Volume 45, Number 2, March-April 2011
|Page(s)||217 - 234|
|Published online||02 August 2010|
- B. Alali and R. Lipton, Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation. IMA preprint, 2241 (2009).
- 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]
- 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]
- 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]
- W. Curtin and R. Miller, Atomistic/continuum coupling methods in multi-scale materials modeling. Mod. Simul. Mater. Sci. Engineering 11 (2003) R33–R68. [CrossRef]
- 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]
- N. Dunford and J. Schwartz, Linear Operators, Part I: General Theory. Interscience, New York (1958).
- 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]
- 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).
- 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]
- 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]
- M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems. Preprint (2009).
- L. Hörmander, Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer, Berlin (1985).
- O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Springer-Verlag, New York (1985).
- 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).
- R.B. Lehoucq and S.A. Silling, Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56 (2008) 1566–1577. [CrossRef] [MathSciNet]
- J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
- 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]
- S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209. [CrossRef] [MathSciNet]
- S.A. Silling, Linearized theory of peridynamic states. Sandia National Laboratories, SAND (2009) 2009–2458.
- S.A. Silling and R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elasticity 93 (2008) 13–37. [CrossRef] [MathSciNet]
- S.A. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid. Preprint (2009).
- 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]
- 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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