# Existence theorem for nonlinear micropolar elasticity

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 1, page 92-110
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topTambača, Josip, and Velčić, Igor. "Existence theorem for nonlinear micropolar elasticity." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 92-110. <http://eudml.org/doc/250735>.

@article{Tambača2010,

abstract = {
In this paper we give an existence theorem for the equilibrium problem for nonlinear micropolar elastic body. We consider the problem in its minimization formulation and apply the direct methods of the calculus of variations. As the main step towards the existence theorem, under some conditions, we prove the equivalence of the sequential weak lower semicontinuity of the total energy and the quasiconvexity, in some variables, of the stored energy function.
},

author = {Tambača, Josip, Velčić, Igor},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Micropolar elasticity; existence theorem; quasiconvexity; micropolar elasticity},

language = {eng},

month = {1},

number = {1},

pages = {92-110},

publisher = {EDP Sciences},

title = {Existence theorem for nonlinear micropolar elasticity},

url = {http://eudml.org/doc/250735},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Tambača, Josip

AU - Velčić, Igor

TI - Existence theorem for nonlinear micropolar elasticity

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/1//

PB - EDP Sciences

VL - 16

IS - 1

SP - 92

EP - 110

AB -
In this paper we give an existence theorem for the equilibrium problem for nonlinear micropolar elastic body. We consider the problem in its minimization formulation and apply the direct methods of the calculus of variations. As the main step towards the existence theorem, under some conditions, we prove the equivalence of the sequential weak lower semicontinuity of the total energy and the quasiconvexity, in some variables, of the stored energy function.

LA - eng

KW - Micropolar elasticity; existence theorem; quasiconvexity; micropolar elasticity

UR - http://eudml.org/doc/250735

ER -

## References

top- I. Aganović, J. Tambača and Z. Tutek, Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. J. Elasticity84 (2006) 131–152. Zbl1158.74401
- A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993). Zbl0781.47046
- J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1976/1977) 337–403. Zbl0368.73040
- P.G. Ciarlet, Mathematical elasticity – Volume I: Three-dimensional elasticity. North-Holland Publishing Co., Amsterdam (1988). Zbl0648.73014
- E. Cosserat and F. Cosserat, Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils [Theory of deformable bodies], Paris (1909). Zbl40.0862.02
- B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin (1989). Zbl0703.49001
- A.C. Eringen, Microcontinuum Field Theories – Volume 1: Foundations and Solids. Springer-Verlag, New York (1999). Zbl0953.74002
- G.B. Folland, Real analysis, Modern techniques and their applications. John Wiley & Sons, Inc., New York (1984). Zbl0549.28001
- I. Hlaváček and M. Hlaváček, On the existence and uniqueness of solution and some variational principles in linear theories of elasticity with couple-stresses. I. Cosserat continuum. Appl. Math.14 (1969) 387–410. Zbl0195.27003
- J. Jeong and P. Neff, Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids (2008) DOI: . Preprint 2550 available at . Zbl1197.74009DOI10.1177/1081286508093581
- P.M. Mariano and G. Modica, Ground states in complex bodies. ESAIM: COCV (2008) published online, DOI: . Zbl1161.74006DOI10.1051/cocv:2008036
- N.G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc.119 (1965) 125–149. Zbl0166.38501
- P. Neff, On Korn's first inequality with nonconstant coefficients. Proc. R. Soc. Edinb. Sect. A132 (2002) 221–243. Zbl1143.74311
- P. Neff, Existence of minimizers for a geometrically exact Cosserat solid. Proc. Appl. Math. Mech.4 (2004) 548–549.
- P. Neff, A geometrically exact Cosserat-shell model including size effects, avoiding degeneracy in the thin shell limit, Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Cont. Mech. Thermodynamics16 (2004) 577–628. Zbl1158.74419
- P. Neff, The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Z. Angew. Math. Mech.86 (2006) 892–912. Preprint 2409 available at . Zbl1104.74007URIhttp://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html
- P. Neff, Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A136 (2006) 997–1012. Preprint 2318 available at http://wwwbib.mathematik.tu-darmstadt.de/Math-Net/Preprints/Listen/pp04.html. Zbl1106.74010
- P. Neff, A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci.44 (2006) 574–594. Zbl1213.74032
- P. Neff, A geometrically exact planar Cosserat shell-model with microstructure. Existence of minimizers for zero Cosserat couple modulus. Math. Meth. Appl. Sci.17 (2007) 363–392. Preprint 2357 available at . Zbl1119.74029URIhttp://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html
- P. Neff and K. Chelminski, A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via $\Gamma $-convergence. Interfaces Free Boundaries9 (2007) 455–492. Zbl1137.74014
- P. Neff and S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elasticity87 (2007) 239–276. Zbl1206.74019
- P. Neff and I. Münch, Curl bounds Grad on SO(3). ESAIM: COCV14 (2008) 148–159. Preprint 2455 available at . Zbl1139.74008URIhttp://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html
- W. Nowacki, Theory of asymmetric elasticity. Oxford, Pergamon (1986). Zbl0604.73020
- W. Pompe, Korn's first inequality with variable coefficients and its generalizations. Commentat. Math. Univ. Carolinae44 (2003) 57–70. Zbl1098.35042
- J. Tambača and I. Velčić, Derivation of a model of nonlinear micropolar plate. (Submitted). Zbl1195.74106

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.