Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T07:15:30.441Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence theorem for nonlinear micropolar elasticity

Published online by Cambridge University Press:  21 October 2008

Josip Tambača and
Igor Velčić
Josip Tambača
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia. tambaca@math.hr; ivelcic@math.hr
Igor Velčić
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia. tambaca@math.hr; ivelcic@math.hr
Get access

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.

Keywords

Micropolar elasticityexistence theoremquasiconvexity

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010, pp. 92 - 110
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.)

Article purchase

Temporarily unavailable

References

Aganović, I., Tambača, J. and Tutek, Z., Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. J. Elasticity 84 (2006) 131–152. CrossRef
A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993).
J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976/1977) 337–403.
P.G. Ciarlet, Mathematical elasticity – Volume I: Three-dimensional elasticity. North-Holland Publishing Co., Amsterdam (1988).
E. Cosserat and F. Cosserat, Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils [Theory of deformable bodies], Paris (1909).
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin (1989).
A.C. Eringen, Microcontinuum Field Theories – Volume 1: Foundations and Solids. Springer-Verlag, New York (1999).
G.B. Folland, Real analysis, Modern techniques and their applications. John Wiley & Sons, Inc., New York (1984).
Hlaváček, I. and Hlaváček, M., 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.
J. Jeong and P. Neff, Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids (2008) DOI: 10.1177/1081286508093581. Preprint 2550 available at http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html.
P.M. Mariano and G. Modica, Ground states in complex bodies. ESAIM: COCV (2008) published online, DOI: 10.1051/cocv:2008036.
Meyers, N.G., Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125–149. CrossRef
Neff, P., Korn's, On first inequality with nonconstant coefficients. Proc. R. Soc. Edinb. Sect. A 132 (2002) 221–243. CrossRef
Neff, P., Existence of minimizers for a geometrically exact Cosserat solid. Proc. Appl. Math. Mech. 4 (2004) 548–549. CrossRef
Neff, P., 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. Thermodynamics 16 (2004) 577–628. CrossRef
Neff, P., 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 http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html. CrossRef
Neff, P., Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A 136 (2006) 997–1012. Preprint 2318 available at http://wwwbib.mathematik.tu-darmstadt.de/Math-Net/Preprints/Listen/pp04.html. CrossRef
Neff, P., A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci. 44 (2006) 574–594. CrossRef
Neff, P., 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 http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html. CrossRef
Neff, P. and Chelminski, K., A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via $\Gamma$ -convergence. Interfaces Free Boundaries 9 (2007) 455–492. CrossRef
Neff, P. and Forest, S., A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elasticity 87 (2007) 239–276. CrossRef
Neff, P. and Münch, I., Curl bounds Grad on SO(3). ESAIM: COCV 14 (2008) 148–159. Preprint 2455 available at http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html. CrossRef
W. Nowacki, Theory of asymmetric elasticity. Oxford, Pergamon (1986).
Pompe, W., Korn's first inequality with variable coefficients and its generalizations. Commentat. Math. Univ. Carolinae 44 (2003) 57–70.
J. Tambača and I. Velčić, Derivation of a model of nonlinear micropolar plate. (Submitted).