A surface piezoelectricity model generalizing the Steigmann–Ogden model for linear elastic solids is described starting from a rigorous variational approach. The boundary and tip conditions on surface-energetic material surfaces are presented. The model is applied to the problem for a one-phase or a two-phase piezoelectric circular fiber in a piezoelectric infinite matrix. The piezoelectric materials are assumed to have a 6-mm hexagonal crystal symmetry. The problem is solved analytically using the complex variable approach. The boundaries between different materials possess surface piezoelectric energy. The matrix is subjected to a far-field loading and contains a piezoelectric screw dislocation. The effective moduli of the piezoelectric composites reinforced by one-phase or two-phase circular fibers are obtained using the Maxwell’s far-field methodology. The results are compared with the generalized self-consistent scheme. An image force on the dislocation is computed using the generalized Peach–Koehler formula. Numerical examples and parametric studies are presented.
RamprasadRShiN.Dielectric properties of nanoscale slabs. Phys Rev B Condens Matter Mater Phys2005; 72(5): 052107.
2.
ZhaoMWangZMaoS.Piezoelectric characterization of individual zinc oxide nanobelt probed by piezoresponse force microscope. Nano Lett2004; 4(4): 587–590.
3.
GurtinMMurdochA.A continuum theory of elastic material surfaces. Arch Rational Mech Anal1975; 57(4): 291–323.
4.
GurtinMMurdochA.Surface stress in solids. Int J Solids Struct1978; 14(6): 431–440.
5.
SteigmannDOgdenR.Plane deformations of elastic solids with intrinsic boundary elasticity. Proc R Soc London Ser A1997; 453: 853–877.
6.
SteigmannDOgdenR.Elastic surface substrate interactions. Proc R Soc London Ser A1999; 455: 437–474.
7.
HuangGYuS.Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys Status Solidi B2006; 243(4): R22–R24.
8.
ChenTChiuMSWengCN.Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids. J Appl Phys2006; 100(7): 074308.
9.
EremeyevVANasedkinA. Mathematical models and finite element approaches for nanosized piezoelectric bodies with uncoulped and coupled surface effects. In: SumbatyanM (ed.) Wave dynamics and composite mechanics for microstructured materials and metamaterials. Singapore: Springer, 2017, pp. 1–18.
10.
NasedkinAVEremeyevVA.Harmonic vibrations of nanosized piezoelectric bodies with surface effects. J Appl Math Mech2014; 94(10): 878–892.
11.
ChenT.Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects. Acta Mech2008; 196(3–4): 205–217.
12.
WangXFanH.A piezoelectric screw dislocation in a bimaterial with surface piezoelectricity. Acta Mech2015; 226(10): 3317–3331.
13.
XiaoJXuYZhangF.Size-dependent effective electroelastic moduli of piezoelectric nanocomposites with interface effect. Acta Mech2011; 222: 59.
14.
XiaoJXuYZhangF.Evaluation of effective electroelastic properties of piezoelectric coated nano-inclusion composites with interface effect under antiplane shear. Int J Eng Sci2013; 69: 61–68.
15.
PakYE.Circular inclusion problem in antiplane piezoelectricity. Int J Solids Struct1992; 29(19): 2403–2419.
16.
DengWMeguidS.Closed form solutions for partially debonded circular inclusion in piezoelectric materials. Acta Mech1999; 137(3): 167–181.
17.
ChaoCChangK.Interacting circular inclusions in antiplane piezoelectricity. Int J Solids Struct1999; 36(22): 3349–3373.
18.
YangBHGaoCF.Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix. Int J Eng Sci2010; 48(5): 518–528.
19.
YangBHGaoCFNodaN.Interactions between n circular cylindrical inclusions in a piezoelectric matrix. Acta Mech2008; 197: 31–42.
20.
LiuYFangQJiangC.A piezoelectric screw dislocation interacting with an interphase layer between a circular inclusion and the matrix. Int J Solids Struct2004; 41(11–12): 3255–3274.
21.
LiuYFangQJiangC.Analysis of a piezoelectric screw dislocation in the interphase layer between a circular inclusion and an unbounded matrix. Mater Chem Phys2006; 98(1): 14–26.
22.
ShenMChenSChenF.A piezoelectric screw dislocation interacting with a nonuniformly coated circular inclusion. Int J Eng Sci2006; 44(1–2): 1–13.
23.
JinBFangQ.Piezoelectric screw dislocations interacting with a circular inclusion with imperfect interface. Arch Appl Mech2008; 78: 105–116.
24.
HanZMogilevskayaSGZemlyanovaAY.On the problem of a Gurtin-Murdoch cylindrical material surface embedded in an infinite matrix. Int J Solids Struct2024; 288: 112617.
25.
HanZZemlyanovaAYMogilevskayaSG.Two-dimensional problem of an infinite matrix reinforced with a Steigmann-Ogden cylindrical surface of circular arc cross-section. Int J Eng Sci2024; 194: 103986.
26.
MogilevskayaSZemlyanovaAMantičV.The use of the Gurtin-Murdoch theory for modeling mechanical processes in composites with two-dimensional reinforcements. Comp Sci Tech2021; 210: 108751.
27.
ZemlyanovaAYMogilevskayaSGSchillingerD.Numerical solution of the two-dimensional Steigmann-Ogden model of material surface with a boundary. Phys D2023; 443: 133531.
28.
PakY.Force on a piezoelectric screw dislocation. J Appl Mech1990; 57(4): 863–869.
29.
BardzokasDFilshtinskyMFilshtinskyL.Mathematical methods in electro-magneto-elasticity. Cham: Springer, 2007.
30.
TierstenH.Linear piezoelectric plate vibrations: elements of the linear theory of piezoelectricity and the vibrations piezoelectric plates. Cham: Springer, 2013.
31.
YangJ.An introduction to the theory of piezoelectricity. Cham: Springer, 2005.
32.
YangJ (ed.) Special topics in the theory of piezoelectricity. Cham: Springer, 2009.
33.
PakY.Linear electro-elastic fracture mechanics of piezoelectric materials. Int J Fract1992; 54: 79–100.
34.
EremeyevV.On dynamic boundary conditions within the linear Steigmann-Ogden model of surface elasticity and strain gradient elasticity. Dyn Proc Gen Cont Struct2019; 103: 195–207.
35.
McCartneyLNKellyA.Maxwell’s far-field methodology applied to the prediction of properties of multi-phase isotropic particulate composites. Proc R Soc A Math Phys Eng Sci2008; 464(2090): 423–446.
36.
McCartneyL.Maxwell’s far-field methodology predicting elastic properties of multiphase composites reinforced with aligned transversely isotropic spheroids. Philos Mag2010; 90(31–32): 4175–4207.
37.
MogilevskayaSGStolarskiHKCrouchSL.On Maxwell’s concept of equivalent inhomogeneity: when do the interactions matter?J Mech Phys Solids2012; 60(3): 391–417.
38.
SevostianovIMogilevskayaSKushchV.Maxwell’s methodology of estimating effective properties: alive and well. Int J Eng Sci2019; 140: 35–88.
39.
ZemlyanovaAMogilevskayaS.Circular inhomogeneity with Steigmann-Ogden interface: local fields and Maxwell’s type approximation formula. Int J Solids Struct2018; 135: 85–98.
40.
MiltonGWSerkovSK.Neutral coated inclusions in conductivity and anti–plane elasticity. Proc R Soc Lond Ser A Math Phys Eng Sci2001; 457(2012): 1973–1997.
41.
AndersonPMHirthJPLotheJ.Theory of dislocations. Cambridge: Cambridge University Press, 2017.
