Isotropic Elastic Medium Research Articles

Indicatrix of TE and TM- polarized wave velocities in crystal of classes 4mm, 3m, 6mm with magneto-electric effect

This work is devoted to the theoretical study of the laws of propagation of electromagnetic waves of TE and TM polarization in anisotropic media belonging to classes 4mm, 3m, 6mm, etc. having a magnetoelectric effect. The directions of the vectors of the phase and group velocities of the polarization waves TE and TM at the boundary of a uniaxial crystal with magnetoelectric properties are considered. In the analytical form, the values of the directions of the phase and group velocities of the TE and TM waves are indicated, depending on the direction of the wave vector of the incident wave. The consequences of the obtained results for uniaxial crystals in the absence of magnetoelectric properties are discussed. The solution of the tasks set in this paper is based on the use of the matrix method. On its basis, various problems of wave processes in an isotropic elastic medium, electromagnetic waves in crystals, the distribution of coupled elastic and electromagnetic waves in piezoelectric and piezomagnetic media with a magnetoelectric effect were previously considered. In the presence of a magnetoelectric effect for electromagnetic waves propagating through uniaxial crystals, the parameters of the wave vector, phase and group velocities are determined. The obtained results are analyzed for electromagnetic waves propagating through uniaxial crystals in the absence of a magnetoelectric effect. The indicatrices of the wave vectors propagating in the plane and the phase velocities of the TM polarization waves are limited ( 0 ) x z. Based on the Rayleigh equation, the values of the group velocity are obtained. The density of electromagnetic energy fluxes and their components are determined for TE and TM waves. The energy transfer rate and its direction are determined. It is shown that the group velocities and directions obtained from the Rayleigh equation and the Umov-Poynting vector do not coincide.

Grain boundary stresses in elastic materials

A simple analytical model of intergranular normal stresses is proposed for a general elastic polycrystalline material with arbitrary shaped and randomly oriented grains under uniform loading. The model provides algebraic expressions for the local grain-boundary-normal stress and the corresponding uncertainties, as a function of the grain-boundary type, its inclination with respect to the direction of external loading and material-elasticity parameters. The knowledge of intergranular normal stresses is a necessary prerequisite in any local damage modeling approach, e.g., to predict the probability for intergranular stress-corrosion cracking or fatigue-crack initiation in structural materials.The model is derived in a perturbative manner, starting with the exact solution of a simple setup and later successively refining it to account for higher order complexities of realistic polycrystalline materials. In the simplest scenario, a bicrystal model is embedded in an isotropic elastic medium and solved for uniaxial loading conditions, assuming 1D Reuss and Voigt approximations on different length scales. In the final iteration, the grain boundary becomes a part of a 3D structure consisting of five 1D chains with arbitrary number of grains and surrounded by an anisotropic elastic medium. Constitutive equations can be solved for arbitrary uniform loading, for any grain-boundary type and choice of elastic polycrystalline material. At each iteration, the algebraic expressions for the local grain-boundary-normal stress, along with the corresponding statistical distributions, are derived and their accuracy systematically verified and validated against the finite element simulation results of different Voronoi microstructures.

Analysis of antiplane problems with singular disturbances for isotropic elastic medium with two elliptical elasitic inclusions

Analysis of antiplane problems with singular disturbances for isotropic elastic medium with two elliptical elasitic inclusions

Transverse shear of an isotropic elastic medium in the case when the binder and inclusions are weakened by crack initiation

An elastic medium is considered, weakened by a doubly periodic system of round holes, filled with absolutely rigid inclusions, soldered along the bypass and has a crack initiation. The medium (binder) is weakened by two periodic systems of rectilinear crack initiation directed collinear to the abscissa and ordinate axes, and their sizes are not the same. General representations are constructed that describe a class of problems with a doubly periodic stress distribution outside circular holes and cracks under transverse shear. The analysis of the limiting equilibrium of cracks in the framework of the end zone model is carried out on the basis of a nonlocal fracture criterion with a force condition for the propagation of the crack tip and a deformation condition for determining the advancement of the edge of the end zone of the crack. Basic resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations in each approximation were solved by the Gaussian method with the choice of the principal element for different values of the order of M, depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing the growth of cracks.

Material Interface in the Finite-Difference Modeling: A Fundamental View

ABSTRACT By analyzing the equations of motion and constitutive relations in the wavenumber domain, we gain important insight into attributes determining the accuracy of finite-difference (FD) schemes. We present heterogeneous formulations of the equations of motion and constitutive relations for four configurations of a wavefield in an elastic isotropic medium. We Fourier-transform the entire equations to the wavenumber domain. Subsequently, we apply the band-limited inverse Fourier transform back to the space domain. We analyze consequences of spatial discretization and wavenumber band limitation. The heterogeneity of the medium and the Nyquist-wavenumber band limitation of the entire equations has important implications for an FD modeling: The grid representation of the heterogeneous medium must be limited by the Nyquist wavenumber. The wavenumber band limitation replaces spatial derivatives both in the homogeneous medium and across a material interface by continuous spatial convolutions. The latter means that the wavenumber band limitation removes discontinuities of the spatial derivatives of the particle velocity and stress at the material interface. This allows to apply proper FD operators across material interfaces. A wavenumber band-limited heterogeneous formulation of the equations of motion and constitutive relations is the general condition for a heterogeneous FD scheme.

Statistical mechanics of phase transitions in elastic media with vanishing thermal expansion.

We consider a minimal spin model for Ising transitions in an isotropic elastic medium in the zero thermal expansion (ZTE) limit. We set up the elastic theory for this system. We use this theory to identify and study the nature of the fluctuations in the system near the second order phase transitions at T_{c} in the ZTE limit given by dT_{c}/dV=0, where V is the system volume, and explore anomalous elasticity. Allowing for the local strain to couple asymmetrically or selectively with the states of the order parameter, we uncover the dramatic effects of these couplings on the fluctuations of the local displacements near T_{c}, and also on the nature of the phase transition itself. Near second-order phase transitions and with weak asymmetry in the order parameter-strain couplings, the variance of the displacement fluctuations in two dimensions scale with the system size L in a universal fashion as [ln(L/a_{0})]^{2/3}; a_{0} is a small-scale cutoff. Likewise, the correlation functions of the difference of the local displacements at two different points separated by r scale as [ln(r/a_{0})]^{2/3} for large r. For stronger selectivity above a finite threshold, this variance diverge as L exceeds beyond a (nonuniversal) size, determined by the model parameters, signaling a transition to a phase with only short-range order or the loss of the positional order of the elastic medium. At dimensions higher than two, for sufficiently weak selectivity, the variance of the displacement fluctuations is L-independent corresponding to long-range order. However, if the selectivity parameters rise beyond a dimension-dependent threshold value, then again the positional order is lost with a concomitant transition to a phase with short-range order. Large values of the order parameter-strain couplings can turn the phase transition into a first order as well. Our theory establishes a one-to-one correspondence between the order of phase transitions and anomalous elasticity near the transitions. Our theory should be a useful guide to possible synthesis of appropriate ZTE materials.

Stiffening or softening of elastic media: Anomalous elasticity near phase transitions.

We present the general theory of Ising transitions in isotropic elastic media with vanishing thermal expansion. By constructing a minimal model with appropriate spin-lattice couplings, we show that in two dimensions near a continuous transition the elasticity is anomalous in unusual ways: the system either significantly stiffens with a hitherto unknown unique, positional order logarithmically stronger than quasi-long-range order, or, as the inversion asymmetry of the order parameter in its coupling with strain increases, it destabilizes when system size L exceeds a finite threshold. In three dimensions, stronger inversion-asymmetric couplings induce instability to the long-range positional order for all L. Sufficiently strong order parameter-displacement couplings can also turn the phase transition first order at all dimensions, concomitant with finite jumps in the elastic modulii across the transition. Our theory establishes a one-to-one correspondence between the order of the phase transitions and anomalous elasticity near the transitions.

Elastic Spin and Orbital Angular Momenta.

Motivated by recent theoretical and experimental interest in the spin and orbital angular momenta of elastic waves, we revisit canonical wave momentum, spin, and orbital angular momentum in isotropic elastic media. We show that these quantities are described by simple universal expressions, which differ from the results of Chaplain etal. [Phys. Rev. Lett. 128, 064301 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.064301] and do not require separation of the longitudinal and transverse parts of the wave field. For cylindrical elastic modes, the normalized z component of the total (spin+orbital) angular momentum is quantized and equals the azimuthal quantum number of the mode, while the orbital and spin parts are not quantized due to the spin-orbit geometric-phase effects. In contrast to the claims of the above article, longitudinal, transverse, and "hybrid" contributions to the angular momenta are equally important in general and cannot be neglected. As another example, we calculate the transverse spin angular momentum of a surface Rayleigh wave.

Plane waves in microstretch elastic solid with voids

A linear theory of microstretch elastic solid containing uniform distribution of uniform voids has been formulated. Constitutive relations are developed by constructing suitable strain energy density function from the basic state variables. Then using the principles of mechanics, the field equations are derived for isotropic homogeneous microstretch elastic medium containing voids. The possibility of propagation of plane waves is explored and found that there may exist four sets of coupled dilatational waves and two sets of coupled transverse waves traveling with distinct speeds. The coupled longitudinal waves are found to be attenuating and dispersive, while the coupled transverse waves are dispersive but non-attenuating in nature. The coupled transverse waves are found to be independent of the presence of stretch and voids in the medium. The attenuation of coupled longitudinal waves arises due to the presence of dissipation coefficient of void volume fraction. In case of Voigt model, only the coupled transverse waves arising due to the microrotation disappear below their respective critical frequency, while in case of non-Voigt model, three sets of coupled longitudinal waves and one set of coupled transverse waves corresponding to predominantly due to the microrotation are non-progressive waves below their respective critical frequencies. Some limiting cases have been discussed and explained. Numerical computations have been carried out for Gauthier composite plate material and the results are displayed graphically.

Identification of Cavities and Inclusions in Linear Elasticity with a Phase-Field Approach

In this paper we deal with the inverse problem of determining cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement’s measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modelled as inclusions with a very small elasticity tensor.

A spectral boundary integral method for the elastic obstacle scattering problem in three dimensions

In this paper, we consider the scattering of a plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in three dimensions. Based on the Helmholtz decomposition, the elastic scattering problem is reduced to a coupled boundary value problem for the Helmholtz and Maxwell equations. A novel system of boundary integral equations is formulated and a spectral boundary integral method is developed for the coupled boundary value problem. Numerical experiments are presented to demonstrate the superior performance of the proposed method.

3D structure-soil-structure interaction in an arbitrary layered half-space

The aim of the study is to evaluate the 3D dynamic response of a finite geological region containing two structures and rested on a semi-infinite elastic layered half-space with a dynamic source radiating transient waves. The hybrid modelling approach is applied. It is based on the decomposition of the whole domain under consideration into two sub-regions: a finite-sized near-field elastic isotropic zone with two containment structures and the open semi-infinite far-field layered region. The far-field zone is a semi-infinite elastic isotropic arbitrary layered medium where the near-field finite geological region is located on. The 3D hybrid computational tool is based on the boundary element method (BEM) for the far-field layered zone and the finite element method (FEM) for the finite near-field domain. The model for the semi-infinite layered zone is further extended by the incorporation of a new condensation algorithm which makes it possible to handle 3D wave propagation through arbitrary layered half-space. The condensation algorithm is developed to avoid high computational memory cost while retaining the compatibility with the hybrid FEM-hosted procedure which facilitates the useful solution for the practical three-dimensional engineering problems. The BEM model of the dynamically active far-field zone is inserted as a macro-finite element (MFE) in the FEM commercial program ABAQUS. The accuracy and convergence study of the hybrid numerical scheme is presented. Numerical simulations convincingly illustrate that the dynamic response of structure-soil-structure system depends on different key factors and their mutual interplay. These factors are arbitrary layering of the far-field geological zone, the characteristics of the dynamic source, the site effects phenomena, the structure-soil-structure dynamic interaction, the type and geometrical disposition of foundations and structures and the 3D features of the dynamic motion.

Effective Medium Theory for Embedded Obstacles in Elasticity with Applications to Inverse Problems

We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.

Unconventional cracks with dislocation-free zones

Using the method of continuously distributed dislocations, we study the dislocation-free zones for unconventional internal cracks in an infinite isotropic elastic medium under uniform remote anti-plane shear stress. The unconventional cracks are represented by a symmetric lip crack with two cusp tips and a symmetric airfoil crack with a single cusp tip. For each case, the equilibrium condition is formulated in terms of a single singular integral equation in the image plane. The resulting singular integral equation is solved numerically to arrive at the dislocation distribution function and the total number of dislocations in the plastic zone, the condition on remote loading and the local mode III stress intensity factor at the cusp tip. In the absence of the dislocation-free zones, the results are valid for the Dugdale model of these unconventional cracks.

A Stable Node-Based Smoothed Finite Element Method with Transparent Boundary Conditions for the Elastic Wave Scattering by Obstacles

In this work, a stable node-based smoothed finite element method with TBC (SNS-FEM-TBC) is proposed to solve the scattering of elastic plane waves by a two-dimensional (2D) homogeneous isotropic elastic medium. First, using Helmholtz decomposition, two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic scattering problem. Second, based on the analytical solutions of Helmholtz equations, TBC operators are deduced. Then, the gradient Taylor expansion is used to construct the stability term to deal with the instability of the original NS-FEM, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing domain as a circle. Finally, based on smoothed Galerkin weak formula, the SNS-FEM-TBC formula of linear algebraic system with linear smoothing gradient is derived. Numerical examples show that SNS-FEM can obtain more stable and accurate solutions than standard FEM. Moreover, the convergence rates of [Formula: see text] and [Formula: see text] semi-norm errors of SNS-FEM are faster.

On mean Green operator and Eshelby tensor of hollow cylinders with length from infinite tubes to flat rings in isotropic elastic media

Using the (RT-IRT) Radon transform and inversion formula method, the author has previously reported the exact formal expressions for the mean shape function (mSF) of a finite solid cylinder of general aspect ratio in any infinite medium and for the related mean Green operator (mGO) and mean Eshelby tensor (mET) in cases of isotropic elastic-like (including dielectric-like) linear properties. Nearly exact solutions were obtained provided an accurate analytical approximation of an elliptic integral. Using the same RT-IRT method, the present work addresses the case of hollow cylinders with finite length in same isotropic property contexts. From examining the general formal mSF expression (and comparing with the limits of infinitely long cylinders and infinitely flat platelets) a simple accurate approximation is provided for the mSF, mGO and mET of "thin" hollow cylinders, from tubes to rings what covers a wide range of practical situations. The same expression being shown to hold on the thick side limit, the conjecture that it may be relevant over the whole thickness range is questioned.

Magnetic-thermal-elastic waves under the impact of induced laser pulses and hyperbolic two temperature theory with memory-dependent derivatives (MDD)

A novel model with theoretical investigations of generalized hyperbolic two-temperature theory in the context of the generalized thermoelasticity theory is studied. The elastic medium is excited at the outer surface by an external magnetic field and pulsed laser. The model is established in the context of the memory-depended derivative (MDD) when the elastic material time-dependent. The governing equations describe the interaction processes for isotropic homogeneous hyperbolic thermoelastic elastic medium in one-dimensional (1D) deformation when the non-Gaussian laser pulse model is applied with time delay. The Laplace transform is used to obtain the analytical solutions of the main physical fields. The medium is loaded when a traction is free and thermally shocked with time-dependent. The complete solutions are obtained in the Laplace time domain when used the inverse of Laplace transform numerically. Numerical results of the main physical quantities are discussed graphically under the effect of hyperbolic two-temperature, magnetic field and laser pulses when the time-delay parameter is taken into consideration.

Analysis of wave scattering at the loosely bonded common interface of piezothermoelastic and isotropic elastic media under LS (Lord-Shulman) and GL (Green-Lindsay) theory of thermoelasticity

The present paper delves into the problem of reflection and refraction of incident plane wave at the loosely bonded common interface of an isotropic elastic medium and piezothermoelastic medium. The mechanical boundary conditions and the non-classical theory of thermo-electrodynamics for a piezothermoelastic material are employed to obtain the amplitude ratios of reflected and refracted waves, respectively. Emphasis has been given to analyze the variation of amplitude ratios of different reflected and transmitted waves under LS (Lord-Shulman) and GL (Green-Lindsay) theories of thermoelasticity. The variation of amplitude ratios corresponding to the alteration of bonding parameter for different waves has been computed numerically in respect to the generalized theory of thermoelasticity. Moreover, the effect of thermal boundaries (isothermal and adiabatic) and electrical boundaries (electrically open and electrically short), on the amplitude ratios for a particular model are presented graphically. The expressions for energy ratios of reflected and refracted waves are derived and energy conservation is validated with the help of obtained graphs under LS and GL theories.

Smart Ultrasound Device for Non-Invasive Real-Time Myocardial Stiffness Quantification of the Human Heart.

Quantitative assessment of myocardial stiffness is crucial to understand and evaluate cardiac biomechanics and function. Despite the recent progresses of ultrasonic shear wave elastography, quantitative evaluation of myocardial stiffness still remains a challenge because of strong elastic anisotropy. In this paper we introduce a smart ultrasound approach for non-invasive real-time quantification of shear wave velocity (SWV) and elastic fractional anisotropy (FA) in locally transverse isotropic elastic medium such as the myocardium. The approach relies on a simultaneous multidirectional evaluation of the SWV without a prior knowledge of the fiber orientation. We demonstrated that it can quantify accurately SWV in the range of 1.5 to 6 m/s in transverse isotropic medium (FA < 0.7) using numerical simulations. Experimental validation was performed on calibrated phantoms and anisotropic ex vivo tissues. A mean absolute error of 0.22 m/s was found when compared to gold standard measurements. Finally, in vivo feasibility of myocardial anisotropic stiffness assessment was evaluated in four healthy volunteers on the antero-septo basal segment and on anterior free wall of the right ventricle (RV) in end-diastole. A mean longitudinal SWV of 1.08 ± 0.20 m/s was measured on the RV wall and 1.74 ± 0.51 m/s on the septal wall with a good intra-volunteer reproducibility (±0.18 m/s). This approach has the potential to become a clinical tool for the quantitative evaluation of myocardial stiffness and diastolic function.

УЧЕТ ТРЕНИЯ НА НАЧАЛЬНОМ ЭТАПЕ ВЕРТИКАЛЬНОГО ВНЕДРЕНИЯ ВЫПУКЛОГО УДАРНИКА В УПРУГУЮ ПОЛУПЛОСКОСТЬ

The initial stage of indentation of a convex rigid punch into isotropic elastic half-plane with friction is considered. A closed mathematical formulation in Cartesian coordinates is presented by equations of motion in the absence of a body force for homogeneous isotropic elastic medium in terms of displacement potentials, Cauchy equations for deformations, Hooke law and translational equation for the punch. Initial conditions are homogeneous. Outside of contact region surface is free from stresses. Inside contact region normal displacements of the surface of a half-space and the surface of a punch are taken to be equal and the relation between tangential and normal stresses in a form of Coulomb friction law is given. Due to the short duration of the supersonic stage and smallness of the contact region radius the tangential stress direction is considered to be point independent. Resolving functional equations are given in a form of convolutions with influence functions. The latter is the solution the original problem for half-plane with a special boundary condition in a form of the Dirac delta function. This solution is obtained in the domain of Laplace transform by time and Fourier transform by spatial coordinate. The original is constructed with the combined Fourier–Laplace inversion method. Solution is given with consideration to the initial stage characteristic property of supersonic velocity of contact region expansion (no smaller than propagation velocity of expansion-compression waves). It is shown that kinematic parameters of the punch, resulting force and discontinuities of the first kind on boundaries of the contact region are independent of friction unlike contact pressure. Calculation examples for different values of friction coefficient are presented.