Boris A. Galanov

Self-similar problems of elastic contact for non-convex punches

Self-similar problems of contact for non-convex punches are considered. The non-convexity of the punch shapes introduces differences from the traditional self-similar contact problems when punch profiles are convex and their shapes are described by homogeneous functions. First, three-dimensional Hertz type contact problems are considered for non-convex punches whose shapes are described by parametric-homogeneous functions. Examples of such functions are numerous including both fractal Weierstrass type functions and smooth log-periodic sine functions. It is shown that the region of contact in the problems is discrete and the solutions obey a non-classical self-similar law. Then the solution to a particular case of the contact problem for an isotropic linear elastic half-space when the surface roughness is described by a log-periodic function, is studied numerically, i.e. the contact problem for rough punches is studied as a Hertz type contact problem without employing additional assumptions of the multi-asperity approach. To obtain the solution, the method of non-linear boundary integral equations is developed. The problem is solved only on the fundamental domain for the parameter of self-similarity because solutions for other values of the parameter can be obtained by renormalization of this solution. It is shown that the problem has some features of chaotic systems, namely the global character of the solution is independent of fine distinctions between parametric-homogeneous functions describing roughness, while the stress field of the problem is sensitive to small perturbations of the punch shape.

Elastic-plastic contact mechanics of indentations accounting for phase transformations

A contact mechanics model is developed which takes into account possible phase transformations in materials induced by hydrostatic and shear stresses associated with indentation. The proposed model allows prediction of the average thickness and approximate shape of the phase transformation zone in semiconductors and ceramics under various types of diamond indenters. The results of theoretical calculation are in good agreement with the available experimental data.

Stress–strain State of Multiwall Carbon Nanotube Under Internal Pressure

A considered application of carbon nanotubes is nanopiping in nanofluidic devices. The use of nanotubes for fluid transport requires large-diameter tubes that can sustain prescribed loading without failure. Two models of the stress–strain state of long multiwall carbon nanotubes, subjected to internal pressure, are described. Cylindrical nanotubes having a Russian doll structure have been considered. It is assumed that the deformations are linear elastic and negligible along the tube axis (in comparison with the radial deformations). This assumption is not restrictive for potential applications of nanotubes, where their deformations must be small and reversible. The distance between the layers is small in comparison to the radii of curvature of graphite layers. In the case of several carbon layers, a discrete model (DM) is proposed. The solutions of DM equations, with corresponding boundary conditions, determine the stresses between the layers, the forces in the layers, and the deformation of the layers. For the case of thick walls built of numerous carbon layers, a continuous model (CM) is proposed. The main CM equation is the Eulers differential equation with corresponding boundary conditions. Its solution defines the continuous distribution of the stresses and strains across the wall thickness of the tube.

Model of oxide scale growth on Si3N4 ceramics: nitrogen diffusion through oxide scale and pore formation

Abstract Model of oxide scale growth on Si 3 N 4 ceramics has been developed. It accounts for the formation of a porous SiO 2 layer between the dense SiO 2 scale and ceramics. Pores are filled with nitrogen, which is formed as a result of a chemical reaction of oxygen with silicon nitride at the interface between the oxide scale and ceramics. Oxygen diffuses through the oxide scale to the chemical reaction site. Differential equations of the model describe the growth of the total thickness of the oxide scale and movement of the boundary between the porous SiO 2 layer and dense SiO 2 . This model can be used to describe oxidation kinetics of silicon nitride ceramics, sialons, other nitrides (TiN, ZrN, etc.) and, in general, materials that form a protective oxide scale having a low permeability for gaseous oxidation products.

An inverse problem for adhesive contact and non-direct evaluation of material properties for nanomechanics applications

Abstract We show how the values of the effective elastic modulus of contacting solids and the work of adhesion, that are the crucial material parameters for application of theories of adhesive contact to nanomechanics, may be quantified from a single test using a non-direct approach (the Borodich-Galanov (BG) method). Usually these characteristics are not determined from the same test, e.g. often sharp pyramidal indenters are used to determine the elastic modulus from a nanoindentation test, while the work of adhesion is determined from a different test by the direct measurements of pull-off force of a sphere. The latter measurements can be greatly affected by roughness of contacting solids and they are unstable due to instability of the load-displacement diagrams at tension. The BG method is based on an inverse analysis of a stable region of the force-displacements curve obtained from the depth-sensing indentation of a sphere into an elastic sample. Various aspects related to solving the inverse problem for adhesive contact and experimental evaluation of material properties for nanomechanics applications are discussed. It is shown that the BG method is simple and robust. Some theoretical aspects of the method are discussed and the BG method is developed by application of statistical approaches to experimental data. The advantages of the BG method are demonstrated by its application to soft polymer (polyvinylsiloxane) samples.

Contact probing of stretched membranes and adhesive interactions: graphene and other two-dimensional materials

Contact probing is the preferable method for studying mechanical properties of thin two-dimensional (2D) materials. These studies are based on analysis of experimental force–displacement curves obtained by loading of a stretched membrane by a probe of an atomic force microscope or a nanoindenter. Both non-adhesive and adhesive contact interactions between such a probe and a 2D membrane are studied. As an example of the 2D materials, we consider a graphene crystal monolayer whose discrete structure is modelled as a 2D isotropic elastic membrane. Initially, for contact between a punch and the stretched circular membrane, we formulate and solve problems that are analogies to the Hertz-type and Boussinesq frictionless contact problems. A general statement for the slope of the force–displacement curve is formulated and proved. Then analogies to the JKR (Johnson, Kendall and Roberts) and the Boussinesq–Kendall contact problems in the presence of adhesive interactions are formulated. General nonlinear relations among the actual force, displacements and contact radius between a sticky membrane and an arbitrary axisymmetric indenter are derived. The dimensionless form of the equations for power-law shaped indenters has been analysed, and the explicit expressions are derived for the values of the pull-off force and corresponding critical contact radius.

Adhesive contact problems for a thin elastic layer: Asymptotic analysis and the JKR theory

Contact problems for a thin compressible elastic layer attached to a rigid support are studied. Assuming that the thickness of the layer is much less than the characteristic dimension of the contact area, a direct derivation of asymptotic relations for displacements and stress is presented. The proposed approach is compared with other published approaches. The cases are established when the leading-order approximation to the non-adhesive contact problems is equivalent to contact problem for a Winkler–Fuss elastic foundation. For this elastic foundation, the axisymmetric adhesive contact is studied in the framework of the Johnson–Kendall–Roberts (JKR) theory. The JKR approach has been generalized to the case of the punch shape being described by an arbitrary blunt axisymmetric indenter. Connections of the results obtained to problems of nanoindentation in the case that the indenter shape near the tip has some deviation from its nominal shape are discussed. For indenters whose shape is described by power-law functions, the explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius.

CONTACT PROBLEMS AND NANOINDENTATION TESTS FOR INDENTERS OF NON- IDEAL SHAPES AND EFFECTS OF MOLECULAR ADHESION

Depth-sensing nanoindentation, when the displacement the indenter is continuously monitored is widely used for analysis and estimations of mechanical properties of materials. Starting from pioneering papers by Bulychev, Alekhin, Shorshorov and their co-workers, nanoindentation tests are connected with Hertzian contact problems and the frictionless BASh relation is commonly used for evaluation of elastic modulus of materials. We discuss further the connections between Hertz type contact problems and nanoindentation tests and derive fundamental relations for depth-sensing nanoindentation for indenters of various shapes and for various boundary conditions within the contact region. For the loading branch, relations are derived among depth of indentation, size of the contact region, load, hardness, and contact area, using authors’ scaling formulae. The relations are valid for indenters of non-ideal shapes, whose shape function is a monomial function an arbitrary degree d, in particular for blunted pyramidal indenters when 1 < d < 2. We show that some uncertainties in nanoindentation measurements, which are sometimes attributed to properties of the material, can be explained and quantitatively described by properly accounting for geometric deviation of the indenter tip from its nominal geometry. Then relation is derived for the slope of the unloading branch of adhesive (no-slip) indentation. The relation is analogous the frictionless BASh relation and it is independent of the geometry of the indenter. Further, the JKR theory of contact the presence of forces of molecular adhesion is extended to describe contact between a monomial indenter of an arbitrary degree d and an elastic sample. Finally, some exact formulae are obtained for adhesive contact (both the no-slip contact and the contact in the presence of molecular adhesive forces) between indenters and isotropic, linear elastic materials. In particular, it is shown that the BASh formula is still valid for contact between a flat punch and a soft elastic sample in the presence of molecular adhesive forces (the Boussinesq-Kendall problem).Copyright

Carbon Nanotubes Under Internal Pressure

A considered application of carbon nanotubes is nanopiping in nanofluidic devices. The use of nanotubes for fluid transport requires large-diameter tubes that can sustain prescribed loading without failure. Two models of the stress-strain state of long multiwall carbon nanotubes, subjected to internal pressure, are described. Cylindrical nanotubes having a Russian doll structure have been considered. It is assumed that the deformations are linear elastic and negligible along the tube axis (in comparison with the radial deformations). This assumption is not restrictive for potential applications of nanotubes, where their deformations must be small and reversible. The distance between the layers is small in comparison to the radii of curvature of graphite layers. In the case of several carbon layers, a discrete model (DM) is proposed. The solutions of DM equations, with corresponding boundary conditions, determine the stresses between the layers, the forces in the layers, and the deformation of the layers. For the case of thick walls built of numerous carbon layers, a continuous model (CM) is proposed. The main CM equation is the Euler’s differential equation with corresponding boundary conditions. Its solution defines the continuous distribution of the stresses and strains across the wall thickness of the tube.