Abstract
We discuss the equations of elastostatics in a Riemannian manifold, which generalize those of classical elastostatics in the three-dimensional Euclidean space. Assuming that the deformation of an elastic body arising in response to given loads should minimize over a specific set of admissible deformations the total energy of the elastic body, we derive the equations of elastostatics in a Riemannian manifold first as variational equations, then as a boundary value problem. We then show that this boundary value problem possesses a solution if the loads are sufficiently small in a specific sense. The proof is constructive and provides an estimation for the size of the loads.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Abraham R, Marsden JE, Ratiu T (1988) Manifolds, tensor analysis, and applications. Springer, New York
Amrouche C, Girault V (1994) Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czech Math J 44:109–140
Andersson L, Beig R, Schmidt BG (2008) Static self-gravitating elastic bodies in Einstein gravity. Commun Pure Appl Math LXI:0988–1023
Aubin T (2010) Some nonlinear problems in Riemannian geometry. Springer, Berlin
Beig R, Schmidt BG (2003) Relativistic elasticity. Class Quantum Gravity 20:889–904
Beig R, Schmidt BG (2003) Static, self-gravitating elastic bodies. Proc R Soc Lond A 459:109–115
Beig R, Schmidt BG (2005) Relativistic elastostatics. I. Bodies in rigid rotation. Class Quantum Gravity 22:2249–2268
Beig R, Wernig-Pichler M (2007) On the motion of a compact elastic body. Commun Math Phys 271:455–465
Carter B, Quintana H (1972) Foundations of general relativistic high-pressure elasticity theory. Proc R Soc Lond A 331:57–83
Chen W, Jost J (2002) A Riemannian version of Korn’s inequality. Calc Var 14:517–530
Ciarlet PG (1988) Mathematical elasticity, volume I: three-dimensional elasticity. North-Holland, Amsterdam
Ciarlet PG (2005) An introduction to differential geometry with applications to elasticity. Springer, Dordrecht
Ciarlet PG, Mardare C (2012) On the Newton-Kantorovich theorem. Anal Appl 10:249–269
Duvaut G, Lions JL (1978) Inequalities in mechanics and physics. Springer, New York
Efrati E, Sharon E, Kupferman R (2009) Elastic theory of unconstrained non-euclidean plates. J Mech Phys Solids 57:762–775
Epstein M, Segev R (1980) Differentiable manifolds and the principle of virtual work in continuum mechanics. J Math Phys 21:1243–1245
Grubic N, LeFloch PG, Mardare C (2014) Mathematical elasticity theory in a Riemannian manifold. J Math Pures Appl 102:1121–1163
Kondrat’ev VA, Oleinik OA (1989) On the dependence of the constant in Korn’s inequality on parameters characterising the geometry of the region. Uspekhi Mat Nauk 44:153–160 (Russ Math Surv 44:187–195)
Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Prentice-Hall, New Jersey
Nomizu K (1960) On local and global existence of Killing vector fields. Ann Math 72:105–120
Segev R (1986) Forces and the existence of stresses in invariant continuum mechanics. J Math Phys 27:163–170
Segev R (2000) The geometry of Cauchy’s fluxes. Arch Ration Mech Anal 154:183–198
Segev R (2001) A correction of an inconsistency in my paper: Cauchy’s theorem on manifolds. J Elast 63:55–59
Segev R, Rodnay G (1999) Cauchy’s theorem on manifolds. J Elast 56:129–144
Simpson HC, Spector SJ (2009) Applications of estimates near the boundary to regularity of solutions in linearized elasticity. SIAM J Math Anal 41:923–935
Valent T (1988) Boundary value problems of finite elasticity: local theorems on existence, uniqueness, and analytic dependence on data. Springer, New York
Wernig-Pichler M (2006) Relativistic elastodynamics. arXiv:gr-qc/0605025v1
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Mardare, C. (2015). Static Elasticity in a Riemannian Manifold. In: Chen, GQ., Grinfeld, M., Knops, R. (eds) Differential Geometry and Continuum Mechanics. Springer Proceedings in Mathematics & Statistics, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-319-18573-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-18573-6_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18572-9
Online ISBN: 978-3-319-18573-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)