Abstract
Magnetoelectroelastic (MEE) materials and structures have been extensively applied in MEE devices such as sensors and transducers, microelectromechanical systems, and smart structures. In order to assess the strength and durability of such materials and structures, exhaustive theoretical and numerical investigations have been conducted over the past two decades. The main purpose of this paper is to present a state-of-the-art review and a critical discussion on the research in the field of the MEE fracture mechanics. Following an introduction, the basic theory of the fracture mechanics in linear magnetoelectroelasticity is explained with special emphasis on the constitutive equations related to different fracture modes, magnetoelectric (ME) crack-face boundary conditions, and fracture parameters for two-dimensional (2D) plane problems. Then, the state of the art of the research on the fracture mechanics of the MEE materials and structures is reviewed and summarized, including 2D antiplane and in-plane as well as three-dimensional (3D) analyses under both static and dynamic loadings. The magnetoelectric effects on the fracture parameters are revealed and discussed. Moreover, numerical investigations based on the finite element method (FEM), boundary element method (BEM), meshless methods, and other novel methods are also reviewed for 2D plane and 3D fracture problems. Finally, some conclusions are drawn with several prospects to open questions and demanding future research topics. In particular, experimental observations are urgently needed to verify the validity of the theoretical predictions of the various fracture criteria. Another great challenge is to tackle the nonlinear phenomena and domain switching in the fracture process zone.
References
1.
Fiebig
, M.
, 2005
, “Revival of the Magnetoelectric Effect
,” J. Phys. D: Appl. Phys.
, 38
(8
), pp. R123
–R152
.10.1088/0022-3727/38/8/R012.
Nan
, C. W.
, Bichurin
, M. I.
, Dong
, S. X.
, Viehland
, D.
, and Srinivasan
, G.
, 2008
, “Multiferroic Magnetoelectric Composites: Historical Perspective, Status, and Future Directions
,” J. Appl. Phys.
, 103
(3
), p. 031101
.10.1063/1.28364103.
Huang
, J. H.
, and Kuo
, W. S.
, 1997
, “The Analysis of Piezoelectric/Piezomagnetic Composite Materials Containing Ellipsoidal Inclusions
,” J. Appl. Phys.
, 81
(3
), pp. 1378
–1386
.10.1063/1.3638744.
Li
, R.
, and Kardomateas
, G. A.
, 2007
, “The Mixed Mode I and II Interface Crack in Piezoelectromagneto-Elastic Anisotropic Bimaterials
,” ASME J. Appl. Mech.
, 74
(4
), pp. 614
–627
.10.1115/1.24244685.
Hu
, K. Q.
, and Li
, G. Q.
, 2005
, “Constant Moving Crack in a Magnetoelectroelastic Material Under Anti-Plane Shear Loading
,” Int. J. Solids Struct.
, 42
(9–10
), pp. 2823
–2835
.10.1016/j.ijsolstr.2004.09.0366.
Parton
, V. Z.
, 1976
, “Fracture Mechanics of Piezoelectric Materials
,” Acta Astronaut.
, 3
(9–10
), pp. 671
–683
.10.1016/0094-5765(76)90105-37.
Gao
, C. F.
, Kessler
, H.
, and Balke
, H.
, 2003
, “Crack Problems in Magnetoelectroelastic Solids. Part II: General Solution of Collinear Cracks
,” Int. J. Eng. Sci.
, 41
(9
), pp. 983
–994
.10.1016/S0020-7225(02)00324-58.
Pak
, Y. E.
, 1990
, “Crack Extension Force in a Piezoelectric Material
,” ASME J. Appl. Mech.
, 57
(3
), pp. 647
–653
.10.1115/1.28970719.
Wang
, B. L.
, and Mai
, Y. W.
, 2003
, “Crack Tip Field in Piezoelectric/Piezomagnetic Media
,” Eur. J. Mech. A - Solids
, 22
(4
), pp. 591
–602
.10.1016/S0997-7538(03)00062-710.
Hao
, T. H.
, and Shen
, Z. Y.
, 1994
, “A New Electric Boundary Condition of Electric Fracture Mechanics and Its Applications
,” Eng. Fract. Mech.
, 47
(6
), pp. 793
–802
.10.1016/0013-7944(94)90059-011.
Zhong
, X. C.
, and Li
, X. F.
, 2007
, “Magnetoelectroelastic Analysis for an Opening Crack in a Piezoelectromagnetic Solid
,” Eur. J. Mech. A - Solids
, 26
(3
), pp. 405
–417
.10.1016/j.euromechsol.2006.08.00212.
Wang
, B. L.
, and Mai
, Y. W.
, 2007
, “Applicability of the Crack-Face Electromagnetic Boundary Conditions for Fracture of Magnetoelectroelastic Materials
,” Int. J. Solids Struct.
, 44
(2
), pp. 387
–398
.10.1016/j.ijsolstr.2006.04.02813.
Feng
, W. J.
, Su
, R. K. L.
, and Pan
, E.
, 2007
, “Fracture Analysis of a Penny-Shaped Magnetically Dielectric Crack in a Magnetoelectroelastic Material
,” Int. J. Fract.
, 146
(3
), pp. 125
–138
.10.1007/s10704-007-9150-x14.
Heyer
, V.
, Schneider
, G. A.
, Balke
, H.
, Drescher
, J.
, and Bahr
, H.-A.
, 1998
, “A Fracture Criterion for Conducting Cracks in Homogeneously Poled Piezoelectric PZT-PIC 151 Ceramics
,” Acta Mater.
, 46
(18
), pp. 6615
–6622
.10.1016/S1359-6454(98)00272-915.
Wang
, T. H.
, and Zhang
, T. Y.
, 2001
, “Electrical Fracture Toughness for Electrically Conductive Deep Notches Driven by Electric Fields in Depoled Lead Zirconate Ceramics
,” Appl. Phys. Lett.
, 79
(25
), pp. 4198
–4200
.10.1063/1.142743716.
Ma
, P.
, Su
, R. K. L.
, and Feng
, W. J.
, 2015
, “Fracture Analysis of an Electrically Conductive Interface Crack With a Contact Zone in a Magnetoelectroelastic Bimaterial System
,” Int. J. Solids Struct.
, 53
, pp. 48
–57
.10.1016/j.ijsolstr.2014.10.02417.
Zhang
, T. Y.
, Zhao
, M. H.
, and Gao
, C. F.
, 2005
, “The Strip Dielectric Breakdown Model
,” Int. J. Fract.
, 132
(4
), pp. 311
–327
.10.1007/s10704-005-2054-818.
Gao
, H.
, Zhang
, T. Y.
, and Tong
, P.
, 1997
, “Local and Global Energy Release Rates for an Electrically Yielded Crack in a Piezoelectric Ceramic
,” J. Mech. Phys. Solids
, 45
(4
), pp. 491
–510
.10.1016/S0022-5096(96)00108-119.
Zhao
, M. H.
, and Fan
, C. Y.
, 2008
, “Strip Electric-Magnetic Breakdown Model in a Magnetoelectroelastic Medium
,” J. Mech. Phys. Solids
, 56
(12
), pp. 3441
–3458
.10.1016/j.jmps.2008.09.00420.
Fan
, C. Y.
, and Zhao
, M. H.
, 2011
, “Nonlinear Fracture of 2D Magnetoelectroelastic Media: Analytical and Numerical Solutions
,” Int. J. Solids Struct.
, 48
(16–17
), pp. 2383
–2392
.10.1016/j.ijsolstr.2011.04.01321.
Fang
, D. N.
, Zhang
, Z. K.
, Soh
, A. K.
, and Lee
, K. L.
, 2004
, “Fracture Criteria of Piezoelectric Ceramics With Defects
,” Mater. Mech.
, 36
(10
), pp. 917
–928
.10.1016/j.mechmat.2003.08.01122.
Fang
, D.
, and Liu
, J.
, 2013
, Fracture Mechanics of Piezoelectric and Ferroelectric Solids
, Springer
, Berlin, Germany
.23.
Lei
, J.
, and Zhang
, C.
, 2018
, “A Simplified Evaluation of the Mechanical Energy Release Rate of Kinked Cracks in Piezoelectric Materials Using the Boundary Element Method
,” Eng. Fract. Mech.
, 188
, pp. 36
–57
.10.1016/j.engfracmech.2017.07.00824.
Qin
, Q.-H.
, and Zhang
, C.
, 2002
, “Book Review on Fracture Mechanics of Piezoelectric Materials
,” ASME Appl. Mech. Rev.
, 55
(1
), pp. B8
–B9
.10.1115/1.144532425.
Park
, S.
, and Sun
, C. T.
, 1995
, “Fracture Criteria for Piezoelectric Ceramics
,” J. Am. Ceram. Soc.
, 78
(6
), pp. 1475
–1480
.10.1111/j.1151-2916.1995.tb08840.x26.
Sih
, G. C.
, Jones
, R.
, and Song
, Z. F.
, 2003
, “Piezomagnetic and Piezoelectric Poling Effects on Mode I and II Crack Initiation Behavior of Magnetoelectroelastic Materials
,” Theor. Appl. Fract. Mech.
, 40
(2
), pp. 161
–186
.10.1016/S0167-8442(03)00044-227.
Li
, X. F.
, and Lee
, K. Y.
, 2004
, “Crack Growth in a Piezoelectric Material With a Griffith Crack Perpendicular to the Poling Axis
,” Philos. Mag.
, 84
(18
), pp. 1789
–1820
.10.1080/1478643041000166322228.
Suo
, Z.
, Kuo
, C.-M.
, Barnett
, D. M.
, and Willis
, J. R.
, 1992
, “Fracture Mechanics for Piezoelectric Ceramics
,” J. Mech. Phys. Solids
, 40
(4
), pp. 739
–765
.10.1016/0022-5096(92)90002-J29.
Dugdale
, D. C.
, 1960
, “Yielding of Steel Sheet Containing Slits
,” J. Mech. Phys. Solids
, 8
(2
), pp. 100
–104
.10.1016/0022-5096(60)90013-230.
Spyropoulos
, C. P.
, Sih
, G. C.
, and Song
, Z. F.
, 2003
, “Magnetoelectroelastic Composite With Poling Parallel to Plane of Line Crack Under Out-of-Plane Deformation
,” Theor. Appl. Fract. Mech.
, 39
(3
), pp. 281
–289
.10.1016/S0167-8442(03)00021-131.
Zhu
, B.
, and Qin
, T.
, 2007
, “Application of Hypersingular Integral Equation Method to Three-Dimensional Crack in Electromagnetothermoelastic Multiphase Composites
,” Int. J. Solids Struct.
, 44
(18–19
), pp. 5994
–6012
.10.1016/j.ijsolstr.2007.02.00732.
Lawn
, B.
, 1993
, Fracture of Brittle Solids
, Cambridge University Press
, Cambridge, UK
.33.
Anderson
, T. L.
, 2005
, Fracture Mechanics: Fundamentals and Applications
, 3rd ed., CRC Press
, Boca Raton, FL
.34.
Wang
, B. L.
, and Mai
, Y. W.
, 2004
, “Fracture of Piezoelectromagnetic Materials
,” Mech. Res. Commun.
, 31
(1
), pp. 65
–73
.10.1016/j.mechrescom.2003.08.00235.
Wang
, B. L.
, Han
, J. C.
, and Mai
, Y. W.
, 2006
, “Mode III Fracture of a Magnetoelectroelastic Layer: Exact Solution and Discussion of the Crack Face Electromagnetic Boundary Conditions
,” Int. J. Fract.
, 139
(1
), pp. 27
–38
.10.1007/s10704-006-6632-136.
Hu
, K. Q.
, Li
, G.-Q.
, and Zhong
, Z.
, 2006
, “Fracture of a Rectangular Piezoelectromagnetic Body
,” Mech. Res. Commun.
, 33
(4
), pp. 482
–492
.10.1016/j.mechrescom.2005.06.01637.
Ma
, L.
, Li
, J.
, Abdelmoula
, R.
, and Wu
, L. Z.
, 2007
, “Mode III Crack Problem in a Functionally Graded Magneto-Electro-Elastic Strip
,” Int. J. Solids Struct.
, 44
(17
), pp. 5518
–5537
.10.1016/j.ijsolstr.2007.01.01238.
Gao
, C. F.
, Tong
, P.
, and Zhang
, T. Y.
, 2004
, “Fracture Mechanics for a Mode III Crack in a Magnetoelectroelastic Solid
,” Int. J. Solids Struct.
, 41
(24–25
), pp. 6613
–6629
.10.1016/j.ijsolstr.2004.06.01539.
Zhou
, Z. G.
, and Wang
, B.
, 2004
, “Two Parallel Symmetry Permeable Cracks in Functionally Graded Piezoelectric/Piezomagnetic Materials Under Anti-Plane Shear Loading
,” Int. J. Solids Struct.
, 41
(16–17
), pp. 4407
–4422
.10.1016/j.ijsolstr.2004.03.00440.
Singh
, B. M.
, Rokne
, J.
, and Dhaliwal
, R. S.
, 2009
, “Closed-Form Solutions for Two Anti-Plane Collinear Cracks in a Magnetoelectroelastic Layer
,” Eur. J. Mech. A - Solids
, 28
(3
), pp. 599
–609
.10.1016/j.euromechsol.2008.10.00441.
Mousavi
, S. M.
, and Paavola
, J.
, 2013
, “Analysis of Functionally Graded Magneto-Electro-Elastic Layer With Multiple Cracks
,” Theor. Appl. Fract. Mech.
, 66
, pp. 1
–8
.10.1016/j.tafmec.2013.11.00742.
Singh
, R.
, and Das
, S.
, 2023
, “Analysis of Multiple Parallel Cracks in a Functionally Graded Magneto-Electro-Elastic Plane Using Boundary Collocation Method
,” Arch. Appl. Mech.
, 93
(12
), pp. 4497
–4516
.10.1007/s00419-023-02506-043.
Guo
, J. H.
, and Lu
, Z. X.
, 2010
, “Anti-Plane Analysis of Multiple Cracks Originating From a Circular Hole in a Magnetoelectroelastic Solid
,” Int. J. Solids Struct.
, 47
(14–15
), pp. 1847
–1856
.10.1016/j.ijsolstr.2010.03.02244.
Rogowski
, B.
, 2011
, “The Mode III Cracks Emanating From an Elliptical Hole in the Piezo-Electro-Magneto-Elastic Materials
,” Arch. Appl. Mech.
, 81
(11
), pp. 1607
–1620
.10.1007/s00419-010-0505-945.
Xiao
, J. H.
, Xu
, Y. L.
, and Zhang
, F. C.
, 2019
, “Fracture Analysis of Magnetoelectroelastic Solid Weakened by Periodic Cracks and Line Inclusions
,” Eng. Fract. Mech.
, 205
, pp. 70
–80
.10.1016/j.engfracmech.2018.11.01946.
Lv
, X.
, and Liu
, G. T.
, 2018
, “Interaction Between Many Parallel Screw Dislocations and a Semi-Infinite Crack in a Magnetoelectroelastic Solid
,” Chin. Phys. B
, 27
(7
), p. 074601
.10.1088/1674-1056/27/7/07460147.
Tupholme
, G. E.
, 2008
, “Shear Crack in a Transversely Isotropic Magnetoelectroelastic Half-Space
,” Mech. Res. Commun.
, 35
(7
), pp. 466
–474
.10.1016/j.mechrescom.2008.01.00148.
Zhou
, Z. H.
, Xu
, X. S.
, and Leung
, A. Y. T.
, 2010
, “Mode III Edge-Crack in Magneto-Electro-Elastic Media by Symplectic Expansion
,” Eng. Fract. Mech.
, 77
(16
), pp. 3157
–3173
.10.1016/j.engfracmech.2010.07.01649.
Liu
, Y. Z.
, Guo
, J. H.
, and Zhang
, X. Y.
, 2019
, “Surface Effect on a Nano-Elliptical Hole or Nano-Crack in Magnetoelectroelastic Materials Under Antiplane Shear
,” ZAMM - Z. Angew. Math. Mech.
, 99
(7
), p. e201900043
.10.1002/zamm.20190004350.
Tupholme
, G. E.
, 2020
, “Stack of Non-Uniformly Loaded Shear Cracks in Magnetoelectroelastic Materials
,” Mech. Adv. Mater. Struct.
, 28
(18
), pp. 1948
–1956
.10.1080/15376494.2020.171611951.
Chue
, C. H.
, and Liu
, T. J. C.
, 2005
, “Magneto-Electro-Elastic Antiplane Analysis of a Bimaterial BaTio3-CoFe2O4 Composite Wedge With an Interface Crack
,” Theor. Appl. Fract. Mech.
, 44
(3
), pp. 275
–296
.10.1016/j.tafmec.2005.09.00452.
Li
, R.
, and Kardomateas
, G. A.
, 2006
, “The Mode III Interface Crack in Piezo-Electromagneto-Elastic Dissimilar Bimaterials
,” ASME J. Appl. Mech.
, 73
(2
), pp. 220
–227
.10.1115/1.207332853.
Mai
, Y. W.
, and Wang
, B. L.
, 2006
, “Closed-Form Solution for an Antiplane Interface Crack Between Two Dissimilar Magnetoelectroelastic Layers
,” ASME J. Appl. Mech.
, 73
(2
), pp. 281
–290
.10.1115/1.208382754.
Su
, R. K. L.
, and Feng
, W. J.
, 2008
, “Fracture Behavior of a Bonded Magneto-Electro-Elastic Rectangular Plate With an Interface Crack
,” Arch. Appl. Mech.
, 78
(5
), pp. 343
–362
.10.1007/s00419-007-0165-655.
Li
, Y. D.
, and Lee
, K. Y.
, 2008
, “Fracture Analysis and Improved Design for a Symmetrically Bonded Smart Structure With Linearly Non-Homogeneous Magnetoelectroelastic Properties
,” Eng. Fract. Mech.
, 75
(10
), pp. 3161
–3172
.10.1016/j.engfracmech.2007.12.00556.
Li
, Y. D.
, and Lee
, K. Y.
, 2010
, “Anti-Plane Shear Fracture of the Interface in a Cylindrical Smart Structure With Functionally Graded Magnetoelectroelastic Properties
,” Acta Mech.
, 212
(1–2
), pp. 139
–149
.10.1007/s00707-009-0251-757.
Bagheri
, R.
, and Mirzaei
, A. M.
, 2019
, “Fracture Analysis in an Imperfect FGM Orthotropic Strip Bonded Between Two Magneto-Electro-Elastic Layers
,” IJST-Trans. Mech. Eng.
, 43
(2
), pp. 253
–271
.10.1007/s40997-017-0129-658.
Hu
, K. Q.
, Zhong
, Z.
, and Chen
, Z. T.
, 2019
, “Interface Crack Between Magnetoelectroelastic and Orthotropic Half-Spaces Under Anti-Plane Loading
,” Theor. Appl. Fract. Mech.
, 99
, pp. 95
–103
.10.1016/j.tafmec.2018.11.01259.
Zhou
, Z. G.
, Wang
, B.
, and Sun
, Y. G.
, 2004
, “Two Collinear Interface Cracks in Magneto-Electro-Elastic Composites
,” Int. J. Eng. Sci.
, 42
(11–12
), pp. 1155
–1167
.10.1016/j.ijengsci.2004.01.00560.
Zhou
, Z. G.
, Chen
, Y.
, and Wang
, B.
, 2007
, “The Behavior of Two Parallel Interface Cracks in Magneto-Electro-Elastic Materials Under an Anti-Plane Shear Stress Loading
,” Compos. Struct.
, 77
(1
), pp. 97
–103
.10.1016/j.compstruct.2005.11.05661.
Wang
, B. L.
, and Han
, J. C.
, 2010
, “Effect of Finite Cracking on the Magneto-Electric Coupling Properties of Magneto-Electro-Elastic Composite Laminates
,” J. Intell. Mater. Syst. Struct.
, 21
(16
), pp. 1669
–1679
.10.1177/1045389X1038612862.
Zhang
, J.
, Jin
, Y.
, and Li
, Y. D.
, 2018
, “Concentrated Force-Induced Fracture of a Multiferroic Composite Cylinder
,” Acta Mech.
, 229
(3
), pp. 1215
–1228
.10.1007/s00707-017-2011-463.
Feng
, F. X.
, Lee
, K. Y.
, and Li
, Y. D.
, 2011
, “Multiple Cracks on the Arc-Shaped Interface in a Semi-Cylindrical Magneto-Electro-Elastic Composite With an Orthotropic Substrate
,” Eng. Fract. Mech.
, 78
(9
), pp. 2029
–2041
.10.1016/j.engfracmech.2011.03.01664.
Shi
, P. P.
, Sun
, S.
, and Li
, X.
, 2013
, “The Cyclically Symmetric Cracks on the Arc-Shaped Interface Between a Functionally Graded Magneto-Electro-Elastic Layer and an Orthotropic Elastic Substrate Under Static Anti-Plane Shear Load
,” Eng. Fract. Mech.
, 105
, pp. 238
–249
.10.1016/j.engfracmech.2013.04.00865.
Hao
, R. J.
, and Liu
, J. X.
, 2006
, “Interaction of a Screw Dislocation With a Semi-Infinite Interfacial Crack in a Magneto-Electro-Elastic Bi-Material
,” Mech. Res. Commun.
, 33
(3
), pp. 415
–424
.10.1016/j.mechrescom.2005.02.02066.
Zheng
, J. L.
, Fang
, Q. H.
, and Liu
, Y. W.
, 2007
, “A Generalized Screw Dislocation Interacting With Interfacial Cracks Along a Circular Inhomogeneity in Magnetoelectroelastic Solids
,” Theor. Appl. Fract. Mech.
, 47
(3
), pp. 205
–218
.10.1016/j.tafmec.2007.01.00567.
Zeng
, X.
, Li
, J.
, Liu
, F.
, and Wen
, P. H.
, 2019
, “Generalized Screw Dislocations Interacting With Interfacial Secondary Crack in Magneto-Electro-Elastic Solid
,” ZAMM - Z. Angew. Math. Mech.
, 99
(8
), p. e201800245
.10.1002/zamm.20180024568.
Li
, X. F.
, Liu
, G. L.
, and Lee
, K. Y.
, 2009
, “Magnetoelectroelastic Field Induced by a Crack Terminating at the Interface of a Bi-Magnetoelectric Material
,” Philos. Mag.
, 89
(5
), pp. 449
–463
.10.1080/1478643080265342869.
Wan
, Y.
, Yue
, Y.
, and Zhong
, Z.
, 2012
, “A Mode III Crack Crossing the Magnetoelectroelastic Bimaterial Interface Under Concentrated Magnetoelectro-Mechanical Loads
,” Int. J. Solids Struct.
, 49
(21
), pp. 3008
–3021
.10.1016/j.ijsolstr.2012.06.00170.
Hu
, K. Q.
, Kang
, Y. L.
, and Qin
, Q. H.
, 2007
, “A Moving Crack in a Rectangular Magnetoelectroelastic Body
,” Eng. Fract. Mech.
, 74
(5
), pp. 751
–770
.10.1016/j.engfracmech.2006.06.01671.
Tupholme
, G. E.
, 2012
, “Magnetoelectroelastic Media Containing a Row of Moving Shear Cracks
,” Mech. Res. Commun.
, 45
, pp. 48
–53
.10.1016/j.mechrescom.2012.07.00272.
Tupholme
, G. E.
, 2020
, “Non-Uniformly Loaded Row of Moving Shear Cracks in Magnetoelectroelastic Media
,” Math. Mech. Solids
, 25
(2
), pp. 374
–388
.10.1177/108128651987823373.
Ayatollahi
, M.
, Mahmoudi
, M. M.
, and Nourazar
, M.
, 2017
, “Analysis of Multiple Moving Mode-III Cracks in a Functionally Graded Magnetoelectroelastic Half-Plane
,” J. Int. Mater. Syst. Struct.
, 28
(19
), pp. 2823
–2834
.10.1177/1045389X1769859374.
Fu
, J.
, Hu
, K.
, Chen
, Z.
, Chen
, L.
, and Qian
, L.
, 2013
, “A Moving Crack Propagating in a Functionally Graded Magnetoelectroelastic Strip Under Different Crack Face Conditions
,” Theor. Appl. Fract. Mech.
, 66
, pp. 16
–25
.10.1016/j.tafmec.2014.01.00775.
Wang
, Y. Z.
, and Kuna
, M.
, 2015
, “General Solutions of Mechanical-Electric-Magnetic Fields in Magneto-Electro-Elastic Solid Containing a Moving Anti-Plane Crack and a Screw Dislocation
,” ZAMM - J. Appl. Math. Mech.
, 95
(7
), pp. 703
–713
.10.1002/zamm.20130027976.
Hu
, K.
, and Chen
, Z.
, 2014
, “An Interface Crack Moving Between Magnetoelectroelastic and Functionally Graded Elastic Layers
,” Appl. Math. Model.
, 38
(3
), pp. 910
–925
.10.1016/j.apm.2013.07.02277.
Zhong
, X. C.
, and Li
, X. F.
, 2006
, “A Finite Length Crack Propagating Along the Interface of Two Dissimilar Magnetoelectroelastic Materials
,” Int. J. Eng. Sci.
, 44
(18–19
), pp. 1394
–1407
.10.1016/j.ijengsci.2006.07.00478.
Ayatollahi
, M.
, and Milan
, A. G.
, 2022
, “Dissimilar Nonhomogeneous Magnetoelectroelastic Layers With Moving Crack at the Interface
,” J. Eng. Math.
, 133
(1
), p. 8
.10.1007/s10665-022-10212-z79.
Chen
, H. S.
, Wei
, W. Y.
, Liu
, J. X.
, and Fang
, D. N.
, 2012
, “Propagation of a Mode-III Interfacial Crack in a Piezoelectric-Piezomagnetic Bi-Material
,” Int. J. Solids Struct.
, 49
(18
), pp. 2547
–2558
.10.1016/j.ijsolstr.2012.05.01380.
Hu
, K.
, Chen
, Z.
, and Fu
, J.
, 2015
, “Moving Dugdale Crack Along the Interface of Two Dissimilar Magnetoelectroelastic Materials
,” Acta Mech.
, 226
(6
), pp. 2065
–2076
.10.1007/s00707-015-1298-281.
Zhou
, Z. G.
, and Wang
, B.
, 2006
, “Dynamic Behavior of Two Parallel Symmetry Cracks in Magneto-Electro-Elastic Composites Under Harmonic Anti-Plane Waves
,” Appl. Math. Mech.
, 27
(5
), pp. 583
–591
.10.1007/s10483-006-0503-y82.
Liang
, J.
, 2008
, “The Dynamic Behavior of Two Parallel Symmetric Cracks in Functionally Graded Piezoelectric/Piezomagnetic Materials
,” Arch. Appl. Mech.
, 78
(6
), pp. 443
–464
.10.1007/s00419-007-0173-683.
Li
, Y. D.
, and Lee
, K. Y.
, 2009
, “Dynamic Responses of a Crack in a Layered Graded Magnetoelectroelastic Sensor Subjected to Harmonic Waves
,” Acta Mech.
, 204
(3–4
), pp. 217
–234
.10.1007/s00707-008-0082-y84.
Zhou
, Z. G.
, Wu
, L. Z.
, and Wang
, B.
, 2005
, “The Dynamic Behavior of Two Collinear Interface Cracks in Magneto-Electro-Elastic Materials
,” Eur. J. Mech. A - Solids
, 24
(2
), pp. 253
–262
.10.1016/j.euromechsol.2004.10.00685.
Zhou
, Z. G.
, and Wang
, B.
, 2008
, “An Interface Crack Between Two Dissimilar Functionally Graded Piezoelectric/Piezomagnetic Material Half Infinite Planes Subjected to the Harmonic Anti-Plane Shear Stress Waves
,” Int. J. Appl. Electromagn. Mech.
, 27
(1–2
), pp. 117
–132
.10.3233/JAE-2008-92186.
Zhong
, X. C.
, and Li
, X. F.
, 2010
, “Diffraction of SH-Waves by an Interfacial Crack Between a Magnetoelectroelastic Solid and an Elastic Material
,” Mech. Adv. Mater. Struct.
, 17
(2
), pp. 134
–144
.10.1080/1537649090324383787.
An
, N.
, Zhang
, J.
, Chen
, Y.
, and Song
, T. S.
, 2022
, “Analysis of Mode III Interface Fracture for Hole-Initiated Cracks in Magnetic-Electro-Elastic Bimaterials
,” Mech. Adv. Mater. Struct.
, 20
, pp. 1
–11
.10.1080/15376494.2022.215033988.
An
, N.
, Song
, T. S.
, and Hou
, G.
, 2023
, “Dynamic Interaction Between Complex Defect and Crack in Functionally Graded Magnetic-Electro-Elastic Bi-Materials
,” Mech. Adv. Mater. Struct.
, 30
(13
), pp. 2748
–2764
.10.1080/15376494.2022.206263189.
Li
, X. F.
, 2005
, “Dynamic Analysis of a Cracked Magnetoelectroelastic Medium Under Antiplane Mechanical and Inplane Electric and Magnetic Impacts
,” Int. J. Solids Struct.
, 42
(11–12
), pp. 3185
–3205
.10.1016/j.ijsolstr.2004.10.02090.
Feng
, W. J.
, and Su
, R. K. L.
, 2006
, “Dynamic Internal Crack Problem of a Functionally Graded Magneto-Electro-Elastic Strip
,” Int. J. Solids Struct.
, 43
(17
), pp. 5196
–5216
.10.1016/j.ijsolstr.2005.07.05091.
Feng
, W. J.
, and Su
, R. K. L.
, 2007
, “Dynamic Fracture Behaviors of Cracks in a Functionally Graded Magneto-Electro-Elastic Plate
,” Eur. J. Mech. A - Solids
, 26
(2
), pp. 363
–379
.10.1016/j.euromechsol.2006.07.00492.
Rogowski
, B.
, 2015
, “The Transient Analysis of a Conducting Crack in Magneto-Electro-Elastic Half-Space Under Anti-Plane Mechanical and In-Plane Electric and Magnetic Impacts
,” Arch. Appl. Mech.
, 85
(1
), pp. 29
–50
.10.1007/s00419-014-0898-y93.
Chen
, X.
, 2009
, “Dynamic Crack Propagation in a Magneto-Electro-Elastic Solid Subjected to Mixed Loads: Transient Mode-III Problem
,” Int. J. Solids Struct.
, 46
(22–23
), pp. 4025
–4037
.10.1016/j.ijsolstr.2009.07.02294.
Ayatollahi
, M.
, Abdel‐Wahab
, M.
, and Hashemi
, S. M. M.
, 2024
, “Transient Analysis of a Cracked Functionally Graded Magneto‐Electro‐Elastic Rectangular Finite Plane Under Anti‐Plane Mechanical and In‐Plane Electric and Magnetic Impacts
,” Fatigue Fract. Eng. Mater. Struct.
, 47
(1
), pp. 220
–239
.10.1111/ffe.1417495.
Arhani
, A. A.
, and Ayatollahi
, M.
, 2022
, “Dynamic Response of Cracked Non‐Homogeneous Magneto‐Electro‐Elastic Layer Sandwiched by Two Dissimilar Orthotropic Layers
,” Fatigue Fract. Eng. Mater. Struct.
, 45
(5
), pp. 1448
–1463
.10.1111/ffe.1367396.
Bagheri
, R.
, and Monfared
, M. M.
, 2018
, “Magneto-Electro-Elastic Analysis of a Strip Containing Multiple Embedded and Edge Cracks Under Transient Loading
,” Acta Mech.
, 229
(12
), pp. 4895
–4913
.10.1007/s00707-018-2289-x97.
Feng
, W. J.
, Xue
, Y.
, and Zou
, Z. Z.
, 2005
, “Crack Growth of an Interface Crack Between Two Dissimilar Magneto-Electro-Elastic Materials Under Anti-Plane Mechanical and In-Plane Electric Magnetic Impact
,” Theor. Appl. Fract. Mech.
, 43
(3
), pp. 376
–394
.10.1016/j.tafmec.2005.03.00898.
Feng
, W. J.
, and Pan
, E.
, 2008
, “Dynamic Fracture Behavior of an Internal Interfacial Crack Between Two Dissimilar Magneto-Electro-Elastic Plates
,” Eng. Fract. Mech.
, 75
(6
), pp. 1468
–1487
.10.1016/j.engfracmech.2007.07.00199.
Wang
, B. L.
, Han
, J. C.
, and Du
, S. Y.
, 2010
, “Transient Fracture of a Layered Magnetoelectroelastic Medium
,” Mech. Mater.
, 42
(3
), pp. 354
–364
.10.1016/j.mechmat.2009.12.002100.
Peng
, X. L.
, and Li
, X. F.
, 2009
, “Transient Response of the Crack Tip Field in a Magnetoelectroelastic Half-Space With a Functionally Graded Coating Under Impacts
,” Arch. Appl. Mech.
, 79
(12
), pp. 1099
–1113
.10.1007/s00419-008-0277-7101.
Hu
, K.
, Chen
, Z.
, and Fu
, J.
, 2014
, “Dynamic Analysis of an Interface Crack Between Magnetoelectroelastic and Functionally Graded Elastic Layers Under Anti-Plane Mechanical and In-Plane Electro-Magnetic Loadings
,” Compos. Struct.
, 107
, pp. 142
–148
.10.1016/j.compstruct.2013.07.057102.
Zhao
, X.
, Qian
, Z.
, Liu
, J.
, and Gao
, C.
, 2020
, “Effects of Electric/Magnetic Impact on the Transient Fracture of Interface Crack in Piezoelectric-Piezomagnetic Sandwich Structure: Anti-Plane Case
,” Appl. Math. Mech.-Engl.
, 41
(1
), pp. 139
–156
.10.1007/s10483-020-2552-5103.
Gao
, C. F.
, Kessler
, H.
, and Balke
, H.
, 2003
, “Crack Problems in Magnetoelectroelastic Solids. Part I: Exact Solution of a Crack
,” Int. J. Eng. Sci.
, 41
(9
), pp. 969
–981
.10.1016/S0020-7225(02)00323-3104.
Song
, Z. F.
, and Sih
, G. C.
, 2003
, “Crack Initiation Behavior in Magnetoelectroelastic Composite Under In-Plane Deformation
,” Theor. Appl. Fract. Mech.
, 39
(3
), pp. 189
–207
.10.1016/S0167-8442(03)00002-8105.
Sih
, G. C.
, and Chen
, E. P.
, 2003
, “Dilatational and Distortional Behavior of Cracks in Magnetoelectroelastic Materials
,” Theor. Appl. Fract. Mech.
, 40
(1
), pp. 1
–21
.10.1016/S0167-8442(03)00031-4106.
Sih
, G. C.
, and Yu
, H. Y.
, 2005
, “Volume Fraction Effect of Magnetoelectroelastic Composite on Enhancement and Impediment of Crack Growth
,” Compos. Struct.
, 68
(1
), pp. 1
–11
.10.1016/j.compstruct.2004.02.015107.
Tian
, W.
, and Rajapakse
, R. K. N. D.
, 2005
, “Fracture Analysis of Magnetoelectroelastic Solids by Using Path Independent Integrals
,” Int. J. Fract.
, 131
(4
), pp. 311
–335
.10.1007/s10704-004-5103-9108.
Hu
, K. Q.
, Chen
, Z. T.
, and Wang
, X. D.
, 2018
, “Boundary Effect on Crack Kinking in a Magnetoelectroelastic Strip With a Central Crack
,” Eng. Fract. Mech.
, 201
, pp. 336
–352
.10.1016/j.engfracmech.2018.07.025109.
Tian
, W. Y.
, and Gabbert
, U.
, 2004
, “Multiple Crack Interaction Problem in Magnetoelectroelastic Solids
,” Eur. J. Mech. A - Solids
, 23
(4
), pp. 599
–614
.10.1016/j.euromechsol.2004.02.002110.
Tian
, W. Y.
, and Gabbert
, U.
, 2005
, “Macrocrack-Microcrack Interaction Problem in Magnetoelectroelastic Solids
,” Mech. Mater.
, 37
(5
), pp. 565
–592
.10.1016/j.mechmat.2004.04.008111.
Zhao
, M. H.
, Wang
, H.
, Yang
, F.
, and Liu
, T.
, 2006
, “A Magnetoelectroelastic Medium With an Elliptical Cavity Under Combined Mechanical-Electric-Magnetic Loading
,” Theor. Appl. Fract. Mech.
, 45
(3
), pp. 227
–237
.10.1016/j.tafmec.2006.03.006112.
Wang
, B. L.
, and Han
, J. C.
, 2007
, “Multiple Cracking of Magnetoelectroelastic Materials in Coupling Thermos-Electro-Magneto-Mechanical Loading Environments
,” Comput. Mater. Sci.
, 39
(2
), pp. 291
–304
.10.1016/j.commatsci.2006.06.008113.
Zhang
, A. B.
, and Wang
, B. L.
, 2014
, “Theoretical Model of Crack Branching in Magnetoelectric Thermoelastic Materials
,” Int. J. Solids Struct.
, 51
(6
), pp. 1340
–1349
.10.1016/j.ijsolstr.2013.12.025114.
Wu
, B.
, Peng
, D.
, and Jones
, R.
, 2024
, “Analysis Method of Collinear Cracks Subjected to Thermo-Magneto-Electro-Elastic Loads
,” Arch. Appl. Mech.
, 94
(1
), pp. 99
–117
.10.1007/s00419-023-02511-3115.
Zhou
, Z. G.
, Zhang
, P. W.
, and Wu
, L. Z.
, 2007
, “Solutions to a Limited-Permeable Crack or Two Limited-Permeable Collinear Cracks in Piezoelectric/Piezomagnetic Materials
,” Arch. Appl. Mech.
, 77
(12
), pp. 861
–882
.10.1007/s00419-007-0135-z116.
Zhou
, Z. G.
, and Chen
, Z. T.
, 2008
, “Basic Solution of a Mode-I Limited-Permeable Crack in Functionally Graded Piezoelectric/Piezomagnetic Materials
,” Int. J. Solids Struct.
, 45
(7–8
), pp. 2265
–2296
.10.1016/j.ijsolstr.2007.11.016117.
Zhong
, X. C.
, 2011
, “Closed-Form Solutions for Two Collinear Dielectric Cracks in a Magnetoelectroelastic Solid
,” Appl. Math. Model.
, 35
(6
), pp. 2930
–2944
.10.1016/j.apm.2010.12.010118.
Zhong
, X. C.
, and Li
, X. F.
, 2008
, “T-Stress Analysis for a Griffith Crack in a Magnetoelectroelastic Solid
,” Arch. Appl. Mech.
, 78
(2
), pp. 117
–125
.10.1007/s00419-007-0143-z119.
Li
, Y. D.
, and Lee
, K. Y.
, 2010
, “Collinear Unequal Crack Series in Magnetoelectroelastic Materials: Mode I Case Solved Via New Real Fundamental Solutions
,” Eng. Fract. Mech.
, 77
(14
), pp. 2772
–2790
.10.1016/j.engfracmech.2010.05.002120.
Rekik
, M.
, El-Borgi
, S.
, and Ounaies
, Z.
, 2012
, “An Embedded Mixed-Mode Crack in a Functionally Graded Magnetoelectroelastic Infinite Medium
,” Int. J. Solids Struct.
, 49
(5
), pp. 835
–845
.10.1016/j.ijsolstr.2011.12.002121.
Zhong
, X. C.
, 2009
, “Analysis of a Dielectric Crack in a Magnetoelectroelastic Layer
,” Int. J. Solids Struct.
, 46
(24
), pp. 4221
–4230
.10.1016/j.ijsolstr.2009.08.011122.
Zhong
, X. C.
, and Lee
, K.
, 2012
, “Dielectric Crack Problem for a Magnetoelectroelastic Strip With Functionally Graded Properties
,” Arch. Appl. Mech.
, 82
(6
), pp. 791
–807
.10.1007/s00419-011-0592-2123.
Wang
, B. L.
, 2012
, “Fracture and Effective Properties of Finite Magnetoelectroelastic Media
,” J. Intell. Mater. Syst. Struct.
, 23
(15
), pp. 1699
–1712
.10.1177/1045389X12451657124.
Feng
, W. J.
, Su
, R. K. L.
, Liu
, J. X.
, and Li
, Y. S.
, 2010
, “Fracture Analysis of Bounded Magnetoelectroelastic Layers With Interfacial Cracks Under Magnetoelectro-Mechanical Loads: Plane Problem
,” J. Intell. Mater. Syst. Struct.
, 21
(6
), pp. 581
–594
.10.1177/1045389X10361630125.
Zhao
, X.
, Liu
, J.
, Qian
, Z.
, and Gao
, C.
, 2022
, “Fracture Behavior of an Interface Crack in a Magnetoelectric Sandwich Structure Under Electric Field: Effects of the Poling Directions
,” J. Intell. Mater. Syst. Struct.
, 33
(15
), pp. 1902
–1915
.10.1177/1045389X211072255126.
Ma
, P.
, Su
, R. K. L.
, and Feng
, W. J.
, 2016
, “Integral Identities Based on Symmetric and Skew-Symmetric Weight Functions for a Semi-Infinite Interfacial Crack in Anisotropic Magnetoelectroelastic Bimaterials
,” Int. J. Solids Struct.
, 88–89
, pp. 178
–191
.10.1016/j.ijsolstr.2016.03.008127.
Cheng
, C. Z.
, Cheng
, X.
, Niu
, Z. R.
, and Recho
, N.
, 2016
, “Singularity Characteristic Analyses for Magneto-Electro-Elastic V-Notches
,” Eur. J. Mech. A - Solids
, 57
, pp. 59
–70
.10.1016/j.euromechsol.2015.12.005128.
Yang
, Y.
, Cheng
, C.
, Yao
, S.
, Li
, J.
, and Niu
, Z.
, 2022
, “Singularity Analysis for the V-Notch in Functionally Graded Piezoelectric/Piezomagnetic Material
,” J. Eng. Math.
, 132
(1
), pp. 17
–33
.10.1007/s10665-021-10198-0129.
Wang
, X.
, Ang
, W. T.
, and Fan
, H.
, 2018
, “Effective Properties of Magnetoelectroelastic Interfaces Weakened by Micro-Cracks
,” ZAMM - J. Appl. Math. Mech.
, 98
(5
), pp. 727
–748
.10.1002/zamm.201700219130.
Hu
, K. Q.
, Chen
, Z. T.
, and Zhong
, Z.
, 2018
, “Interface Crack Between Magnetoelectroelastic and Orthotropic Half-Spaces Under In-Plane Loading
,” Theor. Appl. Fract. Mech.
, 96
, pp. 285
–295
.10.1016/j.tafmec.2018.05.002131.
Gao
, C. F.
, and Noda
, N.
, 2004
, “Thermal-Induced Interfacial Cracking of Magnetoelectroelastic Materials
,” Int. J. Eng. Sci.
, 42
(13–14
), pp. 1347
–1360
.10.1016/j.ijengsci.2004.03.005132.
Feng
, W. J.
, Li
, Y. S.
, and Xu
, Z. H.
, 2009
, “Transient Response of an Interfacial Crack Between Dissimilar Magnetoelectroelastic Layers Under Magnetoelectromechanical Impact Loadings: Mode-I Problem
,” Int. J. Solids Struct.
, 46
(18–19
), pp. 3346
–3356
.10.1016/j.ijsolstr.2009.05.003133.
Hu
, K.
, and Chen
, Z.
, 2013
, “Pre-Kinking of a Moving Crack in a Magnetoelectroelastic Material Under In-Plane Loading
,” Int. J. Solids Struct.
, 50
(16–17
), pp. 2667
–2677
.10.1016/j.ijsolstr.2013.04.016134.
Hu
, K.
, Chen
, Z.
, and Zhong
, Z.
, 2014
, “Pre-Kinking Analysis of a Constant Moving Crack in a Magnetoelectroelastic Strip Under In-Plane Loading
,” Eur. J. Mech. A - Solids
, 43
, pp. 25
–43
.10.1016/j.euromechsol.2013.08.005135.
Ma
, P.
, Su
, K. L.
, and Feng
, W. J.
, 2017
, “Moving Crack With a Contact Zone at Interface of Magnetoelectroelastic Bimaterial
,” Eng. Fract. Mech.
, 181
, pp. 143
–160
.10.1016/j.engfracmech.2017.07.012136.
Ma
, P.
, Su
, R. K. L.
, and Feng
, W.
, 2021
, “Propagation of Conductive Crack Along Interface of Piezoelectric/Piezomagnetic Bimaterials
,” Acta Mech.
, 232
(7
), pp. 2781
–2791
.10.1007/s00707-021-02988-5137.
Ma
, P.
, Su
, R. K. L.
, and Feng
, W. J.
, 2018
, “Singularity of Subsonic and Transonic Crack Propagations Along Interfaces of Magnetoelectroelastic Bimaterials
,” Int. J. Eng. Sci.
, 129
, pp. 21
–33
.10.1016/j.ijengsci.2018.04.005138.
Zhong
, X. C.
, Liu
, F.
, and Li
, X. F.
, 2009
, “Transient Response of a Magnetoelectro-Elastic Solid With Two Collinear Dielectric Cracks Under Impacts
,” Int. J. Solids Struct.
, 46
(14–15
), pp. 2950
–2958
.10.1016/j.ijsolstr.2009.03.023139.
Chen
, Y.
, and Lu
, T. J.
, 2003
, “Recent Developments and Applications of Invariant Integrals
,” ASME Appl. Mech. Rev.
, 56
(5
), pp. 515
–552
.10.1115/1.1582199140.
Chen
, X.
, 2009
, “Energy Release Rate and Path-Independent Integral in Dynamic Fracture of Magneto-Electro-Thermo-Elastic Solids
,” Int. J. Solids Struct.
, 46
(13
), pp. 2706
–2711
.10.1016/j.ijsolstr.2009.03.001141.
Hu
, K.
, and Chen
, Z.
, 2012
, “Pre-Curving Analysis of an Opening Crack in a Magnetoelectroelastic Strip Under In-Plane Impact Loadings
,” J. Appl. Phys.
, 112
(12
), p. 124911
.10.1063/1.4770395142.
Milan
, A. G.
, and Ayatollahi
, M.
, 2020
, “Transient Analysis of Multiple Interface Cracks Between Two Dissimilar Functionally Graded Magneto-Electro-Elastic Layers
,” Arch. Appl. Mech.
, 90
(8
), pp. 1829
–1844
.10.1007/s00419-020-01699-y143.
Eringen
, A. C.
, 1972
, “Nonlocal Polar Elastic Continua
,” Int. J. Eng. Sci.
, 10
(1
), pp. 1
–16
.10.1016/0020-7225(72)90070-5144.
Zhang
, P. W.
, Zhou
, Z. G.
, and Wu
, L. Z.
, 2010
, “Non-Local Theory Solution of a Mode-I Crack in a Piezoelectric/Piezomagnetic Composite Material Plane
,” Int. J. Fract.
, 164
(2
), pp. 213
–229
.10.1007/s10704-010-9477-6145.
Zhou
, Z. G.
, Zhang
, P. W.
, and Wu
, L. Z.
, 2011
, “Non-Local Theory Solution of a Limited-Permeable Mode-I Crack in a Piezoelectric/Piezomagnetic Material Plane
,” ZAMM - J. Appl. Math. Mech.
, 91
(3
), pp. 192
–213
.10.1002/zamm.201000071146.
Jamia
, N.
, El-Borgi
, S.
, Rekik
, M.
, and Usman
, S.
, 2014
, “Investigation of the Behavior of a Mixed-Mode Crack in a Functionally Graded Magneto-Electro-Elastic Material by Use of the Non-Local Theory
,” Theor. Appl. Fract. Mech.
, 74
, pp. 126
–142
.10.1016/j.tafmec.2014.09.002147.
Zhao
, M. H.
, Zhang
, Q. Y.
, Fan
, C. Y.
, and Jia
, J. N.
, 2014
, “Semi-Permeable Crack Analysis in a 2D Magnetoelectroelastic Medium Based on the EMPS Model
,” Mech. Res. Commun.
, 55
, pp. 30
–39
.10.1016/j.mechrescom.2013.10.012148.
Herrmann
, K.
, Loboda
, V.
, and Khodanen
, T.
, 2010
, “An Interface Crack With Contact Zones in a Piezoelectric/Piezomagnetic Bimaterial
,” Arch. Appl. Mech.
, 80
(6
), pp. 651
–670
.10.1007/s00419-009-0330-1149.
Feng
, W. J.
, Ma
, P.
, Pan
, E.
, and Liu
, J. X.
, 2011
, “A Magnetically Impermeable and Electrically Permeable Interface Crack With a Contact Zone in a Magnetoelectroelastic Bimaterial Under Concentrated Magnetoelectromechanical Loads on the Crack Faces
,” Sci. China: Phys., Mech. Astron.
, 54
(9
), pp. 1666
–1679
.10.1007/s11433-011-4403-0150.
Feng
, W. J.
, Ma
, P.
, and Su
, R. K. L.
, 2012
, “An Electrically Impermeable and Magnetically Permeable Interface Crack With a Contact Zone in Magnetoelectroelastic Bimaterials Under a Thermal Flux and Magnetoelectromechanical Loads
,” Int. J. Solids Struct.
, 49
(23–24
), pp. 3472
–3483
.10.1016/j.ijsolstr.2012.07.006151.
Grynevych
, A. A.
, and Loboda
, V. V.
, 2016
, “An Electroded Electrically and Magnetically Charged Interface Crack in a Piezoelectric/Piezomagnetic Bimaterial
,” Acta Mech.
, 227
(10
), pp. 2861
–2879
.10.1007/s00707-016-1654-x152.
Ma
, P.
, Feng
, W. J.
, and Su
, R. K. L.
, 2013
, “Pre-Fracture Zone Model on Electrically Impermeable and Magnetically Permeable Interface Crack Between Two Dissimilar Magnetoelectroelastic Materials
,” Eng. Fract. Mech.
, 102
, pp. 310
–323
.10.1016/j.engfracmech.2013.03.004153.
Viun
, O.
, Lapusta
, Y.
, and Loboda
, V.
, 2015
, “Pre-Fracture Zones Modelling for a Limited Permeable Crack in an Interlayer Between Magneto-Electro-Elastic Materials
,” Appl. Math. Model.
, 39
(21
), pp. 6669
–6684
.10.1016/j.apm.2015.02.015154.
Ma
, P.
, Su
, R. K. L.
, and Feng
, W. J.
, 2015
, “A New Extended Pre-Fracture Zone Model for a Limited Permeable Crack in an Interlayer Between Magnetoelectroelastic Materials
,” Acta Mech.
, 226
(4
), pp. 1045
–1065
.10.1007/s00707-014-1235-9155.
Comninou
, M.
, 1977
, “The Interface Crack
,” ASME J. Appl. Mech.
, 44
(4
), pp. 631
–636
.10.1115/1.3424148156.
Fulton
, C. C.
, and Gao
, H.
, 1997
, “Electrical Nonlinearity in Fracture of Piezoelectric Ceramics
,” ASME Appl. Mech. Rev.
, 50
(11S
), pp. S56
–S63
.10.1115/1.3101851157.
Gao
, C. F.
, Noda
, N.
, and Zhang
, T. Y.
, 2006
, “Dielectric Breakdown Model for a Conductive Crack and Electrode in Piezoelectric Materials
,” Int. J. Eng. Sci.
, 44
(3–4
), pp. 256
–272
.10.1016/j.ijengsci.2005.12.001158.
Herrmann
, K. P.
, Loboda
, V. V.
, and Govorukha
, V. B.
, 2001
, “On Contact Zone Models for an Electrically Impermeable Interface Crack in a Piezoelectric Bimaterial
,” Int. J. Fract.
, 111
(3
), pp. 203
–227
.10.1023/A:1012269616735159.
Loboda
, V.
, Lapusta
, Y.
, and Sheveleva
, A.
, 2007
, “Electro-Mechanical Pre-Fracture Zones for an Electrically Permeable Interface Crack in a Piezoelectric Bimaterial
,” Int. J. Solids Struct.
, 44
(17
), pp. 5538
–5553
.10.1016/j.ijsolstr.2007.01.013160.
Loboda
, V.
, Lapusta
, Y.
, and Sheveleva
, A.
, 2010
, “Limited Permeable Crack in an Interlayer Between Piezoelectric Materials With Different Zones of Electrical Saturation and Mechanical Yielding
,” Int. J. Solids Struct.
, 47
(14–15
), pp. 1795
–1806
.10.1016/j.ijsolstr.2010.03.015161.
Zhao
, M. H.
, Yang
, F.
, and Liu
, T.
, 2006
, “Analysis of a Penny-Shaped Crack in a Magneto-Electro-Elastic Medium
,” Philos. Mag.
, 86
(28
), pp. 4397
–4416
.10.1080/14786430600724439162.
Wang
, B. L.
, Sun
, Y. G.
, and Zhang
, H. Y.
, 2008
, “Analysis of a Penny-Shaped Crack in Magnetoelectroelastic Materials
,” J. Appl. Phys.
, 103
, p. 083530
.10.1063/1.2901180163.
Zheng
, R. F.
, Wu
, T. H.
, Li
, X. Y.
, and Chen
, W. Q.
, 2018
, “Analytical and Numerical Analyses for a Penny-Shaped Crack Embedded in an Infinite Transversely Isotropic Multi-Ferroic Composite Medium: Semi-Permeable Electro-Magnetic Boundary Condition
,” Smart Mater. Struct.
, 27
(6
), p. 065020
.10.1088/1361-665X/aabc30164.
Zhong
, X. C.
, 2010
, “Thickness Effects for a Penny-Shaped Dielectric Crack in a Magnetoelectroelastic Layer
,” Philos. Mag.
, 90
(10
), pp. 1327
–1346
.10.1080/14786430903321404165.
Liu
, L. L.
, and Feng
, W. J.
, 2019
, “Dugdale Plastic Zone Model of a Penny-Shaped Crack in a Magnetoelectroelastic Cylinder Under Magnetoelectroelastic Loads
,” Arch. Appl. Mech.
, 89
(2
), pp. 291
–305
.10.1007/s00419-018-1467-6166.
Chang
, D. M.
, Liu
, X. F.
, Wang
, B. L.
, Liu
, L.
, Wang
, T. G.
, Wang
, Q.
, and Han
, J. X.
, 2020
, “Exact Solutions to Magneto-Electro-Thermo-Elastic Fields for a Cracked Cylinder Composite During Thermal Shock
,” Int. J. Mech. Mater. Des.
, 16
(1
), pp. 3
–18
.10.1007/s10999-019-09456-y167.
Wu
, T. H.
, Li
, X. Y.
, and Tang
, H. P.
, 2021
, “Three-Dimensional Fields in an Infinite Transversely Isotropic Magneto-Electro-Elastic Space With Multiple Coplanar Penny-Shaped Cracks
,” Int. J. Eng. Sci.
, 159
, p. 103434
.10.1016/j.ijengsci.2020.103434168.
Wang
, B. L.
, and Han
, J. C.
, 2006
, “Discussion on Electromagnetic Crack Face Boundary Conditions for the Fracture Mechanics of Magneto-Electro-Elastic Materials
,” Acta Mech. Sin.
, 22
(3
), pp. 233
–242
.10.1007/s10409-006-0102-x169.
Rekik
, M.
, El-Borgi
, S.
, El-Borgi
, Z.
, and Ounaies
, Z.
, 2012
, “The Axisymmetric Problem of a Partially Insulated Mixed-Mode Crack Embedded in a Functionally Graded Pyro Magneto-Electro-Elastic Infinite Medium Subjected to Thermal Loading
,” J. Therm. Stresses
, 35
(11
), pp. 947
–975
.10.1080/01495739.2012.720216170.
Rekik
, M.
, El-Borgi
, S.
, and Ounaies
, Z.
, 2014
, “An Axisymmetric Problem of an Embedded Mixed-Mode Crack in a Functionally Graded Magnetoelectroelastic Infinite Medium
,” Appl. Math. Model.
, 38
(4
), pp. 1193
–1210
.10.1016/j.apm.2013.08.006171.
Zhao
, M. H.
, Guo
, Z. H.
, Fan
, C. Y.
, and Pan
, E.
, 2013
, “Electric and Magnetic Polarization Saturation and Breakdown Models for Penny Shaped Cracks in 3D Magnetoelectroelastic Media
,” Int. J. Solids Struct.
, 50
(10
), pp. 1747
–1754
.10.1016/j.ijsolstr.2013.02.003172.
Zhao
, M. H.
, Guo
, Z. H.
, and Fan
, C. Y.
, 2013
, “Numerical Method for Nonlinear Models of Penny-Shaped Cracks in Three-Dimensional Magnetoelectroelastic Media
,” Int. J. Fract.
, 183
(1
), pp. 49
–61
.10.1007/s10704-013-9874-8173.
Feng
, W. J.
, Pan
, E.
, and Wang
, X.
, 2007
, “Dynamic Fracture Analysis of a Penny-Shaped Crack in a Magnetoelectroelastic Layer
,” Int. J. Solids Struct.
, 44
(24
), pp. 7955
–7974
.10.1016/j.ijsolstr.2007.05.020174.
Zhong
, X. C.
, and Zhang
, K. S.
, 2010
, “Dynamic Analysis of a Penny-Shaped Dielectric Crack in a Magnetoelectroelastic Solid Under Impacts
,” Eur. J. Mech. A - Solids
, 29
(2
), pp. 242
–252
.10.1016/j.euromechsol.2009.10.002175.
Wang
, B. L.
, and Niraula
, O. P.
, 2007
, “Transient Thermal Fracture Analysis of Transversely Isotropic Magneto-Electro-Elastic Materials
,” J. Therm. Stresses
, 30
(3
), pp. 297
–317
.10.1080/01495730601187935176.
Li
, Y. S.
, Jiang
, Y. Y.
, Ren
, J. H.
, and Cai
, Z. Y.
, 2011
, “An Annular Interfacial Crack Between Dissimilar Magnetoelectroelastic Layers
,” Smart Mater. Struct.
, 20
(8
), p. 085011
.10.1088/0964-1726/20/8/085011177.
Li
, Y. S.
, Ren
, J. H.
, Feng
, W. J.
, and Wang
, W.
, 2013
, “Dynamic Fracture Analysis of an Annular Interfacial Crack Between Dissimilar Magnetoelectroelastic Layers
,” Arch. Appl. Mech.
, 83
(1
), pp. 151
–170
.10.1007/s00419-012-0643-3178.
Zheng
, R. F.
, Wu
, T. H.
, and Li
, X. Y.
, 2019
, “Elliptic Crack in Transversely Isotropic Magneto-Electro-Elasticity Under Shear Loading
,” Int. J. Eng. Sci.
, 134
, pp. 47
–65
.10.1016/j.ijengsci.2018.10.006179.
Zheng
, R. F.
, Wu
, T. H.
, Li
, X. Y.
, and Yuan
, J. H.
, 2019
, “Mode I Vertical Elliptic Crack Problem of Multiferroic Composites
,” Int. J. Eng. Sci.
, 141
, pp. 95
–111
.10.1016/j.ijengsci.2019.05.017180.
Zheng
, R. F.
, Wu
, T. H.
, and Li
, X. Y.
, 2019
, “Three-Dimensional Coupling Field for an Electromagnetically Semi-Permeable Elliptical Crack in Multiferroic Composite Media
,” Eng. Fract. Mech.
, 205
, pp. 418
–438
.10.1016/j.engfracmech.2018.10.028181.
Wu
, T. H.
, Li
, X. Y.
, and Chen
, X. H.
, 2020
, “Three-Dimensional Closed-Form Solution to Elliptical Crack Problem in Magneto-Electro-Elasticity: Electrically and Magnetically Induced Maxwell Stress Boundary Condition
,” Int. J. Solids Struct.
, 202
, pp. 729
–744
.10.1016/j.ijsolstr.2020.07.003182.
Zhao
, M. H.
, Fan
, C. Y.
, Yang
, F.
, and Liu
, T.
, 2007
, “Analysis Method of Planar Cracks of Arbitrary Shape in the Isotropic Plane of a Three-Dimensional Transversely Isotropic Magnetoelectroelastic Medium
,” Int. J. Solids Struct.
, 44
(13
), pp. 4505
–4523
.10.1016/j.ijsolstr.2006.11.039183.
Zhao
, M. H.
, Guo
, Z. H.
, Fan
, C. Y.
, Zhang
, R. L.
, and Pan
, E.
, 2013
, “Three-Dimensional Vertical Cracks in Magnetoelectroelastic Media Via the Extended Displacement Discontinuity Boundary Integral Equation Method
,” J. Intell. Mater. Syst. Struct.
, 24
(16
), pp. 1969
–1984
.10.1177/1045389X13488249184.
Zhao
, M. H.
, Li
, N.
, Fan
, C. Y.
, and Xu
, G. T.
, 2008
, “Analysis Method of Planar Interface Cracks of Arbitrary Shape in Three-Dimensional Transversely Isotropic Magnetoelectroelastic Bimaterials
,” Int. J. Solids Struct.
, 45
(6
), pp. 1804
–1824
.10.1016/j.ijsolstr.2007.10.024185.
Zhang
, P. W.
, 2011
, “Dynamic Fracture of a Rectangular Limited-Permeable Crack in Magneto-Electro-Elastic Media Under a Time-Harmonic Elastic P-Wave
,” Int. J. Solids Struct.
, 48
(3–4
), pp. 553
–566
.10.1016/j.ijsolstr.2010.10.020186.
Zhou
, Z. G.
, Zhang
, P. W.
, and Wu
, L. Z.
, 2012
, “Basic Solutions of a 3D Rectangular Limited-Permeable Crack or Two 3D Rectangular Limited-Permeable Cracks in the Piezoelectric/Piezomagnetic Composite Materials
,” Appl. Math. Model.
, 36
(6
), pp. 2404
–2428
.10.1016/j.apm.2011.08.035187.
Bogomol'nyi
, V. M.
, 2007
, “Electroelasticity Relations and Fracture Mechanics of Piezoelectric Structures
,” ASME Appl. Mech. Rev.
, 60
(1
), pp. 21
–36
.10.1115/1.2375142188.
Kaczyński
, A.
, and Kaczyński
, B.
, 2018
, “On 3D Problem of Vertically Uniform Heat Flow Disturbed by an Anticrack in a Transversely Isotropic Magnetoelectroelastic Space
,” Eur. J. Mech. A - Solids
, 69
, pp. 192
–200
.10.1016/j.euromechsol.2018.01.001189.
Rangelov
, T.
, Stoynov
, Y.
, and Dineva
, P.
, 2011
, “Dynamic Fracture Behavior of Functionally Graded Magnetoelectroelastic Solids by BIEM
,” Int. J. Solids Struct.
, 48
(20
), pp. 2987
–2999
.10.1016/j.ijsolstr.2011.06.016190.
Fang
, D. N.
, Li
, F. X.
, and Liu
, B.
, 2013
, “Advances in Developing Electromechanically Coupled Computational Methods for Piezoelectrics/Ferroelectrics at Multiscale
,” ASME Appl. Mech. Rev.
, 65
(6
), p. 060802
.10.1115/1.4025633191.
Wang
, B. L.
, and Mai
, Y. W.
, 2007
, “Self-Consistent Analysis of Coupled Magnetoelectroelastic Fracture-Theoretical Investigation and Finite Element Verification
,” Comput. Methods Appl. Mech. Eng.
, 196
(13–16
), pp. 2044
–2054
.10.1016/j.cma.2006.11.006192.
Rao
, B.
, and Kuna
, M.
, 2008
, “Interaction Integrals for Fracture Analysis of Functionally Graded Magnetoelectroelastic Materials
,” Int. J. Fract.
, 153
(1
), pp. 15
–37
.10.1007/s10704-008-9285-4193.
Belytschko
, T.
, and Black
, T.
, 1999
, “Elastic Crack Growth in Finite Elements With Minimal Remeshing
,” Int. J. Numer. Methods Eng.
, 45
(5
), pp. 601
–620
.10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S194.
Yazid
, A.
, and Abdelmadjid
, H.
, 2008
, “A Survey of the Extended Finite Element
,” Comput. Struct.
, 86
(11
), pp. 1141
–1151
.10.1016/j.compstruc.2007.11.001195.
Mohammadi
, S.
, 2008
, Extended Finite Element Method for Fracture Analysis of Structures
, John Wiley & Sons, Hoboken, NJ.196.
Zhuang
, Z.
, Liu
, Z.
, Cheng
, B.
, and Liao
, J.
, 2014
, Extended Finite Element Method
, Elsevier/Tsinghua University Press
, Beijing, China
.197.
Rojas-Díaz
, R.
, Sukumar
, N.
, Sáez
, A.
, and García-Sánchez
, F.
, 2011
, “Fracture in Magnetoelectroelastic Materials Using the Extended Finite Element Method
,” Int. J. Numer. Methods Eng.
, 88
(12
), pp. 1238
–1259
.10.1002/nme.3219198.
Bhargava
, R. R.
, and Sharma
, K.
, 2012
, “Application of X-FEM to Study Two-Unequal-Collinear Cracks in 2-D Finite Magnetoelectroelastic Specimen
,” Comput. Mater. Sci.
, 60
, pp. 75
–98
.10.1016/j.commatsci.2012.03.013199.
Bui
, T. Q.
, and Zhang
, C.
, 2013
, “Analysis of Generalized Dynamic Intensity Factors of Cracked Magnetoelectroelastic Solids by X-FEM
,” Finite Elem. Anal. Des.
, 69
, pp. 19
–36
.10.1016/j.finel.2013.02.001200.
Yu
, H.
, Wu
, L.
, and Li
, H.
, 2014
, “A Domain-Independent Interaction Integral for Magneto-Electro-Elastic Materials
,” Int. J. Solids Struct.
, 51
(2
), pp. 336
–351
.10.1016/j.ijsolstr.2013.10.005201.
Zhu
, S.
, Yu
, H.
, Zhang
, Y.
, Yan
, H.
, Man
, S.
, and Guo
, L.
, 2023
, “Generalized Dynamic Domain-Independent Interaction Integral in the Transient Fracture Investigation of Magneto-Electro-Elastic Composites
,” Eng. Fract. Mech.
, 292
, p. 109653
.10.1016/j.engfracmech.2023.109653202.
Zhu
, S.
, Yu
, H.
, Hao
, L.
, Shen
, Z.
, Wang
, J.
, and Guo
, L.
, 2023
, “Interaction Integral Method for Thermal Fracture of Nonhomogeneous Magneto-Electro-Elastic Materials
,” Eur. J. Mech. A - Solids
, 98
, p. 104871
.10.1016/j.euromechsol.2022.104871203.
Jena
, J.
, Singh
, S. K.
, and Singh
, I. V.
, 2024
, “A Numerical Study of Semipermeable Crack in Magneto-Electro-Elastic Material With Maxwell Stress
,” Eng. Fract. Mech.
, 299
, p. 109939
.10.1016/j.engfracmech.2024.109939204.
Ma
, P.
, Su
, R. K. L.
, and Feng
, W. J.
, 2016
, “Crack Tip Enrichment Functions for Extended Finite Element Analysis of Two-Dimensional Interface Cracks in Anisotropic Magnetoelectroelastic Bimaterials
,” Eng. Fract. Mech.
, 161
, pp. 21
–39
.10.1016/j.engfracmech.2016.04.038205.
Yan
, Z.
, Feng
, W. J.
, Zhang
, C.
, and Liu
, J. X.
, 2019
, “The Extended Finite Element Method With Novel Crack Tip Enrichment Functions for Dynamic Fracture Analysis of Interfacial Cracks in Piezoelectric–Piezomagnetic Bi-Layered Structures
,” Comput. Mech.
, 64
(5
), pp. 1303
–1319
.10.1007/s00466-019-01709-z206.
Yan
, Z.
, Feng
, W. J.
, and Zhang
, C.
, 2021
, “Interfacial Crack Growth in Piezoelectric-Piezomagnetic Bi-Layered Structures With a Modified Mechanical Energy Release Rate Criterion
,” Compos. Struct.
, 262
, p. 113344
.10.1016/j.compstruct.2020.113344207.
Liu
, Y. J.
, Mukherjee
, S.
, and Nishimura
, N.
, 2012
, “Recent Advances and Emerging Applications of the Boundary Element Method
,” ASME Appl. Mech. Rev.
, 64
(3
), p. 030802
.10.1115/1.4005491208.
Aliabadi
, M. H.
, 1997
, “Boundary Element Formulations in Fracture Mechanics
,” ASME Appl. Mech. Rev.
, 50
(2
), pp. 83
–96
.10.1115/1.3101690209.
Wang
, Y. B.
, and Chau
, K. T.
, 1997
, “A New Boundary Element for Plane Elastic Problems Involving Cracks and Holes
,” Int. J. Fract.
, 89
, pp. 1
–20
.10.1023/A:1007469816603210.
Snyder
, M. D.
, and Cruse
, T. A.
, 1975
, “Boundary Integral Equation Analysis of Cracked Anisotropic Plates
,” Int. J. Fract.
, 11
(2
), pp. 315
–328
.10.1007/BF00038898211.
Crouch
, S. L.
, 1976
, “Solution of Plane Elasticity Problems by the Displacement Discontinuity Method
,” Int. J. Numer. Methods Eng.
, 10
(2
), pp. 301
–343
.10.1002/nme.1620100206212.
Blandford
, G. E.
, Ingraffea
, A. R.
, and Liggett
, J. A.
, 1981
, “Two Dimensional Stress Intensity Factor Computations Using the Boundary Element Method
,” Int. J. Numer. Methods Eng.
, 17
(3
), pp. 387
–404
.10.1002/nme.1620170308213.
Hong
, H.-K.
, and Chen
, J. T.
, 1988
, “Derivations of Integral Equations of Elasticity
,” J. Eng. Mech.
, 114
(6
), pp. 1028
–1044
.10.1061/(ASCE)0733-9399(1988)114:6(1028)214.
Hong
, H.-K.
, and Chen
, J. T.
, 1999
, “Review of Dual Boundary Element Methods With Emphasis on Hypersingular Integrals and Divergent Series
,” ASME Appl. Mech. Rev.
, 52
(1
), pp. 17
–33
.10.1115/1.3098922215.
Portela
, A.
, Aliabadi
, M. H.
, and Rooke
, D. P.
, 1992
, “The Dual Boundary Element Method: Effective Implementation for Crack Problems
,” Int. J. Numer. Methods Eng.
, 33
(6
), pp. 1269
–1287
.10.1002/nme.1620330611216.
Weaver
, J.
, 1977
, “Three Dimensional Crack Analysis
,” Int. J. Solids Struct.
, 13
, pp. 321
–330
.10.1016/0020-7683(77)90016-6217.
Wang
, Y. B.
, and Chen
, W. J.
, 1993
, “Interaction of Two Equal Coplanar Square Cracks in Three Dimensional Elasticity
,” Int. J. Solids Struct.
, 30
(23
), pp. 3315
–3320
.10.1016/0020-7683(93)90116-O218.
Ang
, W. T.
, and Park
, Y. S.
, 1997
, “Hypersinglar Integral Equations for Arbitrarily Located Planar Cracks in an Anisotropic Elastic Bimaterial
,” Eng. Anal. Boundary Elem.
, 20
(2
), pp. 135
–143
.10.1016/S0955-7997(97)00057-X219.
Zhang
, C.
, 2002
, “A 2-D Time-Domain BIEM for Dynamic Analysis of Cracked Orthotropic Solids
,” Comput. Model. Eng. Sci.
, 3
, pp. 381
–398
.10.3970/cmes.2002.003.381220.
Zhang
, C.
, 2002
, “A 2-D Hypersingular Time-Domain Traction BEM for Transient Elastodynamic Crack Analysis
,” Wave Motion
, 35
(1
), pp. 17
–40
.10.1016/S0165-2125(01)00081-6221.
Zhang
, C.
, 1991
, “A Novel Derivation of Non-Hypersingular Time-Domain BIE Formulation for Transient Elastodynamic Crack Analysis
,” Int. J. Solids Struct.
, 28
(3
), pp. 267
–281
.10.1016/0020-7683(91)90193-J222.
Zhang
, C.
, and Gross
, D.
, 1993
, “A Non-Hypersingular Time-Domain BIEM for 3-D Transient Elastodynamic Crack Analysis
,” Int. J. Numer. Methods Eng.
, 36
(17
), pp. 2997
–3017
.10.1002/nme.1620361709223.
Dineva
, P.
, Rangelov
, T.
, and Gross
, D.
, 2005
, “BIEM for 2D Steady-State Problems in Cracked Anisotropic Materials
,” Eng. Anal. Boundary Elem.
, 29
(7
), pp. 689
–698
.10.1016/j.enganabound.2005.02.006224.
Zhao
, M. H.
, Zhang
, Q. Y.
, Li
, X. F.
, Guo
, Y. G.
, Fan
, C. Y.
, and Lu
, C. S.
, 2019
, “An Iterative Approach for Analysis of Cracks With Exact Boundary Conditions in Finite Magnetoelectroelastic Solids
,” Smart Mater. Struct.
, 28
(5
), p. 055025
.10.1088/1361-665X/ab0eb0225.
García-Sánchez
, F.
, Rojas-Díaz
, R.
, Sáez
, A.
, and Zhang
, C.
, 2007
, “Fracture of Magnetoelectroelastic Composite Materials Using Boundary Element Method (BEM)
,” Theor. Appl. Fract. Mech.
, 47
(3
), pp. 192
–204
.10.1016/j.tafmec.2007.01.008226.
Dong
, C. Y.
, Lo
, S. H.
, and Antes
, H.
, 2008
, “Fracture Analysis in 2D Magneto-Electro-Elastic Media by the Boundary Element Method
,” Comput. Mech.
, 41
(2
), pp. 207
–217
.10.1007/s00466-007-0179-5227.
Pasternak
, I.
, 2012
, “Doubly Periodic Arrays of Cracks and Thin Inhomogeneities in an Infinite Magnetoelectroelastic Medium
,” Eng. Anal. Boundary Elem.
, 36
(5
), pp. 799
–811
.10.1016/j.enganabound.2011.12.004228.
Hattori
, G.
, and Sáez
, A.
, 2013
, “Crack Identification in Magnetoelectroelastic Materials Using Neural Networks, Self-Organizing Algorithms and Boundary Element Method
,” Comput. Struct.
, 125
, pp. 187
–199
.10.1016/j.compstruc.2013.05.005229.
Rojas-Díaz
, R.
, Denda
, M.
, García-Sánchez
, F.
, and Sáez
, A.
, 2012
, “Dual BEM Analysis of Different Crack Face Boundary Conditions in 2D Magnetoelectroelastic Solids
,” Eur. J. Mech. A - Solids
, 31
(1
), pp. 152
–162
.10.1016/j.euromechsol.2011.08.002230.
Rojas-Díaz
, R.
, Sáez
, F.
, and García-Sánchez
, A.
, 2010
, “Analysis of Cracked Magnetoelectroelastic Composites Under Time-Harmonic Loading
,” Int. J. Solids Struct.
, 47
(1
), pp. 71
–80
.10.1016/j.ijsolstr.2009.09.011231.
Rojas-Díaz
, R.
, García-Sánchez
, F.
, Sáez
, A.
, Rodríguez-Mayorga
, E.
, and Zhang
, C.
, 2011
, “Fracture Analysis of Plane Piezoelectric/Piezomagnetic Multiphase Composites Under Transient Loading
,” Comput. Methods Appl. Mech. Eng.
, 200
(45–46
), pp. 2931
–2942
.10.1016/j.cma.2011.06.004232.
Wünsche
, M.
, Sáez
, A.
, García-Sánchez
, F.
, and Zhang
, C. Z.
, 2009
, “Cracks in Magnetoelectroelastic Solids Under Impact Loading
,” Key Eng. Mater.
, 417–418
, pp. 377
–380
.10.4028/www.scientific.net/KEM.417-418.377233.
Wünsche
, M.
, Sáez
, A.
, Zhang
, C.
, and García-Sánchez
, F.
, 2011
, “Semi-Permeable Cracks in Magnetoelectroelastic Solids Under Impact Loading
,” Key Eng. Mater.
, 488–489
, pp. 751
–754
.10.4028/www.scientific.net/KEM.488-489.751234.
Wünsche
, M.
, Zhang
, C.
, Sladek
, J.
, Sladek
, V.
, Sáez
, A.
, and García-Sánchez
, F.
, 2013
, “The Influences of Non-Linear Electrical, Magnetic and Mechanical Boundary Conditions on the Dynamic Intensity Factors of Magnetoelectroelastic Solids
,” Eng. Fract. Mech.
, 97
, pp. 297
–313
.10.1016/j.engfracmech.2012.08.006235.
Wünsche
, M.
, Sáez
, A.
, García-Sánchez
, F.
, Zhang
, C. Z.
, and Domínguez
, J.
, 2014
, “Transient Dynamic Analysis of Cracked Multifield Solids With Consideration of Crack-Face Contact and Semi-Permeable Electric/Magnetic Boundary Conditions
,” Key Eng. Mater.
, 618
, pp. 123
–150
.10.4028/www.scientific.net/KEM.618.123236.
Wünsche
, M.
, Zhang
, C.
, Sáez
, A.
, and García-Sánchez
, F.
, 2010
, “Transient Dynamic Analysis of Cracked Magnetoelectroelastic Composites by a Hypersingular Time-Domain BEM
,” Proc. Appl. Math. Mech. (PAMM)
, 10
(1
), pp. 139
–140
.10.1002/pamm.201010062237.
Wünsche
, M.
, Sáez
, A.
, García-Sánchez
, F.
, and Zhang
, C.
, 2012
, “Transient Dynamic Crack Analysis in Linear Magnetoelectroelastic Solids by a Hypersingular Time-Domain BEM
,” Eur. J. Mech. A - Solids
, 32
, pp. 118
–130
.10.1016/j.euromechsol.2011.07.007238.
Lei
, J.
, Zhang
, C.
, and Bui
, T. Q.
, 2015
, “Transient Dynamic Interface Crack Analysis in Magnetoelectroelastic Bi-Materials by a Time-Domain BEM
,” Eur. J. Mech. A - Solids
, 49
, pp. 146
–157
.10.1016/j.euromechsol.2014.07.010239.
Liu
, J. X.
, Liu
, X. L.
, and Zhao
, Y. B.
, 2001
, “Green's Functions for Anisotropic Magnetoelectroelastic Solids With an Elliptical Cavity or a Crack
,” Int. J. Eng. Sci.
, 39
(12
), pp. 1405
–1418
.10.1016/S0020-7225(01)00005-2240.
Qin
, Q. H.
, 2004
, “Green's Functions of Magnetoelectroelastic Solids With a Half-Plane Boundary or Bimaterial Interface
,” Philos. Mag. Lett.
, 84
(12
), pp. 771
–779
.10.1080/09500830500061508241.
Rojas-Díaz
, R.
, Sáez
, A.
, García-Sánchez
, F.
, and Zhang
, C.
, 2008
, “Time-Harmonic Green's Functions for Anisotropic Magnetoelectroelasticity
,” Int. J. Solids Struct.
, 45
(1
), pp. 144
–158
.10.1016/j.ijsolstr.2007.07.024242.
Zhao
, M. H.
, Xu
, G. T.
, and Fan
, C. Y.
, 2010
, “Numerical Method of Crack Analysis in 2D Finite Magnetoelectroelastic Media
,” Comput. Mech.
, 45
(5
), pp. 401
–413
.10.1007/s00466-009-0459-3243.
Feng
, W. J.
, Li
, Y. S.
, Han
, X.
, and Xu
, Z. H.
, 2011
, “Application of the Extended Traction Boundary Element-Free Method to the Fracture of Two-Dimensional Infinite Magnetoelectroelastic Solid
,” Sci. China: Phys., Mech. Astron.
, 54
(6
), pp. 1141
–1153
.10.1007/s11433-010-4243-3244.
Li
, C.
, and Tong
, L.
, 2015
, “2D Fracture Analysis of Magnetoelectroelastic Composites by the SBFEM
,” Compos. Struct.
, 132
, pp. 984
–994
.10.1016/j.compstruct.2015.07.015245.
Zhu
, B. J.
, and Qin
, T. Y.
, 2007
, “Hypersingular Integral Equation Method for a Three Dimensional Crack in Anisotropic Electro-Magneto-Elastic Bimaterials
,” Theor. Appl. Fract. Mech.
, 47
(3
), pp. 219
–232
.10.1016/j.tafmec.2007.01.007246.
Zhu
, B. J.
, Shi
, Y. L.
, Qin
, T. Y.
, Sukop
, M.
, Yu
, S. H.
, and Li
, Y. B.
, 2009
, “Mixed-Mode Stress Intensity Factors of 3D Interface Crack in Fully Coupled Electromagnetothermoelastic Multiphase Composites
,” Int. J. Solids Struct.
, 46
(13
), pp. 2669
–2679
.10.1016/j.ijsolstr.2009.02.010247.
Zhao
, Y.
, Zhao
, M.
, Pan
, E.
, and Fan
, C.
, 2015
, “Green's Functions and Extended Displacement Discontinuity Method for Interfacial Cracks in Three-Dimensional Transversely Isotropic Magneto-Electro-Elastic Bi-Materials
,” Int. J. Solids Struct.
, 52
, pp. 56
–71
.10.1016/j.ijsolstr.2014.09.018248.
Zhao
, Y.
, Pan
, E.
, and Zhao
, M.
, 2015
, “Displacement Discontinuity Analysis of a Nonlinear Interfacial Crack in Three-Dimensional Transversely Isotropic Magneto-Electro-Elastic Bi-Materials
,” Eng. Anal. Boundary Elem.
, 61
, pp. 254
–264
.10.1016/j.enganabound.2015.08.001249.
Lucy
, B. L.
, 1977
, “A Numerical Approach to Testing the Fission Hypothesis
,” Astron. J.
, 82
(12
), pp. 1013
–1024
.10.1086/112164250.
Gingold
, R. A.
, and Monaghan
, J. J.
, 1977
, “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars
,” Mon. Not. R. Astron. Soc.
, 181
(3
), pp. 375
–389
.10.1093/mnras/181.3.375251.
Chen
, Y.
, Lee
, J.
, and Eskandarian
, A.
, 2006
, Meshless Methods in Solid Mechanics
, Springer
, New York
.252.
Atluri
, S. N.
, and Zhu
, T.
, 1998
, “New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics
,” Comput. Mech.
, 22
(2
), pp. 117
–127
.10.1007/s004660050346253.
Sladek
, J.
, Sladek
, V.
, Solek
, P.
, and Pan
, E.
, 2008
, “Fracture Analysis of Cracks in Magneto-Electro-Elastic Solids by the MLPG
,” Comput. Mech.
, 42
(5
), pp. 697
–714
.10.1007/s00466-008-0269-z254.
Sladek
, J.
, Sladek
, V.
, Solek
, P.
, and Zhang
, C.
, 2010
, “Fracture Analysis in Continuously Nonhomogeneous Magneto-Electro-Elastic Solids Under a Thermal Load by the MLPG
,” Int. J. Solids Struct.
, 47
(10
), pp. 1381
–1391
.10.1016/j.ijsolstr.2010.01.025255.
Sladek
, J.
, Sladek
, V.
, Stanak
, P.
, Zhang
, C.
, and Wünsche
, M.
, 2011
, “An Interaction Integral Method for Computing Fracture Parameters in Functionally Graded Magnetoelectroelastic Composites
,” Comput. Mater. Continua
, 23
(1
), pp. 35
–68
.10.3970/cmc.2011.023.035256.
Sladek
, J.
, Sladek
, V.
, Zhang
, C.
, and Wünsche
, M.
, 2012
, “Semi-Permeable Crack Analysis in Magnetoelectroelastic Solids
,” Smart Mater. Struct.
, 21
(2
), p. 025003
.10.1088/0964-1726/21/2/025003257.
Li
, Y. S.
, Feng
, W. J.
, and Xu
, Z. H.
, 2009
, “Fracture Analysis of Cracked 2D Planar and Axisymmetric Problems of Magneto-Electro-Elastic Materials by the MLPG Coupled With FEM
,” Comput. Methods Appl. Mech. Eng.
, 198
(30–32
), pp. 2347
–2359
.10.1016/j.cma.2009.02.021258.
Bui
, T. Q.
, Hirose
, S.
, Zhang
, C.
, Rabczuk
, T.
, Wu
, C. T.
, Saitoh
, T.
, and Lei
, J.
, 2016
, “Extended Isogeometric Analysis for Dynamic Fracture in Multiphase Piezoelectric/Piezomagnetic Composites
,” Mech. Mater.
, 97
, pp. 135
–163
.10.1016/j.mechmat.2016.03.001259.
Singh
, S. K.
, and Singh
, I. V.
, 2021
, “Extended Isogeometric Analysis for Fracture in Functionally Graded Magneto-Electro-Elastic Material
,” Eng. Fract. Mech.
, 247
, p. 107640
.10.1016/j.engfracmech.2021.107640260.
Zhou
, Z. H.
, Yu
, X.
, Yang
, Z. T.
, Xu
, W.
, Lim
, C. W.
, and Xu
, X. S.
, 2019
, “An Isogeometric-Symplectic Coupling Approach for Fracture Analysis of Magnetoelectroelastic Bimaterials With Crack Terminating at the Interface
,” Eng. Fract. Mech.
, 216
, p. 106510
.10.1016/j.engfracmech.2019.106510261.
Tan
, Y.
, Liu
, C.
, Zhao
, J.
, He
, Y.
, Li
, P.
, and Li
, X.
, 2023
, “Phase Field Model for Brittle Fracture in Multiferroic Materials
,” Comput. Methods Appl. Mech. Eng.
, 414
, p. 116193
.10.1016/j.cma.2023.116193262.
Wang
, B.
, Han
, J. C.
, and Du
, S.
, 2008
, Coupled Thermal-Electro-Magneto-Mechanical Cracking of Non-Homogeneous Media
, Nova Science
, New York
.263.
Chen
, X. H.
, and Mai
, Y. W.
, 2013
, Fracture Mechanics of Electromagnetic Material: Nonlinear Field Theory and Applications
, Imperial College Press
, London, UK
.264.
Ritchie
, R. O.
, 1999
, “Mechanisms of Fatigue-Crack Propagation in Ductile and Brittle Solids
,” Int. J. Fract.
, 100
(1
), pp. 55
–83
.10.1023/A:1018655917051265.
Cao
, H.
, and Evans
, A. G.
, 2010
, “Electric-Field-Induced Fatigue Crack Growth in Piezoelectrics
,” J. Am. Ceram. Soc.
, 77
(7
), pp. 1783
–1786
.10.1111/j.1151-2916.1994.tb07051.x266.
Radulovic
, R.
, Bruhns
, O. T.
, and Mosler
, J.
, 2011
, “Effective 3D Failure Simulations by Combining the Advantages of Embedded Strong Discontinuity Approaches and Classical Interface Elements
,” Eng. Fract. Mech.
, 78
(12
), pp. 2470
–2485
.10.1016/j.engfracmech.2011.06.007267.
Armero
, F.
, and Linder
, C.
, 2009
, “Numerical Simulation of Dynamic Fracture Using Finite Elements With Embedded Discontinuities
,” Int. J. Fract.
, 160
(2
), pp. 119
–141
.10.1007/s10704-009-9413-9268.
Rabczuk
, T.
, and Belytschko
, T.
, 2004
, “Cracking Particles: A Simplified Meshfree Method for Arbitrary Evolving Cracks
,” Int. J. Numer. Methods Eng.
, 61
(13
), pp. 2316
–2343
.10.1002/nme.1151269.
Rabczuk
, T.
, Song
, J. H.
, and Belytschko
, T.
, 2009
, “Simulations of Instability in Dynamic Fracture by the Cracking Particles Method
,” Eng. Fract. Mech.
, 76
(6
), pp. 730
–741
.10.1016/j.engfracmech.2008.06.002270.
Miehe
, C.
, Hofacker
, M.
, and Welschinger
, F.
, 2010
, “A Phase Field Model for Rate-Independent Crack Propagation: Robust Algorithmic Implementation Based on Operator Splits
,” Comput. Methods Appl. Mech. Eng.
, 199
(45–48
), pp. 2765
–2778
.10.1016/j.cma.2010.04.011271.
Miehe
, C.
, Welschinger
, F.
, and Hofacker
, M.
, 2010
, “Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field FE Implementations
,” Int. J. Numer. Methods Eng.
, 83
, pp. 1273
–1311
.10.1002/nme.2861272.
Rice
, J. R.
, Beltz
, G. E.
, and Sun
, Y.
, 1992
, Periers Framework for Dislocation Nucleation From a Crack Tip
, Springer-Verlag
, Berlin, Germany
.273.
Tan
, H. L.
, and Yang
, W.
, 1994
, “Atomatic/Continuum Simulation of Interfacial Fracture, Part I: Atomatic Simulation
,” Acta Mech. Sin.
, 10
(2
), pp. 151
–162
.10.1007/BF02486585274.
Yang
, W.
, 1995
, Macroscopic and Microscopic Fracture Mechanics
, National Defence Industry Press
, Beijing, China
(in Chinese).
