Analysis of cell size effects in atomistic crack propagation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1821-1847.

We consider crack propagation in a crystalline material in terms of bifurcation analysis. We provide evidence that the stress intensity factor is a natural bifurcation parameter, and that the resulting bifurcation diagram is a periodic “snaking curve”. We then prove qualitative properties of the equilibria and convergence rates of finite-cell approximations to the “exact” bifurcation diagram.

DOI : 10.1051/m2an/2020005
Classification : 65L20, 70C20, 74A45, 74G20, 74G40, 74G60, 74G65
Mots-clés : Crystal lattices, defects, crack propagation, regularity, bifurcation theory, convergence rates 
@article{M2AN_2020__54_6_1821_0,
     author = {Buze, Maciej and Hudson, Thomas and Ortner, Christoph},
     title = {Analysis of cell size effects in atomistic crack propagation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1821--1847},
     publisher = {EDP-Sciences},
     volume = {54},
     number = {6},
     year = {2020},
     doi = {10.1051/m2an/2020005},
     mrnumber = {4129381},
     zbl = {07357913},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2020005/}
}
TY  - JOUR
AU  - Buze, Maciej
AU  - Hudson, Thomas
AU  - Ortner, Christoph
TI  - Analysis of cell size effects in atomistic crack propagation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2020
SP  - 1821
EP  - 1847
VL  - 54
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2020005/
DO  - 10.1051/m2an/2020005
LA  - en
ID  - M2AN_2020__54_6_1821_0
ER  - 
%0 Journal Article
%A Buze, Maciej
%A Hudson, Thomas
%A Ortner, Christoph
%T Analysis of cell size effects in atomistic crack propagation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2020
%P 1821-1847
%V 54
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2020005/
%R 10.1051/m2an/2020005
%G en
%F M2AN_2020__54_6_1821_0
Buze, Maciej; Hudson, Thomas; Ortner, Christoph. Analysis of cell size effects in atomistic crack propagation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1821-1847. doi : 10.1051/m2an/2020005. http://www.numdam.org/articles/10.1051/m2an/2020005/

R. Alicandro, L. De Luca, A. Garroni and M. Ponsiglione, Metastability and dynamics of discrete topological singularities in two dimensions: a Γ -convergence approach. Arch. Ration. Mech. Anal. 214 (2014) 269–330. | DOI | MR | Zbl

R. Alicandro, L. De Luca, A. Garroni and M. Ponsiglione, Minimising movements for the motion of discrete screw dislocations along glide directions. Calc. Var. Partial Differ. Equ. 56 (2017) 19, Art. 148. | DOI | MR | Zbl

N. Berglund, Kramers’ law: validity, derivations and generalisations. Markov Process. Relat. Fields 19 (2013) 459–490. | MR | Zbl

W.-J. Beyn, A. Champneys, E. Doedel, W. Govaerts, Y.A. Kuznetsov and B. Sandstede, Numerical continuation, and computation of normal forms. In: Handbook of Dynamical Systems III: Towards Applications (2002). | DOI | MR | Zbl

E. Bitzek, J.R. Kermode and P. Gumbsch, Atomistic aspects of fracture. Int. J. Fract. 191 (2015) 13–30. | DOI

J. Braun and C. Ortner, Sharp uniform convergence rate of the supercell approximation of a crystalline defect. Preprint arXiv:1811.08741 (2018). | MR

J. Braun, M.H. Duong and C. Ortner, Thermodynamic limit of the transition rate of a crystalline defect. Preprint arXiv:1810.11643 (2018). | MR

J. Braun, M. Buze and C. Ortner, The effect of crystal symmetries on the locality of screw dislocation cores. SIAM J. Math. Anal. 51 (2019). | DOI | MR | Zbl

F. Brezzi, J. Rappaz and P. Raviart, Finite dimensional approximation of nonlinear problems. Numer. Math. 36 (1980) 1–25. | DOI | MR | Zbl

K.B. Broberg, Cracks and Fracture. Academic Press, San Diego (1999).

M. Buze, T. Hudson and C. Ortner, Analysis of an atomistic model for anti-plane fracture. Math. Models Methods Appl. Sci. 29 (2019) 2469–2521. | DOI | MR | Zbl

J. Chaussidon, M. Fivel and D. Rodney, The glide of screw dislocations in bcc Fe: atomistic static and dynamic simulations. Acta Mater. 54 (2006) 3407–3416. | DOI

K.A. Cliffe, A. Spence and S.J. Tavener, The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numer. 9 (2000) 39–131. | DOI | MR | Zbl

V. Ehrlacher, C. Ortner and A.V. Shapeev, Analysis of boundary conditions for crystal defect atomistic simulations. Preprint arXiv:1306.5334v4 (2016). | MR

V. Ehrlacher, C. Ortner and A.V. Shapeev, Analysis of boundary conditions for crystal defect atomistic simulations. Arch. Ration. Mech. Anal. 222 (2016) 1217–1268. | DOI | MR | Zbl

H. Eyring, The activated complex in chemical reactions. J. Chem. Phys. 3 (1935) 107–115. | DOI

L.B. Freund, Dynamic Fracture Mechanics. Cambridge Monographs on Mechanics. Cambridge University Press (1990). | DOI | MR | Zbl

A. Garg, A. Acharya and C.E. Maloney, A study of conditions for dislocation nucleation in coarser-than-atomistic scale models. J. Mech. Phys. Solids 75 (2015) 76–92. | DOI

E.H. Glaessgen, E. Saether, J. Hochhalter and V. Yamakov, Modeling near-crack-tip plasticity from nano-to micro-scales. Collection of Technical Papers – AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference (2010).

P. Gumbsch and R.M. Cannon, Atomistic aspects of brittle fracture. MRS Bull. 25 (2000) 15–20. | DOI

P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62 (1990) 251–341. | DOI | MR

T. Hudson, Upscaling a model for the thermally-driven motion of screw dislocations. Arch. Ration. Mech. Anal. 224 (2017) 291–352. | DOI | MR | Zbl

T. Hudson and C. Ortner, Existence and stability of a screw dislocation under anti-plane deformation. Arch. Ration. Mech. Anal. 213 (2014) 887–929. | DOI | MR | Zbl

T. Hudson and C. Ortner, Analysis of stable screw dislocation configurations in an anti-plane lattice model. SIAM J. Math. Anal. 41 (2015) 291–320. | DOI | MR | Zbl

F. John and L. Nirenberg, On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961) 415–426. | DOI | MR

H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Applications of bifurcation theory: proceedings of an Advanced Seminar. Mathematics Research Center, the University of Wisconsin at Madison (1977) 359–384. | MR | Zbl

L. Landau, E. Lifshitz and J. Sykes, Theory of elasticity. In: Course of Theoretical Physics. Pergamon Press (1989).

S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics. New York Springer (1999). | DOI | MR | Zbl

B. Lawn, Fracture of brittle solids. 2nd edition. Cambridge Solid State Science Series. Cambridge University Press (1993).

X. Li, A bifurcation study of crack initiation and kinking. Eur. Phys. J. B 86 (2013) 258. | DOI

X. Li, A numerical study of crack initiation in a bcc iron system based on dynamic bifurcation theory. J. Appl. Phys. 116 (2014) 164314. | DOI

C. Ortner and A. Shapeev, Interpolants of lattice functions for the analysis of atomistic/continuum multiscale methods. Preprint arXiv:1204.3705 (2012).

C. Ortner and E. Süli, A note on linear elliptic systems on ℝ$$. Preprint arXiv:1202.3970 (2012).

C. Ortner and F. Theil, Justification of the cauchy–born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207 (2013). | DOI | MR | Zbl

J.R. Rice, Mathematical analysis in the mechanics of fracture, edited by H. Liebowitz. In: Vol. 2 of Mathematical Fundamentals. Fracture: An Advanced Treatise (1968) 191–311. | Zbl

L. Richardson, The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philos. Trans. R. Soc. 210 (1911) 307–357. | JFM

R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer (1998). | DOI | MR | Zbl

W. Rudin, Real and Complex Analysis. McGraw-Hill (1966). | MR | Zbl

C. Sun and Z.-H. Jin, Fracture Mechanics. Academic Press (2012).

C. Taylor and J. Dawes, Snaking and isolas of localised states in bistable discrete lattices. Phys. Lett. A 375 (2010) 14–22. | DOI | Zbl

R. Thomson, C. Hsieh and V. Rana, Lattice trapping of fracture cracks. J. Appl. Phys. 42 (1971) 3154–3160. | DOI

E.P. Wigner, The Transition State Method. Springer Berlin Heidelberg, Berlin, Heidelberg (1997) 163–175.

Cité par Sources :