Karl Schweizerhof

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.

Implementation of Contact Algorithms with High Order FE

The Chapter is showing the implementation of high-order finite elements including the possibility of contact interactions within a special strategy such as contact layer finite element. Some numerical difficulties within the classical Hertz problem are studied (originated to [96]). Later, this example became the benchmark test for iso-geometric finite element, see [175] and [176]. The contact layer finite element is illustrating good correlation with the analytical solution of the famous Hertz contact problem even within a single element covering the contact zone.

Contact Constraints and Constitutive Equations for Contact Tractions

It is required to define the contact conditions including both geometrical conditions such as non-penetration and sticking, and constitutive relations for contact forces and stresses. With regards to the introduced coordinate system and kinematics the contact tractions are naturally split into normal and tangential components. If for the normal components non-penetration conditions can be formulated then for tangential components the constitutive relations are required.

Closest Point Projection Procedure and Corresponding Curvilinear Coordinate System

In this chapter the Closest Point Projection (CPP) procedure as the first important tool to define contact measures is introduced for surfaces and curves. The fundamental issues such as existence and uniqueness of the solution of the Closest Point Projection procedure are intensively studied. In due course the CPP procedure leads to a special curvilinear coordinate system in which later all contact kinematics and weak forms are formulated separately for each contact pair. This chapter creates a basis of the geometrically exact theory of contact interactions for certain geometric pairs such as Surface-To-Surface, Curve-To-Surface and Curve-To-Curve.

Differential Geometry of Surfaces and Curves

In the current chapter we are recalling the necessary information about differential geometry of surfaces and curves: all metrics and curvature parameters which are necessary to describe properties of the surfaces and curves. We also study differential operations in the corresponding curvilinear coordinate systems in a covariant form, such as Weingarten and Gauss-Codazzi formulas for surfaces and Serret-Frenet formula for 3D curves. This information will be fully used further to build all contact kinematics, weak contact forms as well as linearization of operations. The reader who familiar with differential geometry can however easily skip this chapter containing an overview of the formulas from differential geometry of surfaces and curves.

Geometry and Kinematics of Contact

In Chapter 3 is studied the existence and the uniqueness of solutions for the Closest Point Projection (CPP) procedure depending on the geometry of contacting objects. In due course the corresponding CPP procedure leads to a specific curvilinear coordinate system. In those coordinate systems we are going to describe geometry and kinematics for various contact pairs depending on their geometrical description: Point-To-Surface, Point-To-Curve, Curve-To-Curve etc. These cases later on give rise to contact algorithms describing contact between surfaces, contact between curves and surfaces and, finally, contact between curves. In those coordinate systems we are going to build the variational formulation of contact and further on to continue with finite element discretization and formulation of the solution algorithm. Special cases including contact between rigid surfaces and deformable solids are separately studied. The chapter also includes linearization of the necessary kinematic parameters for corresponding contact pairs.