Fracture model for the prediction of the electrical percolation threshold in CNTs/Polymer composites

Yang SHEN, Pengfei HE, Xiaoying ZHUANG

PDF(1917 KB)
PDF(1917 KB)
Front. Struct. Civ. Eng. ›› 2018, Vol. 12 ›› Issue (1) : 125-136. DOI: 10.1007/s11709-017-0396-8
RESEARCH ARTICLE
RESEARCH ARTICLE

Fracture model for the prediction of the electrical percolation threshold in CNTs/Polymer composites

Author information +
History +

Abstract

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.

Keywords

electrical percolation / CNTs/Polymer composites / fracture model / electric conductivity / tunnelling effects

Cite this article

Download citation ▾
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 https://doi.org/10.1007/s11709-017-0396-8

References

[1]
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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[8]
Ramasubramaniam R, Chen J, Liu H. Homogeneous carbon nanotube/polymer composites for electrical applications. Applied Physics Letters, 2003, 83(14): 2928–2930
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[34]
Tadmor E B, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philosophical Magazine A, 1996, 73(6): 1529–1563
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[45]
Belytschko T, Moës N, Usui S, Parimi C. Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering, 2001, 50(4): 993–1013
CrossRef Google scholar
[46]
Sukumar N, Moës N. B Moran, and T Belytschko. Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering, 2000, 48(11): 1549–1570
CrossRef Google scholar
[48]
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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[87]
Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[118]
Belytschko T, Yun Y L, Gu L. Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[127]
Kuna M. Finite element analyses of crack problems in piezoelectric structures. Computational Materials Science, 1998, 13(1): 67–80
CrossRef Google scholar
[128]
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
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar

Acknowledgment

The authors gratefully acknowledge the support from the Sofja Kovalevskaja Programme from AvH, the Ministry of Science and Technology of China (SLDRCE14B-31), Shanghai Qimingxing Program (16QA1404000) and Fundamental Research Funds for the Central Universities.

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg
AI Summary AI Mindmap
PDF(1917 KB)

Accesses

Citations

Detail

Sections
Recommended

/