Solution of contact problems for Gao beam and elastic foundation

2017 ◽  
Vol 23 (3) ◽  
pp. 473-488 ◽  
Author(s):  
Jitka Machalová ◽  
Horymír Netuka
Keyword(s):  
Contact Problem ◽  
Elastic Foundation ◽  
Variational Equation ◽  
Main Idea ◽  
Axial Loading ◽  
Elastic Beam ◽  
Beam Theory ◽  
Contact Problems ◽  
Problem Formulations ◽  
Nonlinear Elastic

This paper presents mathematical formulations and a solution for contact problems that concern the nonlinear beam published by Gao (Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech Res Commun 1996; 23: 11–17) and an elastic foundation. The beam is subjected to a vertical and also axial loading. The elastic deformable foundation is considered at a distance under the beam. The contact is modeled as static, frictionless and using the normal compliance contact condition. In comparison with the usual contact problem formulations, which are based on variational inequalities, we are able to derive for our problem a nonlinear variational equation. Solution of this problem is realized by means of the so-called control variational method. The main idea of this method is to transform the given contact problem to an optimal control problem, which can provide the requested solution. Finally, some results including numerical examples are offered to illustrate the usefulness of the presented solution method.

DYNAMIC STABILITY OF A BEAM ON AN ELASTIC FOUNDATION INCLUDING STRESS WAVE EFFECTS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250017 ◽  
Author(s):  
YING LIU ◽  
G. LU
Keyword(s):  
Dynamic Stability ◽  
Elastic Foundation ◽  
Stress Wave ◽  
Axial Load ◽  
Elastic Beam ◽  
Beam Theory ◽  
Stress Wave Propagation ◽  
Bernoulli Beam ◽  
Wave Effects ◽  
Euler Bernoulli Beam

This paper examines the dynamic stability of an elastic beam on the elastic foundation, in which the stress wave effect is taken into account. Based on Euler–Bernoulli beam theory, the dynamic response of the elastic beam on the elastic foundation to a small transverse perturbation is analyzed. By considering the stress wave propagation in the beam and the constraint of the elastic foundation, the critical bifurcation condition of elastic beam is derived, and the critical axial load of the elastic beam is predicted. Furthermore, the effects of the elastic foundation and the beam length on buckling condition are discussed by using numeric examples. Finally, an approximate solution of critical axial load for elastic beam on the elastic foundation is provided, which may be used to investigate elastic beam buckling problem.

Dynamical contact problems with friction for hemitropic elastic solids

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Avtandil Gachechiladze ◽  
Roland Gachechiladze ◽  
David Natroshvili
Keyword(s):  
Variational Inequality ◽  
Contact Problem ◽  
A Priori Estimates ◽  
Variational Equation ◽  
A Priori ◽  
Three Dimensional ◽  
Contact Problems ◽  
Positive Definiteness ◽  
Elastic Solids ◽  
Dimensional Boundary

Abstract.In the present paper we investigate a three-dimensional boundary-contact problem of dynamics for a homogeneous hemitropic elastic medium with regard to friction. We prove the uniqueness theorem using the corresponding Green formulas and positive definiteness of the potential energy. To analyze the existence of solutions we reduce equivalently the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter and study its solvability by the Faedo–Galerkin method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure the existence theorem for the original contact problem with friction is proved.

scholarly journals Constrained and Unconstrained Optimization Formulations for Structural Elements in Unilateral Contact with an Elastic Foundation

2008 ◽  
Vol 2008 ◽  
pp. 1-15
Author(s):  
Ricardo A. M. Silveira ◽  
Wellington L. A. Pereira ◽  
Paulo B. Gonçalves
Keyword(s):  
Contact Problem ◽  
Elastic Foundation ◽  
Optimization Problem ◽  
Ritz Method ◽  
Inequality Constraints ◽  
Contact Problems ◽  
Unilateral Contact ◽  
Structural Elements ◽  
The Finite Element Method ◽  
Unilateral Contact Problems

In this work, two numerical methodologies are proposed for the solution of unilateral contact problems between a structural member (beam or arch) and an elastic foundation. In the first approach, the finite element method is used to discretize the structure and elastic foundation and the contact problem is formulated as a constrained optimization problem. Only the original variables of the problem are used, subjected to inequality constraints, and the relevant equations are written as a linear complementary problem (LCP). The second approach is based on the Ritz method, where the coordinates defining the limits of the contact regions are considered as additional variables of the problem. The contact problem here is treated as an unconstrained optimum design problem. These proposed methodologies are then tested and compared using results from specific problems involving structures under unilateral contact constraints.

A Laminated Ring on Elastic Foundation Model and a Feedback Contact Algorithm for Tire-Road Contact Analysis

2017 ◽  
Vol 45 (4) ◽  
pp. 307-334
Author(s):  
Chunjian Wang ◽  
Beshah Ayalew ◽  
Timothy Rhyne ◽  
Steve Cron ◽  
John Adcox
Keyword(s):  
Elastic Foundation ◽  
Radial Direction ◽  
Contact Region ◽  
Beam Theory ◽  
Contact Problems ◽  
Euler Beam ◽  
Element Analysis ◽  
Contact Algorithm ◽  
Foundation Model ◽  
Compensation Approach

ABSTRACT This article proposes (1) a two-dimensional tire model that extends the deformable ring on elastic foundation (REF) model by treating the ring as a laminated beam and (2) a feedback compensation approach to solve the tire-road contact problem as facilitated by the laminated REF model. The internal layer of the laminated ring is formulated using the Timoshenko beam theory that can also be easily regressed to an Euler beam. The external layer of the laminated ring is modeled as a circular beam that primarily takes into account the strain energy contributed by the tire tread in the transverse or radial direction. The elastic foundations are assumed to have a predeformation in the radial direction, which can model tire inflation pressure in pneumatic tires or foundation precompression/pretension for nonpneumatic tires. The analytical solution for the static deformation response of this laminated REF model due to an arbitrary external force is detailed first. Then, a feedback p-controller algorithm that penalizes geometry errors in the contact region is outlined as a unified approach that can be used to solve frictionless contact problems between a tire and arbitrary road profiles. The performances of the proposed model and algorithm are compared against those obtained from a detailed finite element analysis. Both flat surface and cleat contact responses are shown to illustrate the utility of this laminated REF model and the contact algorithm.

scholarly journals The Indentation Rolling Resistance in Belt Conveyors: A Model for the Viscoelastic Friction

Lubricants ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 58 ◽  
Author(s):  
Nicola Menga ◽  
Francesco Bottiglione ◽  
Giuseppe Carbone
Keyword(s):  
Contact Problem ◽  
Finite Thickness ◽  
Numerical Procedure ◽  
Contact Problems ◽  
Accurate Solution ◽  
Rolling Contact ◽  
Viscoelastic Layer ◽  
Force Coefficient ◽  
Belt Conveyors ◽  
Energy Dissipated

In this paper, we study the steady-state rolling contact of a linear viscoelastic layer of finite thickness and a rigid indenter made of a periodic array of equally spaced rigid cylinders. The viscoelastic contact model is derived by means of Green’s function approach, which allows solving the contact problem with the sliding velocity as a control parameter. The contact problem is solved by means of an accurate numerical procedure developed for general two-dimensional contact geometries. The effect of geometrical quantities (layer thickness, cylinders radii, and cylinders spacing), material properties (viscoelastic moduli, relaxation time) and operative conditions (load, velocity) are all investigated. Physical quantities typical of contact problems (contact areas, deformed profiles, etc.) are calculated and discussed. Special emphasis is dedicated to the viscoelastic friction force coefficient and to the energy dissipated per unit time. The discussion is focused on the role played by the deformation localized at the contact spots and the one in the bulk of the thin layer, due to layer bending. The model is proposed as an accurate solution for engineering applications such as belt conveyors, in which the energy dissipated on the rolling contact of idle rollers can, in some cases, be by far the most important contribution to their energy consumption.

scholarly journals Analytical corrections for double-cantilever beam tests

2021 ◽  
Author(s):  
T. Chen ◽  
C. M. Harvey ◽  
S. Wang ◽  
V. V. Silberschmidt
Keyword(s):  
Elastic Foundation ◽  
Analytical Solutions ◽  
Data Reduction ◽  
Crack Tip ◽  
Beam Theory ◽  
Dynamic Loads ◽  
Structural Vibration ◽  
Mode I ◽  
Mode I Fracture ◽  
Reduction Methods

AbstractDouble-cantilever beams (DCBs) are widely used to study mode-I fracture behavior and to measure mode-I fracture toughness under quasi-static loads. Recently, the authors have developed analytical solutions for DCBs under dynamic loads with consideration of structural vibration and wave propagation. There are two methods of beam-theory-based data reduction to determine the energy release rate: (i) using an effective built-in boundary condition at the crack tip, and (ii) employing an elastic foundation to model the uncracked interface of the DCB. In this letter, analytical corrections for a crack-tip rotation of DCBs under quasi-static and dynamic loads are presented, afforded by combining both these data-reduction methods and the authors’ recent analytical solutions for each. Convenient and easy-to-use analytical corrections for DCB tests are obtained, which avoid the complexity and difficulty of the elastic foundation approach, and the need for multiple experimental measurements of DCB compliance and crack length. The corrections are, to the best of the authors’ knowledge, completely new. Verification cases based on numerical simulation are presented to demonstrate the utility of the corrections.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

scholarly journals Nonlinear ferro-electro-elastic beam theory

2009 ◽  
Vol 46 (11-12) ◽  
pp. 2397-2406 ◽  
Author(s):  
Uri Kushnir ◽  
Oded Rabinovitch
Keyword(s):  
Elastic Beam ◽  
Beam Theory
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