A non-classical local gradient theory of nonferromagnetic thermoelastic dielectrics is presented, incorporating both the local mass-displacement process and the heat-flux gradient effect. The process of local mass displacement is related to the changes in material microstructure. The nonlocal heat conduction law is also addressed in the model. Thus, the generalized relationship between the higher-grade heat and entropy fluxes is adopted. The gradient-type constitutive relations and governing equations are derived using the fundamental principles of continuum mechanics, non-equilibrium thermodynamics, and electrodynamics. Due to the contribution of higher-grade flux, the nonlocal law of heat conduction is obtained. The constitutive relations for isotropic materials with the corresponding additional material constants are derived. To illustrate the local gradient theory and to show the electro-thermo-mechanical coupling effect in isotropic materials, a straightforward problem is analytically solved for a layered non-piezoelectric structure under non-uniform temperature distribution. The analytical results reveal that the thermal polarization effect can also be pronounced in isotropic materials. To illustrate the model considering the effect of nonlocal heat conduction, the propagation of spherical thermoelastic harmonic waves in a homogeneous and isotropic elastic medium with non-classical heat conduction law is studied.
MindlinRDTierstenHF.Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal1962; 11: 415–448.
9.
KoiterWT.Couple stresses in the theory of elasticity I and II. Proc K Ned Akad Wet B1964; 67: 17–44.
10.
MindlinRD.Micro-structure in linear elasticity. Arch Ration Mech Anal1964; 16: 51–78.
11.
MindlinRD.Second gradient of strain and surface-tension in linear elasticity. Int J Solid Struct1965; 1: 417–438.
12.
YangFChongACMLamDCC, et al. Couple stress based strain gradient theory for elasticity. Int J Solid Struct2002; 39: 2731–2743.
13.
LamDCCYangFChongACM, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids2003; 51: 1477–1508.
14.
EnakoutsaKDella CorteAGiorgioI.A model for elastic flexoelectric materials including strain gradient effects. Math Mech Solids2016; 21(2): 242–254.
15.
TagantsevAK.Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys Rev B1986; 34(7): 5883–5889.
16.
TagantsevAK.Pyroelectric, piezoelectric, flexoelectric, and thermal polarization effects in ionic crystals. Sov Phys Uspekhi1987; 30: 588–603.
17.
YudinPVTagantsevAK.Fundamentals of flexoelectricity in solids. Nanotechnology2013; 24: 432001.
18.
ZubkoPCatalanGTagantsevAK.Flexoelectric effect in solids. Ann Rev Mater Res2013; 43: 387–421.
19.
LiAZhouSQiL, et al. A reformulated flexoelectric theory for isotropic dielectrics. J Phys D Appl Phys2015; 48(12): 465502.
20.
MindlinRD.Polarization gradient in elastic dielectrics. Int J Solid Struct1968; 4: 637–642.
21.
KafadarCB.The theory of multipoles in classical electromagnetism. Int J Eng Sci1971; 9(9): 831–853.
22.
DemirayHEringenAC.On the constitutive relations of polar elastic dielectrics. Lett Appl Eng Sci1973; 1: 517–527.
23.
YangXMHuYTYangJS.Electric field gradient effects in antiplane problems of polarized ceramics. Int J Solid Struct2004; 41: 6801–6811.
24.
SahinEDostS.A strain-gradient theory of elastic dielectrics with spatial dispersion. Int J Eng Sci1988; 26(12): 1231–1245.
25.
KalpakidesVKAgiasofitouEK.On material equations in second order gradient electroelasticity. J Elast2002; 67: 205–227.
26.
MarangantiRSharmaNDSharmaP.Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s functions and embedded inclusions. Phys Rev B2006; 74: 1411014.
27.
HuSLShenSP.Electric field gradient theory with surface effect for nano-dielectrics. Comput Mater Contin2009; 13: 63–87.
28.
HuSShenS.Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci Chin Phys Mech Astron2010; 53: 1497–1504.
29.
EringenAC.Microcontinuum field theories I: foundation and solids. New York: Springer, 1999.
30.
YangJ.A review of a few topics in piezoelectricity. Appl Mech Rev2006; 59: 335–345.
31.
YanZJiangL.Modified continuum mechanics modeling on size-dependent properties of piezoelectric nanomaterials: a review. Nanomaterials2017; 7: 27.
32.
BeranMJMcCoyJJ.The use of strain gradient theory for analysis of random media. Int J Solid Struct1970; 6: 1267–1275.
33.
MindlinRD.Elasticity, piezoelectricity and crystal lattice dynamics. J Elast1972; 2(4): 217–282.
34.
MauginGA.Nonlinear waves in elastic crystals. Oxford: Oxford University Press, 1999.
35.
ChenJ.Micropolar theory of flexoelectricity. J Adv Math Appl2013; 1: 269–2746.
36.
AttiaMAMahmoudFF.Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. Int J Mech Sci2016; 105: 126–134.
37.
BurakY.Constitutive equations of locally gradient thermomechanics. Proc Acad Sci USSR1987; 12: 19–23 (in Ukrainian).
38.
BurakYYKondratVFHrytsynaOR.Subsurface mechanoelectromagnetic phenomena in thermoelastic porized bodies in the case of local displacements of mass. Mater Sci2007; 43(4): 449–463.
39.
BurakYKondratVHrytsynaO.An introduction of the local displacements of mass and electric charge phenomena into the model of the mechanics of polarized electromagnetic solids. J Mech Mater Struct2008; 3(6): 1037–1046.
40.
HrytsynaOKondratV.Local gradient theory for thermoelastic dielectrics: accounting for mass and electric charge transfer due to structural changes. J Mech Mater Struct2019; 14(4): 549–568.
41.
HrytsynaOR.Applications of the local gradient elasticity to the description of the size effect of shear modulus. SN Appl Sci2020; 2: 1453.
42.
HrytsynaOR.Bernoulli-Euler beam model based on local gradient theory of elasticity. J Mech Mater Struct2020; 15(4): 471–487.
43.
ChaplaYKondratSHrytsynaO, et al. On electromechanical phenomena in thin dielectric films. Task Quart2009; 13(1–2): 145–154.
44.
HrytsynaO.Influence of subsurface inhomogeneity on the propagation of SH waves in isotropic materials. Mater Sci2017; 53(2): 273–281.
45.
HrytsynaO.Electromechanical fields in a hollow piezoelectric cylinder under non-uniform load: flexoelectric effect. Math Mech Solids2022; 27(2): 262–280.
46.
HrytsynaO.The effect of local mass displacement on coupled fields in dielectrics. Appl Nanosci2022; 12: 711–723.
47.
LebonGTorrisiMValentiA.A non-local thermodynamic analysis of second sound propagation in crystalline dielectrics. J Phys Condens Matter1995; 7: 1461–1474.
48.
YuYJTianXGXiongQL.Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity. Eur J Mech A Solids2016; 60: 238–253.
49.
LebonG.Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J NonEquilib Thermodyn2014; 39(1): 35–59.
50.
PovstenkoY.Fractional thermoelasticity problem for an infinite solid with a cylindrical hole under harmonic heat flux boundary condition. Acta Mech2019; 230: 2137–2144.
51.
LandauLDLifshitzEM.Electrodynamics of continuum media. 2nd ed.Oxford: Butterworth-Heinemann, 1984.
52.
BredovMMRumyantsevVVToptyhinIN.Classic electrodynamics (Классическая электродинамика). Moscow: Nauka, 1985 (in Russian).
53.
NowackiJP.Electro-elastic fields of a plane thermal inclusion in isotropic dielectrics with polarization gradient. Arch Mech2004; 56(1): 33–57.
54.
NowackiW.Teoria sprężystości. Warsaw: Państwowe Wydawnictwo Naukowe, 1970 (in Polish).
55.
KovalenkoAD.Thermoelasticity: basic theory and applications. Groningen: Wolters-Noordhoff, 1970.
