Fracture model for the prediction of the electrical percolation threshold in CNTs/Polymer composites
Yang SHEN1, Pengfei HE1, Xiaoying ZHUANG2,3,4()
1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China 2. Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai, China 3. Institute of Continuum Mechanics, Leibniz-Universität Hannover, Germany 4. State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China
In this paper, we propose a 3D stochastic model to predict the percolation threshold and the effective electric conductivity of CNTs/Polymer composites. We consider the tunneling effect in our model so that the unrealistic interpenetration can be avoided in the identification of the conductive paths between the CNTs inside the polymer. The results are shown to be in good agreement with reported experimental data.
. [J]. Frontiers of Structural and Civil Engineering, 2018, 12(1): 125-136.
Yang SHEN, Pengfei HE, Xiaoying ZHUANG. Fracture model for the prediction of the electrical percolation threshold in CNTs/Polymer composites. Front. Struct. Civ. Eng., 2018, 12(1): 125-136.
Sandler J, Shaffer M S P, Prasse T, Bauhofer W, Schulte K, Windle A H. Development of a dispersion process for carbon nanotubes in an epoxy matrix and the resulting electrical properties. Polymer, 1999, 40(21): 5967–5971 https://doi.org/10.1016/S0032-3861(99)00166-4
2
Michael G H, Tejas N. Radiofrequency interaction with conductive colloids: permittivity and electrical conductivity of single-wall carbon nanotubes in sallne. Bioelectromagnetics, 2010, 31(8): 582–588
3
Martin C A, Sandler J K W, Shaffer M S P, Schwarz M K, Bauhofer W, Schulte K, Windle A H. Formation of percolating networks in multi-wall carbon-nanotube–epoxy composites. Composites Science and Technology, 2004, 64(15): 2309–2316 https://doi.org/10.1016/j.compscitech.2004.01.025
4
Bauhofer W, Kovacs J Z. A review and analysis of electrical percolation in carbon nanotube polymer composites. Composites Science and Technology, 2009, 69(10): 1486–1498 https://doi.org/10.1016/j.compscitech.2008.06.018
5
Bryning M B, Islam M F, Kikkawa J M, Yodh A G. Very low conductivity threshold in bulk isotropic single-walled carbon nanotube–epoxy composites. Advanced Materials, 2005, 17(9): 1186–1191 https://doi.org/10.1002/adma.200401649
6
Ounaies Z, Park C, Wise K E, Siochi E J, Harrison J S. Electrical properties of single wall carbon nanotube reinforced polyimide composites. Composites Science and Technology, 2003, 63(11): 1637–1646 https://doi.org/10.1016/S0266-3538(03)00067-8
7
Kymakis E, Amaratunga G A J. Electrical properties of single-wall carbon nanotube-polymer composite films. Journal of Applied Physics, 2006, 99(8): 56 https://doi.org/10.1063/1.2189931
8
Ramasubramaniam R, Chen J, Liu H. Homogeneous carbon nanotube/polymer composites for electrical applications. Applied Physics Letters, 2003, 83(14): 2928–2930 https://doi.org/10.1063/1.1616976
9
Griebel M, Hamaekers J. Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites. Computer Methods in Applied Mechanics and Engineering, 2004, 193(17): 1773–1788
10
Frankland S J V, Caglar A, Brenner D W, Griebel M. Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube-polymer interfaces. Journal of Physical Chemistry B, 2002, 106(12): 3046–3048 https://doi.org/10.1021/jp015591+
11
Zhu R, Pan E, Roy A K. Molecular dynamics study of the stress–strain behavior of carbon-nanotube reinforced epon 862 composites. Materials Science and Engineering A, 2007, 447(1): 51–57 https://doi.org/10.1016/j.msea.2006.10.054
12
Arash B, Park H S, Rabczuk T. Mechanical properties of carbon nanotube reinforced polymer nanocomposites: A coarse-grained model. Composites. Part B, Engineering, 2015, 80: 92–100 https://doi.org/10.1016/j.compositesb.2015.05.038
13
Quayum M S, Zhuang X, Rabczuk T. Computational model generation and rve design of self-healing concrete. Journal of Contemporary Physics, 2015, 50(4): 383–396
14
Mortazavi B, Baniassadi M, Bardon J, Ahzi S. Modeling of two-phase random composite materials by finite element, mori–tanaka and strong contrast methods. Composites. Part B, Engineering, 2013, 45(1): 1117–1125 https://doi.org/10.1016/j.compositesb.2012.05.015
15
Mortazavi B, Bardon J, Ahzi S. Interphase effect on the elastic and thermal conductivity response of polymer nanocomposite materials: 3d finite element study. Computational Materials Science, 2013, 69: 100–106 https://doi.org/10.1016/j.commatsci.2012.11.035
17
Hamdia K M, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190 https://doi.org/10.1016/j.compstruct.2015.08.051
18
Silani M, Talebi H, Ziaei-Rad S, Kerfriden P, Bordas S P A, Rabczuk T. Stochastic modelling of clay/epoxy nanocomposites. Composite Structures, 2014, 118: 241–249 https://doi.org/10.1016/j.compstruct.2014.07.009
19
Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (pncs). Composites. Part B, Engineering, 2014, 59: 80–95 https://doi.org/10.1016/j.compositesb.2013.11.014
20
Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimization of fiber distribution in fiber reinforced composite by using nurbs functions. Computational Materials Science, 2014, 83: 463–473 https://doi.org/10.1016/j.commatsci.2013.11.032
22
Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464 https://doi.org/10.1016/j.compositesb.2014.09.008
23
Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimal fiber content and distribution in fiber-reinforced solids using a reliability and nurbs based sequential optimization approach. Structural and Multidisciplinary Optimization, 2015, 51(1): 99–112 https://doi.org/10.1007/s00158-014-1114-y
25
Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31 https://doi.org/10.1016/j.advengsoft.2016.06.005
26
Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96(PB):520–535
27
Msekh M A, Silani M, Jamshidian M, Areias P, Zhuang X, Zi G, He P, Rabczuk T. Predictions of j integral and tensile strength of clay/epoxy nanocomposites material using phase field model. Composites. Part B, Engineering, 2016, 93: 97–114 https://doi.org/10.1016/j.compositesb.2016.02.022
28
Hamdia K M, Zhuang X, He P, Rabczuk T. Fracture toughness of polymeric particle nanocomposites: Evaluation of models performance using bayesian method. Composites Science and Technology, 2016, 126: 122–129 https://doi.org/10.1016/j.compscitech.2016.02.012
30
Ben Dhia H. Multiscale mechanical problems: the arlequin method. Comptes Rendus de l’Academie des Sciences Series IIB Mechanics Physics Astronomy, 1998, 326(12): 899–904
31
Dhia H B, Rateau G. The arlequin method as a flexible engineering design tool. International Journal for Numerical Methods in Engineering, 2005, 62(11): 1442–1462 https://doi.org/10.1002/nme.1229
32
Xiao S P, Belytschko T. A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering, 2004, 193(17): 1645–1669 https://doi.org/10.1016/j.cma.2003.12.053
33
Wagner G J, Liu W K. Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal of Computational Physics, 2003, 190(1): 249–274 https://doi.org/10.1016/S0021-9991(03)00273-0
34
Tadmor E B, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philosophical Magazine A, 1996, 73(6): 1529–1563 https://doi.org/10.1080/01418619608243000
35
Shenoy V B, Miller R, Tadmor E B, Rodney D, Phillips R, Ortiz M. An adaptive finite element approach to atomic-scale mechanicsthe quasicontinuum method. Journal of the Mechanics and Physics of Solids, 1999, 47(3): 611–642 https://doi.org/10.1016/S0022-5096(98)00051-9
36
Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071 https://doi.org/10.1007/s00466-013-0948-2
37
Silani M, Talebi H, Hamouda A M, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15:18-23
38
Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143 https://doi.org/10.1016/j.tafmec.2013.12.004
39
Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74(1): 30–38 https://doi.org/10.1016/j.tafmec.2014.06.009
40
Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92 https://doi.org/10.1016/j.advengsoft.2014.09.016
43
Budarapu P R, Gracie R, Bordas S P A, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148 https://doi.org/10.1007/s00466-013-0952-6
44
Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541 https://doi.org/10.1615/IntJMultCompEng.2013005838
Ghasemi H, Park H S, Rabczuk T. A level-set based iga formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258 https://doi.org/10.1016/j.cma.2016.09.029
49
Nanthakumar S S, Valizadeh N, Park H S, Rabczuk T. Surface effects on shape and topology optimization of nanostructures. Computational Mechanics, 2015, 56(1): 97–112 https://doi.org/10.1007/s00466-015-1159-9
53
Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92-93: 242–246 https://doi.org/10.1016/j.compstruc.2011.10.021
55
Kumar S, Singh I V, Mishra B K, Rabczuk T. Modeling and simulation of kinked cracks by virtual node xfem. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 1425–1466 https://doi.org/10.1016/j.cma.2014.10.019
56
Chen L, Rabczuk T, Bordas S P A, Liu G R, Zeng K Y, Kerfriden P. Extended finite element method with edge-based strain smoothing (ESM-XFEM) for linear elastic crack growth. Computer Methods in Applied Mechanics and Engineering, 2012, 209-212: 250–265 https://doi.org/10.1016/j.cma.2011.08.013
58
Bordas S P A, Natarajan S, Kerfriden P, Augarde C E, Mahapatra D R, Rabczuk T, Pont S D. On the performance of strain smoothing for quadratic and enriched finite element approximations (xfem/gfem/pufem). International Journal for Numerical Methods in Engineering, 2011, 86(4-5): 637–666 https://doi.org/10.1002/nme.3156
61
Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143 https://doi.org/10.1016/j.engfracmech.2015.10.042
62
Rabizadeh E, Saboor Bagherzadeh A, Rabczuk T. Goal-oriented error estimation and adaptive mesh refinement in dynamic coupled thermoelasticity. Computers & Structures, 2016, 173: 187–211 https://doi.org/10.1016/j.compstruc.2016.05.024
63
Areias P, Rabczuk T, Msekh M A. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350 https://doi.org/10.1016/j.cma.2016.01.020
64
Areias P, Rabczuk T, de Sá J C. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Computational Mechanics, 2016, 58(6): 1003–1018 https://doi.org/10.1007/s00466-016-1328-5
65
Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315 https://doi.org/10.1007/s00466-015-1172-z
66
Areias P, Rabczuk T, Camanho P P. Finite strain fracture of 2d problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72(1): 50–63 https://doi.org/10.1016/j.tafmec.2014.06.006
67
Areias P, Dias-Da-Costa D, Sargado J M, Rabczuk T. Element-wise algorithm for modeling ductile fracture with the rousselier yield function. Computational Mechanics, 2013, 52(6): 1429–1443 https://doi.org/10.1007/s00466-013-0885-0
68
Areias P, Rabczuk T, Dias-da Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137 https://doi.org/10.1016/j.engfracmech.2013.06.006
69
Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122 https://doi.org/10.1002/nme.4477
70
Nguyen-Xuan H, Liu G R, Bordas S, Natarajan S, Rabczuk T. An adaptive singular es-fem for mechanics problems with singular field of arbitrary order. Computer Methods in Applied Mechanics and Engineering, 2013, 253: 252–273 https://doi.org/10.1016/j.cma.2012.07.017
71
Rabczuk T, Belytschko T, Xiao S P. Stable particle methods based on lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14): 1035–1063 https://doi.org/10.1016/j.cma.2003.12.005
73
Rabczuk T, Areias P. A meshfree thin shell for arbitrary evolving cracks based on an extrinsic basis. Computer Modeling in Engineering & Sciences, 2006, 16(2): 115–130
74
Zi G, Rabczuk T, Wall W. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382 https://doi.org/10.1007/s00466-006-0115-0
75
Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760 https://doi.org/10.1007/s00466-006-0067-4
76
Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548 https://doi.org/10.1002/nme.2013
78
Amiri F, Millán D, Arroyo M, Silani M, Rabczuk T. Fourth order phase-field model for local max-ENT approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 254–275 https://doi.org/10.1016/j.cma.2016.02.011
79
Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109 https://doi.org/10.1016/j.tafmec.2013.12.002
80
Amiri F, Anitescu C, Arroyo M, Bordas S P A, Rabczuk T. Xlme interpolants, a seamless bridge between xfem and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57 https://doi.org/10.1007/s00466-013-0891-2
82
Talebi H, Samaniego C, Samaniego E, Rabczuk T. On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods. International Journal for Numerical Methods in Engineering, 2012, 89(8): 1009–1027 https://doi.org/10.1002/nme.3275
83
Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71
84
Nguyen V P, Rabczuk T, Bordas S, Duflot M. Meshless methods: A review and computer implementation aspects. Mathematics and Computers in Simulation, 2008, 79(3): 763–813 https://doi.org/10.1016/j.matcom.2008.01.003
86
Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically nonlinear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758 https://doi.org/10.1016/j.engfracmech.2008.06.019
87
Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476 https://doi.org/10.1002/nme.5257
88
Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343 https://doi.org/10.1002/nme.1151
89
Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40): 2437–2455 https://doi.org/10.1016/j.cma.2010.03.031
91
Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29-30): 2777–2799 https://doi.org/10.1016/j.cma.2006.06.020
93
Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on kirchhoff-love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291 https://doi.org/10.1016/j.cma.2014.08.025
94
Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based xiga for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146 https://doi.org/10.1016/j.compstruc.2014.09.017
96
Thai T Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604 https://doi.org/10.1016/j.cma.2016.02.031
97
Chan C L, Anitescu C, Rabczuk T. Volumetric parametrization from a level set boundary representation with pht-splines. CAD Computer Aided Design, 2017, 82: 29–41 https://doi.org/10.1016/j.cad.2016.08.008
100
Nguyen V P, Anitescu C, Bordas S P A, Rabczuk T. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 2015, 117: 89–116 https://doi.org/10.1016/j.matcom.2015.05.008
103
Thai C H, Ferreira A J M, Bordas S P A, Rabczuk T, Nguyen-Xuan H. Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. European Journal of Mechanics. A, Solids, 2014, 43: 89–108 https://doi.org/10.1016/j.euromechsol.2013.09.001
106
Thai C H, Nguyen-Xuan H, Nguyen-Thanh N, Le T H, Nguyen-Thoi T, Rabczuk T. Static, free vibration, and buckling analysis of laminated composite reissner-mindlin plates using nurbs-based isogeometric approach. International Journal for Numerical Methods in Engineering, 2012, 91(6): 571–603 https://doi.org/10.1002/nme.4282
107
Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wchner R, Bletzinger K U, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using pht-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47-48): 3410–3424 https://doi.org/10.1016/j.cma.2011.08.014
108
Nguyen-Thanh N, Nguyen-Xuan H, Bordas S P A, Rabczuk T. Isogeometric analysis using polynomial splines over hierarchical t-meshes for two-dimensional elastic solids. Computer Methods in Applied Mechanics and Engineering, 2011, 200(21-22): 1892–1908 https://doi.org/10.1016/j.cma.2011.01.018
111
Zhuang X, Huang R, Liang C, Rabczuk T. A coupled thermohydro-mechanical model of jointed hard rock for compressed air energy storage. Mathematical Problems in Engineering, 2014, 2014
113
Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947 https://doi.org/10.1007/s00466-013-0855-6
116
Frédéric Feyel, Jean-Louis Chaboche. Fe 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Computer Methods in Applied Mechanics and Engineering, 2000, 183(3): 309–330
117
Zeng X, Xu X, Prathamesh M. Shenai, Eugene Kovalev, Charles Baudot, Nripan Mathews, and Yang Zhao. Characteristics of the electrical percolation in carbon nanotubes/polymer nanocomposites. Journal of Physical Chemistry C, 2011, 115(44): 21685–21690 https://doi.org/10.1021/jp207388n
118
Belytschko T, Yun Y L, Gu L. Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256 https://doi.org/10.1002/nme.1620370205
119
Belytschko T, Lu Y Y, Gu L, Tabbara M. Element-free galerkin methods for static and dynamic fracture. International Journal of Solids and Structures, 1995, 32(17): 2547–2570 https://doi.org/10.1016/0020-7683(94)00282-2
120
Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3d: framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998 https://doi.org/10.1002/nme.4365
121
Zhuang X, Zhu H, Augarde C. An improved mesh-less shepard and least squares method possessing the delta property and requiring no singular weight function. Computational Mechanics, 2014, 53(2): 343–357 https://doi.org/10.1007/s00466-013-0912-1
122
Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495 https://doi.org/10.1007/s00466-006-0122-1
123
Bordas S, Rabczuk T, Zi G. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Engineering Fracture Mechanics, 2008, 75(5): 943–960 https://doi.org/10.1016/j.engfracmech.2007.05.010
124
Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23): 1391–1411 https://doi.org/10.1016/j.compstruc.2008.08.010
125
Song J H, Areias P, Belytschko T. A method for dynamic crack and shear band propagation with phantom nodes. International Journal for Numerical Methods in Engineering, 2006, 67(6): 868–893 https://doi.org/10.1002/nme.1652
126
Enderlein M, Ricoeur A, Kuna M. Finite element techniques for dynamic crack analysis in piezoelectrics. International Journal of Fracture, 2005, 134(3-4): 191–208 https://doi.org/10.1007/s10704-005-0522-9
Shang F, Kuna M, Abendroth M. Meinhard Kuna, and Martin Abendroth. Finite element analyses of three-dimensional crack problems in piezoelectric structures. Engineering Fracture Mechanics, 2003, 70(2): 143–160 https://doi.org/10.1016/S0013-7944(02)00039-5
129
Béchet E, Scherzer M, Kuna M. Application of the x-fem to the fracture of piezoelectric materials. International Journal for Numerical Methods in Engineering, 2009, 77(11): 1535–1565 https://doi.org/10.1002/nme.2455
130
Nanthakumar S S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 1: 153–176
133
Gupta S S, Bosco F G, Batra R C. Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration. Computational Materials Science, 2010, 47(4): 1049–1059 https://doi.org/10.1016/j.commatsci.2009.12.007