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.

The method of caustics for the determination of normal loads acting on surfaces of elastic bodies

1980 ◽  
Vol 15 (1) ◽  
pp. 37-41 ◽  
Author(s):  
P S Theocaris ◽  
N I Ioakimidis
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Elastic Bodies ◽  
Plate And Shell ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Quantities Of Interest

The optical method of caustics constitutes an efficient experimental technique for the determination of quantities of interest in elasticity problems. Up to now, this method has been applied only to two-dimensional elasticity problems (including plate and shell problems). In this paper, the method of caustics is extended to the case of three-dimensional elasticity problems. The particular problems of a concentrated force and a uniformly distributed loading acting normally on a half-space (on a circular region) are treated in detail. Experimentally obtained caustics for the first of these problems were seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of various, more complicated, three-dimensional elasticity problems, including contact problems, by the method of caustics is also possible.

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 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.

A discussion of the method of dimensionality reduction

2015 ◽  
Vol 230 (9) ◽  
pp. 1424-1431 ◽  
Author(s):  
Ivan Argatov
Keyword(s):  
Analytical Solution ◽  
Dimensionality Reduction ◽  
Contact Area ◽  
Exact Analytical Solution ◽  
Contact Problems ◽  
Base Case ◽  
Winkler Foundation ◽  
Method Of Dimensionality Reduction ◽  
Local Contact ◽  
Wide Range

The Method of Dimensionality Reduction (MDR) can be regarded as a formalism for analytical solution of some commonly encountered classes of contact problems using a “mechanical intuition” based on the Winkler foundation model. Such an approach makes it much easier to account for a wide range of physical effects associated with contact interaction (e.g. friction, adhesion, and damping). However, there is still a controversy about the method and its applications (see, e.g., the comment on validity of the MDR-based model of rough contact) – which we believe comes from a misunderstanding of the method itself, and which, in turn, can be reconsidered in view of the recently published book on the MDR. The MDR was originally introduced for Hertz’s problem of axisymmetric frictionless local contact and was generalized subsequently for arbitrary axisymmetric geometry of linearly elastic bodies in unilateral local contact. The latter problem, for which the MDR yields the exact analytical solution, can be viewed as a base case that is used to extend, in a unified manner, the model of local contact by taking into account adhesion, friction, and viscous damping. In what follows, we overview the main concepts of the method starting with the base-case contact problem in which the MDR is rooted, and discuss limitations of the MDR as well. For the sake of their completeness, some criticisms that apply equally to conventional contact mechanics solutions are also considered. It is emphasized that the axisymmetric Hertz-type contact problems with a circular contact area constitute the proven range of validity of the MDR, while the extension of the method to other types of contact (e.g. axisymmetric with a multiply-connected contact area, non-axisymmetric) is a field ripe for research.

scholarly journals FROM WINKLER’S FOUNDATION TO POPOV’S FOUNDATION

2019 ◽  
Vol 17 (2) ◽  
pp. 181 ◽  
Author(s):  
Ivan Argatov
Keyword(s):  
Contact Problems ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
Elastic Contact ◽  
Contact Interface ◽  
Modeling Framework ◽  
One Dimensional ◽  
Method Of Dimensionality Reduction ◽  
Elastic Bodies ◽  
Contact Configuration

In recent years, the method of dimensionality reduction (MDR) has started to figure as a very convenient tool for dealing with a wide class of elastic contact problems. The MDR modeling framework introduces an equivalent punch profile and a one-dimensional Winkler-type elastic foundation, called henceforth Popov’s foundation. While the former mainly accounts for the geometry of contact configuration, the Popov foundation inherits the main characteristics of both the contact interface (like friction and adhesion) and the contacting elastic bodies (e.g., anisotropy, viscoelasticity or inhomogeneity). The discussion is illustrated with an example of the Kendall-type adhesive contact for an isotropic elastic half-space.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

Reduction of Free-Edge Stress Concentration

1985 ◽  
Vol 52 (4) ◽  
pp. 801-805 ◽  
Author(s):  
P. R. Heyliger ◽  
J. N. Reddy
Keyword(s):  
Finite Element ◽  
Stress Concentration ◽  
Finite Element Model ◽  
Three Dimensional ◽  
Element Model ◽  
Free Edge ◽  
Shear Stresses ◽  
Normal Stresses ◽  
Interlaminar Shear ◽  
Three Dimensional Elasticity

A quasi-three dimensional elasticity formulation and associated finite element model for the stress analysis of symmetric laminates with free-edge cap reinforcement are described. Numerical results are presented to show the effect of the reinforcement on the reduction of free-edge stresses. It is observed that the interlaminar normal stresses are reduced considerably more than the interlaminar shear stresses due to the free-edge reinforcement.

Stiffness Constants for Composite Beams Including Large Initial Twist and Curvature Effects

1995 ◽  
Vol 48 (11S) ◽  
pp. S61-S67 ◽  
Author(s):  
Carlos E. S. Cesnik ◽  
Dewey H. Hodges
Keyword(s):  
Wave Length ◽  
Three Dimensional ◽  
Radius Of Curvature ◽  
Cross Sectional ◽  
Curvature Effects ◽  
Three Dimensional Representation ◽  
Stiffness Model ◽  
Sectional Analysis ◽  
Three Dimensional Elasticity ◽  
Coupling Phenomena

An asymptotically exact methodology, based on geometrically nonlinear, three-dimensional elasticity, is presented for cross-sectional analysis of initially curved and twisted, nonhomogeneous, anisotropic beams. Through accounting for all possible deformation in the three-dimensional representation, the analysis correctly accounts for the complex elastic coupling phenomena in anisotropic beams associated with shear deformation. The analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross section is small relative to the wave length of the deformation and to the minimum radius of curvature and/or twist. The resulting cross-sectional elastic constants exhibit second-order dependence on the initial curvature and twist. As is well known, the associated geometrically-exact, one-dimensional equilibrium and kinematical equations also depend on initial twist and curvature. The corrections to the stiffness model derived herein are also necessary in general for proper representation of initially curved and twisted beams.

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