The effective piezoelectric and flexoelectric properties of heterogeneous solid bodies with constituents obeying a piezoelectric behavior are evaluated in full generality, based on the asymptotic expansion method. The successive situations of materials obeying a piezoelectric and flexoelectric behavior at the macroscale is envisaged in the present work. Closed-form expressions for the effective flexoelectric properties are obtained for stratified materials. A general theory for laminated piezoelectric plates is formulated on the basis of the formulated asymptotic models, and the response of the homogeneous substitution plate is evaluated for a loading consisting of a pure bending moment, triggering electric fields and strain and electric fields gradients within the plate thickness. The local mechanical and electric fields at the microscopic level within the initial heterogeneous stratified domain are evaluated by solving unit cell boundary value problems for the localization operators. An effective flexoelectric plate model for a stratified composite is constructed, showing the generation of the gradient of an electric field under application of a pure bending moment.
MalikanM.Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory. Appl Math Model2017; 48: 196–207.
2.
MalikanM. Temperature influences on shear stability of a nanosize plate with piezoelectricity effect, Multidiscip Model Mater Struct2018; 14: 125–142.
3.
MalikanM.Electro-thermal buckling of elastically supported double-layered piezoelectric nanoplates affected by an external electric voltage. Multidiscip Model Mater Struct.
4.
MalikanMEremeyevVA.On the dynamics of a visco–piezo–flexoelectric nanobeam. Symmetry (Basel)2020; 12: 643.
5.
OliveMAuffrayN.Symmetry classes in piezoelectricity from second-order symmetries. Math Mech Complex Syst2021; 9: 77–105.
6.
MauriniCPougetJDell’IsolaF.On a model of layered piezoelectric beams including transverse stress effect. Int J Solids Struct2004; 41: 4473–4502.
7.
GiorgioIGalantucciLDella CorteA, et al. Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications. Int J Appl Electromagn Mech2015; 47: 1051–1084.
8.
GrekovAAKramarovSOKuprienkoAA.Anomalous behavior of the two-phase lamellar piezoelectric texture. Ferroelectrics1987; 76: 43–48.
9.
BenvenisteYDvorakGJ.Uniform fields and universal relations in piezoelectric composites. J Mech Phys Solids1992; 40: 1295–1312.
10.
CastilleroJBOteroJARamosRR, et al. Asymptotic homogenization of laminated piezocomposite materials. Int J Solids Struct1998; 35: 527–541.
11.
GałkaATelegaJJWojnarR.Some computational aspects of homogenization of thermopiezoelectric composites. Comput Assist Mech Eng Sci1996; 3: 133–154.
12.
BacigalupoA.Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits. Meccanica2014; 49: 1407–1425.
13.
BakhvalovNPanasenkoG. Homogenization: averaging processes in periodic media. Math Comput Simul1990; 32: 425–426.
14.
BensoussanALionsJ-LPapanicolaouG (eds). Asymptotic Analysis for Periodic Structures. Elsevier. Epub ahead of print. 1978.
15.
BacigalupoAGambarottaL.Computational dynamic homogenization for the analysis of dispersive waves in layered rock masses with periodic fractures. Comput Geotech2014; 56: 61–68.
16.
PeerlingsRHJFleckNA.Computational evaluation of strain gradient elasticity constants. Int J Multiscale Comput Eng2004; 2: 599–620.
17.
SmyshlyaevVPCherednichenkoKD.On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. J Mech Phys Solids2000; 48: 1325–1357.
18.
AddessiDDe BellisMLSaccoE.Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains. Mech Res Commun2013; 54: 27–34.
19.
BaccaMBigoniDDal CorsoF, et al. Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites. Part I: Closed form expression for the effective higher-order constitutive tensor. Int J Solids Struct2013; 50: 4010–4019.
20.
BacigalupoAGambarottaL.Second-gradient homogenized model for wave propagation in heterogeneous periodic media. Int J Solids Struct2014; 51: 1052–1065.
21.
BigoniDDruganWJ.Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech2007; 74: 741–753.
22.
ForestSTrinhDK.Generalized continua and non-homogeneous boundary conditions in homogenisation methods. Z Angew Math Mech2011; 91: 90–109.
23.
MieheCSchröderJSchotteJ.Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng1999; 171: 387–418.
24.
KouznetsovaVGGeersMGDBrekelmansWAM. Multi-scale second-order computational homogenization of multi-phase materials: A nested finite element solution strategy. Comput Methods Appl Mech Eng2004; 193: 5525–5550.
25.
PanasenkoG. Multi-scale Modelling for Structures and Composites. Berlin/Heidelberg: Springer-Verlag. Epub ahead of print. 2005.
26.
ZhikovVVKozlovSMOleinikOA, et al. Averaging and G-convergence of differential operators. Russ Math Surv1979; 34: 69–147.
27.
RahaliYDos ReisFGanghofferJ-F.Multiscale homogenization schemes for the construction of second-order grade anisotropic continuum media of architectured materials. Int J Multiscale Comput Eng2017; 15: 35–78.
28.
MawassyNRedaHGanghofferJ-F, et al. A variational approach of homogenization of piezoelectric composites towards piezoelectric and flexoelectric effective media. Int J Eng Sci2021; 158: 103410.
29.
EnakoutsaKDellaCAGiorgioI.A model for elastic flexoelectric materials including strain gradient effects. Math Mech Solids2016; 21: 242–254.
30.
EnakoutsaKDel VescovoDScerratoD.Combined polarization field gradient and strain field gradient effects in elastic flexoelectric materials. Math Mech Solids2017; 22: 938–951.
