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.

Bioinspired materials that act like living tissues and can repair internal damage by themselves, i.e. self-healing materials, are an active field of research. Here a methodology for experimental testing of self-healing ability of soft polymer materials is described. The methodology is applied to a recently synthesized polyurethane material Smartpol (ADHTECH Smart Polymers & Adhesives S.L., Alicante, Spain). Series of tests showed that the material demonstrated self-healing ability. The tests included the following steps: each Smartpol specimen was cut in halves, then it was put together under compression, and after specified amount of time, it was pulled apart while monitoring the force in contact. The test conditions were intentionally chosen to be non-ideal. These non-idealities simultaneously included: (1) separation time was rather long (minutes and dozens of minutes), (2) there was misalignment of specimen parts when they were put together, (3) contacting surfaces were non-flat, and (4) repeated testing of the same specimens was performed and, therefore, repeated damage was simulated. Despite the above, the recovery of structural integrity (self-healing) of the material was observed which demonstrated the remarkable features of Smartpol. Analysis of the experimental results showed clear correlation between adhesion forces (observed through the values of maximum pull-off force) and the time in contact which is a clear indicator of self-healing ability of material. It is argued that the factors contributing to self-healing of the tested material at macro-scale were high adhesion and strong viscoelasticity. The results of fitting the force relaxation data by means of mathematical model containing multiple exponential terms suggested that the material behaviour may be adequately described by the generalized Maxwell model.

Classical methods of material testing become extremely complicated or impossible at micro-/nanoscale. At the same time, depth-sensing indentation (DSI) can be applied without much change at various length scales. However, interpretation of the DSI data needs to be done carefully, as length-scale dependent effects, such as adhesion, should be taken into account. This review paper is focused on different DSI approaches and factors that can lead to erroneous results, if conventional DSI methods are used for micro-/nanomechanical testing, or testing soft materials. We also review our recent advances in the development of a method that intrinsically takes adhesion effects in DSI into account: the Borodich–Galanov (BG) method, and its extended variant (eBG). The BG/eBG methods can be considered a framework made of the experimental part (DSI by means of spherical indenters), and the data processing part (data fitting based on the mathematical model of the experiment), with such distinctive features as intrinsic model-based account of adhesion, the ability to simultaneously estimate elastic and adhesive properties of materials, and non-destructive nature.

The new method of the forced resonance vibrations construction in mechanical systems with internal resonance is represented. According to this approach, the generalized theory of non-linear normal vibration modes by Shaw and Pierre, the modified Rauscher method and the harmonic balance method are combined with a new iterative computation procedure. The proposed approach is used in analysis of the single-disk rotor system with the isotropic-elastic shaft and the non-linear supports of Duffing type. Gyroscopic effects, asymmetrical disposition of the disk on the shaft and internal resonance are also taken into account. The NNM approach allows reducing the 8-DOF problem of the rotor dynamics to the 2-DOF non-linear system for each non-linear normal mode. Both the model of massless supports and the model of supports with inertial effects are considered. It is shown that in last case all resonance regimes are separated into two different kinds. First kind corresponds to cyclic symmetric trajectories in a system's configuration space; the second kind corresponds to centrally symmetric ones. Regimes of the first kind can be evaluated by the use of the simplified mathematical model proposed in this work. Simplified model consists only of four generalized coordinates instead of the eight initial ones.

The classic Johnson–Kendall–Roberts (JKR) contact theory was developed for frictionless adhesive contact between two isotropic elastic spheres. The advantage of the classical JKR formalism is the use of the principle of superposition of solutions to non-adhesive axisymmetric contact problems. In the recent years, the JKR formalism has been extended to other cases, including problems of contact between an arbitrary-shaped blunt axisymmetric indenter and a linear elastic half-space obeying rotational symmetry of its elastic properties. Here the most general form of the JKR formalism using the minimal number of a priori conditions is studied. The corresponding condition of energy balance is developed. For the axisymmetric case and a convex indenter, the condition is reduced to a set of expressions allowing explicit transformation of force–displacement curves from non-adhesive to corresponding adhesive cases. The implementation of the developed theory is demonstrated by presentation of a two-term asymptotic adhesive solution of the contact between a thin elastic layer and a rigid punch of arbitrary axisymmetric shape. Some aspects of numerical implementation of the theory by means of Finite-Element Method are also discussed. This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.