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Derivation of the nonlinear bending–torsion model for a junction of elastic rods

Published online by Cambridge University Press:  07 June 2012

Josip Tambača
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (tambaca@math.hr; ivelcic@math.hr)
Igor Velčić
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (tambaca@math.hr; ivelcic@math.hr)

Abstract

We derive the one-dimensional bending–torsion equilibrium model for the junction of straight rods. The starting point is a three-dimensional nonlinear elasticity equilibrium problem written as a minimization problem for a union of thin, rod-like bodies. By taking the limit as the thickness of the three-dimensional rods tends to zero, and by using ideas from the theory of Γ-convergence, we find that the resulting model consists of the union of the usual one-dimensional nonlinear bending–torsion rod models which satisfy the following transmission conditions at the junction point: continuity of displacement and rotation of the cross-sections; balance of contact forces and contact couples.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2012

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