# Mathematical analysis for the peridynamic nonlocal continuum theory

- Volume: 45, Issue: 2, page 217-234
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topDu, Qiang, and Zhou, Kun. "Mathematical analysis for the peridynamic nonlocal continuum theory." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.2 (2011): 217-234. <http://eudml.org/doc/273242>.

@article{Du2011,

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.},

author = {Du, Qiang, Zhou, Kun},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {peridynamic model; nonlocal continuum theory; well-posedness; Navier equation; spring network system; Cauchy problem; classical elastic models},

language = {eng},

number = {2},

pages = {217-234},

publisher = {EDP-Sciences},

title = {Mathematical analysis for the peridynamic nonlocal continuum theory},

url = {http://eudml.org/doc/273242},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Du, Qiang

AU - Zhou, Kun

TI - Mathematical analysis for the peridynamic nonlocal continuum theory

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2011

PB - EDP-Sciences

VL - 45

IS - 2

SP - 217

EP - 234

AB - 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.

LA - eng

KW - peridynamic model; nonlocal continuum theory; well-posedness; Navier equation; spring network system; Cauchy problem; classical elastic models

UR - http://eudml.org/doc/273242

ER -

## References

top- [1] B. Alali and R. Lipton, Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation. IMA preprint, 2241 (2009). Zbl1320.74029
- [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.
- [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. Zbl1189.35067MR2485435
- [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. Zbl1079.74509MR2069430
- [5] W. Curtin and R. Miller, Atomistic/continuum coupling methods in multi-scale materials modeling. Mod. Simul. Mater. Sci. Engineering 11 (2003) R33–R68.
- [6] K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids54 (2006) 1811–1842. Zbl1120.74690MR2244039
- [7] N. Dunford and J. Schwartz, Linear Operators, Part I: General Theory. Interscience, New York (1958). Zbl0084.10402MR1009162
- [8] E. Emmrich and O. Weckner, Analysis and numerical approximation of an integrodifferential equation modelling non-local effects in linear elasticity. Math. Mech. Solids12 (2005) 363–384. Zbl1175.74013MR2349159
- [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). Zbl1133.35098
- [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. Zbl1133.35098MR2375050
- [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. Zbl1173.74303MR2354453
- [12] M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems. Preprint (2009). Zbl1210.35057MR2728700
- [13] L. Hörmander, Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer, Berlin (1985). Zbl1115.35005
- [14] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Springer-Verlag, New York (1985). Zbl0588.35003MR793735
- [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. Solids56 (2008) 1566–1577. Zbl1171.74319MR2404022
- [17] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). Zbl0189.40603
- [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.
- [19] S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids48 (2000) 175–209. Zbl0970.74030MR1727557
- [20] S.A. Silling, Linearized theory of peridynamic states. Sandia National Laboratories, SAND (2009) 2009–2458. Zbl1188.74008MR2592410
- [21] S.A. Silling and R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elasticity93 (2008) 13–37. Zbl1159.74316MR2430855
- [22] S.A. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid. Preprint (2009). Zbl05774098
- [23] O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids53 (2005) 705–728. Zbl1122.74431MR2116266
- [24] K. Zhou and Q. Du, Mathematical and Numerical Analysis of Peridynamic Models with Nonlocal Boundary Conditions. SIAM J. Numer. Anal. (submitted). Zbl1220.82074MR2733097

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.