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 : https://doi.org/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},
     zbl = {1161.74006},
     mrnumber = {2513091},
     language = {en},
     url = {www.numdam.org/item/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/item/COCV_2009__15_2_377_0/

[1] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. | MR 475169 | Zbl 0368.73040

[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 1919825 | Zbl 1054.74008

[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 2035124 | Zbl 1140.74523

[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 1205144 | Zbl 0793.58011

[5] E. Binz, M. De Leon and D. Socolescu, Global dynamics of media with microstructure. Extracta Math. 14 (1999) 99-125. | MR 1758956 | Zbl 0988.74009

[6] G. Capriz, Continua with latent microstructure. Arch. Rational Mech. Anal. 90 (1985) 43-56. | MR 792882 | Zbl 0569.73001

[7] G. Capriz, Continua with Microstructure. Springer-Verlag, Berlin (1989). | MR 985585 | Zbl 0676.73001

[8] G. Capriz, Smectic liquid crystals as continua with latent microstructure. Meccanica 30 (1994) 621-627. | MR 1360975 | Zbl 0836.76004

[9] G. Capriz and P. Biscari, Special solutions in a generalized theory of nematics. Rend. Mat. 14 (1994) 291-307. | MR 1310564 | Zbl 0813.76010

[10] G. Capriz and P. Giovine, On microstructural inertia. Math. Models Methods Appl. Sci. 7 (1997) 211-216. | MR 1440606 | Zbl 0869.73065

[11] P. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171-188. | MR 862546 | Zbl 0628.73043

[12] C. De Fabritiis and P.M. Mariano, Geometry of interactions in complex bodies. J. Geom. Phys. 54 (2005) 301-323. | MR 2139085 | Zbl 1145.74004

[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 134065

[17] J.L. Ericksen, Conservation laws for liquid crystals. Trans. Soc. Rheol. 5 (1961) 23-34. | MR 158610

[18] J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Rational Mech. Anal. 113 (1991) 97-120. | MR 1079183 | Zbl 0729.76008

[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 99135 | Zbl 0081.39303

[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 1981861 | Zbl 1090.74010

[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 2244798 | Zbl 1101.74015

[22] M. Frémond, Non-Smooth Thermomechanics. Springer-Verlag, Berlin (2000). | MR 1885252 | Zbl 0990.80001

[23] M. Giaquinta and G. Modica, On sequences of maps with equibounded energies. Calc. Var. Partial Differ. Equ. 12 (2001) 213-222. | MR 1825872 | Zbl 1013.49030

[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 2262657 | Zbl 1111.49001

[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 84059 | Numdam | MR 1050333 | Zbl 0713.49014

[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 155454 | MR 1019705 | Zbl 0678.49006

[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 980756 | Zbl 0712.73009

[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 165005 | MR 1183409 | Zbl 0762.49018

[29] M. Giaquinta, G. Modica and J. Souček, A weak approach to finite elasticity. Calc. Var. Partial Differ. Equ. 2 (1994) 65-100. | MR 1384395 | Zbl 0806.49007

[30] M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations, Vol. I. Springer-Verlag, Berlin (1998). | MR 1645086 | Zbl 0914.49001

[31] M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations, Vol. II. Springer-Verlag, Berlin (1998). | MR 1645086 | Zbl 0914.49001

[32] R. Hardt and F.H. Lin, A remark on H 1 mappings. Manuscripta Math. 56 (1986) 1-10. | EuDML 155153 | Zbl 0618.58015

[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 1114.37308

[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 0159.57101

[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 1071.76007

[40] P.M. Mariano, Mechanics of quasi-periodic alloys. J. Nonlinear Sci. 16 (2006) 45-77. | Zbl 1138.74309

[41] P.M. Mariano, Cracks in complex bodies: covariance of tip balances. J. Nonlinear Sci. 18 (2008) 99-141. | MR 2386717 | Zbl 1135.74004

[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 2201460 | Zbl 1152.74315

[43] J. Miekisz, Stable quasicrystals ground states. J. Stat. Phys. 88 (1997) 691-711. | MR 1467626 | Zbl 0945.82569

[44] R.D. Mindlin, Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16 (1964) 51-78. | MR 160356 | Zbl 0119.40302

[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 78330 | Numdam | MR 1267368 | Zbl 0863.49002

[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 545517 | Zbl 0444.73018

[47] Y.G. Reshetnyak, General theorems on semicontinuity and on convergence with a functional. Sibir. Math. 8 (1967) 801-816. | Zbl 0179.20902

[48] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Sibir. Math. 9 (1968) 1039-1045. | Zbl 0176.44402

[49] Y.G. Reshetnyak, Space Mappings with Bounded Distorsion, Translations of Mathathematical Monographs 73. American Mathematical Society, Providence (1989). | MR 994644 | Zbl 0667.30018

[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 1305562 | Zbl 0816.73051

[52] M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media. Springer-Verlag, Berlin (1997). | MR 1423807 | Zbl 0870.73004

[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.