Alexander Konyukhov

On a Geometrically Exact Theory for Contact Interactions

The focus of the contribution is on the developments concerning an unified geometrical formulation of contact algorithms in a covariant form for various geometrical situations of contacting bodies leading to contact pairs: surface-to-surface, line-to-surface, point-to-surface, line-to-line, point-to-line, point-to-point. The computational contact algorithm will be considered in accordance with the geometry of contact bodies in a covariant form. This combination forms a geometrically exact theory of contact interaction. The contribution focuses on an overview of the literature and then presents in a review type the contributions of the authors on the topic.

Symmetrization of Various Friction Models Based on an Augmented Lagrangian Approach

The standard implementation of the classical Coulomb friction model together with the Newton iterative method for the finite element method leads to non-symmetric tangent matrices for sliding zones of contact surfaces. This fact is known in literature as consequence of the non-associativity of the friction law. Considering anisotropic models for friction, especially including coupling of adhesion and friction, leads to additional non-symmetries due to anisotropy. Since, non-symmetry of matrices is a non-desirable feature of most engineering problems, various proposals for symmetrization are known in computational mechanics. A further suggestion is made in this contribution. The covariant approach for both isotropic and anisotropic frictional contact problems leads to a very simple structure of the tangent matrices. This allows to obtain very robust tangent matrices within the symmetrized Augmented Lagrangian method. In the current contribution, the nested Uzawa algorithm is applied for symmetrization within the Augmented Lagrangian approach for an anisotropic friction model including adhesion and friction. The numerical examples show the good convergence behavior for various problems such as small and large sliding problems.

On a Continuous Transfer of History Variables for Frictional Contact Problems Based on Interpretations of Covariant Derivatives as a Parallel Translation

Regularization methods based on the penalization of the tangent displacements are among the most exploited techniques in combination with finite element methods to model frictional interactions. Usually the global tangent displacements are described via convective coordinates which are e.g. used in a finite element discretization of the contact surface. These displacements serve to compute the tangent tractions in the case of sticking via a regularization procedure as well as in the case of sliding via a return-mapping scheme. The convective coordinates of the contact point as well as the corresponding tangent tractions are considered as history variables and have to be stored for each load step. In this contribution, we discuss the particular issue of continuous transfer for history variables in the case of large deformation problems adapted for the covariant contact description developed in Konyukhov and Schweizerhof [4]. Some specific examples are chosen to illustrate the effect of incorrect transfer for both non-frictional and frictional problems and, therefore, the necessity of the proposed techniques.

Contact Modelling of Large Radius Air Bending with Geometrically Exact Contact Algorithm

Usage of high-strength steels in conventional air bending is restricted due to limited bendability of these metals. Large-radius punches provide a typical approach for decreasing deformations during the bending process. However, as deflection progresses the loading scheme changes gradually. Therefore, modelling of the contact interaction is essential for an accurate description of the loading scheme. In the current contribution, the authors implemented a plane frictional contact element based on the penalty method. The geometrically exact contact algorithm is used for the penetration determination. The implementation is done using the OOFEM - open source finite element solver. In order to verify the simulation results, experiments have been conducted on a bending press brake for 4 mm Weldox 1300 with a punch radius of 30 mm and a die opening of 80 mm. The maximum error for the springback calculation is 0.87° for the bending angle of 144°. The contact interaction is a crucial part of large radius bending simulation and the implementation leads to a reliable solution for the springback angle.

Frictional Interaction of a Spiral Rope and a Cylinder – 3D-Generalization of the Euler-Eytelwein Formula Considering Pitch

Historically, following e.g. Holzer (2005) the problem of the definition of frictional rope forces appearing in a rope sliding over a cylinder has been formulated in the dissertation of Gautier (1717). The solution of this problem was reported by Euler in his Remarks on the effect of friction on equilibrium published by the Berlin Academy of science, see Euler (1769). Since the first time publishing the Euler solution by Eytelwein (1808) in his Handbuch der Statik fester Korper in 1808 the problem is spread through practical applications and became known as Euler-Eytelwein problem in standard books of technical mechanics see e.g. in Gross et.al. (2004), and is also known as belt or coil friction formula see e.g. in Maurer and Roark (1944), or as capstan friction problem, see e.g. in Meriam (1978).

Anisotropic adhesion-friction models – Some particular details of implementation and numerical examples

This chapter is focusing on the finite element implementation of coupled anisotropic adhesion-friction model which has been developed and analyzed in Sections 6.2, 6.3 and 6.4. The model, first, is implemented for the full Newton method and then the Augmented Lagrangian method. The tangent matrices is then based on the linearization derived in Sect. 7.3.

Various Aspects of Implementation of the Curve-To-Curve Contact Model

There are several questions in further applications of the curve-to-curve approach to contact between bodies:

Linearization of the Weak Forms – Tangent Matrices in a Covariant Form

The full Newton iterative method will be applied to solve the global equilibrium equations on the stage of the numerical solution. This requires the full linearization of the corresponding weak forms representing the equilibrium conditions on the contact boundaries described in Chapter 5 for all contact cases. Linearization is obtained using the covariant derivatives in the corresponding local surface coordinate system, where derivatives of contact tractions are taken in covariant forms as described in Chapter 6 and derivatives of corresponding convective coordinates are described in Chapter 4. Linearized weak forms are the basis to create tangent matrices in the form independent of any approximations of the object (surfaces, curves, beams etc.). All tangent matrices are formulated then via the abstract approximation operator A. The matrices are split into several parts and possessing a clear geometrical structure for all studied geometrical contact cases – Surface-To-Surface, Point-To-Curve, Curve-To-Curve including also various constitutive relations for contact tractions.

Surface-To-Surface Contact – Various Aspects for Implementations within the Finite Element Method

The current Chapter is devoted to the finite implementation of various algorithms for Surface-To-Surface contact pair. Newton iterative scheme is the main solution method for the most algorithms. Thus, the result of linearization developed in Sect. 7.1 is directly employed to construct the tangent matrices. Diversity of contact approaches leading to the corresponding contact elements are depending on the type of approximation involved into the discretization within the finite element method. First, the standard Node-To-Surface (NTS) approach for non-frictional and frictional problems are considered. Implementation of the Mortar method within the Segment-To-Segment (STS) type of contact element together with integration by subdivision is shown to be effective to satisfy the Contact Patch Test . The simplest smoothing technique for contact surfaces is shown as the implementation of the Segment-To-Segment type of contact element – this technique is the basis for the isogeometric implementation. A special Segment-To-Analytical Surface (STAS) contact approach is illustrated for the contact with rigid surfaces described analytically. The Large Penetration algorithm is presented as additional technique to accelerate the global solution. It is shown to be effective for some problems where the large penetration and as a result large load steps are applicable. Finally, two versions of implementation of the Nitsche method are shown.

Experimental validations of the coupled anistropic adhesion-friction model

The coupled anisotropic adhesion-friction model studied numerically in chapter 11 is verified in this chapter experimentally for some structured surfaces (originally published in [98]). The necessity to apply a coupled contact interface model including anisotropy for both adhesion and friction is shown in the current chapter via a set of experiments for a rubber surface possessing a periodical waviness, and therefore, an obvious anisotropic structure. The focus of experimental investigations is placed upon the measurements of the global macro characteristics such as global forces and trajectories of a sliding block in order to validate the proposed computational model. The theoretical results presented in Sect. 6.4 is taken here for validation.