The Firefighter Problem: A Structural Analysis

We consider the complexity of the firefighter problem where b>=1 firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al.,2007) and on trees of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget b>=1. We then show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph

WTC2005-64085 MODELS OF FRICTION AND WEAR OF DLC FILMS

ABSTRACT Wear and friction of DLC (diamond-like carbon) covered counterparts are under consideration. Experiments showed that abrasion is the leading wear mechanism at the beginning of the wear process. However, the ability of the surface to wear away the counterpart reduces very rapidly, often as a power law function of the cycle numbers. This phenomenon was explained assuming that the initial abrasiveness of a coating is determined by the number of the nano-sharp asperities that were in contact with the counterpart, i.e. by the number of the sharp asperities within the nominal region of contact. On this basis, a model of abrasive wear was developed, using the concept of statistical self-similarity of distribution of the nano-sharp asperities within the current contact region. After the sharp asperities were blunted or removed from the contact region, the wear is related either to phase transformations or to chemical mechanisms. Recent experimental studies of dry sliding between two hydrogenated DLC coated counterparts in low oxygen environment showed that adsorbates have considerable influence on friction and the friction coefficient increases with the increasing of the time interval between contacts. The observed friction phenomena are assumed caused by a reaction between the adsorbate and carbon atoms of the coatings, and when the slider passes a point on the track, it removes mechanically some adsorbate from the surface. The mechanical action leads to re-exposure of the surface to gases in the environment. We assume that there is a transient short-life high temperature field at the vicinities of contacting protuberances that may cause various transformations of the surface. We suppose that first an adsorbate molecule becomes physically adsorbed to the surface and then chemisorbtion may occur between the carbon atoms of the coating and the 'sticky' oxygen atoms. The atoms or molecules of adsorbate interact with the conterpart. Our modeling established a direct connection between this kind of molecular friction and gradual wear. Using the new adsorption-desorption model, the numerical analyses of the friction coefficient were compared with experimental DLC friction results. INTRODUCTION Carbon-containing thin films such as DLC and boron carbide have the ability to enhance the fatigue resistance of heavily loaded steel component

Analytical study of fundamental nanoindentation test relations for indenters of non-ideal shapes

Nanoindentation techniques provide a unique opportunity to obtain mechanical properties of materials of very small volumes. The load–displacement and load–area curves are the basis for nanoindentation tests, and their interpretation is usually based on the main assumptions of the Hertz contact theory and formulae obtained for ideally shaped indenters. However, real indenters have some deviation from their nominal shapes leading researchers to develop empirical 'area functions' to relate the apparent contact area to depth. We argue that for both axisymmetric and three-dimensional cases, the indenter shape near the tip can be well approximated by monomial functions of radius. In this case problems obey the self-similar laws. Using Borodich's similarity considerations of three-dimensional contact problems and the corresponding formulae, fundamental relations are derived for depth of indentation, size of the contact region, load, hardness, and contact area, which are valid for both elastic and non-elastic, isotropic and anisotropic materials. For loading the formulae depend on the material hardening exponent and the degree of the monomial function of the shape. These formulae are especially important for shallow indentation (usually less than 100 nm) where the tip bluntness is of the same order as the indentation depth. Uncertainties in nanoindentation measurements that arise from geometric deviation of the indenter tip from its nominal geometry are explained and quantitatively described