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Uspekhi Mat. Nauk, 2008, Volume 63, Issue 1(379), Pages 37–110 (Mi umn8545)  

This article is cited in 41 scientific papers (total in 41 papers)

Korn inequalities for elastic junctions of massive bodies, thin plates, and rods

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: Korn inequalities have been obtained for junctions of massive elastic bodies, thin plates, and rods in many different combinations. These inequalities are asymptotically sharp thanks to the introduction of various weight factors in the $L_2$-norms of the displacements and their derivatives. Since thin bodies display different reactions to stretching and bending, such Korn inequalities are necessarily anisotropic. Junctions of elastic bodies with contrasting stiffness are allowed, but the constants in the inequalities obtained are independent of both the relative thickness $h\in(0,1]$ and the relative rigidity $\mu\in(0,+\infty)$. The norms corresponding to rigidly clamped elements of a structure are essentially different from the norms corresponding to hard-movable or movable elements that are not fastened directly, but only by means of neighbouring elements; therefore, an adequate structure of the weighted anisotropic norms is determined by the geometry of the whole junction. Each variant of Korn inequality is supplied with an example confirming the optimal choice of the weight factors.


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English version:
Russian Mathematical Surveys, 2008, 63:1, 35–107

Bibliographic databases:

UDC: 517.946
MSC: Primary 74K30; Secondary 35B45, 35Q72, 74B05, 74E10, 74K10, 74K20
Received: 15.10.2007

Citation: S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Uspekhi Mat. Nauk, 63:1(379) (2008), 37–110; Russian Math. Surveys, 63:1 (2008), 35–107

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper Korn inequalities for elastic junctions of massive bodies, thin plates, and rods
\jour Uspekhi Mat. Nauk
\yr 2008
\vol 63
\issue 1(379)
\pages 37--110
\jour Russian Math. Surveys
\yr 2008
\vol 63
\issue 1
\pages 35--107

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    This publication is cited in the following articles:
    1. Leugering G., Nazarov S.A., Slutskij A.S., Taskinen J., “Asymptotic Analysis of a BIT Brace Shaped Junction of Thin Rods”, ZAMM-Z. Angew. Math. Mech., UNSP e201900227  crossref  isi
    2. S. A. Nazarov, “Rayleigh waves in an elastic half-layer with partly jammed periodic boundary”, Dokl. Phys., 53:11 (2008), 600–604  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    3. Nazarov S.A., “A gap in the continuous spectrum of an elastic waveguide”, C. R., Méc., Acad. Sci. Paris, 336:10 (2008), 751–756  crossref  zmath  isi  elib  scopus
    4. S. A. Nazarov, “Opening a gap in the essential spectrum of the elasticity problem in a periodic semi-layer”, St. Petersburg Math. J., 21:2 (2010), 281–307  mathnet  crossref  mathscinet  zmath  isi
    5. S. A. Nazarov, “A Gap in the Essential Spectrum of the Neumann Problem for an Elliptic System in a Periodic Domain”, Funct. Anal. Appl., 43:3 (2009), 239–241  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. G. Cardone, A. Corbo Esposito, S. A. Nazarov, “Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain”, St. Petersburg Math. J., 21:4 (2010), 601–634  mathnet  crossref  mathscinet  zmath  isi
    7. Cardone G., Nazarov S.A., Taskinen J., “A criterion for the existence of the essential spectrum for beak-shaped elastic bodies”, J. Math. Pures Appl. (9), 92:6 (2009), 628–650  crossref  mathscinet  zmath  isi  elib  scopus
    8. S. A. Nazarov, G. H. Sweers, A. S. Slutskij, “Plate reinforcement with periodic families of disconnected rigid rods”, Dokl. Phys., 54:8 (2009), 397–401  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    9. S. A. Nazarov, K. Ruotsalainen, J. Taskinen, “Gaps in the essential spectrum of infinite periodic necklace-shaped elastic waveguide”, C. R. Méc. Acad. Sci. Paris, 337:3 (2009), 119–123  crossref  isi  elib  scopus
    10. S. A. Nazarov, “Gap detection in the spectrum of an elastic periodic waveguide with a free surface”, Comput. Math. Math. Phys., 49:2 (2009), 323–333  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    11. Durante T., Cardone G., Nazarov S.A., “Modeling junctions of plates and beams by means of self-adjoint extensions”, Vestn. St. Petersbg. Univ. Math., 42:2 (2009), 67–75  crossref  mathscinet  mathscinet  zmath  elib
    12. Cardone G., Corbo Esposito A., Nazarov S.A., “Korn's inequality for periodic solids and convergence rate of homogenization”, Appl. Anal., 88:6 (2009), 847–876  crossref  mathscinet  zmath  isi  elib
    13. G. V. Alekseev, A. M. Khludnev, “Treschina v uprugom tele, vykhodyaschaya na granitsu pod nulevym uglom”, Vestn. NGU. Ser. matem., mekh., inform., 9:2 (2009), 15–29  mathnet
    14. S. A. Nazarov, K. Ruotsalainen, J. Taskinen, “Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps”, Appl. Anal., 89:1 (2010), 109–124  crossref  mathscinet  zmath  isi  elib  scopus
    15. S. A. Nazarov, “An example of multiple gaps in the spectrum of a periodic waveguide”, Sb. Math., 201:4 (2010), 569–594  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. Nazarov S.A., “Gap in a continuous spectrum of an elastic waveguide with a partly clamped surface”, J. Appl. Mech. Tech. Phys., 51:1 (2010), 114–124  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    17. Nazarov S.A., Sweers G.H., Slutskii A.S., “The flexural rigidity of a thin plate reinforced with periodic systems of separated rods”, J. Appl. Math. Mech., 74:3 (2010), 313–322  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    18. Campbell A., Nazarov S.A., Sweers G.H., “Spectra of two-dimensional models for thin plates with sharp edges”, SIAM J. Math. Anal., 42:6 (2010), 3020–3044  crossref  mathscinet  zmath  isi  elib  scopus
    19. Nazarov S.A., Sokolowski J., “On asymptotic analysis of spectral problems in elasticity”, Latin American Journal of Solids and Structures, 8:1 (2011), 27–54  crossref  mathscinet  isi  scopus
    20. S. A. Nazarov, G. H. Sweers, A. S. Slutskij, “Homogenization of a thin plate reinforced with periodic families of rigid rods”, Sb. Math., 202:8 (2011), 1127–1168  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Nazarov S.A., “Localized elastic fields in periodic waveguides with defects”, J. Appl. Mech. Tech. Phys., 52:2 (2011), 311–320  crossref  mathscinet  zmath  adsnasa  isi  isi  elib  elib  scopus
    22. Nazarov S.A., “Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps”, J. Math. Sci., 181:5 (2012), 632–667  crossref  mathscinet  zmath  elib
    23. Nazarov S.A., Slutskij A.S., Sweers G.H., “Korn inequalities for a reinforced plate”, J. Elasticity, 106:1 (2012), 43–69  crossref  mathscinet  zmath  isi  scopus
    24. S. A. Nazarov, “Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler”, Siberian Math. J., 53:2 (2012), 274–290  mathnet  crossref  mathscinet  isi
    25. S. A. Nazarov, “Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle”, Izv. Math., 77:1 (2013), 87–142  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    26. S. A. Nazarov, A. S. Slutskij, Jari Taskinen, “Korn inequality for a thin rod with rounded ends”, Math. Meth. Appl. Sci, 2014, n/a  crossref  mathscinet  isi
    27. S. A. Nazarov, “Umov-Mandelshtam radiation conditions in elastic periodic waveguides”, Sb. Math., 205:7 (2014), 953–982  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. R. Bunoiu, G. Cardone, S. A. Nazarov, “Scalar boundary value problems on junctions of thin rods and plates”, ESAIM: M2AN, 48:5 (2014), 1495  crossref  mathscinet  zmath  isi
    29. Yury Grabovsky, Davit Harutyunyan, “Exact Scaling Exponents in Korn and Korn-Type Inequalities for Cylindrical Shells”, SIAM J. Math. Anal, 46:5 (2014), 3277  crossref  mathscinet  zmath  isi  scopus
    30. Bare D.Z., Orlik J., Panasenko G., “Asymptotic Dimension Reduction of a Robin-Type Elasticity Boundary Value Problem in Thin Beams”, Appl. Anal., 93:6 (2014), 1217–1238  crossref  mathscinet  zmath  isi  scopus
    31. Yury Grabovsky, Davit Harutyunyan, “Rigorous Derivation of the Formula for the Buckling Load in Axially Compressed Circular Cylindrical Shells”, J Elast, 2015  crossref  mathscinet  isi  scopus
    32. S.A.. Nazarov, Jari Taskinen, “Spectral gaps for periodic piezoelectric waveguides”, Z. Angew. Math. Phys, 2015  crossref  mathscinet  isi  scopus
    33. Yu. S. Naishtut, “Reshenie kraevykh zadach teorii tonkikh uprugikh obolochek metodom Neimana”, Kompyuternye issledovaniya i modelirovanie, 7:6 (2015), 1143–1153  mathnet  crossref
    34. Lazarev N.P., Itou H., Neustroeva N.V., “Fictitious Domain Method For An Equilibrium Problem of the Timoshenko-Type Plate With a Crack Crossing the External Boundary At Zero Angle”, 33, no. 1, 2016, 63–80  crossref  mathscinet  zmath  isi  scopus
    35. Harutyunyan D., “Sharp Weighted Korn and Korn-Like Inequalities and an Application to Washers”, J. Elast., 127:1 (2017), 59–77  crossref  mathscinet  zmath  isi  scopus
    36. Nazarov S.A., Slutskij A.S., “A Folded Plate Clamped Along One Side Only”, C. R. Mec., 345:12 (2017), 903–907  crossref  isi
    37. Grabovsky Yu., Harutyunyan D., “Korn Inequalities For Shells With Zero Gaussian Curvature”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 35:1 (2018), 267–282  crossref  mathscinet  zmath  isi  scopus
    38. Kozlov V.A. Nazarov S.A., “Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies”, J. Elast., 132:1 (2018), 103–140  crossref  mathscinet  zmath  isi  scopus
    39. Nazarov S.A., Slutskii A.S., “Asymptotics of Natural Oscillations of Elastic Junctions With Readily Movable Elements”, Mech. Sol., 53:1 (2018), 101–115  crossref  isi  scopus
    40. Leugering G., Nazarov S.A., Slutskij A.S., “The Asymptotic Analysis of a Junction of Two Elastic Beams”, ZAMM-Z. Angew. Math. Mech., 99:1 (2019), UNSP e201700192  crossref  isi  scopus
    41. Harutyunyan D. Mikayelyan H., “Weighted Asymptotic Korn and Interpolation Korn Inequalities With Singular Weights”, Proc. Amer. Math. Soc., 147:8 (2019), 3635–3647  crossref  isi
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