The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).     

A model of a general elastic curved rod

We indicate a new approach to the deformation of three‐dimensional curved rods with variable cross‐section. The model consists of a system of nine ordinary differential equations for which we prove existence and uniqueness via the coercivity of the association bilinear form. From the geometrical point of view, we are using the Darboux frame or a new local frame requiring just a C¹‐parameterization of the curve. Our model also describes the deformation occurring in the cross‐sections of the rod. Copyright © 2002 John Wiley & Sons, Ltd.     

On a model of viscoelastic rod in unilateral contact with a rigid wall

^** Corresponding author. Email: atanackovic{at}uns.ns.ac.yu We study translatory motion of a body to which a viscoelasticrod with the constitutive equation with fractional derivativesis attached. The body with a rod impacts against a rigid wall.It is shown that the problem is described with a coupled systemof differential equations having integer and fractional derivativeshaving the form x⁽²⁾ = –f; f + af⁽⁾ = x + bx⁽⁾, x(0) =0, x⁽¹⁾(0) = 1. The unique solvability in S'₊ is proved andinterpretation of solutions is given. Also, some a priori estimatesof the solution are given. In particular, we showed that restrictionson coefficients that follow from the second law of thermodynamicsimply that the velocity after the impact is smaller than thevelocity before the impact.     

A variational characterization of a hyperelastic rod with hard self-contact

We consider an elastic rod, modeled as a curve in space with an impenetrable surrounding tube of radius ρ, subject to a general class of boundary conditions. The impossibility of self-intersection is then imposed as a family of scalar constraints on the physical separation of nonlocal pairs of points on the rod. Thus, the usual variational formulation of energy minimization is considered in a context of nonconvex, nonsmooth optimization. We show the existence of minimizers within suitably defined homotopy classes associated with both the centerline and the frame along the rod. The principle results are then concerned with derivation of first-order necessary conditions for optimality and some consequences of these for the contact forces and for regularity.     

A new model for the analysis of reinforced concrete members with a coupled HdBNM/FEM

As a truly boundary-type meshless method, the hybrid boundary node method (HdBNM) does not require ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. In this paper, the HdBNM is coupled with the finite element method (FEM) for predicting the mechanical behaviors of reinforced concrete. The steel bars are considered as body forces in the concrete. A bond model is presented to simulate the bond-slip between the concrete and steels using fictitious spring elements. The computational scale and cost for meshing can be further reduced. Numerical examples, in 2D and 3D cases, demonstrate the efficiency of the proposed approach.     

The rolling of a wheel with a pneumatic tyre on a planet

A model of a pneumatic tyre as a system with an infinite number of degrees of freedom is proposed, when its surface is represented by the deformed surface of a torus. Using a number of hypotheses a functional of the potential energy of the deformations of the tyre is obtained as a function of the deformations of its tread. A complete system of equations of motion is obtained, assuming that the wheel rolls without slipping in the area of contact of the tread with the plane, with respect to the previously unknown part of the tread. In two special cases of the rolling of a wheel with breakaway and on a banking, all the characteristics of the motion (the contact area, the tyre deformation, and the forces and moments applied to the disc of the wheel) are obtained.     

A consistent model for adhesion: The case of the rod

A one-dimensional curvilinear model of adhesion is established in the framework of continuum mechanics and thermodynamics. In addition to the usual kinematic variables, the contact zone is characterized by an internal variable b\beta, representing the intensity of adhesion. The model is constructed by using a first gradient theory associated with the principle of virtual powers. An application is given showing the qualitative behavior of the debonding process.     

A constitutive model for continuous fiber reinforced polymer plies

In the present paper a constitutive model is reviewed which can be used to predict the non-linear behavior of continuous fiber reinforced laminates with polymeric matrix materials. The constitutive model considers stiffness degradation and plastic strain accumulation at the length scale of the individual plies (laminae). These effects are modeled via two different phenomenological approaches, however, their interaction is considered when the constitutive equations are solved by an implicit integration scheme. To demonstrate the predictive capabilities of the individual model parts, examples are given where the above mentioned effects are decoupled. This way, their impact on the laminate's response can be studied independently. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.     

The vibrations of a free and loaded tyre

Using the model of a wheel with a reinforced tyre [Vil’ke VG, Kozhevnikov IF. A model of a wheel with a reinforced tyre. Vestnik MGU. Ser.1. Matematika Mekhanika 2004;4:37–45], the natural frequencies and natural forms of vibrations of a free or loaded tyre in the neighbourhood of the equilibrium position are determined. The spectrum of natural frequencies and natural forms of vibration are found analytically for a free (unloaded) tyre with a fixed disc. A similar problem is solved numerically in the case of a loaded tyre. The results of this analysis can be used to estimate the level of noise which occurs when a vehicle moves on an uneven surface.     

A three-dimensional progressive damage model for fiber reinforced composites with an implicit-explicit integration scheme

This contribution proposes a fully three dimensional “continuum damage model” (CDM) to describe the interlaminar and intralaminar failure mechanisms of transversely isotropic elastic-brittle materials under static loading. The constitutive model is derived from an energy function with independent damage variables for each damage mode. The evolution law is based on energy dissipation within the damage process, taking into account the critical energy release rate to weaken the effect of mesh dependent outcome. The onset of damage can be predicted with Cuntze's failure mode concept [1] as well as with Hashin's failure criteria. In this model linear stress decreasing is assumed. In addition, an implicit-explicit integration scheme, first proposed by Oliver [3] for isotropic damage models, is adapted to increase the stability and robustness of numerical simulations and to decrease the computational cost of material failure analyses. By comparing the results from implicit-explicit integration schemes and standard implicit integration schemes, a high level of agreement is found. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Waves induced by the longitudinal impact of a rod against a stepped rod in contact with a rigid barrier

The longitudinal elastic central impact of a rod system, consisting of a uniform rod having a pre-impact velocity against a stepped rod which is in a state of rest and interacts by means of unilateral constraints with a rigid barrier, is modelled.     

Scale factor in an experimental model of a highly reinforced composite

Formation of shrinkage defects during isothermal curing of network polymers is considered. The cure was carried out under conditions of adhesive interaction of the polymer matrix with the rigid filler present in the composite. The dependence of the relative mean distance between cohesive defects on the tube diameter in the tube model developed earlier was investigated over more than two orders of magnitude of tube diameters. This distance was found to be constant in the range considered. A similar process of shrinkage defect formation was observed for a plate model.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 163–168, March–April, 1995.The work was financed by the Russian Fund for Basic Research (Project 94-03-09253).     

Optimal two-sided boundary control of heat transmission in a rod. Hyperbolic model

We consider the mixed problem for one-dimensional hyperbolic system of thermal conductivity equations. We construct a class of boundary controls that provide given distribution on phase vector (T, q) in a given moment of time. From this class we choose a control by the Lagrange method that minimizes a square functional of loss.     

Three-dimensional non-linear oscillations of a rod with hinged supports

The free and forced flexural oscillations of a rod with hinged supports are investigated analytically and numerically. The geometrical non-linearity due to the change in the length of the central line of the rod accompanying its three-dimensional motion is taken into account. The oscillations of a rod with different natural frequencies in two mutually perpendicular directions as a consequence of the variance in the flexural stiffnesses of the rod or the stiffnesses of the supports in the different directions, are considered. It is shown in the case of natural oscillations that, together with two planar forms of motion, a form exists when a certain threshold value is exceeded, which corresponds to the motion of the cross-sections of the rod in a circle. The amplitude-frequency and phase-frequency characteristics of the system are constructed and qualitatively investigated in the neighbourhood of the principal resonance.