sibadiНаучный рецензируемый журнал "Вестник СибАДИ"The Russian Automobile and Highway Industry Journal2071-72962658-5626The Siberian State Automobile and Highway University10.26518/2071-7296-2026-23-1-130-157LMWXDAsibadi-2159Research ArticleСТРОИТЕЛЬСТВО И АРХИТЕКТУРАCONSTRUCTION AND ARCHITECTUREУчет влияния содержания воздушных пустот при расчете критической длины трещины в асфальтобетонеAccounting for the influence of air void content in the calculation of critical crack length in asphalt concreteГагаринаТ. Ю.GagarinaT. Yu.

Гагарина Татьяна Юрьевна – магистрант кафедры «Строительство и эксплуатация дорог» ФГБОУ ВО «СибАДИ», работает над магистерской диссертацией.

644050, Омск, просп. Мира, 5

Gagarina Tatyana Yu. – Master’s student, working on her Master’s thesis, “Road Construction and Operation» Department, “Automobile and Road, Industrial and Civil Engineering” Institute, SibADI.

5, Prospect Mira, Omsk, 644050

https://orcid.org/0000-0002-5534-6338АлександроваН. П.AleksandrovaN. P.

Александрова Наталья Павловна – канд. техн. наук, доц., доц. кафедры «Строительство и эксплуатация дорог» института «Автомобильно-дорожное, промышленное и гражданское строительство» СибАДИ. Author ID (РИНЦ): 257218, Author ID (Scopus): 57191525817.

644050, Омск, просп. Мира, 5

Aleksandrova Natalya P. – Candidate of Technical Sciences, Associate Professor, “Road Construction and Operation» Department, “Automobile and Road, Industrial and Civil Engineering” Institute, SibADI. Author ID (Scopus): 57191525817.

(5, Prospect Mira, Omsk, 644050

nata26.74@mail.ruhttps://orcid.org/0000-0003-2009-5361АлександровА. С.AleksandrovA. S.

Александров Анатолий Сергеевич – канд. техн. наук, доц., доц. кафедры «Строительство и эксплуатация дорог» ФГБОУ ВО «СибАДИ». Researcher ID: I-8860-2018, Author ID (РИНЦ): 639655, Author ID (Scopus): 57191531014.

644050, Омск, просп. Мира, 5

Aleksandrov Anatoliy S. – Candidate of Technical Sciences, Associate Professor, “Road Construction and Operation» Department, “Automobile and Road, Industrial and Civil Engineering” Institute, SibADI. Researcher ID: I-8860-2018, Author ID (Scopus): 57191531014.

5, Prospect Mira, Omsk, 644050

aleksandrov00@mail.ruСибирский государственный автомобильно-дорожный университет (СибАДИ)РоссияThe Siberian State Automobile and Highway University (SibADI)Russian Federation202624022026231130157Copyright © Гагарина Т.Ю., Александрова Н.П., Александров А.С., 20262026Гагарина Т.Ю., Александрова Н.П., Александров А.С.Gagarina T.Y., Aleksandrova N.P., Aleksandrov A.S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.sibadi.org/jour/article/view/2159Введение

Введение. Параметры прочности и деформативности асфальтобетона существенно зависят от его температуры. В широком диапазоне варьирования температуры асфальтобетон проявляет упруго-вязко-пластические свойства. Это приводит к тому, что на макроуровне при увеличении температуры прочность асфальтобетона и его сопротивляемость деформации уменьшается. В условиях отрицательных температур асфальтобетон проявляет свойства хрупкого тела, а при положительных температурах асфальтобетон нужно рассматривать как квазихрупкий материал. Следовательно, в практику расчета нежестких дорожных одежд и проектирование состава асфальтобетонов необходимо внедрять материальные константы микроуровня (поверхностную энергию, энергию разрушения, предельные значения коэффициентов интенсивности напряжений или трещиностойкость, вязкость разрушения). Выполнен анализ методов расчета дорожных одежд, применяемых в практике дорожного строительства. Сформулирована цель работы.

Методы и материалы

Методы и материалы. Приведены сведения о концепциях хрупкого и квазихрупкого разрушения А. Гриффится и Дж. Ирвина, описан критерий роста трещины в виде J-интеграла Черепанова – Райса. Сделан вывод, что одним из вариантов расчета асфальтобетонных слоев дорожной одежды при нулевой и отрицательной температуре является применение теории хрупкого разрушения А. Гриффитса. Применение механики хрупкого разрушения позволяет определить критическое напряжение при заданном размере дефектов в структуре асфальтобетона и, наоборот, критическую длину трещины при заданном напряжении. Следующим этапом должен стать расчет по коэффициентам интенсивности напряжений или энергии разрушения, применяемые в рамках линейно-упругой механики разрушения, но учитывающей образование в вершине трещины пластической зоны с малыми необратимыми деформациями. Классические формулы А. Гриффитса содержат материальные константы, в том числе модуль упругости, величина которого зависит от содержания воздушных пустот. На микроуровне воздушные пустоты представляют собой концентраторы напряжений. Поэтому учет содержания воздушных пустот при определении модуля упругости асфальтобетона, применяемого в расчете дорожной одежды, является актуальной задачей, имеющей практическую значимость. Выполнен обзор научных работ по определению энергетических констант горячих асфальтобетонов в зависимости от вариации различных факторов.

Результаты

Результаты. Приведены результаты расчета критической длины трещины для горячих асфальтобетонов на битумах марок БНД, соответствующие допуску по содержанию воздушных пустот. Из анализа результатов расчета следует, что увеличение содержания воздушных пустот приводит к снижению модуля упругости асфальтобетона и уменьшению критической длины трещины. Расчеты выполнены для трех величин удельной поверхностной энергии.

Заключение

Заключение. Полученные результаты позволяют более детально производить расчет дорожной одежды

Introduction

Introduction. The strength and deformation parameters of asphalt concrete are significantly dependent on its temperature. Over a wide temperature range, asphalt concrete shows elastic, viscous and plastic properties. This leads to the fact that at the macrolevel, with the rise of temperature, asphalt concrete strength declines and its deformation resistance decreases. Under temperature conditions below zero, asphalt concrete behaves as a brittle material, while at temperatures above zero, it should be considered a quasi-brittle material. Consequently, it is necessary to implement microlevel material constants (surface energy, fracture energy, critical stress intensity factors or crack toughness, fracture viscosity) into calculation practice for flexible pavements and content design of asphalt concrete mixes. The analysis of pavement design methods currently used in road construction practice has been performed. The objective of the work has been formulated.

Materials and methods

Materials and methods. Information on the concepts of brittle and quasi-brittle fracture by A. Griffith and J. Irwin has been provided, and the criterion of crack growth in the form of the Cherepanov-Rice J-integral has been described. It is concluded that one approach for calculating asphalt concrete layers in pavements at zero and subzero temperatures is the application of A. Griffith’s theory of brittle fracture. The application of brittle fracture mechanics allows to determine the critical stress for the given defect size in the asphalt concrete structure, and conversely, the critical crack length for the given stress. The next stage should involve calculations based on stress intensity factors or fracture energy, applied within the framework of linear elastic fracture mechanics, but taking into the account the formation of a plastic zone with small irreversible deformations at the crack tip. Classical Griffith formulas contain material constants, including the elastic modulus, the magnitude of which depends on the air void content. At the microlevel, air voids act as stress concentrators. Therefore, attention to the air void content in determining the elastic modulus of asphalt concrete used in pavement design is a relevant task with a practical significance. A review of scientific works on determining the energy constants of hot mix asphalt concrete in accordance with the variation of different factors has been conducted.

Results

Results. Critical crack length calculation results for hot mix asphalt based on BND bitumen grades, corresponding to permissible air void content standards, are presented. Analysis of the calculation results shows that the increase in air void content leads to the decrease in the elastic modulus of asphalt concrete and to the reduction in the critical crack length. Calculations have been performed for three values of specific surface energy.

Conclusion

Conclusion. The obtained results allow to make more detailed calculation of the road pavement design.

асфальтобетонсодержание воздушных пустоткритическая длина трещиныэнергия разрушениярасчет асфальтобетонных покрытийasphalt concreteair void contentcritical crack lengthfracture energycalculation of asphalt-concrete pavementsReferencesГорский М.Ю., Кадыров Г.Ф., Стрельцов А.В., Симчук Е.Н. Совершенствование методики расчета нежестких дорожных одежд с учетом применения решения задачи теории упругости для многослойного полупространства // Дороги и мосты. 2021. Т. 46, № 2. С. 53–74.Gorsky M.Y., et al. Improving the Calculation Methodology of Flexible Road Pavements Using the Application of the Elastic Theory Solution for a Multilayer Half-Space. Roads and Bridges. 2021; 46(2): 53-74. (In Russ.)Радовский Б.С., Мерзликин А.Е. Оценка погрешностей, возникающих при расчете нежестких дорожных одежд // Дороги и мосты. 2016. Т. 35, № 1. С. 59–69.Radovsky B.S., Merzlikin A.E. Estimation of Errors Occurring in the Calculation of Non-Rigid Road Pavements. Roads and Bridges. 2016; 35(1): 59-69. (In Russ.)Симчук Е.Н., Кадыров Г.Ф., Горский М.Ю., Медведев Д.В., Стрельцов А.В. Актуальные подходы к моделированию напряженно-деформированного состояния нежестких дорожных одежд от статических и динамических нагрузок // Дороги и мосты. 2025. Т. 53, № 1. С. 55–71. Https://doi.org/10.70991/1815-896X-2025-1-53-55-71Simchuk E.N., et al. Actual approaches to modeling the stress-strain state of non-rigid road pavements from static and dynamic loads. Roads and Bridges. 2025; 53(1): 55-71. (In Russ.) Https://doi.org/10.70991/1815-896X-2025-1-53-55-71Мерзликин А.Е., Корчажников Я.Н. Продление сроков службы нежестких дорожных одежд при проектировании: тривиальные и нетривиальные способы // Дороги и мосты. 2018. Т. 39, № 1. С. 105–117.Merzlikin A.E., Korchazhnikov Ya.N. Extension of the service life of non-rigid road pavements during design: trivial and non-trivial methods. Roads and Bridges. 2018; 39(1): 105-117. (In Russ.)Косенко Н.В., Горячев М.Г. Обоснование кратковременных расчетных модулей упругости асфальтобетонов для проектирования дорожных одежд улично-дорожной сети // Наука и техника в дорожной отрасли. 2025. № 2. С. 20–22.Kosenko N.V., Goryachev M.G. Justification of the loading duration of the street and road network pavements. Advanced Science and Technology for Highways. 2025; 2: 20–22. (In Russ.)Горячев М.Г. Оценка поправочного коэффициента для определения минимального требуемого значения модуля упругости нежестких дорожных одежд // Транспортное строительство. 2018. № 5. С. 10–12.Goryachev M.G. Determination of the correction factor for the minimum required modulus of elasticity of flexible pavements. Science, Technology and Practice Journal Transport construction. 2018; 5: 10–12. (In Russ.)Ekwulo E.O., Eme D.B. Expected traffic, pavement thickness, fatigue and rutting strain relationship for low volume asphalt pavement // The International Journal Of Engineering And Science. 2013. Т. 2. № 8. С. 62–77.Ekwulo, E.O., Eme, D.B. Expected traffic, pavement thickness, fatigue and rutting strain relationship for low volume asphalt pavement. The International Journal Of Engineering And Science. 2013; 2(8): 62–77.Owais M. Analysing Witczak 1-37A, Witczak 1-40D and Modified Hirsch Models for asphalt dynamic modulus prediction using global sensitivity analysis // International Journal of Pavement Engineering. 2023. Т. 24. № 1. Article No 2268808. Https://doi.org/10.1080/10298436.2023.2268808Owais M. Analysing Witczak 1-37A, Witczak 1-40D and Modified Hirsch Models for asphalt dynamic modulus prediction using global sensitivity analysis. International Journal of Pavement Engineering. 2023; 24(1): Article No 2268808. Https://doi.org/10.1080/10298436.2023.2268808Asadi B., Hajj R., Al-Qadi I.L. Asphalt concrete dynamic modulus prediction: Bayesian neural network approach. // International Journal of Pavement Engineering. 2023. Т. 24. No 2. Article № 2270569, Https://doi.org/10.1080/10298436.2023.2270569Asadi B., Hajj R., Al-Qadi I.L. Asphalt concrete dynamic modulus prediction: Bayesian neural network approach. International Journal of Pavement Engineering. 2023; 24(2):. Article No 2270569, Https://doi.org/10.1080/10298436.2023.2270569Belhaj M., et al. Evaluating Factors Influencing Dynamic Modulus Prediction: GRA-MLR Compared with Sigmoidal Modelling for Asphalt Mixtures with Reclaimed Asphalt // Infrastructures. 2025. №. 10. Article № 269. Https://doi.org/10.3390/infrastructures10100269Belhaj, M., et al. Evaluating Factors Influencing Dynamic Modulus Prediction: GRA-MLR Compared with Sigmoidal Modelling for Asphalt Mixtures with Reclaimed Asphalt. Infrastructures. 2025; 10: Article No 269. https://doi.org/10.3390/infrastructures10100269Hanandeh S., et al. Prediction the Dynamic Modulus of Hot Asphalt Mix Using Genetic Algorithms and Neural Network Modeling // Civil Engineering Journal. 2025. Т. 11. № 7. Pp. 2765–2781. Https://doi.org/10.28991/CEJ-2025-011-07-08Hanandeh S., et al. Prediction the Dynamic Modulus of Hot Asphalt Mix Using Genetic Algorithms and Neural Network Modeling. Civil Engineering Journal. 2025; 11(7): Pp. 2765–2781. Https://doi.org/10.28991/CEJ-2025-011-07-08Aleksandrova N.P., Chysow V.V. The usage of integral equations hereditary theories for calculating changes of measures of the theory of damage when exposed to repeated loads // Magazine of Civil Engineering. 2016. Т. 62. № 2. С. 69–82. Https://doi.org/10.5862/MCE.62.7.Aleksandrova N.P., Chysow V.V. The usage of integral equations hereditary theories for calculating changes of measures of the theory of damage when exposed to repeated loads. Magazine of Civil Engineering. 2016; 62(2): 69-82. Https://doi.org/10.5862/MCE.62.7.Углова Е.В. Прогнозирование остаточного ресурса асфальтобетонных покрытий с учетом реальных условий эксплуатации. // Вестник ВолгГАСУ. Серия Строительство и архитектура. 2010. № 17(36). С.43–47.Chusov V.V., Murtazin R.Kh., Aleksandrov A.S. The consideration of air void content effect on asphalt concrete elastic modulus in pavement design. The Russian Automobile and Highway Industry Journal. 2025;22(6):1000-1017. (In Russ.) Https://doi.org/10.26518/2071-7296-2025-22-6-1000-1017.Чусов В.В., Муртазин Р.Х., Александров А.С. Учет влияния содержания воздушных пустот на модуль упругости асфальтобетонов для расчета дорожных одежд // Вестник СибАДИ. 2025. № 22(6). С. 1000–1017. Https://doi.org/10.26518/2071-7296-2025-22-6-1000-1017.Uglova E.V. Forecasting of the residual resource asphalt concrete coverings with the account real working conditions. Bulletin of Volgograd state university of architecture and civil engineering. Series Construction and Architecture. 2017; 36(17): 43–47 (in Russ.)Uglova E.V., Tiraturjan A.N., Eganyan G.V. Calibration of the prediction model for fatigue damage accumulation in asphalt courses of flexible pavements for the conditions specific to the Russian Federation // IOP Conf. Series: Materials Science and Engineering. 2019. Т. 698. Article No 077010. Https://doi.org/10.1088/1757-899X/698/7/077010Uglova E.V., Tiraturjan A.N., Eganyan G.V. Calibration of the prediction model for fatigue damage accumulation in asphalt courses of flexible pavements for the conditions specific to the Russian Federation. IOP Conf. Series: Materials Science and Engineering. 2019; 698: Article No 077010. Https://doi.org/10.1088/1757-899X/698/7/077010Углова Е.В., Тиратурян А.Н., Шило О.А. Прогнозирование накопления усталостных разрушений в асфальтобетонных слоях нежестких дорожных одежд // Научный журнал строительства и архитектуры. 2019. № 3 (55). С. 52–61. Https://doi.org/10.25987/VSTU.2019.55.3.006Uglova E.V., Tiraturyan A.N., Shilo O.A. Prediction of Failure Fatigue Accumulation in Asphalt Concrete Layers of Flexible Pavements. Russian Journal of Building Construction and Architecture. 2019; 55(3): 52–61. (In Russ). Https://doi.org/0.25987/VSTU.2019.55.3.006Пегин П.А., Капский Д.В., Буртыль Ю.В. Разработка методики оценки продольной ровности при изменении прочности дорожных конструкций // Бюллетень результатов научных исследований. 2022. Вып. 4. С. 37–47. Https://doi.org/10.20295/2223-9987-2022-4-37-47Pegin P.A., Kapski D.V., Burtyl Yu.V. Development of Assessment Methodology for Pavement Longitudinal Evenness When Road Construction Durability Changes. Bulletin of scientific research results. 2022; 4: 37–47. (In Russ.) Https://doi.org/10.20295/2223-9987-2022-4-37-47Буртыль Ю.В., Капский Д.В. Моделирование взаимосвязи ровности и прочности нежестких дорожных одежд на основании теоретическо-практических исследований // Вестник СибАДИ. 2022. Т.19, № 4 (86). С. 570–583. Https://doi.org/10.26518/2071-7296-2022-19-4-570-583Burtyl Y. V., Kapski D. V. Modelling the relationship of smoothness and resistibility in non-rigid pavements based on theoretical and practical studies. The Russian Automobile and Highway Industry Journal. 2022; 19(4): 570-583. https://doi.org/10.26518/2071-7296-2022-19-4-570-583Iskakbayev A.I., Teltayev B.B., Rossi C.O. Deformation and strength of asphalt concrete under static and step loadings // In book: Transport Infrastructure and Systems. 2017. С. 3–8. Https://doi.org/10.1201/9781315281896-1Iskakbayev A.I., Teltayev B.B. Rossi C.O. Deformation and strength of asphalt concrete under static and step loadings. In book: Transport Infrastructure and Systems. 2017; 3-8. Https://doi.org/10.1201/9781315281896-1Elnashar G., Bhat R.B., Sedaghati R. Modeling pavement damage and predicting fatigue cracking of fexible pavements based on a combination of deterministic method with stochastic approach using Miner’s hypothesis // Applied Sciences. 2019. Т. 1. Article No 229. Https://doi.org/10.1007/s42452-019-0238-5Elnashar G., Bhat R.B., Sedaghati R. Modeling pavement damage and predicting fatigue cracking of fexible pavements based on a combination of deterministic method with stochastic approach using Miner’s hypothesis. Applied Sciences. 2019; 1: Article No 229. doi.org/10.1007/s42452-019-0238-5Fahad M., Nagy R. Fatigue damage analysis of pavements under autonomous truck tire passes. // Pollack Periodica. 2022. Т. 17. № (3). С 59–64. Https://doi.org/10.1556/606.2022.00588Fahad M., Nagy R. Fatigue damage analysis of pavements under autonomous truck tire passes. Pollack Periodica. 2022; 17(3): 59–64. Https://doi.org/10.1556/606.2022.00588Olexa T., Mandula J. Comparison of complex modulus and elasticity modulus of bitumen bonded materials // Pollack Periodica. 2016; Т. 11. №3. С. 131–140. Https://doi.org//10.1556/606.2016.11.3.12Olexa T., Mandula J. Comparison of complex modulus and elasticity modulus of bitumen bonded materials. Pollack Periodica. 2016; Т. 11. №3. С. 131–140. Https://doi.org//10.1556/606.2016.11.3.12Tiraturyan A.N., Lyapin A.A. Analysis of the deformation energy dissipation in a layered medium under dynamic loading (on the example of highways) // Soil mechanics and foundation engineering. 2024. Т. 61. № 5. С. 445–451. Https://doi.org/10.1007/s11204-024-09995-3Tiraturyan A.N., Lyapin A.A. Analysis of the deformation energy dissipation in a layered medium under dynamic loading (on the example of highways). Soil mechanics and foundation engineering. 2024; 61(5): 445–451. Https://doi.org/10.1007/s11204-024-09995-3Тиратурян А.Н., Акулов В.В. Энергия деформирования в слоистой среде при ударном нагружении (на примере автомобильных дорог) // Геология и геофизика Юга России. 2024. Т. 14, № 4. С. 128–141. Https://doi.org/10.46698/VNC.2024.52.44.011Tiraturyan A.N., Akulov V.V. Deformation energy in a layered half-space under impact loading (on the example of highways). Geology and Geophysics of Russian South. 2024; 14(4): 128-141. (in Russ.) Https://doi.org/10.46698/VNC.2024.52.44.011Tiraturyan A.N. Forecasting of the Residual Life of Pavements on Highways Based on the Analysis of Energy Dissipation under the Dynamic Influence of Transport // Journal of friction and wear. 2023. Т. 4. С. 91–96 Https://doi.org/10.3103/S1068366623020113Tiraturyan A.N. Forecasting of the Residual Life of Pavements on Highways Based on the Analysis of Energy Dissipation under the Dynamic Influence of Transport. // Journal of friction and wear. 2023. Т. 4. С. 91–96 https://doi.org/10.3103/S1068366623020113Тиратурян А.Н., Углова Е.В., Ляпин А.А. Исследование распределения энергии динамического воздействия транспортных средств в слоях нежесткой дорожной конструкции // Вестник Пермского национального исследовательского политехнического университета. Механика. 2017. № 2. С. 178–194. Https://doi.org/10.15593/perm.mech/2017.2.10Tiraturyan A.N., Uglova E.V., Lyapin A.A. Studying the energy distribution of the dynamic influences of road transport on the layers of nonrigid pavements. PNRPU Mechanics Bulletin. 2017; No. 2: 178-194. Https://doi.org/10.15593/perm.mech/2017.2.10Kadyrov G.F., Simchuk Ye.N., Tiraturyan А. N. Comparative testing of asphalt concrete for fatigue life using various modern laboratory methods // Russian journal of building construction and architecture. 2024. Т. 63. № 3. С. 65–75. Https://doi.org/10.36622/2542-0526.2024.63.3.006Kadyrov G.F., Simchuk Ye.N., Tiraturyan А. N. Comparative testing of asphalt concrete for fatigue life using various modern laboratory methods. Russian journal of building construction and architecture. 2024; 63(3): 65–75. Https://doi.org/10.36622/2542-0526.2024.63.3.006Tiraturyan A.N. Modelling of stress-strain state of asphalt concrete layers in pavements taking into account the results of laboratory four-point bending tests // Construction Materials and Products. 2024. Т. 7. № 4. Статья № 5. Https://doi.org/10.58224/2618-7183-2024-7-4-5Tiraturyan A.N. Modelling of stress-strain state of asphalt concrete layers in pavements taking into account the results of laboratory four-point bending tests. Construction Materials and Products. 2024; 7(4). Article No 5. Https://doi.org/10.58224/2618-7183-2024-7-4-5Gao H, Yang X, Zhang C. Experimental and numerical analysis of three-point bending fracture of pre-notched asphalt mixture beam // Construction and Building Materials. 2015. Т. 90. 1–10. Https://doi.org/10.1016/j.conbuildmat.2015.04.047Gao H, Yang X, Zhang C. Experimental and numerical analysis of three-point bending fracture of pre-notched asphalt mixture beam. Construction and Building Materials. 2015. Т. 90. 1–10. Https://doi.org/10.1016/j.conbuildmat.2015.04.047Aliha M.R.M., et al. Study of characteristic specification on mixed mode fracture toughness of asphalt mixtures // Construction and Building Materials. 2014. Т. 54. С. 623–635. Https://doi.org/10.1016/j.conbuildmat.2013.12.097Aliha M.R.M., et al. Study of characteristic specification on mixed mode fracture toughness of asphalt mixtures. Construction and Building Materials. 2014; 54: 623–635. Https://doi.org/10.1016/j.conbuildmat.2013.12.097Pirmohammad S., Ayatollahi M. Asphalt concrete resistance against fracture at low temperatures under different modes of loading // Cold Regions Science and Technology. 2015. Т. 110. С.149–159. Https://doi.org/10.1016/j.coldregions.2014.11.001Pirmohammad S., Ayatollahi M. Asphalt concrete resistance against fracture at low temperatures under different modes of loading. Cold Regions Science and Technology. 2015; 110: 149–159. Https://doi.org/10.1016/j.coldregions.2014.11.001Li X., et al. Effect of factors affecting fracture energy of asphalt concrete at low temperature // Road Materials and Pavement Design. 2008. Т. 9. № 1. С. 397–416. Https://doi.org/10.1080/14680629.2008.9690176Li X., et al. Effect of factors affecting fracture energy of asphalt concrete at low temperature. Road Materials and Pavement Design. 2008; 9(1): 397–416. Https://doi.org/10.1080/14680629.2008.9690176Li X., Marasteanu M. Using semi circular bending test to evaluate low temperature fracture resistance for asphalt concrete // Experimental Mechanics. 2010. Т. 50. № 7. С. 867–876. Https://doi.org/10.1007/s11340-009-9303-0Li X., Marasteanu M. Using semi circular bending test to evaluate low temperature fracture resistance for asphalt concrete. Experimental Mechanics. 2010; 50.(7): 867–87.6 Https://doi.org/10.1007/s11340-009-9303-0Li X, et al. Factors study in low-temperature fracture resistance of asphalt concrete // Journal of Materials in Civil Engineering. 2010. Т. 22. № 2. С. 145–152. Https://doi.org/10.1061/(ASCE)0899-1561(2010)22:2(145)Li X, et al. Factors study in low-temperature fracture resistance of asphalt concrete. Journal of Materials in Civil Engineering. 2010; 22(2). 145–152. Https://doi.org/10.1061/(ASCE)0899-1561(2010)22:2(145)Artamendi I., Khalid H.A. A comparison between beam and semi-circular bending fracture tests for asphalt // Road Materials and Pavement Design. 2006. Т. 7. № 1. С. 163–180. Https://doi.org/10.1080/14680629.2006.9690063Artamendi I., Khalid H.A. A comparison between beam and semi-circular bending fracture tests for asphalt. Road Materials and Pavement Design. 2006; 7(1): 163–180. Https://doi.org/10.1080/14680629.2006.9690063Mansourian A., Razmi A., Razavi M. Evaluation of fracture resistance of warm mix asphalt containing jute fibers // Construction and Building Materials. 2016. Т. 117. С. 37–46. Https://doi.org/10.1016/j.conbuildmat.2016.04.128Mansourian A., Razmi A., Razavi M. Evaluation of fracture resistance of warm mix asphalt containing jute fibers. Construction and Building Materials. 2016; 117: 37–46. Https://doi.org/10.1016/j.conbuildmat.2016.04.128Pirmohammad S., Abdi M., Ayatollahi M.R. Effect of support type on the fracture toughness and energy of asphalt concrete at different temperature conditions // Engineering Fracture Mechanics. 2021. Т. 254. № 7.: Article No 107921. Https://doi.org/10.1016/j.engfracmech.2021.107921Pirmohammad S., Abdi M., Ayatollahi M.R. Effect of support type on the fracture toughness and energy of asphalt concrete at different temperature conditions. Engineering Fracture Mechanics. 2021; 254(7): Article No 107921. Https://doi.org/10.1016/j.engfracmech.2021.107921Pirmohammad S., Kiani A. Impact of temperature cycling on fracture resistance of asphalt concretes // Computers and Concrete. 2016. Т. 17. №4. С. 541–551. Https://doi.org/10.12989/cac.2016.17.4.541Pirmohammad S., Kiani A. Impact of temperature cycling on fracture resistance of asphalt concretes. Computers and Concrete. 2016; 17(4): 541–551. Https://doi.org/10.12989/cac.2016.17.4.541Amin I., et al. Laboratory evaluation of asphalt binder modified with carbon nanotubes for Egyptian climate // Construction and Building Materials. 2016. Т. 121.№ 8. С. 361–372. Https://doi.org/10.1016/j.conbuildmat.2016.05.168Amin I., et al. Laboratory evaluation of asphalt binder modified with carbon nanotubes for Egyptian climate. Construction and Building Materials. 2016; 121(8): 361–372. Https://doi.org/10.1016/j.conbuildmat.2016.05.168Arabani M., Faramarzi M. Characterization of CNTs-modified HMA’s mechanical properties // Construction and Building Materials. 2015. Т. 83. № 1. С. 207–215. Https://doi.org/10.1016/j.conbuildmat.2015.03.035Arabani M., Faramarzi M. Characterization of CNTs-modified HMA’s mechanical properties. Construction and Building Materials. 2015; 83(1): 207–215. Https://doi.org/10.1016/j.conbuildmat.2015.03.035Ashish P.K., Singh D., Bohm S. Evaluation of rutting, fatigue and moisture damage performance of nanoclay modified asphalt binder // Construction and Building Materials 2016. Т. 113. С. 341–350. Https://doi.org/10.1016/j.conbuildmat.2016.03.057Ashish P.K., Singh D., Bohm S. Evaluation of rutting, fatigue and moisture damage performance of nanoclay modified asphalt binder. Construction and Building Materials. 2016; 113: 341–350. Https://doi.org/10.1016/j.conbuildmat.2016.03.057Kordi. Z, Shafabakhsh G. Evaluating mechanical properties of stone mastic asphalt modified with Nano Fe2O3 // Construction and Building Materials. 2017. Т. 134. С. 530–539 Https://doi.org/10.1016/j.conbuildmat.2016.12.202Kordi Z., Shafabakhsh G. Evaluating mechanical properties of stone mastic asphalt modified with Nano Fe2O3. Construction and Building Materials. 2017; 134: 530-539. Https://doi.org/10.1016/j.conbuildmat.2016.12.202Pirmohammad S, Majd-Shokorlou Y, Amani B Experimental investigation of fracture properties of asphalt mixtures modified with Nano Fe2O3 and carbon nanotubes // Road Mater Pavement Des 2019. Т. № 1. С. 1–23. Https://doi.org/10.1080/14680629.2019.1608289Pirmohammad S., Majd-Shokorlou Y., Amani B. Experimental investigation of fracture properties of asphalt mixtures modified with Nano Fe2O3 and carbon nanotubes. Road Mater Pavement Des. 2019; 21(1):1-23. Https://doi.org/10.1080/14680629.2019.1608289Shafabakhsh G., Ani O.J. Experimental investigation of effect of Nano TiO2/SiO2 modified bitumen on the rutting and fatigue performance of asphalt mixtures containing steel slag aggregates // Construction and Building Materials 2015. Т. 98. С. 692–702. Https://doi.org/10.1016/j.conbuildmat.2015.08.083Shafabakhsh G., Ani O.J. Experimental investigation of effect of Nano TiO2/SiO2 modified bitumen on the rutting and fatigue performance of asphalt mixtures containing steel slag aggregates. Construction and Building Materials. 2015: 98: 692-702. Https://doi.org/10.1016/j.conbuildmat.2015.08.083Zhang H.L., et al. High and low temperature properties of nano-particles/polymer modified asphalt // Construction and Building Materials. 2016. Т. 114. № 1. С. 323–332. Https://doi.org/10.1016/j.conbuildmat.2016.03.118Zhang H.L., et al. High and low temperature properties of nano-particles/polymer modified asphalt. Construction and Building Materials. 2016; 114(1): 323–332. Https://doi.org/10.1016/j.conbuildmat.2016.03.118Ziari. H, et al. The investigation of the impact of carbon nano tube on bitumen and HMA performance // Petroleum Science and Technology. 2014. Т. 32. № 17. С 2102–2108. Https://doi.org/10.1080/10916466.2013.763827Ziari. H, et al. The investigation of the impact of carbon nano tube on bitumen and HMA performance. Petroleum Science and Technology. 2014; 32(17): 2102–2108. Https://doi.org/10.1080/10916466.2013.763827Телтаев Б.Б. Новый взгляд на низкотемпературное трещинообразование в асфальтобетонном покрытии // Известия Национальной академии наук Республики Казахстан. Серия Геологии и Технических наук. 2016. Т. 419, № 5. С.161–178.Teltayev B.B. Fresh approach to low temperature cracking in asphalt concrete pavement News of the National Academy of Sciences of the Republic Kazakhstan. Series ofGeology and Technical sciences. 2016; 419(5): 161–178. (In Russ.)Мозговой В.В. Направленное регулирование термовязкоупругих свойств асфальтобетона // Вестник ХНАДУ. 2017. Т. 79. С. 89–93.Mozhovyi V. Targeted regulation of termoviscoelasticity properties of asphalt-concrete. Bulletin of the Kharkiv national university of automobile and road engineering. 2017; 79: 89 –93. (In Russ.)Колесников Г.Н., Гаврилов Т.А. Моделирование условий появления низкотемпературных трещин в асфальтобетонном слое автомобильной дороги // Вестник Томского государственного университета. Математика и механика. 2018. Т. 56. С. 57–66.Kolesnikov G.N., Gavrilov T.A. Simulation of the conditions for a lowtemperature crack appearance in the asphalt concrete layer of a road. Tomsk State University Journal of Mathematics and Mechanics. 2018; 56: 57-66. (In Russ.)

The authors declare that there are no conflicts of interest present.