Homogeneous Solids Research Articles

The calculation of elastic constants from displacement fluctuations

We present a methodology for the accurate and efficient extraction of elastic constants in homogeneous solids via the calculation of the atomic displacement correlation function. This approach is validated for cubic solids parametrized by both Lennard-Jones and embedded-atom method potentials. Finally, we also discuss the extension of this method to obtain the elastic properties of inhomogeneous solids.

Inverse determination of heat generation sources in two-dimensional homogeneous solids: Application to orthotropic medium

An implementation of the Network Simulation Method for determining the time-dependent heat generation of a source in a 2-D heat conduction problem under first and second type boundary conditions is presented. The thermal properties of the orthotropic solid are temperature-dependent. No prior information of the functional dependence of the time-dependent source was required. Input data for this inverse problem comprise the temperature history (measurements) at a particular location of the boundary. The common iterative least-squares approach, typical for this kind of ill-posed problem, is used in the estimation, while a piecewise continuous function is proposed as an approximate solution for the estimations. An application to estimate the triangular time dependence of the heat source generation, with error-affected measurements, is shown.

Combined methane reforming in presence of CO 2 and O 2 over LaFe 1− xCo xO 3 mixed-oxide perovskites as catalysts precursors

The paper deals with the effect of modification with cobalt of a series of LaFe 1− x Co x O 3 mixed-oxide perovskites as catalyst precursors in the combined reforming of methane with CO 2 and O 2 for the production of syngas. The perovskite type oxides were synthesized by fine chemical using the citrate sol–gel method and characterized by techniques such as FT-IR, BET surface area, X-ray photoelectron spectroscopy (XPS), X-ray diffraction (XRD), TPR and inductively coupled plasma emission spectroscopy (ICP), under the condition of as-synthesized, reduced and used samples. The results showed that through this synthesis it is possible to obtain highly crystalline, homogeneous and pure solids, with well-defined structures. After reduction, the stabilization by the presence of iron of highly dispersed Co metallic particles from the cation on position B over the corresponding oxides was evidenced, decreasing deactivation by avoiding the sintering of metals and coke formation on the surface of the catalysts. The existence of strong Fe–Co interaction in the LaFe 1− x Co x O 3 with x = 0.4–0.6 produces a synergetic effect improving methane conversion and H 2 production.

Thermal stresses in microstretch elastic plates

We consider the linear theory of homogeneous and isotropic thermo-microstretch elastic solids. First, we present the basic equations which characterize the bending of thin plates. Then we establish a uniqueness result with no definiteness assumption on constitutive coefficients. Existence of solutions is proved under assumption that the internal energy density is positive definite. In the equilibrium theory we present a theorem of minimum potential energy. Finally, the effects of a concentrated heat source in an infinite plate are studied.

Radiative Ignition and Extinction Dynamics of Energetic Solids

The role of heat transfer in radiant ignition/extinction behavior of homogeneous energetic solids has been studied computationally. A model has been developed based on simplified chemical kinetics, unsteady heat transfer processes in the solid phase, and quasi-steady heat transfer in the gas phase. The general behavior of ignition and extinction dynamics for a spatially uniform incident radiant flux has been simulated and explained in terms of unsteady heat transfer phenomena. Two critical heat flux levels for the incident radiation are identified. For fluxes between the critical values, there is an ignition corridor with upper and a lower limits on the allowable time for radiation exposure to achieve ignition. Below the lower critical flux there is no upper limit on the allowable exposure time to achieve ignition. Above the upper critical flux, stable ignition (self-sustained combustion on removal of the radiant flux) is not possible, that is, the ignition corridor becomes vanishingly small. The model explains the ignition corridor in terms of unsteady conductive, advective, and in-depth radiative heat transfer processes in the solid and quasi-steady conductive/advective processes in the gas-phase flame zone. Comparison is made with experimental data for go/no-go ignition behavior of cyclotetramethylene-tetranitramine (HMX).

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6 mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.

Finite amplitude transverse waves in materials with memory

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids with memory. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part depends not only on current but also on previous deformations. The body is first subjected to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that finite amplitude circularly polarized waves may propagate along either n + or n −, where n +, n − are the normals to the planes of the central circular section of the ellipsoid x · B −1 x = 1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.

Congratulations to Akusticheskii Zhurnal on the Occasion of Its 50th Anniversary. Vibrations of Spherical Inclusions in Elastic Solids

Variational principles are derived for the analysis of dynamical phenomena associated with spherical inclusions embedded in homogeneous isotropic elastic solids. The starting point is Hamilton’s principle, with the material properties assumed to vary only with the radial distance r from the origin. Attention is restricted to disturbances that are symmetric about the polar (z) axis, such that the nonzero displacement components in spherical coordinates, u r and uθ, are independent of the polar coordinate φ. The symmetry allows for a decoupling of the polar components, the nth of which is described by U r, n (r, t)P n (cosθ) and Uθ, n(r, t)dP n /dθ. A variational principle is subsequently derived for the field quantities U r, n and Uθ, n. Concepts analogous to those of the theory of matched asymptotic expansions are used to embellish the principle in order to allow for the damping associated with the outward radiation of elastic waves. Examples illustrating the use of the variational principle for formulating plausible lumped-parameter models are given for the cases of n = 0 and n = 1.

Numerical analysis of the cutting interaction between indenters acting on disordered materials

In this paper, an attempt is made to find some general relations for the microcutting process in brittle or quasi-brittle materials, under different hypotheses of microscopic failure behavior. Fracture beneath the indenters and sudden chip formation are the main dissipation mechanisms taken into consideration. Fracture patterns in more homogeneous brittle solids are obtained by the Finite Element Method in the framework of Linear Elastic Fracture Mechanics (LEFM). On the other hand, the quasi-brittle response due to microstructural heterogeneities is taken into account by Lattice Model simulations. The analysis is not limited to the more common study of a single indenter. When two indenters are acting in parallel, their mutual distance plays an important role. If the indenters are very close, they behave like a unique larger indenter, whereas if the distance is relatively large, their mechanical interaction vanishes. In addition, when the distance is approximately three to four times their dimension, the mechanism of chipping (with formation of secondary chip between the two parallel grooves) can take place, improving the ratio of removed volume to spent energy and then the demolition ability of the two indenters. Some comparisons are proposed between the presented approach and more sophisticated and computationally demanding models from the literature, as well as with experimental data. The analysis should provide useful hints for the optimal design of super-abrasive tools.

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.

Finite Volume Method for 2D Elastic Wave Propagation

This article is concerned with a finite volume formulation of elastic wave propagation. It considers a homogeneous domain as well as a domain composed of two homogeneous elastic solids in perfect contact. The interface is parallel to the free surface. In the case of a homogeneous domain, it investigates the usefulness of the method and compares the results to a number of existing methods. In the case of a two-layer domain, it compares the results to an approximate calculation.

FINITE DIFFERENCE METHODS FOR ELASTIC WAVE PROPAGATION IN LAYERED MEDIA

This paper is concerned with the numerical modeling of elastic wave propagation in layered media. It considers two isotropic homogeneous elastic solids in perfect contact. The interface is parallel to the free surface. Two finite difference methods are developed. The usefulness of the methods are investigated for long time simulations and the accuracy of the results are compared with the response from an approximate model.

Non-Newtonian annular vertical flow of sand suspensions in aqueous solutions of guar gum

In this work we present an experimental study of non-Newtonian solid–liquid suspension flow in the annular space of a concentric-pipe system that simulates the geometry of wells during drilling for petroleum exploration and production, excluding effects of drill rotation. The liquids used were solutions of biodegradable guar gum at various concentrations and the solid phase was siliceous sand. We studied how the mean annular velocity, polymer concentration and degree of polymer cross-linking affect axial solids distribution. The results show that the solids suspension capacity of the solutions is enhanced by the presence of polymer: the addition of guar gum decreases the minimum annular velocity needed to achieve a homogeneous axial solids distribution in the annular region. However, the amount of solids that is carried by the solutions is not always increased in the presence of polymer. The chemical cross-linking of guar gum with borax promotes a substantial increase of transported solids in the annular space. Overall, the results presented show that relatively high viscosities, high polymer relaxation times under shear flow and non-zero residual shear stresses are desired rheological characteristics for a drilling fluid to obtain a uniform solids distribution in the annulus, as well as high solids carrying capacity.

Non-Newtonian annular vertical flow of sand suspensions in aqueous solutions of guar gum

In this work we present an experimental study of non-Newtonian solid–liquid suspension flow in the annular space of a concentric-pipe system that simulates the geometry of wells during drilling for petroleum exploration and production, excluding effects of drill rotation. The liquids used were solutions of biodegradable guar gum at various concentrations and the solid phase was siliceous sand. We studied how the mean annular velocity, polymer concentration and degree of polymer cross-linking affect axial solids distribution. The results show that the solids suspension capacity of the solutions is enhanced by the presence of polymer: the addition of guar gum decreases the minimum annular velocity needed to achieve a homogeneous axial solids distribution in the annular region. However, the amount of solids that is carried by the solutions is not always increased in the presence of polymer. The chemical cross-linking of guar gum with borax promotes a substantial increase of transported solids in the annular space. Overall, the results presented show that relatively high viscosities, high polymer relaxation times under shear flow and non-zero residual shear stresses are desired rheological characteristics for a drilling fluid to obtain a uniform solids distribution in the annulus, as well as high solids carrying capacity.

Response of orthotropic micropolar elastic medium due to time harmonic source

The present paper is concerned with the plane strain problem in homogeneous micropolar orthotropic elastic solids. The disturbance due to time harmonic concentrated source is investigated by employing eigen-value approach. The integral transforms have been inverted by using a numerical technique to obtain the component of displacement, force stress and couple stress in the physical domain. The results of these quantities are given and illustrated graphically.

Thermodynamically consistent variational principles with applications to electrically and magnetically active systems

We propose a theoretical framework to derive thermodynamically consistent equilibrium equations and kinetic driving forces to describe the time evolution for electrically and magnetically active materials. This procedure starts from the combined statement of the first and second laws of thermodynamics, naturally incorporates Maxwell’s equations, and accommodates the description of continuous phase transformations for conserved and non-conserved order parameters. The kinetics of conserved and non-conserved ordered parameters are introduced, the adequate gradient flow is identified, thus the appropriate kinetics (e.g., Allen–Cahn, Cahn–Hilliard) are derived. Example applications of this theory include the electromechanical fields of piezoelectric materials and the wave equation in the limit of chemically homogeneous solids. Moreover, we derive a thermodynamically consistent set of partial differential equations which describe the transport of charged species in conductive, non-polarizable, magnetizable solids, and in polarizable, electrically insulating, non-magnetizable solids.

Elasticity theory connection rules for epitaxial interfaces

Elasticity theory provides an accurate description of the long-wavelength vibrational dynamics of homogeneous crystalline solids, and with supplemental boundary conditions on the displacement field can also be applied to abrupt heterojunctions and interfaces. The conventional interface boundary conditions, or connection rules, require that the displacement field and its associated stress field be continuous through the interface. We argue, however, that these boundary conditions are generally incorrect for epitaxial interfaces, and we give the general procedure for deriving the correct conditions, which depend essentially on the detailed microscopic structure of the interface. As a simple application of our theory we analyze in detail a one-dimensional model of an inhomogeneous crystal, a chain of harmonic oscillators with an abrupt change in mass and spring stiffness parameters. Our results have implications for phonon dynamics in nanostructures such as superlattices and nanoparticles, as well as for the thermal boundary resistance at epitaxial interfaces.

Optimal dip based on dissipation of back thrusts and hinges in fold‐and‐thrust belts

Back thrusts and hinges are two types of transition between rigid sections of hanging walls observed in fold‐and‐thrust belts. Back thrusts are typical of frictional and homogeneous solids and hinges of creeping and layered materials. Our objective is to study the orientation of these transitions for the special case of a lower flat‐ramp transition in a fault‐bend fold with the following general two‐step methodology. In the first step, the forces acting on the transition are determined using equilibrium of each rigid section. In the second step, the optimal dip of the transition is obtained by minimizing the total dissipation of the structure. The three sources of dissipation, of comparable magnitude, are at the transition, on the flat, and on the ramp. For frictional material flows, the back thrusts are velocity discontinuities with optimal dips always less than half the complementary ramp angle, leading to hanging wall thickening. The optimal dip agrees well with the results of physical analogue and numerical experiments. For creeping and layered materials, it is shown that a destabilizing deformation mechanism, selected to be flexural slip, is necessary for the strain to localize and the existence of hinges to be justified. Activation of flexural slip reduces dissipation at the transition and affects the optimal transition dip. The two‐step methodology proposed here could be seen as a first attempt in producing mechanically balanced cross sections accounting for material rheology. This approach should complement the now classical kinematic models of folding.

Electronic noise due to multiple trap levels in homogeneous solids and in space-charge layers

A near-exact expression is derived for carrier-density fluctuations associated with multiple trap levels, interacting with one band only (say, the conduction band), using the many-variate Master Equation. The canonical constraint is handled with the Darwin–Fowler method. For a homogeneous solid with a discrete number of traps the spectrum consists of a sum of Lorentzians, as expected. For the case of a continuously distributed energy range, the results do not support a Bernamont–Surdin–McWhorter 1/f-like envelope spectrum; on the contrary, no more than two decades of 1/f noise can be expected. The situation is different, however, when band-bending occurs. Employing a Green’s function approach, developed by us previously, we give a formal complete result, which can be evaluated when Ec(r) is known.