In this paper, we establish the transfer matrix for an orthotropic elastic layer made of nanomaterial that is modeled by the weakly nonlocal elasticity model. This model is proposed recently and different from other existing nonlocal models of elasticity it has been proved to be well-posed for any problem of plane waves. Since the established transfer matrix is totally explicit, it is a convenient tool in investigating various problems of plane waves propagating in layered nonlocal elastic media. To prove this point, we use it to derive the dispersion equation of Lamb waves propagating in a composite plate consisting of two weakly nonlocal orthotropic layers. The dispersion equation in explicit form of Lamb waves has been derived, and it is shown that the derivation of this equation is much more simple than the one using the traditional technique. The obtained explicit dispersion equation will be used to establish techniques monitoring the health of layered nanostructures.
EringenAC.Nonlocal continuum field theories. New York: Springer, 2002.
2.
EringenAC.Plane waves in nonlocal micropolar elasticity. Int J Eng Sci1984; 22: 1113–1121.
3.
ChenA-LYanD-JWangY-S, et al. Anti-plane transverse waves propagation in nanoscale periodic layered piezoelectric structures. Ultrasonics2016; 65: 154–164.
4.
SinghDKaurGTomarSK.Waves in nonlocal elastic solids with voids. J Elasticity2017; 128: 85–114.
5.
BachherMSarkarN.Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer. Waves Random Complex Media2019; 29: 595–613.
6.
TongLYuYHuW, et al. On wave propagation characteristics in fluid saturated porous materials by nonlocal Biot theory. J Sound Vibration2016; 379: 106–118.
7.
LiCGuoHTianX, et al. Nonlocal diffusion-elasticity based on nonlocal mass transfer and nonlocal elasticity and its application in shock-induced responses analysis. Mech Adv Mater Struct2021; 28: 827–838.
8.
IijimaS.Helical microtubules of graphitic carbon. Nature1991; 354: 56–58.
9.
MarottideSciarraFRussoP.Experimental characterization, predictive mechanical and thermal modeling of nanostructures and their polymer composites. Amsterdam: Elsevier, 2019.
10.
ChenJZhuYJiangW.A stretchable and transparent strain sensor based on sandwich-like PDMS/CNTs/PDMS composite containing an ultrathin conductive CNT layer. Comp Sci Tech2020; 186: 107938.
11.
ZhaoBChenZChengZ, et al. Piezoelectric nanogenerators based on electrospun PVDF-coated mats composed of multilayer polymer-coated BaTiO3 nanowires. ACS Appl Nano Mater2022; 5: 8417–8428.
12.
VermaKL. Nonlocal wave surfaces in elastic solids. Int J Eng2 : 41–46.
13.
SinghB.Propagation of waves in an incompressible rotating transversely isotropic nonlocal solid. Vietnam J Mech2021; 43: 237–252.
14.
DasNSarkarNLahiriA.Reflection of plane waves from the stress-free isothermal and insulated boundaries of a nonlocal thermoelastic solid, Appl Math Model2019; 73: 526–544.
15.
PramanikASBiswasS.Surface waves in nonlocal thermoelastic medium with state space approach. J Therm Stress2020; 43: 667–686.
16.
ZhangQSunYYangJ.Propagation and reflection of plane waves in biological tissue based on nonlocal TPL thermoelasticity. Int Comm Heat Mass Transfer2021; 128: 105587.
17.
YanD-JChenA-LWangY-S, et al. Propagation of guided elastic waves in nanoscale layered periodic piezoelectric composites. Eur J Mech A/Solids2017; 66: 158–167.
18.
YanD-JChenA-LWangY-S, et al. In-plane elastic wave propagation in nanoscale periodic layered Piezoelectric structuresInt J Mech Sci2018; 142–143: 276–288.
19.
LiuCYuJWangX, et al. Reflection and transmission of elastic waves through nonlocal piezoelectric plates sandwiched in two solid half-spaces. Thin Wall Struc2021; 168: 108306.
20.
KhuranaATomarSK.Reflection of plane longitudinal waves from the stress-free boundary of a nonlocal, micropolar solid half-space. J Mech Mater Struct2013; 8: 95–107.
21.
KhuranaATomarSK.Rayleigh-type waves in nonlocal micropolar solid half-space. Ultrasonics2017; 73: 162–168.
22.
KhuranaATomarSK.Waves at interface of dissimilar nonlocal micropolar elastic half-spaces. Mech Adv Mater Struct2019;26: 825–833.
23.
KalkaKKSheoranDDeswalS.Reflection of plane waves in a nonlocal micropolar thermoelastic medium under the effect of rotation. Acta Mech2020; 231: 2849–2866.
24.
KumarSTomarSK.Plane waves in nonlocal micropolar thermoelastic material with voids. J Therm Stress2020; 43: 1355–1378.
25.
DeswalSSheoranDThakranS, et al. Reflection of plane waves in a nonlocal microstretch thermoelastic medium with temperature dependent properties under three-phase-lag model. Mech Adv Mater Struct2022; 29: 1692–1707.
26.
ChakrabortyA.Prediction of negative dispersion by a nonlocal poroelastic theory. J Acoust Soc Am2008; 123: 56–67.
27.
TongLHLaiSKZengLL, et al. Nonlocal scale effect on Rayleigh wave propagation in porous fluid-saturated materials. Int J Mech Sci2018; 148: 459–466.
28.
KaurGSinghDTomarSK.Rayleigh-type wave in a nonlocal elastic solid with voids. Eur J Mech/A Solids2018; 71: 134–150.
29.
KaurGSinghDTomarSK.Love waves in a nonlocal elastic media with voids. J Vib Control2019; 25: 1470–1483.
30.
SarkarNTomarSK.Plane waves in nonlocal thermoelastic solid with voids. J Therm Stress2019; 42: 580–606.
31.
KaurGSinghDTomarSK.Lamb waves in nonlocal elastic with voids. J Mech Mater Struct2020; 16: 389–405.
32.
EringenAC.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983; 54: 4703–4710.
33.
Abd-AllaAMAbo-DahabSMAhmedSM, et al. Effect of magnetic field and voids on Rayleigh waves in a nonlocal thermoelastic half-space. J Strain Anal2022; 57: 61–72.
34.
GillKSSawhneyS.Rayleigh waves with impedance boundary conditions in a nonlocal micropolar thermoelastic material. J Phys Conference Series2020; 1531: 012048.
BiswasS.Rayleigh waves in a nonlocal thermoelastic layer lying over a nonlocal thermoelastic half-space. Acta Mech2020; 231: 4129–4144.
37.
ChakrabortyA.Wave propagation in anisotropic media with non-local elasticity. Int J Solids Struct2007; 44: 5723–5741.
38.
EringenAC.Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci1972; 10: 425–435.
39.
EringenAC.Theory of nonlocal elasticity and some applications. Res Mech1987; 21: 313–342.
40.
ZhuXLiL.Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Compos Struct2017; 178: 87–96.
41.
CeballesSLarkinKRojasE, et al. Nonlocal elasticity and boundary condition paradoxes: a review. J Nanopart Res2021; 23: 66.
42.
RomanoGBarrettaR.Nonlocal elasticity in nano-beams: the stress-driven integral model. Int J Eng Sci2017; 115: 14–27
43.
LimCZhangGReddyJ.A high order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids2015; 78: 298–313.
44.
RomanoGBarrettaRDiacoM, et al. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci2017; 121: 151–156.
45.
KaplunovJPrikazchikovDAPrikazchikovaL.On non-locally elastic Rayleigh wave. Phil Trans R Soc A2022; 380(2231): 20210387.
46.
KaplunovJPrikazchikovDAPrikazchikovaL.On integral and differential formulations in nonlocal elasticity. Eur J Mech A Solids2023; 100: 104497.
47.
AnhVTNVinhPC.Expressions of nonlocal quantities and application to Stoneley waves in weakly nonlocal orthotropic elastic half-spaces. Math Mech Solids2023; 28: 2420–2435.
48.
AnhVTNVinhPCTuanTT, et al. Weakly nonlocal Rayleigh waves with impedance boundary conditions. Continuum Mech Thermodyn2023; 35: 2081–2094.
49.
AnhVTNVinhPC.The incompressible limit method and Rayleigh waves in incompressible layered nonlocal orthotropic elastic media. Acta Mech2023; 234: 403–421.
50.
RaghuPRajagopalAReddyJN.Nonlocal transient dynamic analysis of laminated composite plates. Mech Adv Mater Struct2020; 27: 1076–1084.
51.
RokhlinSIWangL.Stable recursive algorithm for elastic wave propagation in layered anisotropic media: stiffness matrix method. J Acoust Soc Am2002; 112: 822–834.
52.
HostenBCastaingsM.Surface impedance matrices to model the propagation in multilayered media. Ultrasonics2003; 41: 501–507.
53.
ChenX.A systematic and efficient method of computing normal modes for multilayered half-space. Geophys J Int1993; 115: 391–409.
LoweMJS.Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans Ultrason Ferroelect Freq Contr1995; 42: 525–542.
56.
ThomsonWT.Transmission of elastic waves through a stratified solid medium. J Appl Phys1950; 21: 89–93.
57.
HaskellNA.The dispersion of surface waves on multilayered media. Bull Seism Soc Am1953; 43: 17–34.
58.
GreenWAGreenER.Stress vibration due to an impact line load on a four-ply fibre composite plate. Int J Solids Struct1991; 28: 567–595.
59.
SolyanikFI.Transmission of plane waves through a layered medium of anisotropic materials. Sov Phys Acoust1977; 23: 533–536.
60.
RokhlinSIWangYJ.Equivalent boundary conditions for thin orthotropic layer between two solids. Reflection, refraction, and interface waves. J Acoust Soc Am1992; 91: 1875–1887.
61.
VinhPCAnhVTNLinhNTK. On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer. Waves Random Complex Media2016; 26: 176–188.
62.
AnhVTNVinhPCLinhNTK, et al. Explicit transfer matrix for an incompressible orthotropic elastic layer and applications. Z Angew Math Phys2021; 72: 145.
63.
RogersonGASandifordKJ.Some comments on the dispersion relation for periodically layered pre-stressed elastic media. Int J Eng Sci2002; 40: 23–49.
VinhPCAnhVTNMerodioJ, et al. Explicit transfer matrices of pre-stressed elastic layers. Int J Non linear Mech2018; 106: 288–296.
66.
ChebakovRKaplunovJRogersonGA.Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects. Proc R Soc A Math Phys Eng Sci2016; 472(2186): 20150800.
67.
ChebakovRKaplunovJRogersonGA.A non-local asymptotic theory for thin elastic plates. Proc R Soc A Math Phys Eng Sci2017; 473: 20170249.
68.
SahinO.An asymptotic formulation for boundary value problems in a non-local elastic half-space. Acta Mech2024; 235: 2061–2075.
69.
LinhNTKVinhPCThangLT, et al. Lamb waves in sandwich orthotropic elastic plates. Waves Random Complex Media. Epub ahead of print 17June2021. DOI: 10.1080/17455030.2021.1940352.
70.
NobiliAErbasBSignoriniC.Veering of Rayleigh-Lamb waves in orthorhombic materials. Math Mech Solids2022; 27: 1783–1799.
