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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Notes
- 1.
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 \(\xi ^\alpha \).
- 2.
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.
- 3.
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).
- 4.
Since the parameters \(a_i\) describing the projection never enter the formulation explicitly the name virtual elements was introduced.
- 5.
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.
- 6.
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.
- 7.
This interpolation is the same as in (34) and thus consistent with the VEM formulation.
- 8.
The same constitutive relation is used for the stabilization term (45), however with different Lame constants.
- 9.
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.
- 10.
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 \(\omega _{min}\rightarrow 0\).
References
Beirão da Veiga L., Brezzi F., Cangiani A., Manzini G., Marini L. and Russo A. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01):199–214 (2013).
Beirão Da Veiga L., Brezzi F. and Marini L. Virtual Elements for linear elasticity problems. SIAM Journal on Numerical Analysis, 51(2):794–812 (2013).
Beirão da Veiga L., Brezzi F., Marini L.D. and Russo A. The Hitchhiker’s Guide to the Virtual Element Method. Mathematical Models and Methods in Applied Sciences, 24(08):1541–1573 (2014).
Beirão Da Veiga L., Lovadina C. and Mora D. A Virtual Element Method for elastic and inelastic problems on polytope meshes. Computer Methods in Applied Mechanics and Engineering, 295:327–346 (2015).
Chi H., Beirão da Veiga L. and Paulino G. Some basic formulations of the virtual element method (VEM) for finite deformations. Computer Methods in Applied Mechanics and Engineering, 318:148–192 (2017).
Cottrell J.A., Hughes T.J.R. and Bazilevs Y. Isogeometric Analysis. Wiley (2009).
Dimitri R., De Lorenzis L., Scott M., Wriggers P., Taylor R. and Zavarise G. Isogeometric large deformation frictionless contact using t-splines. Computer Methods in Applied Mechanics and Engineering, 269:394–414 (2014).
de Lorenzis L., Temizer I., Wriggers P. and Zavarise G. A large deformation frictional contact formulation using NURBS-based isogeometric analysis. International Journal for Numerical Methods in Engineering, 87:1278–1300 (2011).
de Lorenzis L., Wriggers P. and Zavarise G. Isogeometric analysis of 3d large deformation contact problems with the augmented lagrangian formulation. Computational Mechanics, 49:1–20 (2012).
Eterovic A.L. and Bathe K.J. An interface interpolation scheme for quadratic convergence in the finite element analysis of contact problems. In Computational Methods in Nonlinear Mechanics, pages 703–715. Springer-Verlag, Berlin, New York (1991).
Fischer K.A. and Wriggers P. Frictionless 2d contact formulations for finite deformations based on the mortar method. Computational Mechanics, 36:226–244 (2005).
Fischer K.A. and Wriggers P. Mortar based frictional contact formulation for higher order interpolations using the moving friction cone. Comput. Methods Appl. Mech. Engrg., 2006:5020–5036 (2006).
Franke D., Düster A., Nübel V. and Rank E. A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2D Hertzian contact problem. Computational Mechanics, 45:513–522 (2010).
Gain A.L., Talischi C. and Paulino G.H. On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Computer Methods in Applied Mechanics and Engineering, 282:132–160 (2014).
Hartmann S., Oliver J., Weyler R., Cante J. and Hernndez J. A contact domain method for large deformation frictional contact problems. part 2: Numerical aspects. Computer Methods in Applied Mechanics and Engineering, 198:2607–2631 (2009).
Hesch C. and Betsch P. A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems. Int. J. Numer. Meth. Engng., 77:1468–1500 (2009).
Hüeber S. and Wohlmuth B.I. Thermo-mechanical contact problems on non-matching meshes. Comput. Methods Appl. Mech. Engrg., 198:1338–1350 (2009).
Hughes T.J.R., Reali A. and Sangalli G. Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Engrg., 199:301–313 (2010).
Johnson K.L. Contact Mechanics. Cambridge University Press (1985).
Krstulovic-Opara L., Wriggers P. and Korelc J. A \({C}^1\)-continuous formulation for 3d finite deformation frictional contact. Computational Mechanics, 29:27–42 (2002).
Krysl P. Mean-strain eight-node hexahedron with optimized energy-sampling stabilization for large-strain deformation. International Journal for Numerical Methods in Engineering, 103:650–670 (2015).
Labra C., Rojek J., Onate E. and Zarate F. Advances in discrete element modelling of underground excavations. Acta Geotechnica, 3:317–322 (2008).
Laursen T.A. Computational Contact and Impact Mechanics. Springer, Berlin, New York, Heidelberg (2002).
Oliver J., Hartmann S., Cante J., Weyler R. and Hernndez J. A contact domain method for large deformation frictional contact problems. part 1: Theoretical basis. Computer Methods in Applied Mechanics and Engineering, 198:2591–2606 (2009).
Onate E., Idelsohn S.R., Celigueta M.A. and Rossi R. Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. Computer Methods in Applied Mechanics and Engineering, 197:1777–1800 (2008).
Padmanabhan V. and Laursen T. A framework for development of surface smoothing procedures in large deformation frictional contact analysis. Finite Elements in Analysis and Design, 37:173–198 (2001).
Piegl L. and Tiller W. The NURBS Book. Springer, Berlin Heidelberg New York, \(2^{\text{nd}}\) edition (1996).
Pietrzak G. and Curnier A. Large deformation frictional contact mechanics: continuum formulation and augmented lagrangean treatment. Computer Methods in Applied Mechanics and Engineering, 177:351–381 (1999).
Popp A., Gee M.W. and Wall W.A. A finite deformation mortar contact formulation using a primal-dual active set strategy. Int. J. Numer. Meth. Engng., 79:1354–1391 (2009).
Puso M.A., Laursen T.A. and j. Solberg. A segment-to-segment mortar contact method for quadratic elements and large deformations. Comput. Methods Appl. Mech. Engrg., 197:555–566 (2008).
Schröder J. Anisotropic polyconvex energies. In J. Schröder, editor, Polyconvex Analysis, pages 1–53. CISM, Springer, Wien (2009). 62.
Temizer I., Wriggers P. and Hughes T.J.R. Contact treatment in isogeometric analysis with NURBS. Computer Methods in Applied Mechanics and Engineering, 200:1100–1112 (2011).
Temizer I., Wriggers P. and Hughes T.J.R. Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS. Computer Methods in Applied Mechanics and Engineering, 209-211:115–128 (2012).
Tur M., Fuenmayor F.J. and Wriggers P. A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Comput. Methods Appl. Mech. Engrg., 198:2860–2873 (2009).
Wriggers P. Computational Contact Mechanics. \(2^{nd}\) ed., Springer, Berlin, Heidelberg, New York (2006).
Wriggers P. Nonlinear Finite Elements. Springer, Berlin, Heidelberg, New York (2008).
Wriggers P. and Hudobivnik B. A low order virtual element formulation for finite elasto-plastic deformations. Computer Methods in Applied Mechanics and Engineering, 327:459–477 (2017).
Wriggers P., Krstulovic-Opara L. and Korelc J. Smooth \({C}^1\)- interpolations for two-dimensional frictional contact problems. International Journal for Numerical Methods in Engineering, 51:1469–1495 (2001).
Wriggers P., Reddy B., Rust W. and Hudobivnik B. Efficient virtual element formulations for compressible and incompressible finite deformations. Computational Mechanics, 60:253–268 (2017).
Wriggers P., Rust W. and Reddy B. A virtual element method for contact. Computational Mechanics, 58:1039–1050 (2016).
Wriggers P., Schröder J. and Schwarz A. A finite element method for contact using a third medium. Computational Mechanics, 52:837–847 (2013).
Ziefle M. and Nackenhorst U. Numerical techniques for rolling rubber wheels: treatment of inelastic material properties and frictional contact. Computational Mechanics, 42:337–356 (2008).
Zienkiewicz O.C. and Taylor R.L. The Finite Element Method, volume 1. Butterworth-Heinemann, Oxford, UK, 5th edition (2000).
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Wriggers, P. (2018). Advanced Discretization Methods for Contact Mechanics. In: Popp, A., Wriggers, P. (eds) Contact Modeling for Solids and Particles. CISM International Centre for Mechanical Sciences, vol 585. Springer, Cham. https://doi.org/10.1007/978-3-319-90155-8_2
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