Conical differentiability for bone remodeling contact rod models
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 382-400.

We prove the conical differentiability of the solution to a bone remodeling contact rod model, for given data (applied loads and rigid obstacle), with respect to small perturbations of the cross section of the rod. The proof is based on the special structure of the model, composed of a variational inequality coupled with an ordinary differential equation with respect to time. This structure enables the verification of the two following fundamental results: the polyhedricity of a modified displacement constraint set defined by the obstacle and the differentiability of the two forms associated to the variational inequality.

DOI: 10.1051/cocv:2005011
Classification: 49J15,  49J20,  74B20,  74K10,  74L15,  90C31
Keywords: adaptive elasticity, functional spaces, polyhedric set, rod
Figueiredo, Isabel N. ; Leal, Carlos F. ; Pinto, Cecília S. 1

1 Departamento de Matemática, Escola Superior de Tecnologia de Viseu, Campus Politécnico 3504-510 Viseu, Portugal;
@article{COCV_2005__11_3_382_0,
     author = {Figueiredo, Isabel N. and Leal, Carlos F. and Pinto, Cec{\'\i}lia S.},
     title = {Conical differentiability for bone remodeling contact rod models},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {382--400},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {3},
     year = {2005},
     doi = {10.1051/cocv:2005011},
     zbl = {1085.49005},
     mrnumber = {2148850},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2005011/}
}
TY  - JOUR
AU  - Figueiredo, Isabel N.
AU  - Leal, Carlos F.
AU  - Pinto, Cecília S.
TI  - Conical differentiability for bone remodeling contact rod models
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2005
DA  - 2005///
SP  - 382
EP  - 400
VL  - 11
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2005011/
UR  - https://zbmath.org/?q=an%3A1085.49005
UR  - https://www.ams.org/mathscinet-getitem?mr=2148850
UR  - https://doi.org/10.1051/cocv:2005011
DO  - 10.1051/cocv:2005011
LA  - en
ID  - COCV_2005__11_3_382_0
ER  - 
%0 Journal Article
%A Figueiredo, Isabel N.
%A Leal, Carlos F.
%A Pinto, Cecília S.
%T Conical differentiability for bone remodeling contact rod models
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2005
%P 382-400
%V 11
%N 3
%I EDP-Sciences
%U https://doi.org/10.1051/cocv:2005011
%R 10.1051/cocv:2005011
%G en
%F COCV_2005__11_3_382_0
Figueiredo, Isabel N.; Leal, Carlos F.; Pinto, Cecília S. Conical differentiability for bone remodeling contact rod models. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 382-400. doi : 10.1051/cocv:2005011. http://www.numdam.org/articles/10.1051/cocv:2005011/

[1] P.G. Ciarlet, Mathematical Elasticity, Vol. 1: Three-Dimensional Elasticity. Stud. Math. Appl., North-Holland, Amsterdam 20 (1988). | MR | Zbl

[2] S.C. Cowin and D.H. Hegedus, Bone remodeling I: theory of adaptive elasticity. J. Elasticity 6 (1976) 313-326. | Zbl

[3] S.C. Cowin and R.R. Nachlinger, Bone remodeling III: uniqueness and stability in adaptive elasticity theory. J. Elasticity 8 (1978) 285-295. | Zbl

[4] L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island (1998). | MR | Zbl

[5] I.N. Figueiredo and L. Trabucho, Asymptotic model of a nonlinear adaptive elastic rod. Math. Mech. Solids 9 (2004) 331-354. | Zbl

[6] A. Haraux, 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. | Zbl

[7] D.H. Hegedus and S.C. Cowin, Bone remodeling II: small strain adaptive elasticity. J. Elasticity 6 (1976) 337-352. | Zbl

[8] F. Mignot, Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. | Zbl

[9] J. Monnier and L. Trabucho, An existence and uniqueness result in bone remodeling theory. Comput. Methods Appl. Mech. Engrg. 151 (1998) 539-544. | Zbl

[10] M. Pierre and J. Sokolowski, Differentiability of projection and applications, E. Casas Ed. Marcel Dekker, New York. Lect. Notes Pure Appl. Math. 174 (1996) 231-240. | Zbl

[11] M. Rao and J. Sokolowski, Sensitivity analysis of unilateral problems in H 0 2 (Ω) and applications. Numer. Funct. Anal. Optim. 14 (1993) 125-143. | Zbl

[12] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Shape Sensitivity Analysis. Springer-Verlag, New York, Springer Ser. Comput. Math. 16 (1992). | MR | Zbl

[13] 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. | Zbl

[14] T. Valent, Boundary Value Problems of Finite Elasticity. Springer Tracts Nat. Philos. 31 (1988). | MR | Zbl

Cited by Sources: