Thermo-viscoelastic transversely isotropic rotating hollow cylinder based on three-phase lag thermoelastic model and fractional Kelvin-Voigt type. (English) Zbl 1494.74010
Summary: In the presented work, a modified fractional Kelvin-Voigt model (KVM) is introduced to explain the time-dependent behavior of viscoelastic materials based on thermo-viscoelasticity theory and fractional calculus. The fractional KVM involves mechanical viscoelastic parameters and differential operators of fractional orders. In addition, the governing system equations are constructed within the framework of generalized thermoelasticity with three different delay times (TPL). The resulting derived model is used to study the thermo-viscoelastic interactions in a rotating transversely hollow cylinder under the influence of thermal loads and due to a magnetic field of constant intensity. The system of equations is solved, and solutions for non-dimensional physical quantities such as temperature, radial displacement, and thermal stresses are derived by applying the Laplace transform technique. To assess the influences of fractional order, relaxation factors and delay time parameters using different models of thermo-viscoelasticity in the presence and absence of fractional orders, several comparisons are made in tables and figures. Finally, according to the discussion of the numerical results, the prominent role of these parameters in the propagation behavior of mechanical and thermal waves is revealed.
MSC:
| 74D05 | Linear constitutive equations for materials with memory |
| 74F05 | Thermal effects in solid mechanics |
| 74S40 | Applications of fractional calculus in solid mechanics |
| 74F15 | Electromagnetic effects in solid mechanics |
Keywords:
generalized thermoelasticity; magnetic field; radial displacement; thermal stress; Laplace transform; delay time parameterReferences:
| [1] | Lord, HW; Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids., 15, 299-309 (1967) · Zbl 0156.22702 · doi:10.1016/0022-5096(67)90024-5 |
| [2] | Green, AE; Lindsay, KA, Thermoelasticity, J. Elast., 2, 1-7 (1972) · Zbl 0775.73063 · doi:10.1007/BF00045689 |
| [3] | Green, AE; Naghdi, PM, A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. A. Math. Phys. Eng. Sci., 432, 171-194 (1991) · Zbl 0726.73004 |
| [4] | Green, AE; Naghdi, PM, On undamped heat waves in an elastic solid, J. Therm. Stress., 15, 253-264 (1992) · doi:10.1080/01495739208946136 |
| [5] | Green, AE; Naghdi, PM, Thermoelasticity without energy dissipation, J. Elast., 31, 189-208 (1993) · Zbl 0784.73009 · doi:10.1007/BF00044969 |
| [6] | Tzou, DY, A unified field approach for heat conduction from macro- to micro-scales, J. Heat Transf., 117, 8-16 (1995) · doi:10.1115/1.2822329 |
| [7] | Chandrasekharaiah, DS, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev., 51, 705-729 (1998) · doi:10.1115/1.3098984 |
| [8] | Roy Choudhuri, SK, On a thermoelastic three-phase-lag model, J. Therm. Stress., 30, 231-238 (2007) · doi:10.1080/01495730601130919 |
| [9] | Quintanilla, R.; Racke, R., A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transf., 51, 24-29 (2008) · Zbl 1140.80363 · doi:10.1016/j.ijheatmasstransfer.2007.04.045 |
| [10] | Abouelregal, AE, Two-temperature thermoelastic model without energy dissipation including higher order time-derivatives and two phase-lags, Mater. Res. Exp., 6, 11, 116535 (2019) · doi:10.1088/2053-1591/ab447f |
| [11] | Abouelregal, AE, A novel model of nonlocal thermoelasticity with time derivatives of higher order, Math. Method Appl. Sci., 43, 6746-6760 (2020) · Zbl 1454.35374 · doi:10.1002/mma.6416 |
| [12] | Abouelregal, AE, Generalized mathematical novel model of thermoelastic diffusion with four phase lags and higher-order time derivative, Eur. Phys. J. Plus, 135, 2 (2020) · doi:10.1140/epjp/s13360-020-00282-2 |
| [13] | Abouelregal, AE, A novel generalized thermoelasticity with higher-order time-derivatives and three-phase lags, Multidisc. Model Mater. Struct., 16, 4, 689-711 (2019) · doi:10.1108/MMMS-07-2019-0138 |
| [14] | Tiwari, R.; Mukhopadhyay, S., Analysis of wave propagation in presence of a continuous line heat source under heat transfer with memory dependent derivatives, Math. Mech. Solids, 23, 5, 820-834 (2017) · Zbl 1395.74043 · doi:10.1177/1081286517692020 |
| [15] | Tiwari, R.; Mukhopadhyay, S., On harmonic plane wave propagation under fractional order thermoelasticity: an analysis of fractional order heat conduction equation, Math. Mech. Solids, 22, 4, 782-797 (2015) · Zbl 1371.74142 · doi:10.1177/1081286515612528 |
| [16] | Tiwari, R.; Kumar, R.; Abouelregal, AE, Analysis of magneto-thermoelastic problem in piezo-elastic medium under the theory of non-local memory dependent heat conduction with three phase lags, Mech. Time Depend. Mater. (2021) · doi:10.1007/s11043-021-09487-z |
| [17] | Tiwari, R.; Kumar, R., Investigation of thermal excitation induced by laser pulses and thermal shock in the half space medium with variable thermal conductivity, Wave Random Compl. (2020) · Zbl 1507.74109 · doi:10.1080/17455030.2020.1851067 |
| [18] | Tiwari, R.; Misra, JC, Magneto-thermoelastic excitation induced by a thermal shock: a study under the purview of three phase lag theory, Wave Random Compl. (2020) · Zbl 1505.74043 · doi:10.1080/17455030.2020.1800861 |
| [19] | Tiwari, R.; Kumar, R., Analysis of Plane wave propagation under the purview of three phase lag theory of thermoelasticity with non-local effect, Eur. J. Mech. A/Solids (2021) · Zbl 1484.74039 · doi:10.1016/j.euromechsol.2021.104235 |
| [20] | Abouelregal, AE, Thermo-viscoelastic properties in a non-simple three dimensional material based on fractional derivative Kelvin-Voigt model, Indian J. Phys. (2021) · doi:10.1007/s12648-020-01979-x |
| [21] | Biswas, S.; Mukhopadhyay, B.; Shaw, S., Effect of rotation in magneto-thermoelastic transversely isotropic hollow cylinder with threephase-lag model, Mech. Based Des. Struct. Mach. (2019) · doi:10.1080/15397734.2018.1545587 |
| [22] | Sherief, HH; Raslan, WE, Thermoelastic interactions without energy dissipation in an unbounded body with a cylindrical cavity, J. Therm. Stress., 39, 3, 326-332 (2016) · doi:10.1080/01495739.2015.1125651 |
| [23] | Abouelregal, AE; Abo-Dahab, SM, A two-dimensional problem of a mode-I crack in a rotating fibre-reinforced isotropic thermoelastic medium under dual-phase-lag model, Sadhana, 43, 1, 13 (2018) · Zbl 1391.74224 · doi:10.1007/s12046-017-0769-7 |
| [24] | Othman, MIA; Abbas, IA, Generalized thermoelasticity of thermal shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation, Int. J. Thermophys., 33, 5, 913-923 (2012) · doi:10.1007/s10765-012-1202-4 |
| [25] | Maugin, GA, Continuum mechanics of electromagnetic solids (1988), Amsterdam: Elsevier, Amsterdam · Zbl 0652.73002 |
| [26] | Eringen, AC; Maugin, GA, Electrodynamics of continua I, foundations and solid media (1990), New York: Springer, New York · doi:10.1007/978-1-4612-3226-1 |
| [27] | Othman, MIA; Elmaklizi, YD; Ahmed, EAA, Effect of magnetic field on piezo-thermoelastic medium with three theories, Results Phys., 7, 3361-3368 (2017) · doi:10.1016/j.rinp.2017.08.058 |
| [28] | Paria, G., Magneto-elasticity and magneto-thermoelasticity, Adv. Appl. Mech., 10, 73-112 (1966) · doi:10.1016/S0065-2156(08)70394-6 |
| [29] | Abouelregal, AE, Modified fractional photo-thermoelastic model for a rotating semiconductor half-space subjected to a magnetic field, SILICON, 12, 2837-2850 (2020) · doi:10.1007/s12633-020-00380-x |
| [30] | Tiwari, R.; Mukhopadhyay, S., On electro-magneto-thermoelastic plane waves under Green- Naghdi theory of thermoelasticity-II, J. Therm. Stress., 40, 8, 1040-1062 (2017) · doi:10.1080/01495739.2017.1307094 |
| [31] | Xiong, C., Guo, Y.: Effect of variable properties and moving heat source on magnetothermoelastic problem under fractional order thermoelasticity. Adv. Mater. Sci. Eng. (2016). |
| [32] | Abouelregal, AE; Mohammad-Sedighi, H.; Shirazi, AH; Malikan, M.; Eremeyev, VA, Computational analysis of an infinite magneto-thermoelastic solid periodically dispersed with varying heat flow based on non-local Moore-Gibson-Thompson approach, Con. Mech. Thermodyn. (2021) · doi:10.1007/s00161-021-00998-1 |
| [33] | Abouelregal, AE; Ahmad, H., Response of thermoviscoelastic microbeams affected by the heating of laser pulse under thermal and magnetic fields, Phys. Scr., 95, 125501 (2020) · doi:10.1088/1402-4896/abc03d |
| [34] | Mondal, S., Memory response in a magneto-thermoelastic rod with moving heat source based on Eringen’s nonlocal theory under dual-phase lag heat conduction, Int. J. Comp. Meth., 17, 9, 1950072 (2020) · Zbl 07342703 · doi:10.1142/S0219876219500725 |
| [35] | Abouelregal, AE; Abo-Dahab, SM, Dual phase lag model on magneto-thermoelasticity infinite non-homogeneous solid having a spherical Cavity, J. Therm. Stress., 35, 9, 820-841 (2012) · doi:10.1080/01495739.2012.697838 |
| [36] | Zuo, P.; Cheng, Y.; Wang, Z.; Dou, X.; Liu, J., Tension and bending of the particle raft driven by a magnet Author links open overlay, Colloid Interf. Sci. Commun., 45, 100528 (2021) · doi:10.1016/j.colcom.2021.100528 |
| [37] | Ingman, D.; Suzdalnitsky, J., Response of viscoelastic plate to impact, ASME J. Vib. Acous., 130, 011010 (2008) · doi:10.1115/1.2731416 |
| [38] | Katsikadelis, JT, Nonlinear dynamic analysis of viscoelastic membranes described with fractional differential models, J. Theor. Appl. Mech., 50, 3, 743-753 (2012) |
| [39] | Farno, E.; Baudez, JC; Parthasarathy, R.; Eshtiaghi, N., Rheological characterisation of thermally-treated anaerobic digested sludge: impact of temperature and thermal history, Water Res., 56, 156-161 (2014) · doi:10.1016/j.watres.2014.02.048 |
| [40] | Nicolle, S.; Vezin, P.; Palierne, JF, A strain-hardening bi-power law for the nonlinear behaviour of biological soft tissues, J. Biomech., 43, 927-932 (2010) · doi:10.1016/j.jbiomech.2009.11.002 |
| [41] | Ng Trevor, SK; McKinley, GH, Power law gels at finite strains: the nonlinear rheology of gluten gels, J. Rheol., 52, 417-449 (2008) · doi:10.1122/1.2828018 |
| [42] | Jóźwiak, B.; Orczykowska, M.; Dziubiński, M., Fractional generalizations of Maxwell and Kelvin-Voigt models for biopolymer characterization, Pub. Lib. Sci. One, 10, e0143090 (2015) |
| [43] | Farno, E.; Baudez, JC; Eshtiaghi, N., Comparison between classical Kelvin-Voigt and fractional derivative Kelvin-Voigt models in prediction of linear viscoelastic behaviour of waste activated sludge, Sci. Total Environ., 1, 613-614 (2018) |
| [44] | Bonfanti, A.; Kaplan, JL; Charras, G.; Kabla, AJ, Fractional viscoelastic models for power-law materials, Soft Matter, 16, 6002-6020 (2020) · doi:10.1039/D0SM00354A |
| [45] | Schiessel, H.; Metzler, R.; Blumen, A.; Nonnenmacher, T., Generalized viscoelastic models: their fractional equations with solutions, J. Phys. A Math. Gen., 28, 6567 (1995) · Zbl 0921.73154 · doi:10.1088/0305-4470/28/23/012 |
| [46] | Gemant, A., A method of analyzing experimental results obtained from elasto-viscous bodies, Physics, 7, 8, 311-317 (1936) · doi:10.1063/1.1745400 |
| [47] | Bagley, RL; Torvik, PJ, Fractional calculus: a different approach to the analysis of viscoelastically damped structures, Am. Inst. Aeronaut. Astronaut. J., 21, 5, 741-748 (1983) · Zbl 0514.73048 · doi:10.2514/3.8142 |
| [48] | Bagley, RL; Torvik, PJ, On the fractional calculus model of viscoelastic behaviour, J. Rheol., 30, 1, 133-155 (1986) · Zbl 0613.73034 · doi:10.1122/1.549887 |
| [49] | Zhang, C.; Zhu, H.; Shi, B.; Liu, L., Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation, J. Rock Mech. Geotech. Eng., 6, 4, 373-379 (2014) · doi:10.1016/j.jrmge.2014.04.007 |
| [50] | Dikmen, U., Modeling of seismic wave attenuation in soil structures using fractional derivative scheme, J. Balkan Geophysical. Soc., 8, 4, 175e88 (2005) |
| [51] | Zhang, X.; Li, Z.; Wang, X.; Yu, J., The fractional Kelvin-Voigt model for circumferential guided waves in a viscoelastic FGM hollow cylinder, Appl. Math. Model, 89, 299-313 (2021) · Zbl 1485.74044 · doi:10.1016/j.apm.2020.06.077 |
| [52] | Ren, D.; Shen, X.; Li, C., The fractional Kelvin-Voigt model for Rayleigh surface waves in viscoelastic FGM infinite half space, Mech. Res. Commun., 87, 53-58 (2017) · doi:10.1016/j.mechrescom.2017.12.004 |
| [53] | Xue, Z.; Liu, J.; Tian, X.; Yu, Y., Thermal shock fracture associated with a unified fractional heat conduction, Eur. J. Mech. A/Solids, 85 (2021) · Zbl 1476.74137 · doi:10.1016/j.euromechsol.2020.104129 |
| [54] | Xue, Z.; Tian, X.; Liu, J., Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model, Appl. Math. Mod., 80, 840-858 (2020) · Zbl 1481.74681 · doi:10.1016/j.apm.2019.11.021 |
| [55] | Povstenko, YZ, Fractional heat conduction equation and associated thermal stress, J. Therm. Stress., 28, 83-102 (2005) · doi:10.1080/014957390523741 |
| [56] | Povstenko, YZ, Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses, Mech. Res. Commun., 37, 4, 436-440 (2010) · Zbl 1272.74140 · doi:10.1016/j.mechrescom.2010.04.006 |
| [57] | Sherief, H.; Sayed, AE; Latief, AE, Fractional order theory of thermoelasticity, Int. J. Solids Struct., 47, 269-275 (2010) · Zbl 1183.74051 · doi:10.1016/j.ijsolstr.2009.09.034 |
| [58] | Jumarie, G., Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio, Comput. Math. Appl., 59, 3, 1142-1164 (2010) · Zbl 1189.91230 · doi:10.1016/j.camwa.2009.05.015 |
| [59] | Ezzat, MA, Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Phys. B Cond. Mat., 406, 1, 30-35 (2011) · doi:10.1016/j.physb.2010.10.005 |
| [60] | Abouelregal, AE, A modified law of heat conduction of thermoelasticity with fractional derivative and relaxation time, J. Mol. Eng. Mat., 08, 2050003 (2020) · doi:10.1142/S2251237320500033 |
| [61] | Abouelregal, AE, Fractional derivative Moore-Gibson-Thompson heat equation without singular kernel for a thermoelastic medium with a cylindrical hole and variable properties, J. Appl. Math. Mech. (2021) · Zbl 07815146 · doi:10.1002/zamm.202000327 |
| [62] | Abouelregal, AE, Modified fractional thermoelasticity model with multi-relaxation times of higher order: application to spherical cavity exposed to a harmonic varying heat, Waves Rand. Comp. Media, 31, 5, 812-832 (2021) · Zbl 1497.74009 · doi:10.1080/17455030.2019.1628320 |
| [63] | Mainardi, F.; Spada, G., Creep, relaxation and viscosity properties for basic fractional models in rheology, Eur. Phys. J. Spec. Top., 193, 133-160 (2011) · doi:10.1140/epjst/e2011-01387-1 |
| [64] | Miller, KS; Ross, B., An introduction to the fractional calculus and fractional differential equations (1993), Hoboken: Wiley, Hoboken · Zbl 0789.26002 |
| [65] | Honig, G.; Hirdes, U., A method for the numerical inversion of the Laplace transform, J. Comput. Appl. Math., 10, 13-132 (1984) · Zbl 0535.65090 · doi:10.1016/0377-0427(84)90075-X |
| [66] | Farno, E.; Baudez, JC; Parthasarathy, R.; Eshtiaghi, N., The viscoelastic characterisation of thermally-treated waste activated sludge, Chem. Eng. J., 304, 362-368 (2016) · doi:10.1016/j.cej.2016.06.082 |
| [67] | Zhu, HH; Liu, LC; Pei, HF; Shi, B., Settlement analysis of viscoelastic foundation under vertical line load using a fractional Kelvin-Voigt model, Geomech. Eng., 4, 1, 67-78 (2012) · doi:10.12989/gae.2012.4.1.067 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
