Infinite Elastic Medium Research Articles

Plane strain problem of an elastic matrix containing multiple Gurtin–Murdoch material surfaces along straight segments

This paper presents the study of the plane strain problem of an infinite isotropic elastic medium subjected to far-field load and containing multiple Gurtin–Murdoch material surfaces located along straight segments. Each material segment represents a membrane of vanishing thickness characterized by its own elastic stiffness and residual surface tension. The governing equations, the jump conditions, and the surface tip conditions are reviewed. The displacements in the matrix are sought as the sum of complex variable single-layer elastic potentials whose densities are equal to the jumps in complex tractions across the segments. The densities are found by solving the system of coupled hypersingular boundary integral equations. The approximations by a series of Chebyshev’s polynomials of the second kind are used with the square root weight functions chosen to satisfy the tip conditions automatically. Numerical examples are presented to illustrate the influence of dimensionless parameters and to study the effects of interactions.

A modified Moore-Gibson-Thompson fractional model for mass diffusion and thermal behavior in an infinite elastic medium with a cylindrical cavity

<p>This article discussed a new fractional model that included governing equations describing mass and thermal diffusion in elastic materials. We formulated the thermal and mass diffusion equations using the Atangana-Baleanu-Caputo (ABC) fractional derivative and the Moore-Gibson-Thomson (MGT) equation. In addition to the fractional operators, this improvement included incorporating temperature and diffusion relaxation periods into the Green and Naghdi model (GN-Ⅲ). To verify the proposed model and analyze the effects of the interaction between temperature and mass diffusion, an infinite thermoelastic medium with a cylindrical hole was considered. We analyzed the problem under boundary conditions where the concentration remained constant, the temperature fluctuated and decreased, and the surrounding cavity was free from any external forces. We applied Laplace transform techniques and Mathematica software to generate calculations and numerical results for various field variables. We then compared the obtained results with those from previous relevant models. We have graphically depicted the results and extensively examined and evaluated them to understand the effects of the relationship between temperature and mass diffusion in the system.</p>

Analytical solution for diffraction of magnetoelastic plane waves by a rigid strip in self-reinforced medium: a contour integration method

This study analytically addresses the diffraction of magnetoelastic plane waves by a rigid strip in a self-reinforced infinite elastic medium. Formulation of the studied model includes mixed type boundary value problem and has been solved by the method of integral equation. The expressions of vertical diffracted displacement and normal stress are established in the closed form by employing the contour integration technique. For the sake of validation, the deduced expressions of vertical diffracted displacement and normal stress are reduced to the special cases, such as without magnetoelastic coupling as well as anisotropy and matched with the pre-established results. The diffraction patterns for the vertical displacement component in the considered material medium are calculated numerically, and its dependence on various affecting parameters viz. magnetoelastic coupling parameter, phase velocity, wave number of the propagating magnetoelastic plane wave, and distance has been depicted graphically for self-reinforced and isotropic materials. Moreover, the influence of anisotropy of the infinite medium has been portrayed remarkably by the comparative study.

An accurate semi-analytical method for the treatment of an eccentric annular crack embedded in an infinite isotropic elastic medium under arbitrary internal pressure

This work provides a semi-analytical method for the calculation of the stress intensity factor (SIF) associated with an internally pressurized eccentric annular crack embedded in an infinite isotropic elastic solid. The inner and outer edges of the crack consist of the circumferences of two eccentric circles and the internal pressure exerted on the crack faces has an arbitrary distribution. Using Somigliana formulation a hypersingular integral equation governing the proposed crack problem is obtained. Next, by the employment of a conformal mapping the considered crack is mapped on to a concentric annular crack. In the mapped domain, with the aids of Fourier expansion of the displacement discontinuity function across the crack faces together with the notion of the Hadamard finite-part integral, the hypersingular integral equation is converted to a system of linear algebraic equations for the unknown constants of the expansion. Upon the determination of these constants, the angular dependent SIFs along both the inner and outer edges of the crack are obtained accurately. The accuracy of the current formulation is showcased through the re-examination of the simple concentric annular crack problems considered previously in the literature. In the presence of eccentricity, the effects of such geometrical parameters as the ratio of the radii of the inner to outer eccentric circular edges of the crack and the normalized eccentricity on the values of the SIFs along both the inner and outer edges of the crack are studied. As it will be seen the material properties of the elastic matrix do not affect the values of SIFs.

Mode-III external circular crack in an infinite orthotropic medium under torsional loading

This paper contains an investigation of interaction of torsional wave with a mode-III outer circular crack in an infinite orthotropic elastic medium. Using mixed boundary condition, the problem reduced into a dual integral equation. After some manipulation and use of Abel’s singular integral equation, the dual integral equation transformed into the Fredholm second kind integral equation, which is solved numerically. The obtained solution has been adopted to interpret the physical quantities like Crack Torsional Displacement (CTD) and mode-III Stress Intensity Factor (SIF) which are shown graphically.

Plane waves in Moore–Gibson–Thompson thermoelasticity considering nonlocal elasticity effect

The Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory together with the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in an infinite elastic medium that conducts heat. Two sets of coupled longitudinal thermoelastic waves have been initiated. These waves are dispersive in nature and experience attenuation. In conjunction with the coupled waves, there also exists one uncoupled vertically shear–type (SV-type) wave. The SV-type wave is dispersive in nature but without any attenuation. The elastic nonlocality parameter influences all these waves. Furthermore, the SV-type wave faces critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. We also explore the problem of reflection of thermoelastic waves at a stress-free insulated and isothermal plane boundary of the medium considered here. The reflection coefficients and their respective energy ratios of the reflected waves to that of incident wave are determined analytically and numerically by considering an appropriate material. The numerical results are presented graphically to highlight the effects of various parameters of interest. Some important observations about the prediction of the MGT model in comparison to the other existing models (Lord–Shulman (LS) and the Green–Nagdhi theory of type III (GN-III) models) are highlighted.

Variation of Displacement Fields for Great Earthquakes due to Rectangular Fault System

The expressions for the displacement fields produced by a finite rectangular fault associated with great earthquakes are explained by elastic theory of dislocation. The variation of displacement fields has been estimated by considering the elastic half-space in contact with traction free boundary and varying depth, distance from top to bottom of the fault. It is also carried out in an infinite elastic medium. Also, the variation of displacement fields with the distance from fault especially for strike slip fault shows that even if different fault length and approximately equal magnitudes, the variation of displacement fields with the distance from the fault is almost the same in the case of great earthquakes viz. San Francisco (1906) and Bihar-Nepal (1934).

A Boundary-Field Formulation for Elastodynamic Scattering

An incoming elastodynamic wave impinges on an elastic obstacle is embedded in an infinite elastic medium. The objective of the paper is to examine the subsequent elastic fields scattered by and transmitted into the elastic obstacle. By applying a boundary-field equation method, we are able to formulate a nonlocal boundary problem (NBP) in the Laplace transformed domain, using the field equations inside the obstacle and boundary integral equations in the exterior elastic medium. Existence, uniqueness and stability of the solutions to the NBP are established in Sobolev spaces for two different integral representations. The corresponding results in the time domain are obtained. The stability bounds are translated into time domain estimates that can serve as the starting point for a numerical discretization based on Convolution Quadrature.

Hyperbolic two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to the non-Gaussian laser beam

The present work is devoted to a study of the induced temperature and stress fields in an elastic infinite medium with a cylindrical cavity under the purview of hyperbolic two-temperature thermoelasticity. The medium is an isotropic homogeneous thermoelastic material. The bounding plane surface of the cavity is loaded thermally by a non-Gaussian laser beam with a pulse duration of two ps. An exact solution to the problem is obtained in the Laplace transform space and the inversions of the Laplace transform have been carried out numerically. The derived expressions are computed numerically for copper and the results are presented in graphical form. The two-temperature parameter has significant effects on all the studied functions and plays a vital role in the speed propagation of the thermomechanical waves.

Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions.

We present formal asymptotic approximations of fields representing the in-plane dynamic response of elastic solids containing clusters of closely interacting small rigid inclusions. For finite densely perforated bodies, the asymptotic scheme is developed to approximate the eigenfrequencies and the associated eigenmodes of the elastic medium with clamped boundaries. The asymptotic algorithm is also adapted to address the scattering of in-plane waves in infinite elastic media containing dense clusters. The results are accompanied by numerical simulations that illustrate the accuracy of the asymptotic approach.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

Dispersion of torsional surface waves in a threefold concentric compounded cylinder with imperfect interface

The present paper investigates the torsional wave propagation in a threefold concentric pre-stressed compounded cylinder with imperfect contact conditions. The three-dimensional linearized theory of elastic waves and the piecewise homogeneous body model has been employed to formulate the problem. The mathematical modeling has been carried out in two independent cases. In the first case, a solid cylinder encased in a hollow cylinder embedded in an infinite elastic medium has been considered. Whereas the second case comprises a hollow cylinder of finite thickness in place of a solid cylinder. By means of Murnaghan potential, the mechanical characteristics of the three materials have been used. Further, the dispersion relations for both the cases have been obtained in terms of the Bessel and modified Bessel functions. In order to validate the present findings, two particular cases have been derived that matches with the previous works. The first case is obtained by removing the outermost cylinder, while the second case has been derived by removing the imperfection in addition to that. To summarize the computations, a complete numerical simulation has been carried out, and graphical illustrations have been shown to aid the mathematical analyses.

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.

Analytical solution for a vibrating rigid sphere with an elastic shell in an infinite linear elastic medium

The analytical solution is given for a vibrating rigid core sphere, oscillating up and down without volume change, situated at the center of an elastic material spherical shell, which in turn is situated inside an infinite (possible different) elastic medium. The solution is based on symmetry considerations and the continuity of the displacement both at the core and the shell-outer medium boundaries as well as the continuity of the stress at the outer edge of the shell. Furthermore, a separation into longitudinal and transverse waves is used. Analysis of the solution shows that a surprisingly complex range of physical phenomena can be observed when the frequency is changed while keeping the material parameters the same, especially when compared to the case of a core without any shell. With a careful choice of materials, shell thickness and vibration frequency, it is possible to filter out most of the longitudinal waves and generate pure tangential waves in the infinite domain (and vice-versa, we can filter out the tangential waves and generate longitudinal waves). When the solution is applied to different frequencies and with the help of a fast Fourier transform (FFT), a pulsed vibration is shown to exhibit the separation of the longitudinal (L) and transverse (T) waves (often called P- and S-waves in earthquake terminology).

Features of modeling of stress and strain waves in an anisotropic medium on the example of a wooden element

This article describes the hypotheses of the occurrence, propagation, and modification of stress and strain waves caused by external loads in isotropic and anisotropic infinite and finite elastic media. A model of an infinite elastic medium experiencing a point external impulse is presented. The model demonstrates the propagation of a longitudinal plane wave. Compaction and rarefaction of the medium are observed in the plane with wave propagation. A graph of changes in the amplitude of a longitudinal plane wave is presented in the same coordinate system. The problem is posed of expanding the numerical model of a finite elastic medium in the form of an anisotropic wooden rod experiencing a plane external impulse. The model should demonstrate the propagation of longitudinal and transverse waves and describe the volumetric deformation of an anisotropic material. Compaction and rarefaction of the medium are shown in the plane, coinciding with the direction of wave propagation. A graph of the change in the shear wave amplitude is presented in the same coordinate system. The combination of these two graphsreveals the difference in wave propagation velocities and the combination of amplitudes. The model will make it possible to identify the presence of Rayleigh waves and to describe the reflection of waves from the boundary of the medium.

Numerical Fracture mechanics based prediction for the roughening of brittle cracks in 2D disordered solids

The behavior of fractures in 2D brittle disordered materials results from the competition between the microstructural disorder that roughens cracks and the material elasticity that straightens them. The experimental fracture surfaces left behind are generally scale invariant and their complex geometry are still largely unexplained. Here, this issue is addressed numerically using the boundary element method that predicts incrementally the trajectory followed by cracks in an infinite elastic medium with a heterogeneous field of fracture energy. The predicted fracture profiles are characterized using their height-height correlation function and show scale invariant properties. In particular, simulated cracks are shown to follow a random walk with roughness exponent $$\zeta = 0.5$$ , irrespective of the level of microstructural disorder, indicating that the crack behavior is neither persistent nor anti-persistent, but instead, that the orientation of the next propagation increment is chosen randomly, independently of the past trajectory. This behavior is then interpreted from fracture mechanics calculations that show that the restoring force emerging from elasticity that generally straightens cracks vanishes in the limit of large specimens with respect to the characteristic size of the material heterogeneity. Our findings shed light on some experimental fracture patterns recently reported and suggests an explanation for the observation of two different types of fracture morphologies in 2D disordered brittle solids.

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.

Dispersion of Longitudinal Waves by Four Coplanar Mode-I Cracks in an Infinite Elastic Medium

Dispersion of Longitudinal Waves by Four Coplanar Mode-I Cracks in an Infinite Elastic Medium

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.

The Ether Made Explicit Through Light, A Possible Universal Informational Matrix

Abstract Supossing that the ether is an infinite elastic medium, the constitutive material law that describes such a behaviour is determined. In a totally particular case, corresponding both to the stress tensor, and also to the strain tensor, it is shown that this constitutive material equation can be identified with an electromagnetic field, and, by extension, the Universe is of a informational type (universal informational matrix).

Misfit Stresses Due to a Cylindrical Dilatational Inclusion of Annular-Sector Cross-Section in an Infinite Elastic Medium

An elastic model for a cylindrical dilatational inclusion of annular-sector cross-section in an infinite elastic medium is considered. The stress fields are found in a closed analytical form and are illustrated by stress maps. Specific features in the stress distribution are revealed and discussed in detail. It is shown that the stress magnitude can be so high that various mechanisms of stress relaxation can be activated.