Homogeneous Elastic Layer Research Articles

Analytical solution for axially loaded floating pipe piles in homogeneous elastic soil layer

An analytical solution for the analysis of open-ended floating pipe piles in a homogeneous elastic soil layer is developed in this paper, based on the Tajimi-type elastic continuum model. Solving the governing equations of the outer and inner soil results in closed-form expressions for the external and internal frictional resistances developing at the soil-pile interfaces. The one-dimensional rod theory is adopted to model the pipe pile and its underlying soil hollow column. The external and internal frictional resistances are substituted into the governing equations of the pile and soil column to derive closed-form solutions for the vertical displacement and head stiffness of the pipe pile. The accuracy of the proposed model is verified by comparisons against existing numerical studies. Finally, parametric analyses are presented to discuss the sensitivity of the behavior of axially loaded floating pipe piles to the main problem parameters.

A Receding Contact Problem of Two Layers one of Functionally Graded, Loaded by Circular Rigid Block and Resting on a Pasternak Foundation

In this study, frictionless receding contact problem of two elastic layers which one is functionally graded material (FGM) resting on a Pasternak foundation is considered. The external load is applied to the homogeneous elastic layer by means of a circular rigid block and the functionally graded layer rests on a Pasternak foundation. The effect of gravity forces is neglected, and only compressive normal tractions can be transmitted through the interfaces. Displacement and stress expressions for the layers are obtained using the theory of elasticity and integral transformation technique. By applying the boundary conditions for the problem, reduced to two integral equations in which the contact stresses and contact lengths are unknown. The system of integral equations is numerically solved by making use of appropriate Gauss Chebyshev integration formulas. The equilibrium conditions are satisfied in the solution and the contact stresses and contact distances related to the problem are obtained for various dimensionless quantities.

Analytical solution for axial load transfer of floating piles

This paper presents an analytical solution for the analysis of axially-loaded compressible floating piles embedded in a homogeneous elastic soil layer of finite thickness. The mathematical model is based on a Tajimi-type elastic continuum formulation, that allows deriving an accurate closed-form expression for the frictional stresses developing at the soil-pile interface. The pile and its underlying soil column are considered as one-dimensional axially deforming rods, and solution of their governing equations leads to closed-form expressions for pile and soil displacements and stresses. The proposed model provides results of similar accuracy to boundary integral solutions for the special case of floating piles in elastic half-space, and to numerical analyses for the general problem of floating piles embedded in a layer of finite thickness. As such, the model is used to prepare charts for the settlement analysis of floating piles in practice, based on a parametric analysis that unveils the sensitivity of the axial response of floating piles to the different problem parameters.

Consolidation of multilayered soil with fractional derivative viscoelasticity due to surface loading and internal pumping

Classical Terzaghi’s solution provides a useful tool to predict the one-dimensional consolidation of homogeneous elastic soil layer with fully drained/undrained boundary condition under instantaneous loading. Due to the sedimentation process and complex internal structure, natural soils usually exhibit multilayered inhomogeneity and viscoelastic behavior. In practical cases, the drainage capacity of the boundaries can change with time. Thus, these practical factors should be well in considered for accurate prediction of the consolidation. This paper extends the classical Terzaghi’s solution to multilayered viscoelastic soil with continuous drainage boundaries and subjected to time dependent loading. The fractional derivative Kelvin-Voigt model is used to describe the viscoelasticity of soil. Two loading cases namely surface loading and internal pumping are considered. The consolidation problem is solved by Laplace transform technique and a transfer matrix formulation. Analytical solutions for excess pore water pressure, effective stress and settlement are given. The present solution is general and can reduce to existing solutions available in the literatures. Numerical studies are conducted to investigate the effects of viscoelastic parameters, boundary drainage capacity and loading rate on the consolidation behavior. It is shown that the multilayered soil system with large viscosity coefficient and small fractional order can consolidate fast. For the consolidation due to internal pumping, the drainage parameters have no influence on the consolidation process.

Estimation of contact lengths using deep learning neural network

One of the most common problems in engineering is contact problems. In recent years, researchers have turned to alternative methods that can offer effective solutions in a shorter time, instead of solutions containing complex and long mathematical expressions. This study focuses on the estimation of the contact lengths in a homogeneous elastic layer suppressed by two elastic punches with two solution methods. Firstly, a new model was designed for estimation using Deep Learning Neural Network (DNN), one of the deep learning structures. Estimation of contact lengths was provided with the output of the DNN model, which was fed with the homogeneous elastic layer, the ratio of shear modules of the punches and the input parameters of punch radii. The finite element method was used as the second solution method. The problem was modeled in the ANSYS programme and the solution was made with the same parameters used in DNN modeled. The results obtained from both solutions were compared with the solutions obtained by the theory of elasticity and classical NN in the literature. It had been seen that the results obtained with DNN and ANSYS were compatible with the results obtained with analytical and classical NN and the margin of error was smaller.

К исследованию влияния дефекта покрытия на поверхностный волновой процесс

The paper presents an analytical and numerical factorization method for studying the interaction of coating plates in the contact area on a rectilinear fault. Using the method of eigenfunctions and the block element method, we obtained new analytical representations of the solution for the problem, which makes it possible to estimate the behavior of the surface of the medium. We have calculated the wave field on the surface of a homogeneous elastic layer with a pinched bottom face and a composite coating with varying physical and mechanical characteristics of the plates and substrate. We have also illustrated the results of the numerical implementation for the developed algorithm and given a brief discussion about the influence of the fault boundary conditions, as well as the physical and mechanical properties of the inhomogeneous coating and substrate on the wave field formed under the influence of a vertical localized surface source.

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.

The Effect of Mining Remnants on Elastic Strain Energy Arising in the Tremor-Inducing Layer

A vast majority of hard coal deposits in Poland have a multi-seam structure, hence the presence of mining remnants left from previous operations. The impact of those remnants (exploitation edges or residual pillars) can further intensify geomechanical phenomena occurring in the rock mass, leading to changes in the original state of stress. This applies to all layers within the rock strata, including thick and coherent ones (referred to as tremor-inducing layers) where the impacts of mining remnants are likely to trigger tremors, thus enhancing the rock bursts hazard. In the light of the geomechanical model of rock strata recalled in the study, it is assumed that homogeneous and isotropic elastic layers are found between the considered mining remnant (which is revealed as the stress distribution), and the rock medium modelled as a homogeneous and isotropic half-plane. Development of the state of stress in the bedded medium was brought down to the analysis of interacting elastic layers, where the biharmonic equation is satisfied for each layer and for each respective half-plane. This equation can be solved by the integral Fourrier transform method. The impacts of the exploitation edge and the residual pillar on the elastic strain energy in the tremor-inducing layer is illustrated by recalling the Burzyński’s stress criterion. Strain energy in the tremor-inducing layer was analysed for various deformation properties of the surrounding strata and for various methods of coal extraction from the seam underneath the tremor-inducing layer. The results of the study evidence that a change in deformation properties of strata in the vicinity of the tremor-inducing layer may affect the state of stress and strain energy, which impacts on the tremor hazard levels in the vicinity of mining remnants areas.

2D Green’s functions for an elastic layer on a rigid support loaded by an internal point force

The problem of a homogeneous isotropic elastic layer resting on a rigid base and loaded by an internal point force is analytically investigated under plane strain conditions. The displacement field is sought as the superposition of a fundamental solution (doublet state) and a homogeneous solution which allows satisfying the boundary conditions at the upper and lower boundaries of the layer. The displacement field is represented through convergent Fourier integral transforms. Once the closed form solution in the transformed domain is found, the displacement and stress fields are assessed by numerical inversion of transforms. Results concerning both the displacements and stresses for different positions of the load application point are reported and compared to FEM solution, finding very good agreement. The solutions obtained for horizontal and vertical point forces define the Green’s functions for the layer, which can be used to describe the mechanical interaction between the layer and an embedded body. As a striking application, the interaction between an elastic layer and an embedded laterally loaded wall or diaphragm is finally addressed, finding the contact pressure and, in turn, the internal forces in the diaphragm.

Modeling of earthquake source parameters on December 12, 2018 (08:49:56,16; 36,4478° N; 140,5788° E; H = 62,0 km; Mw = 4,3, Japan)

Modeling of earthquake source parameters, such as the orientation of the fault plane and the direction of the fault slip, is important for understanding the physics of earthquake source processes, determining the stress-strain state of the geological medium and seismic hazard estimation. For modeling source parameters of the earthquake on December 12, 2018 at 08:49:56,16 (UTC) in Japan (36,4478° N, 140,5788° E, Northern Ibaraki Pref region) at a depth of 62 km with a magnitude of Mw = 4.3, the waveforms inversion was used to determine seismic moment tensor and representation it through a focal mechanism. The earthquake source is considered as a point source of seismic waves which propagate in a medium represented by a set of horizontally homogeneous elastic layers. An algorithm for determining seismic tensor components based on the forward problem solved by the matrix method, and using the generalized inverse solution, selecting only direct waves is applied. The input data for determining seismic moment components are data of only direct P waves selected from the observed records at six seismic stations of the Japanese local network NIED F-net: TSK, YMZ, ASI, ONS, SBT, KSK. The seismic moment tensor components were determined through waveform inversion using the matrix method. The obtained results, presented through a focal mechanism, are compared to the results obtained by the National Research Institute of Earth Sciences and Resistance to Natural Disasters (NIED CMT solutions). As a result of focal mechanisms comparison, it is concluded that the proposed algorithm for determining seismic moment tensor components can be used if it is impossible to use another method, or requires some refinement for another method. This approach is especially relevant for regions with low seismicity and insufficient number of stations. In addition, this method reduces the effects of an inaccurate medium model, because direct waves are much less distorted than reflected and converted, and that increases the accuracy and reliability of the method.

On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Singular electro-elastic fields at the tip of the crack normal to the interface between a functionally graded piezoelectric material strip and a homogeneous elastic layer induced by an electric load

Investigation of the reversible response of track ballast considering complementary approaches

This paper aims at showing that the reversible behavior of track ballast can be reasonably modeled by a homogeneous elastic layer in the computation of the reversible dynamic response of railway structures subjected to moving train loads. To accomplish this, 2D simulations by means of the discrete element method (DEM) and triaxial tests were carried out. Importantly, these two approaches were performed placing a soft support under the granular specimen. A small number of loading cycles has been considered in both approaches since the long-term settlement of the ballast layer is not the focus of this study, unlike most of the existing research on this topic. In the DEM part, the transfer of the mechanical loads applied to the sample through force chains in the granular medium has been illustrated and discussed based on the numerical responses obtained. In addition, the analysis of the overall response of the grain samples using a homogenized constitutive law shows that the reversible behavior of a ballast layer can be modeled by a continuum approach. However, investigating the effect of the subbase stiffness on the mechanical response of the ballast layer indicates that the homogenized behavior is not intrinsic but depends on the sublayer. These results were confirmed by triaxial tests performed in the laboratory on samples of crushed stones resting on an elastomer mat with different values of the stiffness. Finally, the results from the DEM simulations and the triaxial tests attest that the reversible response of railways can be computed using an elastic layer for the ballast bed provided that an appropriate choice for its apparent modulus is made. The approach that consists of deducing a reversible homogenized behavior of track ballast based on DEM simulations and/or triaxial tests taking into account the support layer looks promising. In future research, more realistic DEM simulations could be envisaged to develop more accurate constitutive laws that could integrate nonlinear and/or anisotropic aspects.

Cracked elastic layer with surface elasticity under antiplane shear loading

A mode-III crack embedded in a homogeneous isotropic elastic layer of nanoscale finite thickness is studied in this article. The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem, where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation. Using the Fourier transform, we convert the problem to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces. By expanding the out-of-plane displacement as series of Chebyshev polynomials, the Galerkin method is invoked to reduce the singular integro-differential equation with Cauchy kernel to a set of algebraic linear equations for the unknown coefficients. An approximate solution is determined, and the influences of surface elasticity on the elastic field and stress intensity factor are examined and displayed graphically. It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus.

К исследованию состояния системы множественных штолен под действием нормальных и сдвиговых нагрузок

The method for studying the boundary value problems modeling multilayer structures with multiple interlayer bulkheads, developed in the works of scientists of the SSC RAS and KubSU, is generalized in this paper to the vector case of a general spatial problem.A complementary approach is presented, allowing analyzing the result of vertical and shearing effectson partitionsof ore formation. As a substrate and a coating layer, we consider homogeneous elastic layers that possess the same physical and mechanical properties but have different thicknesses.The boundary of the coating layer is considered free of stress. Linear static equations for the Kirchhoff plate system are used to describethe state of the ore formation. The block element method used in the work allowed us to reduce the boundary value problem for two elastic layers separated by a plate with a set of parallel linear holes to a system of integral equations of the Fredholm type solvableby the integral factorization method. The described approach can be applied to the study of different-sized block structures on other scales.

To the study of vibration of materials with coatings in the presence of defects

In the paper a new method for studying the properties of deformable bodies with coatings is developed. We considered two models of the foundation: a homogeneous elastic layer with clamped lower edge and a two-layer packet containing a flat rigid inclusion located in the plane of the substrate elastic properties. We used the method of eigenfunctions in the study of the problem concerning steady-state oscillations of a deformable material with a cracked coating. For a foundation that does not contain defects, we described a method for the determination of main parameters characterizing the stress-strain state of a structure with a cracked coating composed of two long plates. We considered a continuous coating for the model of defective foundation. In the course of our study we obtained the relations for determining the characteristics of the stress-strain state of coating/substrate systems as well as presented an example of the application of the method for studying edge effects near the junction boundaries of a composite coating.

Green’s function technique to study the influence of heterogeneity on horizontally polarised shear-wave propagation due to a line source in composite layered structure

The present paper investigates Green’s function technique to study the impact of the non-linear heterogeneity on the propagation of the horizontally polarised shear (SH) wave in a composite layered structure due to the line source. The composite structure is comprised of two distinct isotropic homogeneous elastic layers of finite width lying over an isotropic heterogeneous semi-infinite elastic medium. A general quadratic heterogeneity (including the linear heterogeneity term) in the rigidity of the semi-infinite medium has been taken into account. An efficient analytical treatment involving Green’s function technique along with the application of Fourier transformation has been employed to establish the closed form of the dispersion equation for the propagating wave. As a special case of the problem, the closed form of the dispersion equation has been found in well agreement with the pre-established and classical results. The influence of linear and non-linear heterogeneities on the phase velocity of the horizontally polarised shear-wave has been analysed comparatively for numerous cases associated with the considered model and the corresponding single layer half-space model. It is reported that the phase velocity of the horizontally polarised shear-wave is more pronounced in the case of the corresponding single layer half-space structure as compared to the considered structure. Also, the impact of the general quadratic heterogeneity (supported by the linear heterogeneity) enhances the phase velocity most favourably as compared to the case of the simple quadratic heterogeneity, linear heterogeneity and homogeneity in the semi-infinite medium.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

Stress State of an Inhomogeneous Elastic Layer Made of Heterogeneous Materials with a Coin-Like Rigid Inclusion under Torsion

Problems on cracks and inclusions closely relate to problems of determination of the stress-strain state of homogeneous and inhomogeneous elastic solids that contain stress concentrators. Due to their theoretical and practical importance to the issues of construction structures, machines and their parts in machinery, in the calculation of hydro-technical structures as well as in various other spheres of applied mechanics, these problems have become a subject of investigation for many authors. This paper studies the problem of determination of the stress-strain state of a piecewise homogeneous elastic layer under torsion. The upper and lower surfaces of the compound layer are loaded with tangential forces and the interface of the heterogeneous layers contains a thin absolutely rigid coin-like inclusion. It is required to determine stress jumps on the boundary of the inclusion as well as intensity coefficients of the stresses (ICS).