##
**A mixed two-phase stress/strain driven elasticity: in applications on static bending, vibration analysis and wave propagation.**
*(English)*
Zbl 1493.74012

Summary: In recent years, there are numerous papers that deal with two-phase strain-driven model, including softening effect, and two-phase stress-driven model, with stiffening effect, and also the nonlocal strain gradient model, with both the softening and stiffening effects, to seek the impacts of length scales on the mechanics of structures in small scales. The current paper is a novel well-posed mixture model to cover all previous theories through employing various proposed small scales. Concurrent accounting of stiffening and softening effects is performed through employing a combination of the two-phase strain-driven elasticity with the two-phase stress-driven elasticity, with two additional local phase fraction factors and two various type of nonlocal parameters. Accordingly, in this model the total strain or stress at a certain point is regarded as a function of the stress or strain of all neighboring points in addition to the point of interest. Compatibility between integral and differential relations without any conflict of restrictions is doable through considering constitutive boundary conditions as key point of this match. To demonstrate its application values, the proposed mixture model as an efficient theoretical tool is hired for static bending, vibration and wave propagation in a mixed two-phase stress/strain driven elasticity system and the new essential relations are developed through examples for static bending, free vibration and wave propagating in Euler-Bernoulli nanobeams. The results are obtained by proposing an efficient exact solution, and the integrity as well as reliability of the present constitutive differential relations are evaluated through some validation studies. The output results in the framework of new suggested mixture model address some points about static bending, free vibration and wave propagation that is more comprehensive than the previous results of the contemporary continuum theories. Thus, based on the observations, this mixed two-phase stress/strain driven elasticity could be cover a widespread range of dynamic response in the nano-systems that the contemporary continuum theories can be only investigated some aspects of those problems.

### MSC:

74B99 | Elastic materials |

74A20 | Theory of constitutive functions in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74H45 | Vibrations in dynamical problems in solid mechanics |

74J05 | Linear waves in solid mechanics |

74M25 | Micromechanics of solids |

### Keywords:

two-phase stress-driven local/nonlocal elasticity; two-phase strain-driven local/nonlocal elasticity; integral/differential constitutive law; Euler-Bernoulli nanobeam; harmonic wave
PDF
BibTeX
XML
Cite

\textit{S. Behdad} and \textit{M. Arefi}, Eur. J. Mech., A, Solids 94, Article ID 104558, 15 p. (2022; Zbl 1493.74012)

Full Text:
DOI

### References:

[1] | Abdelrahman, A. A.; Esen, I.; Özarpa, C.; Eltaher, M. A., Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory, Appl. Math. Model. (2021) · Zbl 1481.74423 |

[2] | Alam, M.; Mishra, S. K., Nonlinear vibration of nonlocal strain gradient functionally graded beam on nonlinear compliant substrate, Compos. Struct., 263, 113447 (2021) |

[3] | Aria, A.; Friswell, M.; Rabczuk, T., Thermal vibration analysis of cracked nanobeams embedded in an elastic matrix using finite element analysis, Compos. Struct., 212, 118-128 (2019) |

[4] | Aria, A. I.; Rabczuk, T.; Friswell, M. I., A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams, Eur. J. Mech. Solid. (2019) · Zbl 1473.74033 |

[5] | Awrejcewicz, J.; Kudra, G.; Mazur, O., Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory, Nonlinear Dynam., 1-21 (2021) |

[6] | Barretta, R.; de Sciarra, F. M., Variational nonlocal gradient elasticity for nano-beams, Int. J. Eng. Sci., 143, 73-91 (2019) · Zbl 1476.74008 |

[7] | Barretta, R.; Čanađija, M.; Luciano, R.; de Sciarra, F. M., Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams, Int. J. Eng. Sci., 126, 53-67 (2018) · Zbl 1423.74457 |

[8] | Barretta, R.; Faghidian, S. A.; Luciano, R., Longitudinal vibrations of nano-rods by stress-driven integral elasticity, Mech. Adv. Mater. Struct., 1-9 (2018) |

[9] | Barretta, R.; Faghidian, S. A.; Luciano, R.; Medaglia, C.; Penna, R., Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models, Compos. B Eng., 154, 20-32 (2018) |

[10] | Barretta, R.; Faghidian, S. A.; Luciano, R.; Medaglia, C.; Penna, R., Stress-driven two-phase integral elasticity for torsion of nano-beams, Compos. B Eng., 145, 62-69 (2018) |

[11] | Barretta, R.; Caporale, A.; Faghidian, S. A.; Luciano, R.; de Sciarra, F. M.; Medaglia, C. M., A stress-driven local-nonlocal mixture model for Timoshenko nano-beams, Compos. B Eng., 164, 590-598 (2019) |

[12] | Behdad, S.; Fakher, M.; Hosseini-Hashemi, S., Dynamic stability and vibration of two-phase local/nonlocal VFGP nanobeams incorporating surface effects and different boundary conditions, Mech. Mater., 153, 103633 (2021) |

[13] | Behdad, S.; Fakher, M.; Naderi, A.; Hosseini-Hashemi, S., Vibrations of defected local/nonlocal nanobeams surrounded with two-phase Winkler-Pasternak medium: non-classic compatibility conditions and exact solution, Waves Random Complex Media, 1-36 (2021) |

[14] | Challamel, N.; Wang, C., The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology, 19, 345703 (2008) |

[15] | Challamel, N.; Zhang, Z.; Wang, C.; Reddy, J.; Wang, Q.; Michelitsch, T., On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach, Arch. Appl. Mech., 84, 1275-1292 (2014) · Zbl 1341.74094 |

[16] | Chu, Y.-M.; Nazir, U.; Sohail, M.; Selim, M. M.; Lee, J-R., Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach, Fractal Fract., 5, 3, Article 119 pp. (2021) |

[17] | Chu, Y.-M.; Shankaralingappa, B. M.; Gireesha, B. J.; Alzahrani, F.; Ijaz Khan, M.; Khan, S. U., Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano-material surface, Appl. Math. Comput. (2021) · Zbl 07483705 |

[18] | Darban, H.; Fabbrocino, F.; Feo, L.; Luciano, R., Size-dependent buckling analysis of nanobeams resting on two-parameter elastic foundation through stress-driven nonlocal elasticity model, Mech. Adv. Mater. Struct., 1-9 (2020) |

[19] | Eringen, A. C., Nonlocal Continuum Field Theories (2002), Springer Science & Business Media · Zbl 1023.74003 |

[20] | Eringen, A. C.; Edelen, D., On nonlocal elasticity, Int. J. Eng. Sci., 10, 233-248 (1972) · Zbl 0247.73005 |

[21] | Fakher, M.; Hosseini-Hashemi, S., Bending and free vibration analysis of nanobeams by differential and integral forms of nonlocal strain gradient with Rayleigh-Ritz method, Mater. Res. Express, 4, 125025 (2017) |

[22] | Fakher, M.; Hosseini-Hashemi, S., Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution, Eng. Comput., 1-15 (2020) |

[23] | Fakher, M.; Hosseini-Hashemi, S., Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method, J. Vib. Control (2020), 1077546320927619 |

[24] | Fakher, M.; Hosseini-Hashemi, S., On the vibration of nanobeams with consistent two-phase nonlocal strain gradient theory: exact solution and integral nonlocal finite-element model, Eng. Comput., 1-24 (2020) |

[25] | Fakher, M.; Rahmanian, S.; Hosseini-Hashemi, S., On the carbon nanotube mass nanosensor by integral form of nonlocal elasticity, Int. J. Mech. Sci., 150, 445-457 (2019) |

[26] | Fakher, M.; Behdad, S.; Hosseini-Hashemi, S., Vibration analysis of stress-driven nonlocal integral model of viscoelastic axially FG nanobeams, Eur. Phys. J. Plus, 135, 1-21 (2020) |

[27] | Fakher, M.; Behdad, S.; Naderi, A.; Hosseini-Hashemi, S., Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium, Int. J. Mech. Sci., 171, 105381 (2020) |

[28] | Fang, J.; Zheng, S.; Xiao, J.; Zhang, X., Vibration and thermal buckling analysis of rotating nonlocal functionally graded nanobeams in thermal environment, Aero. Sci. Technol., 106, 106146 (2020) |

[29] | Farajpour, M.; Shahidi, A.; Farajpour, A., Elastic waves in fluid-conveying carbon nanotubes under magneto-hygro-mechanical loads via a two-phase local/nonlocal mixture model, Mater. Res. Express, 6, Article 0850a8 pp. (2019) |

[30] | Fernández-Sáez, J.; Zaera, R., Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory, Int. J. Eng. Sci., 119, 232-248 (2017) · Zbl 1423.74476 |

[31] | Fernández-Sáez, J.; Zaera, R.; Loya, J.; Reddy, J., Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved, Int. J. Eng. Sci., 99, 107-116 (2016) · Zbl 1423.74477 |

[32] | Gao, T.; Li, C.; Yang, M., Mechanics analysis and predictive force models for the single-diamond grain grinding of carbon fiber reinforced polymers using CNT nano-lubricant, J. Mater. Proc. Technol. 2021, 290, 116976 (2021) |

[33] | Guo, S.; Li, C.; Zhang, Y., Experimental evaluation of the lubrication performance of mixtures of castor oil with other vegetable oils in MQL grinding of nickel-based alloy, J. Clean. Prod., 140, 3, 1060-1076 (2017) |

[34] | Hosseini-Hashemi, S.; Behdad, S.; Fakher, M., Vibration analysis of two-phase local/nonlocal viscoelastic nanobeams with surface effects, Eur. Phys. J. Plus, 135, 190 (2020) |

[35] | Huang, Y.; Fu, J.; Liu, A., Dynamic instability of Euler-Bernoulli nanobeams subject to parametric excitation, Compos. B Eng., 164, 226-234 (2019) |

[36] | Iqbal, M. A.; Wang, Y.; Miah, M. M.; Osman, M. S., Study on Date-Jimbo-Kashiwara-Miwa Equation with Conformable Derivative Dependent on Time Parameter to Find the Exact Dynamic Wave Solutions,, Fractal Fract, 6, 1, Article 4 pp. (2022), In press |

[37] | Jalaei, M.; Civalek, Ӧ., On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam, Int. J. Eng. Sci., 143, 14-32 (2019) · Zbl 1476.82018 |

[38] | Khaniki, H. B., On vibrations of nanobeam systems, Int. J. Eng. Sci., 124, 85-103 (2018) · Zbl 1423.74396 |

[39] | Khaniki, H. B., On vibrations of FG nanobeams, Int. J. Eng. Sci., 135, 23-36 (2019) · Zbl 1423.74397 |

[40] | Khodabakhshi, P.; Reddy, J., A unified integro-differential nonlocal model, Int. J. Eng. Sci., 95, 60-75 (2015) · Zbl 1423.74133 |

[41] | Kiani, K.; Żur, K. K., Vibrations of double-nanorod-systems with defects using nonlocal-integral-surface energy-based formulations, Compos. Struct., 256, 113028 (2020) |

[42] | Kröner, E., Elasticity theory of materials with long range cohesive forces, Int. J. Solid Struct., 3, 731-742 (1967) · Zbl 0163.19402 |

[43] | Krumhansl, J., Some Considerations of the Relation between Solid State Physics and Generalized Continuum Mechanics. Mechanics of Generalized Continua, 298-311 (1968), Springer · Zbl 0188.58902 |

[44] | Kunin, I., The Theory of Elastic Media with Microstructure and the Theory of Dislocations. Mechanics of Generalized Continua, 321-329 (1968), Springer |

[45] | Li, L.; Hu, Y., Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Comput. Mater. Sci., 112, 282-288 (2016) |

[46] | Li, X.; Li, L.; Hu, Y.; Ding, Z.; Deng, W., Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Compos. Struct., 165, 250-265 (2017) |

[47] | Li, B.; Li, C.; Zhang, Y., Heat transfer performance of MQL grinding with different nanofluids for Ni-based alloys using vegetable oil, J. Clean. Prod., 154, 1-11 (2017) |

[48] | Lim, C.; Zhang, G.; Reddy, J., A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, J. Mech. Phys. Solid., 78, 298-313 (2015) · Zbl 1349.74016 |

[49] | Liu, X.; Zhang, G.; Li, J.; Shi, G.; Zhou, M.; Huang, B.; Yang, W., Deep learning for Feynman’s path integral in strong-field time-dependent dynamics, Phys. Rev. Lett., 124, 11, 113202 (2020) |

[50] | Mahmoudpour, E.; Hosseini-Hashemi, S.; Faghidian, S., Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model, Appl. Math. Model., 57, 302-315 (2018) · Zbl 1480.74118 |

[51] | Martin, O., Nonlocal effects on the dynamic analysis of a viscoelastic nanobeam using a fractional Zener model, Appl. Math. Model., 73, 637-650 (2019) · Zbl 1481.74455 |

[52] | Mousavi, A. A.; Zhang, C.; Masri, S. F.; Gholipour, G., Structural damage detection method based on the complete ensemble empirical mode decomposition with adaptive noise: a model steel truss bridge case study, Struct. Health Monit., 84049609 (2021) |

[53] | Naderi, A.; Behdad, S.; Fakher, M.; Hosseini-Hashemi, S., Vibration analysis of mass nanosensors with considering the axial-flexural coupling based on the two-phase local/nonlocal elasticity, Mech. Syst. Signal Process., 145, 106931 (2020) |

[54] | Naderi, A.; Fakher, M.; Hosseini-Hashemi, S., On the local/nonlocal piezoelectric nanobeams: vibration, buckling, and energy harvesting, Mech. Syst. Signal Process., 151, 107432 (2021) |

[55] | Nazeer, M.; Hussain, F.; Ijaz Khan, M., Theoretical study of MHD electro-osmotically flow of third-gradefluid in micro channel, Appl. Math. Comput., (2021) |

[56] | Penna, R.; Feo, L.; Lovisi, G., Hygro-thermal bending behavior of porous FG nano-beams via local/nonlocal strain and stress gradient theories of elasticity, Compos. Struct., 263, 113627 (2021) |

[57] | Polyanin, A. D.; Manzhirov, A. V., Handbook of Integral Equations (1998), CRC press · Zbl 0896.45001 |

[58] | Qi, X.; Zhang, D.; Guo, H.; Chen, Y., The fairing arrangement for vortex induced vibration suppression effect in soliton current, J. Coast Res., 103, 293-298 (2020) |

[59] | Radić, N., On buckling of porous double-layered FG nanoplates in the Pasternak elastic foundation based on nonlocal strain gradient elasticity, Compos. B Eng., 153, 465-479 (2018) |

[60] | Romano, G.; Barretta, R., Nonlocal elasticity in nanobeams: the stress-driven integral model, Int. J. Eng. Sci., 115, 14-27 (2017) · Zbl 1423.74512 |

[61] | Romano, G.; Barretta, R.; Diaco, M.; de Sciarra, F. M., Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., 121, 151-156 (2017) |

[62] | Şimşek, M., Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory, Compos. Struct., 224, 111041 (2019) |

[63] | Sourani, P.; Hashemian, M.; Pirmoradian, M.; Toghraie, D., A comparison of the bolotin and incremental harmonic balance methods in the dynamic stability analysis of an Euler-Bernoulli nanobeam based on the nonlocal strain gradient theory and surface effects, Mech. Mater., 103403 (2020) |

[64] | Talebitooti, R.; Rezazadeh, S. O.; Amiri, A., Comprehensive semi-analytical vibration analysis of rotating tapered AFG nanobeams based on nonlocal elasticity theory considering various boundary conditions via differential transformation method, Compos. B Eng., 160, 412-435 (2019) |

[65] | Thota, S., A new root-finding algorithm using exponential series, Ural Math. J., 5, 83-90 (2019) · Zbl 1450.65044 |

[66] | Trabelssi, M.; El-Borgi, S.; Fernandes, R.; Ke, L.-L., Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation, Compos. B Eng., 157, 331-349 (2019) |

[67] | Tuna, M.; Kirca, M., Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method, Compos. Struct., 179, 269-284 (2017) |

[68] | Vaccaro, M. S.; Pinnola, F. P.; Marotti de Sciarra, F.; Barretta, R., Dynamics of stress-driven two-phase elastic beams, Nanomaterials, 11, 1138 (2021) |

[69] | Wang, Y.; Li, C.; Zhang, Y., Experimental evaluation of the lubrication properties of the wheel/workpiece interface in MQL grinding with different nanofluids, Tribol. Int., 99, 1, 198-210 (2016) |

[70] | Wang, F.-Z.; Khan, M. N.; Ahmad, I.; Ahmad, H.; Abu-Zinadah, H.; Chu, Y.-M., Numerical solution of traveling waves in chemical kinetics: time-fractional fishers equations, Fractals, 30, 2, Article 22400051 pp. (2022) · Zbl 07507535 |

[71] | Xiang, G.; Zhang, Y.; Gao, X.; Li, H.; Huang, X., Oblique detonation waves induced by two symmetrical wedges in hydrogen-air mixtures, Fuel, 295 (2021) |

[72] | Xing, M.; Shijie, W., Design and study on vibration characteristics of self-excited vibration layered subsoiler for coastal soil, J. Coast Res., 103, 318-322 (2020) |

[73] | Xu, X.-J.; Meng, J.-M., A size-dependent elastic theory for magneto-electro-elastic materials, Eur. J. Mech. Solid., 86, 104198 (2021) · Zbl 1484.74023 |

[74] | Xu, X.-J.; Deng, Z.-C.; Zhang, K.; Xu, W., Observations of the softening phenomena in the nonlocal cantilever beams, Compos. Struct., 145, 43-57 (2016) |

[75] | Yang, M.; Li, C.; Zhang, Y., Predictive model for minimum chip thickness and size effect in single diamond grain grinding of zirconia ceramics under different lubricating conditions, Ceram. Int., 45, 12, 14908-14920 (2019) |

[76] | Yang, W.; Lin, Y.; Chen, X.; Xu, Y.; Song, X., Wave mixing and high-harmonic generation enhancement by a two-color field driven dielectric metasurface, Chin. Opt Lett., 19, 12 (2021) |

[77] | Yang, M.; Li, C.; Luo, L., Predictive model of convective heat transfer coefficient in bone micro-grinding using nanofluid aerosol cooling, Int. Commun. Heat Mass Tran., 125, 105317 (2021) |

[78] | Zaera, R.; Serrano, Ó.; Fernández-Sáez, J., On the consistency of the nonlocal strain gradient elasticity, Int. J. Eng. Sci., 138, 65-81 (2019) · Zbl 1425.74053 |

[79] | Zaera, R.; Serrano, Ó.; Fernández-Sáez, J., Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity, Meccanica, 55, 469-479 (2020) · Zbl 1485.74013 |

[80] | Zeighampour, H.; Beni, Y. T.; Dehkordi, M. B., Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory, Thin-Walled Struct., 122, 378-386 (2018) |

[81] | Zhou, G.; Deng, R.; Zhou, X.; Long, S.; Li, W.; Lin, G.; Li, X., Gaussian inflection point selection for LiDAR hidden echo signal decomposition, Geosci. Rem. Sens. Lett. IEEE, 1-5 (2021) |

[82] | Zhu, X.; Li, L., Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity, Int. J. Mech. Sci., 133, 639-650 (2017) |

[83] | Zhu, X.; Li, L., Closed form solution for a nonlocal strain gradient rod in tension, Int. J. Eng. Sci., 119, 16-28 (2017) · Zbl 1423.74334 |

[84] | Zhu, X.; Li, L., On longitudinal dynamics of nanorods, Int. J. Eng. Sci., 120, 129-145 (2017) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.