Research Papers

Mixed Finite Elements for Flexoelectric Solids

[+] Author and Article Information
Feng Deng

State Key Laboratory for Strength
and Vibration of Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: dengfeng12399@stu.xjtu.edu.cn

Qian Deng

State Key Laboratory for Strength
and Vibration of Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: tonydqian@mail.xjtu.edu.cn

Wenshan Yu

State Key Laboratory for Strength
and Vibration of Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China

Shengping Shen

State Key Laboratory for Strength
and Vibration of Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: sshen@mail.xjtu.edu.cn

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 18, 2017; final manuscript received May 24, 2017; published online June 14, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(8), 081004 (Jun 14, 2017) (12 pages) Paper No: JAM-17-1207; doi: 10.1115/1.4036939 History: Received April 18, 2017; Revised May 24, 2017

Flexoelectricity (FE) refers to the two-way coupling between strain gradients and the electric field in dielectric materials, and is universal compared to piezoelectricity, which is restricted to dielectrics with noncentralsymmetric crystalline structure. Involving strain gradients makes the phenomenon of flexoelectricity size dependent and more important for nanoscale applications. However, strain gradients involve higher order spatial derivate of displacements and bring difficulties to the solution of flexoelectric problems. This dilemma impedes the application of such universal phenomenon in multiple fields, such as sensors, actuators, and nanogenerators. In this study, we develop a mixed finite element method (FEM) for the study of problems with both strain gradient elasticity (SGE) and flexoelectricity being taken into account. To use C0 continuous elements in mixed FEM, the kinematic relationship between displacement field and its gradient is enforced by Lagrangian multipliers. Besides, four types of 2D mixed finite elements are developed to study the flexoelectric effect. Verification as well as validation of the present mixed FEM is performed through comparing numerical results with analytical solutions for an infinite tube problem. Finally, mixed FEM is used to simulate the electromechanical behavior of a 2D block subjected to concentrated force or voltage. This study proves that the present mixed FEM is an effective tool to explore the electromechanical behaviors of materials with the consideration of flexoelectricity.

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Yudin, P. , and Tagantsev, A. , 2013, “ Fundamentals of Flexoelectricity in Solids,” Nanotechnology, 24(43), p. 432001. [CrossRef] [PubMed]
Kawai, H. , 1969, “ The Piezoelectricity of Poly (Vinylidene Fluoride),” J. Appl. Phys., 8(7), p. 975. [CrossRef]
Deng, Q. , Liu, L. , and Sharma, P. , 2014, “ Flexoelectricity in Soft Materials and Biological Membranes,” J. Mech. Phys. Solids, 62, pp. 209–227. [CrossRef]
Petrov, A. G. , 2002, “ Flexoelectricity of Model and Living Membranes,” Biochim. Biophys. Acta, Biomembr., 1561(1), pp. 1–25. [CrossRef]
Ma, W. , and Cross, L. E. , 2006, “ Flexoelectricity of Barium Titanate,” Appl. Phys. Lett., 88(23), p. 232902. [CrossRef]
Narvaez, J. , and Catalan, G. , 2014, “ Origin of the Enhanced Flexoelectricity of Relaxor Ferroelectrics,” Appl. Phys. Lett., 104(16), p. 162903. [CrossRef]
Shu, L. , Wei, X. , Jin, L. , Li, Y. , Wang, H. , and Yao, X. , 2013, “ Enhanced Direct Flexoelectricity in Paraelectric Phase of Ba(Ti0.87Sn0.13)O3 Ceramics,” Appl. Phys. Lett., 102(15), p. 152904. [CrossRef]
Tagantsev, A. , 1986, “ Piezoelectricity and Flexoelectricity in Crystalline Dielectrics,” Phys. Rev. B, 34(8), p. 5883. [CrossRef]
Čepič, M. , and Žekš, B. , 2001, “ Flexoelectricity and Piezoelectricity: The Reason for the Rich Variety of Phases in Antiferroelectric Smectic Liquid Crystals,” Phys. Rev. Lett., 87(8), p. 085501. [CrossRef] [PubMed]
Prost, J. , and Pershan, P. S. , 1976, “ Flexoelectricity in Nematic and Smectic—A Liquid Crystals,” J. Appl. Phys., 47(6), pp. 2298–2312. [CrossRef]
Kogan, S. M. , 1964, “ Piezoelectric Effect During Inhomogeneous Deformation and Acoustic Scattering of Carriers in Crystals,” Sov. Phys. Solid State, 5(10), pp. 2069–2070.
Nguyen, T. D. , Mao, S. , Yeh, Y. W. , Purohit, P. K. , and McAlpine, M. C. , 2013, “ Nanoscale Flexoelectricity,” Adv. Mater., 25(7), pp. 946–974. [CrossRef] [PubMed]
Chu, B. , and Salem, D. , 2012, “ Flexoelectricity in Several Thermoplastic and Thermosetting Polymers,” Appl. Phys. Lett., 101(10), p. 103905. [CrossRef]
Lu, J. , Lv, J. , Liang, X. , Xu, M. , and Shen, S. , 2016, “ Improved Approach to Measure the Direct Flexoelectric Coefficient of Bulk Polyvinylidene Fluoride,” J. Appl. Phys., 119(9), p. 094104. [CrossRef]
Petrov, A. G. , 2006, “ Electricity and Mechanics of Biomembrane Systems: Flexoelectricity in Living Membranes,” Anal. Chim. Acta, 568(1), pp. 70–83. [CrossRef] [PubMed]
Maranganti, R. , Sharma, N. , and Sharma, P. , 2006, “ Electromechanical Coupling in Nonpiezoelectric Materials Due to Nanoscale Nonlocal Size Effects: Green's Function Solutions and Embedded Inclusions,” Phys. Rev. B, 74(1), p. 014110. [CrossRef]
Abdollahi, A. , and Arias, I. , 2015, “ Constructive and Destructive Interplay Between Piezoelectricity and Flexoelectricity in Flexural Sensors and Actuators,” ASME J. Appl. Mech., 82(12), p. 121003. [CrossRef]
Mashkevich, V. , and Tolpygo, K. , 1957, “ Electrical, Optical and Elastic Properties of Diamond Type Crystals,” Sov. Phys. JETP-USSR, 5(3), pp. 435–439.
Majdoub, M. , Sharma, P. , and Cagin, T. , 2008, “ Enhanced Size-Dependent Piezoelectricity and Elasticity in Nanostructures Due to the Flexoelectric Effect,” Phys. Rev. B, 77(12), p. 125424. [CrossRef]
Sharma, N. , Landis, C. , and Sharma, P. , 2010, “ Piezoelectric Thin-Film Superlattices Without Using Piezoelectric Materials,” J. Appl. Phys., 108(2), p. 024304. [CrossRef]
Sharma, N. , Maranganti, R. , and Sharma, P. , 2007, “ On the Possibility of Piezoelectric Nanocomposites Without Using Piezoelectric Materials,” J. Mech. Phys. Solids, 55(11), pp. 2328–2350. [CrossRef]
Hu, S. , and Shen, S. , 2010, “ Variational Principles and Governing Equations in Nano-Dielectrics With the Flexoelectric Effect,” Sci. China: Phys., Mech. Astron., 53(8), pp. 1497–1504. [CrossRef]
Shen, S. , and Hu, S. , 2010, “ A Theory of Flexoelectricity With Surface Effect for Elastic Dielectrics,” J. Mech. Phys. Solids, 58(5), pp. 665–677. [CrossRef]
Mohammadi, P. , Liu, L. , and Sharma, P. , 2014, “ A Theory of Flexoelectric Membranes and Effective Properties of Heterogeneous Membranes,” ASME J. Appl. Mech., 81(1), p. 011007. [CrossRef]
Liu, L. , 2014, “ An Energy Formulation of Continuum Magneto-Electro-Elasticity With Applications,” J. Mech. Phys. Solids, 63, pp. 451–480. [CrossRef]
Deng, Q. , Kammoun, M. , Erturk, A. , and Sharma, P. , 2014, “ Nanoscale Flexoelectric Energy Harvesting,” Int. J. Solids Struct., 51(18), pp. 3218–3225. [CrossRef]
Liang, X. , Zhang, R. , Hu, S. , and Shen, S. , 2017, “ Flexoelectric Energy Harvesters Based on Timoshenko Laminated Beam Theory,” J. Intell. Mater. Syst. Struct., epub.
Abdollahi, A. , Peco, C. , Millán, D. , Arroyo, M. , and Arias, I. , 2014, “ Computational Evaluation of the Flexoelectric Effect in Dielectric Solids,” J. Appl. Phys., 116(9), p. 093502. [CrossRef]
Ma, W. , and Cross, L. E. , 2002, “ Flexoelectric Polarization of Barium Strontium Titanate in the Paraelectric State,” Appl. Phys. Lett., 81(18), pp. 3440–3442. [CrossRef]
Ma, W. , and Cross, L. E. , 2005, “ Flexoelectric Effect in Ceramic Lead Zirconate Titanate,” Appl. Phys. Lett., 86(7), p. 072905. [CrossRef]
Zubko, P. , Catalan, G. , and Tagantsev, A. K. , 2013, “ Flexoelectric Effect in Solids,” Annu. Rev. Mater. Res., 43(1), pp. 387–421. [CrossRef]
Ahmadpoor, F. , and Sharma, P. , 2015, “ Flexoelectricity in Two-Dimensional Crystalline and Biological Membranes,” Nanoscale, 7(40), pp. 16555–16570. [CrossRef] [PubMed]
Krichen, S. , and Sharma, P. , 2016, “ Flexoelectricity: A Perspective on an Unusual Electromechanical Coupling,” ASME J. Appl. Mech., 83(3), p. 030801. [CrossRef]
Mao, S. , and Purohit, P. K. , 2014, “ Insights Into Flexoelectric Solids From Strain-Gradient Elasticity,” ASME J. Appl. Mech., 81(8), p. 081004. [CrossRef]
Ray, M. , 2014, “ Exact Solutions for Flexoelectric Response in Nanostructures,” ASME J. Appl. Mech., 81(9), p. 091002. [CrossRef]
Ahluwalia, R. , Tagantsev, A. K. , Yudin, P. , Setter, N. , Ng, N. , and Srolovitz, D. J. , 2014, “ Influence of Flexoelectric Coupling on Domain Patterns in Ferroelectrics,” Phys. Rev. B, 89(17), p. 174105. [CrossRef]
Chen, H. , Soh, A. K. , and Ni, Y. , 2014, “ Phase Field Modeling of Flexoelectric Effects in Ferroelectric Epitaxial Thin Films,” Acta Mech., 225(4–5), pp. 1323–1333. [CrossRef]
Gu, Y. , Hong, Z. , Britson, J. , and Chen, L.-Q. , 2015, “ Nanoscale Mechanical Switching of Ferroelectric Polarization Via Flexoelectricity,” Appl. Phys. Lett., 106(2), p. 022904. [CrossRef]
Chen, W. , Zheng, Y. , Feng, X. , and Wang, B. , 2015, “ Utilizing Mechanical Loads and Flexoelectricity to Induce and Control Complicated Evolution of Domain Patterns in Ferroelectric Nanofilms,” J. Mech. Phys. Solids, 79, pp. 108–133. [CrossRef]
Yvonnet, J. , and Liu, L. , 2017, “ A Numerical Framework for Modeling Flexoelectricity and Maxwell Stress in Soft Dielectrics at Finite Strains,” Comput. Methods Appl. Mech. Eng., 313, pp. 450–482. [CrossRef]
Xia, Z. C. , and Hutchinson, J. W. , 1996, “ Crack Tip Fields in Strain Gradient Plasticity,” J. Mech. Phys. Solids, 44(10), pp. 1621–1648. [CrossRef]
Shu, J. , and Fleck, N. , 1998, “ The Prediction of a Size Effect in Microindentation,” Int. J. Solids Struct., 35(13), pp. 1363–1383. [CrossRef]
Herrmann, L. , 1983, “ Mixed Finite Elements for Couple-Stress Analysis,” Hybrid and Mixed FEM, S. N. Atluri, R. H. Gallagher, and O. C. Zienkiewicz , eds., Wiley, New York.
Mindlin, R. D. , 1964, “ Micro-Structure in Linear Elasticity,” Arch. Ration. Mech. Anal., 16(1), pp. 51–78. [CrossRef]
Shu, J. Y. , King, W. E. , and Fleck, N. A. , 1999, “ Finite Elements for Materials With Strain Gradient Effects,” Int. J. Numer. Methods Eng., 44(3), pp. 373–391. [CrossRef]
Amanatidou, E. , and Aravas, N. , 2002, “ Mixed Finite Element Formulations of Strain-Gradient Elasticity Problems,” Comput. Methods Appl. Mech. Eng., 191(15), pp. 1723–1751. [CrossRef]
Mao, S. , Purohit, P. K. , and Aravas, N. , 2016, “ Mixed Finite-Element Formulations in Piezoelectricity and Flexoelectricity,” Proc. R. Soc. A, 472(2190), p. 20150879. [CrossRef]
Aravas, N. , 2011, “ Plane-Strain Problems for a Class of Gradient Elasticity Models—A Stress Function Approach,” J. Elasticity, 104(1–2), pp. 45–70. [CrossRef]
Zienkiewicz, O. C. , Taylor, R. L. , and Taylor, R. L. , 1977, The Finite Element Method, McGraw-Hill, London.
Gao, X.-L. , and Park, S. , 2007, “ Variational Formulation of a Simplified Strain Gradient Elasticity Theory and Its Application to a Pressurized Thick-Walled Cylinder Problem,” Int. J. Solids Struct., 44(22), pp. 7486–7499. [CrossRef]
Askes, H. , and Aifantis, E. C. , 2011, “ Gradient Elasticity in Statics and Dynamics: An Overview of Formulations, Length Scale Identification Procedures, Finite Element Implementations and New Results,” Int. J. Solids Struct., 48(13), pp. 1962–1990. [CrossRef]
Boffi, D. , and Lovadina, C. , 1997, “ Analysis of New Augmented Lagrangian Formulations for Mixed Finite Element Schemes,” Numer. Math., 75(4), pp. 405–419. [CrossRef]


Grahic Jump Location
Fig. 4

Comparison of FEM results with analytical solution for (a) radial displacement, (b) electric potential, (c) radial strain, and (d) circumferential strain versus radius. Considering the axisymmetry of model, all results are extracted from the FEM results along the 45 deg axis (marked by black lines in the contours).

Grahic Jump Location
Fig. 3

Schematic of quadrilateral FEM Mesh for a quarter of the model shown in Fig. 1. Total numbers of quadrilateral elements (Q47 and Q59) are 360. Each quadrilateral element can be divided into two triangular elements. Thus, such a model can be further meshed with 720 triangular elements (T37, T45).

Grahic Jump Location
Fig. 2

An infinite length tube with an axisymmetric cross section. The inner and outer radii of model are ri = 10 μm and ro=20μm, respectively. On the inner and outer surfaces, the radial displacements are ui = 0.045 μm and uo = 0.05 μm. Voltage difference across the internal and external surface is 1.0 V.

Grahic Jump Location
Fig. 1

Schematic of four types of elements in the triangular (T) and quadrilateral (Q) shapes. (a) T37, (b) Q47, (c) T45, and (d) Q59. Elements T37 and T45 have seven nodes (three corner nodes, three midside nodes, and one inner node). Elements Q47 and Q59 have nine nodes (four corner nodes, four midside nodes, and one inner node). “○” denotes components of displacement (u1, u2) and electric potential (φ). “+” and “×” are DOFs of displacement gradient (ψ11, ψ12, ψ21, ψ22) and Lagrangian multiplier (α11, α12, α21, α22).

Grahic Jump Location
Fig. 6

Plane strain model of a block subjected to a concentrated (a) force and (b) voltage. The width and height of the block are 20 μm and 10 μm, respectively. At the bottom of block, displacements in the horizontal and vertical directions are fixed to be zero and so is electric potential. Concentrated force F and voltage V are 100 μN and 5 V, respectively.

Grahic Jump Location
Fig. 5

Comparison of FEM results with exact solution of five special cases for (a) volume strain, (b) radial gradient of volume strain, (c) electric potential, and (d) electric field versus radius. Scatter and solid lines denote FEM results using element Q59 and analytical solution, respectively. Note that FEM results using elements T37, T45, and Q47 are not shown since they are equal to those for element Q59.

Grahic Jump Location
Fig. 7

FEM meshes using quadrilateral elements for the model in Fig. 6

Grahic Jump Location
Fig. 8

Distributions of (a) electric potential and (b) electric field component E2 for block subjected to concentrated force

Grahic Jump Location
Fig. 9

Variation of electric potential and electric field component E2 at top surface of block

Grahic Jump Location
Fig. 10

Distributions of strains (a) ε11 and (b) ε22 generated by applied voltage via flexoelectricity

Grahic Jump Location
Fig. 11

Variations of (a) electric potential along symmetry axis (x1 = 0) and (b) electric field component E2 at the bottom surface of the block. Results from SGE&FE are compared with those from the electrostatic theory.



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