Interaction and multiscale mechanics

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Localized particle boundary condition enforcements for the state-based peridynamics

  • Wu, C.T. (Livermore Software Technology Corporation) ;
  • Ren, Bo (Department of Civil and Environmental Engineering, University of California)
  • Received : 2013.10.23
  • Accepted : 2013.11.11
  • Published : 2014.03.25
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Abstract

The state-based peridynamics is considered a nonlocal method in which the equations of motion utilize integral form as opposed to the partial differential equations in the classical continuum mechanics. As a result, the enforcement of boundary conditions in solid mechanics analyses cannot follow the standard way as in a classical continuum theory. In this paper, a new approach for the boundary condition enforcement in the state-based peridynamic formulation is presented. The new method is first formulated based on a convex kernel approximation to restore the Kronecker-delta property on the boundary in 1-D case. The convex kernel approximation is further localized near the boundary to meet the condition that recovers the correct boundary particle forces. The new formulation is extended to the two-dimensional problem and is shown to reserve the conservation of linear momentum and angular momentum. Three numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed approach.

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References

  1. Askari, E., Bobaru, F., Lehoucq, R.B., Parks, M.L., Silling, S.A. and Weckner, O. (2008), "Peridynamics for multi-scale materials modeling", J. Phys: Conference Series 125: 012078. Doi:10.1088/1742-6596/125/012078.
  2. Bazant, Z. and Jirasek, M. (2002), "Nonlocal integration formulations of plasticity and damage: survey and progress", J. Engrg. Mech., 128, 1119-1149. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119)
  3. Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Engrg., 37, 229-256. https://doi.org/10.1002/nme.1620370205
  4. Belytschko, T., Liu, W.K., Moran, B. and Elkhodary, K. (2014), Nonlinear finite elements for continua and structures, 2nd edition, Wiley.
  5. Bessa, M.A., Foster, J.T., Belytschko, T. and Liu, W.K. (2014), "A meshfree unification: reproducing kernel peridynamics", Comp. Mech., DOI: 10.1007/s00466-013-0969-x.
  6. Chen, J.S., Wu, C.T. and Belytschko, T. (2000), "Regularization of Material Instabilities by Meshfree Approximations with Intrinsic Length Scales", Int. J. Numer. Methods Engrg., 47, 1303-1322. https://doi.org/10.1002/(SICI)1097-0207(20000310)47:7<1303::AID-NME826>3.0.CO;2-5
  7. Chen, J.S., Wu, C.T., Yoon, S. and You, Y. (2000), "A stabilized conforming nodal integration for Galerkin mesh-free methods", Int. J. Numer. Meth. Engrg., 50, 435-466.
  8. Chen, X. and Gunzburger, M. (2011), "Continuous and discontinuous finite element methods for a peridynamics model of mechanics", Comput. Meth. Appl. Mech. Eng., 200, 1237-1250. https://doi.org/10.1016/j.cma.2010.10.014
  9. Curtin, W.A. and Miller, R.E. (2003), "Atomistic/continuum coupling in computational materials science", Model. Simul. Mater. Sci. Eng., 11, 33-68. https://doi.org/10.1088/0965-0393/11/3/201
  10. de Pablo. J.J. and Curtin, W.A. (2007), Multi-scale modeling in advanced materials research: Challenges, novel methods, and emerging applications, MRS Bulletin vol. 32, Material Research Society.
  11. Kilic, B. and Madenci, E. (2010), "An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory", Theor. Appl. Fract. Mech., 53, 194-204. https://doi.org/10.1016/j.tafmec.2010.08.001
  12. 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
  13. McVeigh, C. and Liu, W.K. (2008), "Multi-resolution modeling of ductile reinforced brittle composites", J. Mech. Phys. Solids , 57 (2), 244-267.
  14. Oterkus, E., Madenci, E., Weckner, O., Silling, S., Bogert, P. and Tessler, A. (2012), "Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot", Compos. Struct., 94, 839-850. https://doi.org/10.1016/j.compstruct.2011.07.019
  15. Parks, M.L., Lehoucq, R.B., Plimption, S. and Silling, S. (2008), "Implementing peridynamics withn a molecular dynamics code", Comp. Phys. Comm. 179, 777-783. https://doi.org/10.1016/j.cpc.2008.06.011
  16. Park, C.K., Wu, C.T. and Kan, C.D. (2011), "On the analysis of dispersion property and stable time step in meshfree method using the generalized meshfree approximation", Finite Elements in Analysis and Design, 47, 683-697. https://doi.org/10.1016/j.finel.2011.02.001
  17. Ren, B. and Li, S.F. (2013), "A three-dimensional atomistic-based process zone model simulation of fragmentation in polycrystalline solids", Int. J. Numer. Meth. Engrg., 93, 989-1014. https://doi.org/10.1002/nme.4430
  18. Seleson, P., Parks, M.L., Gunzburger, M. and Lehoucq, R.B. (2010), "Peridynamics as an upscaling of molecular dynamics", Multiscale Model. Siml., 8, 204-227.
  19. Seleson, P. and Gunzburger, M. (2010), "Bridging methods for atomistic-to-continuum coupling and their implementation", Comm. Comp. Phys., 7, 831-876.
  20. Silling, S.A. (2000), "Reformulation of elasticity theory for discontinuities and long-range forces", J. Mech. Phys. Solids, 48, 175-209. https://doi.org/10.1016/S0022-5096(99)00029-0
  21. Silling, S.A. and Askari, E. (2005), "A meshfree method based on the peridynamic model of solid mechanics", Comp. Struct., 83, 1526-1535. https://doi.org/10.1016/j.compstruc.2004.11.026
  22. Silling, S.A., Epton, M., Weckner, O., Xu, J. and Askari, E. (2007), "Peridynamic states and constitutive modeling", J. Elasticity, 88, 151-184. https://doi.org/10.1007/s10659-007-9125-1
  23. Tupek, M.R., Rimoli, J.J. and Radovitzky, R. (2013), "An approach for incorporating classical continuum damage models in state-based peridynamics", Comput. Meth. Appl. Mech. Eng., 263, 20-26. https://doi.org/10.1016/j.cma.2013.04.012
  24. Watanabe, I. and Terada, K.A. (2010), "A method of predicting macroscopic yield strength of polycrystalline metals subjected to plastic forming by micro-macro de-coupling scheme", Int. J. Mech. Sci., 52, 343-355. https://doi.org/10.1016/j.ijmecsci.2009.10.006
  25. Wu, C.T. and Koish, M. (2009), "A meshfree procedure for the microscopic analysis of particle-reinforced rubber compounds", Interact. Multiscale Mech., 2(2), 147-169.
  26. Wu, C.T., Park, C.K. and Chen, J.S. (2011), "A generalized approximation for the meshfree analysis of solids", Int. J. Numer. Meth. Engrg., 85, 693-722. https://doi.org/10.1002/nme.2991
  27. Wu, C.T. and Hu, W. (2011), "Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids", Comp. Meth. Appl. Mech. Eng., 200, 2991-3010. https://doi.org/10.1016/j.cma.2011.06.013
  28. Wu, C.T. and Koishi, M. (2012), "Three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites", Int. J. Numer. Meth. Engrg., 91, 1137-1157. https://doi.org/10.1002/nme.4306
  29. Wu, C.T., Hu, W. and Chen, J.S. (2012), "Meshfree-enriched finite element methods for the compressible and near-incompressible elasticity", Int. J. Numer. Meth. Engrg., 90, 882-914. https://doi.org/10.1002/nme.3349
  30. Wu, C.T., Guo, Y. and Askari, E. (2013), "Numerical modeling of composite solids using an immersed meshfree Galerkin method", Composit. B, 45, 1397-1413. https://doi.org/10.1016/j.compositesb.2012.09.061
  31. Wu, C.T. and Hu, W. (2013), "Multi-scale finite element analysis of acoustic waves using global residual-free meshfree enrichments, Interact. Multiscale Mech., 6(2), 83-105. https://doi.org/10.12989/imm.2013.6.2.083
  32. Zhang, T. and Gunzburger, M. (2010), "Quadrature-rule type approximations to the quasi-continuum method for long-range interatomic interactions", Comput. Meth. Appl. Mech. Eng., 199, 648-659. https://doi.org/10.1016/j.cma.2009.10.015
  33. Zhou, K. and Du, Q. (2011), "Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions", SIAM J. Numer. Anal., 48, 1759-1780.

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