×

A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects. I: Reconsideration of curvature-based flexoelectricity theory. (English) Zbl 07589909

Summary: A new non-classical theory of elastic dielectrics is developed using the couple stress and electric field gradient theories that incorporates the couple stress, quadrupole and curvature-based flexoelectric effects. The couple stress theory and an extended Gauss’s law for elastic dielectrics with quadrupole polarization are applied to derive the constitutive relations of this new theory through energy conservation. The governing equations and the complete boundary conditions are simultaneously obtained through a variational formulation based on the Gibbs-type variational principle. The constitutive relations of general anisotropic and isotropic materials with the corresponding independent material constants are also provided, respectively. To illustrate the newly proposed theory and to show the flexoelectric effect in isotropic materials, one pure bending problem of a simply supported beam is analytically solved by directly applying the formulas derived. The analytical results reveal that the flexoelectric effect is present in isotropic materials. In addition, the incorporation of both the couple stress and flexoelectric effects always leads to increased values of the beam bending stiffness.

MSC:

74-XX Mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lam, DCC, Yang, F, Chong, ACM, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids 2003; 51: 1477-1508. · Zbl 1077.74517
[2] McFarland, AW, Colton, JS. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 2005; 15: 1060-1067.
[3] Zhang, GY, Gao, X-L. A new Bernoulli-Euler beam model based on a reformulated strain gradient elasticity theory. Math Mech Solids 2020; 25: 630-643. · Zbl 1446.74156
[4] Zhang, GY, Qu, YL, Gao, X-L, et al. A transversely isotropic magneto-electro-elastic Timoshenko beam model incorporating microstructure and foundation effects. Mech Mater 2020; 149: 103412-1∼13.
[5] Maranganti, R, Sharma, ND, Sharma, P. Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys Rev B 2006; 74: 014110-1∼14.
[6] Qu, YL, Jin, F, Yang, JS. Effects of mechanical fields on mobile charges in a composite beam of flexoelectric dielectrics and semiconductors. J Appl Phys 2020; 127: 194502-1∼6.
[7] Liu, W, Deng, F, Xie, S, et al. Electromechanical analysis of direct and converse flexoelectric effects under a scanning probe tip. J Mech Phys Solids 2020; 142: 104020-1∼16.
[8] Wang, L, Liu, S, Feng, X, et al. Flexoelectronics of centrosymmetric semiconductors. Nat Nanotechnol 2020; 15: 661-667.
[9] Maugin, GA. Nonlocal theories or gradient-type theories: a matter of convenience. Arch Mech 1979; 31: 15-26. · Zbl 0424.73006
[10] Yang, JS. A review of a few topics in piezoelectricity. Appl Mech Rev 2006; 59: 335-345.
[11] Eringen, AC. Theory of nonlocal piezoelectricity. J Math Phys 1984; 25: 717-727. · Zbl 0547.73093
[12] Mindlin, RD. Polarization gradient in elastic dielectrics. Int J Solids Struct 1968; 4: 637-642. · Zbl 0159.57001
[13] Kafadar, CB. Theory of multipoles in classical electromagnetism. Int J Eng Sci 1971; 9: 831-853. · Zbl 0223.31012
[14] Demiray, H, Eringen, CA. On the constitutive relations of polar elastic dielectrics. Lett Appl Eng Sci 1973; 1: 517-527.
[15] Ma, WH, Cross, LE. Observation of the flexoelectric effect in relaxor \(Pb(Mg_{1/3}Nb_{2/3})O_3\) ceramics. Appl Phys Lett 2001; 78: 2920-2921.
[16] Krichen, S, Sharma, P. Flexoelectricity: a perspective on an unusual electromechanical coupling. J Appl Mech 2016; 83: 030801-1∼5.
[17] Enakoutsa, K, Corte, AD, Giorgio, I. A model for elastic flexoelectric materials including strain gradient effects. Math Mech Solids 2016; 21: 242-254. · Zbl 1333.74034
[18] Mindlin, RD. Microstructure in linear elasticity. Arch Ration Mech Anal 1964; 16: 51-78. · Zbl 0119.40302
[19] Fleck, NA, Hutchinson, JW. Strain gradient plasticity. Adv Appl Mech 1997; 33: 296-361. · Zbl 0894.73031
[20] Toupin, RA. Elastic materials with couple-stresses. Arch Ration Mech Anal 1962; 11: 385-414. · Zbl 0112.16805
[21] Mindlin, RD. Influence of couple-stresses on stress concentrations. Exp Mech 1963; 3: 1-7.
[22] Koiter, WT. Couple-stresses in the theory of elasticity. Proc K Ned Akad Wet B 1964; 67: 17-44. · Zbl 0124.17405
[23] Yang, F, Chong, ACM, Lam, DCC, et al. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002; 39: 2731-2743. · Zbl 1037.74006
[24] Park, SK, Gao, X-L. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z Angew Math Phys 2008; 59: 904-917. · Zbl 1157.74014
[25] Hadjesfandiari, AR, Dargush, GF. Couple stress theory for solids. Int J Solids Struct 2011; 48: 2496-2510.
[26] Gad, AI, Gao, X-L. Two versions of the extended Hill’s lemma for non-Cauchy continua based on the couple stress theory. Math Mech Solids 2021; 26: 244-262. · Zbl 07357401
[27] Qu, YL, Li, P, Zhang, GY, et al. A microstructure-dependent anisotropic magneto-electro-elastic Mindlin plate model based on an extended modified couple stress theory. Acta Mech 2020; 231: 4323-4350. · Zbl 1451.74153
[28] Wang, GF, Yu, SW, Feng, XQ. A piezoelectric constitutive theory with rotation gradient effects. Eur J Mech A Solid 2004; 23: 455-466. · Zbl 1060.74542
[29] Hadjesfandiari, AR. Size-dependent piezoelectricity. Int J Solids Struct 2013; 50: 2781-2791.
[30] Qu, YL, Jin, F, Yang, JS. Magnetically induced charge motion in the bending of a beam with flexoelectric semiconductor and piezomagnetic dielectric layers. J Appl Phys 2021; 129: 064503.
[31] Qu, YL, Jin, F, Yang, JS. Flexoelectric effects in second-order extension of rods. Mech Res Commun 2021; 111: 103625.
[32] El Dhaba, AR, Gabr, ME. Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion. Math Mech Solids 2020; 25: 820-837. · Zbl 1446.74119
[33] Qu, YL, Jin, F, Yang, JS. Torsion of a flexoelectric semiconductor rod with a rectangular cross section. Arch Appl Mech. Epub ahead of print 7 January 2021. DOI: 10.1007/s00419-020-01867-0.
[34] Li, A, Zhou, S, Qi, L, et al. A flexoelectric theory with rotation gradient effects for elastic dielectrics. Model Simul Mater Sci Eng 2016; 24: 015009.
[35] Nikolov, S, Han, C-S, Raabe, D. On the origin of size effects in small-strain elasticity of solid polymers. Int J Solids Struct 2007; 44: 1582-1592. · Zbl 1125.74007
[36] Griffiths, DJ. Introduction to electrodynamics. 4th ed. Glenview: Cambridge University Press, 2017. · Zbl 1377.78001
[37] Qu, YL, Li, P, Jin, F. A general dynamic model based on Mindlin’s high-frequency theory and the microstructure effect. Acta Mech 2020; 231: 3847-3869. · Zbl 1451.74042
[38] Yang, JS. An introduction to the theory of piezoelectricity. New York: Springer, 2005. · Zbl 1066.74001
[39] Steigmann, DJ. The variational structure of a nonlinear theory for spatial lattices. Meccanica 1996; 31: 441-455. · Zbl 0868.73090
[40] Gao, X-L, Mall, S. Variational solution for a cracked mosaic model of woven fabric composites. Int J Solids Struct 2001; 38: 855-874. · Zbl 1045.74019
[41] Yang, JFC, Lakes, RS. Experimental study of micropolar and couple stress elasticity in compact bone in bending. J Biomech 1982; 15: 91-98.
[42] Sharma, ND, Maranganti, R, Sharma, P. On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J Mech Phys Solids 2007; 55: 2328-2350. · Zbl 1171.74016
[43] Park, SK, Gao, X-L. Bernoulli-Euler beam model based on a modified couple stress theory. J Micromech Microeng 2006; 16: 2355-2359.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.