Infinite Medium Research Articles

Propagation of acoustic waves in 3D periodic media: Dispersion, clusters, and local gaps

We present a theoretical study of the propagation of acoustic waves in a 3D infinite medium containing a periodic array of small identical inclusions of arbitrary shape with transmission conditions on their interfaces. The inclusion size a is much smaller than the array period. We present the dispersion relation and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in several different directions. Different clusters may contain waves with the same direction, and the frequencies of the waves depend on the clusters but not on the direction of waves. We show that global gaps do not exist if a is small enough. The notion of local gaps which depends on the choice of the wavevector k, is introduced and discussed. The location of local gaps for a medium with a simple cubic lattice of identical inclusions is determined.

New Methodology for Evaluating Strength Degradation from Temperature Increase in Concrete Hydration under Adiabatic Conditions

Cement-based construction materials, commonly known as “cement concrete”, result from the hydration reaction of cement, which releases heat. Numerous studies have examined the heat of cement hydration and other thermal properties of these materials. However, a significant gap in the literature is the assessment of the impact of the hydration temperature on the material’s strength, particularly compressive strength. This work presents an experimental methodology that consistently estimates the temperature evolution of a mixture used to manufacture concrete or mortar during the first hours of Portland cement hydration. The methodology aims to ensure results that correspond to an infinite medium (adiabatic conditions), where there are no heat losses to the surroundings. Results obtained under adiabatic conditions (simulating an infinite medium) indicate that a ready-made mortar (Portland cement: sand: water; 1:2.5:0.5) can reach temperatures of approximately 100 °C after 48 h of hydration. Under these conditions, compressive strength decreases by up to 20%.

Eringen’s nonlocal elasticity theory for the analysis of two temperature generalized thermoelastic interactions in an anisotropic medium with memory

PurposeThe purpose of this paper is to investigate the thermoelastic interactions in a homogeneous, transversely isotropic infinite medium with a spherical cavity in the context of two temperature Lord-Shulman (2TLS) generalized theory of thermoelasticity considering Eringen’s nonlocal theory and memory dependent derivative (MDD). Memory-dependent derivative is found to be better than fractional calculus for reflecting the memory effect which leads us to the current investigation.Design/methodology/approachThe governing field equations of the problem are solved analytically using the eigenvalue approach in the transformed domain of Laplace when the cavity’s boundary is being loaded thermomechanically. Using MATLAB software the numerical solution in real space-time domain is obtained by Stehfest method.FindingsNumerical results for the different thermophysical quantities are presented in graphs and the effects of delay time parameter, non-local parameter and two temperature parameters are studied thereafter. The outcomes of this study convince that the displacement u, conductive temperature ϕ, thermodynamic temperature θ are concave upward whereas radial stress τrr is concave downward for every choice of delay time parameter ω, two temperature parameter η and non-local parameter “ζ”. As a specific instance of our findings, the conclusions of an equivalent problem involving integer order thermoelasticity theory can be obtained, and the corresponding results of this article can be readily inferred for isotropic materials.Originality/valueThe novelty of this research lies in the adoption of generalized thermoelastic theory with memory dependent derivative and Eringen’s nonlocality for analyzing the thermoelastic interactions in an infinite body with spherical cavity by employing eigenvalue approach. It has applications to many thermo-dynamical systems.

Effects of spin torque within ferromagnetic infinite medium: The short-wave approximation and Painlevé analysis

In this paper, we investigate the effects of spin-transfer torques within the ferromagnetic infinite medium through the short-wave approximation method. As a result, we have derived the new (1+1) dimensional nonlinear evolution system, which describes the propagation of electromagnetic short waves within the ferromagnet in the presence of electric current density. Using the Painlevé analysis and Hirota’s bilinearization, we unearth the integrability properties of this new evolution system. In the wake of such an analysis, the typical class of excitations and its physical implications are presented. We remark that the current density acts on magnetization like an effective magnetic damping, which is important for the stabilization of magnetic information storage and data process elements.

Elastodynamic behaviors of steady moving straight dislocation within thin nano film

The elastodynamic dislocation behaviors are of great interest for understanding the performances of structural alloys under intense dynamic loading conditions. The formation, propagations, and interactions of dislocations (such as injected dislocation, accelerating dislocation, steady moving dislocation at high constant speed) are quite different from static dislocations. For steady-moving dislocation within the isotropic infinite medium, the effects of surface and interface on steady-moving dislocations within limited space are still known. In this paper, we investigate the elastodynamic image stress simulation of steady moving dislocation within film of limited thickness at constant speed using Eigenstrain theory, Lorentz transformation, and steady dynamic equilibrium equations. We propose an efficient solution method that involves complex Fourier series, transforming partial differential equations into ordinary differential equations, and ultimately into a set of algebraic equations in spectral space. The effects of dislocation speed and position near the free surface on the image stress of steady-moving climbing and gliding dislocations within the thin film are examined. The results show that relativistic effects are significant for certain dislocation configurations and stress components, whereas other stress components are less sensitive to relativistic effects near the transonic speed region.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Solution of a Dugdale–Barenblatt crack in an infinite strip by a hyper-singular integral equation

This work treats the case of a Dugdale–Barenblatt crack within an infinite strip through the resolution of a hyper singular integral equation. The crack is perpendicular to the strip boundaries and located at its center. The solution approach is based on second order Chebyshev polynomials and requires meticulous treatment of the jump discontinuities within the loading distribution along the crack faces. The relationship between the width of the strip and the length of the cohesive zone has been established. The variation in applied load with the increase in crack length, considering different ratios of the initial crack length to the strip width is illustrated. Furthermore, the crack propagation is simulated. Validation of our approach is achieved through comparison with both the infinite medium case and the work of H. Tada et al. “The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, Pennsylvania. 1973”.

A review on the application of the boundary element method for solving acoustic eigenvalues in infinite media

Vibration analysis is fundamental in the analysis and design of rotating machinery designs, such as turbines, pumps and bearings. It is equally important in the theory of sound and is also related to the properties of electromagnetic waveguides. Its application extends to vehicle structures, equipment and design machines, both for reference and comfort. However, a vibrational analysis also has more daring purposes, directed to the definition of seismic attributes, to the dispersion of phenomena related to acoustics in infinite media. This solution can be adapted as the frequency of vibrations and how outdoor environments are designed for self-balancing devices are designed for Helmholtz self-balancing devices, which are directly adjusted. Typical pressure media situations, Boundary Element Method (BEM) is considered as one of the technical circumstances or more considered, due to its pressure field accuracy, especially differentiated by barriers and scattering of general circumstances, whose frequencies are complex numbers. This work aims to present the various particularities of BEM in approaching these problems, which involve determining the acoustic eigenvalues in open and closed domains, highlighting its operational advantages and numerical difficulties.

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L ≫ l around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of l and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that l ≫ 1 ). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion. This work extends, the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of Gloria and Otto.

Shock Response of Water-Protected Spherical Explosion Containment Vessels

Abstract Explosion containment vessels (ECVs) are important equipment used for underwater explosion experiment research. In this paper, a method is proposed to improve the blast resistance of the ECV by placing it in the water medium. The shock response caused by the underwater explosion in a spherical ECV is studied. The single-layer thin-walled spherical vessel shell submerged in an infinite domain water medium is deduced and simplified as a single-degree-of-freedom elastic vibration system with viscous damping. The underwater explosion shock wave is simplified to an exponential shock loading, and the analytical solutions of the radial shock vibration of the spherical shell are obtained using Duhamel's integrals. Compared with the numerical simulation results, the accuracy of the theoretical model is verified. The study results show that the water medium radiates out vibration energy and plays an important role in eliminating vibration through damping. It is found that the vibration damping ratio can be controlled by adjusting the vessel radius and thickness, so that the vibration of the shell can be controlled within three periods and the impact fatigue can be reduced. In addition, the radiation damping of the water medium greatly reduces the maximum radial displacement of the spherical shell, which significantly improves the blast resistance of the spherical shell.

An adiabatic boundary condition with cylindrical perfect conductors for improved approximations from heat-pulse measurement near the soil-atmosphere interface

The cylindrical-perfect-conductors (CPC) theory, which assumes an infinite and homogeneous soil medium, can be used to analyze heat pulse (HP) measurements to estimate soil thermal property values. However, when a HP sensor is positioned near the soil surface, the CPC theory is not valid because of the change in media properties at the soil-atmosphere interface. In this study, a CPC solution considering an adiabatic boundary condition (CPC-ABC) is presented to account for the soil-atmosphere interface effect. Compared with the results from numerical simulations, the CPC-ABC solution gave more accurate soil temperature values at the sensing probe than did a CPC model. When a HP sensor was positioned horizontally at a depth of 1 mm below a sand soil surface, the relative errors (RE) of CPC estimated thermal property values were as large as 52%, while the RE values based on the CPC-ABC solution were about 8%. Results from numerical simulations and laboratory experiments both showed that the CPC model worked well for horizontally positioned HP sensors with probe lengths of 70-mm at burial depths greater than 15 mm. Soil-atmosphere interface effect was largely dependent on HP sensor dimensions and measurement volumes. Overall, the extended CPC-ABC theory provided accurate soil thermal property estimates by considering the effects of finite probe properties and the presence of the soil-atmosphere interface.

Physical model of neutron scattering by clathrate hydrate and C 60 hosting paramagnetic oxygen molecules

This paper describes the physical modelling of neutron scattering in two polycrystalline inclusion compounds, fully deuterated clathrate hydrate and C60 , each with paramagnetic oxygen as guest molecules. For studying the suitability of these materials for neutron moderation to very low energies, the model includes, in addition to the magnetic neutron scattering by the oxygen, the nuclear scattering by all constituents. The theoretical total cross sections are calculated based on the phonon density of states obtained by density functional theory and molecular dynamics simulations. At low temperatures, the developed scattering kernels are in good agreement with experimental neutron scattering data reported in the literature. At 20 K and above, a Lorentzian distribution for the zero-field splitting of the magnetic substates of the spin triplet of the oxygen molecules helps to reproduce magnetic peaks observed in inelastic neutron scattering experiments better than the original theory based on a single-valued splitting constant. Neutron spectra obtained by Monte Carlo simulations in infinite media are presented, highlighting the potential use of O2 -containing fully deuterated clathrate hydrate as a neutron moderator for the production of very cold neutrons.

Mode I sub- and super-shear rupture and forerunning in porous media

Numerical simulations demonstrate the occurrence of both sub- and supershear ruptures in dry and saturated porous media during mode I fracture. Initially, the Finite Element Method is applied to investigate a case of sub-shear velocity in a dry beam on an elastic foundation subjected to sinusoidal loading. Subsequently, the Extended Finite Element Method is employed for fluid injection in both an infinite saturated porous medium and a plate. The results reveal a transition from smooth behavior to a stepwise pattern and eventually to forerunning as the injection rate increases. Notably, this progression has been observed for the first time. Finally, a hybrid Finite Element/Peridynamic model is utilized to explore both dry and fully saturated media under mechanical loading. In the case of saturated media, fluid injection is also considered. Both methods document velocities surpassing the shear wave. The noteworthy aspect is that these results, obtained through entirely different numerical methods and constitutive relations, justify confidence in the assertion that, in mode I fracture, speeds exceeding the shear wave and even the dilatation wave speeds can be achieved through forerunning. The loading condition investigated may be relevant in geophysics.

Analysis of hole wall stability in saturated clay using undrained solution for cylindrical cavity contraction

To investigate the stress state of soil during the instability of the hole wall (hole shrinkage) in saturated clay, the theory of cylindrical cavity contraction in an infinite medium is applied, utilizing the modified Cam-Clay (MCC) model. By employing intermediate variables, the issue of cylindrical cavity contraction is reformulated into a first-order ordinary differential equation in Lagrangian form, with three effective stress components and pore pressure as the unknowns. In MATLAB, the stress conditions at the elastoplastic boundary serve as the initial values for solving the initial value problem of the differential equation system. After determining the correlation between the stress distribution around the cavity and the diameter reduction ratio, the analysis proceeds to evaluate the internal support pressure required for stabilizing the hole wall according to various stability criteria. This pressure is then integrated into the hole wall stability coefficient formula within the soil pressure model, facilitating the evaluation of hole wall stability. The effectiveness and accuracy of the proposed semi-analytical elastic–plastic solutions were subsequently confirmed through comparison with finite element analysis results, underscoring its relevance for addressing hole wall stability challenges.

Mott waves and fragment size in viscoplastic metals

Previous ductile fragmentation analysis involves unloading wave (Mott wave) propagations in rate-independent rigid-plastic materials. This paper studies the dynamic fragmentations of a rigid-viscoplastic material in which there exists a rigid unloading zone and a viscoplastic unloading zone. A linear diffusion equation incorporating a moving boundary condition for the viscoplastic unloading zone is presented. The fracture process of an isolated crack in the infinite medium, and that of an array of equally spaced cracks, are investigated numerically. The crack array problem reveals an optimal crack spacing upon which the material fails and unloads most rapidly. Based on the 'principle of the rapidest unloading (PRU)', this crack spacing is regarded as the measure of fragment size. Systematic numerical calculations are conducted to obtain the fragment size data as functions of strain rate and viscosity coefficient. A modified Grady-Kipp fragment size formula is proposed to accommodate the viscous effect. In the case of non-viscous material, the formula returns to the classical Grady-Kipp fragment size model. Finite element simulations using rate-dependent constitutive model show similar trend of viscous effect on the fragment sizes, and the modified fragment size formula yields better agreement with the simulated data.

Temperature Field Around Nearly Spherical Cavities in Uniform Heat Flow

Abstract The problem of determining temperature distribution near a spherical cavity in an otherwise infinite medium which is under uniform heat flow is a classical problem of linear heat conduction theory. Extensive reviews of relevant work can be found in [1-5]. It is clear from these studies is that what has been achieved in this regard is largely related to cavities with highly canonical shapes, mostly spherical. Solutions of similar problems for cavities with non-canonical shapes are rare. In the present article, we consider the case of a cavity in an infinite solid in a uniform heat flow, whose shape deviates slightly from a perfectly spherical shape (hereafter called nearly spherical cavity). To the first order in the small parameter characterizing the boundary perturbation, we are able to derive closed-form expressions for the temperature field around the nearly spherical cavity for sufficiently smooth boundary perturbations that are arbitrary functions of the azimuthal and polar angles.

Simultaneous estimation of heat source strength of hot wire and thermal conductivity of material

The hot-wire (HW) technique is a widely used method for measuring the thermal conductivity (k) of insulation materials. However, there are some limitations to the applicability. For instance, inaccuracies may arise when the wire is assumed to be infinitely thin and long, or when the sample material is assumed to be an infinite medium. Additionally, the accuracy of k measurements relies on accurately specifying the heat source strength (QW ) generated by the HW. The modified conjugate gradient method (CGM) provides a solution to these limitations of the HW technique. In the modified CGM technique, a scaling factor based on the Levenberg-Marquardt method is utilized to address the low derivative and divergence issues of the traditional CGM. The current work simultaneously estimates QW generated by the HW and k of the sample materials by using modified CGM. The investigation looked into the effect of sensor positioning relative to the HW and the introduction of artificial Gaussian error (noise) with a standard deviation of 0.01 °C, 0.02 °C and 0.1 °C into the simulated temperature measurements. The modified CGM estimates the unknown parameters accurately compared to the traditional CGM. The measurement location should be closer to the HW to obtain accurate estimation results. The noise level of 0.1 °C results in low accuracy of estimation of k and QW for Plexiglas with one sensor, which can be enhanced by employing two sensors. Overall, the modified CGM algorithm proved to be a robust inverse algorithm that could be employed to estimate unknown parameters.

Thermal Properties of Backfill Material for Underground Heat Exchange Applications

We are investigating the effect of the thermal properties of backfill material on the efficiency of underground thermal energy systems. We have used aluminum to increase the thermal properties of clay-bentonite and to investigate its potential as a backfill material in underground heat exchange applications. The measurements of thermal properties were made at room temperature using the Transient Plane Source (TPS) technique. The measurements were based on an infinite homogeneous medium using different composites of aluminum and clays in amounts of aluminum concentration ranging from 6 to 25 percent of the sample’s weight. The results confirmed an overall relative increase of 170% and 135% in the thermal conductivity and diffusivity, respectively as the Al content increased by 25%. The estimated average of the thermal transmittance (U-value ) of these samples was in the range from 0.40 to 1.1 W/m2K, which was directly proportional to the relative increase in the measured properties. Our findings can be applied to underground thermal energy systems, ground source heat pumps, and solid dielectric underground transmission lines.

WELL-POSEDNESS OF A NONLINEAR INTERFACE PROBLEM DESCRIBED BY HEMIVARIATIONAL INEQUALITIES VIA BOUNDARY INTEGRAL OPERATORS

This paper is devoted to the well-posedness of a novel nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results and, moreover, establish a stability result for extended real-valued HVIs with respect to the extended real-valued function as a parameter. Based on the latter general stability result, we provide various stability results for the interface problem, as well as the stability of a related bilateral obstacle interface problem with respect to the obstacles.

On optimal control in a nonlinear interface problem described by hemivariational inequalities

Abstract This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.