Geodesic
Differentiation of energy functional with respect to delamination's length in problem of contact between plate and beam
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 935-949.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper considers a model which describes a contact between an elastic plate and an elastic beam. In this model, there may be a delamination, i. e. the displacements of the plate and the beam may not coincide on a part of beam's midline. The purposes of the research are to prove the differentiability of the energy functional with respect to the delamination's length and to find the explicit formula for the corresponding derivative.
Keywords: contact problem, plate, beam, delamination, unilateral constraints, shape sensitivity analysis, energy release rate, energy functional derivative. 
@article{SEMR_2018_15_a96,
     author = {A. I. Furtsev},
     title = {Differentiation of energy functional with respect to delamination's length in problem of contact between plate and beam},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {935--949},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a96/}
}
TY  - JOUR
AU  - A. I. Furtsev
TI  - Differentiation of energy functional with respect to delamination's length in problem of contact between plate and beam
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2018
SP  - 935
EP  - 949
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2018_15_a96/
LA  - ru
ID  - SEMR_2018_15_a96
ER  - 
%0 Journal Article
%A A. I. Furtsev
%T Differentiation of energy functional with respect to delamination's length in problem of contact between plate and beam
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2018
%P 935-949
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2018_15_a96/
%G ru
%F SEMR_2018_15_a96
A. I. Furtsev. Differentiation of energy functional with respect to delamination's length in problem of contact between plate and beam. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 935-949. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a96/

[1] L. A. Caffarelli, A. Friedman, “The obstacle problem for the biharmonic operator”, Ann. Scuola Norm. Sup. Pisa, 6:1 (1979), 151–184 | MR | Zbl

[2] L. A. Caffarelli, A. Friedman, A. Torelli, “The two-obstacle problem for the biharmonic operator”, Pacific J. Math., 103:3 (1982), 325–335 | DOI | MR | Zbl

[3] B. Schild, “On the coincidence set in biharmonic variational inequalities with thin obstacles”, Ann. Scuola Norm. Sup. Pisa, 13:4 (1986), 559–616 | MR | Zbl

[4] G. Dal Maso, G. Paderni, “Variational inequalities for the biharmonic operator with varying obstacles”, Ann. Mat. Pura Appl., 153:1 (1988), 203–227 | DOI | MR | Zbl

[5] A. M. Khludnev, J. Sokolowski, Modelling and control in solid mechanics, Birkhauser Verlag, Basel–Boston–Berlin, 1997 | MR | Zbl

[6] A. M. Khludnev, K.-H. Hoffmann, N. D. Botkin, “The variational contact problem for elastic objects of different dimensions”, Sib. Math. J., 47:3 (2006), 584–593 | DOI | MR | Zbl

[7] A. M. Khludnev, “On unilateral contact of two plates aligned an angle to each other”, J. Appl. Mech. and Tech. Phys., 49:4 (2008), 553–-567 | DOI | MR | Zbl

[8] A. M. Khludnev, G. Leugering, “Unilateral contact problems for two perpendicular elastic structures”, J. Analysis and its Applications, 27:2 (2008), 157–177 | MR | Zbl

[9] A. M. Khludnev, A. Tani, “Unilateral contact problem for two inclined elastic bodies”, Eur. J. Mech. A Solids, 27:3 (2008), 365–377 | DOI | MR | Zbl

[10] T. A. Rotanova, “On the statements and solvability of the problems on the contact of two plates containing rigid inclusions”, Sib. Zh. Ind. Mat., 15:2 (2012), 107–118 (in Russian) | MR | Zbl

[11] N. V. Neustroeva, “Unilateral Contact of Elastic Plates to Rigid Inclusion”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 9:4 (2009), 51–64 (in Russian) | Zbl

[12] A. I. Furtsev, “On contact of thin obstacle and plate, containing thin inclusion”, Sib. J. Pure and Applied Math., 17:4 (2017), 94–111 (in Russian)

[13] A. M. Khludnev, V. A. Kovtunenko, Analysis of Cracks in Solids, WIT Press, Southampton–Boston, 2000

[14] A. M. Khludnev, “Theory of cracks with a possible contact of the crack edges”, Usp. Mekh., 3:4 (2005), 41–82

[15] J. Sokolowski, J. P. Zolesio, Introduction to shape optimization, Springer Verlag, Berlin, 1992 | MR | Zbl

[16] J. Sokolowski, A. A. Novotny, Topological derivatives in shape optimization, Springer Science Business Media, 2012 | MR

[17] A. M. Khludnev, J. Sokolowski, “Griffith formulae for elasticity systems with unilateral conditions in domains with cracks”, Eur. J. Mech. A Solids, 19:1 (2000), 105–119 | DOI | MR | Zbl

[18] M. Bach, A. M. Khludnev, V. A. Kovtunenko, “Derivatives of the energy functional for 2D-problems with a crack under signorini and friction conditions”, Math. Meth. Appl. Sci., 23:6 (2000), 515–534 | 3.0.CO;2-S class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[19] E. M. Rudoy, “The Griffiths formula for a plate with a crack”, Sib. Zh. Ind. Mat., 5:3 (2002), 155–161 (in Russian) | MR

[20] V. A. Kovtunenko, “Nonconvex problem for crack with nonpenetration”, Z. Angew. Math. Mech., 85:4 (2005), 242–251. MR2131983 | DOI | MR | Zbl

[21] V. A. Kovtunenko, “Primal–dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration”, J. Appl. Math., 71:5 (2006), 635–657 | MR | Zbl

[22] E. M. Rudoy, “Differentiation of energy functionals in the problem on a curvilinear crack with possible contact between the shores”, Mech. Solids, 42:6 (2008), 935–946 | DOI | MR

[23] E. M. Rudoy, “Differentiation of energy functionals in the problem of a curvilinear crack in a plate with a possible contact of the crack faces”, J. Appl. Mech. Tech. Phys., 49:5 (2008), 832–845 | DOI | MR

[24] N. P. Lazarev, E. M. Rudoy, “Shape sensitivity analysis of Timoshenko’s plate with a crack under the nonpenetration condition”, Z. Angew. Math. Mech., 94:9 (2014), 730–739 | DOI | MR | Zbl

[25] E. M. Rudoy, “The Griffith formula and Cherepanov–Rice integral for a plate with a rigid inclusion and a crack”, J. Math. Sci., 186:3 (2012), 511–529 | DOI | MR

[26] N. P. Lazarev, “Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion”, Z. Angew. Math. Phys., 66:4 (2015), 2025–2040 | DOI | MR | Zbl

[27] E. M. Rudoy, “Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body”, Z. Angew. Math. Phys., 66:4 (2015), 1923–1937 | DOI | MR | Zbl

[28] V. V. Shcherbakov, “The Griffith formula and J-integral for elastic bodies with Timoshenko inclusions”, Z. Angew. Math. Mech., 96:11 (2016), 1306–1317 | DOI | MR

[29] A. M. Khludnev, V. V. Shcherbakov, “Singular path-independent energy integrals for elastic bodies with Euler–Bernoulli inclusions”, Math. Mech. Solids, 22:11 (2017), 2180–2195 | DOI | MR | Zbl

[30] V. V. Shcherbakov, “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions”, Z. Angew. Math., 68:1 (2017), 26 | DOI | MR | Zbl