Homogeneous Elastic Medium Research Articles

A novel Newton method for inverse elastic scattering problems

This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier–Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.

Analysis of thermo-mechanical incompatibility of fullerenes with polymer matrices of composites

Unique strength and rigidity properties of carbon nanotubes and fullerenes do them very perspective reinforcing admixtures into composites produced on the basis of polymeric and metallic matrices. It is established that even relatively small their dopes into composites can essentially enlarge physico-mechanical characteristics of polymers. At the same time, comparatively low value of the linear thermal expansion coefficient of carbon nanomolecules results in the possibility of the emergence of the internal mechanism of the additional intrastructural stresses origination in the polymer conditioned due to thermomechanical incompatibility of the composite fractions. To examine this possibility, the theoretical modelling of the thermomechanical deforming of the considered heterogeneous structures under thermal change action was carried out. In doing so, the fullerenes were simulated as a thin elastic shell with adduced (effective) thickness, elasticity modulus, Poissonn’s coefficient and linear thermal expansion coefficient, the polymer matrix was supposed to be a homogeneous elastic medium with the prescribed parameters of thermoelasticity. The system of constitutive ordinary differential equations was deduced, which described the thermoelastic stress-strainedstate of the elastic fragment chosen for consideration. These equations solutions were deduced in the closed form for the case when the system temperature changed in. Fenolformaldegid, epoxy rezin, polycarbonate, polyamide, polystirol, polyester, and polypropylene, possessing increased values of the linear thermal expansion coefficients, were chosen as the matrix materials. With these initial data, the values of the fullerene radial displacements were calculated for two magnitudes of the sphere radius. It is demonstrated that owing to the high value of the fullerene effective elasticity modulus, the thermal deflections of its wall poorly depend on the properties of the encompassing matrix and practically are equal to its free thermal displacements. The thermal stresses of the polymer matrices have the largest values in the zones of the interface surfaces and they decrease proportionally to cube of the radial coordinate, so, the damages provoked by them have localized character.

Non-classical theory of electro-thermo-elasticity incorporating local mass displacement and nonlocal heat conduction

A non-classical local gradient theory of nonferromagnetic thermoelastic dielectrics is presented, incorporating both the local mass-displacement process and the heat-flux gradient effect. The process of local mass displacement is related to the changes in material microstructure. The nonlocal heat conduction law is also addressed in the model. Thus, the generalized relationship between the higher-grade heat and entropy fluxes is adopted. The gradient-type constitutive relations and governing equations are derived using the fundamental principles of continuum mechanics, non-equilibrium thermodynamics, and electrodynamics. Due to the contribution of higher-grade flux, the nonlocal law of heat conduction is obtained. The constitutive relations for isotropic materials with the corresponding additional material constants are derived. To illustrate the local gradient theory and to show the electro-thermo-mechanical coupling effect in isotropic materials, a straightforward problem is analytically solved for a layered non-piezoelectric structure under non-uniform temperature distribution. The analytical results reveal that the thermal polarization effect can also be pronounced in isotropic materials. To illustrate the model considering the effect of nonlocal heat conduction, the propagation of spherical thermoelastic harmonic waves in a homogeneous and isotropic elastic medium with non-classical heat conduction law is studied.

Study of Waves Propagating in Anisotropic Homogeneous Microstretch Elastic Medium

Study of Waves Propagating in Anisotropic Homogeneous Microstretch Elastic Medium

Transverse vibrations of underground pipeline with pinched ends

The study of transverse oscillations of extended underground structures from the action of constant harmonic forces is of practical interest. Transverse vibrations of an underground cylindrical structure of finite length under the action of a system of concentrated forces that change according to a harmonic law are considered. The environment surrounding the structure is assumed to be elastic, while the interaction of the “environment-structure” system is studied. When considering the problem, the Fourier method and the results of solving the problem of vibrations of an unbounded cylindrical structure in an elastic medium from the action of concentrated harmonic forces were used. The problem of transverse vibrations of a cylindrical rod of finite length interacting with an elastic homogeneous medium is solved by the method of compensating loads. In this case, the structure is considered a source of waves radiated into space and satisfying the conditions of radiation to infinity. The resulting solution is approximate and reliable within the length of the rod. The analysis of the influence of soil conditions and geometric dimensions on the amplitude of oscillations of an underground structure was carried out according to the graphs of the amplitude-frequency characteristic of the object.

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.

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.

Secondary growth and closure behavior of planar hydraulic fractures during shut-in

The evolution of fractures during shut-in after hydraulic fracturing is an important topic in petroleum engineering. Considering the closure and secondary expansion of fractures during shut-in can further improve the accuracy of predicting hydraulic fracture parameters. Based on linear elastic fracture mechanics, elastic mechanics, and the implicit level set algorithm, a numerical model of planar hydraulic fracture propagation was established in this study. In the proposed model, the reservoir was considered as a homogeneous elastic medium, and the fluid loss was characterized using Carter's filtration model. The secondary propagation and fracture closure behavior during shut-in could be simulated by solving the strongly nonlinear system of fluid and stress coupling. Based on this model, the evolution behavior of hydraulic fractures during injection and shut-in was analyzed. According to the variation characteristics of the fracture length, the evolution process could be divided into three stages: fracture propagation during injection, secondary growth, and fracture stopping (but not closing). According to the variation characteristics of the fracture opening and net pressure, the evolution process could be divided into three stages: nonlinear increase, linear decrease, and gradual decrease. For low-permeability reservoirs, fracture closure and secondary propagation may take several times the injection time after shut-in, and the secondary propagation length may reach tens of meters. The results show that the larger the filtration coefficient, the faster the fracture closure will be, and the shorter the secondary propagation distance. The higher the viscosity of the fracturing fluid, the slower the fracture closure will be, while the secondary propagation distance is almost unaffected. The fracture toughness has little effect on the fracture closure and secondary propagation. The crack will close first in the region with large interlayer stress during shut-in. The findings of this study can help for better understanding of the evolution mechanism of fractures during shut-in and closure.

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.

Modeling of hydraulic fracture in a medium containing a hollow inclusion

In this work, the problem of the development of a hydraulic fracture crack near a circular cavity in a three-dimensional homogeneous elastic medium is considered. The results of numerical calculations performed using the extended finite element method (XFEM) implemented in the ABAQUS software package are presented. Various variants of the position of the initial disk crack relative to the cavity are considered. Some regularities of the development of a three-dimensional hydraulic fracture near the cavity have been found.

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.

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.

Optimal efficiency of high-frequency chest wall oscillations and links with resistance and compliance in a model of the lung

Chest physiotherapy is a set of techniques used to help the draining of the mucus from the lung in pathological situations. The choice of the techniques and their adjustment to the patients or to the pathologies remain as of today largely empirical. High-frequency chest wall oscillation (HFCWO) is one of these techniques, performed with a device that applies oscillating pressures on the chest. However, there is no clear understanding of how HFCWO devices interact with the lung biomechanics. Hence, we study idealized HFCWO manipulations applied to a mathematical and numerical model of the biomechanics of the lung. The lung is represented by a fluid–structure interaction model based on an airway tree that is coupled to a homogeneous elastic medium. We show that our model is driven by two dimensionless numbers that drive the effect of the idealized HFCWO manipulation on the model of the lung. Our model allows us to analyze the stress applied to an idealized mucus by the air–mucus interaction and by the airway walls deformation. This stress behaves as a buffer and has the effect of reducing the stress needed to overcome the idealized mucus yield stress. Moreover, our model predicts the existence of an optimal range of the working frequencies of HFCWO. This range is in agreement with the frequencies actually used by practitioners during HFCWO maneuvers. Finally, our model suggests that analyzing the mouth airflow during HFCWO maneuvers could allow us to estimate the compliance and the hydrodynamic resistance of the lung of a patient.

Solving an inverse acousto-elastic scattering problems by combining full-waveform redatuming and time reversal

The aim of this paper is to propose a new method to solve the inverse scattering problem in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine the Young modulus of elastic unknown “inclusions”, i.e. not observable solid objects, located in the elastic medium, from partial aperture boundary measurements. This approach combines a time-reversal approach and redatuming - a technique that consists in virtually moving the sensors from the original acquisition location to an arbitrary position - which allows to reduce the size of the inspected medium. Thanks to this strategy, we can recover the properties of the scatterer, this technique being also relatively insensitive to noise in the data.

A highly accurate boundary integral method for the elastic obstacle scattering problem

Consider the scattering of a time-harmonic plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in two dimensions. In this paper, a novel boundary integral formulation is proposed and its highly accurate numerical method is developed for the elastic obstacle scattering problem. More specifically, based on the Helmholtz decomposition, the model problem is reduced to a coupled boundary integral equation with singular kernels. A regularized system is constructed in order to handle the degenerated integral operators. The semi-discrete and full-discrete schemes are studied for the boundary integral system by using the trigonometric collocation method. Convergence is established for the numerical schemes in some appropriate Sobolev spaces. Numerical experiments are presented for both smooth and nonsmooth obstacles to demonstrate the superior performance of the proposed method.

Elastic properties of solid nematics

The general approach to study the properties of the mechanical deformations of solid nematics, which are the macroscopic homogeneous elastic media having the rotational symmetry of the nematic liquid crystals is proposed. The stress tensor, the Young modulus and the Poisson ratios for the parallel and perpendicular homogeneous orientations of nematic molecules relative to the axis of external forces influence are obtained by the varying of the free energy of mechanical deformation. It is shown that these constants have the anisotropic character and the experiments for the direct measurement of five elasticity coefficients entering the free energy expression are proposed.

An axisymmetric problem for a penny-shaped crack under the influence of the Steigmann–Ogden surface energy

A problem for a nanosized penny-shaped fracture in an infinite homogeneous isotropic elastic medium is considered. The fracture is opened by applying an axisymmetric normal traction to its surface. The surface energy in the Steigmann–Ogden form is acting on the boundary of the fracture. The problem is solved by using the Boussinesq potentials represented by the Hankel transforms of certain unknown functions. With the help of these functions, the problem can be reduced to a system of two singular integro-differential equations. The numerical solution to this system can be obtained by expanding the unknown functions into the Fourier–Bessel series. Then the approximations of the unknown functions can be obtained by solving a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. Parametric studies of the dependence of the solutions on the mechanical and the geometric parameters of the system are undertaken. It is shown that the surface parameters have a significant influence on the behaviour of the material system. In particular, the presence of surface energy leads to the size-dependency of the solutions and smoother behaviour of the solutions near the tip of the crack.

Multiscale seismic imaging with inverse homogenization

SummarySeismic imaging techniques such as elastic full waveform inversion (FWI) have their spatial resolution limited by the maximum frequency present in the observed waveforms. Scales smaller than a fraction of the minimum wavelength cannot be resolved, and only a smoothed, effective version of the true underlying medium can be recovered. These finite-frequency effects are revealed by the upscaling or homogenization theory of wave propagation. Homogenization aims at computing larger scale effective properties of a medium containing small-scale heterogeneities. We study how this theory can be used in the context of FWI. The seismic imaging problem is broken down in a two-stage multiscale approach. In the first step, called homogenized FWI (HFWI), observed waveforms are inverted for a smooth, fully anisotropic effective medium, that does not contain scales smaller than the shortest wavelength present in the wavefield. The solution being an effective medium, it is difficult to directly interpret it. It requires a second step, called downscaling or inverse homogenization, where the smooth image is used as data, and the goal is to recover small-scale parameters. All the information contained in the observed waveforms is extracted in the HFWI step. The solution of the downscaling step is highly non-unique as many small-scale models may share the same long wavelength effective properties. We therefore rely on the introduction of external a priori information, and cast the problem in a Bayesian formulation. The ensemble of potential fine-scale models sharing the same long wavelength effective properties is explored with a Markov chain Monte Carlo algorithm. We illustrate the method with a synthetic cavity detection problem: we search for the position, size and shape of void inclusions in a homogeneous elastic medium, where the size of cavities is smaller than the resolving length of the seismic data. We illustrate the advantages of introducing the homogenization theory at both stages. In HFWI, homogenization acts as a natural regularization helping convergence towards meaningful solution models. Working with fully anisotropic effective media prevents the leakage of anisotropy induced by the fine scales into isotropic macroparameters estimates. In the downscaling step, the forward theory is the homogenization itself. It is computationally cheap, allowing us to consider geological models with more complexity (e.g. including discontinuities) and use stochastic inversion techniques.

The use of the Gurtin-Murdoch theory for modeling mechanical processes in composites with two-dimensional reinforcements

This paper explores the possibility of using the Gurtin-Murdoch theory of material surface for modeling mechanical processes in nanomaterials reinforced by two-dimensional flexible and extensible nanoplatelets. In accordance with the theory, reinforcement is modeled by a vanishing thickness prestressed membrane embedded in an isotropic elastic matrix material. The governing equations for the model are reviewed with a detailed discussion of the conditions at the membrane tips. Plane strain assumption is made and with the purpose of representing the displacements in the bulk material, a single layer elastic potential is employed, with the density representing the jump in tractions across the membrane. Expressions for the remaining elastic fields are provided in terms of complex integral representations. The case of a rectilinear membrane of finite length is considered in detail. Numerical solution for this case is based on the use of approximations that automatically incorporate the membrane-tip conditions into the resulting system of boundary integral equations. Numerical examples illustrate the influence of governing dimensionless parameters and present simulations of the local elastic fields in the materials under study. Additionally, it is shown that, in absence of surface tension, the problem considered here is related to that of a membrane type elastic inhomogeneity embedded into a homogeneous elastic medium.

Arrest of a radial hydraulic fracture upon shut-in of the injection

We investigate the propagation and arrest of a radial hydraulic fracture upon the end of the injection. Depending on the regime of propagation at the time of shut-in of the injection, excess elastic energy may be stored in the surrounding medium. Once the injection has stopped, the hydraulic fracture will arrest when the energy release rate falls under the material fracture energy. Fluid leaking-off to the surrounding medium acts as an energy sink such that the available excess energy for fracture growth decreases faster and as a result impacts the arrest (actually controls it in the zero toughness limit). Under the assumption of a homogeneous elastic medium and the Carter’s leak-off model, we show that the post shut-in propagation of the hydraulic fracture depends on the dimensionless toughness Ks and leak-off Cs coefficient at the time of shut-in. Our investigation highlights that for an impermeable rock, the arrest radius is independent of the dimensionless toughness at shut-in. In the limit of a permeable rock with zero fracture toughness, the arrest radius is independent of the dimensionless leak-off coefficient only for Cs<0.25. For larger values of Cs, the radius of arrest reduces with increasing Cs. We delineate the limit above which the arrest is immediate upon shut-in. This limit is given by a critical leak-off coefficient at shut-in Cs,c≈0.53 for the large leak-off/small toughness cases and by the relation Cs,cKs,c≈0.78-0.313·Ks,c for small leak-off/large toughness (where Ks,c is equivalently the critical dimensionless toughness at shut-in). Immediate arrest in the impermeable limit is observed for Ks,c≈2.5. If both (Ks and Cs) are smaller than their critical value for immediate arrest, post shut-in propagation occurs and a self-similar pulse viscosity storage solution emerges. Scaling arguments combined with numerical simulations, show that the propagation post shut-in scales as 1.23Ks-2/5 in the impermeable and small leak-off cases, and as 0.75Cs-2/13 in the zero toughness limit. The growth post shut-in can be significant in impermeable rocks - with a final radius up to twice larger than the radius at shut-in for realistic material and injection parameters.