Abstract
The direct flexoelectricity in dielectrics, as an electromechanical mechanism coupling electric field and strain gradient, exhibits strong size dependence and structures associated. In the present paper, the effective properties of particulate flexoelectric composites are predicted by the flexoelectric theory. Numerical finite element simulations are realized for representative volume elements (RVE) of the isotropic matrix filled with a spherical flexoelectric inclusion by using mixed variational principle and finite element method (FEM). The elastic fields inside the RVE of different inclusion sizes and volume fractions are studied. The influences of flexoelectricity on the mechanical properties of the composites are discussed. Effective properties of the composites are estimated based on the obtained numerical results. It is shown that flexoelectricity has a great influence on the bulk modulus of composites with a nanoscale inclusion. Due to its size dependence, the flexoelectricity can be neglected in the model with a micron-scale inclusion. At the same time, the selection of length scale will affect the changing trend of effective material properties with volume fraction. Our results suggest that the influence of flexoelectricity on the properties of nanoscale dielectric composites should be emphasized.
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Chu, L., Dui, G., Mei, H., Liu, L., Li, Y.: An analysis of flexoelectric coupling associated electroelastic fields in functionally graded semiconductor nanobeams. J. Appl. Phys. 130, 115701 (2021)
Yudin, P., Tagantsev, A.: Fundamentals of flexoelectricity in solids. Nanotechnology 24, 432001 (2013)
Deng, F., Deng, Q., Yu, W., Shen, S.: Mixed finite elements for flexoelectric solids. J. Appl. Mech.-T. ASME 84, 081004 (2017)
Mao, S., Purohit, P.K., Aravas, N.: Mixed finite-element formulations in piezoelectricity and flexoelectricity. Proc. R. Soc. A Math. Phys. 472, 20150879 (2016)
Sharma, S., Vaish, R., Kumar, R.: An isogeometric analysis-based investigation of the flexocaloric effect in functionally graded dielectrics. Acta Mech. 232, 4261–4271 (2021)
Qu, Y.L., Jin, F., Yang, J.S.: Stress-induced electric potential barriers in thickness-stretch deformations of a piezoelectric semiconductor plate. Acta Mech. 232, 4533–4543 (2021)
Qu, Y.L., Jin, F., Yang, J.S.: Buckling of flexoelectric semiconductor beams. Acta Mech. 232, 2623–2633 (2021)
Deng, Q., Liu, L., Sharma, P.: Flexoelectricity in soft materials and biological membranes. J. Mech. Phys. Solids 62, 209–227 (2014)
Petrov, A.G.: Flexoelectricity of model and living membranes. Biochim. Biophys. Acta Biomembr. 1561, 1–25 (2002)
Krichen, S., Sharma, P.: Flexoelectricity: a perspective on an unusual electromechanical coupling. J. Appl. Mech. 83, 030801 (2016)
Sharma, N., Landis, C., Sharma, P.: Piezoelectric thin-film superlattices without using piezoelectric materials. J. Appl. Phys. 108, 024304 (2010)
Mashkevich, V., Tolpygo, K.: Electrical, optical and elastic properties of diamond type crystals. Sov. Phys. JETP 5, 435–439 (1957)
Scott, J.F.: Lattice perturbations in CaWO4 and CaMoO4. J. Chem. Phys. 48, 874–876 (1968)
Maranganti, R., Sharma, N., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys. Rev. B 74, 014110 (2006)
Sharma, N., Maranganti, R., Sharma, P.: On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J. Mech. Phys. Solids 55, 2328–2350 (2007)
Majdoub, M.S., Sharma, P., Cagin, T.: Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Rev. B 77, 125424 (2008)
Hu, S., Shen, S.: Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China Phys. Mech. Astron. 53, 1497–1504 (2010)
Shen, S., Hu, S.: A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids 58, 665–677 (2010)
Milton, G.W.: The Theory of Composites. Cambridge University Press, Oxford (2004)
Koizumi, M.: The concept of FGM. Ceram. Trans. 34, 3–10 (1993)
Miyamoto, Y., Kaysser, W., Rabin, B., Kawasaki, A., Ford, R.: Functionally Graded Materials: Design, Processing and Applications. Kluwer Academic Publication, Hague (1999)
Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60, 195–216 (2007)
Xin, L., Dui, G., Yang, S., Zhang, J.: An elasticity solution for functionally graded thick-walled tube subjected to internal pressure. Int. J. Mech. Sci. 89, 344–349 (2014)
Chu, L., Li, Y., Dui, G.: Size-dependent electromechanical coupling in functionally graded flexoelectric nanocylinders. Acta Mech. 230, 3071–3086 (2019)
Chu, L., Dui, G.: Exact solutions for functionally graded micro-cylinders in first gradient elasticity. Int. J. Mech. Sci. 148, 366–373 (2018)
Raju, B., Hiremath, S.R., Mahapatra, D.R.: A review of micromechanics based models for effective elastic properties of reinforced polymer matrix composites. Compos. Struct. 204, 607–619 (2018)
Kundalwal, S.I.: Review on micromechanics of nano- and micro-fiber reinforced composites. Polym. Compos. 39, 4243–4274 (2018)
Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222 (1965)
Kerner, E.: The elastic and thermo-elastic properties of composite media. Proc. Phys. Soc. Lond. Sect. B 69, 808 (1956)
Solyaev, Y., Lurie, S., Korolenko, V.: Three-phase model of particulate composites in second gradient elasticity. Eur. J. Mech. A Solids 78, 103853 (2019)
Ke, L., Yang, J., Kitipornchai, S., Wang, Y.: Axisymmetric postbuckling analysis of size-dependent functionally graded annular microplates using the physical neutral plane. Int. J. Eng. Sci. 81, 66–81 (2014)
Xin, L., Yang, S., Zhou, D., Dui, G.: An approximate analytical solution based on the Mori-Tanaka method for functionally graded thick-walled tube subjected to internal pressure. Compos. Struct. 135, 74–82 (2016)
Kundalwal, S.I., Choyal, V.K., Choyal, V.: Flexoelectric effect in boron nitride-graphene heterostructures. Acta Mech. 232, 3781–3800 (2021)
Kundalwal, S.I., Meguid, S.A., Weng, G.J.: Strain gradient polarization in graphene. Carbon 117, 462–472 (2017)
Chen, W., Zheng, Y., Feng, X., Wang, B.: Utilizing mechanical loads and flexoelectricity to induce and control complicated evolution of domain patterns in ferroelectric nanofilms. J. Mech. Phys. Solids 79, 108–133 (2015)
Abdollahi, A., Peco, C., Millan, D., Arroyo, M., Arias, I.: Computational evaluation of the flexoelectric effect in dielectric solids. J. Appl. Phys. 116, 093502 (2014)
Yvonnet, J., Liu, L.: A numerical framework for modeling flexoelectricity and Maxwell stress in soft dielectrics at finite strains. Comput. Method. Appl. M. 313, 450–482 (2017)
Zheng, Y., Chu, L., Dui, G., Zhu, X.: Modeling and simulation of functionally graded flexoelectric micro-cylinders based on the mixed finite element method. Appl. Phys. A 127, 228 (2021)
Zheng, Y., Chu, L., Dui, G., Zhu, X.: Numerical predictions for the effective electrical properties of flexoelectric composites with a single inclusion. Appl. Phys. A 127, 686 (2021)
Aravas, N.: Plane-strain problems for a class of gradient elasticity models—a stress function approach. J. Elasticity 104, 45–70 (2011)
Thorpe, M., Jasiuk, I.: New results in the theory of elasticity for two-dimensional composites. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 438, 531–544 (1992)
Ostoja-Starzewski, M.: Microstructural Randomness and Scaling in Mechanics of Materials. CRC Press, Boca Raton (2007)
Deng, Q., Kammoun, M., Erturk, A., Sharma, P.: Nanoscale flexoelectric energy harvesting. Int. J. Solids Struct. 51, 3218–3225 (2014)
Chu, B., Salem, D.R.: Flexoelectricity in several thermoplastic and thermosetting polymers. Appl. Phys. Lett. 101, 2069 (2012)
Liang, X., Hu, S., Shen, S.: Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart Mater. Struct. 24, 105012 (2015)
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The authors acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 11772041).
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Zheng, Y., Chu, L., Dui, G. et al. Numerical predictions for the effective properties of flexoelectric composites with spherical inclusion. Acta Mech 233, 2093–2106 (2022). https://doi.org/10.1007/s00707-022-03207-5
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DOI: https://doi.org/10.1007/s00707-022-03207-5