Kai Willner

Elasto-Plastic Normal Contact of Three-Dimensional Fractal Surfaces Using Halfspace Theory

The elasto-plastic normal contact of fractal surfaces is investigated. To study the influence of several surface parameters like fractal dimension and resolution, the surfaces are numerically generated using a special form of the structure function which is motivated by measurements of real rough surfaces. The contact simulation uses an iterative elastic halfspace solution based on a variational principle. A simple modification allows also the approximative solution of elasto-plastic contact problems. The influence of different surface parameters is studied with respect to the load-area relationship and the load-gap relationship. The simulations show that for realistic surface parameters the deformation is always in the plastic range.

UNCERTAINTY MODELING USING FUZZY ARITHMETIC BASED ON SPARSE GRIDS: APPLICATIONS TO DYNAMIC SYSTEMS

Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadehs extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to compute expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In this paper, we illustrate the general applicability of this new method by computing two dynamic systems subjected to uncertain parameters as well as uncertain initial conditions. The first model consists of a system of delay differential equations simulating the periodic outbreak of a disease. In the second model, we consider a multibody mechanism described by an algebraic differential equation system.

Fully Coupled Frictional Contact Using Elastic Halfspace Theory

The effect of dry metallic friction can be attributed to two major mechanisms: adhesion and ploughing. While ploughing is related to severe wear and degradation, adhesion can be connected to pure elastic deformations of the contacting bodies and is thus the predominant mechanism in a stable friction pair. The transmitted friction force is then proportional to the real area of contact. Therefore, a lot of effort has been put into the determination of the fraction of real area of contact under a given load. A broad spectrum of analytical and numerical models has been employed. However, it is quite common to employ the so-called Mindlin assumptions, where the contact area is determined by the normal load only, disregarding the influence of friction. In the subsequent tangential loading, usually the contact pressure distribution is kept fixed such that the coupling between the tangential and normal solutions is neglected. Here, a numerical solution scheme based on elastic halfspace theory for frictional contact problems is presented where full coupling between the normal and tangential tractions and displacements is taken into account. Several examples show the influence of the coupling effects, but also the limitations for the analysis of rough contacts.

On applying fuzzy arithmetic to finite element problems

Fuzzy arithmetic, based on Zadehs (1973) extension principle, is applied to solve finite element problems with uncertain parameters. As an example, a rather simple, one-dimensional static problem consisting of a two-component massless rod under tensile load is considered. Application of fuzzy arithmetic directly to the traditional techniques for the numerical solution of finite elements, i.e. primarily on the algorithms for solving systems of linear equations, however turns out to be impracticable in all circumstances. In contrast to the use of exclusively crisp numbers, the results for the calculations including fuzzy numbers usually differ to a large extent depending on the solution technique applied. The uncertainties expressed in the different calculation results are then basically twofold. On one hand, uncertainty is caused by the presence of parameters with fuzzy value, whilst on the other, an additional undesirable uncertainty is artificially created by the solution technique itself. For this reason, an overview of the most common techniques for solving finite element problems is offered, rating them with respect to minimizing the occurence of artificial uncertainties. Moreover a special technique is outlined which leads to modified solution procedures with reduced artificial uncertainties.

Experimental Verification of a Benchmark Forming Simulation

Forming of near-net-shaped and load-adapted functional components, as it is developed in the TransregionalCollaborative Research Centre on Sheet-Bulk Metal Forming SFB/TR73, causes different problems, which lead to non-optimal manufacturing results. For these high complex processes the prediction of forming effects can only be realized by simulations. A stamping process of pressing eight punches into a circular blank is chosen for the considered investigations. This reference process is designed to reflect the main aspects, which strongly affect the final outcome of forming processes. These are the orthotropic material behaviour, the optimal design of the initial blank and the influences of different contact and friction laws. The aim of this work is to verify the results of finite element computations for the proposed forming process by experiments. Evaluation methods are presented to detect the influence of the anisotropy and also to quantify the optimal blank design, which is determined by inverse form finding. The manufacturing accuracy of the die plate and the corresponding roughness data of the milled surface are analysed, whereas metrological investigations are required. This is accomplished by the help of advanced measurement techniques like a multi-sensor fringe projection system and a white light interferometer. Regarding the geometry of the punches, micromilling of the die plate is also a real challenge, especially due to the hardness of the high-speed steel ASP 2023 (approx. 63 HRC). The surface roughness of the workpiece before and after the forming process is evaluated to gain auxiliary data for enhancing the friction modelling and to characterise the contact behaviour.

Investigation of Jointed Structures Using the Multiharmonic Balance Method

This paper deals with the investigation of a jointed structure in the frequency domain. Because of the nonlinearity of most jointed systems the steady state response shall be calculated utilizing the Multiharmonic Balance Method (MHBM). In the framework of this method the response displacements of the system are assumed to be periodic and are approximated by a Fourier-type series expansion considering a limited number of harmonic parts. The MHBM is presented for a system containing a bolted lap joint. For the contacting bodies small deformations and linear elastic material behavior are assumed. The system will be modeled with the Finite Element Method (FEM) using so called Zero Thickness (ZT) elements for discretizing the contact plane. Since this contact area is known and only small relative displacements occur, the suggested contact element is well suited for the present problem. The ZT elements contain a three-dimensional nonlinear contact law considering dry friction effects. A model updating process is used to fit the calculation results with respect to the measured data of a real system.

Elasto-plastic Contact Of Rough Surfaces

If two rough surfaces are brought into contact, asperities will deform either elastically or plastically. While the mode of initial deformation will depend on geometrical and material properties at any single asperity, the deformation upon unloading and reloading will be in the elastic regime. As it is not feasible to model the behaviour of each single asperity in a structural mechanics application, global contact laws are needed. In the present paper such contact laws are derived for several deformation models.

Indentation Of A Functionally Graded Elastic Solid:Application Of An Adhesively Bonded Plate Model

This paper examines the influence of adhesive bonding on the flexural interaction between a thin plate and an isotropic elastic halfspace. The modelling is developed for the possible application of the results to the study of functionally stiffness graded elastic media where the grading is restricted to the near surface region.

Numerical Continuation Methods for the Concept of Non-linear Normal Modes

Non-linear normal modes (NNMs) can be considered as a non-linear analogon to the description of linear systems with linear normal modes (LNMs). The definition of NNMs can be found in Vakakis (Normal modes and localization in nonlinear systems, Wiley, New York, 1996). Small systems with a low number of degrees of freedom and non-linear couplings (cubic springs) are investigated here. With increasing energy in the system the progressive non-linearity leads to a hardening effect. One typical dynamical property of non-linear systems is the frequency-energy dependency of the resulting oscillations. A good graphic illustration is to plot such a dependency in a so called frequency-energy plot (FEP). A NNM branch can be calculated by a numerical continuation method with starting at low energy level in a quasi linear regime and increasing the energy and reducing the period of the oscillation iteratively. Thereby a branch is a family of NNM oscillations with qualitatively equal motion properties (Peeters et al., Mech Syst Signal Process 23:195–216, 2009). In non-linear systems internal resonances and other phenomena can occur. Several tongues can bifurcate from a NNM branch. Therefore ordinary continuation methods fail at such bifurcation points. Here a predictor-corrector-method is used and different corrector algorithms are discussed for the branch continuation.

Epistemic and Aleatoric Uncertainty in Modeling

One major difficulty that exists in reconciling model predictions of a system with experimental measurements is assessing and treating the uncertainties in the system. There are several enumerated sources of uncertainty in model prediction of physical phenomena, the primary ones being: model form error, aleatoric uncertainty of model parameters, epistemic uncertainty of model parameters, and model solution error. These forms of uncertainty can have insidious consequences for modeling if not properly identified and accounted for. In particular, confusion between aleatoric and epistemic uncertainty can lead to a fundamentally incorrect model being inappropriately fit to data such that the model seems to be correct. As a consequence, the model will not be able to be predictive outside of the regime in which it is fit as it is a model of the incorrect physics. This chapter looks at the effects of aleatoric and epistemic uncertainty in order to make recommendations for properly accounting for them in a modeling framework. Two perspectives in particular are examined: first, the propagation in uncertainty in predictions as parameters fit to all test regimes are assimilated, and second, the informational entropy in parameter space as data are fitted to experiments in all test regimes.