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Atomistic to Continuum limits for computational materials science

Published online by Cambridge University Press:  16 June 2007

Xavier Blanc ,
Claude Le Bris  and
Pierre-Louis Lions
Xavier Blanc
Affiliation:
Laboratoire J.-L. Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris, France. blanc@ann.jussieu.fr
Claude Le Bris
Affiliation:
CERMICS, École Nationale des Ponts et Chaussées, 6-8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée. MICMAC, Inria Rocquencourt, 78153 Le Chesnay, France. lebris@cermics.enpc.fr
Pierre-Louis Lions
Affiliation:
Collège de France, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France. CEREMADE, Université Paris Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France. lions@ceremade.dauphine.fr
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Abstract

The present article is an overview of some mathematical results, which provide elements of rigorous basis for some multiscale computations in materials science. The emphasis is laid upon atomistic to continuum limits for crystalline materials. Various mathematical approaches are addressed. The setting is stationary. The relation to existing techniques used in the engineering literature is investigated.

Keywords

Problems of mechanicsvariational problemsdiscrete to continuum limitmultiscale modelshomogenization theoryΓ-limitquasiconvexitygradient flowsquasicontinuum methodadaptivity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 391 - 426
Copyright
© EDP Sciences, SMAI, 2007

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References

Alicandro, R. and Cicalese, M., A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1–37. CrossRef
M. Anitescu, D. Negrut, P. Zapol and A. El-Azab, A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approach. Technical report ANL/MCS-P1303-1105, Argonne National Laboratory, Argonne, Illinois (2005). Available at http://www-unix.mcs.anl.gov/~anitescu/PUBLICATIONS/quasicont.pdf.
N. Antonic, C.J. van Duijn, W. Jäger and A. Mikelic, Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Springer (2002).
Arndt, M. and Griebel, M., Derivation of higher order gradient continuum models from atomistic models for crystalline solids. SIAM J. Multiscale Model. Simul. 4 (2005) 531–562. CrossRef
Arroyo, M. and Belytshko, T., A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes. Mech. Mater. 35 (2003) 175–622. CrossRef
N.W. Ashcroft and N.D. Mermin, Solid-State Physics. Saunders College Publishing (1976).
A. Askar, Lattice dynamical foundations of continuum theories. World Scientific, Philadelphia (1985).
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63 (1977) 337–403. CrossRef
J.M. Ball, Singularities and computation of miminizers for variational problems, in Foundations of Computational Mathematics, R. DeVore, A. Iserles and E. Suli Eds., Cambridge University Press London Mathematical Society Lecture Note Series 284 (2001) 1–20.
J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics. Springer (2002) 3–59.
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13–52. CrossRef
Ball, J.M. and James, R.D., Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Royal Soc. London A 338 (1992) 389–450. CrossRef
Ball, J.M. and Murat, F., W1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. CrossRef
T.J. Barth, T. Chan and R. Haimes Eds., Multiscale and multiresolution methods, Lecture notes in computational science and engineering 20. Springer (2002).
Bénilan, P., Brezis, H. and Crandall, M., A semilinear equation in $L^1({\mathbb R}^N)$ . Ann. Sc. Norm. Sup. Pisa 2 (1975) 523–555.
A. Bensoussan, J.-L. Lions and G. Papnicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications 5. North-Holland (1978).
F. Bethuel, G. Huisken, S. Müller and K. Steffen, Variational models for microstructures and phase transition, in Calculus of Variations and Geometric Evolution Problems, Lecture Notes in Mathematics 1713. Springer (1999) 85–210.
K. Bhattacharya, Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford Series on Materials Modelling, Oxford University Press (2003).
Bhattacharya, K. and Dolzmann, G., Relaxation of some multi-well problems. Proc. Royal Soc. Edinburgh A 131 (2001) 279–320. CrossRef
X. Blanc, A mathematical insight into ab initio simulations of solid phase, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, M. Defranceschi and C. Le Bris Eds., Lect. Notes Chem. 74. Springer (2000) 133–158.
Blanc, X., Geometry optimization for crystals in Thomas-Fermi type theories of solids. Comm. P.D.E. 26 (2001) 651–696. CrossRef
Blanc, X., Unique solvability for system of nonlinear elliptic PDEs arising in solid state physics. SIAM J. Math. Anal. 38 (2006) 1235–1248. CrossRef
Blanc, X. and Le Bris, C., Optimisation de géométrie dans le cadre des théories de Thomas-Fermi pour les cristaux périodiques [Geometry optimization for Thomas-Fermi type theories of solids]. Note C.R. Acad. Sci. Sér. 1 329 (1999) 551–556.
Blanc, X. and Le Bris, C., Thomas-Fermi type models for polymers and thin films. Adv. Diff. Equ. 5 (2000) 977–1032.
X. Blanc and C. Le Bris, Periodicity of the infinite-volume ground-state of a one-dimensional quantum model. Nonlinear Anal., T.M.A 48 (2002) 791–803.
Blanc, X. and Le Bris, C., Définition d'énergies d'interfaces à partir de modèles atomiques. Note C.R. Acad. Sci. Sér. 1 340 (2005) 535–540.
Blanc, X., Le Bris, C. and Legoll, F., Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN 39 (2005) 797–826. CrossRef
X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sinica (to appear).
Blanc, X., Le Bris, C. and Lions, P.-L., Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus [From molecular models to continuum mechanics]. Note C.R. Acad. Sci. Sér. 1 332 (2001) 949–956.
Blanc, X., Le Bris, C. and Lions, P.-L., From molecular models to continuum mechanics. Arch. Rat. Mech. Anal. 164 (2002) 341–381. CrossRef
Blanc, X., Le Bris, C. and Lions, P.-L., A definition of the ground state energy for systems composed of infinitely many particles. Comm. P.D.E 28 (2003) 439–475. CrossRef
Blanc, X., Le Bris, C. and Lions, P.-L., Du discret au continu pour des modèles de réseaux aléatoires d'atomes [Discrete to continuum limit for some models of stochastic lattices of atoms]. Note C.R. Acad. Sci. Sér. 1. 342 (2006) 627–633.
Blanc, X., Le Bris, C. and Lions, P.-L., On the energy of some microscopic stochastic lattices. Arch. Rat. Mech. Anal. 184 (2007) 303–339. CrossRef
X. Blanc, C. Le Bris and P.-L. Lions (in preparation).
A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002).
Braides, A., Non-local variational limits of discrete systems. Commun. Contemp. Math. 2 (2000) 285–297.
Braides, A. and Gelli, M.S., Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41–66. CrossRef
Braides, A. and Gelli, M.S., Limits of discrete systems with long-range interactions. J. Convex Anal. 9 (2002) 363–399.
A. Braides and M.S. Gelli, The passage from discrete to continuous variational problems: a nonlinear homogenization process. Preprint of the Scuola Normale Superiore di Pisa (2003). Available at http://cvgmt.sns.it/cgi/get.cgi/papers/bragel03/
Braides, A., Dal Maso, G. and Garroni, A., Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rat. Mech. Anal. 146 (1999) 23–58. CrossRef
Braides, A., Gelli, M.S. and Sigalotti, M., The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case. Proc. Steklov Inst. Math. 236 (2002) 395–414.
L. Breimana, Probability, Classics in Applied Mathematics. SIAM, Philadelphia (1992).
Brezis, H., Semilinear equations in ${\mathbb R}^N$ without condition at infinity. Appl. Math. Optim. 12 (1984) 271–282. CrossRef
V.V. Bulatov and T. Diaz de la Rubia, Multiscale modelling of materials. MRS Bulletin 26 (2001).
D. Caillerie, A. Mourad and A. Raoult, Discrete homogenization in graphene sheet modeling, J. Elasticity 84 (2006) 33–68. CrossRef
C. Carstensen, Numerical Analysis of Microstructure, in Theory and Numerics of Differential Equations, J.F. Blowey, J.P. Coleman and A.W. Craig Eds., Springer (2001) 59–126.
Carstensen, C. and Roubíček, T., Numerical approximation of young measuresin non-convex variational problems. Numer. Math. 84 (2000) 395–415. CrossRef
Catto, I., Le Bris, C. and Lions, P.-L., Limite thermodynamique pour des modèles de type Thomas-Fermi. Note C.R.A.S. Sér. 1 322 (1996) 357–364.
Catto, I., Le Bris, C. and Lions, P.-L., Sur la limite thermodynamique pour des modèles de type Hartree et Hartree-Fock [On the thermodynamic limit for Hartree and Hartree-Fock type models]. Note C.R.A.S. Sér. 1 327 (1998) 259–266.
I. Catto, C. Le Bris and P.-L. Lions, Mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford University Press (1998).
I. Catto, C. Le Bris and P.-L. Lions, On the thermodynamic limit for Hartree-Fock type models. Ann. Inst. H. Poincaré, Anal. Non Linéaire 18 (2001) 687–760. CrossRef
I. Catto, C. Le Bris and P.-L. Lions, On some periodic Hartree-type models for crystals. Ann. Inst. H. Poincaré, Anal. Non Linéaire 19 (2002) 143–190. CrossRef
Catto, I., Le Bris, C. and Lions, P.-L., From atoms to crytals: a mathematical journey. Bull. Amer. Math. Soc. 42 (2005) 291–363.
Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237–277. CrossRef
P.G. Ciarlet, Mathematical elasticity, Vol. 1. North Holland (1993).
G. Csányi, T. Albaret, G. Moras, M.C. Payne and A. De Vita, Multiscale hybrid simulation methods for material systems J. Phys. Condens. Matt. 17 (2005) R691.
R. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag Berlin (1989).
G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston, Inc., Boston, MA (1993).
P. Deák, T. Frauenheim and M.R. Pederson, Eds., Computer simulation of materials at atomic level. Wiley (2000).
Delaunay, B.N., Dolbilin, N.P., Shtogrin, M.I. and Galiulin, R.V., A local criterion for regularity of a system of points. Sov. Math. Dokl. 17 (1976) 319–322.
G. Dolzmann, Variational Methods for Crystalline Microstructure – Analysis and Computation. Springer-Verlag (2003).
E, W. and Engquist, B., The Heterogeneous Multi-Scale Methods. Comm. Math. Sci. 1 (2003) 87–132. CrossRef
E, W. and Huang, Z., Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87 (2001) 135501. CrossRef
E, W. and Huang, Z., A dynamic atomistic-continuum method for the simulation of crystalline materials. J. Comp. Phys. 182 (2002) 234–261. CrossRef
W. E and P.B. Ming, Atomistic and continuum theory of solids, I. Preprint (2003).
E, W. and Ming, P.B., Analysis of multiscale methods. J. Comp. Math. 22 (2004) 210–219.
E, W. and Ming, P.B., Cauchy-Born rule and stability of crystals: static problems. Arch. Rat. Mech. Anal. 183 (2007) 241–297. CrossRef
Fago, M., Hayes, R.L., Carter, E.A. and Ortiz, M., Density-functional-theory-based local quasicontinuum method: Prediction of dislocation nucleation. Phys. Rev. B 70 (2004) 100102(R). CrossRef
Fonseca, I., Variational methods for elastic crystals. Arch. Rat. Mech. Anal. 97 (1987) 187–220. CrossRef
Fonseca, I., The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988) 175–195.
Friesecke, G. and James, R.D., A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48 (2000) 1519–1540. CrossRef
Friesecke, G., James, R.D. and Müller, S., Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris Sér. I 334 (2002) 173–178. CrossRef
Friesecke, G. and Theil, F., Validity and failure of the Cauchy-Born hypothesis in a Two-Dimensional Mass-Spring Lattice. J. Nonlinear Sci. 12 (2002) 445–478. CrossRef
Gardner, C.S. and Radin, C., The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys. 20 (1979) 719–724. CrossRef
Geymonat, G., Krasucki, F. and Lenci, S., Analyse asymptotique du comportement d'un assemblage collé [Asymptotic analysis of the behaviour of a bonded joint]. C.R. Acad. Sci. Paris Sér. I 322 (1996) 1107–1112.
Geymonat, G., Krasucki, F. and Lenci, S., Mathematical analysis of a bonded joint with a soft thin adhesive. Math. Mech. Solids 4 (1999) 201–225. CrossRef
WJ. Hehre, L. Radom, P.V.R. Shleyer and J. Pople, Ab initio molecular orbital theory. Wiley (1986).
Iosifescu, O., Licht, C. and Michaille, G., Variational limit of a one dimensional discrete and statistically homogeneous system of material points. Asymptot. Anal. 28 (2001) 309–329.
Iosifescu, O., Licht, C. and Michaille, G., Variational limit of a one-dimensional discrete and statistically homogeneous system of material points. C.R. Acad. Sci. Paris Sér. I Math. 332 (2001) 575–580. CrossRef
John, F., Rotation and strain. Comm. Pure Appl. Math. 14 (1961) 391–413. CrossRef
F. John, Bounds for deformations in terms of average strains, in Inequalities III, O. Shisha Ed. (1972) 129–144.
D. Kinderlehrer, Remarks about equilibrium configurations of crystals, in Material instabilities in contiuum mechanics and related mathematical problems, J.M. Ball Ed., Oxford University Press (1998) 217–242.
Kinderlehrer, D. and Pedregal, P., Characterization of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329–365. CrossRef
Kinderlehrer, D. and Pedregal, P., Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–90. CrossRef
O. Kirchner, L.P. Kubin and V. Pontikis Eds., Computer simulation in materials science, Kluwer (1996).
H. Kitagawa, T. Aihara Jr. and Y. Kawazoe Eds., Mesoscopic dynamics of fracture, Advances in Materials Research. Springer (1998).
C. Kittel, Introduction to Solid State Physics. 7th edn. Wiley (1996).
Knap, J. and Ortiz, M., Analysis, An of the QuasiContinuum Method. J. Mech. Phys. Solids 49 (2001) 1899. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems. I-II-III. Comm. Pure Appl. Math. 39 (1986) 113–137, 139–182, 353–377. CrossRef
U. Krengel, Ergodic theorems, Studies in Mathematics 6. de Gruyter (1985).
J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 16 (1999) 1–13. CrossRef
C. Le Bris, Computational Chemistry, in Handbook of numerical analysis, Vol. X, P.G. Ciarlet Ed., North-Holland (2003).
C. Le Bris, Computational chemistry from the perspective of numerical analysis, Acta Numer. 14 (2005) 363–444. CrossRef
Li, J., Van Vliet, K.J., Zhu, T., Suresh, S. and Yip, S., Atomistic mechanisms governing elastic limit and incipient plasticity in crystals. Nature 418 (2002) 307. CrossRef
Licht, C., Comportement asymptotique d'une bande dissipative mince de faible rigidité [Asymptotic behaviour of a thin dissipative layer with low stiffness]. C.R. Acad. Sci. Paris Sér. I 317 (1993) 429–433.
Licht, C. and Michaille, G., Une modélisation du comportement d'un joint collé élastique [A modelling of elastic adhesively bonding joints]. C.R. Acad. Sci. Paris Sér. I 322 (1996) 295–300.
Lieb, E.H., Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys. 53 (1981) 603–641 . CrossRef
Lieb, E.H. and Simon, B., The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23 (1977) 22–116. CrossRef
P. Lin, A nonlinear wave equation of mixed type for fracture dynamics. Research report No. 777, Department of Mathematics, The National University of Singapore, August 2000. Available at http://www.math.nus.edu.sg/ matlinp/WWW/linsiap.pdf
Lin, P., Theoretical and numerical analysis of the quasi-continuum approximation of a material particle model. Math. Comput. 72 (2003) 657–675. CrossRef
P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. Preprint 2005-80 of the Institute for mathematical sciences, National University of Singapore (2005). Available at http://www.ims.nus.edu.sg/preprints/2005-80.pdf
Lin, P. and Shu, C.W., Numerical solution of a virtual internal bond model for material fracture. Physica D 167 (2002) 101–121. CrossRef
Liu, W.K., Qian, D. and Horstemeyer, M.F., Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials. Comp. Meth. Appl. Mech. Eng. 193 (2004) 17–20.
Luskin, M., On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191–258. CrossRef
M. Luskin, Computational modeling of microstructure, in Proceedings of the International Congress of Mathematicians, ICM, Beijing (2002) 707–716.
Miller, R. and Tadmor, E.B., The Quasicontinuum Method: Overview, applications and current directions. J. Computer-Aided Materials Design 9 (2002) 203–239. CrossRef
Miller, R., Tadmor, E.B., Phillips, R. and Ortiz, M., Quasicontinuum simulation of fracture at the atomic scale. Modelling Simul. Mater. Sci. Eng. 6 (1998) 607. CrossRef
Morrey Jr, C.B.., Quasi-convexity and the lower semi-continuity of multiple integrals. Pacific J. Math. 2 (1952) 25–53. CrossRef
S. Müller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems. Lect. Notes Math. 1713. Springer Verlag, Berlin (1999) 85–210.
Nijboer, B.R.A. and Ventevogel, W.J., On the configuration of systems of interacting particles with minimum potential energy per particle. Physica 98A (1979) 274.
Nijboer, B.R.A and Ventevogel, W.J., On the configuration of systems of interacting particles with minimum potential energy per particle. Physica 99A (1979) 569.
C. Ortner, Continuum limit of a one-dimensional atomistic energy based on local minimization. Technical report 05/11, Oxford University Computing Laboratory (2005).
Pagano, S. and Paroni, R., A simple model for phase transitions: from the discrete to the continuum problem. Quart. Appl. Math. 61 (2003) 89–109. CrossRef
P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser (1997).
P. Pedregal, Variational Methods in Nonlinear Elasticity. SIAM (2000).
C. Pisani Ed., Quantum mechanical ab initio calculation of the properties of crystalline materials, Lecture Notes in Chemistry 67. Springer (1996).
D. Raabe, Computational Material Science. Wiley (1998).
Radin, C., Ground states for soft disks. J. Stat. Phys. 26 (1981) 365. CrossRef
Reshetnyak, Y.G., Liouville's theory on conformal mappings under minimal regularity assumptions. Sibirskii Math. 8 (1967) 69–85. CrossRef
M.O. Rieger and J. Zimmer, Young measure flow as a model for damage, SIAM J. Math. Anal. (2005) (to appear).
R.E. Rudd and J.Q. Broughton, Concurrent coupling of length scales in solid state system, in [59] 251–291.
B. Schmidt, On the passage form atomic to continuum theory for thin films. Preprint 82/2005 of the Max Planck Institute of Leipzig (2005). Available at http://www.mis.mpg.de/preprints/2005/prepr2005_82.html
B. Schmidt, Qualitative properties of a continuum theory for thin films. Preprint 83/2005 of the Max Planck Institute of Leipzig (2005). Available at http://www.mis.mpg.de/preprints/2005/prepr2005_83.html
B. Schmidt, A derivation of continuum nonlinear plate theory form atomistic models. Preprint 90/2005 of the Max Planck Institute of Leipzig (2005). Available at http://www.mis.mpg.de/preprints/2005/prepr2005_90.html
Shenoy, V.B., Miller, R., Tadmor, E.B., Phillips, R. and Ortiz, M., Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80 (1998) 742. CrossRef
Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R. and Ortiz, M., An adaptative finite element approach to atomic-scale mechanics – the QuasiContinuum Method. J. Mech. Phys. Solids 47 (1999) 611. CrossRef
Solovej, J.P., Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules. Comm. Math. Phys. 129 (1990) 561–598. CrossRef
V. Šveràk, On regularity for Monge-Ampère equations. Preprint, Heriott-Watt University (1991).
Šveràk, V., Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh A 120 (1992) 185–189. CrossRef
V. Šveràk, On the problem of two wells, in Microstructure and phase transition, IMA Vol. Math. Appl. 54. Springer, New York, (1993) 183–189.
A. Szabo and N.S. Ostlund, Modern quantum chemistry: an introduction. Macmillan (1982).
Tadmor, E.B. and Phillips, R., Mixed atomistic and continuum models of deformation in solids. Langmuir 12 (1996) 4529. CrossRef
Tadmor, E.B., Ortiz, M. and Phillips, R., Quasicontinuum analysis of defects in solids. Phil. Mag. A. 73 (1996) 1529–1563. CrossRef
Tadmor, E.B., Smith, G.S., Bernstein, N. and Kaxiras, E., Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59 (1999) 235. CrossRef
Theil, F., A proof of crystallization in two dimensions. Comm. Math. Phys. 262 (2006) 209–236. CrossRef
L. Truskinovsky, Fracture as a phase transformation, in Contemp. Res. in Mech. and Math. of Materials, Ericksen's symposium, R. Batra and M. Beatty Eds., CIMNE, Barcelone (1996) 322–332.
Ventevogel, W.J., On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Physica 92A (1978) 343. CrossRef
Yip, S., Synergistic materials science. Nature Mater. 2 (2003) 3–5. CrossRef
L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia-London-Toronto (1969).
Zaittouni, F., Lebon, F. and Licht, C., Étude théorique et numérique du comportement d'un assemblage de plaques [Theoretical study of the behaviour of bonded plates]. C.R. Mécanique 330 (2002) 359–364. CrossRef