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2018 | OriginalPaper | Chapter

Advanced Discretization Methods for Contact Mechanics

Author : Peter Wriggers

Published in: Contact Modeling for Solids and Particles

Publisher: Springer International Publishing

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Abstract

Modeling of contact problems is essential for many problems in engineering in order to predict the behaviour and response of various systems. One can think of pile driving, complex bearings, connections in Civil Engineering, of vehicle road interaction, machines and forming processes in Mechanical Engineering and of MEMS and electrical circuits in Electrical Engineering. All these systems need predictions of the behaviour, durability and efficiency. Hence models are needed that have to be solved by numerical methods due to their complexity. This contribution is aimed at modeling of contact in solid mechanics. Due to the necessity to use numerical methods for the solution of most contact applications this paper will focus mainly on numerical simulation models. Here especially new methodologies are considered that are non-standard and open the possibility for more general application ranges when compared to conventional approaches.

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previous chapter State-of-the-Art Computational Methods for Finite Deformation Contact Modeling of Solids and Structures

next chapter Finite Wear and Soft Elasto-Hydrodynamic Lubrication: Beyond the Classical Frictional Contact of Soft Solids

Here we use the standard relations $\mathbf a_\alpha ^2 \cdot \mathbf a^{2\beta } = \delta _\alpha ^\beta $ and $\mathbf A_\alpha ^2 \cdot \mathbf A^{2\beta } = \delta _\alpha ^\beta $. Furthermore $(\,\,)_{,\alpha }$ denotes differentiation with respect to the convective coordinate $\xi ^\alpha $.

For the analysis of small deformation problems the kinematical relation (1) can be linearized which yields

$$\begin{aligned} \, \Delta {g}_{N+} \ = \ [\, {\mathbf u}^1 - \hat{\mathbf u}^2(\bar{\varvec{\xi } }) \, ] \cdot \bar{\mathbf N}^2 + g_0\,. \qquad \qquad {(4)} \end{aligned}$$

$\mathbf u^\gamma $ represents the displacement field which is introduced in the kinematically linear case to connect the current and the reference configuration via: $\mathbf x^\gamma = \mathbf X^\gamma + \mathbf u^\gamma $. The variable $g_0$ denotes the initial gap between the two bodies which is given by $g_0 = [\mathbf X^1 - \hat{\mathbf X}^2 (\bar{\varvec{\xi }})] \cdot \overline{\mathbf N}^2$ and the normal $\bar{\mathbf N}^2=(\bar{\mathbf A}^2_1 \times \bar{\mathbf A}^2_2) \,/\, \Vert \bar{\mathbf A}^2_1 \times \bar{\mathbf A}^2_2 \Vert $ is related to the reference configuration.

Note that this simple and efficient computation of the virtual ansatz space is only valid for linear interpolations. For quadratic interpolations one has to use the weak form (10).

Since the parameters $a_i$ describing the projection never enter the formulation explicitly the name virtual elements was introduced.

In general the virtual element method leads to stiffness matrices that have the same nodal degrees of freedom as finite elements. Thus VEM fits in the standard FEM framework and hence the VEM can easily be combined with standard finite elements. This can be additionally explored to create a node-to-node contact approach for contact situations with non-matching meshes that is very simple to formulate.

For large sliding, one has to start this update algorithm at every incremental step of the Newton procedure in order to obtain the current local contact connections.

This interpolation is the same as in (34) and thus consistent with the VEM formulation.

The same constitutive relation is used for the stabilization term (45), however with different Lame constants.

In case of friction two additional different structural tensors have to be defined that are associated with the tangents at the deformed surface of bodies ${B}^\alpha $, e.g, $\varvec{M}_1^\alpha = \varvec{\varphi }^\alpha _{,1}\otimes \varvec{\varphi }^\alpha _{,1}$ and $\varvec{M}_2^\alpha = \varvec{\varphi }^\alpha _{,2}\otimes \varvec{\varphi }^\alpha _{,2}$ are defined.

Note that in case of $\omega _{min}\rightarrow 0$ the third medium approaches a two dimensional contact interface. Hence the eigenvector $\varvec{e}_{min}$ will be equivalent to the normal vector $ \varvec{n}$ of the contact interface which means that the new formulation converges to a classical contact setting for $\omega _{min}\rightarrow 0$.

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Title: Advanced Discretization Methods for Contact Mechanics
Author: Peter Wriggers
Publisher: Springer International Publishing
Book: Contact Modeling for Solids and Particles
Print ISBN: 978-3-319-90154-1

Electronic ISBN: 978-3-319-90155-8

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-90155-8_2

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