A comparison between the material point method and the finite element method in view of extrusion problems

An appropriate method for the simulation of continuous forming processes is the material point method (MPM) [1],[2] which combines the viewpoints of fluid dynamics and solid mechanics. The MPM and related methods [3] are derived from the particle‐in‐cell methods [4]. Bodies are discretised by Lagragian particles with pointwise mass distributions. The differential equations in their weak form are solved on temporary meshes built of standard finite elements. At the end of each time step the particle positions are updated and the mesh is replaced by a new mesh with a regular shape. The state variables at the nodes of the new mesh are extracted from the state variables at the particles by a transfer algorithm. When particles pass element boundaries, numerical difficulties might be observed. These are eliminated by a smooth approximation of nodal data from material point data. The modified MPM has been implemented together with the FEM in one programme because the similarities of the methods outbalance the differences. On the basis of numerical examples the results of both methods are compared. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Excitation of Rolling Tires from the Road Surface Texture

Rolling tires are excited from the contact with the rough road surface to vibrations, which cause rolling noise. A two scale approach is suggested, where at the macro–scale the vibration of the rolling tire structure is modeled by quite detailed finite element methods. The road surface is described using measured textures. A fine resolution finite element discretization of the tread rubber is performed in order to resolve the asperity contact. The material properties are described by a non–linear viscoelastic rubber model. The tread patch is enforced to approach the rough surface in a transient dynamics manner. From these investigations an enveloping surface profile is reconstructed to be used for the excitation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An operator based notation of the Lagrange equations for systems with non-material boundary conditions

Problems with non-material boundary conditions are very important in many applications of continuum mechanics: for example rolling contact, calenders, breaking disks, band saws and belts. This paper deals with the LAGRANGE-equation for non-material boundary conditions. Such boundary conditions can be described by non-material control volumes that cannot be assigned for one and only one set of particles over time. To define these volumes, a fictitious body is used. Within the scope of this paper several configurations are introduced for the definition of the location of the original and fictitious particles. With the help of the used operator notation it is possible to demonstrate a consistent representation of derivative operators and assign operators as well as entities to their configurations. On this basis, the LAGRANGE-equation is derived for material and non-material control volumes in the EULERIAN and LAGRANGIAN representation. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact creep as the principal source of the losses associated with the rolling of a plastic ball

An engineering method is proposed for calculating the contact creep of plastic balls used in rolling bearings. The method is based on empirical expressions relating the relative creep strain with the specific load and its duration of action. The complex contact-creep characteristics of the material are obtained by means of simple laboratory tests.Moscow Bauman Higher Technical College. Translated from Mekhanika Polimerov, No. 3, pp. 498–504, May–June, 1969.     

Shape and topology optimization of contact problems by level set methods

This paper deals with the numerical solution of a topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is considered. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. In the paper shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. Level set based numerical algorithm for the solution of the shape optimization problem is proposed. Level set method is used to describe the position of the boundary of the body and its evolution on a fixed mesh. This evolution is governed by Hamilton – Jacobi equation. The speed vector field driving the propagation of the boundary of the body is given by the shape derivative of a cost functional with respect to the free boundary. Numerical examples are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material-Force-Based Refinement Indicators in Adaptive Strategies

A material-force-based refinement indicator for adaptive finite element strategies for finite elasto-plasticity is proposed. Starting from the local format of the spatial balance of linear momentum, a dual material counterpart in terms of Eshelby's energy-momentum tensor is derived. For inelastic problems, this material balance law depends on the material gradient of the internal variables. In a global format the material balance equation coincides with an equilibrium condition of material forces. For a homogeneous body, this condition corresponds to vanishing discrete material nodal forces. However, due to insufficient discretization, spurious material forces occur at the interior nodes of the finite element mesh. These nodal forces are used as an indicator for mesh refinement. Assigning the ideas of elasticity, where material forces have a clear energetic meaning, the magnitude of the discrete nodal forces is used to define a relative global criterion governing the decision on mesh refinement. Following the same reasoning, in a second step a criterion on the element level is computed which governs the local h-adaptive refinement procedure. The mesh refinement is documented for a representative numerical example of finite elasto-plasticity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On wheel–rail contact problem with material properties varying with depth

Numerous laboratory experiments indicate that the use of a layer or a coating material attached to the conventional steel body reduce the magnitude of contact stress. Therefore in this paper we solve numerically the wheel–rail contact problem with friction and wear assuming the existence of a small elastic layer on the rail surface. Material properties of this layer are changing with its depth. The friction between the bodies is governed by Coulomb law. In contact zone Archard's law of wear is assumed. We take special features of this rolling contact problem and use so-called quasistatic approach to solve this contact problem. Finite element method is used as a discretization method. The numerical results including the distribution of normal stress along the contact boundary are provided and discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of a viscoelastic matrix in linear frictional contact

The paper is devoted to study of acoustic wave propagation in a partially consolidated composite material containing loose particles. Friction of particles against the consolidated part of the material causes mechanical energy dissipation. This situation is modelled by assuming that the medium has a periodic microstructure changing rapidly on the small scale ε. Each of the periodic microscopic cells is composed of a viscoelastic matrix containing a rigid particle in frictional contact with the matrix. We use the methods of two‐scale convergence to obtain effective acoustic equations for the homogenized material. The effective equations are history‐dependent and contain the body force term, reminiscent of the well‐known Stokes drag force. Copyright © 2004 John Wiley & Sons, Ltd.     

Rolling Contact Problem with the Generalized Coulomb Friction

This paper deals with the numerical solution of the wheel - rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermoelastic rolling contact problems for multilayer materials

The paper deals with the existence of solutions to the thermoelastic rolling contact problems for nonhomogeneous materials. One of the contacting surfaces is assumed to be covered with a graded material coating. The thermal and mechanical features of the coating material depend on its depth. The thermoelastic contact problem is governed by the system of mildly coupled evolutionary boundary value problems with discontinuous coefficients. Quasistatic approach is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The Faedo–Galerkin approach combined with the penalization and smoothing approach are used to show the existence of solutions to this contact problem. The operator splitting method is used to solve the problem numerically. Numerical results indicating the reduction of mechanically and/or thermally induced stresses are provided.     

Numerical Solution of Contact Problems using Level Set Methods on Voxel Discretizations

Fast solvers performing on a regular grid are often used for problems in elasticity, in order to avoid expensive mesh generations. However, if overlaps between solid materials occur without any interactions, these might deteriorate the results, especially for the stresses. Therefore, we want to present an approach for developing numerical methods for contact problems on a regular mesh with the help of signed distance data and multi-material voxels. In this contribution we will focus on problems in linear elastostatics with contact between several different bodies. Finally, we present the results from a numerical test for the two dimensional Hertz problem, solved on a triangular regular mesh. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)