Über die exakte Abbildung ausgewählter dreidimensionaler Kontakte auf Systeme mit niedrigerer räumlicher Dimension (English shop)

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In 2007, the foundation was laid for a new theory of contact and friction, the so-called method of dimensionality reduction. V.L. Popov and T. Geike succeeded in mapping the three-dimensional Hertzian contact exactly onto a one-dimensional model. Building on this, they developed a 1D model, excellently suited to typical tribological systems, for simulating the 3D contact of rough surfaces, coupled with an enormous saving in computing time. Following the basic idea of the reduction method, the present work deals primarily with the exact mapping of three-dimensional contact problems onto systems of lower spatial dimension. Starting from classical elasticity theory, analytical proof is first provided that every conformal, frictionless, axially symmetric normal contact can be mapped onto a one-dimensional model that exactly reproduces the relationships between normal force, indentation depth and contact radius of the original. In addition, various possibilities are demonstrated by means of which the real contact stresses can be exactly extracted from the dynamics of the substitute system. The generalisation of the adhesion theory of Johnson, Kendall and Roberts to arbitrarily shaped axially symmetric contacts dates back to 2005. The dissertation demonstrates that this theory can likewise be mapped exactly and in a very simple manner by means of a one-dimensional model. Furthermore, a correspondence principle is derived from certain form invariances, valid for the normal contact and the axially symmetric tangential contact. It permits the exact conversion between the field quantities of plane and axially symmetric systems. The stresses and displacements within the axially symmetrically loaded half-space can thus be exactly reproduced from a state of plane strain or plane stress. The correspondence principle is equally applicable to layered or to inhomogeneous half-spaces. An interface between this 2D reduction and the 1D model is presented, and the exactness of the reduction algorithm is corroborated on the basis of selected numerical simulations. The principle is not tied to any numerical discretisation method and can be implemented without difficulty in any commercial software. In practice, two-dimensional models are frequently employed to simulate three-dimensional tribological systems. All of these accept a certain error, since the nature of plane and spatial elastic solids is fundamentally different. If, by contrast, elastically inhomogeneous two-dimensional media are considered, in particular the Gibson half-plane, various characteristics of the homogeneous three-dimensional continuum can be reproduced exactly. Such media are likewise a subject of the work; particular attention is devoted to the tangential contact of a sphere in the state of partial slip. In addition, the dissertation contains a systematically structured chapter on the isotropy of elastic lattices. With a view to representing the isotropic plane continuum, the existing models are compared under kinematic-dynamic and energetic aspects. From the standpoint of contact mechanics, an error analysis based on numerical simulations appears difficult, because satisfying all boundary conditions in the plane case constitutes a problem in its own right. Departing from the principle of the work of seeking to map contact problems exactly, the contact of self-affine fractal surfaces is finally investigated numerically with the aid of a three-dimensional hierarchical lattice model. The in part very strong assumptions lead to a considerable reduction in degrees of freedom and thus to a saving in computing time. The extent to which acceptable results can be achieved with this model with regard to contact area, pressure distribution, relative approach of the surfaces as well as topography and tightness on different scales is discussed.