Abstract
The present work attempts to develop the Lord–Shulman (LS) generalized thermoelasticity for flexoelectric materials. To do this, the energy equation and entropy inequality were developed so that all energy components were included as well. The relation between heat flux and temperature was considered as LS assumption and in the same manner as LS theory, and the final forms of constitutive relations and heat conduction equation were extracted. Then by putting the extracted constitutive relations into the dynamic equations of flexoelasticity, the final dynamic governing generalized equations of thermo-flexoelasticity based on LS thermoelasticity model were derived. The generalized thermo-flexoelasticity equations were extracted in the present work for the first time and can be applied to general flexoelectric materials. The derived thermo-flexoelasticity equations showed new couplings between polarization and temperature and also between heat conduction equation and polarization, in such a way that by putting the LS relaxation time equal to zero, the developed generalized thermo-flexoelasticity model was reduced to classical equations of flexoelasticity with thermal effects. As a case study, a one-dimensional flexoelectric layer was considered and the generalized governing thermo-flexoelasticity equations were derived. Finally, a consistent finite element approach was employed to solve the governing equations. The results under both temperature and traction shock loading were also presented. The numerical results showed that all thermal, elastic, and electrical parameters propagated as waves with finite speed.
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References
Maranganti, R., Sharma, N.D., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B 74(1), 014110 (2006)
Majdoub, M.S., Sharma, P., Cagin, T.: Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Rev. B 77, 125424 (2008)
Fu, J.Y., Zhu, W.Y., Li, N., Cross, L.E.: Experimental studies of the converse flexoelectric effect induced by the inhomogeneous electric field in a barium strontium titanate composition. J. Appl. Phys. 100(2), 024112 (2006). https://doi.org/10.1063/1.2219990
Ma, W.: A study of flexoelectric coupling associated internal electric field and stress in thin film ferroelectrics. Physica Status Solidi (b) 245(4), 761–768 (2008)
Hetnarski, R.B., Eslami, M.R., Gladwell, G.M.L.: Thermal stresses: advanced theory and applications (2009)
Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)
Othman, M.I., Fekry, M., Marin, M.: Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating. Struct. Eng. Mech. 73(6), 621–629 (2020). https://doi.org/10.12989/sem.2020.73.6.621
Abbas, I., Hobiny, A., Marin, M.: Photo-thermal interactions in a semi-conductor material with cylindrical cavities and variable thermal conductivity. J. Taibah Univ. Sci. 14, 1369–1376 (2020). https://doi.org/10.1080/16583655.2020.1824465
Abouelregal, A.E., Mohammad-Sedighi, H., Shirazi, A.H., Malikan, M., Eremeyev, V.A.: Computational analysis of an infinite magneto-thermoelastic solid periodically dispersed with varying heat flow based on non-local Moore–Gibson–Thompson approach. Contin. Mech. Thermodyn. 34(4), 1067–1085 (2022)
Singh, B.: On the theory of generalized thermoelasticity for piezoelectric materials. Appl. Math. Comput. 171, 398–405 (2005)
Jabbari, M., Yooshi, A.: Theory of generalized piezoporo thermoelasticity. J. Solid Mech. 4(4), 327–338 (2012)
Ma, Y., He, T.: Dynamic response of a generalized piezoelectric-thermoelastic problem under fractional order theory of thermoelasticity. Mech. Adv. Mater. Struct. 23(10), 1173–1180 (2016). https://doi.org/10.1080/15376494.2015.1068397
Ma, Y., He, T.: the transient response of a functionally graded piezoelectric rod subjected to a moving heat source under fractional order theory of thermoelasticity. Mech. Adv. Mater. Struct. 24(9), 789–796 (2017)
Taghizadeh, A., Kiani, Y.: Generalized thermoelasticity of a piezoelectric layer. J. Therm. Stress. 42, 863–873 (2019)
Marin, M., Abouelregal, A., Mohamed, B.: The theory of thermoelasticity with a memory-dependent dynamic response for a thermo-piezoelectric functionally graded rotating rod. (2023) https://doi.org/10.21203/rs.3.rs-2750537/v1
Samani, M.S.E., Beni, Y.T.: Size dependent thermo-mechanical buckling of the flexoelectric nanobeam. Mater. Res. Expr. 5(8), 085018 (2018). https://doi.org/10.1088/2053-1591/aad2ca
Jani, S.M.H., Kiani, Y.: Generalized thermo-electro-elasticity of a piezoelectric disk using Lord-Shulman theory. J. Therm. Stress. (2020). https://doi.org/10.1080/01495739.2020.1718044
Jani, S.M.H., Kiani, Y.: Symmetric thermo-electro-elastic response of piezoelectric hollow cylinder under thermal shock using lord-shulman theory. Int. J. Struct. Stab. Dyn. (2020). https://doi.org/10.1142/S0219455420500595
Bazarra, N., Fernández, J., Quintanilla, R.: Lord-Shulman thermoelasticity with microtemperatures. Appl. Math. Optim. (2020). https://doi.org/10.1007/s00245-020-09691-2
Ghobadi, A., Golestanian, H., Beni, Y.T., Żur, K.K.: On the size-dependent nonlinear thermo-electro-mechanical free vibration analysis of functionally graded flexoelectric nano-plate. Commun. Nonlinear Sci. Numer. Simul. 95, 105585 (2021). https://doi.org/10.1016/j.cnsns.2020.105585
Beni, Y.T.: Size dependent torsional electro-mechanical analysis of flexoelectric micro/nanotubes. Eur. J. Mech. A/Solids 95, 104648 (2022)
Babadi, A.F., Beni, Y.T., Żur, K.K.: On the flexoelectric effect on size-dependent static and free vibration responses of functionally graded piezo-flexoelectric cylindrical shells. Thin-Walled Struct. 179, 109699 (2022)
Ragab, M., Abo-Dahab, S.M., Abouelregal, A.E., Kilany, A.A.: A thermoelastic piezoelectric fixed rod exposed to an axial moving heat source via a dual-phase-lag model. Complexity 2021, 1–11 (2021). https://doi.org/10.1155/2021/5547566
Hosseini, S.M.H., Beni, Y.T.: Free vibration analysis of rotating piezoelectric/flexoelectric microbeams. Appl. Phys. A 129, 330 (2023). https://doi.org/10.1007/s00339-023-06615-z
Hosseini, S.M.H., Beni, Y.T.: On the vibration of size-dependent rotating flexoelectric microbeams. Appl. Phys. A 130, 58 (2024). https://doi.org/10.1007/s00339-023-07207-7
Ghobadi, A., Beni, Y.T., Golestanian, H.: Nonlinear thermo-electromechanical vibration analysis of size-dependent functionally graded flexoelectric nano-plate exposed magnetic field. Arch. Appl. Mech. 90, 2025–2070 (2020). https://doi.org/10.1007/s00419-020-01708-0
Abouelregal, A.E., Askar, S.S., Marin, M., Mohamed, B.: The theory of thermoelasticity with a memory-dependent dynamic response for a thermo-piezoelectric functionally graded rotating rod. Sci. Rep. 13(1), 9052 (2023). https://doi.org/10.1038/s41598-023-36371-2
Abdoul-Anziz, H., Auffray, N., Desmorat, B.: Symmetry classes and matrix representations of the 2D flexoelectric law. Symmetry 12(4), 674 (2020)
Guinovart-Sanjuán, D., Mohapatra, R., Rodríguez-Ramos, R., Espinosa-Almeyda, Y., Rodríguez-Bermúdez, P.: Influence of nonlocal elasticity tensor and flexoelectricity in a rod: an asymptotic homogenization approach. Int. J. Eng. Sci. 193, 103960 (2023). https://doi.org/10.1016/j.ijengsci.2023.103960
Yurkov, A.S., Yudin, P.V.: Continuum model for converse flexoelectricity in a thin plate. Int. J. Eng. Sci. 182, 103771 (2023)
Malikan, M., Dastjerdi, S., Eremeyev, V., Sedighi, H.M.: On a 3D material modelling of smart nanocomposite structures. Int. J. Eng. Sci. 193(2023), 103966 (2023)
Malikan, M., Eremeyev, V.A.: On a flexomagnetic behavior of composite structures. Int. J. Eng. Sci. 175, 103671 (2022)
Malikan, M., Eremeyev, V.A.: On dynamic modeling of piezomagnetic/flexomagnetic microstructures based on Lord-Shulman thermoelastic model. Arch. Appl. Mech. 93, 181–196 (2023). https://doi.org/10.1007/s00419-022-02149-7
Beni, Y.T.: Size-dependent electro-thermal buckling analysis of flexoelectric microbeams. Int. J. Struct. Stab. Dyn. (2023). https://doi.org/10.1142/S0219455424500937
Tadi Beni, Y.: Size dependent coupled electromechanical torsional analysis of porous FG flexoelectric micro/nanotubes. Mech. Syst. Signal Process. 178(2022), 109281 (2022)
Alihemmati, J., Beni, Y.T.: Generalized thermoelasticity of microstructures: Lord-Shulman theory with modified strain gradient theory. Mech. Mater. 172, 104412 (2022)
Li, A., Zhou, S., Qi, L., Chen, X.: A reformulated flexoelectric theory for isotropic dielectrics. J. Phys. D Appl. Phys. 48(46), 465502 (2015)
Truesdell, C., Noll, W.: The non-linear field theories of mechanics, Encyclopedia of Physics, Vol. III/3, Springer, Berlin (1965)
Awad, E., El Dhaba, A.R., Fayik, M.: A unified model for the dynamical flexoelectric effect in isotropic dielectric materials. Eur. J. Mech. A. Solids 95, 104618 (2022)
Segerlind, L.J.: Applied Finite Element Analysis. Wiley (1991)
Rao, S.S.: Mechanical Vibrations, 6th edition, Pearson, (2016)
Alihemmati, J., Tadi Beni, Y., Kiani, Y.: Application of chebyshev collocation method to unified generalized thermoelasticity of a finite domain. J. Therm. Stress. 44(5), 547–565 (2021). https://doi.org/10.1080/01495739.2020.1867941
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Kheibari, F., Beni, Y.T. & Golestanian, H. On the generalized flexothermoelasticity of a microlayer. Acta Mech (2024). https://doi.org/10.1007/s00707-024-03884-4
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DOI: https://doi.org/10.1007/s00707-024-03884-4