TY - JOUR

T1 - Asymptotic analysis of an elastic rod with rounded ends

AU - Nazarov, Sergey A.

AU - Slutskij, Andrey S.

AU - Taskinen, Jari

PY - 2020/7/15

Y1 - 2020/7/15

N2 - We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4.

AB - We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4.

KW - elliptic equations and systems

KW - Korn inequality

KW - linear elasticity system

KW - mechanics of deformable solids

KW - roundness exponent

KW - thin rod

KW - EQUATIONS

UR - http://www.scopus.com/inward/record.url?scp=85083796154&partnerID=8YFLogxK

U2 - 10.1002/mma.6380

DO - 10.1002/mma.6380

M3 - Article

AN - SCOPUS:85083796154

VL - 43

SP - 6396

EP - 6415

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 10

ER -