epub"><day>30</day><month>06</month><year>2023</year></pub-date><volume>0</volume><issue>2</issue><fpage>60</fpage><lpage>71</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Филяков А.Д., Сарычев В.Д., Чумачков И.И., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Филяков А.Д., Сарычев В.Д., Чумачков И.И.</copyright-holder><copyright-holder xml:lang="en">Filyakov A., Sarychev V., Chumachkov I.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://vestnik.sibsiu.ru/jour/article/view/151">https://vestnik.sibsiu.ru/jour/article/view/151</self-uri><abstract><p>К механическим свойствам и эксплуатационным характеристикам рельсовых сталей предъявляются повышенные требования. В связи с этим разработка новых марок рельсовых сталей на сегодняшний день является актуальной задачей. Бейнитные стали, не содержащие карбидов, являются своего рода потенциальными кандидатами для применения на железных дорогах, благодаря своей более высокой вязкости разрушения, сопротивлению усталости и износостойкости. Процесс термической обработки оказывает существенное влияние на механические свойства бейнитных рельсовых сталей. Решающим фактором, определяющим технологию производства рельсовых сталей, является прогнозирование микроструктуры. Для того, чтобы отойти от метода перебора, необходимо создавать математические модели охлаждения рельсов, учитывающие химический состав стали. Для моделирования структурно-фазовых превращений в рельсах для неизотермических условий разрабатываются два подхода: Аврами-Колмогорова с учетом правила Шейля и модель фазового поля. Настоящий обзор посвящен современным исследованиям по компьютерному моделированию структурно-фазовых превращений по этим двум подходам. Первый подход был предложен в 30-е годы XX века для изотермических условий, позднее был развит для неизотермического случая в рамках правила Шейля. В настоящее время к этому подходу интерес со стороны исследователей не ослабевает из-за малого времени расчета. Однако описать пространственное распределение фаз и структур этот метод не может, поэтому активно развиваются методы фазового поля, расчеты по которому могут занимать от нескольких часов до нескольких суток. В представленном обзоре проведен анализ указанных подходов, а также продемонстрированы их ограничения.</p></abstract><trans-abstract xml:lang="en"><p>For rail steels, there are increased requirements for the mechanical properties and performance characteristics. Therefore, the development of new grades of rail steel is an urgent task. Continuously cooled carbide-free bainitic steels are potential candidates for railway applications owing to their high fracture toughness, fatigue resistance, and wear resistance. The heat-treatment process has a significant impact on the limiting mechanical properties of bainitic rail steels. Predicting the microstructure is a decisive factor in reshaping the production technology. To move away from the enumeration method, it is necessary to create mathematical models of rail cooling that consider the chemical composition. To model structural phase transformations in rails under non-isothermal conditions, two approaches are being developed: Avrami-Kolmogorov with the Scheil rule and the phase field model. This review focuses on modern research on computer simulations of structural phase transformations using these two approaches. The first approach was proposed in the 30 s of the 20th century for isothermal conditions and was later developed for the non-isothermal case in the framework of Scheil's rule. At present, the interest of researchers in this approach has not weakened owing to the short calculation time. However, this method cannot describe the spatial distribution of phases and structures; therefore, methods of the phase field are being actively developed, the calculations of which can range from several hours to several days. This review analyses these approaches and demonstrates their limitations.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>модель фазового поля</kwd><kwd>JMAK</kwd><kwd>фазовые превращения</kwd><kwd>бейнитная сталь</kwd><kwd>рельсы</kwd></kwd-group><kwd-group xml:lang="en"><kwd>phase field model</kwd><kwd>JMAK</kwd><kwd>phase transformation</kwd><kwd>bainite steel</kwd><kwd>rails</kwd></kwd-group><funding-group><funding-statement xml:lang="ru">Исследование выполнено за счет гранта Российского научного фонда № 22-79-10229, https://rscf.ru/project/22-79-10229/?CODE=22-79-10229#</funding-statement><funding-statement xml:lang="en">This work was supported by the Russian Science Foundation no. 22-79-10229, https://rscf.ru/project/22-79-10229/?CODE=22-79-10229#</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Garnham J.E., Davis C.L. Rail materials. In: Wheel-rail interface handbook. 2009. P. 125–171. https://doi.org/10.1533/9781845696788.1.125</mixed-citation><mixed-citation xml:lang="en">Garnham J.E., Davis C.L. Rail materials. In: Wheel-rail interface handbook. 2009, pp. 125–171. https://doi.org/10.1533/9781845696788.1.125</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Vickerman R. International Encyclopedia of Transportation. Elsevier, 2021. P. 4569–4418.</mixed-citation><mixed-citation xml:lang="en">Vickerman R. International Encyclopedia of Transportation. Elsevier, 2021, pp. 4569–4418.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Mićić M., Brajović L., Lazarević L., Popović Z. Inspection of RCF rail defects – Review of NDT methods // Mechanical Systems and Signal Processing. 2023. Vol. 182. Article 109568.</mixed-citation><mixed-citation xml:lang="en">Mićić M., Brajović L., Lazarević L., Popović Z. Inspection of RCF rail defects–Review of NDT methods. Mechanical Systems and Signal Processing. 2023, vol. 182, article 109568.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Królicka A., Lesiuk G., Radwański K., Ku-ziak R., et al. Comparison of fatigue crack growth rate: Pearlitic rail versus bainitic rail // International Journal of Fatigue. 2021. Vol. 149. Article 106280. http://dx.doi.org/10.1016/j.ijfatigue. 2021.106280</mixed-citation><mixed-citation xml:lang="en">Królicka A., Lesiuk G., Radwański K., Kuziak R., et al. Comparison of fatigue crack growth rate: Pearlitic rail versus bainitic rail. International Journal of Fatigue. International Journal of Fatigue. 2021, vol. 149, article 106280. http://dx.doi.org/10.1016/j.ijfatigue.2021.106280</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Aglan H.A. Fatigue crack growth and frac-ture behavior of bainitic rail steels. United States. Federal Railroad Administration. Office of Railroad Policy and Development, 2011.</mixed-citation><mixed-citation xml:lang="en">Aglan H.A. Fatigue crack growth and fracture behavior of bainitic rail steels. United States. Federal Railroad Administration. Office of Railroad Policy and Development, 2011.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Ruijie Z., Chunlei Z., Bo L., Xubiao W., Xiaofeng L., et al. Research progress on rolling contact fatigue damage of bainitic rail steel // Engineering Failure Analysis. 2022. Article 106875. http://dx.doi.org/10.56748/ejse. 131621</mixed-citation><mixed-citation xml:lang="en">Ruijie Z,. Chunlei Z., Bo L., Xubiao W., Xiaofeng L., et al. Research progress on rolling contact fatigue damage of bainitic rail steel. Engineering Failure Analysis. 2022,               article 106875. http://dx.doi.org/10.56748/ejse. 131621</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Hasan S.M., Chakrabarti D., Singh S.B. Dry rolling/sliding wear behaviour of pearlitic rail and newly developed carbide-free bainitic rail steels // Wear. 2018. Vol. 408. P. 151–159. https://doi.org/10.1016/j.wear.2018.05.006</mixed-citation><mixed-citation xml:lang="en">Hasan S.M., Chakrabarti D., Singh S.B. Dry rolling/sliding wear behaviour of pearlitic rail and newly developed carbide-free bainitic rail steels. Wear. 2018, vol. 408, pp. 151–159. https://doi.org/10.1016/j.wear.2018.05.006</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Tung P.Y., Zhou X., Morsdorf L., Morsdorf L., et al. Formation mechanism of brown etching layers in pearlitic rail steel // Mate-rialia. 2022. Vol. 26. Article 101625. http://dx.doi.org/ 10.1016/j.mtla.2022.101625</mixed-citation><mixed-citation xml:lang="en">Tung P.Y., Zhou X., Morsdorf L., Morsdorf L., et al. Formation mechanism of brown etching layers in pearlitic rail steel. Materialia. 2022, vol. 26, article 101625. http://dx.doi.org/10. 1016/j.mtla.2022.101625</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Wang K., Tan Z., Gao G., Gui X., Misra, Bingzhe Baia. 2016. “Ultrahigh strength-toughness combination in Bainitic rail steel: The determining role of austenite stability during tempering” // Materials Science and Engineering A..2016. Vol. 662. P. 162–168. http://dx.doi.org/10.1016/j.msea.2016.03.043</mixed-citation><mixed-citation xml:lang="en">Wang K., Tan Z., Gao G., Gui X., et al. Misra, Bingzhe Baia. 2016. “Ultrahigh strength-toughness combination in Bainitic rail steel: The determining role of austenite stability during tempering”. Materials Science and Engineering A. 2016, vol. 662, pp. 162–168. http://dx.doi.org/10.1016/j.msea.2016.03.043</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Zhang R., Zheng C., Chen C., Lv B., et al. Study on fatigue wear competition mecha-nism and microstructure evolution on the surface of a bainitic steel rail // Wear. 2021. Vol. 482. Article 203978. http://dx.doi.org/10.1016/j. wear.2021.203978</mixed-citation><mixed-citation xml:lang="en">Zhang R., Zheng C., Chen C., Lv B., et al. Study on fatigue wear competition mechanism and microstructure evolution on the surface of a bainitic steel rail. Wear. 2021, vol. 482, article 203978. http://dx.doi.org/10.1016/j. wear.2021.203978</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Stock R., Pippan R. RCF and wear in theory and practice-The influence of rail grade on wear and RCF // Wear. 2011. Vol. 271. No. 1-2. P. 125–133. http://dx.doi.org/10.1016/j. wear.2010.10.015</mixed-citation><mixed-citation xml:lang="en">Stock R., Pippan R. RCF and wear in theory and practice-The influence of rail grade on wear and RCF. Wear. 2011, vol. 271, no. 1-2, pp. 125–133. http://dx.doi.org/10.1016/j.wear. 2010.10.015</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Ueda M., Matsuda K. Effects of carbon content and hardness on rolling contact fatigue resistance in heavily loaded pearlitic rail steels // Wear. 2020. Vol. 444. Article 203120. https://doi.org/10.1016/j.wear.2019.203120</mixed-citation><mixed-citation xml:lang="en">Ueda M., Matsuda K. Effects of carbon content and hardness on rolling contact fatigue resistance in heavily loaded pearlitic rail steels. Wear. 2020, vol. 444, article 203120. https://doi.org/10.1016/j.wear.2019.203120</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A., Dutta A., Makineni S.K., Herbig M., et al. In-situ observation of strain partitioning and damage development in continuously cooled carbide-free bainitic steels using micro digital image correlation // Materials Science and Engineering: A. 2019. Vol. 757. P. 107–116. http://dx.doi.org/10.1016/j.msea.2019.04.098</mixed-citation><mixed-citation xml:lang="en">Kumar A., Dutta A., Makineni S.K., Herbig M., et al. In-situ observation of strain partitioning and damage development in continuously cooled carbide-free bainitic steels using micro digital image correlation. Materials Science and Engineering: A. 2019, vol. 757, pp. 107–116. http://dx.doi.org/10.1016/j.msea.2019.04.098</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Sourmail T., Caballero F.G., García-Mateo C., Smanio V., et al. Evaluation of potential of high Si high C steel nanostructured bainite for wear and fa-tigue applications // Materials Science and Technol-ogy. 2013. Vol. 29. No. 10. P. 1166–1173. http://dx.doi.org/10.1179/1743284713Y.0000000242</mixed-citation><mixed-citation xml:lang="en">Sourmail T., Caballero F.G., García-Mateo C., Smanio V., et al. Evaluation of potential of high Si high C steel nanostructured bainite for wear and fatigue applications. Materials Science and Technology. 2013, vol. 29, no. 10, pp. 1166–1173. http://dx.doi.org/10.1179/1743284 713Y.0000000242</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A., Makineni S.K., Dutta A., Goulas C., et al. Design of high-strength and dam-age-resistant carbide-free fine bainitic steels for railway crossing applications // Materials Science and Engineering: A. 2019. Vol. 759. P. 210–223.</mixed-citation><mixed-citation xml:lang="en">Kumar A., Makineni S.K., Dutta A., Goulas C., et al. Design of high-strength and damage-resistant carbide-free fine bainitic steels for railway crossing applications. Materials Science and Engineering: A. 2019, vol. 759,        pp. 210–223.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Leiro A., Kankanala A., Vuorinen E., Pra-kash B. Tribological behaviour of carbide-free bainitic steel under dry rolling/sliding conditions // Wear. 2011. Vol. 273. No. 1. P. 2–8. http://dx.doi.org/10.1016/j.wear.2011.03.025</mixed-citation><mixed-citation xml:lang="en">Leiro A., Kankanala A., Vuorinen E., Prakash B. Tribological behaviour of carbide-free bainitic steel under dry rolling/sliding conditions. Wear. 2011, vol. 273, no. 1, pp. 2–8. http://dx.doi.org/10.1016/j.wear.2011.03.025</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Hu F., Wu K.M., Hodgson P.D. Effect of retained austenite on wear resistance of nanostructured dual phase steels // Materials Science and Technology. 2016. Vol. 32. No. 1. P. 40–48.</mixed-citation><mixed-citation xml:lang="en">Hu F., Wu K.M., Hodgson P.D. Effect of retained austenite on wear resistance of nanostructured dual phase steels. Materials Science and Technology. 2016, vol. 32, no. 1, pp. 40–48.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Liu J., Li Y., Zhang Y., Hu Y., et al. Dry rolling/sliding wear of bainitic rail steels under different contact stresses and slip ratios // Materials. 2020. Vol. 13. No. 20. P. 4678. http://dx.doi.org/10.3390/ma13204678</mixed-citation><mixed-citation xml:lang="en">Liu J., Li Y., Zhang Y., Hu Y., et al. Dry rolling/sliding wear of bainitic rail steels under different contact stresses and slip ratios. Materials. 2020, vol. 13, no. 20, pp. 4678. http://dx.doi.org/10.3390/ma13204678</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Masoumi M., Ariza E.A., Sinatora A., Goldenstein H. Role of crystallographic orientation and grain boundaries in fatigue crack propagation in used pearlitic rail steel // Materials Science and Engineering: A. 2018. Vol. 722. P. 147–155. http://dx.doi.org/10.1016/j.msea. 2018.03.028</mixed-citation><mixed-citation xml:lang="en">Masoumi M., Ariza E. A., Sinatora A., Gold-enstein H. Role of crystallographic orientation and grain boundaries in fatigue crack propagation in used pearlitic rail steel. Materials Science and Engineering: A. 2018, vol. 722, pp. 147–155. http://dx.doi.org/10.1016/j.msea.2018.03.028</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Zhou Y., Wang S., Wang T., Xu Y., Li Z. Field and laboratory investigation of the re-lationship between rail head check and wear in a heavy-haul railway // Wear. 2014. Vol. 315. No. 1–2. P. 68–77. http://dx.doi.org/10.1016/j.wear. 2014.04.004</mixed-citation><mixed-citation xml:lang="en">Zhou Y., Wang S., Wang T., Xu Y., Li Z. Field and laboratory investigation of the relationship between rail head check and wear in a heavy-haul railway. Wear. 2014, vol. 315, no. 1-2,   pp. 68–77. http://dx.doi.org/10.1016/j.wear. 2014.04.004</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Ding H.H., Fu Z.K., Wang W.J., Guo J., et al. Investigation on the effect of rotational speed on rolling wear and damage behaviors of wheel/rail materials // Wear. 2015. Vol. 330. P. 563–570. http://dx.doi.org/10.1016/j. wear.2014.12.043</mixed-citation><mixed-citation xml:lang="en">Ding H.H., Fu Z.K., Wang W.J., Guo J., et al. Investigation on the effect of rotational speed on rolling wear and damage behaviors of wheel/rail materials. Wear. 2015, vol. 330, pp. 563–570. http://dx.doi.org/10.1016/j.wear.2014.12.043</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Huang Y.B., Shi L.B., Zhao X.J., Cai Z.B., et al. On the formation and damage mecha-nism of rolling contact fatigue surface cracks of wheel/rail under the dry condition // Wear. 2018. Vol. 400. P. 62–73. http://dx.doi.org/ 10.1016/j.wear.2017.12.020</mixed-citation><mixed-citation xml:lang="en">Huang Y.B., Shi L.B., Zhao X.J., Cai Z.B., et al. On the formation and damage mechanism of rolling contact fatigue surface cracks of wheel/rail under the dry condition. Wear. 2018, vol. 400, pp. 62–73. http://dx.doi.org/ 10.1016/j.wear.2017.12.020</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Guo L.C., Zhu W.T., Shi L.B., Liu Q.Y., et al. Study on wear transition mechanism and wear map of CL60 wheel material under dry and wet conditions // Wear. 2019. Vol. 426. P. 1771–1780. http://dx.doi.org/10.1016/j.wear.2018. 12.049</mixed-citation><mixed-citation xml:lang="en">Guo L.C., Zhu W.T., Shi L.B., Liu Q.Y., et al. Study on wear transition mechanism and wear map of CL60 wheel material under dry and wet conditions. Wear. 2019, vol. 426, pp. 1771–1780. http://dx.doi.org/10.1016/j.wear. 2018.12.049</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Maya-Johnson S., Santa J. F., Toro A. Dry and lubricated wear of rail steel under rolling contact fatigue-Wear mechanisms and crack growth // Wear. 2017. Vol. 380. P. 240–250. http://dx.doi.org/10.1016/j.wear.2017.03.025</mixed-citation><mixed-citation xml:lang="en">Maya-Johnson S., Santa J.F., Toro A. Dry and lubricated wear of rail steel under rolling contact fatigue-Wear mechanisms and crack growth. Wear. 2017, vol. 380, pp. 240–250. http://dx.doi.org/10.1016/j.wear.2017.03.025</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Benoît D., Salima B., Marion R. Multiscale characterization of head check initiation on rails under rolling contact fatigue: Mechan-ical and microstructure analysis // Wear. 2016. Vol. 366. P. 383–391. http://dx.doi.org/10.1016/j. wear.2016.06.019</mixed-citation><mixed-citation xml:lang="en">Benoît D., Salima B., Marion R. Multiscale characterization of head check initiation on rails under rolling contact fatigue: Mechanical and microstructure analysis. Wear. 2016, vol. 366, pp. 383–391. http://dx.doi.org/ 10.1016/j.wear.2016.06.019</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Messaadi M., Oomen M., Kumar A. Friction modifiers effects on tribological behaviour of bainitic rail steels // Tribology international. 2019. Vol. 140. Article 105857.</mixed-citation><mixed-citation xml:lang="en">Messaadi M., Oomen M., Kumar A. Friction modifiers effects on tribological behaviour of bainitic rail steels. Tribology international. 2019, vol. 140, article 105857.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Lee K.M., Polycarpou A.A. Wear of conventional pearlitic and improved bainitic rail steels // Wear. 2005. Vol. 259. No. 1–6. P. 391–399. http://dx.doi.org/10.1016/j.wear.2005.02.058</mixed-citation><mixed-citation xml:lang="en">Lee K.M., Polycarpou A.A. Wear of conventional pearlitic and improved bainitic rail steels. Wear. 2005, vol. 259, no. 1-6, pp. 391–399. http://dx.doi.org/10.1016/j.wear.2005.02.058</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Hasan S.M., Chakrabarti D., Singh S.B. Dry rolling/sliding wear behaviour of pearlitic rail and newly developed carbide-free bainitic rail steels // Wear. 2018. Vol. 408. P. 151–159. http://dx.doi.org/10.1016/j.wear.2018.05.006</mixed-citation><mixed-citation xml:lang="en">Hasan S.M., Chakrabarti D., Singh S.B. Dry rolling/sliding wear behaviour of pearlitic rail and newly developed carbide-free bainitic rail steels. Wear. 2018, vol. 408, pp. 151–159. http://dx.doi.org/10.1016/j.wear.2018.05.006</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Liu J.P., Li Y.Q., Zhou Q.Y., Zhang Y. H., et al. New insight into the dry rolling-sliding wear mechanism of carbide-free bainitic and pearlitic steel // Wear. 2019. Vol. 432. Article 202943. https://doi.org/10.1016/j.wear. 2019.202943</mixed-citation><mixed-citation xml:lang="en">Liu J.P., Li Y.Q., Zhou Q.Y., Zhang Y. H., et al. New insight into the dry rolling-sliding wear mechanism of carbide-free bainitic and pearlitic steel. Wear. 2019, vol. 432, article 202943. https://doi.org/10.1016/j.wear. 2019.202943</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Chen Y., Ren R., Pan J., Pan R., et al. Microstructure evolution of rail steels under different dry sliding conditions: A compari-son between pearlitic and bainitic microstructures // Wear. 2019. Vol. 438. Article 203011. https://doi.org/10.1016/j.wear.2019.203011</mixed-citation><mixed-citation xml:lang="en">Chen Y., Ren R., Pan J., Pan R., et al. Microstructure evolution of rail steels under different dry sliding conditions: A comparison between pearlitic and bainitic microstructures. Wear. 2019, vol. 438, article 203011. https://doi.org/ 10.1016/j.wear.2019.203011</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Li Q., Guo J., Zhao A. Effect of upper bain-ite on wear behaviour of high-speed wheel steel // Tribology Letters. 2019. Vol. 67. P. 1–9.</mixed-citation><mixed-citation xml:lang="en">Li Q., Guo J., Zhao A. Effect of upper bainite on wear behaviour of high-speed wheel steel. Tribology Letters. 2019, vol. 67, pp. 1–9.</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Kuziak R., Pidvysotskyy V., Pernach M., Rauch L., et al. Selection of the best phase transformation model for optimization of manufacturing processes of pearlitic steel rails //Archives of Civil and Mechanical Engineering. 2019. Vol. 19. P. 535–546. http://dx.doi. org/10.1016/j.acme.2018.12.004</mixed-citation><mixed-citation xml:lang="en">Kuziak R., Pidvysotskyy V., Pernach M., Rauch L., et al. Selection of the best phase transformation model for optimization of manufacturing processes of pearlitic steel rails. Archives of Civil and Mechanical Engineering. 2019, vol. 19, pp. 535–546. http://dx.doi.org/ 10.1016/j.acme.2018.12.004</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Tehler M. Modeling phase transformations and volume changes during cooling of case hardening steels. KTH, 2009.</mixed-citation><mixed-citation xml:lang="en">Tehler M. Modeling phase transformations and volume changes during cooling of case hardening steels. KTH, 2009.</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">Bzowski K., Rauch L., Pietrzyk M. Applica-tion of statistical representation of the microstructure to modeling of phase trans-formations in DP steels by solution of the diffusion equation // Procedia Manufactur-ing. 2018. Vol. 15. P. 1847–1855. http://dx.doi.org/10.1016/j. promfg.2018.07.205</mixed-citation><mixed-citation xml:lang="en">Bzowski K., Rauch L., Pietrzyk M. Applica-tion of statistical representation of the microstructure to modeling of phase transformations in DP steels by solution of the diffusion equation. Procedia Manufacturing. 2018, vol. 15, pp. 1847–1855. http://dx.doi.org/10.1016/j. promfg.2018.07.205</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Rauch L., Kuziak R., Pietrzyk M. From high accuracy to high efficiency in simulations of processing of Dual-Phase steels // Metallurgical and Materials transactions B. 2014 Vol. 45. P. 497–506. http://dx.doi.org/10.1007/s11663-013-9926-5</mixed-citation><mixed-citation xml:lang="en">Rauch L., Kuziak R., Pietrzyk M. From high accuracy to high efficiency in simulations of processing of Dual-Phase steels. Metallurgical and Materials transactions B. 2014, vol. 45, pp. 497–506. http://dx.doi.org/10.1007/ s11663-013-9926-5</mixed-citation></citation-alternatives></ref><ref id="cit36"><label>36</label><citation-alternatives><mixed-citation xml:lang="ru">Rheingans B., Mittemeijer E.J. Phase trans-formation kinetics: advanced modeling strategies // JOM. 2013. Vol. 65. P. 1145–1154.</mixed-citation><mixed-citation xml:lang="en">Rheingans B., Mittemeijer E.J. Phase trans-formation kinetics: advanced modeling strategies. JOM. 2013, vol. 65, pp. 1145–1154.</mixed-citation></citation-alternatives></ref><ref id="cit37"><label>37</label><citation-alternatives><mixed-citation xml:lang="ru">Starink M.J. On the meaning of the impinge-ment parameter in kinetic equations for nucleation and growth reactions // Journal of Materials Science. 2001. Vol. 36. No. 18. P. 4433–4441.</mixed-citation><mixed-citation xml:lang="en">Starink M.J. On the meaning of the impinge-ment parameter in kinetic equations for nucleation and growth reactions. Journal of Materials Science. 2001, vol. 36, no. 18, pp. 4433–4441.</mixed-citation></citation-alternatives></ref><ref id="cit38"><label>38</label><citation-alternatives><mixed-citation xml:lang="ru">Todinov M.T. On some limitations of the Johnson–Mehl–Avrami–Kolmogorov equa-tion // Acta materialia. 2000. Vol. 48. No. 17. P. 4217–4224.</mixed-citation><mixed-citation xml:lang="en">Todinov M.T. On some limitations of the Johnson–Mehl–Avrami–Kolmogorov equation. Acta materialia. 2000, vol. 48, no. 17,            pp. 4217–4224.</mixed-citation></citation-alternatives></ref><ref id="cit39"><label>39</label><citation-alternatives><mixed-citation xml:lang="ru">Kempen A.T.W., Sommer F., Mittemeijer E.J. Determination and interpretation of iso-thermal and non-isothermal transformation kinetics; the effective activation energies in terms of nucleation and growth // Journal of materials science. 2002. Vol. 37. P. 1321–1332. https://doi.org/ 10.1023/a:1014556109351</mixed-citation><mixed-citation xml:lang="en">Kempen A.T.W., Sommer F., Mittemeijer E.J. Determination and interpretation of isothermal and non-isothermal trans-formation kinetics; the effective activation energies in terms of nucleation and growth. Journal of materials science. 2002, vol. 37, pp. 1321–1332. https://doi.org/ 10.1023/a:1014556109351</mixed-citation></citation-alternatives></ref><ref id="cit40"><label>40</label><citation-alternatives><mixed-citation xml:lang="ru">Liu F., Sommer F., Bos C., Mittemeijer E.J. Analysis of solid state phase transformation kinetics: models and recipes // International materials reviews. 2007. Vol. 52. No. 4. P. 193–212. http://dx.doi.org/10.1179/174328007X160308</mixed-citation><mixed-citation xml:lang="en">Liu F., Sommer F., Bos C., Mittemeijer E.J. Analysis of solid state phase transformation kinetics: models and recipes. International materials reviews. 2007, vol. 52, no. 4, pp. 193–212. http://dx.doi.org/10.1179/174328007X160308</mixed-citation></citation-alternatives></ref><ref id="cit41"><label>41</label><citation-alternatives><mixed-citation xml:lang="ru">Robson J.D. Modeling competitive continu-ous and discontinuous precipitation // Acta Materialia. 2013. Vol. 61. No. 20. P. 7781–7790. http://dx.doi.org/10.1016/j.actamat.2013.09.017</mixed-citation><mixed-citation xml:lang="en">Robson J. D. Modeling competitive con-tinuous and discontinuous precipitation. Acta Materialia. 2013, vol. 61, no. 20, pp. 7781–7790. http://dx.doi.org/10.1016/j.actamat.2013.09.017</mixed-citation></citation-alternatives></ref><ref id="cit42"><label>42</label><citation-alternatives><mixed-citation xml:lang="ru">Ranganathan S., Von Heimendahl M. The three activation energies with isothermal transformations: applications to metallic glasses // Journal of Materials Science.1981. Vol. 16. P. 2401–2404.</mixed-citation><mixed-citation xml:lang="en">Ranganathan S., Von Heimendahl M. The three activation energies with isothermal transformations: applications to metallic glasses. Journal of Materials Science. 1981, vol. 16, pp. 2401–2404.</mixed-citation></citation-alternatives></ref><ref id="cit43"><label>43</label><citation-alternatives><mixed-citation xml:lang="ru">Liu F., Sommer F., Mittemeijer E.J. Analysis of the kinetics of phase transformations; roles of nucleation index and temperature dependent site saturation, and recipes for the extraction of kinetic parameters // Journal of Materials Science. 2007. Vol. 42. P. 573–587.</mixed-citation><mixed-citation xml:lang="en">Liu F., Sommer F., Mittemeijer E. J. Analysis of the kinetics of phase transformations; roles of nucleation index and temperature dependent site saturation, and recipes for the extraction of kinetic parameters. Journal of Materials Science. 2007, vol. 42, pp. 573–587.</mixed-citation></citation-alternatives></ref><ref id="cit44"><label>44</label><citation-alternatives><mixed-citation xml:lang="ru">Dill E.D., Folmer J.C.W., Martin J.D. Crystal growth simulations to establish physically relevant kinetic parameters from the empirical Kolmogorov–Johnson–Mehl–Avrami model // Chemistry of Materials. 2013. Vol. 25. No. 20. P. 3941–3951.</mixed-citation><mixed-citation xml:lang="en">Dill E.D., Folmer J.C.W., Martin J.D. Crystal growth simulations to establish physically relevant kinetic parameters from the empirical Kolmogorov–Johnson–Mehl–Avrami model. Chemistry of Materials. 2013, vol. 25, no. 20, pp. 3941–3951.</mixed-citation></citation-alternatives></ref><ref id="cit45"><label>45</label><citation-alternatives><mixed-citation xml:lang="ru">Weinberg M.C. A test of the Johnson-Mehl-Avrami equation // Journal of crystal growth. 1987. Vol. 82. No. 4. P. 779–780. https://doi.org/10.1016/S0022-0248(87)80025-8</mixed-citation><mixed-citation xml:lang="en">Weinberg M.C. A test of the Johnson-Mehl-Avrami equation. Journal of crystal growth. 1987, vol. 82, no. 4, pp. 779–780. https://doi.org/10.1016/S0022-0248(87)80025-8</mixed-citation></citation-alternatives></ref><ref id="cit46"><label>46</label><citation-alternatives><mixed-citation xml:lang="ru">Weinberg M.C., Birnie III D.P., Shneidman V.A. Crystallization kinetics and the JMAK equation // Journal of non-crystalline solids. 1997. Vol. 219. P. 89–99. https://doi.org/10.1016/S0022-3093%2897%2900261-5</mixed-citation><mixed-citation xml:lang="en">Weinberg M.C., Birnie III D.P., Shneidman V.A. Crystallization kinetics and the JMAK equation. Journal of non-crystalline solids. 1997, vol. 219, pp. 89–99. https://doi.org/10.1016/S0022-3093%2897% 2900261-5</mixed-citation></citation-alternatives></ref><ref id="cit47"><label>47</label><citation-alternatives><mixed-citation xml:lang="ru">Kelton K.F., Spaepen F.A study of the devitrification of Pd82Si18 over a wide temperature range // Acta Metallurgica. 1985. Vol. 33. No. 3. P. 455–464.</mixed-citation><mixed-citation xml:lang="en">Kelton K.F., Spaepen F.A study of the devitrification of Pd82Si18 over a wide temperature range. Acta Metallurgica. 1985, vol. 33, no. 3, pp. 455–464.</mixed-citation></citation-alternatives></ref><ref id="cit48"><label>48</label><citation-alternatives><mixed-citation xml:lang="ru">Yinnon H., Uhlmann D.R. Applications of thermoanalytical techniques to the study of crystallization kinetics in glass-forming liq-uids, part I: theory // Journal of Non-Crystalline Solids. 1983. Vol. 54. No. 3. P. 253–275.</mixed-citation><mixed-citation xml:lang="en">Yinnon H., Uhlmann D.R. Applications of thermoanalytical techniques to the study of crystallization kinetics in glass-forming liq-uids, part I: theory. Journal of Non-Crystalline Solids. 1983, vol. 54, no. 3, pp. 253–275.</mixed-citation></citation-alternatives></ref><ref id="cit49"><label>49</label><citation-alternatives><mixed-citation xml:lang="ru">Levine L.E., Narayan K.L., Kelton K.F. Fi-nite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation // Journal of Materials Research. 1997. Vol. 12. P. 124–132.</mixed-citation><mixed-citation xml:lang="en">Levine L.E., Narayan K.L., Kelton K.F. Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation. Journal of Materials Research. 1997, vol. 12, pp. 124–132</mixed-citation></citation-alternatives></ref><ref id="cit50"><label>50</label><citation-alternatives><mixed-citation xml:lang="ru">Chen L.Q. Annu Rev Mater Res // Annu. Rev. Mater. Res. 2002. Vol. 32. P. 113–140.</mixed-citation><mixed-citation xml:lang="en">Chen L.Q. Annu Rev Mater Res. Annu. Rev. Mater. Res. 2002, vol. 32, pp. 113–140.</mixed-citation></citation-alternatives></ref><ref id="cit51"><label>51</label><citation-alternatives><mixed-citation xml:lang="ru">Fallah V., Amoorezaei M., Provatas N., Corbin S.F., et al. Phase-field simulation of solidification morphology in laser powder deposition of Ti–Nb alloys // Acta Materi-alia. 2012. Vol. 60. No. 4. P. 1633–1646. http://dx.doi.org/ 10.1016/j.actamat.2011.12.009</mixed-citation><mixed-citation xml:lang="en">Fallah V., Amoorezaei M., Provatas N., Corbin S. F., et al. Phase-field simulation of solidification morphology in laser powder deposition of Ti–Nb alloys. Acta Materialia. 2012, vol. 60, no. 4, pp. 1633–1646. http://dx.doi.org/10.1016/ j.ac-tamat.2011.12.009</mixed-citation></citation-alternatives></ref><ref id="cit52"><label>52</label><citation-alternatives><mixed-citation xml:lang="ru">Beckermann C., Diepers H.J., Steinbach I., Karma A., et al. Modeling melt convection in phase-field simulations of solidification // Journal of Computational Physics. 1999. Vol. 154. No. 2. P. 468–496.</mixed-citation><mixed-citation xml:lang="en">Beckermann C., Diepers H. J., Steinbach I., Karma A., et al. Modeling melt convection in phase-field simulations of solidification. Journal of Computational Physics. 1999, vol. 154, no. 2, pp. 468–496.</mixed-citation></citation-alternatives></ref><ref id="cit53"><label>53</label><citation-alternatives><mixed-citation xml:lang="ru">Böttger B., Apel M., Eiken J., Schaffnit P., et al. Phase‐Field simulation of solidification and solid‐state transformations in mul-ticomponent steels // Steel Research International. 2008. Vol. 79. No. 8. P. 608–616. http://dx.doi.org/ 10.2374/SRI08SP021-79-2008-608</mixed-citation><mixed-citation xml:lang="en">Böttger B., Apel M., Eiken J., Schaffnit P., et al. Phase‐Field simulation of solidification and solid‐state transformations in multicomponent steels. Steel Research International. 2008, vol. 79, no. 8, pp. 608–616. http://dx.doi.org/ 10.2374/SRI08SP021-79-2008-608</mixed-citation></citation-alternatives></ref><ref id="cit54"><label>54</label><citation-alternatives><mixed-citation xml:lang="ru">Tiaden J. Phase-field simulations of the peritectic solidification of Fe-C // Journal of Crystal Growth. 1999. Vol. 198. P. 1275–1280.</mixed-citation><mixed-citation xml:lang="en">Tiaden J. Phase-field simulations of the peritectic solidification of Fe-C. Journal of Crystal Growth. 1999, vol. 198, pp. 1275–1280.</mixed-citation></citation-alternatives></ref><ref id="cit55"><label>55</label><citation-alternatives><mixed-citation xml:lang="ru">Steinbach I., Pezzolla F., Nestler B., Seeßel-berg M., et al. A phase field concept for multiphase systems // Physica D: Nonlinear Phenomena. 1996. Vol. 94. No. 3. P. 135–147.</mixed-citation><mixed-citation xml:lang="en">Steinbach I., Pezzolla F., Nestler B., Seeßel-berg M., et al. A phase field concept for multiphase systems. 1996, vol. 94, no. 3, pp. 135–147.</mixed-citation></citation-alternatives></ref><ref id="cit56"><label>56</label><citation-alternatives><mixed-citation xml:lang="ru">Militzer M. Phase field modeling of micro-structure evolution in steels // Current Opin-ion in Solid State and Materials Science. 2011. Vol. 15. No. 3. P. 106–115. https://doi.org/10.1016/j.cossms.2010.10.001</mixed-citation><mixed-citation xml:lang="en">Militzer M. Phase field modeling of micro-structure evolution in steels. Current Opinion in Solid State and Materials Science. 2011, vol. 15, no. 3, pp. 106–115. https://doi.org/ 10.1016/j.cossms.2010.10.001</mixed-citation></citation-alternatives></ref><ref id="cit57"><label>57</label><citation-alternatives><mixed-citation xml:lang="ru">Pariser G., Schaffnit P., Steinbach I., Bleck, W. Simulation of the γ–α‐transformation using the phase‐field method // Steel Research. 2001. Vol. 72. No. 9. P. 354–360. http://dx.doi.org/10.1002/srin.200100130</mixed-citation><mixed-citation xml:lang="en">Pariser G., Schaffnit P., Steinbach I., Bleck, W. Simulation of the γ–α‐transformation using the phase‐field method. Steel Research. 2001, vol. 72, no. 9, pp. 354–360. http://dx.doi.org/ 10.1002/srin.200100130</mixed-citation></citation-alternatives></ref><ref id="cit58"><label>58</label><citation-alternatives><mixed-citation xml:lang="ru">Elder K.R., Grant M., Provatas N., Kosterlitz J. M. Sharp interface limits of phase-field models // Physical Review E. 2001. Vol. 64. No. 2. Article 021604. http://dx.doi.org/10.1103/ PhysRevE.64.021604</mixed-citation><mixed-citation xml:lang="en">Elder K.R., Grant M., Provatas N., Kosterlitz J. M. Sharp interface limits of phase-field modelsю Physical Review E. 2001, vol. 64, no. 2, article 021604. http://dx.doi.org/10.1103/ PhysRevE.64.021604</mixed-citation></citation-alternatives></ref><ref id="cit59"><label>59</label><citation-alternatives><mixed-citation xml:lang="ru">Jou H.J., Lusk M.T. Comparison of John-son-Mehl-Avrami-Kologoromov kinetics with a phase-field model for microstructural evolution driven by substructure energy // Physical Review B. 1997. Vol. 55. No. 13. P. 8114. http://dx.doi.org/10.1103/PhysRevB.55.8114</mixed-citation><mixed-citation xml:lang="en">Jou H. J., Lusk M. T. Comparison of Johnson-Mehl-Avrami-Kologoromov kinetics with a phase-field model for microstructural evolution driven by sub-structure energy. Physical Review B. 1997, vol. 55, no. 13, pp. 8114. http://dx.doi.org/10.1103/PhysRevB.55.8114</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
