Mathematical analysis for the peridynamic nonlocal continuum theory
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 217-234.

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.

DOI : 10.1051/m2an/2010040
Classification : 45A05, 46N20, 74B99
Mots clés : peridynamic model, nonlocal continuum theory, well-posedness, Navier equation
@article{M2AN_2011__45_2_217_0,
     author = {Du, Qiang and Zhou, Kun},
     title = {Mathematical analysis for the peridynamic nonlocal continuum theory},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {217--234},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {2},
     year = {2011},
     doi = {10.1051/m2an/2010040},
     mrnumber = {2804637},
     zbl = {1269.45005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010040/}
}
TY  - JOUR
AU  - Du, Qiang
AU  - Zhou, Kun
TI  - Mathematical analysis for the peridynamic nonlocal continuum theory
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2011
SP  - 217
EP  - 234
VL  - 45
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010040/
DO  - 10.1051/m2an/2010040
LA  - en
ID  - M2AN_2011__45_2_217_0
ER  - 
%0 Journal Article
%A Du, Qiang
%A Zhou, Kun
%T Mathematical analysis for the peridynamic nonlocal continuum theory
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 217-234
%V 45
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010040/
%R 10.1051/m2an/2010040
%G en
%F M2AN_2011__45_2_217_0
Du, Qiang; Zhou, Kun. Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 217-234. doi : 10.1051/m2an/2010040. http://www.numdam.org/articles/10.1051/m2an/2010040/

[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.

[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. | MR | Zbl

[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. | MR | Zbl

[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. Solids 54 (2006) 1811-1842. | MR | Zbl

[7] N. Dunford and J. Schwartz, Linear Operators, Part I: General Theory. Interscience, New York (1958). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems. Preprint (2009). | MR | Zbl

[13] L. Hörmander, Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Springer, Berlin (1985). | Zbl

[14] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Springer-Verlag, New York (1985). | MR | Zbl

[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. | MR | Zbl

[17] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | Zbl

[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. Solids 48 (2000) 175-209. | MR | Zbl

[20] S.A. Silling, Linearized theory of peridynamic states. Sandia National Laboratories, SAND (2009) 2009-2458. | MR | Zbl

[21] S.A. Silling and R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elasticity 93 (2008) 13-37. | MR | Zbl

[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. | MR | Zbl

[24] K. Zhou and Q. Du, Mathematical and Numerical Analysis of Peridynamic Models with Nonlocal Boundary Conditions. SIAM J. Numer. Anal. (submitted). | MR | Zbl

Cité par Sources :