Geodesic
Investigation of different influence functions in peridynamics
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 4, pp. 342-360.

Voir la notice de l'article provenant de la source Math-Net.Ru

Peridynamics is a non–local numerical method for solving fracture problems based on integral equations. It is assumed that particles in a continuum are endowed with volume and interact with each other at a finite distance, as in molecular dynamics. The influence function in peridynamic models is used to limit the force acting on a particle and to adjust the bond strength depending on the distance between the particles. It satisfies certain continuity conditions and describes the behavior of non-local interaction. The article investigates various types of influence function in peridynamic models on the example of three-dimensional problems of elasticity and fracture. In the course of the work done, the bond-based and state-based fracture models used in the Sandia Laboratory are described, 6 types of influence functions for the bond-based model and 2 types of functions for the state-based model are presented, and the corresponding formulas for calculating the stiffness of the bond are obtained. For testing, we used the problem of propagation of a spherically symmetric elastic wave, which has an analytical solution, and a qualitative problem of destruction of a brittle disk under the action of a spherical impactor. Graphs of radial displacement are given, raster images of simulation results are shown.
Keywords: peridynamics, molecular dynamics, influence function, bond stiffness function, interaction horizon, bond
Mots-clés : nonlocal interactions 
@article{SVMO_2023_25_4_a8,
     author = {Yu. N. Deryugin and M. V. Vetchinnikov and D. A. Shishkanov},
     title = {Investigation of different influence functions in peridynamics},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {342--360},
     publisher = {mathdoc},
     volume = {25},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2023_25_4_a8/}
}
TY  - JOUR
AU  - Yu. N. Deryugin
AU  - M. V. Vetchinnikov
AU  - D. A. Shishkanov
TI  - Investigation of different influence functions in peridynamics
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2023
SP  - 342
EP  - 360
VL  - 25
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2023_25_4_a8/
LA  - ru
ID  - SVMO_2023_25_4_a8
ER  - 
%0 Journal Article
%A Yu. N. Deryugin
%A M. V. Vetchinnikov
%A D. A. Shishkanov
%T Investigation of different influence functions in peridynamics
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2023
%P 342-360
%V 25
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2023_25_4_a8/
%G ru
%F SVMO_2023_25_4_a8
Yu. N. Deryugin; M. V. Vetchinnikov; D. A. Shishkanov. Investigation of different influence functions in peridynamics. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 4, pp. 342-360. http://geodesic.mathdoc.fr/item/SVMO_2023_25_4_a8/

[1] F. Bobaru, P.H. Geubelle, J.T. Foster, S. A. Silling, Handbook of peridynamic modeling, Taylor Francis, NY, 2016, 586 pp. | MR

[2] E. Madenci, E. Oterkus, Peridynamic theory and its applications, New York: Springer, 2014, 297 pp. | Zbl

[3] D. A. Shishkanov, M. V. Vetchinnikov, Yu. N. Deryugin, “Peridynamics method for problems solve of solids destruction.”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 24:4 (2022), 452–468 (In Russ.) | DOI

[4] V. N. Sofronov, M. V. Vetchinnikov, M. A. Dyemina, “Use of Hamiltonian dynamics methods in computational continuum mechanics”, Zhurnal VANT, 2020, no. 4, 17 pp. (In Russ.) | MR

[5] M.L. Parks, P. Seleson, S. J. Plimpton, R.B. Lehoucq , S.A. Siling, Peridynamics with LAMMPS: A User Guide v0.2 Beta, New Mexico, 2008, 28 pp.

[6] A. N. Anisimov, S. A. Grushin, B. L. Voronin, S. V. Kopkin, A. M. Yerofeev, D. A. Demin, M. A. Demina, M. V. Zdorova, M. V. Vetchinnikov, N. S. Ericheva, N. O. Kovalenko, I. A. Kryuchkov, A. G. Kechin, V. A. Degtyarev, Certificate of state registration of the computer program No. 2010614974. A complex of molecular dynamic modeling programs (MoDyS), 2010 (In Russ.)

[7] M. L. Parks, Lehoucq, R. B., S. J. Plimpton, S. A. Silling, “Implementing peridynamics within a molecular dynamics code”, Computer Physics Communications, 179 (2008), 777–783 | DOI | Zbl

[8] S. A. Silling, “Reformulation of elasticity theory for discontinuities and long-range forces”, Journal of Mechanics and Physics of Solids, 48:1 (2000), 175–209 | DOI | MR | Zbl

[9] S. A. Silling, E. Askari, “A meshfree method based on the peridynamic model of solid mechanics”, Computers Structures, 93:17 (2005), 1526–1535 | DOI

[10] S. A. Silling, M. Zimmermann, R. Abeyaratne, “Deformation of a peridynamic bar”, Journal of Elasticity, 73 (2003), 173–190 | DOI | MR | Zbl

[11] Z. Chen, J. W. Ju, G. Su, X. Huang, S. Li., L. Zhai, “Influence of micro-modulus functions on peridynamics simulation of crack propagation and branching in brittle materials”, Engineering Fracture Mechanics, 216 (2019) | DOI

[12] S. A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, “Peridynamic states and constitutive modeling”, Journal of Elasticity, 88 (2007), 151–184 | DOI | MR | Zbl

[13] A. J. Mitchell, S. A. Silling, D. J. Littlewood, “A position-aware linear solid constitutive model for peridynamics”, Journal of Mechanics of Materials and Structures, 20:5 (2015), 539–557 | DOI | MR

[14] J. A. Mitchell, “On the ‘DSF’ and the ‘dreaded surface effect’, studies of presentation at Workshop on Nonlocal Damage and Failure”, Sandia National Laboratories, 2013

[15] S. A. Silling, R. B. Lehoucq, “Convergence of Peridynamics to Classical Elasticity Theory”, Journal of Elasticity, 93 (2008), 13–37 | DOI | MR | Zbl