scholarly journals A Review of the Method of Dimensionality Reduction in Contact Mechanics: Applications for Structural Damping, Wear and Adhesion

2015 ◽  
Author(s):  
V.L. Popov ◽  
M. Heß ◽  
M. Popov
Keyword(s):  
Dimensionality Reduction ◽  
Three Dimensional ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Tangential Direction ◽  
Fretting Wear ◽  
Normal Contact ◽  
Method Of Dimensionality Reduction ◽  
Frictional Damping ◽  
Arbitrary Axis

In the method of dimensionality reduction (MDR), contacts of three-dimensional bodies are mapped to the contact problem with a one-dimensional elastic or viscoelastic foundation. This is valid for the normal contact, the tangential contact and the normal contact of viscoelastic bodies. For the above classes of contact problems, several examples are considered and discussed in detail. This includes: (a) Fretting wear for arbitrary histories of loading (for simultaneous oscillations both in normal and horizontal directions); (b) Frictional damping under the influence of oscillations in normal and tangential direction as well as normal and torsional loading; (c) Adhesion of bodies of arbitrary axis-symmetric shape with extension to the adhesive contact of elastomers.

Linear Complementary Formulations Involving Frictional Contact for Elasto-Plastic Deformable Bodies

1997 ◽  
Vol 64 (1) ◽  
pp. 80-89 ◽  
Author(s):  
Maocheng Li ◽  
Desong Sha ◽  
K. K. Tamma
Keyword(s):  
Frictional Contact ◽  
Three Dimensional ◽  
Nonlinear Behavior ◽  
Small Deformation ◽  
Contact Problems ◽  
Tangential Direction ◽  
Normal Direction ◽  
Contact Forces ◽  
Contact Boundary ◽  
Normal Contact

In the present study, an incremental variational inequality is described for frictional contact problems with material non linear behavior assumed to be elasto-plastic for the contacting bodies. On the contacting boundaries, the constraint conditions include noninterpenetration along the normal direction of the contact boundary and Coulomb friction law in the sliding direction. After numerical discretization using the finite element method, an effective linear complementary formulation is then established with two unknown variables and two complementary variables for each contact nodal pair. The proposed developments permit a reduced number of unknown variables which are chosen as the gap function for the normal direction and the norm of the incremental sliding displacements for the tangential direction; and the complementary variables are taken as the normal contact forces and slack variables in the tangential directions. The resulting linear complementary equations are then solved employing an explicit Conjugate Gradient Based Projection (CGBP) method in conjunction with a generalized Newton-Raphson iteration procedure to account for the material nonlinear behavior. The methodology is valid for three-dimensional frictional contact representations; however, for purposes of illustration of the proposed approaches, attention is confined to applications involving two-dimensional static elasto-plastic problems under small deformation. Numerical examples are presented which clearly show that the developments satisfy the problem physics and contact conditions with features to include high accuracy and reduced computational costs.

scholarly journals METHOD OF DIMENSIONALITY REDUCTION IN CONTACT MECHANICS AND FRICTION: A USER’S HANDBOOK. III. VISCOELASTIC CONTACTS

2018 ◽  
Vol 16 (2) ◽  
pp. 99 ◽  
Author(s):  
Valentin L. Popov ◽  
Emanuel Willert ◽  
Markus Heß
Keyword(s):  
Dimensionality Reduction ◽  
Three Dimensional ◽  
Contact Problems ◽  
Method Of Dimensionality Reduction ◽  
Mapping Rules ◽  
Elastic Bodies ◽  
Elementary Calculus ◽  
Algebraic Operations ◽  
Three Dimensional Elasticity ◽  
Large Groups

Until recently the analysis of contacts in tribological systems usually required the solution of complicated boundary value problems of three-dimensional elasticity and was thus mathematically and numerically costly. With the development of the so-called Method of Dimensionality Reduction (MDR) large groups of contact problems have been, by sets of specific rules, exactly led back to the elementary systems whose study requires only simple algebraic operations and elementary calculus. The mapping rules for axisymmetric contact problems of elastic bodies have been presented and illustrated in the previously published parts of The User's Manual, I and II, in Facta Universitatis series Mechanical Engineering [5, 9]. The present paper is dedicated to axisymmetric contacts of viscoelastic materials. All the mapping rules of the method are given and illustrated by examples.

An Extension of CEB Elastic-Plastic Contact Model

2007 ◽  
Author(s):  
A. Sepehri ◽  
K. Farhang
Keyword(s):  
Contact Force ◽  
Tangential Force ◽  
Three Dimensional ◽  
Tangential Direction ◽  
Asperity Contact ◽  
Force Component ◽  
Normal Contact ◽  
Elastic Plastic ◽  
Force Components ◽  
Plastic Parts

Three dimensional elastic-plastic contact of two nominally flat rough surfaces is by developing the equations governing the shoulder-shoulder contact of asperities based on the Chang, Etsion and Bogy (CEB) model of contact in which volume conservation is assumed in the plastic flow regime. Shoulder-shoulder asperity contact yields a slanted contact force consisting of both tangential (parallel to mean plane) and normal components. Each force component comprises elastic and elastic-plastic parts. Statistical summation of normal force components leads to the derivation of the normal contact force for the elastic-plastic contact akin to the CEB model. Half-plane tangential force due to elastic-plastic contact is derived through the statistical summation of tangential force component along an arbitrary tangential direction.

A General Three-Dimensional Numerical Technique for Determining the Contact Area of an Arbitrary Punch on an Elastic Half-Space

2016 ◽  
Vol 08 (01) ◽  
pp. 1650005 ◽  
Author(s):  
Jing Jin Shen ◽  
Feng Yu Xu ◽  
Guo Ping Jiang
Keyword(s):  
Numerical Method ◽  
Contact Area ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Contact Problems ◽  
Linear Interpolation ◽  
Contact Boundary ◽  
Reciprocal Theorem ◽  
Normal Contact ◽  
Indentation Force

The paper presents a numerical method for determining the contact area in three-dimensional elastostatic normal contact without friction. The method makes use of the theorem developed by Barber, the contact area is that over which the total indentation force achieves its maximum value. By approximating the punch by linear interpolation, the analytical expression for the indentation force is derived by virtue of the reciprocal theorem. The physical meaning of the parameter which determines the contact boundary is discussed, and its feasible range corresponding to the contact area is found. Then, the numerical algorithm for determining the parameter is developed and applied to solve several normal contact problems. The results show that the proposed numerical method possesses a good property on accuracy and convergency.

Modeling of Fretting Wear Under Gross Slip and Partial Slip Conditions

2007 ◽  
Vol 129 (3) ◽  
pp. 528-535 ◽  
Author(s):  
L. Gallego ◽  
D. Nélias
Keyword(s):  
Contact Problem ◽  
Three Dimensional ◽  
Fretting Wear ◽  
Contact Model ◽  
Partial Slip ◽  
Tangential Displacement ◽  
Normal Contact ◽  
Slip Conditions ◽  
Static Contact ◽  
Wear Law

The paper presents a numerical model to investigate fretting wear either under partial or gross slip conditions. An efficient three-dimensional elastic–static contact model to solve both the normal contact problem and the tangential contact problem is presented. The contact model is validated with analytical solutions for a sphere on flat geometry. A wear law issued from the literature and based on the friction energy is used to simulate surface wear. Numerical friction logs are obtained and the wear rate evolution is found to be highly dependent on the tangential displacement.

scholarly journals SIMULATION OF FRICTIONAL DISSIPATION UNDER BIAXIAL TANGENTIAL LOADING WITH THE METHOD OF DIMENSIONALITY REDUCTION

2017 ◽  
Vol 15 (2) ◽  
pp. 295
Author(s):  
Andrey V. Dimaki ◽  
Roman Pohrt ◽  
Valentin L. Popov
Keyword(s):  
Dimensionality Reduction ◽  
Normal Load ◽  
Superposition Principle ◽  
Three Dimensional ◽  
Method Of Dimensionality Reduction ◽  
Elastic Bodies ◽  
Frictional Dissipation ◽  
Order Of Magnitude ◽  
Constant Normal Load ◽  
Numeric Solution

The paper is concerned with the contact between the elastic bodies subjected to a constant normal load and a varying tangential loading in two directions of the contact plane. For uni-axial in-plane loading, the Cattaneo-Mindlin superposition principle can be applied even if the normal load is not constant but varies as well. However, this is generally not the case if the contact is periodically loaded in two perpendicular in-plane directions. The applicability of the Cattaneo-Mindlin superposition principle guarantees the applicability of the method of dimensionality reduction (MDR) which in the case of a uni-axial in-plane loading has the same accuracy as the Cattaneo-Mindlin theory. In the present paper we investigate whether it is possible to generalize the procedure used in the MDR for bi-axial in-plane loading. By comparison of the MDR-results with a complete three-dimensional numeric solution, we arrive at the conclusion that the exact mapping is not possible. However, the inaccuracy of the MDR solution is on the same order of magnitude as the inaccuracy of the Cattaneo-Mindlin theory itself. This means that the MDR can be also used as a good approximation for bi-axial in-plane loading.

scholarly journals NUMERICAL IMPLEMENTATION OF FRETTING WEAR IN THE FRAMEWORK OF THE MDR

2019 ◽  
Vol 17 (1) ◽  
pp. 87
Author(s):  
Qiang Li ◽  
Fabian Forsbach ◽  
Justus Benad
Keyword(s):  
Numerical Calculation ◽  
Stress Distribution ◽  
Three Dimensional ◽  
Fretting Wear ◽  
Numerical Implementation ◽  
One Dimensional ◽  
Method Of Dimensionality Reduction ◽  
Improve Accuracy ◽  
The One ◽  
Archard’S Law Of Wear

Two numerical methods are proposed to improve accuracy of the numerical calculation of fretting wear in the framework of the Method of Dimensionality Reduction (MDR). Due to the singularity of the transformation equations, instabilities appear at the border between the stick and slip regions after many transformations from the one-dimensional to the three-dimensional contact and back. In these two methods, the transformation equations are reformulated to weaken the singularity of the integrals and a stable simulation of fretting wear is realized even with the wear models which go beyond the classical Archard law. With an example of dual-oscillation, we show the change in the worn profile of a parabolic indenter as well as the stress distribution on the contacting surface during the oscillating cycles under the Archard’s law of wear and Coulomb’s law of friction.

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