Ground states in complex bodies
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 377-402.

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end.

DOI : 10.1051/cocv:2008036
Classification : 74A30, 49J45, 74A60, 49Q15, 74A99
Mots clés : cartesian currents, complex bodies, ground states, multifield theories
@article{COCV_2009__15_2_377_0,
     author = {Mariano, Paolo Maria and Modica, Giuseppe},
     title = {Ground states in complex bodies},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {377--402},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {2},
     year = {2009},
     doi = {10.1051/cocv:2008036},
     mrnumber = {2513091},
     zbl = {1161.74006},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008036/}
}
TY  - JOUR
AU  - Mariano, Paolo Maria
AU  - Modica, Giuseppe
TI  - Ground states in complex bodies
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 377
EP  - 402
VL  - 15
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008036/
DO  - 10.1051/cocv:2008036
LA  - en
ID  - COCV_2009__15_2_377_0
ER  - 
%0 Journal Article
%A Mariano, Paolo Maria
%A Modica, Giuseppe
%T Ground states in complex bodies
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 377-402
%V 15
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008036/
%R 10.1051/cocv:2008036
%G en
%F COCV_2009__15_2_377_0
Mariano, Paolo Maria; Modica, Giuseppe. Ground states in complex bodies. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 377-402. doi : 10.1051/cocv:2008036. http://www.numdam.org/articles/10.1051/cocv:2008036/

[1] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. | MR | Zbl

[2] J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics and Dynamics, P. Newton, P. Holmes and A. Weinstein Eds., Springer-Verlag, New York (2002) 3-59. | MR | Zbl

[3] B. Bernardini and T.J. Pence, A multifield theory for the modeling of the macroscopic behavior of shape memory materials, in Advances in Multifield Theories for Continua with Substructure, G. Capriz and P.M. Mariano Eds., Birkhäuser, Boston (2004) 199-242. | MR | Zbl

[4] F. Bethuel, H. Brezis and J.M. Coron, Relaxed energies for harmonic maps, in Variational methods, H. Berestycki, J. Coron and I. Ekeland Eds., Birkhäuser, Basel (1990) 37-52. | MR | Zbl

[5] E. Binz, M. De Leon and D. Socolescu, Global dynamics of media with microstructure. Extracta Math. 14 (1999) 99-125. | MR | Zbl

[6] G. Capriz, Continua with latent microstructure. Arch. Rational Mech. Anal. 90 (1985) 43-56. | MR | Zbl

[7] G. Capriz, Continua with Microstructure. Springer-Verlag, Berlin (1989). | MR | Zbl

[8] G. Capriz, Smectic liquid crystals as continua with latent microstructure. Meccanica 30 (1994) 621-627. | MR | Zbl

[9] G. Capriz and P. Biscari, Special solutions in a generalized theory of nematics. Rend. Mat. 14 (1994) 291-307. | MR | Zbl

[10] G. Capriz and P. Giovine, On microstructural inertia. Math. Models Methods Appl. Sci. 7 (1997) 211-216. | MR | Zbl

[11] P. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171-188. | MR | Zbl

[12] C. De Fabritiis and P.M. Mariano, Geometry of interactions in complex bodies. J. Geom. Phys. 54 (2005) 301-323. | MR | Zbl

[13] P.-G. De Gennes and J. Prost, The Physics of Liquid Crystals. Oxford University Press, Oxford (1993).

[14] M. Deneau, F. Dunlop and C. Ogney, Ground states of frustrated Ising quasicrystals. J. Phys. A 26 (1993) 2791-2802.

[15] A.R. Denton and J. Hafner, Thermodynamically stable one-component metallic quasicrystals. Europhys. Lett. 38 (1997) 189-194.

[16] J.L. Ericksen, Theory of anisotropic fluids. Trans. Soc. Rheol. 4 (1960) 29-39. | MR

[17] J.L. Ericksen, Conservation laws for liquid crystals. Trans. Soc. Rheol. 5 (1961) 23-34. | MR

[18] J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Rational Mech. Anal. 113 (1991) 97-120. | MR | Zbl

[19] J.L. Ericksen and C.A. Truesdell, Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1 (1958) 295-323. | MR | Zbl

[20] M. Foss, W.J. Hrusa and V.J. Mizel, The Lavrentiev gap phenomenon in nonlinear elasticity. Arch. Rational Mech. Anal. 167 (2003) 337-365. | MR | Zbl

[21] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55-91. | MR | Zbl

[22] M. Frémond, Non-Smooth Thermomechanics. Springer-Verlag, Berlin (2000). | MR | Zbl

[23] M. Giaquinta and G. Modica, On sequences of maps with equibounded energies. Calc. Var. Partial Differ. Equ. 12 (2001) 213-222. | MR | Zbl

[24] M. Giaquinta and D. Mucci, Maps into manifolds and currents: area and W 1,2 , W 1 2 , BV energies. CRM series, Scuola Normale Superiore, Pisa (2006). | MR | Zbl

[25] M. Giaquinta, G. Modica and J. Souček, Cartesian currents and variational problems for mappings into spheres. Ann. Scuola Normale Superiore 14 (1989) 393-485. | EuDML | Numdam | MR | Zbl

[26] M. Giaquinta, G. Modica and J. Souček, The Dirichlet energy of mappings with values into the sphere. Manuscripta Mat. 65 (1989) 489-507. | EuDML | MR | Zbl

[27] M. Giaquinta, G. Modica and J. Souček, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 106 (1989) 97-159. Erratum and addendum. Arch. Rational Mech. Anal. 109 (1990) 385-392. | MR | Zbl

[28] M. Giaquinta, G. Modica and J. Souček, The Dirichlet integral for mappings between manifolds: Cartesian currents and homology. Math. Ann. 294 (1992) 325-386. | EuDML | MR | Zbl

[29] M. Giaquinta, G. Modica and J. Souček, A weak approach to finite elasticity. Calc. Var. Partial Differ. Equ. 2 (1994) 65-100. | MR | Zbl

[30] M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations, Vol. I. Springer-Verlag, Berlin (1998). | MR | Zbl

[31] M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations, Vol. II. Springer-Verlag, Berlin (1998). | MR | Zbl

[32] R. Hardt and F.H. Lin, A remark on H 1 mappings. Manuscripta Math. 56 (1986) 1-10. | EuDML | Zbl

[33] D.D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in Geometry, Mechanics and Dynamics, P. Newton, P. Holmes and A. Weinstein Eds., Springer-Verlag, New York (2002) 113-167. | Zbl

[34] C. Hu, R. Wang and D.-H. Ding, Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63 (2000) 1-39.

[35] H.-C. Jeong and P.J. Steinhardt, Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys. Rev. B 48 (1993) 9394-9403.

[36] F.M. Leslie, Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28 (1968) 265-283. | Zbl

[37] C.N. Likos, Effective interactions in soft condensed matter physics. Phys. Rep. 348 (2001) 267-439.

[38] P.M. Mariano, Multifield theories in mechanics of solids. Adv. Appl. Mech. 38 (2002) 1-93.

[39] P.M. Mariano, Migration of substructures in complex fluids. J. Phys. A 38 (2005) 6823-6839. | Zbl

[40] P.M. Mariano, Mechanics of quasi-periodic alloys. J. Nonlinear Sci. 16 (2006) 45-77. | Zbl

[41] P.M. Mariano, Cracks in complex bodies: covariance of tip balances. J. Nonlinear Sci. 18 (2008) 99-141. | MR | Zbl

[42] P.M. Mariano and F.L. Stazi, Computational aspects of the mechanics of complex bodies. Arch. Comp. Meth. Eng. 12 (2005) 391-478. | MR | Zbl

[43] J. Miekisz, Stable quasicrystals ground states. J. Stat. Phys. 88 (1997) 691-711. | MR | Zbl

[44] R.D. Mindlin, Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16 (1964) 51-78. | MR | Zbl

[45] S. Müller, Q. Tang and B.S. Yan, On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 217-243. | EuDML | Numdam | MR | Zbl

[46] J.W. Nunziato and S.C. Cowin, A nonlinear theory of elastic materials with voids. Arch. Rational Mech. Anal. 72 (1979) 175-201. | MR | Zbl

[47] Y.G. Reshetnyak, General theorems on semicontinuity and on convergence with a functional. Sibir. Math. 8 (1967) 801-816. | Zbl

[48] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Sibir. Math. 9 (1968) 1039-1045. | Zbl

[49] Y.G. Reshetnyak, Space Mappings with Bounded Distorsion, Translations of Mathathematical Monographs 73. American Mathematical Society, Providence (1989). | MR | Zbl

[50] E.K.H. Salje, Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press, Cambridge (1993).

[51] R. Segev, A geometrical framework for the statics of materials with microstructure. Mat. Models Methods Appl. Sci. 4 (1994) 871-897. | MR | Zbl

[52] M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media. Springer-Verlag, Berlin (1997). | MR | Zbl

[53] J.J. Slawianowski, Quantization of affine bodies. Theory and applications in mechanics of structured media, in Material substructures in complex bodies: from atomic level to continuum, G. Capriz and P.M. Mariano Eds., Elsevier (2006) 80-162.

[54] A.P. Tsai, J.Q. Guo, E. Abe, H. Takakura and T.J. Sato, Alloys - A stable binary quasicrystals. Nature 408 (2000) 537-538.

Cité par Sources :