1683-3414 1814-0807 Владикавказский математический журнал Vladikavkaz Mathematical Journal Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН) Градиентная модель нестационарной термоупругости для составного стержня Gradient Model of Non-Stationary Thermoelasticity for a Composite Rod 10.46698/v0156-6205-6438-l 19874 09 2026 28 3 91 107 https://vmj.ru/archive/detail.php?ELEMENT_ID=19874&SECTION_ID=673 Нестеров С. А. Nesterov S. A. 1079@list.ru Южный математический институт - филиал ВНЦ РАН, РОССИЯ, 362025, Владикавказ, ул. Ватутина, 53 Southern Mathematical Institute VSC RAS, 53 Vatutina St., Vladikavkaz 362025, Russia Моделирование поведения стержневых микроразмерных конструкций имеет большое практическое значение для проектирования термоэлектрических устройств. Ранее проведенные исследования не учитывали совместного влияния теплового и механического масштабных параметров на распределение температуры и смещений составных тел. Это приводило к большим расхождениям расчетов и экспериментальных данных. Мы исследуем задачу уточнения нестационарного термонапряженного состояния составного микростержня. Один торец стержня защемлен и находится при нулевой температуре, а на другом задан нестационарный тепловой поток. Получена обобщенная (градиентная) модель термоупругости для составного микростержня в квазистатической постановке. В определяющие соотношения данной модели вводим тепловой масштабный параметр, а в выражении для теплового потока учитываем градиенты температуры второго порядка. Постановка градиентной модели включает уравнения теплопроводности и равновесия более высокого порядка по сравнению с классической теорией и расширенный список граничных условий. В рамках работы построены численно-аналитическое и асимптотическое решения задачи в случае слабой связанности. Задача сведена к последовательному решению задач градиентной теплопроводности и градиентной теории упругости в трансформантах Лапласа. Обращение трансформант осуществлено на основе модифицированного метода Дурбина. Решение сингулярно возмущенной задачи теплопроводности построено с помощью асимптотического подхода. Выражения для температуры представлены в виде суммы классического решения и погранслойных функций. Обнаружены следующие эффекты, не описанные ранее: 1) с увеличением масштабного параметра значения температуры возрастают на малых временах; 2) происходит более плавное изменение температуры в окрестности точки сопряжения. Точность асимптотических формул проверена путем сравнения с численно-аналитическим решением. Определены границы их применимости. Представленный подход позволит более точно проводить расчеты на прочность составных микроразмерных тел, например, микробалки и микроцилиндра. Modeling the behavior of micro-sized rod structures is of great practical importance for the design of thermoelectric devices. Previous studies did not take into account the combined effect of thermal and mechanical scaling parameters on the distribution of temperature and displacements of composite bodies. This led to large discrepancies between calculations and experimental data. We study the problem of refining the non-stationary thermal stress state of a composite micro-rod. One end of the rod is clamped and is at zero temperature, and a non-stationary heat flux is specified at the other end. A generalized (gradient) model of thermoelasticity for a composite micro-rod in a quasi-static formulation is obtained. We introduce a thermal scaling parameter into the defining relations of this model, and take into account second-order temperature gradients in the expression for the heat flux. The formulation of the gradient model includes heat conduction and equilibrium equations of a higher order compared to the classical theory as well as an extended list of boundary conditions. In this work, numerical-analytical and asymptotic solutions are constructed for the problem in the case of weak coupling. The problem is reduced to a sequential solution of the problems of gradient heat conductivity and gradient theory of elasticity in Laplace transforms. The inversion of transforms is carried out on the basis of the modified Durbin method. The solution of the singularly perturbed problem of heat conductivity is constructed using the asymptotic approach. Expressions for the temperature are presented as the sum of the classical solution and boundary layer functions. The following effects, not described earlier, are discovered: 1) with an increase in the scale parameter, the temperature values at small times increase; 2) the temperature changes more smoothly in the vicinity of the conjugation point. The accuracy of the asymptotic formulas is verified by comparing with the numerical-analytical solution. The limits of applicability of the asymptotic formulas are determined. The presented approach will allow more accurate calculations of the strength of composite micro-sized bodies, such as a microbeam and a microcylinder. микроразмерные структуры градиентная термоупругость нестационарная задача составной стержень масштабный эффект асимптотика метод Вишика - Люстерника моментные напряжения полные напряжения полный тепловой поток micro-sized structures gradient thermoelasticity non-stationary problem composite rod scale effect asymptotics Vishik-Lyusternik method couple stresses total stresses total heat flux 1 Roncaglia A., Ferri M. Thermoelectric materials in MEMS and NEMS: A review // Sci. Adv. Mater. 2008. Vol. 3, № 3. P. 401–419. DOI: 10.1166/sam.2011.1168. Roncaglia, A. and Ferri, M. Thermoelectric Materials in MEMS and NEMS: A Review, Science of Advanced Materials, 2008, vol. 3, no. 3, pp. 401-419. DOI: 10.1166/sam.2011.1168. 2 Mbarek S. B., Alcheikh N., Yonis M. I. Recent advances on MEMS based Infrared Thermopile detectors // Microsyst. Technol. 2022. Vol. 28, № 8. P. 1751–1764. DOI: 10.1007/s00542-022-05306-8. Mbarek, S. B., Alcheikh, N. and Yonis, M. I. Recent Advances on MEMS Based Infrared Thermopile Detectors, Microsystem Technologies, 2022, vol. 28, no. 8, pp. 1751-1764. DOI: 10.1007/s00542-022-05306-8. 3 Ganwani M. M., Shaikh B., Sinha S. Design and Investigation of MEMS-Based Thermopile Detector Arrays for Infrared Sensing Applications // 2024 IEEE Applied Sensing Conference (APSCON 2024). 2024. P. 193–196. DOI: 10.1109/APSCON60364.2024.1046607. Ganwani, M. M., Shaikh, B. and Sinha, S. Design and Investigation of MEMS-Based Thermopile Detector Arrays for Infrared Sensing Applications, 2024 IEEE Applied Sensing Conference (APSCON), 2024. DOI: 10.1109/APSCON60364.2024.1046607. 4 Dhar A. Heat transport in low-dimensional systems // Adv. Phys. 2008. Vol. 57, № 5. P. 457–537. DOI: 10.1080/00018730802538522. Dhar, A. Heat Transport in Low-Dimensional Systems, Advances in Physics, 2008, vol. 57, no. 5, pp. 457-537. DOI: 10.1080/00018730802538522. 5 Zhang G., Li B. W. Impacts of doping on thermal and thermoelectric properties of nanomaterials // Nanoscale. 2010. Vol. 2, № 7. P. 1058–1068. DOI: 10.1039/C0NR00095G. Zhang, G. and Li, B. W. Impacts of Doping on Thermal and Thermoelectric Properties of Nanomaterials, NanoScale, 2010, vol. 2, no. 7, pp. 1058-1068. DOI: 10.1039/C0NR00095G. 6 Stewart D. A., Norris P. M. Size effects on the thermal conductivity of thin metallic wires: micro-scale implications // Microscale Thermophys. Eng. 2000. Vol. 4, № 2. P. 89–101. DOI: 10.1080/108939500404007. Stewart, D. A. and Norris, P. M. Size Effects on the Thermal Conductivity of Thin Metallic Wires: Micro-Scale Implications, Microscale Thermophysical Engineering, 2000, vol. 4, no. 2, pp. 89-101. DOI: 10.1080/108939500404007. 7 Koh S. J. A., Lee H. P. Molecular dynamics simulation of size and strain rate dependent mechanical response of FCC metallic nanowires // Nanotechnology. 2006. Vol. 17, № 14. P. 3451–3467. DOI: 10.1088/0957-4484/17/14/018. Koh, S. J. A. and Lee, H. P. Molecular Dynamics Simulation of Size and Strain Rate Dependent Mechanical Response of FCC Metallic Nanowires, Nanotechnology, 2006, vol. 17, pp. 3451-3467. DOI: 10.1088/0957-4484/17/14/018. 8 Ahmadi G., Firoozbakhsh K. First strain gradient theory of thermoelasticity // Int. J. Solids Struct. 1975. Vol. 11, № 3. P. 339–345. DOI: 10.1016/0020-7683(75)90073-6. Ahmadi, G. and Firoozbakhsh, K. First Strain Gradient Theory of Thermoelasticity, International Journal of Solids and Structures, 1975, vol. 11, no. 3, pp. 339-345. DOI: 10.1016/0020-7683(75)90073-6. 9 Forest S., Amestoy M. Hypertemperature in thermoelastic solids // C. R. Mecanique. 2008. Vol. 336, № 4. P. 347–353. DOI: 10.1016/j.crme.2008.01.007. Forest, S. and Amestoy, M. Hypertemperature in Thermoelastic Solids, Comptes Rendus Mecanique, 2008, vol. 336, no. 4, pp. 347-353. DOI: 10.1016/j.crme.2008.01.007. 10 Forest S., Aifantis E. C. Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua // Int. J. Solids Struct. 2010. Vol. 47, № 25–26. P. 3367–3376. DOI: 10.1016/j.ijsolstr.2010.07.009. Forest, S. and Aifantis, E. C. Some Links Between Recent Gradient Thermo-Elasto-Plasticity Theories and the Thermomechanics of Generalized Continua, International Journal of Solids and Structures, 2010, vol. 47, no. 25-26, pp. 3367-3376. DOI: 10.1016/j.ijsolstr.2010.07.009. 11 Alnefaie K. A note on gradient thermoelasticity // J. Mech. Behav. Mater. 2014. Vol. 23, № 3–4. P. 107–108. DOI: 10.1515/jmbm-2014-0013. Alnefaie, K. A Note on Gradient Thermoelasticity, Journal of the Mechanical Behavior of Materials, 2014, vol. 23, no. 3-4, pp. 107-108. DOI: 10.1515/jmbm-2014-0013. 12 Nika G., Muntean A. Hypertemperature effects in heterogeneous media and thermal flux at small-length scales // Netw. Heterog. Media. 2023. Vol. 18, № 3. P. 1207–1225. DOI: 10.3934/nhm.2023052. Nika, G. and Muntean, A. Hypertemperature Effects in Heterogeneous Media and Thermal Flux at Small-Length Scales, Networks and Heterogeneous Media, 2023, vol. 18, no. 3, pp. 1207-1225. DOI: 10.3934/nhm.2023052. 13 Iesan D., Quintanilla R. A second gradient theory of thermoelasticity // J. Elasticity. 2023. Vol. 154. P. 629–643. DOI: 10.1007/s10659-023-10020-1. Iesan, D. and Quintanilla, R. A Second Gradient Theory of Thermoelasticity, Journal of Elasticity, 2023, vol. 154, pp. 629-643. DOI: 10.1007/s10659-023-10020-1. 14 Bazarra N., Fernandez J. R., Quintanilla R. A. A strain gradient problem with a fourth-order thermal law // J. Comput. Appl. Math. 2024. Vol. 445. Art. 115818. DOI: 10.1016/j.cam.2024.115818. Bazarra, N., Fernandez, J. R. and Quintanilla, R. A Strain Gradient Problem with a Fourth-Order Thermal Law, Journal of Computational and Applied Mathematics, 2024, vol. 445, article no. 115818. DOI: 10.1016/j.cam.2024.115818. 15 Dong Y., Cao B. Y., Guo Z. Y. Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics // Physica E Low Dimens. Syst. Nanostruct. 2014. Vol. 56. P. 256–262. DOI: 10.1016/j.physe.2013.10.006. Dong, Y., Cao, B. Y. and Guo, Z. Y. Size Dependent Thermal Conductivity of Si Nanosystems Based on Phonon Gas Dynamics, Physica E: Low-Dimensional Systems and Nanostructures, 2014, vol. 56, pp. 256-262. DOI: 10.1016/j.physe.2013.10.006. 16 Лурье С. А., Фам Т., Соляев Ю. О. Градиентная модель термоупругости и ее приложения к моделированию тонкослойных композитных структур // Мех. композиц. материалов и конструкций. 2012. Т. 18, № 3. С. 440–449. Lurie, S. A., Pham, T. and Soliaev, J. O. Gradient Model of Thermoelasticity and Its Applications for the Modelling of Thin Layered Structures, Mekhanika Kompozitsionnykh Materialov i Konstruktsiy, 2012, vol. 18, no. 3, pp. 440-449 (in Russian). 17 Lurie S. A., Belov Р. A., Volkov-Bogorodskii D. B. Variational models of coupled gradient thermoelasticity and thermal conductivity // Mater. Phys. Mech. 2019. Vol. 42, № 5. P. 564–581. DOI: 10.18720/MPM.4252019_10. Lurie, S. A., Belov, P. A. and Volkov-Bogorodskii, D. B. Variational Models of Coupled Gradient Thermoelasticity and Thermal Conductivity, Materials Physics and Mechanics, 2019, vol. 42, no. 5, pp. 564-581. DOI: 10.18720/MPM.4252019_10. 18 Lurie S. A., Volkov-Bogorodskii D. B., Belov Р. A. Analytical solution of stationary coupled thermoelasticity problem for inhomogeneous structures // Mathematics. 2022. Vol. 10, № 1. Art. 90. DOI: 10.3390/math10010090. Lurie, S. A., Volkov-Bogorodskii, D. B. and Belov, P. A. Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures, Mathematics, 2022, vol. 10, no. 1, article no. 90. DOI: 10.3390/math10010090. 19 Волков-Богородский Д. Б., Лурье С. А. Общие решения связанных задач нестационарной термоупругости и теплообмена в неоднородных средах // Композиты и наноструктуры. 2024. Т. 16, № 2. С. 135–149. DOI: 10.36236/1999-7590-2024-16-2-135-149. Volkov-Bogorodskii, D. B. and Lurie, S. A. General Solutions for Non-Stationary Coupled Problems of Thermoelasticity and Thermal Conductivity in Inhomogeneous Media, Kompozity i nanostruktury [Composites and Nanostructures], 2024, vol. 16, no. 2, pp. 135-149 (in Russian). DOI: 10.36236/1999-7590-2024-16-2-135-149. 20 Вишик М. И., Люстерник Л. А. Регулярное вырождение и пограничный слой для линейных дифференциальных уравнений с малым параметром // Успехи мат. наук. 1957. Т. 12, № 5 (77). С. 3–122. Vishik, M. I. and Lyusternik, L. A. Regular Degeneration and Boundary Layer for Linear Differential Equations with Small Parameter, Uspekhi Matematicheskikh Nauk, 1957, vol. 12, no. 5 (77), pp. 3-122 (in Russian). 21 Бауэр С. М., Смирнов А. Л., Товстик П. Е., Филиппов С. Б. Асимптотические методы в механике твердого тела. Москва–Ижевск: Регулярная и хаотическая динамика, 2007. 360 с. Bauer, S. M., Smirnov, A. L., Tovstik, P. E. and Filippov, S. B. Asymptotic Methods in Mechanics of Solids [Asimptoticheskie metody v mekhanike tverdogo tela], Moscow, Izhevsk, Regulyarnaya i khaoticheskaya dinamika, 2007, 360 p. (in Russian). 22 Нестеров С. А. Градиентная модель свободных продольных колебаний упругого стержня // Наука Юга России. 2025. Т. 21, № 1. С. 15–23. DOI: 10.7868/S25000640250103. Nesterov, S. A. Gradient Model of Free Longitudinal Vibrations of an Elastic Rod, Nauka Yuga Rossii [Science of the South of Russia], 2025, vol. 21, no. 1, pp. 15-23 (in Russian). DOI: 10.7868/S25000640250103. 23 Vatulyan А. О., Nesterov S. А. On the deformation of a composite rod in the framework of gradient thermoelasticity // Mater. Phys. Mech. 2020. Vol. 46, № 1. P. 27–41. DOI: 10.18149/MPM.4612020_3. Vatulyan, A. O. and Nesterov, S. A. On the Deformation of a Composite Rod in the Framework of Gradient Thermoelasticity, Materials Physics Mechanics, 2020, vol. 46, no. 1, pp. 27-41. DOI: 10.18149/MPM.4612020_3. 24 Ватульян А. О., Нестеров С. А. Решение задачи градиентной термоупругости для полосы с покрытием // Учен. зап. Казан. ун-та. Сер. Физ.-мат. науки. 2021. Т. 163, № 2. С. 181–196. DOI: 10.26907/2541-7746.2021.2.181-196. Vatulyan, A. O. and Nesterov, S. A. Solution of the Problem of Gradient Thermoelasticity for a Coated Strip, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2021, vol. 163, no. 2, pp. 181-196 (in Russian). DOI: 10.26907/2541-7746.2021.2.181-196. 25 Vatulyan A. O., Nesterov S. A., Yavruyan O. V. The Models of gradient mechanics and singularly perturbed boundary value problems // Lobachevskii J. Math. 2023. Vol. 44. P. 3604–3612. DOI: 10.1134/S1995080223080565. Vatulyan, A. O., Nesterov, S. A. and Yavruyan, O. V. The Models of Gradient Mechanics and Singularly Perturbed Boundary Value Problems, Lobachevskii Journal of Mathematics, 2023, vol. 44, pp. 3604-3612. DOI: 10.1134/S1995080223080565. 26 Sladek J., Sladek V., Repka M. The heat conduction in nanosized structures // Fiz. Mezomekh. 2021. Vol. 24, № 5. P. 122–129. DOI: 10.24412/1683-805X-2021-5-122-129. Sladek, J., Sladek, V. and Repka, M. The Heat Conduction in Nanosized Structures, Fizicheskaya Mezomekhanika, 2021, vol. 24, no. 5, pp. 122-129. DOI: 10.24412/1683-805X-2021-5-122-129. 27 Zenkour A. M., Abouelregal A. E., Alnefaie K. A., Zhang X., Aifantis E. C. Nonlocal thermoelasticity theory for thermal-shock nanobeams with temperature-dependent thermal conductivity // J. Therm. Stresses. 2015. Vol. 38, № 9. P. 1049–1067. DOI: 10.1080/01495739.2015.1038490. Zenkour, A. M., Abouelregal, A. E., Alnefaie, K. A., Zhang, X. and Aifantis, E. C. Nonlocal Thermoelasticity Theory for Thermal-Shock Nanobeams with Temperature-Dependent Thermal Conductivity, Journal of Thermal Stresses, 2015, vol. 38, no. 9, pp. 1049-1067. DOI: 10.1080/01495739.2015.1038490. 28 Sarkar N. Thermoelastic responses of a finite rod due to nonlocal heat conduction // Acta Mech. 2020. Vol. 231. P. 947–955. DOI: 10.1007/s00707-019-02583-9. Sarkar, N. Thermoelastic Responses of a Finite Rod Due to Nonlocal Heat Conduction, Acta Mechanica, 2020, vol. 231, pp. 947-955. DOI: 10.1007/s00707-019-02583-9. 29 Yang W. Z., Chen Z. T. Nonlocal dual-phase-lag heat conduction and the associated nonlocal thermal-viscoelastic analysis // Int. J. Heat Mass Transf. 2020. Vol. 156. Art. 119752. DOI: 10.1016/j.ijheatmasstransfer.2020.119752. Yang, W. Z. and Chen, Z. T. Nonlocal Dual-Phase-Lag Heat Conduction and the Associated Nonlocal Thermal-Viscoelastic Analysis, International Journal of Heat and Mass Transfer, 2020, vol. 156. DOI: 10.1016/j.ijheatmasstransfer.2020.119752. 30 Awrejcewicz J., Krysko V. A., Pavlov S. P., Zhigalov M. V., Kalutsky L. A., Krysko A. V. Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress theory // Nonlinear Dyn. 2020. Vol. 99. P. 919–943. DOI: 10.1007/s11071-019-04976-w. Awrejcewicz, J., Krysko, V. A., Pavlov, S. P., Zhigalov, M. V., Kalutsky, L. A. and Krysko, A. V. Thermoelastic Vibrations of a Timoshenko Microbeam Based on the Modified Couple Stress Theory, Nonlinear Dynamics, 2019, vol. 99, pp. 919-943. DOI: 10.1007/s11071-019-04976-w. 31 Awrejcewicz J., Krysko V. A., Sopenk A. A., Zhigalov M. V., Kirichenko A. V., Krysko A. V. Mathematical modelling of physically/geometrically non-linear micro-shells with account of coupling of temperature and deformation fields // Chaos Solit. Fractals. 2017. Vol. 104. P. 635–654. DOI: 10.1016/j.chaos.2017.09.008. Awrejcewicz, J., Krysko, V. A., Sopenk, A. A., Zhigalov, M. V., Kirichenko, A. V. and Krysko, A. V. Mathematical Modelling of Physically/Geometrically Non-Linear Micro-Shells With Account of Coupling of Temperature and Deformation Fields, Chaos, Solitons and Fractals, 2017, vol. 104, pp. 635-654. DOI: 10.1016/j.chaos.2017.09.008. 32 Kirichenko V. F., Awrejcewicz J., Kirichenko A. V., Krysko A. V., Krysko V. A. On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections // Int. J. Non-Linear Mech. 2015. Vol. 74. P. 51–72. DOI: 10.1016/j.ijnonlinmec.2015.03.011. Kirichenko, V. F., Awrejcewicz, J., Kirichenko, A. V., Krysko, A. V. and Krysko, V. A. On the Non-Classical Mathematical Models of Coupled Problems of Thermo-Elasticity for Multi-Layer Shallow Shells with Initial Imperfections, International Journal of Non-Linear Mechanics, 2015, vol. 74, pp. 51-72. DOI: 10.1016/j.ijnonlinmec.2015.03.011. 33 Han W., Hua B., Xiulin R. Perspective: Predicting and optimizing thermal transport properties with machine learning methods // Energy AI. 2022. Vol. 8, № 12. Art. 100153. DOI: 10.1016/j.egyai.2022.100153. Han, W., Hua, B. and Xiulin, R. Perspective: Predicting and Optimizing Thermal Transport Properties with Machine Learning Methods, Energy and AI, 2022, vol. 8, no. 12, article no. 100153. DOI: 10.1016/j.egyai.2022.100153. 34 Zobeiry N., Humfeld K. D. A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications // Eng. Appl. Artif. Intell. 2021. Vol. 101. Art. 104232. DOI: 10.1016/j.engappai.2021.104232. Zobeiry, N. and Humfeld, K. D. A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications, Engineering Applications of Artificial Intelligence, 2021, vol. 101, article no. 104232. DOI: 10.1016/j.engappai.2021.104232. 35 Schneider T., Bedrikow A. B., Stahl K. Enhanced prediction of thermomechanical systems using machine learning, PCA, and finite element simulation // Adv. Model. Simul. Eng. Sci. 2024. Vol. 11. Art. 14. DOI: 10.1186/s40323-024-00268-0. Schneider, T., Bedrikow, A. B. and Stahl, K. Enhanced Prediction of Thermomechanical Systems Using Machine Learning, PCA, and Finite Element Simulation, Advanced Modeling and Simulation in Engineering Sciences, 2024, vol. 11, article no. 14. DOI: 10.1186/s40323-024-00268-0. 36 Shotaro S. F. Thermodynamics and feature extraction by machine learning // Phys. Rev. Research. 2020. Vol. 2, № 3. DOI: 10.1103/PhysRevResearch.2.033415. Shotaro, S. F. Thermodynamics and Feature Extraction by Machine Learning, Physical Review Research, 2020, vol. 2, no. 3. DOI: 10.1103/PhysRevResearch.2.033415. 37 Li Z., He Y., Lei J., Guo S., Liu D., Wang L. A standard experimental method for determining the material length scale based on modified couple stress theory // Int. J. Mech. Sci. 2018. Vol. 141. P. 198–205. DOI: 10.1016/j.ijmecsci.2018.03.035. Li, Z., He, Y., Lei, J., Guo, S., Liu, D. and Wang, L. A Standard Experimental Method for Determining the Material Length Scale Based on Modified Couple Stress Theory, International Journal of Mechanical Science, 2018, vol. 141, pp. 198-205. DOI: 10.1016/j.ijmecsci.2018.03.035. 38 Li Z., He Y., Zhang B., Lei J., Guo S., Liu D. Experimental investigation and theoretical modelling on nonlinear dynamics of cantilevered microbeams // Eur. J. Mech. A Solids. 2019. Vol. 78. Art. 103834. DOI: 10.1016/j.euromechsol.2019.103834. Li, Z., He, Y., Zhang, B., Lei, J., Guo, S. and Liu, D. Experimental Investigation and Theoretical Modelling on Nonlinear Dynamics of Cantilevered Microbeams, European Journal of Mechanics — A / Solids, 2019, vol. 78, article no. 103834. DOI: 10.1016/j.euromechsol.2019.103834. 39 Feher A., Lukacs N., Somlai L., Fodor T., Szucs M., Fulop T., Van P., Kovacs R. Size effects and beyond-Fourier heat conduction in room-temperature experiments // J. Non-Equilib. Thermodyn. 2021. Vol. 46, № 4. P. 403–411. DOI: 10.1515/jnet-2021-0033. Feher, A., Lukacs, N., Somlai, L., Fodor, T., Szucs, M., Fulop, T., Van P. and Kovacs R. Size Effects and Beyond-Fourier Heat Conduction in Room-Temperature Experiments, Journal of Non-Equilibrium Thermodynamics, 2021, vol. 46, pp. 403-411. DOI: 10.1515/jnet-2021-0033. 40 Lam D. C. C., Yang F., Chong A. C. M., Wang J., Tong P. Experiments and theory in strain gradient elasticity // J. Mech. Phys. Solids. 2003. Vol. 51, № 8. P. 1477–1508. DOI: 10.1016/S0022-5096(03)00053-X. Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J. and Tong, P. Experiments and Theory in Strain Gradient Elasticity, Journal of the Mechanics and Physics of Solids, 2003, vol. 51, no. 8, pp. 1477-1508. DOI: 10.1016/S0022-5096(03)00053-X. 41 Liebold C., Muller W. H. Comparison of gradient elasticity models for the bending of micromaterials // Comput. Mater. Sci. 2003. Vol. 116. P. 52–61. DOI: 10.1016/j.commatsci.2015.10.031. Liebold, C. and Muller, W. H. Comparison of Gradient Elasticity Models for the Bending of Micromaterials, Computational Materials Science, 2003, vol. 116, pp. 52-61. DOI: 10.1016/j.commatsci.2015.10.031. 42 Gusev A. A., Lurie S. A. Strain-gradient elasticity for bridging continuum and atomistic estimates of stiffness of binary Lennard-Jones crystals // Adv. Eng. Mater. 2010. Vol. 12, № 6. P. 529–533. DOI: 10.1002/adem.201000004. Gusev, A. A. and Lurie, S. A. Strain-gradient Elasticity for Bridging Continuum and Atomistic Estimates of Stiffness of Binary Lennard-Jones Crystals, Advanced Engineering Materials, 2010, vol. 12, no. 6, pp. 529-533. DOI: 10.1002/adem.201000004. 43 Zhao X. An efficient approach for the numerical inversion of Laplace transform and its application in dynamic fracture analysis of a piezoelectric laminate // Int. J. Solids Struct. 2004. Vol. 41, № 3. P. 3653–3674. DOI: 10.1016/j.ijsolstr.2004.01.006. Zhao, X. An Efficient Approach for the Numerical Inversion of Laplace Transform and Its Application in Dynamic Fracture Analysis of a Piezoelectric Laminate, International Journal of Solids and Structures, 2004, vol. 41, no. 3, pp. 3653-3674. DOI: 10.1016/j.ijsolstr.2004.01.006. 44 Вестяк В. А., Федотенков Г. В. Алгоритм численного обращения преобразования Лапласа в классе обобщенных функций, образующих алгебру со сверткой // Прикл. матем. и матем. физ. 2015. Т. 1, № 1. С. 67–76. DOI: 10.18262/ammp.2015.0101-05. Vestyak, V. A. and Fedotenkov, G. V. Algorithm for the Numerical Laplace Transform Inversion in the Class of Generalized Functions Forming an Algebra with Convolution, Prikladnaya matematika i matematicheskaya fizika [Applied Mathematics and Mathematical Physics], 2015, vol. 1, no. 1, pp. 67-76 (in Russian). DOI: 10.18262/ammp.2015.0101-05.