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Modeling of the junction between two rods. (English) Zbl 0743.73020

In the present paper the problem of modeling the junction of two linear elastic straight slender rods made of homogeneous isotropic material meeting at a right angle is studied. It is shown that the solution for the associated three-dimensional (\(3D\)) model converges in an appropriate sense towards the solution of a \(1D\) model formed by two fourth-order rod equations coupled by a set of junction conditions, provided that the thickness goes to 0. The solution of this problem is based on the idea of dual scaling, i.e. independent scaling of both rods forming the \(L\)- shaped junction. Then new unknowns that consist of a pair of scaled displacements defined on each rod are considered, which satisfy an appropriate \(H^ 1\)-bound. This allows to find one-dimensional limit displacements of Bernoulli-Navier type for each rod, which in turn satisfy the equations of an associated limit variational problem. As compared to \(3D-2D\) and \(2D-2D\) junctions the analyzed model problem of a \(1D-1D\) junction requires a thorough investigation of torsional effects. The results of the paper may be generalized to other geometries of junctions. A brief outline is given for the special cases of two rods with circular cross-section, junctions at an arbitrary angle, and junctions of three rods.

MSC:

74E30 Composite and mixture properties
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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