Elastic response of a free half-space to random force excitations applied on the boundary

A response of an elastic half-space to random forces applied normally to the free boundary is studied. This paper is the second part of the study we presented in [I.A. Shalimova, K.K. Sabelfeld, The response of an elastic three-dimensional half-space to random correlated displacement perturbations on the boundary, Physica A 389 (21) (2010) 4436–4449] where the case of random displacements on the boundary was considered. We analyze the white noise excitations in detail, and derive explicitly the mean of the elastic energy, the strain and displacement correlation tensors. Simulation algorithms are constructed both for displacement and strain random fields, which enables us to calculate any desired statistical characteristics.     

Phase structure function of random wave fields

The phase properties of a complex Gaussian field, which can arise following scattering from random phase screens and through extended random media, are investigated. It is shown that the unwrapped phase of a complex Gaussian field constitutes a non-stationary process, such that the phase autocorrelation function does not exist. However, the phase structure function remains finite, allowing analytical results to be obtained for various field correlations. Methods for numerical simulation are discussed and their results found to be in excellent agreement with analytical predictions. More general considerations reveal that the phase structure function of a complex Gaussian field increases linearly with large separation distance. The results are relevant to phase-sensitive detection in fields undergoing strong intensity fluctuations.     

The influence of porosity on vibrations of elastic solids

Wave scattering due to inhomogeneities in a solid leads to damping of wave motion in a random mixture of elastic materials. In particular, the phenomenon is observed in a material containing random porosity. A general dissipative model is presented to evaluate the influence of small randomly distributed pores on the dynamic response of elastic structures. The numerical results are given for transverse wave propagation and for vibrations of a beam. It is shown that analysis of vibrations of elastic solids containing random porosity can be carried out by the methods of viscoelasticity.     

A memory function model for the velocity autocorrelation function and the self-diffusion coefficient in simple dense fluids

A memory-function model is used to compute the velocity autocorrelation function and the self-diffusion coefficient of a dense Lennard-Jones fluid from the zero-time correlation functions of the molecular velocity and its first two time derivatives. It is shown that these zero-time correlation functions can be evaluated in terms of the radial distribution function and the pair potential only, i.e. without considering higher order correlation functions. Since molecular dynamics results are available for the radial distribution function as well as the velocity autocorrelation function and the self-diffusion coefficient, a rigorous test of the chosen memory function is possible. The agreement is reasonable, although generally not within the error bands of the molecular dynamics results.     

Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity

This paper considers the scattering of a plane, time-harmonic wave by an inclusion with heterogeneous nonlinear elastic properties embedded in an otherwise homogeneous linear elastic solid. When the inclusion and the surrounding matrix are both isotropic, the scattered second harmonic fields are obtained in terms of the Green's function of the surrounding medium. It is found that the second harmonic fields depend on two independent acoustic nonlinearity parameters related to the third order elastic constants. Solutions are also obtained when these two acoustic nonlinearity parameters are given as spatially random functions. An inverse procedure is developed to obtain the statistics of these two random functions from the measured forward and backscattered second harmonic fields.     

The non-stationary freefield response for a moving load with a random amplitude

This paper presents a stochastic solution procedure for the calculation of the non-stationary freefield response due to a moving load with a random amplitude. In this case, a non-stationary autocorrelation function and a time-dependent spectral density are required to characterize the response at a fixed point in the freefield. The non-stationary solution is derived from the solution in the case of a moving load with a deterministic amplitude. It is shown how the deterministic solution can be calculated in an efficient way by means of integral transformation methods if the problem geometry exhibits a translational invariance in the direction of the moving load. A key ingredient is the transfer function between the source and the receiver that represents the fundamental response in the freefield due to an impulse load at a fixed location. The solution in the case of a moving load with a random amplitude is formulated in terms of the double forward Fourier transform of the non-stationary autocorrelation function. The solution procedure is illustrated with an example where the non-stationary autocorrelation function and the time-dependent standard deviation of the freefield response are computed for a moving harmonic load with a random phase shift. The results are compared with the response in the deterministic case.     

Stochastic Analysis of an Elastic 3D Half-Space Respond to Random Boundary Displacements: Exact Results and Karhunen-Loéve Expansion

A stochastic response of an elastic 3D half-space to random displacement excitations on the boundary plane is studied. We derive exact results for the case of white noise excitations which are then used to give convolution representations for the case of general finite correlation length fluctuations of displacements prescribed on the boundary. Solutions to these elasticity problem are random fields which appear to be horizontally homogeneous but inhomogeneous in the vertical direction. This enables us to construct explicitly the Karhunen-Loève (K-L) series expansion by solving the eigen-value problem for the correlation operator. Simulation results are presented and compared with the exact representations derived for the displacement correlation tensor. This paper is a complete 3D generalization of the 2D case study we presented in Sabelfeld and Shalimova (J. Stat. Phys. 132(6):1071–1095, 2008). This work is supported partly by the RFBR Grants N 06-01-00498, N 09-01-00152, the WIAS Institute, Berlin, and DFG, Germany, under Grant SA 861/6-1 of 2008.     

A new statistical mechanics approach to dense granular media

A new statistical mechanics approach to dense granular media is presented. The thermodynamic formalism is set out directly in terms of elastic potential energy, such that the configurational temperature (the intensive property which defines the steady state) relates to a quadratic function of the stresses (rather than other linear functions used in recent developments). Dense granular media are considered as canonical ensembles of noninteracting clusters, which can be identified with repeatable equilibrium configurations. Then, particles can be located in a new proposed phase space (conceived to separate the elastic potential energy levels). Although the importance of this paper lies in the method itself, it has been illustratively applied to the simple case of two-dimensional (2D) dense granular media (an arrangement of frictionless monodisperse elastic disks under isotropic horizontal stress compression). In this case, the temperature is directly replaced by the squared external pressure, and the packing ratio of the most probable microstate is close to the reported value of random close packing. Moreover, some interesting general conclusions arise.     

Microscopic 3-D displacement field measurements using digital speckle photography

A technique to measure object shape and 3-D displacement fields in micro-scale is offered by microscopic stereo digital speckle photography. The displacement of the random features that are often present on many engineering surfaces when viewed in a microscope is measured with the system, using image correlation. In this paper the equipment, physical model and calibration routines are described. The technique can be applied for sub-mm sized objects of arbitrary shape for small deformation fields. As a verifying experiment, an in-plane rotation of a flat calibration plate is presented. The expected in-plane errors are shown to be less than 0.1 μm and the corresponding out-of-plane errors about three times larger. As a pilot experiment, micro-structural paper expansion is studied, when exposed to humidity. The scaling properties of the microscope as well as the sampling criteria and reliability of the system are discussed in detail.     

Long range density correlations in disordered solids

Although the structure of a glass closely resembles that of a dense liquid over short length scales, its solidity implies that the two-particle distribution function is long range. In this paper the explicit form of this long range behaviour is derived in terms of the elastic constants of the glass and the relationship is shown between density fluctuations and the microscopic form of the displacement vector u(r). With minor modifications these results are valid also for a crystal. Several ramifications of these results are explored.     

Pattern formation in Ferroelastic Transitions

We study pattern formation in ferroelastic materials using the Ginzburg–Landau approach. Since ferroelastic transitions are driven by strain, the nonlinear elastic free energy is expressed as an expansion in the appropriate (i.e., order parameter) strain variables. However, the displacement fields are the real independent variables, whereas the components of the strain tensor are related to each other through elastic compatibility relations. These constraints manifest as an anisotropic long-range interaction which drastically influences the underlying microstructure. The evolution of the microstructure is demonstrated for (i) a hexagonal-to-orthorhombic transition using a strain-based approach with explicit long-range interactions; and (ii) a cubic-to-tetragonal transition by solving the force-balance equations for the displacement fields.     

Spectral velocity estimation in ultrasound using sparse data sets

Velocity distributions in blood vessels can be displayed using ultrasound scanners by making a Fourier transform of the received signal and then showing spectra in an M-mode display. It is desired to show a B-mode image for orientation, and data for this have to be acquired interleaved with the flow data. This either halves the effective pulse repetition frequency f(prf) or gaps appear in the spectrum from B-mode emissions. This paper presents a technique to maintain the highest possible f(prf) and at the same time show a B-mode image. The power spectrum can be calculated from the Fourier transform of the autocorrelation function, and it is shown that the autocorrelation function can be calculated for a sparse set of data where flow and B-mode emissions are interspaced. Both short deterministic sequences of emissions and full random sequences can be used. The dynamic range of the sparse sequence is reduced compared to a full sequence. Typically, a reduction of 20 dB is found when using 66% of the data compared to using all data. The theory of the method and examples from simulations of flow in arteries are presented. The audio signal can also be generated from the spectrogram.     

Nonergodicity of a Time Series Obeying Lévy Statistics

Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied. Such a process, called a Lévy walk, describes dynamical behaviors of many physical systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time. Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed, and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The considered situation is in full contrast to the usual assumptions of ergodicity and stationarity.     

Discrete and non-discrete mixed methods applied to eigenvalue problems of plates

A mixed variational formulation for eigenvalue problems of plates is presented. Spline functions with multiple nodes are used to interpolate the displacement and moment fields. The solution procedure can be applied in either discrete or non-discrete forms. In contrast with displacement methods, the specified boundary conditions can be considered very easily by introducing multiplicity in the boundary nodes. Numerical examples include buckling and free vibration, of rectangular plates, with in-plane loading and or elastic foundations. The accuracy of the results obtained and the superiority of the mixed methods presented to conventional displacement approaches are discussed.     

弱散射屏的像面散斑自相关函数特性的实验研究

在对随机弱散射屏进行表面参数的原子力显微镜测量和建立了门积分取样平均的随机光强自相关函数测量系统的基础上,对弱散射屏在严格像面和离焦像面上产生的散斑自相关函数进行了测量。发现在严格像面上,散斑平均颗粒的大小随表面粗糙度增加而减小,且光强自相关函数次极大的相关间隔宽度随粗糙度增加而减小;而次极大的超伏随粗糙度的增大而增大;在离焦像面上,离焦量的增加使光强的自相关函数下降变得平滑,并使极小值点和次极大点变得不明显或者消失。     

Elastic wave field computation in multilayered nonplanar solid structures: a mesh-free semianalytical approach

Multilayered solid structures made of isotropic, transversely isotropic, or general anisotropic materials are frequently used in aerospace, mechanical, and civil structures. Ultrasonic fields developed in such structures by finite size transducers simulating actual experiments in laboratories or in the field have not been rigorously studied. Several attempts to compute the ultrasonic field inside solid media have been made based on approximate paraxial methods like the classical ray tracing and multi-Gaussian beam models. These approximate methods have several limitations. A new semianalytical method is adopted in this article to model elastic wave field in multilayered solid structures with planar or nonplanar interfaces generated by finite size transducers. A general formulation good for both isotropic and anisotropic solids is presented in this article. A variety of conditions have been incorporated in the formulation including irregularities at the interfaces. The method presented here requires frequency domain displacement and stress Green's functions. Due to the presence of different materials in the problem geometry various elastodynamic Green's functions for different materials are used in the formulation. Expressions of displacement and stress Green's functions for isotropic and anisotropic solids as well as for the fluid media are presented. Computed results are verified by checking the stress and displacement continuity conditions across the interface of two different solids of a bimetal plate and investigating if the results for a corrugated plate with very small corrugation match with the flat plate results.     

On hydrodynamical fluctuations

The microscopic fluctuations of mass and momentum densities are used as Mori variables in a generalized Langevin scheme. The random force-force correlation function is calculated, at very short times, for a classical many-body system. Though restricted to short times, our result is rigorous and new. The autocorrelation function for the random force is shown to be spatially non-local and biased by the presence of an external field.