Free access
Issue
ESAIM: M2AN
Volume 45, Number 2, March-April 2011
Page(s) 217 - 234
DOI http://dx.doi.org/10.1051/m2an/2010040
Published online 02 August 2010
  1. B. Alali and R. Lipton, Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation. IMA preprint, 2241 (2009).
  2. E. Askari, F. Bobaru, R.B. Lehoucq, M.L. Parks, S.A. Silling and O. Weckner, Peridynamics for multiscale materials modeling. J. Phys. Conf. Ser. 125 (2008) 012078. [CrossRef]
  3. G. Aubert and P. Kornprobst, Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems? SIAM J. Numer. Anal. 47 (2009) 844–860. [CrossRef] [MathSciNet]
  4. T. Belytschko and S.P. Xiao, A bridging domain method for coupling continua with molecular dynamics. Int. J. Mult. Comp. Eng. 1 (2003) 115–126. [CrossRef]
  5. W. Curtin and R. Miller, Atomistic/continuum coupling methods in multi-scale materials modeling. Mod. Simul. Mater. Sci. Engineering 11 (2003) R33–R68. [CrossRef]
  6. K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54 (2006) 1811–1842. [CrossRef] [MathSciNet]
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  8. E. Emmrich and O. Weckner, Analysis and numerical approximation of an integrodifferential equation modelling non-local effects in linear elasticity. Math. Mech. Solids 12 (2005) 363–384. [CrossRef]
  9. E. Emmrich and O. Weckner, The peridynamic equation of motion in nonlocal elasticity theory, in III European Conference on Computational Mechanics – Solids, Structures and Coupled Problems in Engineering, C.A. Mota Soares, J.A.C. Martins, H.C. Rodrigues, J.A.C. Ambrosio, C.A.B. Pina, C.M. Mota Soares, E.B.R. Pereira and J. Folgado Eds., Lisbon, Springer (2006).
  10. E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5 (2007) 851–864. [MathSciNet]
  11. J. Fish, M.A. Nuggehally, M.S. Shephard, C.R. Picu, S. Badia, M.L. Parks and M. Gunzburger, Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comp. Meth. Appl. Mech. Eng. 196 (2007) 4548–4560. [CrossRef]
  12. M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems. Preprint (2009).
  13. L. Hörmander, Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer, Berlin (1985).
  14. O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Springer-Verlag, New York (1985).
  15. R.B. Lehoucq and S.A. Silling, Statistical coarse-graining of molecular dynamics into peridynamics. Technical Report, SAND2007-6410, Sandia National Laboratories, Albuquerque and Livermore (2007).
  16. R.B. Lehoucq and S.A. Silling, Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56 (2008) 1566–1577. [CrossRef] [MathSciNet]
  17. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
  18. R.E. Miller and E.B. Tadmor, The quasicontinuum method: Overview, applications, and current directions. J. Comp.-Aided Mater. Des. 9 (2002) 203–239. [CrossRef]
  19. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209. [CrossRef] [MathSciNet]
  20. S.A. Silling, Linearized theory of peridynamic states. Sandia National Laboratories, SAND (2009) 2009–2458.
  21. S.A. Silling and R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elasticity 93 (2008) 13–37. [CrossRef] [MathSciNet]
  22. S.A. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid. Preprint (2009).
  23. O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53 (2005) 705–728. [CrossRef] [MathSciNet]
  24. K. Zhou and Q. Du, Mathematical and Numerical Analysis of Peridynamic Models with Nonlocal Boundary Conditions. SIAM J. Numer. Anal. (submitted).

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