Abstract
Natural defects such as nano inclusions and nanocracks are inevitable in dielectric materials. When materials are subjected to mechanical loading, the strain gradient around crack tips and inclusions would become large and induce significant flexoelectric fields. In contrast to classical crack–inclusion problems, the interactions between these flexoelectric fields may locally change the electromechanical behaviors of materials and result in some interesting phenomena. To better understand the crack–inclusion interactions in flexoelectric solids, in this work, we use a collocation mixed finite element method to model and analyze the flexoelectric fields around the crack tip and inclusion. On the basis of the J-integral, we analyze how the flexoelectric effect affect the interaction energy between nanocracks and nearby nano inclusions. This work proposes a new coupling mechanism in crack–inclusion problems and may inspire future experiments in flexoelectric solids.
Issue Section:
Research Papers
References
1.
Chandratre
, S.
, and Sharma
, P.
, 2012
, “Coaxing Graphene to Be Piezoelectric
,” Appl. Phys. Lett.
, 100
(2
), p. 023114
. 2.
Mozaffari
, K.
, Ahmadpoor
, F.
, Deng
, Q.
, and Sharma
, P.
, 2023
, “A Minimal Physics-Based Model for Musical Perception
,” Proc. Natl. Acad. Sci. USA
, 120
(5
), p. e2216146120
. 3.
Deng
, Q.
, Ahmadpoor
, F.
, Brownell
, W. E.
, and Sharma
, P.
, 2019
, “The Collusion of Flexoelectricity and Hopf Bifurcation in the Hearing Mechanism
,” J. Mech. Phys. Solids.
, 130
(1
), pp. 245
–261
. 4.
Zubko
, P.
, Catalan
, G.
, and Tagantsev
, A.
, 2013
, “Flexoelectric Effect in Solids
,” Ann. Rev. Materials Res.
, 43
(7
), pp. 387
–421
. 5.
Wang
, B.
, Gu
, Y.
, Zhang
, S.
, and Chen
, L.-Q.
, 2019
, “Flexoelectricity in Solids: Progress, Challenges, and Perspectives
,” Prog. Mater. Sci.
, 106
(1
), p. 100570
. 6.
Wen
, X.
, Li
, D.
, Tan
, K.
, Deng
, Q.
, and Shen
, S.
, 2019
, “Flexoelectret: An Electret With Tunable Flexoelectric-Like Response
,” Phys. Rev. Lett.
, 19
(4
), p. 148001
. 7.
Bhaskar
, U.
, Banerjee
, N.
, Abdollahi
, A.
, Wang
, Z.
, Schlom
, D.
, Rijnders
, G.
, and Catalan
, G.
, 2016
, “A Flexoelectric Microelectromechanical System on Silicon
,” Nat. Nanotechnol.
, 11
(3
), pp. 263
–266
. 8.
Huang
, W.
, Kwon
, S.-R.
, Zhang
, S.
, Yuan
, F.-G.
, and Jiang
, X.
, 2014
, “A Trapezoidal Flexoelectric Accelerometer
,” J. Intell. Mater. Syst. Struct.
, 25
(3
), pp. 271
–277
. 9.
Jiang
, X.
, Huang
, W.
, and Zhang
, S.
, 2013
, “Flexoelectric Nano-Generator: Materials, Structures and Devices
,” Nano Energy
, 2
(6
), pp. 1079
–1092
. 10.
Abdollahi
, A.
, Peco
, C.
, Millán
, D.
, Arroyo
, M.
, Catalan
, G.
, and Arias
, I.
, 2015
, “Fracture Toughening and Toughness Asymmetry Induced by Flexoelectricity
,” Phys. Rev. B
, 92
(9
), p. 094101
. 11.
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
. 12.
Sladek
, J.
, Sladek
, V.
, Stanak
, P.
, Zhang
, C.
, and Tan
, C.-L.
, 2017
, “Fracture Mechanics Analysis of Size-Dependent Piezoelectric Solids
,” Int. J. Solids. Struct.
, 113–114
(1
), pp. 1
–9
. 13.
Xu
, M.
, Tian
, X.
, Deng
, Q.
, Li
, Q.
, and Shen
, S.
, 2023
, “Directly Observing the Evolution of Flexoelectricity at the Tip of Nanocracks
,” Nano. Lett.
, 23
(1
), pp. 66
–72
. 14.
Mao
, S.
, and Purohit
, P.
, 2015
, “Defects in Flexoelectric Solids
,” J. Mech. Phys. Solids
, 84
(7
), pp. 95
–115
. 15.
Wang
, H.
, Jiang
, X.
, Wang
, Y.
, Stark
, R. W.
, van Aken
, P. A.
, Mannhart
, J.
, and Boschker
, H.
, 2020
, “Direct Observation of Huge Flexoelectric Polarization Around Crack Tips
,” Nano. Lett.
, 20
(1
), pp. 88
–94
. 16.
Cordero
, K.
, Kianirad
, H.
, Canalias
, C.
, Sort
, J.
, and Catalan
, G.
, 2019
, “Flexoelectric Fracture-Ratchet Effect in Ferroelectrics
,” Phys. Rev. Lett.
, 122
(13
), p. 135502
.17.
Sladek
, J.
, Sladek
, V.
, Wünsche
, M.
, and Kasala
, J.
, 2018
, “Gradient Piezoelectricity for Cracks Under an Impact Load
,” Int. J. Fracture
, 210
(1
), pp. 95
–111
. 18.
Sladek
, J.
, Sladek
, V.
, Wünsche
, M.
, and Zhang
, C.
, 2018
, “Effects of Electric Field and Strain Gradients on Cracks in Piezoelectric Solids
,” Eur. J. Mech. – A/Solids
, 71
(1
), pp. 187
–198
. 19.
Ma
, L.
, Lu
, T. J.
, and Korsunsky
, A. M.
, 2005
, “Vector J-Integral Analysis of Crack Interaction With Pre-existing Singularities
,” J. Appl. Mech.
, 73
(5
), pp. 876
–883
. 20.
Li
, R.
, and Chudnovsky
, A.
, 1993
, “Energy Analysis of Crack Interaction With an Elastic Inclusion
,” Int. J. Fracture
, 63
(10
), pp. 247
–261
. 21.
Li
, R.
, and Chudnovsky
, A.
, 1993
, “Variation of the Energy Release Rate as a Crack Approaches and Passes Through an Elastic Inclusion
,” Int. J. Fract.
, 59
(4
), pp. R69
–R74
. 22.
Li
, Z.
, and Yang
, L.
, 2004
, “The Near-Tip Stress Intensity Factor for a Crack Partially Penetrating an Inclusion
,” J. Appl. Mech.
, 71
(4
), pp. 465
–469
. 23.
Li
, Q.
, and Lv
, J.
, 2017
, “Invariant Integrals of Crack Interaction With an Inhomogeneity
,” Eng. Fract. Mech.
, 171
(1
), pp. 76
–84
. 24.
Tamate
, O.
, 1968
, “Effect of a Circular Inclusion on Stresses Around a Line Crack in a Sheet Under Tension
,” Int. J. Fracture
, 4
(1
), pp. 257
–266
. 25.
Erdogan
, F.
, Gupta
, G.
, and Ratwani
, M.
, 1975
, “Interaction Between a Circular Inclusion and Arbitrarily Oriented Crack
,” J. Appl. Mech.
, 41
(4
), pp. 1007
–1013
. 26.
Hsu
, Y.
, and Shivakumar
, V.
, 1976
, “Interaction Between an Elastic Circular Inclusion and Two Symmetrically Placed Collinear Cracks
,” Int. J. Fracture
, 12
(4
), pp. 619
–630
. 27.
Xiao
, Z.
, and Pae
, K.
, 1991
, “The Interaction Between a Penny-Shaped Crack and a Spherical Inhomogeneity in an Infinite Solid Under Uniaxial Tension
,” Acta Mech.
, 90
(3
), pp. 91
–104
. 28.
Guo
, Y.
, and Li
, Q.
, 2017
, “Material Configurational Forces Applied to Mixed Mode Crack Propagation
,” Theor. Appl. Fract. Mec.
, 89
(1
), pp. 147
–157
. 29.
Liang
, X.
, Hu
, S.
, and Shen
, S.
, 2014
, “A New Bernoulli-Euler Beam Model Based on a Simplified Strain Gradient Elasticity Theory and Its Applications
,” Composite Struct.
, 111
(1
), pp. 317
–323
. 30.
Zhang
, R.
, Liang
, X.
, and Shen
, S.
, 2015
, “A Timoshenko Dielectric Beam Model With Flexoelectric Effect
,” Meccanica
, 51
(5
), pp. 1181
–1188
. 31.
Deng
, Q.
, 2017
, “Size and Shape Effects on the Electromechanical Behavior of a Flexoelectric Truncated Cone
,” J. Appl. Mech.
, 84
(10
), p. 101007
. 32.
Sladek
, J.
, Stanak
, P.
, Han
, Z.
, Sladek
, V.
, and Atluri
, S.
, 2013
, “Applications of the Mlpg Method in Engineering & Sciences: A Review
,” CMES – Computer Model. Eng. Sci.
, 92
(6
), pp. 423
–475
.33.
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
. 34.
Codony
, D.
, Marco
, O.
, Fernández-Méndez
, S.
, and Arias
, I.
, 2019
, “An Immersed Boundary Hierarchical B-spline Method for Flexoelectricity
,” Comput. Methods. Appl. Mech. Eng.
, 354
(1
), pp. 750
–782
. 35.
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
(1
), pp. 450
–482
. 36.
Mao
, S.
, Purohit
, P.
, and Aravas
, N.
, 2016
, “Mixed Finite-Element Formulations in Piezoelectricity and Flexoelectricity
,” Proc. Royal Soc. A: Math. Phys. Engi. Sci.
, 472
(6
), p. 20150879
. 37.
Deng
, F.
, Deng
, Q.
, Yu
, W.
, and Shen
, S.
, 2017
, “Mixed Finite Elements for Flexoelectric Solids
,” J. Appl. Mech.
, 84
(8
), p. 081004
. 38.
Tian
, X.
, Sladek
, J.
, Sladek
, V.
, Deng
, Q.
, and Li
, Q.
, 2021
, “A Collocation Mixed Finite Element Method for the Analysis of Flexoelectric Solids
,” Int. J. Solids. Struct.
, 217–218
(1
), pp. 27
–39
. 39.
Beattie
, A.
, and Samara
, G.
, 2003
, “Pressure Dependence of the Elastic Constants of SrTiO3
,” J. Appl. Phys.
, 42
(6
), pp. 2376
–2381
. 40.
Sumigawa
, T.
, Shimada
, T.
, HUANG
, K.
, Mizuno
, Y.
, Hagiwara
, Y.
, Ozaki
, N.
, and Kitamura
, T.
, 2022
, “Ultrasmall-Scale Brittle Fracture Initiated From a Dislocation in SrTiO3
,” Nano. Lett.
, 22
(5
), pp. 2077
–2084
. 41.
Lu
, Y.
, Jia
, D.
, Gao
, F.
, Chen
, Z.
, and Hu
, T.
, 2014
, “First-Principles Study on the Elastic Properties of Sr–Ti–O Ceramics
,” Solid State Commun.
, 182
(1
), pp. 43
–46
. 42.
Hong
, J.
, and Vanderbilt
, D.
, 2013
, “First-Principles Theory and Calculation of Flexoelectricity
,” Phys. Rev. B
, 88
(17
), p. 174107
. 43.
Noether
, E.
, 1971
, “Invariant Variation Problems
,” Transp. Statist. Phys.
, 1
(3
), pp. 186
–207
. 44.
Li
, Q.
, Lv
, J.
, Guo
, Y.
, and Tian
, X.
, 2018
, “A Consistent Framework of Material Configurational Mechanics in Piezoelectric Materials
,” Acta Mech.
, 229
(1
), pp. 299
–322
. 45.
Tian
, X.
, Xu
, M.
, Zhou
, H.
, Deng
, Q.
, Li
, Q.
, Sladek
, J.
, and Sladek
, V.
, 2022
, “Analytical Studies on Mode III Fracture in Flexoelectric Solids
,” ASME J. Appl. Mech.
, 89
(4
), p. 041006
. 46.
Tian
, X.
, Li
, Q.
, and Deng
, Q.
, 2019
, “The J-Integral in Flexoelectric Solids
,” Int. J. Fracture
, 215
(1
), pp. 67
–76
. 47.
Moran
, B.
, and Shih
, C.
, 1987
, “A General Treatment of Crack Tip Contour Integrals
,” Int. J. Fracture
, 35
(4
), pp. 295
–310
. 48.
Shivakumar
, K.
, and Raju
, I.
, 1992
, “An Equivalent Domain Integral Method for Three-Dimensional Mixed-Mode Fracture Problems
,” Eng. Fract. Mech.
, 42
(6
), pp. 935
–959
. Copyright © 2023 by ASME
You do not currently have access to this content.
