Isotropic Elastic Half-space Research Articles

МАТЕМАТИЧНЕ ТА КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ КОНТАКТНОЇ ВЗАЄМОДІЇ ТРАНСВЕРСАЛЬНО-ІЗОТРОПНИХ ПРУЖНИХ ПІВПРОСТОРІВ ЗА ПРИСУТНОСТІ МІЖ НИМИ ЖОРСТКОГО ПЛОСКОГО ВКЛЮЧЕННЯ ДОВІЛЬНОЇ ФОРМИ

Mathematical and computer modeling of the contact interaction of two transversely isotropic elastic half-spaces with different properties in the presence of a rigid flat inclusion between them under compression is carried out. Based on the representation of the general solution of the system of equilibrium equations for a transversally isotropic body in terms of harmonic functions, a correspondence is established between the contact interaction parameters for two transversely isotropic and two elastic isotropic half-spaces (in the presence of a rigid planar inclusion of an arbitrary shape between them), which generalizes Gladwell's result to the case interactions of transversely isotropic half-spaces. By means of the mathematical and computer modeling, the contact interaction of transversely isotropic half-spaces with a disk-like inclusion is studied. The analysis of the numerical results is carried out, the influence of the elastic properties of the half-space, the geometric dimensions of the inclusion on the parameters of the contact interaction is studied.

Tensile Fault Dislocation in an Irregular-Layered Elastic Half-Space

In the present paper, an analytical solution for the static deformation of a two dimensional model consisting of an infinite homogeneous isotropic elastic layer of uniform thickness placed over an irregular isotropic elastic half-space due to movement of a long tensile fault has been obtained. The rectangular shaped irregularity is assumed to be present in the lower half-space and assuming that the fault lies in the elastic layer at a finite depth say ’<i>h</i>’ to the upper surface of the layer. For numerical computation, the expressions of displacements and stresses are calculated by using Sneddon’s method and the effect of source depth and irregularity on the displacements and stresses has been investigated graphically.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

Influence of boundary conditions on the reflection and transmission of qP-wave at an interface between two nonlocal transversely isotropic elastic half-spaces

This work is concerned with the reflection and transmission of quasi P wave incident at an imperfect interface between two nonlocal transversely isotropic elastic half-spaces. The linear spring model is used to describe the imperfection of bonding behavior at the interface. Reflection, transmission coefficients (RTC) have been derived analytically for when a longitudinal displacement wave strikes for both imperfect and perfect interface. Finally, numerical examples are provided to show the effect of the imperfect interface, nonlocal parameter and incident angle on the reflection and transmission coefficients.

Mathematical modeling of the contact interaction of two elastic transversely isotropic half-spaces, one of which contains a near-surface groove of an elliptical section

On the basis of a mathematical model, the problem of compression of two elastic transversely isotropic half-spaces, one of which contains a shallow near-surface groove of an elliptical section, is considered. The solution to the problem is obtained using the Elliott representation for a transversely isotropic body in terms of harmonic functions, classical harmonic potentials and reducing the boundary value problem to considering an integro-differential equation with an unknown domain of integration. As a special case, the obtained analytical expressions yield the basic parameters of the contact of transversely isotropic half-spaces in the presence of an axisymmetric groove in one of them, as well as the parameters of the contact interaction of two elastic isotropic half-spaces, one of which contains an elliptical cross-section groove. Numerical results are obtained, the influence of elastic properties of half-spaces, geometrical parameters of groove and loading on contact interaction and closing of the gap between bodies is studied.

Inhomogeneous transversely isotropic space under influence of concentrated power and temperature sources

The problem of constructing fundamental solutions to the thermoelasticity problem for a piecewise-homogeneous transversely isotropic space is reduced to the matrix Riemann problem in the space of generalized slow growth functions. As a result of the solution of which, were obtained expressions in explicit form for the components of the stress tensor and the displacement vector in plane of connection of transversely isotropic elastic half-spaces containing concentrated stationary heat sources. The temperature distribution is investigated depending on the thermophysical characteristics of the half-space materials.

Interaction of circular cavity and crack near anisotropic bi-material interface under SH-waves incidence

AbstractAn anisotropic bi-material interface with a circular hollow and a crack was used to explore the scattering of SH waves using complex variables and Green's function techniques. A horizontal interface in a bi-material media forms an elastic anisotropic half-space with a circular cavity and a linear crack, whereas the other half-space is an elastic isotropic half-space. The Green's function approach is used to derive the relevant Green's functions, which are then used in the “crack division” procedure to create a crack. During the joining of the two pieces, unknown anti-plane forces are applied to the horizontal surfaces to ensure the interface maintains its continuity criteria. For weakly singular integral equations, Fredholm integral equations are inferred and solved using the direct discrete technique. The simulation results were represented graphically to show how different parameters affect the influence of dynamic stress concentration factors around the cavity and dynamic stress intensity factors toward the crack tip. For example, a cavity's wave number and angle of incidence, as well as the distance between the cavity's center and its horizontal interface and its radius, are all elements that must be taken into consideration.

Study of SH-Wave Propagation in an Initially Stressed Triclinic Layer Sandwiched between Transversely Isotropic Elastic and Heterogeneous Poroelastic Half-Spaces

Study of SH-Wave Propagation in an Initially Stressed Triclinic Layer Sandwiched between Transversely Isotropic Elastic and Heterogeneous Poroelastic Half-Spaces

Linear interface crack under harmonic shear: Effects of crack's faces closure and friction

The linear crack between two dissimilar elastic isotropic half-spaces under normal harmonic shear loading is considered taking the crack's faces contact interaction and friction into account. The problem is solved by the boundary integral equations method and the components of the solution are represented by the Fourier series. The numerical convergence of the method is analysed. The results are validated through the comparison with the classical model solutions obtained for the static problems with and without friction. The effects of material properties and values of the friction coefficient on the distribution of the stress intensity factors (normal opening and transverse shear modes) are presented and analysed.

Three-Dimensional Dynamic Response Analysis of Rigid Foundation Embedded in Layered Transversely Isotropic Half-Space

ABSTRACT This paper employs an indirect boundary element formulation for 3D soil-structure interaction analysis in layered transversely isotropic (TI) elastic half-space under incident qP, qSV, and SH waves. Transformation matrix and dynamic Green’s function are employed to compute dynamic impedance coefficients, and foundation displacement is computed by superposition of two cases (with and without mass of foundation and upper structure). Upper structure displacement is solved from dynamic equilibrium equation. An embedded foundation with a single-point structure is simulated to study the influence of the ratio of transverse and vertical Young’s modulus of TI medium and incident angle on soil-structure system response.

The surface tension effect revealed via the indentation scaling index

The axisymmetric frictionless indentation problem for an isotropic elastic half-space with the uniform surface tension is considered. The first-order regular and singular asymptotic models are constructed for the limit situations of relatively small (in the membrane limit) or large (in the Hertzian limit) values of the contact radius. By using the method of collocation, a two-parameter analytical approximate solution is developed with coefficients evaluated in triple quadratures. The accuracy of the obtained analytical models is investigated in the case of a paraboloidal indenter by comparison with numerical solutions available in the literature. Based on the asymptotic behavior of the solution in the limit situations, the so-called indentation scaling index is introduced for the analysis of the depth-sensing indentation data. By making use of the novel contact parameter it is possible to extract the surface tension effect independently of determining the reduced elastic modulus of a tested semi-infinite sample. It is shown that by employing an appropriate normalization, the contact problem can be reduced to that parameterized by only one parameter (relative contact radius), which implies that irrespectively of the combination of the material parameters, the force–displacement curve drawn in the normalized variables will be universal.

Green’s function analysis of shear wave propagation in heterogeneous poroelastic sandwiched layer influenced by an impulsive source

The present analysis confers the propagation characteristics of horizontally polarized​ shear (SH) wave through a heterogeneous transversely isotropic fluid-saturated poroelastic sandwiched layer of finite width embedded between two heterogeneous isotropic elastic half-spaces due to the impact of an impulsive line source. The dissipation of energy caused by the relative fluid flow is neglected in transversely isotropic fluid-saturated poroelastic sandwiched layer. The heterogeneities in layer and half-spaces are considered with the respective spatial quadratic and exponential type variations in the elastic constants and densities. Green’s functions are obtained for this sandwiched layered structure by considering a charge-density/line-force (Dirac-delta function) at lower bonding surface of layer and half-space. Dispersion relation for the propagation of SH wave in aforesaid structure has been obtained with well-known properties of Green’s functions, Fourier’s transformation and appropriate boundary conditions. The obtained dispersion relation is reduced for a structure with a heterogeneous transversely isotropic fluid-saturated poroelastic layer of finite width overlying a heterogeneous isotropic elastic half-space in the particular case of problem. Moreover, the obtained dispersion relations are validated and matched with pre-established standard and classical results. The effect of various affecting parameters viz. anisotropy of layer, porosity of layer and heterogeneities of layer and half-spaces on phase velocity of propagating wave in said structure are shown graphically through numerical computations. Further, a comparative study of distinct cases of heterogeneous/homogeneous structures is also performed which serves as the major highlight of the present study.

2-D Analysis of Generalized Thermoelastic Porous Medium under the Effect of Laser Pulse and Microtemperature

The paper presents the analytical solutions for a generalized thermoelastic medium consisting of microtemperatures and voids subjected to a laser pulse loading the medium thermally. The 0.02 ps pulse duration of the non-Gaussian laser beam is apt for heating a homogenous isotropic elastic half-space. A method called the normal mode analysis is employed to evaluate numerically the effects of various variables such as the micro-temperature vector, variation in the fraction field of the volume, first heat flux moment tensor, temperature distribution on the stresses and displacement components of the medium. In addition, the graphical illustration of the physical response of the medium has been presented in the presence and absence of void parameters, as well as in the presence of laser pulse with two different acting periods.

Waves at the imperfect boundary of elastic and bio-thermoelastic diffusive media

The present investigation deals with the reflection and transmission phenomena in isotropic elastic half-space (ES) and bio-thermoelastic diffusive half-space (BDS) for the study of plane harmonic waves. The two half-spaces are separated by an imperfect interface and the governing equations for BDS are modified by introducing phase lag parameters in Fourier’s law of heat conduction and Fick’s law of mass diffusion. After expressing the equations in two-dimension, the potential functions are used to simplify the equations for further calculation. The plane P or SV wave is incident from ES which results in two reflected waves in ES and four transmitted waves in BDS. The amplitude ratios are obtained for these reflected and transmitted waves concerening the incident P or SV wave for stiffness forces. The obtained amplitude ratios are further used to drive the energy ratios of various reflected and transmitted waves. These ratios are impacted by the characteristics of the wave and material properties of the media. The resulting energy ratios are computed numerically by developing a numerical algorithm and displayed in the form of graphs to show the effects of blood perfusion rate, lagging times, and stiffness parameters.

Interior-stress fields produced by a general axisymmetric punch

This work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile. A novel set of mathematical procedures is introduced to process the basic elastic solutions (obtained by the method of Hankel transform, which was pioneered by Sneddon) and the solution of the dual integral equations. These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space. The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study, and the deduced results are verified by comparing them with the classical results. Finally, these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

Thermoelastic medium in the context of four theories subjected to gravity field and laser pulse

The paper aims at studying the impact of gravity and laser pulse on the overall structure of the generalized thermoelasticity equations concerning a standardized elastic isotropic half-space. Using four generalized thermoelasticity theories, i.e. the Green-Naghdi (GN) theory, Green-Lindsay (GL) theory, Lord-Schulman (LS) theory, and Coupled Theory (CT), the authors applied the formulation. In the physical sphere, they attained the analytical expressions for the (Mechanical and Maxwell’s) stresses distribution, displacement components, and temperature through utilizing the normal mode analysis. Moreover, they calculated these expressions numerically and plotted the conforming graphs to demonstrate and make comparisons between theoretical findings. The paper concludes that the impact of gravity and laser pulse field is highly significant.

A Mode-I crack for a rotational fibre-reinforced thermoelastic medium with thermal relaxation time

A general model of the equations with coupled theory (CD) and Lord-Shulman (L-S) theory with one relaxation time is applied to study the influence of reinforcement on the total deformation for an infinite space weakened by a finite linear opening Mode-I Crack is solving. We study the influence of reinforcement on the total deformation of rotating thermoelastic half-space and the interaction with each other. The material is homogeneous isotropic elastic half space. The crack is subjected to prescribed temperature and stress distribution. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses and temperature. The variations of the considered variables with the horizontal distance are illustrated graphically. Comparisons are made with the results in two theories in the presence and absence of rotation. A comparison also is made between the two theories for different depths.

Near-Resonant Regimes of a Moving Load on a Pre-Stressed Incompressible Elastic Half-Space

Abstract The article is concerned with the analysis of the problem for a concentrated line load moving at a constant speed along the surface of a pre-stressed, incompressible, isotropic elastic half-space, within the framework of the plane-strain assumption. The focus is on the near-critical regimes, when the speed of the load is close to that of the surface wave. Both steady-state and transient regimes are considered. Implementation of the hyperbolic–elliptic asymptotic formulation for the surface wave field allows explicit approximate solution for displacement components expressed in terms of the elementary functions, highlighting the resonant nature of the surface wave. Numerical illustrations of the solutions are presented for several material models.

Mathematical Modeling of Vibration Dampers of Vibration-Insulated Structures

Vibration dampers are installed on the machine foundations in order to reduce the vibration level. Such technological solutions are most expedient in the case of a harmonic load with a low instability of the vibration frequency. Unfortunately, dampers do not provide such a large reduction in the dynamic effect on the base, as vibration isolation, but in some cases their efficiency turns out to be quite sufficient with a relatively simple implementation and low manufacturing cost. The use of dynamic vibration dampers gives a great effect when an increased vibration of foundations occurs during the operation of equipment in metallurgical production, for example, when processing materials by pressure, reconstructing enterprises and replacing heavy equipment. During the operation of heavy forging equipment and manipulators for various purposes, the foundations of these devices can be considered as a rigid body. The model soil on which this foundation is installed can be considered a homogeneous elastic isotropic half-space. When calculating with such mathematical models, one can use solutions of the corresponding dynamic contact problems. A comparative analysis of the effectiveness of damping foundation vibrations using different foundation models, including the model of an elastic, homogeneous half-space and a system of semi-infinite rods, the modulus of elasticity of which increases with depth according to the quadratic law, shows a fairly close agreement.

The fundamental solution of the problem of thermoelasticity for a piecewise homogeneous transversely isotropic elastic space

The problem of constructing fundamental solutions to the thermoelasticity problem for a piecewise-homogeneous transversely isotropic space is reduced to the matrix Riemann problem in the space of generalized slow growth functions. As a result of the solution of which, were obtained expressions in explicit form for the components of the vector of the fundamental solution of the heat conduction problem, as well as simple representations for the components of the stress tensor and the displacement vector in plane of connection of transversely isotropic elastic half-spaces containing concentrated stationary heat sources. The temperature distribution is investigated depending on the thermophysical characteristics of the half-space materials.