A new boundary element method (BEM) is developed for three-dimensional analysis of fiber-reinforced composites based on a rigid-inclusion model. Elasticity equations are solved in an elastic domain containing inclusions which can be assumed much stiffer than the host elastic medium. Therefore the inclusions can be treated as rigid ones with only six rigid-body displacements. It is shown that the boundary integral equation (BIE) in this case can be simplified and only the integral with the weakly-singular displacement kernel is present. The BEM accelerated with the fast multipole method is used to solve the established BIE. The developed BEM code is validated with the analytical solution for a rigid sphere in an infinite elastic domain and excellent agreement is achieved. Numerical examples of fiber-reinforced composites, with the number of fibers considered reaching above 5800 and total degrees of freedom above 10 millions, are solved successfully by the developed BEM. Effective Young’s moduli of fiber-reinforced composites are evaluated for uniformly and “randomly” distributed fibers with two different aspect ratios and volume fractions. The developed fast multipole BEM is demonstrated to be very promising for large-scale analysis of fiber-reinforced composites, when the fibers can be assumed rigid relative to the matrix materials.
Issue Section:
Technical Papers
1.
Mackerle
, J.
, 1994
, “Finite Element and Boundary Element Library for Composites—A Bibliography (1991–1993)
,” Finite Elem. Anal. Design
, 17
, pp. 155
–165
.2.
Achenbach
, J. D.
, and Zhu
, H.
, 1989
, “Effect of Interfacial Zone on Mechanical Behavior and Failure of Fiber-Reinforced Composites
,” J. Mech. Phys. Solids
, 37
, pp. 381
–393
.3.
Zhu
, H.
, and Achenbach
, J. D.
, 1991
, “Effect of Fiber-Matrix Interphase Defects on Microlevel Stress States at Neighboring Fibers
,” J. Compos. Mater.
, 25
, pp. 224
–238
.4.
Gulrajani
, S. N.
, and Mukherjee
, S.
, 1993
, “Sensitivities and Optimal Design of Hexagonal Array Fiber Composites With Respect to Interphase Properties
,” Int. J. Solids Struct.
, 30
, pp. 2009
–2026
.5.
Pan
, L.
, Adams
, D. O.
, and Rizzo
, F. J.
, 1998
, “Boundary Element Analysis for Composite Materials and a Library of Green’s Functions
,” Comput. Struct.
, 66
, pp. 685
–693
.6.
Liu
, Y. J.
, Xu
, N.
, and Luo
, J. F.
, 2000
, “Modeling of Interphases in Fiber-Reinforced Composites Under Transverse Loading Using the Boundary Element Method
,” J. Appl. Mech.
, 67
, pp. 41
–49
.7.
Liu
, Y. J.
, and Xu
, N.
, 2000
, “Modeling of Interface Cracks in Fiber-Reinforced Composites With the Presence of Interphases Using the Boundary Element Method
,” Mech. Mater.
, 32
, pp. 769
–783
.8.
Chen
, X. L.
, and Liu
, Y. J.
, 2001
, “Multiple-Cell Modeling of Fiber-Reinforced Composites With the Presence of Interphases Using the Boundary Element Method
,” Comput. Mater. Sci.
, 21
, pp. 86
–94
.9.
Dundurs
, J.
, and Markenscoff
, X.
, 1989
, “A Green’s Function Formulation of Anticracks and Their Interaction With Load-Induced Singularities
,” J. Appl. Mech.
, 56
, pp. 550
–555
.10.
Hu
, K. X.
, and Chandra
, A.
, 1993
, “Interactions Among General Systems of Cracks and Anticracks—An Integral-Equation Approach
,” J. Appl. Mech.
, 60
, pp. 920
–928
.11.
Hu
, K. X.
, and Huang
, Y.
, 1993
, “A Microcracked Solid Reinforced by Rigid-Line Fibers
,” Compos. Sci. Technol.
, 49
, pp. 145
–151
.12.
Hu
, K. X.
, Chandra
, A.
, and Huang
, Y.
, 1994
, “On Crack, Rigid-Line Fiber, and Interface Interactions
,” Mech. Mater.
, 19
, pp. 15
–28
.13.
Chandra
, A.
, Huang
, Y.
, Wei
, X.
, and Hu
, K. X.
, 1995
, “A Hybrid Micro-Macro BEM Formulation for Micro-Crack Clusters in Elastic Components
,” Int. J. Numer. Methods Eng.
, 38
, pp. 1215
–1236
.14.
Huang
, Y.
, Hu
, K. X.
, and Chandra
, A.
, 1995
, “Stiffness Evaluation for Solids Containing Dilute Distributions of Inclusions and Microcracks
,” J. Appl. Mech.
, 62
, pp. 71
–77
.15.
Leite
, L. G. S.
, Coda
, H. B.
, and Venturini
, W. S.
, 2003
, “Two-Dimensional Solids Reinforced by Thin Bars Using the Boundary Element Method
,” Eng. Anal. Boundary Elem.
, 27
, pp. 193
–201
.16.
Dong
, C. Y.
, Lo
, S. H.
, and Cheung
, Y. K.
, 2003
, “Interaction Between Cracks and Rigid-Line Inclusions by an Integral Equation Approach
,” Comput. Mech.
, 31
, pp. 238
–252
.17.
Nishimura, N., and Liu, Y. J., 2004, “Thermal Analysis of Carbon-Nanotube Composites Using a Rigid-Line Inclusion Model by the Boundary Integral Equation Method,” Comput. Mech., (in press).
18.
Ingber
, M. S.
, and Papathanasiou
, T. D.
, 1997
, “A Parallel-Supercomputing Investigation of the Stiffness of Aligned, Short-Fiber-Reinforced Composites Using the Boundary Element Method
,” Int. J. Numer. Methods Eng.
, 40
, pp. 3477
–3491
.19.
Primo
, A. R. M.
, Wrobel
, L. C.
, and Power
, H.
, 2000
, “Boundary Integral Formulation for Slow Viscous Flow in a Deforming Region Containing a Solid Inclusion
,” Eng. Anal. Boundary Elem.
, 24
, pp. 53
–63
.20.
Kit
, H. S.
, Mykhas’skiv
, V. V.
, and Khaj
, O. M.
, 2002
, “Analysis of the Steady Oscillations of a Plane Absolutely Rigid Inclusion in a Three-Dimensional Elastic Body by the Boundary Element Method
,” J. Appl. Math. Mech.
, 66
, pp. 817
–824
.21.
Nishimura
, N.
, 2002
, “Fast Multipole Accelerated Boundary Integral Equation Methods
,” Appl. Mech. Rev.
, 55
, pp. 299
–324
.22.
Greengard
, L.
, and Helsing
, J.
, 1998
, “On the Numerical Evaluation of Elastostatic Fields in Locally Isotropic Two-Dimensional Composites
,” J. Mech. Phys. Solids
, 46
, pp. 1441
–1462
.23.
Greengard
, L.
, Kropinski
, M. C.
, and Mayo
, A.
, 1996
, “Integral Equation Methods for Stokes Flow and Isotropic Elasticity in the Plane
,” J. Comput. Phys.
, 125
, pp. 403
–414
.24.
Helsing
, J.
, 1995
, “An Integral Equation Method for Elastostatics of Periodic Composites
,” J. Mech. Phys. Solids
, 43
, pp. 815
–828
.25.
Greengard
, L.
, and Rokhlin
, V.
, 1987
, “A Fast Algorithm for Particle Simulations
,” J. Comput. Phys.
, 73
, pp. 325
–348
.26.
Fu
, Y.
, Klimkowski
, K. J.
, Rodin
, G. J.
, Berger
, E.
, Browne
, J. C.
, Singer
, J. K.
, Geijn
, R. A. V. D.
, and Vemaganti
, K. S.
, 1998
, “A Fast Solution Method for Three-Dimensional Many-Particle Problems of Linear Elasticity
,” Int. J. Numer. Methods Eng.
, 42
, pp. 1215
–1229
.27.
Peirce
, A. P.
, and Napier
, J. A. L.
, 1995
, “A Spectral Multipole Method for Efficient Solution of Large-Scale Boundary Element Models in Elastostatics
,” Int. J. Numer. Methods Eng.
, 38
, pp. 4009
–4034
.28.
Popov
, V.
, and Power
, H.
, 2001
, “An O(N) Taylor Series Multipole Boundary Element Method for Three-Dimensional Elasticity Problems
,” Eng. Anal. Boundary Elem.
, 25
, pp. 7
–18
.29.
Nishimura
, N.
, Yoshida
, K.
, and Kobayashi
, S.
, 1999
, “A Fast Multipole Boundary Integral Equation Method for Crack Problems in 3D
,” Eng. Anal. Boundary Elem.
, 23
, pp. 97
–105
.30.
Yoshida
, K.
, Nishimura
, N.
, and Kobayashi
, S.
, 2001
, “Application of Fast Multipole Galerkin Boundary Integral Equation Method to Crack Problems in 3D
,” Int. J. Numer. Methods Eng.
, 50
, pp. 525
–547
.31.
Lai
, Y.-S.
, and Rodin
, G. J.
, 2003
, “Fast Boundary Element Method for Three-Dimensional Solids Containing Many Cracks
,” Eng. Anal. Boundary Elem.
, 27
, pp. 845
–852
.32.
Rizzo
, F. J.
, 1967
, “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics
,” Q. Appl. Math.
, 25
, pp. 83
–95
.33.
Rizzo
, F. J.
, Shippy
, D. J.
, and Rezayat
, M.
, 1985
, “A Boundary Integral Equation Method for Radiation and Scattering of Elastic Waves in Three Dimensions
,” Int. J. Numer. Methods Eng.
, 21
, pp. 115
–129
.34.
Mukherjee, S., 1982, Boundary Element Methods in Creep and Fracture, Applied Science Publishers, NY.
35.
Banerjee, P. K., 1994, The Boundary Element Methods in Engineering, 2nd ed., McGraw–Hill, NY.
36.
Brebbia, C. A., and Dominguez, J., 1989, Boundary Elements—An Introductory Course, McGraw–Hill, NY.
37.
Kane, J. H., 1994, Boundary Element Analysis in Engineering Continuum Mechanics, Prentice–Hall, Englewood Cliffs, NJ.
38.
Takahashi, T., Kobayashi, S., and Nishimura, N., 1999, “Fast Multipole BEM Simulation of Overcoring in an Improved Conical-End Borehole Strain Measurement Method,” in Mechanics and Engineering—In Honor of Professor Qinghua Du’s 80th Anniversary, edited by Yao, Z. H., Tsinghua University Press, Beijing, pp. 120–127.
39.
Yoshida, K., 2001, “Applications of Fast Multipole Method to Boundary Integral Equation Method,” Ph.D. dissertation, Department of Global Environment Engineering, Kyoto University.
40.
Timoshenko, S. P., and Goodier, J. N., 1987, Theory of Elasticity, 3rd ed., McGraw–Hill, NY.
41.
Mura, T., 1987, Micromechanics of Defects in Solids, 2nd revised ed., Kluwer Academic, Dordrecht.
42.
Thostenson
, E. T.
, Ren
, Z. F.
, and Chou
, T.-W.
, 2001
, “Advances in the Science and Technology of Carbon Nanotubes and Their Composites: A Review
,” Compos. Sci. Technol.
, 61
, pp. 1899
–1912
.43.
Ruoff
, R. S.
, and Lorents
, D. C.
, 1995
, “Mechanical and Thermal Properties of Carbon Nanotubes
,” Carbon
, 33
, pp. 925
–930
.44.
Lu
, J. P.
, 1997
, “Elastic Properties of Single and Multilayered Nanotubes
,” J. Phys. Chem. Solids
, 58
, pp. 1649
–1652
.45.
Qian
, D.
, Dickey
, E. C.
, Andrews
, R.
, and Rantell
, T.
, 2000
, “Load Transfer and Deformation Mechanisms in Carbon Nanotube-Polystyrene Composites
,” Appl. Phys. Lett.
, 76
, pp. 2868
–2870
.46.
Fisher
, F. T.
, Bradshaw
, R. D.
, and Brinson
, L. C.
, 2002
, “Effects of Nanotube Waviness on the Modulus of Nanotube-Reinforced Polymers
,” Appl. Phys. Lett.
, 80
, pp. 4647
–4649
.47.
Fisher
, F. T.
, Bradshaw
, R. D.
, and Brinson
, L. C.
, 2003
, “Fiber Waviness in Nanotube-Reinforced Polymer Composites—I: Modulus Predictions Using Effective Nanotube Properties
,” Compos. Sci. Technol.
, 63
, pp. 1689
–1703
.48.
Liu
, Y. J.
, and Chen
, X. L.
, 2003
, “Evaluations of the Effective Materials Properties of Carbon Nanotube-Based Composites Using a Nanoscale Representative Volume Element
,” Mech. Mater.
, 35
, pp. 69
–81
.49.
Chen
, X. L.
, and Liu
, Y. J.
, 2004
, “Square Representative Volume Elements for Evaluating the Effective Material Properties of Carbon Nanotube-Based Composites
,” Comput. Mater. Sci.
, 29
, pp. 1
–11
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