Slip Waves Research Articles

Metastable core-shell precipitation strengthened high-entropy alloys fabricated by direct energy deposition with multi-stage terrace-like slip wave toughening

This study investigates the development of novel high-entropy alloys (HEAs) with enhanced mechanical properties through an innovative fabrication method of direct energy deposition (DED). The focus is on the creation of metastable core-shell precipitation-strengthened HEAs that exhibit a unique multi-stage terrace-like slip wave toughening mechanism, a novel approach to improving both strength and ductility simultaneously. Mechanical testing reveals that the developed HEAs exhibit superior mechanical properties, including high yield strength, ultimate tensile strength, and exceptional ductility. The improvement in these properties is attributed to the multi-stage terrace-like slip wave toughening mechanism activated by the unique microstructural features. This toughening mechanism involves the sequential activation of slip systems, facilitated by the stress concentration around the core-shell precipitates and the subsequent propagation of slip waves across the material. The terrace-like pattern of these slip waves enhances the material's ability to deform plastically, providing a significant toughening effect while maintaining high strength levels. Furthermore, the study delves into the fundamental interactions between the microstructural elements and the deformation mechanisms. It elucidates how the core-shell precipitates and the matrix cooperate to distribute stress uniformly, delay the onset of necking, and prevent premature failure. This synergistic interaction between the microstructural features and the slip wave toughening mechanism is central to the remarkable balance of strength and ductility achieved in the HEAs. The introduction of a multi-stage terrace-like slip wave toughening mechanism offers a new pathway to designing HEAs with an exceptional amalgamation of strength and ductility.

Frictional slip wave solutions for dynamic sliding between a layer and a half-space

Interfacial waves propagate along the interface between two elastic solids, and are the subject of basic interest to diverse scientific fields, like, seismology, geophysics, material science and composite structures. The focus of the present work is to analyse the interfacial waves arising due to frictional slipping between an elastic layer and an elastic half-space having dissimilar elastic properties, wherein the deformations are confined to the plane of the solids. The interfacial friction is modelled by employing a slip-dependent friction law. The elastodynamic relations between the displacement discontinuities across the fractured interface and the associated traction components of stress for pure slip events (no crack opening along the interface) are obtained. This allows us to derive a secular dispersion relation governing the interfacial wave solutions. The dispersion relation relating the phase velocity and wavenumber is analysed numerically. A family of solutions are shown to exist depending on the critical slip distance, δc, a parameter governing the interfacial friction law, and on the contrast in material properties. We demonstrate the existence of a number of wave modes, all being dispersive in nature. The fundamental wave mode with the lowest phase velocity exists over the largest range of wavenumbers, while other wave modes exist only at sufficiently high non-dimensional wavenumber. The dispersion curves for the frictional slip waves are found to lie between the dispersion curves for the cases of the smooth and the bonded interface. As a case study, the present model is used to study the dispersion of slip waves propagating in Earth's crust-mantle geophysical model. The field observations of surface wave dispersion appear to be in good agreement with that of the frictional slip waves studied here. Based on our results, we suggest an alternative explanation for the observed surface wave dispersion at the regional and global scales. It is suggested that frictional slip waves at the crust-mantle interface could propagate with phase velocities similar to the Rayleigh and Love surface waves.

Detachment waves in frictional contact: I — A two-mass system

This work studies a mathematical model for the motion of a two mass–spring–damper system with friction, and has a two-fold purpose. The first is to start a mathematical study of the way detachment or slip waves, when the system transits from stick to slip motion, propagate in discrete and continuous systems. The introduction of friction changes the problems into systems of differential set-valued inclusions, which are mathematically interesting, but rather complex. The second is to use such a ‘simple’ system as an example of a very general existence and uniqueness theorem for systems described by set-valued pseudomonotone operators, Kuttler and Shillor (1999) and Andrews et al. (2020).

Spatial Heterogeneity of Fault Slip and the Radiated Spectra of Ground Motions

ABSTRACTKinematic models of simulating earthquake radiation in seismic hazard analysis typically require prescribing the distribution of final slip over the hypothetical fault planes. The spatial spectra of heterogeneous slip affect the frequency spectra of the seismic body waves. The representation integral of elasticity provides a convenient analytical tool by which the relationships between the slip spectra in the wavenumber domain and the wave spectra in the frequency domain can be scrutinized. In the limit of the waves from a small source in the far field, the Fourier spectrum of wave displacement is the spectrum of the slip-rate function multiplied by the spatial slip spectrum representing fault directivity. A popular model for the latter is the k-square slip distribution. Classic results prescribe that for a typical ω-square source time function, such multiplication, conversely to a common assumption that the k-square slip distribution always leads to the ω-square decay of the high-frequency seismic spectra, can result in the ω−4 power-law decay. Such steep fall-off rates are highly unusual in observations, suggesting that the k-square heterogeneous slip in certain cases may significantly underpredict the realistic high-frequency ground motions, including peak velocities and accelerations. An alternative use of heterogeneous slip distributions would be to explain the additional high-frequency diminution of the observed spectra that is usually attributed to ad hoc cutoff (“fmax” or “kappa”) filters. The simple asymptotic relationships between heterogeneous fault slip and body wave spectra may not hold true in the vicinity of large earthquakes, at distances of main interest to hazard calculations.

Love Wave or Slip Wave?

Abstract Conventional models of the structure of the Earth, such as the Preliminary Reference Earth Model (PREM), assume a bonded interface between the crust and the upper mantle. The bonded contact model is consistent with the observation of Love waves during an earthquake. However, anomalies in the Love wave dispersion have been reported in the literature. When slip occurs at the crust-mantle interface, another kind of an interfacial wave, called the slip wave can exist. It is shown that the dispersion relation of the slip wave, with a slip weakening friction law, appears to be in agreement with the observations at seismic frequencies. This suggests that slip could occur at the crust-mantle interface.

Fast Solution of Rotor Losses in Inverter-Fed Cage Induction Motors With Skewed Slots

The conventional calculation of rotor losses in cage induction motors by the time-stepping finite element method requires the full slip waves of the rotor flux densities and currents. The calculations of the full slip waves are extremely expensive in CPU time, memory, and hard disk space for the induction motors with skewed slots and pulsewidth modulated supply. This article proposes a technique that reproduces those full slip waves by utilizing both the time varying and the spatial information of the rotor electromagnetic quantities and, hence, significantly saves the calculation cost. The more rotor bars per pair of poles the induction motor has, the more calculation cost this technique saves. The calculation costs of the proposed and conventional methods are compared for the rotor losses of a 5.5-kW inverter-fed skewed induction motor under load condition, and the calculated losses are validated by tests.

On the transient planar contact problem in the presence of dry friction and slip

This article models a plane strain dynamic contact problem for an infinite elastic body. Contact is established for x > 0 under the action of a time-dependent remote compressive load, and the system is subjected to a time-dependent remote tangential load. The two faces of the contact interface can slide relative to one other to accommodate mismatches between the shear stresses and the existing Coulomb friction. This is shown to be a cause of interfacial waves of slip, which this article models by deriving the general expression for the corrective traction due to an arbitrary slip distribution. This is achieved using a variant of the Wiener-Hopf technique. Combined with existing closed-form expressions for the interfacial tractions due to compressive and shear loads, this enables the formulation of the elastodynamic extension to the stick-and-slip problem of the Cattaneo-Mindlin type. Exploiting self-similarity a simple numerical algorithm is detailed for solving the resulting Volterra integral equations of the first kind. The solution is shown to display a number of features entirely missed in static problems: slip waves are shown to exist irrespective of the magnitude of the applied loads and the friction coefficient; a regime of reverse and forward slip is also shown to exist for low friction coefficients, brought about by the interfacial Rayleigh waves. The practical implications of these solutions are discussed.

A phase-plane analysis of localized frictional waves.

Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

A slip wave solution in antiplane elasticity

AbstractIt is shown that a slip wave solution exists for antiplane sliding of an elastic layer on an elastic half-space. It is a companion solution to the well-known Love wave solution.

Explicit expression of polarization vector for surface waves, slip waves, Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space x2≥0 does not depend explicitly on the material property in the half-space x2≤0. It depends on the material property in the half-space x2≤0 implicitly through the interfacial slip wave speed υ. The same is true for the polarization vector for the material in the half-space x2≤0.

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.

Microstructural self-organization triggered by twin boundaries during dry sliding wear

A novel microstructural self-organization reaction is observed near the surface of a Cu–Ni–Sn bronze subjected to dry sliding wear. Scanning electron microscopy using electron-backscattered diffraction reveals the formation of near periodic patterns comprising domains with alternating orientation separated by narrow boundaries. Remarkably, orientation patterning takes place only near twin boundaries intercepting the sliding surface, and only on a specific side of the boundaries. The domain period, which ranges from a few to tens of micrometers, increases with the applied load. This orientation patterning is suppressed by promoting steady sliding using solid lubricants such as Ag and self-generated oxide-bearing tribolayers. Transmission electron microscopy reveals planar dislocation glide in the orientation domains, with multiple active slip systems. A rationalization of these results is offered based on an instability triggered by the interaction of elastic stick–slip waves with pre-existing twin boundaries.

Self-organization at the frictional interface for green tribology

Despite the fact that self-organization during friction has received relatively little attention from tribologists so far, it has the potential for the creation of self-healing and self-lubricating materials, which are important for green or environment-friendly tribology. The principles of the thermodynamics of irreversible processes and of the nonlinear theory of dynamical systems are used to investigate the formation of spatial and temporal structures during friction. The transition to the self-organized state with low friction and wear occurs through destabilization of steady-state (stationary) sliding. The criterion for destabilization is formulated and several examples are discussed: the formation of a protective film, microtopography evolution and slip waves. The pattern formation may involve self-organized criticality and reaction-diffusion systems. A special self-healing mechanism may be embedded into the material by coupling the corresponding required forces. The analysis provides the structure-property relationship, which can be applied for the design optimization of composite self-lubricating and self-healing materials for various ecologically friendly applications and green tribology.

Existence of one-component Rayleigh waves, Stoneley waves, Love waves, slip waves and one-component waves in a plate or layered plate

It is known that one-component surface (Rayleigh) waves exist in an anisotropic elastic half-space. Since the solution shows that the displacement normal to the free surface vanishes everywhere, a onecomponent surface wave is also a one-component slip wave in the half-space if the boundary of the half-space is a slippery surface. We show that no other one-component slip waves exist for the halfspace. As to steady waves in a bimaterial that consists of two dissimilar anisotropic elastic materials, one-component slip waves can be constructed from two one-component surface waves. There are no other one-component slip waves for a bimaterial. By imposing the continuity of the displacement at the interface on the one-component slip wave, a one-component Stoneley wave is obtained. Although one-component waves for the half-space can also propagate in a homogeneous plate, we present new one-component waves in a plate for which the Stroh eigenvalue p is real. By superposition of the onecomponent waves in the layer and in the half-space, one-component Love waves can be constructed. Finally, we show that one-component waves can propagate in a layered plate.

Thermodynamics of surface degradation, self-organization and self-healing for biomimetic surfaces

Friction is a dissipative irreversible process; therefore, entropy is produced during frictional contact. The rate of entropy production can serve as a measure of degradation (e.g. wear). However, in many cases friction leads to self-organization at the surface. This is because the excess entropy is either driven away from the surface, or it is released at the nanoscale, while the mesoscale entropy decreases. As a result, the orderliness at the surface grows. Self-organization leads to surface secondary structures either due to the mutual adjustment of the contacting surfaces (e.g. by wear) or due to the formation of regular deformation patterns, such as friction-induced slip waves caused by dynamic instabilities. The effect has practical applications, since self-organization is usually beneficial because it leads to friction and wear reduction (minimum entropy production rate at the self-organized state). Self-organization is common in biological systems, including self-healing and self-cleaning surfaces. Therefore, designing a successful biomimetic surface requires an understanding of the thermodynamics of frictional self-organization. We suggest a multiscale decomposition of entropy and formulate a thermodynamic framework for irreversible degradation and for self-organization during friction. The criteria for self-organization due to dynamic instabilities are discussed, as well as the principles of biomimetic self-cleaning, self-lubricating and self-repairing surfaces by encapsulation and micro/nanopatterning.

Subsonic slip waves along the interface between two piezoelectric solids in sliding contact with local separation

The Stroh formalism of piezoelectric crystals and singular integral equation technique are applied to study the propagation of possible slip waves in presence of local separation at the interface between two frictionless contact piezoelectric solids, which are pressed together by uniaxial pressure and laid in the electric field. The problem is cast into a set of singular integral equations of which the closed solutions are obtained. Discussion on the existence of such slip waves is presented. The results show that such slip waves, which have square-root singularities at both ends of the local separation zones, can propagate in some special material combinations. And the existence of such slip waves is related with the applied mechanical and electric fields.

Subsonic slip waves and steady sliding at the interface between two anisotropic elastic half-spaces in frictional contact with stick–slip

The theoretical study is presented on the existence and behaviors of slip waves along a frictionally contact interface between two similar or dissimilar anisotropic solids which are pressed together by remote pressure and meanwhile sheared by remote shearing traction. We consider the wave motion in a symmetry plane perpendicular to the interface and thus the in-pane motion is uncoupled with the anti-plane motion. The external loads may or may not lead to steady rigid sliding between two solids, while the separation of the interface is excluded by assuming the applied pressure is large enough. The local stick–slip motion without local separation at the frictionally contact interface caused by the perturbed slip waves is studied. The Stroh formalism, together with the concept of the surface impedance tensor is employed. The boundary value problem involving unknown slip/stick zones is cast to a Hilbert singular integral equation with an unknown integral interval. The explicit solutions representing the slip waves are obtained. The existence and behaviors of such slip waves are discussed based on theoretical and numerical analysis. For the case of no initial steady sliding between two solids (i.e., the applied shearing traction is lower than the interface friction force), slip waves which are singular at one end of the slip zone may propagate. For the case of initial steady sliding (i.e., the applied shearing traction can break the interface friction), no such slip waves exist.

Interface waves on the sliding contact between identical piezoelectric crystals of general anisotropy

The theory is developed of acoustic waves guided by the perfect sliding contact between identical piezoelectric crystals. The wave propagation under two types of electric boundary conditions has been investigated: (1) non-metallized interface; (2) metallized interface. The main attention has been paid to the so-called subsonic slip waves. A number of theorems has been proved concerning the existence of such waves in structures of arbitrary symmetry. A thorough study of the existence conditions for slip waves has been carried out as applied to structures assuming certain particular orientations. The results of analytic considerations have been illustrated by numerical computations.

Analysis of dynamic instability of interfacial slip waves based on the surface impedance tensor

A new method relying on the Stroh formulism and the theory of the surface impedance tensor was developed to investigate the dynamic instability of interfacial slip waves. The concept of the surface impedance tensor was extended to the case where the wave speed is of a complex value, and the boundary conditions at the frictionally contacting interface were expressed by the surface impedance tensor. Then the boundary value problem was transformed to searching for zeroes of a complex polynomial in the unit circle. As an example, the steady frictional sliding of an elastic half-space in contact with a rigid flat surface was considered in details. A quartic complex characteristic equation was derived and its solution behavior in the unit circle was discussed. An explicit expression for the instability condition of the interfacial slip waves was presented.

Vibration and Stability of Frictional Sliding of Two Elastic Bodies With a Wavy Contact Interface

Abstract The stability of steady sliding, with Amontons-Coulomb friction, of two elastic bodies with a rough contact interface is analyzed. The bodies are modeled as elastic half-spaces, one of which has a periodic wavy surface. The steady-state solution yields a periodic set of contact and separation zones, but the stability analysis requires consideration of dynamic effects. By considering a spatial Fourier decomposition of the vibration modes, the dynamic problem is reduced to a singular integral equation for determining the eigenvectors (modes) and eigenvalues (frequencies). A pure imaginary root for an eigenvalue corresponds to a standing wave confined to the interface, while a positive/negative real part of the eigenvalue indicates instability/dissipation. A complex eigenvector indicates a complex mode of vibration. Two types of modes are considered—periodic symmetric modes with period equal to the surface waviness period and periodic antisymmetric modes with the period equal to twice the surface waviness. The singular integral equation is solved by reducing it to a system of linear algebraic equations using a Jacobi polynomial series and a collocation method. For the limit of zero friction it can be demonstrated analytically that the problem is self-adjoint and the eigenvalues, if they exist, are pure imaginary (no energy dissipation). These roots are found for a wide range of material properties and ratios of separation to contact zones lengths. For the limiting case of complete contact, the solution found corresponds to a superposition of two slip waves (generalized Rayleigh waves) traveling in opposite directions and forming a standing wave. With increasing separation zone length, the vibration frequency decreases from the slip wave frequency to the smaller surface wave frequency of the two bodies. With a nonzero separation zone, solutions can exist for material combinations which do not allow slip waves. For nonzero friction and sliding velocities, unstable solutions are found. The degree of instability is proportional to the product of the friction coefficient and the sliding velocity. These instabilities may contribute to the formation of friction-induced vibrations at high sliding speeds.