Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T04:14:59.659Z Has data issue: false hasContentIssue false
  • English
  • Français

Conical differentiability for bone remodelingcontact rod models

Published online by Cambridge University Press:  15 July 2005

Isabel N. Figueiredo ,
Carlos F. Leal  and
Cecília S. Pinto
Isabel N. Figueiredo
Affiliation:
Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal; isabel.figueiredo@mat.uc.pt; carlosl@mat.uc.pt
Carlos F. Leal
Affiliation:
Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal; isabel.figueiredo@mat.uc.pt; carlosl@mat.uc.pt
Cecília S. Pinto
Affiliation:
Departamento de Matemática, Escola Superior de Tecnologia de Viseu, Campus Politécnico 3504-510 Viseu, Portugal; cagostinho@mat.estv.ipv.pt
Get access

Abstract

We prove the conical differentiability of the solution to a boneremodeling contact rod model, for given data (applied loads andrigid obstacle), with respect to small perturbations of the crosssection of the rod. The proof is based on the special structure ofthe model, composed of a variational inequality coupled with anordinary differential equation with respect to time. Thisstructure enables the verification of the two followingfundamental results: the polyhedricity of a modified displacementconstraint set defined by the obstacle and the differentiabilityof the two forms associated to the variational inequality.

Keywords

Adaptive elasticityfunctional spacespolyhedric setrod.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 3 , July 2005 , pp. 382 - 400
Copyright
© EDP Sciences, SMAI, 2005

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

P.G. Ciarlet, Mathematical Elasticity, Vol. 1: Three-Dimensional Elasticity. Stud. Math. Appl., North-Holland, Amsterdam 20 (1988).
Cowin, S.C. and Hegedus, D.H., Bone remodeling I: theory of adaptive elasticity. J. Elasticity 6 (1976) 313–326. CrossRef
Cowin, S.C. and Nachlinger, R.R., Bone remodeling III: uniqueness and stability in adaptive elasticity theory. J. Elasticity 8 (1978) 285–295. CrossRef
L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island (1998).
Figueiredo, I.N. and Trabucho, L., Asymptotic model of a nonlinear adaptive elastic rod. Math. Mech. Solids 9 (2004) 331–354. CrossRef
Haraux, A., How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29 (1977) 615–631. CrossRef
Hegedus, D.H. and Cowin, S.C., Bone remodeling II: small strain adaptive elasticity. J. Elasticity 6 (1976) 337–352. CrossRef
Mignot, F., Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22 (1976) 130–185. CrossRef
Monnier, J. and Trabucho, L., An existence and uniqueness result in bone remodeling theory. Comput. Methods Appl. Mech. Engrg. 151 (1998) 539–544. CrossRef
Pierre, M. and Sokolowski, J., Differentiability of projection and applications, E. Casas Ed. Marcel Dekker, New York. Lect. Notes Pure Appl. Math. 174 (1996) 231–240.
Rao, M. and Sokolowski, J., Sensitivity analysis of unilateral problems in $H^2_0(\Omega)$ and applications. Numer. Funct. Anal. Optim. 14 (1993) 125–143. CrossRef
J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Shape Sensitivity Analysis. Springer-Verlag, New York, Springer Ser. Comput. Math. 16 (1992).
L. Trabucho and J.M. Viaño, Mathematical Modelling of Rods, P.G. Ciarlet and J.L Lions Eds. North-Holland, Amsterdam, Handb. Numer. Anal. 4 (1996) 487–974.
T. Valent, Boundary Value Problems of Finite Elasticity. Springer Tracts Nat. Philos. 31 (1988).