On the Oval Shapes of Beach Stones

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

CURRENT FLOW DYNAMICS IN PLASMA OPENING SWITCH

The electromagnetic fields and processes of current flow in the plasma channel of the plasma opening switch were considered in the electron hydrodynamics approximation. Taking into account the nature and features of the current flow through the plasma filled region of plasma opening switch allowed to obtain the structure of the current channel and its dynamics along the plasma. The current flow is a wave process. When this wave reaches the boundary of plasma it seems to be the break of current at this moment. The criterion of this break is obtained.

Effect of Wave Process of Plastic Deformation at Forging on the Fatigue Fracture Mechanism of Titanium Compressor Disks of Gas Turbine Engine

The low-cycle fatigue behavior of the VT3-1 titanium alloy (Ti–6Al–3Mo–2Cr alloy) under loading with a triangular and trapezoidal shape of cycle waveform was studied on round specimens prepared from forged compressor disks of a gas turbine engine. The filament type structure after forging has alternating filaments with the ductile and quasi-brittle state of the metal as a result of the wave process of plastic deformation during the metal forging process. The crack propagation, regardless of the cyclic waveform shape, occurs by the crack meso-tunneling mechanism: initially, the cracks propagate along the filaments by a quasi-brittle mechanism with the formation of a facetted pattern relief on the fracture surface reflecting the two-phase structure of the titanium alloy, and then, the bridge between the meso-tunnels is fractured with the formation of fatigue striations. The part of the crack growth duration Np/Nf in the durability Nf is determined on the basis of measuring the fatigue striation spacing, and it depends on the crack path with respect to the material filaments. The growth of a fatigue crack in the case of in-service failure of a compressor disk of a gas turbine engine is considered, taking into account the crack meso-tunneling effect, and the fatigue crack growth duration in the disk is determined on the basis of quantitative fractography.

Main directions of development of the theory of seismic resistance of building and structures

The article discusses the main directions of the development of the theory of seismic resistance of buildings and structures. Various models of the «building-foundation-foundation» system are presented. Much attention is paid to the influence of the foundation on seismic effects and the wave process of earthquakes.

Asymptotic analysis of the Korteweg-de Vries equation by the nonlinear WKB technique

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The non-linear WKB technique has been used to construct the asymptotic step-like solution to the equation. Such a solution contains regular and singular parts of the asymptotics. The regular part of the solution describes the background of the wave process, while its singular part reflects specific features associated with soliton properties. The singular part of the searched asymp\-totic solution has the main term that, like the soliton solution, is the quickly decreasing function of the phase variable $\tau$. In contrast, other terms do not possess this property. An algorithm of constructing asymptotic step-like solutions to the singularly perturbed Korteweg--de Vries equation with variable coefficients is presented. In some sense, the constructed asymptotic solution is similar to the soliton solution to the Korteweg-de Vries equation $u_t+uu_x+u_{xxx}=0$. Statement on the accuracy of the main term of the asymptotic solution is proven.