In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

Keywords: multiscale methods, variational problems, continuum mechanics, discrete mechanics

@article{M2AN_2005__39_4_797_0, author = {Blanc, Xavier and Bris, Claude Le and Legoll, Fr\'ed\'eric}, title = {Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {797--826}, publisher = {EDP-Sciences}, volume = {39}, number = {4}, year = {2005}, doi = {10.1051/m2an:2005035}, mrnumber = {2165680}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2005035/} }

TY - JOUR AU - Blanc, Xavier AU - Bris, Claude Le AU - Legoll, Frédéric TI - Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2005 SP - 797 EP - 826 VL - 39 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2005035/ DO - 10.1051/m2an:2005035 LA - en ID - M2AN_2005__39_4_797_0 ER -

%0 Journal Article %A Blanc, Xavier %A Bris, Claude Le %A Legoll, Frédéric %T Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2005 %P 797-826 %V 39 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2005035/ %R 10.1051/m2an:2005035 %G en %F M2AN_2005__39_4_797_0

Blanc, Xavier; Bris, Claude Le; Legoll, Frédéric. Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 (2005) no. 4, pp. 797-826. doi : 10.1051/m2an:2005035. http://www.numdam.org/articles/10.1051/m2an:2005035/

[1] A note on the theory of SBV functions. Bollettino U.M.I. Sez. B 7 (1997) 375-382. | Zbl

and ,[2] Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (2000). | MR | Zbl

, and ,[3]

, and , work in preparation, and Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics, Preprint, Laboratoire Jacques-Louis Lions, Université Paris 6 (2004), available at http://www.ann.jussieu.fr/publications/2004/R04029.html[4] From molecular models to continuum mechanics. Arch. Rational Mech. Anal. 164 (2002) 341-381. | Zbl

, and ,[5] Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rational Mech. Anal. 146 (1999) 23-58. | Zbl

, and ,[6] The Mathematical Theory of Finite Element Methods. Springer (1991). | Zbl

and ,[7] Concurrent coupling of length scales: Methodology and application. Phys. Rev. B 60 (1999) 2391-2403.

, , and ,[8] An $O\left({h}^{2}\right)$ method for a non-smooth boundary value problem. Aequationes Math. 2 (1968) 39-49. | Zbl

,[9] Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, North Holland (1988). | MR | Zbl

,[10] Basic Error Estimates for Elliptic Problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet, J.-L. Lions, Eds., North-Holland (1991) 17-351. | Zbl

,[11] W. E and P. Ming, private communication.

[12] An analysis of the Quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899-1923. | Zbl

and ,[13] Numerical Methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis, Vol. III, P.G. Ciarlet, J.-L. Lions, Eds., North-Holland (1994) 465-622. | Zbl

,[14] Méthodes moléculaires et multi-échelles pour la simulation numérique des matériaux

,[15] Mathematical foundations of Elasticity. Dover (1994). | MR

and ,[16] Quasicontinuum simulation of fracture at the atomic scale. Model. Simul. Mater. Sci. Eng. 6 (1998) 607-638.

, , and ,[17] Numerical Approximation of Partial Differential Equations. Springer (1997). | MR | Zbl

and ,[18] Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.

, and ,[19] Mixed atomistic and continuum models of deformation in solids. Langmuir 12 (1996) 4529-4534.

and ,[20] Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59 (1999) 235-245.

, , and ,[21] Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80 (1998) 742-745.

, , , and ,[22] An adaptative finite element approach to atomic-scale mechanics - the Quasicontinuum method, J. Mech. Phys. Solids 47 (1999) 611-642. | Zbl

, , , , and ,[23] The nonlinear field theories of mechanics theory of elasticity. Handbuch der Physik, III/3, Springer Berlin (1965) 1-602. | Zbl

and ,[24] Fracture as a phase transformation, in Contemporary research in mechanics and mathematics of materials, Ericksen's Symposium, R. Batra and M. Beatty, Eds., CIMNE, Barcelona (1996) 322-332.

,[25] Quantifying the early stages of plasticity through nanoscale experiments and simulations. Phys. Rev. B 67 (2003) 104105.

, , , and ,[26] Numerical simulation of cohesive fracture by the virtual-internal-bond model. Comput. Model. Engrg. Sci. 3 (2002) 263-289. | Zbl

, , , and ,*Cited by Sources: *