Flexible belt hanging on two pulleys: Contact problem at non-material kinematic description

For two-and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two-and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The twoand three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory. The author acknowledges the support of the Government of the Russian Federation (Contract No. 14.Z50.31.0046).

Section: Introductionmentioning

confidence: 99%

Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

Eremeyev

2019

“…In the contact problem for strings and rods, particular attention should be paid to the transition conditions at the boundaries between the contact and free zones, which we call “touching points” in the following. It is known that concentrated contact forces at these touching points result with Kirchhoff (shear undeformable) rod models even in static problems with frictionless contact; see [2, 5, 6, 11, 19, 32]. The perfect adhesion (no-slip) condition between the moving belt and the surface of a rotating pulley also results in concentrated tangential contact forces in the case of a string model of the belt [14, 30, 33].…”

Section: Introductionmentioning

confidence: 99%

Steady state motion of a shear deformable beam in contact with a traveling surface

Oborin

Vetyukov

2019

Self Cite

A shear deformable beam moving along a straight path is considered as an idealization of the problem of stationary operation of a belt drive. The partial contact with a traveling surface results in the shear deformation of the beam. The tangential contact force grows near the end of the contact zone. Assuming perfect adhesion of the lower fiber of the beam to the traveling surface (no slip), we analytically demonstrate the necessity of accounting for concentrated contact forces and jump conditions, which is important for modeling the belt–pulley interaction. Along with dynamic effects, we further consider a frictional model with zones of stick and slip contact and demonstrate its convergence to the results with perfect adhesion at growing maximal friction force.

“…Many essentially nonlinear solutions to problems of in-plane deformations of rod structures are available in the literature. Thus, a contact of a closed elastic rod with cylindrical pulleys is analyzed in [14][15][16]. Denoël and Detournay [17] used Eulerian kinematic formulation in the problem of deformation of an elastica within a channel contacting both its walls.…”

Section: Introductionmentioning

confidence: 99%

A rod model for large bending and torsion of an elastic strip with a geometrical imperfection

Vetyukov

Schmidrathner

2019

Self Cite

We consider an initially horizontal curved elastic strip, which bends and twists under the action of the varying length of the span between the clamped ends and of the gravity force. Equations of the theory of rods, linearized in the vicinity of a largely pre-deformed state, allow for semi-analytical (or sometimes closedform) solutions. A nonlinear boundary value problem determines the vertical bending of a perfect beam, while the small natural curvature additionally leads to torsion and out-of-plane deflections described by the linear equations of the incremental theory. Numerical experiments demonstrate perfect correspondence to the finite element rod model of the strip. Comparisons to the predictions of the shell model allow estimating the range of applicability of the three-dimensional theory of rods. Practically relevant conclusions follow for the case of high pre-tension of the strip.