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A Meshfree procedure for the microscopic analysis of particle-reinforced rubber compounds

  • Wu, C.T. (Livermore Software Technology Corporation) ;
  • Koishi, M. (CAE Laboratory, The Yokohama Rubber Co., Ltd.)
  • Received : 2009.03.19
  • Accepted : 2009.05.04
  • Published : 2009.06.25

Abstract

This paper presents a meshfree procedure using a convex generalized meshfree (GMF) approximation for the large deformation analysis of particle-reinforced rubber compounds on microscopic level. The convex GMF approximation possesses the weak-Kronecker-delta property that guarantees the continuity of displacement across the material interface in the rubber compounds. The convex approximation also ensures the positive mass in the discrete system and is less sensitive to the meshfree nodal support size and integration order effects. In this study, the convex approximation is generated in the GMF method by choosing the positive and monotonic increasing basis function. In order to impose the periodic boundary condition in the unit cell method for the microscopic analysis, a singular kernel is introduced on the periodic boundary nodes in the construction of GMF approximation. The periodic boundary condition is solved by the transformation method in both explicit and implicit analyses. To simulate the interface de-bonding phenomena in the rubber compound, the cohesive interface element method is employed in corporation with meshfree method in this study. Several numerical examples are presented to demonstrate the effectiveness of the proposed numerical procedure in the large deformation analysis.

Keywords

References

  1. Arroyo, M. and Ortiz, M. (2006), "Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods", Int. J. Numer. Meth. Eng., 65, 2167-2202. https://doi.org/10.1002/nme.1534
  2. Belikov, V. V. and Semenov, A. Y. (2000), "Non-Sibsonian interpolation on arbitrary system of points in Euclidean space and adaptive isolines generation", Appl. Number. Math., 32(4), 371-387. https://doi.org/10.1016/S0168-9274(99)00058-6
  3. Belytschko, T., Lu, Y. Y., and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Eng., 37(2), 229-256. https://doi.org/10.1002/nme.1620370205
  4. Belytschko, T., Organ, D. and Krongauz, Y. (1995), "A coupled finite element-element-free Galerkin method", Comput. Mech., 17, 186-195. https://doi.org/10.1007/BF00364080
  5. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P. (1996), "Meshless methods: An overview and recent developments", Comput. Meth. Appl. Mech. Eng., 139, 3-47. https://doi.org/10.1016/S0045-7825(96)01078-X
  6. Belytschko, T., Loehnert, S. and Song, J. H. (2008), "Multiscale aggregating discontinuities: A method for circumventing loss of material stability", Int. J. Numer. Meth. Eng., 73, 869-894. https://doi.org/10.1002/nme.2156
  7. Chen, J. S., Pan, C., Wu, C. T. and Liu, W. K. (1996), "Reproducing kernel particle methods for large deformation analysis of non-linear structures", Comput. Meth. Appl. Mech. Eng., 139, 195-227. https://doi.org/10.1016/S0045-7825(96)01083-3
  8. Chen, J. S. and Wang, H. P. (2000), "New boundary condition treatments in meshfree computation of contact problems", Comput. Meth. Appl. Mech. Eng., 187, 441-468. https://doi.org/10.1016/S0045-7825(00)80004-3
  9. Chen, J. S. and Mehraeen, S. (2004), "Variationally consistent multi-scale modeling and homogenization of stressed grain growth", Comput. Meth. Appl. Mech. Eng., 193, 1825-1848. https://doi.org/10.1016/j.cma.2003.12.038
  10. Chen, J. S. and Mehraeen, S. (2005), "Multi-scale modeling of heterogeneous materials with fixed and evolving microstructures", Model. Simul. Mater. Sc., 13, 95-121. https://doi.org/10.1088/0965-0393/13/1/007
  11. Chen, J. S., Hu, W. and Hu, H. Y. (2008), "Reproducing kernel enhanced local radial basis collocation method", Int. J. Numer. Meth. Eng., 75, 600-627. https://doi.org/10.1002/nme.2269
  12. Cook, R. D., Malkus, D. S. and Plesha, M. E. (1998) Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York.
  13. Duchon, J. (1977), "Splines minimizing rotation-invariant semi-norms in Sobolev spaces", in Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and K. Zeller (eds.), Springer Lecture Notes in Math. 571, Springer-Verlag, Berlin, 85-100.
  14. Gu, L. (2003), "Moving kriging interpolation and element-free Galerkin method", Int. J. Numer. Meth. Eng., 56, 1-11. https://doi.org/10.1002/nme.553
  15. Hill, R. (1963), "Elastic properties of reinforced solids: some theoretical principles", J. Mech. Phys Solid, 11, 357-372. https://doi.org/10.1016/0022-5096(63)90036-X
  16. Hu, H. Y., Chen, J. S. and Hu, W. (2007), "Weighted radial basis collocation method for boundary value problems", Int. J. Numer. Meth. Eng., 69, 2736-2757. https://doi.org/10.1002/nme.1877
  17. Jaynes, E. T. (1957), "Information theory and statistical mechanics", Physical Review, 106(4), 620-630. https://doi.org/10.1103/PhysRev.106.620
  18. Kansa, E. J. (1992a), 'Multiqudrics - a scattered data approximation scheme with applications to computational fluid dynamics - I. Surface approximations and partial derivatives", Comput. Math. Appl., 19, 127-145.
  19. Kansa, E. J. (1992b), "Multiqudrics - a scattered data approximation scheme with applications to computational fluid dynamics - II. Solutions to parabolic, hyperbolic and elliptic partial differential equations", Comput. Math. Appl., 19, 147-161.
  20. Lancaster, P. and Salkauskas, K. (1981), "Surfaces generated by moving least squares methods", Math. Comput., 37, 141-158. https://doi.org/10.1090/S0025-5718-1981-0616367-1
  21. Li, S. and Liu, W. K. (2002), "Meshfree and particle methods and their applications", Appl. Mech. Review., 55, 1-34. https://doi.org/10.1115/1.1431547
  22. Liu, G. R. and Gu, Y. T. (2001), "A point interpolation method for two-dimensional solids", Int. J. Numer. Meth. Eng., 50, 937-951. https://doi.org/10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X
  23. Liu, W. K., Jun, S. and Zhang, Y. F. (1995), "Reproducing kernel particle methods", Int. J. Numer. Meth. Fluids., 20, 1081-1106. https://doi.org/10.1002/fld.1650200824
  24. Lucy, L. (1977), "A numerical approach to testing the fission hypothesis", Astron. J., 82, 1013-1024. https://doi.org/10.1086/112164
  25. Nemat-Nasser, S., Hori, M. (1993), Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam.
  26. Ortiz, M. and Pandolfi, A. (1999), "Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis", Int. J. Numer. Meth. Fluids., 44, 1267-1282. https://doi.org/10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0.CO;2-7
  27. Powell, M. J. D. (1992), "The theory of radial basis function approximation in 1990", in Advances in Numerical Analysis, Light FW (edn), Oxford University Press, Oxford, 303-322.
  28. Rabczuk, T., Xiao, S. P. and Sauer, M. (2006), "Coupling of mesh-free methods with finite elements: basic concepts and test results, Commun. Numer. Meth. Eng., 22, 1031-1065. https://doi.org/10.1002/cnm.871
  29. Shannon, C. E. (1948), "A mathematical theory of communication", Bell Syst. Tech. J., 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
  30. Smit, R. J. M, Brekelmans, W. A. M. and Meijer, H. E. H. (1998), "Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling", Comput. Meth. Appl. Mech. Eng., 155, 181-192. https://doi.org/10.1016/S0045-7825(97)00139-4
  31. Sukumar, N. (2004), "Construction of polygonal interpolants: a maximum entropy approach", Int. J. Numer. Meth. Eng., 61, 2159-2181. https://doi.org/10.1002/nme.1193
  32. Terada, K, Hori, M., Kyoya, T. and Kikuchi, N. (2000), "Simulation of the multi-scale convergence in computational homogenization approaches", Int. J. Solids Struct., 37, 2285-2311. https://doi.org/10.1016/S0020-7683(98)00341-2
  33. Terada, K., Asai, M. and Yamagishi, M. (2003), "Finite cover method for linear and nonlinear analysis of heterogeneous solids", Int. J. Numer. Meth. Eng., 58, 1321-1346. https://doi.org/10.1002/nme.820
  34. Van der Sluis, O., Schreurs, P. J. G. and Meijer, H. E. H. (2001), "Homogenization of structured elastoviscoplastic solids at finite strains", Mech. Mater., 33, 499-522. https://doi.org/10.1016/S0167-6636(01)00066-7
  35. Wang, D, Chen, J. S. and Sun, L. (2003), "Homogenization of magnetostrictive particle-filled elastomers using an interface-enriched reproducing kernel particle method", Finite Elem. Anal. Design., 39, 765-782. https://doi.org/10.1016/S0168-874X(03)00058-1
  36. Wu, C. T., Park, C. K. and Chen, J. S. (2009), "A generalized meshfree approximation for the meshfree analysis of solid", Int. J. Numer. Meth. Eng., submitted.
  37. Zavattieri, P. D. and Espinosa, H. D. (2001), "Grain level analysis of ceramic microstructures subjected to normal impact loading", Acta Materialia, 49(20), 4291-4311. https://doi.org/10.1016/S1359-6454(01)00292-0

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