Vladislav Aleshin

Acoustic probing of the jamming transition in an unconsolidated granular medium

Experimentally determined dispersion relations for acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through nonaffine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure.

Acoustic modes propagating along the free surface of granular media

In unconsolidated granular materials under gravity there exist acoustical waves propagating along the surface with anomalously low sound velocity. The presented theory describes these guided surface acoustic modes (GSAM) confined between the surface of the granular materials and in-depth layers with increasing rigidity. The analysis is based on the obtained original analytical solution of the Helmholtz equation that has never been used both in classical and quantum mechanics. This solution is valid for a particular rigidity profile, whereas the general case of grains with or without adhesion has been analyzed numerically. In contrast to the Rayleigh wave polarized in the sagittal (vertical) plane, which is the unique localized mode in a homogeneous solid, an infinite number of modes with sagittal polarization as well as an infinite number of shear horizontal modes have been found. The difference in physical mechanisms of localization is discussed, and the transformation of the GSAMs into the Rayleigh wave at the increasing adhesion is demonstrated: The first sagittal mode transforms into the Rayleigh one, while the others delocalize. The theory explains the experimentally observed magnitude of velocity for the acoustic waves in sand elliptically polarized in the sagittal plane.

Strain wave evolution equation for nonlinear propagation in materials with mesoscopic mechanical elements

Nonlinear wave propagation in materials, where distribution function of mesoscopic mechanical elements has very different scales of variation along and normally to diagonal of Preisach-Mayergoyz space, is analyzed. An evolution equation for strain wave, which takes into account localization of element distribution near the diagonal and its slow variation along the diagonal, is proposed. The evolution equation provides opportunity to model propagation of elastic waves with strain amplitudes comparable to and even higher than characteristic scale of element localization near Preisach-Mayergoyz space diagonal. Analytical solutions of evolution equation predict nonmonotonous dependence of wave absorption on its amplitude in a particular regime. The regime of self-induced absorption for small-amplitude nonlinear waves is followed by the regime of self-induced transparency for high-amplitude waves. The developed theory might be useful in seismology, in high-pressure nonlinear acoustics, and in nonlinear acoustic diagnostics of damaged and fatigued materials.

Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part II: Numerical implementation

HIGHLIGHTSWe present a 2D model for wave propagation in samples with internal contacts.The model contains two units: an elastic wave propagation model and a crack model.The crack model considers roughness, friction, memory dependence and hysteresis.Three contact states can be described: contact loss, total sliding and partial slip.We discuss an instructive numerical example of nonlinear wave‐crack interaction. ABSTRACT Our study aims at the creation of a numerical toolbox that describes wave propagation in samples containing internal contacts (e.g. cracks, delaminations, debondings, imperfect intergranular joints) of known geometry with postulated contact interaction laws including friction. The code consists of two entities: the contact model and the solid mechanics module. Part I of the paper concerns an in‐depth description of a constitutive model for realistic contacts or cracks that takes into account the roughness of the contact faces and the associated effects of friction and hysteresis. In the crack model, three different contact states can be recognized: contact loss, total sliding and partial slip. Normal (clapping) interactions between the crack faces are implemented using a quadratic stress‐displacement relation, whereas tangential (friction) interactions were introduced using the Coulomb friction law for the total sliding case, and the Method of Memory Diagrams (MMD) in case of partial slip. In the present part of the paper, we integrate the developed crack model into finite element software in order to simulate elastic wave propagation in a solid material containing internal contacts or cracks. We therefore implemented the comprehensive crack model in MATLAB® and introduced it in the Structural Mechanics Module of COMSOL Multiphysics®. The potential of the approach for ultrasound based inspection of solids with cracks showing acoustic nonlinearity is demonstrated by means of an example of shear wave propagation in an aluminum sample containing a single crack with rough surfaces and friction.

Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part I: Theoretical background

HIGHLIGHTSWe present a 2D model for wave propagation in samples with internal contacts.The crack model considers roughness, friction, memory dependence and hysteresis.Three contact states can be described: contact loss, total sliding and partial slip.Sliding is described by Coulombs friction law or the Method of Memory Diagrams.The full crack model algorithm is illustrated by means of a numerical example. ABSTRACT Our study aims at the creation of a numerical toolbox that describes wave propagation in samples containing internal contacts (e.g. cracks, delaminations, debondings, imperfect intergranular joints) of known geometry with postulated contact interaction laws including friction. The code consists of two entities: the contact model and the solid mechanics module. Part I of the paper concerns the modeling of internal contacts (called cracks for brevity), while part II is related to the integration of the developed contact model into a solid mechanics module that allows the description of wave propagation processes. The contact model is used to produce normal and tangential load‐displacement relationships, which in turn are used by the solid mechanics module as boundary conditions at the internal contacts. Due to friction, the tangential reaction curve is hysteretic and memory‐dependent. In addition, it depends on the normal reaction curve. An essential feature of the proposed contact model is that it takes into account the roughness of the contact faces. On one hand, accounting for roughness makes the contact model more complicated since it gives rise to a partial slip regime when some parts on the contact area experience slip and some do not. On the other hand, as we will show, the concept of contact surfaces covered by asperities receding under load makes it possible to formulate a consistent contact model that provides nonlinear load‐displacement relationships for any value of the drive displacements and their histories. This is a strong advantage, since this way, the displacement‐driven model allows for a simple explicit procedure of data exchange with the solid mechanics module, while more traditional flat‐surface contacts driven by loads generate a complex iterative procedure. More specifically, the proposed contact model is based on the previously developed method of memory diagrams that allows one to automatically obtain memory‐dependent solutions to frictional contact problems in the particular case of partial slip. Here we extend the solution onto cases of total sliding and contact loss which is possible while using the displacement‐driven formulation. The method requires the knowledge of the normal contact response obtained in our case as a result of statistical consideration of roughness of contact faces.

Reflection of Nonlinear Acoustic Waves from the Mechanically Free Surface of an Unconsolidated Granular Medium

The problem of the nonlinear reflection of acoustic waves from a mechanically free surface of an unconsolidated granular layer under gravity is solved analytically using the successive approximations method. The theory revealed specific dependencies of the characteristics of the generated acoustic harmonics of longitudinal and shear waves on frequency and the thickness of the granular layer, which are related to a power-law gravity-induced depth stratification of linear and nonlinear mechanical properties of the granular layer. The developed theory could be useful for the analysis of the acoustic experiments directed to the investigation of fundamental mechanical properties of unconsolidated granular media near the jamming transition taking place at zero confining pressure.

Physical constitutive equations for nonlinear acoustics of materials with internal contacts

A physical theory for the acoustics of microdamaged materials is presented based on the analysis of internal contact roughness. As a start, we adopt the Whitehouse and Archard contact mechanics yielding a decomposition of continuously roughs surfaces into a set of spherical asperities of different sizes and heights. Coupled with a model for the force‐displacement relationship for microcontacts, taking into account adhesion hysteresis, it provides a description for an individual intergranular contact. We upscale this model using continuum mechanics of solids with cracks and derive a set of equations for the evolution of acoustical waves. We use this model to predict results for nonlinear acoustical experiments and derive amplitude dependences of resonant frequency and harmonics, and slow dynamics phenomena.

Simulation Study of the Localization of a Near-Surface Crack Using an Air-Coupled Ultrasonic Sensor Array

The importance of Non-Destructive Testing (NDT) to check the integrity of materials in different fields of industry has increased significantly in recent years. Actually, industry demands NDT methods that allow fast (preferably non-contact) detection and localization of early-stage defects with easy-to-interpret results, so that even a non-expert field worker can carry out the testing. The main challenge is to combine as many of these requirements into one single technique. The concept of acoustic cameras, developed for low frequency NDT, meets most of the above-mentioned requirements. These cameras make use of an array of microphones to visualize noise sources by estimating the Direction Of Arrival (DOA) of the impinging sound waves. Until now, however, because of limitations in the frequency range and the lack of integrated nonlinear post-processing, acoustic camera systems have never been used for the localization of incipient damage. The goal of the current paper is to numerically investigate the capabilities of locating incipient damage by measuring the nonlinear airborne emission of the defect using a non-contact ultrasonic sensor array. We will consider a simple case of a sample with a single near-surface crack and prove that after efficient excitation of the defect sample, the nonlinear defect responses can be detected by a uniform linear sensor array. These responses are then used to determine the location of the defect by means of three different DOA algorithms. The results obtained in this study can be considered as a first step towards the development of a nonlinear ultrasonic camera system, comprising the ultrasonic sensor array as the hardware and nonlinear post-processing and source localization software.

Frictional contact of two spheres for arbitrary 2D loading: memory diagrams and Preisach analysis

Frictional interaction of two elastic spheres is a fundamental problem of contact mechanics. The tangential displacement depends on the normal and tangential force in a complicated and, in general, hysteretic way. The key characteristic of the system is the traction distribution in the contact zone that has a complex piecewise-smooth character. We propose a method of memory diagrams that enables to extract all memory information from traction distributions and store it in a compact manner. Then, instead on considerations on the case-to-case basis of a particular loading history, a general algorithm applicable for any loading history can be formulated. Finally, we used the obtained solution to prove that friction of two spheres is the Preisach system. The Preisach formulation of the problem enables to replace the custom-made algorithm of the memory diagrams method by a standard routine used over more than 70 years in ferromagnetism.

General solutions to the mechanical contact problem

A theory for mechanical contact of solids with friction is developed for a wide range of contacting profiles and loading histories. The starting point of the approach is the normal load-displacement dependency that can be obtained via analytical solutions for regular e.g. axisymmetric profiles and via modeling for random surface topologies. Then, the Jaeger elastic principle based on similarity of equations for normal and tangential deformation allows us to obtain the tangential load-displacement relation for simple loading, i.e. for constant normal and tangential actions, and, moreover, to replace rough profiles by equivalent axisymmetric ones. Since in the presence of tangential action a zone of stick and a zone of slip appear in the contact area, the problem becomes memory dependent. An original general scheme of memory organization called memory diagram is presented. Evolutions of memory diagrams are governed by a special set of rules; the solution for a given memory diagram is purely analytical, alth...