Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T11:14:59.814Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical analysis for the peridynamic nonlocal continuum theory*

Published online by Cambridge University Press:  02 August 2010

Qiang Du  and
Kun Zhou
Qiang Du
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. [email protected]; [email protected]
Kun Zhou
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. [email protected]; [email protected]
Get access

Abstract

We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided.

Keywords

Peridynamic modelnonlocal continuum theorywell-posednessNavier equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 2 , March 2011 , pp. 217 - 234
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

B. Alali and R. Lipton, Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation. IMA preprint, 2241 (2009).
Askari, E., Bobaru, F., Lehoucq, R.B., Parks, M.L., Silling, S.A. and Weckner, O., Peridynamics for multiscale materials modeling. J. Phys. Conf. Ser. 125 (2008) 012078. CrossRef
Aubert, G. and Kornprobst, P., 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
Belytschko, T. and Xiao, S.P., A bridging domain method for coupling continua with molecular dynamics. Int. J. Mult. Comp. Eng. 1 (2003) 115–126. CrossRef
Curtin, W. and Miller, R., Atomistic/continuum coupling methods in multi-scale materials modeling. Mod. Simul. Mater. Sci. Engineering 11 (2003) R33–R68. CrossRef
Dayal, K. and Bhattacharya, K., Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54 (2006) 1811–1842. CrossRef
N. Dunford and J. Schwartz, Linear Operators, Part I: General Theory. Interscience, New York (1958).
Emmrich, E. and Weckner, O., 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).
Emmrich, E. and Weckner, O., 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. CrossRef
Fish, J., Nuggehally, M.A., Shephard, M.S., Picu, C.R., Badia, S., Parks, M.L. and Gunzburger, M., 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).
Lehoucq, R.B. and Silling, S.A., Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56 (2008) 1566–1577. CrossRef
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
Miller, R.E. and Tadmor, E.B., The quasicontinuum method: Overview, applications, and current directions. J. Comp.-Aided Mater. Des. 9 (2002) 203–239. CrossRef
Silling, S.A., Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209. CrossRef
S.A. Silling, Linearized theory of peridynamic states. Sandia National Laboratories, SAND (2009) 2009–2458.
Silling, S.A. and Lehoucq, R.B., Convergence of peridynamics to classical elasticity theory. J. Elasticity 93 (2008) 13–37. CrossRef
S.A. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid. Preprint (2009).
Weckner, O. and Abeyaratne, R., The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53 (2005) 705–728. CrossRef
K. Zhou and Q. Du, Mathematical and Numerical Analysis of Peridynamic Models with Nonlocal Boundary Conditions. SIAM J. Numer. Anal. (submitted).