×

On Timoshenko thin elastic inclusions inside elastic bodies. (English) Zbl 1327.74100

Summary: The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Finally, exemplary networks of Timoshenko beams are considered as inclusions as well.

MSC:

74K30 Junctions
74R10 Brittle fracture
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Hoshino T, Am J Physiol Heart Circ Physiol 297 (2) pp H802– (2009)
[2] Bessoud A-L, Comptes Rendus Mathematiques 346 pp 697– (2008) · Zbl 1138.74034
[3] Bessoud A-L, Asymptotic Anal 61 (1) pp 1– (2009)
[4] Pasternak IM, J Mathem Sci 186 (1) pp 31– (2012)
[5] Savula YH, J Comput Appl Math 5 (1) pp 129– (2004)
[6] Vynnytska L, Comput Mech 50 (5) pp 533– (2012) · Zbl 1312.74014
[7] Grisvard P, Singularities in Boundary Value Problems (1992) · Zbl 0766.35001
[8] Kozlov VA, Proc Royal Soc Edinburgh 117 pp 31– (1991) · Zbl 0728.73057
[9] Nazarov SA, Elliptic Problems in Domains with Piecewise Smooth Boundaries (1991)
[10] Khludnev AM, Analysis of Cracks in Solids (2000)
[11] Khludnev AM, Elasticity Problems in Nonsmooth Domains (2010)
[12] Kovtunenko VA, J Appl Math Mechs 67 (1) pp 109– (2003)
[13] Lazarev NP, J Appl Mech Tech Phys 53 (2) pp 299– (2012) · Zbl 1298.74093
[14] Lazarev NP, Sib Zh Ind Mat 15 (3) pp 58– (2012)
[15] Rudoy EM, Izvestiya RAS, Mech of Solids 6 pp 113– (2007)
[16] Khludnev AM, Europ J Mech A/Solids 29 (3) pp 392– (2010)
[17] Khludnev AM, Izvestiya RAS, Mech of Solids 5 pp 98– (2010)
[18] Khludnev AM, Europ J Mech A/Solids 32 pp 69– (2012) · Zbl 1278.74096
[19] Khludnev AM, Math Method Appl Sci 33 (16) pp 1955– (2010)
[20] Khludnev AM, Z Angew Math Mech 91 (2) pp 125– (2011) · Zbl 1370.74136
[21] Leblond JB, Europ J Mech A/Solids 19 pp 633– (2000) · Zbl 0966.74006
[22] Neustroeva NV, Vestn Novosibirsk State U Math, Mech, Inform 9 (4) pp 51– (2009)
[23] Rotanova TA, Vestn Novosibirsk State U Math, Mech, Inform 11 (1) pp 87– (2011)
[24] Rudoy EM, Appl Math Mechs 75 (2) pp 719– (2011)
[25] DOI: 10.1002/zamm.201100137 · Zbl 1322.74065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.