Skip to main content Accessibility help
×
Home
Hostname: page-component-558cb97cc8-fjc52 Total loading time: 0.912 Render date: 2022-10-07T16:41:06.976Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Ground states in complex bodies

Published online by Cambridge University Press:  30 May 2008

Paolo Maria Mariano
Affiliation:
DICeA, University of Florence, via Santa Marta 3, 50139 Firenze, Italy; paolo.mariano@unifi.it
Giuseppe Modica
Affiliation:
Dipartimento di Matematica Applicata “G. Sansone”, University of Florence, via Santa Marta 3, 50139 Firenze, Italy; giuseppe.modica@unifi.it
Get access

Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337403. CrossRef
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.
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.
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.
Binz, E., de Leon, M. and Socolescu, D., Global dynamics of media with microstructure. Extracta Math. 14 (1999) 99125.
G. Capriz, Continua with latent microstructure. Arch. Rational Mech. Anal. 90 (1985) 43–56.
G. Capriz, Continua with Microstructure. Springer-Verlag, Berlin (1989).
Capriz, G., Smectic liquid crystals as continua with latent microstructure. Meccanica 30 (1994) 621627. CrossRef
Capriz, G. and Biscari, P., Special solutions in a generalized theory of nematics. Rend. Mat. 14 (1994) 291307.
Capriz, G. and Giovine, P., On microstructural inertia. Math. Models Methods Appl. Sci. 7 (1997) 211216. CrossRef
Ciarlet, P. and Nečas, J., Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171188. CrossRef
de Fabritiis, C. and Mariano, P.M., Geometry of interactions in complex bodies. J. Geom. Phys. 54 (2005) 301323. CrossRef
P.-G. De Gennes and J. Prost, The Physics of Liquid Crystals. Oxford University Press, Oxford (1993).
Deneau, M., Dunlop, F. and Ogney, C., Ground states of frustrated Ising quasicrystals. J. Phys. A 26 (1993) 27912802.
Denton, A.R. and Hafner, J., Thermodynamically stable one-component metallic quasicrystals. Europhys. Lett. 38 (1997) 189194. CrossRef
Ericksen, J.L., Theory of anisotropic fluids. Trans. Soc. Rheol. 4 (1960) 2939. CrossRef
Ericksen, J.L., Conservation laws for liquid crystals. Trans. Soc. Rheol. 5 (1961) 2334. CrossRef
Ericksen, J.L., Liquid crystals with variable degree of orientation. Arch. Rational Mech. Anal. 113 (1991) 97120. CrossRef
Ericksen, J.L. and Truesdell, C.A., Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1 (1958) 295323. CrossRef
Foss, M., Hrusa, W.J. and Mizel, V.J., The Lavrentiev gap phenomenon in nonlinear elasticity. Arch. Rational Mech. Anal. 167 (2003) 337365. CrossRef
Francfort, G. and Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 5591.
M. Frémond, Non-Smooth Thermomechanics. Springer-Verlag, Berlin (2000).
Giaquinta, M. and Modica, G., On sequences of maps with equibounded energies. Calc. Var. Partial Differ. Equ. 12 (2001) 213222. CrossRef
M. Giaquinta and D. Mucci, Maps into manifolds and currents: area and W $^{1,2}$ , W $^{\frac{1}{2}}$ , BV energies. CRM series, Scuola Normale Superiore, Pisa (2006).
M. Giaquinta, G. Modica and J. Souček, Cartesian currents and variational problems for mappings into spheres. Ann. Scuola Normale Superiore 14 (1989) 393–485.
Giaquinta, M., Modica, G. and Souček, J., The Dirichlet energy of mappings with values into the sphere. Manuscripta Mat. 65 (1989) 489507. CrossRef
Giaquinta, M., Modica, G. and Souček, J., Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 106 (1989) 97159. Erratum and addendum. Arch. Rational Mech. Anal. 109 (1990) 385–392. CrossRef
Giaquinta, M., Modica, G. and Souček, J., The Dirichlet integral for mappings between manifolds: Cartesian currents and homology. Math. Ann. 294 (1992) 325386. CrossRef
Giaquinta, M., Modica, G. and Souček, J., A weak approach to finite elasticity. Calc. Var. Partial Differ. Equ. 2 (1994) 65100. CrossRef
M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations, Vol. I. Springer-Verlag, Berlin (1998).
M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations, Vol. II. Springer-Verlag, Berlin (1998).
Hardt, R. and Lin, F.H., A remark on H 1 mappings. Manuscripta Math. 56 (1986) 110. CrossRef
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.
Hu, C., Wang, R. and Ding, D.-H., Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63 (2000) 139. CrossRef
H.-C. Jeong and P.J. Steinhardt, Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys. Rev. B 48 (1993) 9394–9403.
Leslie, F.M., Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28 (1968) 265283.
Likos, C.N., Effective interactions in soft condensed matter physics. Phys. Rep. 348 (2001) 267439. CrossRef
Mariano, P.M., Multifield theories in mechanics of solids. Adv. Appl. Mech. 38 (2002) 193. CrossRef
Mariano, P.M., Migration of substructures in complex fluids. J. Phys. A 38 (2005) 68236839. CrossRef
Mariano, P.M., Mechanics of quasi-periodic alloys. J. Nonlinear Sci. 16 (2006) 4577. CrossRef
Mariano, P.M., Cracks in complex bodies: covariance of tip balances. J. Nonlinear Sci. 18 (2008) 99141. CrossRef
Mariano, P.M. and Stazi, F.L., Computational aspects of the mechanics of complex bodies. Arch. Comp. Meth. Eng. 12 (2005) 391478. CrossRef
Miekisz, J., Stable quasicrystals ground states. J. Stat. Phys. 88 (1997) 691711. CrossRef
Mindlin, R.D., Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16 (1964) 5178.
Müller, S., Tang, Q. and Yan, B.S., On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 217243. CrossRef
Nunziato, J.W. and Cowin, S.C., A nonlinear theory of elastic materials with voids. Arch. Rational Mech. Anal. 72 (1979) 175201.
Reshetnyak, Y.G., General theorems on semicontinuity and on convergence with a functional. Sibir. Math. 8 (1967) 801816. CrossRef
Reshetnyak, Y.G., Weak convergence of completely additive vector functions on a set. Sibir. Math. 9 (1968) 10391045. CrossRef
Y.G. Reshetnyak, Space Mappings with Bounded Distorsion, Translations of Mathathematical Monographs 73. American Mathematical Society, Providence (1989).
E.K.H. Salje, Phase transitions in ferroelastic and co-elastic crystals. Cambridge University Press, Cambridge (1993).
Segev, R., A geometrical framework for the statics of materials with microstructure. Mat. Models Methods Appl. Sci. 4 (1994) 871897. CrossRef
M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media. Springer-Verlag, Berlin (1997).
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.
Tsai, A.P., Guo, J.Q., Abe, E., Takakura, H. and Sato, T.J., Alloys – A stable binary quasicrystals. Nature 408 (2000) 537538. CrossRef

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Ground states in complex bodies
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Ground states in complex bodies
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Ground states in complex bodies
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *