Magneto-Thermoelastic Fracture in Nonlocal Cracked Nanobeams using Modified Couple Stress Model with Memory Effect

Document Type : Research Paper

Authors
1 Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore 641114, Tamil Nadu, India
2 Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania
3 Academy of Romanian Scientists, Ilfov Street, 3, 050045 Bucharest, Romania
Abstract
The study of thermal characteristics in cracked nanobeams is vital for the reliable functioning of micro and nanoelectromechanical systems. Classical solid mechanics and Fourier heat law fail to accurately predict size-dependent elastic deformation and thermal hysteresis effects in such structures. To address this, a new computational model rooted in memory dependent derivatives is constructed to assess viscoelastic rotating cracked nanobeams subjected to a magnetic field and heat source. The model incorporates nonlocal generalized couple stress theory and generalized thermoelasticity considering memory dependent effects. The governing equations are derived based on the Euler-Bernoulli beam model, Maxwell's formulations, and a fractional-order Kelvin-Voigt viscoelastic framework. The deflection, temperature field, displacement, and bending moment of the cracked nanobeam are numerically determined using the Laplace transform and its inverse.
Keywords
Subjects

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[1] Currano, L. J., Yu, M., Balachandran, B., Latching in a MEMS shock sensor: Modeling and experiments, Sensors and Actuators A: Physical, 159(1), 2010, 41-50.
[2] Eom, K., Park, H. S., Yoon, D. S., Kwon, T., Nanomechanical resonators and their applications in biological/chemical detection: Nano mechanics principles, Physics Reports, 503(4–5), 2011, 115–163. 
[3] Xia, D., Yu, C., Kong, L., The development of micromachined gyroscope structure and circuitry technology, Sensors, 14(1), 2014, 1394–1473.  
[4] Farajpour, A., Ghayesh, M. H., Farokhi, H., A review on the mechanics of nanostructures, International Journal of Engineering Science, 133, 2018, 231–263. 
[5] Nadeem, M., Ain, Q. T., Almakayeel, N., Shao, Y., Wang, S., Shutaywi, M., Analysis of nanobeam-based microstructure in n/mems system using van der waals forces, Facta Universitatis, Series: Mechanical Engineering, 22(4), 2024, 673-688.
[6] Chandel, V. S., Wang, G., Talha, M., Advances in modelling and analysis of nano structures: a review, Nanotechnology Reviews, 9(1), 2020, 230–258. 
[7] Wen, D. L., Deng, H. T., Liu, X., Li, G. K., Zhang, X. R., Zhang, X. S., Wearable multi-sensing double-chain thermoelectric generator, Microsystems & Nanoengineering, 6(1), 2020, 68. 
[8] Michihata, M., Surface sensing principle of microprobe system for micro-scale coordinate metrology: a review, Metrology, 2(1), 2022, 46–72. 
[9] Fleck, N. A., Muller, G. M., Ashby, M. F., Hutchinson, J. W., Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia, 42(2), 1994, 475–487. 
[10] Shetty, S., Palkar, V. R., Pinto, R., Size effect study in magnetoelectric BiFeO₃ system, Pramana, 58, 2002, 1027–1030. 
[11] Borjalilou, V., Asghari, M., Bagheri, E., Small-scale thermoelastic damping in micro-beams utilizing the modified couple stress theory and the dual-phase-lag heat conduction model, Journal of Thermal Stresses, 42(7), 2019, 801–814. 
[12] Li, M., Cai, Y., Fan, R., Wang, H., Borjalilou, V., Generalized thermoelasticity model for thermoelastic damping in asymmetric vibrations of nonlocal tubular shells, Thin-Walled Structures, 174, 2022, 109142. 
[13] Biot, M. A., Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, 1956, 27(3), 240–253. 
[14] Pellam, J. R., Theory of second sound absorption in rotating helium, Physical Review Letters, 9(7), 1962, 281. 
[15] Lord, H. W., Shulman, Y., A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, 15(5), 1967, 299–309. 
[16] Green, A. E., Lindsay, K. A., Thermoelasticity, Journal of Elasticity, 2(1), 1972, 1–7. 
[17] Green, A. E., Naghdi, P. M., Thermoelasticity without energy dissipation, Journal of Elasticity, 31(3), 1993, 189–208. 
[18] Sherief, H. H., El-Sayed, A. M. A., Abd El-Latief, A., Fractional order theory of thermoelasticity, International Journal of Solids and Structures, 47(2), 2010, 269–275. 
[19] Kumar, R., Pathania, V., Gupta, V., Barak, M. S., Ahmad, H., Thermoelastic modeling with dual porosity interacting with an inviscid liquid, Journal of Applied and Computational Mechanics, 10(1), 2024, 111-124.
[20] Ezzat, M. A., El-Karamany, A. S., El-Bary, A. A., Generalized thermo-viscoelasticity with memory-dependent derivatives, International Journal of Mechanical Sciences, 89, 2014, 470–475. 
[21] Wang, J. L., Li, H. F., Surpassing the fractional derivative: Concept of the memory-dependent derivative, Computers & Mathematics with Applications, 62(3), 2011, 1562–1567. 
[22] Wang, J. L., Li, H. F., Memory-dependent derivative versus fractional derivative (I): Difference in temporal modeling, Journal of Computational and Applied Mathematics, 384, 2021, 112923.
[23] Kumar, R., Analysis of the quality factor of micromechanical resonators using memory-dependent derivative under different models, Archive of Applied Mechanics, 91(6), 2021, 2735–2745. 
[24] Tiwari, R., Kumar, R., Non-local effect on quality factor of micro-mechanical resonator under the purview of three-phase-lag thermoelasticity with memory-dependent derivative, Applied Physics A, 128(3), 2022, 190. 
[25] Abouelregal, A. E., Mohammad-Sedighi, H., Faghidian, S. A., Shirazi, A. H., Temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load, Facta Universitatis, Series: Mechanical Engineering, 19(4), 2021, 633–656. 
[26] Abouelregal, A. E., Marin, M., Ochsner, A., The influence of a non-local Moore–Gibson–Thompson heat transfer model on an underlying thermoelastic material under the model of memory-dependent derivatives, Continuum Mechanics and Thermodynamics, 35(2), 2023, 545–562. 
[27] Abouelregal, A. E., Mohammed, F. A., Benhamed, M., Zakria, A., Ahmed, I. E., Vibrations of axially excited rotating micro-beams heated by a high-intensity laser in light of a thermo-elastic model including the memory-dependent derivative, Mathematics and Computers in Simulation, 199, 2022, 81–99. 
[28] Li, Y., He, T., The transient response of a functionally graded half-space heated by a laser pulse based on the generalized thermoelasticity with memory-dependent derivative, Mechanics of Advanced Materials and Structures, 28(22), 2021, 2299–2309. 
[29] Tiwari, R., Abouelregal, A. E., Kumari, K., Kumar, P., Memory impacts on skin tissue responses exposed to harmonic heat during thermal therapy, Archive of Applied Mechanics, 94(10), 2024, 3119–3134. 
[30] Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., Tong, P., Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51(8), 2003, 1477–1508. 
[31] Liebold, C., Müller, W. H., Comparison of gradient elasticity models for the bending of micromaterials, Computational Materials Science, 0116, 2016, 52–61. 
[32] Eringen, A. C., Wegner, J. L., Nonlocal continuum field theories, Applied Mechanics Reviews, 56(2), 2002, B20–B22. 
[33] Gurtin, M. E., Murdoch, A. I., A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis, 57, 1975, 291–323. 
[34] Mindlin, R. D., Tiersten, H., Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11, 1962, 415–448. 
[35] Yang, F. A. C. M., Chong, A. C. M., Lam, D. C. C., Tong, P., Couple stress-based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10), 2002, 2731–2743. 
[36] Adeli Mahdavi, M., Mahmoudi, R., Soleimani, A., Pull-in behaviour of a micro switch actuated by the electrostatic under a uniform longitudinal magnetic field based on nonlocal couple stress theory, Journal of Computational and Applied Mechanics, 54(4), 2023, 577–587. 
[37] Mahmoudi, R., Barati, A., Hosseini, M., Hadi, A., Torsional vibration of functionally porous nanotube based on nonlocal couple stress theory, International Journal of Applied Mechanics, 13(10), 2021, 2150122. 
[38] Kharnoob, M. M., Cepeda, L. C., Jacome, E., Choto, S., Alazbjee, A. A. A., Sapaev, I. B., Hussein, A. M., Yacin, Y., Alawadi, A. H. R., Alsalamy, A., Analysis of thermoelastic damping in a microbeam following a modified strain gradient theory and the Moore–Gibson–Thompson heat equation, Mechanics of Time-Dependent Materials, 28(4), 2024, 2367–2393. 
[39] Malikan, M., Eremeyev, V. A., On time-dependent nonlinear dynamic response of micro-elastic solids, International Journal of Engineering Science, 182, 2023, 103793. 
[40] Hill, R. M., Dissado, L. A., Viscoelastic response, Rheologica Acta, 24, 1985, 537–539. 
[41] Buhariwala, K. J., Hansen, J. S., Dynamics of viscoelastic structures, Rheologica Acta, 26(2), 1988, 220–227. 
[42] Bagley, R., On the equivalence of the Riemann–Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials, Fractional Calculus and Applied Analysis, 10(2), 2007, 123–126. 
[43] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. 
[44] Ezzat, M. A., Ezzat, S. M., Alduraibi, N. S., On size-dependent thermo-viscoelasticity theory for piezoelectric materials, Waves in Random and Complex Media, 2022, 1–23. 
[45] Lee, H. P., Dynamic response of a beam with intermediate point constraints subject to a moving load, Journal of Sound and Vibration, 171(3), 1994, 361–368. 
[46] Mishra, J. C., Samanta, S. C., Chakrabarty, A. K., Magneto-thermo-mechanical interaction in an aeolotropic viscoelastic cylinder permeated by magnetic field subjected to a periodic loading, International Journal of Engineering Science, 29(10), 1991, 1209–1216. 
[47] Allam, M. N., Elsibai, K. A., Abouelregal, A. E., Electro magneto-thermoelastic problem in a thick plate using Green and Naghdi theory, International Journal of Engineering Science, 47(5–6), 2009, 680–690. 
[48] Kim, K., Xu, X., Guo, J., Fan, D. L., Ultrahigh-speed rotating nanoelectromechanical system devices assembled from nanoscale building blocks, Nature Communications, 5(1), 2014, 3632. 
[49] Selvamani, R., Prabhakaran, T., Ebrahimi, F., Damping characteristics of nonlocal strain gradient waves in thermoviscoelastic graphene sheets subjected to nonlinear substrate effects, Physical Mesomechanics, 27(4), 2024, 461-471. 
[50] Tiwari, R., Kumar, R., Kumar, R., Analysis of magnetic field effect in micro-beam resonators at distinct boundary conditions, Waves in Random and Complex Media, 33(2), 2023, 312–328. 
[51] Barati, A., Norouzi, S., Nonlocal elasticity theory for static torsion of the bi-directional functionally graded microtube under magnetic field, Journal of Computational and Applied Mechanics, 51(1), 2020, 30–36. 
[52] Abouelregal, A. E., Elmasry, Y., Thermomagnetic modeling of a nonlocal viscoelastic half-space exposed to an internal heat source through a two-phase delay model, Waves in Random and Complex Media, 34(3), 2024, 1923–1944. 
[53] Abouelregal, A. E., Ahmad, H., Nofal, T. A., Abu-Zinadah, H., Thermo-viscoelastic fractional model of rotating nanobeams with variable thermal conductivity due to mechanical and thermal loads, Modern Physics Letters B, 35(18), 2021, 2150297. 
[54]  Zeeshan, A., Khan, M.I., Ellahi, R., Marin, M., Computational intelligence approach for optimising MHD Casson ternary hybrid nanofluid over the shrinking sheet with the effects of radiation, Applied Sciences, 13(17), 2023,  9510. 
[55] Malikan, M., Eremeyev, V. A., Effect of surface on the flexomagnetic response of ferroic composite nanostructures; nonlinear bending analysis, Composite Structures, 271, 2021, 114179. 
[56] Milic, P., Marinkovic, D., Klinge, S., Cojbasic, Z., Reissner-Mindlin based isogeometric finite element formulation for piezoelectric active laminated shells, Tehnicki Vjesnik, 30(2), 2023, 416-425.
[57] Selvamani, R., Rubine, L., Prabhakaran, T., Yaylaci, M.U.R.A.T., Free Vibration Analysis of a Functionally Graded Magneto-Piezo-Thermoelastic Ceramic-Metal Nanobeam Using Modified Nonlocal State-Space Strain Gradient Theory, Physical Mesomechanics, 28(2), 2025, 263-274. 
[58] Adams, R. D., Cawley, P., Pye, C. J., Stone, B. J., A vibration technique for non-destructively assessing the integrity of structures, Journal of Mechanical Engineering Science, 20(2), 1978, 93–100. 
[59] Diethelm, K., The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin, 2010. 
[60] Loya, J. A., Rubio, L., Fernández-Sáez, J., Natural frequencies for bending vibrations of Timoshenko cracked beams, Journal of Sound and Vibration, 290(3–5), 2006, 640–653. 
[61] Loya, J., López-Puente, J., Zaera, R., Fernández-Sáez, J., Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model, Journal of Applied Physics, 105(4), 2009, 043514.