Multilayered Elastic Media Research Articles

A Boundary-Element Analysis of Crack Problems in Multilayered Elastic Media: A Review

Crack problems in multilayered elastic media have attracted extensive attention for years due to their wide applications in both a theoretical analysis and practical industry. The boundary element method (BEM) is widely chosen among various numerical methods to solve the crack problems. Compared to other numerical methods, such as the phase field method (PFM) or the finite element method (FEM), the BEM ensures satisfying accuracy, broad applicability, and satisfactory efficiency. Therefore, this paper reviews the state-of-the-art progress in a boundary-element analysis of the crack problems in multilayered elastic media by concentrating on implementations of the two branches of the BEM: the displacement discontinuity method (DDM) and the direct method (DM). The review shows limitation of the DDM in applicability at first and subsequently reveals the inapplicability of the conventional DM for the crack problems. After that, the review outlines a pre-treatment that makes the DM applicable for the crack problems and presents a DM-based method that solves the crack problems more efficiently than the conventional DM but still more slowly than the DDM. Then, the review highlights a method that combines the DDM and the DM so that it shares both the efficiency of the DDM and broad applicability of the DM after the pre-treatment, making it a promising candidate for an analysis of the crack problems. In addition, the paper presents numerical examples to demonstrate an even faster approximation with the combined method for a thin layer, which is one of the challenges for hydraulic-fracturing simulation. Finally, the review concludes with a comprehensive summary and an outlook for future study.

Analysis of Crack Problems in Multilayered Elastic Medium by a Consecutive Stiffness Method

We propose a boundary-element-based method for crack problems in multilayered elastic medium that consists of a set of individually homogeneous strata. The method divides the medium along the slit-like crack surface so that the effects of the elements placed along one crack surface become distinguishable from those placed along the other surface. As a result, the direct method that cannot be directly applied for crack problems turns out to be applicable. After that, we derive a recursive formula that obtains a “stiffness matrix” for each layer by exploiting the chain-like structure of the system, enabling a sequential computation to solve the displacements on the crack surface in each layer “consecutively” in a descending order from the very top layer to the very bottom one. In our method, the final system of equations only contains the unknown displacements on the crack surface, ensuring the efficiency of the method. The numerical examples demonstrate better accuracy and broader applicability of our method compared to the displacement discontinuity method and more-acceptable efficiency of our method compared to the conventional direct method.

Complete solution for the axisymmetric problem of a penny-shaped crack near and parallel to an arbitrarily graded interface in FGMs

Previous analytical treatments to the crack problems in functionally graded materials (FGMs) were mostly based on the assumption that only the shear modulus is variable according to special functions while the Poisson’s ratio is constant. However, the material inhomogeneity in real FGMs is more complex than that can be well approximated using the assumption. This paper examines the axisymmetric problem of a penny-shaped crack near and parallel to a graded interface with arbitrarily variable shear modulus and Poisson’ s ratio. To consider the general material inhomogeneity, a new efficient solution of general axisymmetric edge dislocation loop in multilayered elastic medium is derived first using the General Kelvin’s Solution (GKS) based method. This method allows the approximation of general material inhomogeneity using large number of homogenous sublayers without the loss of computational efficiency, accuracy and stability. The challenging issue in multilayered elasticity associated with the calculation of near-source elastic field is also tackled, which enables the high precision evaluation of crack tip field. Exact solutions are given for the full elastic field. Numerical studies are conducted to explore the fracture mechanics of FGMs. The results show that the gradation pattern can have certain effect on the fracture response if the crack is very closed to the graded interface.

Rayleigh-like waves in multilayered elastic media containing voids: Use of the Haskell matrix method

Using the Haskell matrix method, the dispersion relation of Rayleigh-like surface waves propagating through a multilayered elastic solid half-space is derived. Each layer as well as the half-space is assumed to have voids (pores) distributed evenly throughout. This dispersion relation is then reduced for a 2-layered model (single layer over a half-space) to study the characteristics of phase speed of Rayleigh-like wave. For a particular model, the numerical computations are performed to observe the effect of voids on the fundamental mode of Rayleigh-like waves for n = 2 and 3. For the 2-layered model, it is also shown that the particle motion remains elliptical but influenced by the presence of voids. In the absence of voids from the model, the dispersion relation earlier obtained by Haskell (1953) for the case n = 2 is recovered successfully.

Stress and displacement fields around an arbitrary shape tunnel surrounded by a multilayered elastic medium subjected to harmonic waves under plane strain conditions

This study focuses on the propagation of harmonic mechanical waves around both circular and non-circular tunnels. The excavation disturbed zone is explicitly modeled with a multilayered description of the surroundings of the tunnel. The purpose of this paper is thus to establish closed-form solutions for stress and displacement fields for circular and non-circular geometries taking into account the mechanical heterogeneity of the area adjacent to the tunnel. Particularly, the multilayered non-circular shape configuration is a great step not treated previously. The classical approach based on the wave function expansion method is used to determine the expressions of stresses and displacements around circular geometries. To handle non-circular tunnel shapes, the complex variable method based on the Muskhelishvili approach and conformal mapping functions is introduced. Results indicate that the excavation damaged zone strongly modifies the distribution of stress and displacement fields around the tunnel. Geometrical tunnel shapes also affect stress and displacement polar diagrams.

Mechanics of multilayered or functionally graded material pressed onto an imperfect rigid base with a circular opening by localized surface loading

Loaded materials constrained by imperfect boundary can be found in many engineering scenarios, including blanking process, contact pressure sensor and soil/rock foundation subjected to underground mining. This paper examines the surface loading problem of a multilayered or functionally graded material (FGM) resting on an imperfect rigid smooth base weakened by a circular opening. The surface loading is axisymmetric and coaxial with the circular opening. This mixed boundary value problem is analogous to a Mode-I penny-shaped crack in multilayered elastic medium. With the aid of the GKS-based method for N-layered elasticity, the problem is formulated as a system of dual integral equations and then reduced to a Fredholm integral equation of the second kind. Solutions of the SIF, COD and full elastic field are expressed in terms of the solution of the integral equation. Numerical results obtained from the present solutions are compared with the those from finite element method. Using the new solutions, numerical studies are also conducted for four examples, including a homogenous layer, a multilayered material, a material with FGM coating and an asphalt pavement structure with FGM properties. These numerical results demonstrate that (1) a small size of surface loading can result in high SIF value at the crack tip; (2) placing the hard layer as the top surface is most effective for reducing ground settlement; (3) the gradation in Poisson’s ratio can have non-negligible effect on the SIF for thin FGM coating; (4) effect of the circular opening to the ground settlement can be significant if its radius is comparable to the total thickness of the upper layers.

Extension of the kinematic approximations to the multilayered elastic orthorhombic medium

Understanding the kinematics of horizontally layered reservoir rocks is important to their proper characterization; to accomplish this, it is necessary to specify the explicit model for these kinematic properties. Accurate approximations for traveltime and relative geometric spreading in an elastic homogeneous orthorhombic (ORT) model have been investigated with different forms: shifted hyperbola form (SHF), Taylor series, and the rational form. We have extended these approximations to the multilayered ORT model by adopting composite coefficients and effective model parameters. The multilayered model is characterized without and with the azimuthal variation among layers. There is an overdetermined problem when the azimuthal variation exists; to address that case, the least-squares method is adopted. To check the feasibility of the expansion, we select the SHF approximation specified in the homogeneous elastic ORT model for the calculation in the numerical example. Four groups of examples are analyzed to investigate the influence on the accuracy of the approximation with the change in rotation angle, degree of anisotropy, and the direction of the orientation. The results indicate that, for the multilayer, the accuracy of the approximation is proportional to the degree of anisotropy and the value of the angle of rotation. The relative errors in traveltime and relative geometric spreading in this multilayered extension are very small and can be implemented in practical applications.

A unified mathematical treatment of interfacial edge dislocations in three-dimensional functionally graded materials

Mechanical behavior of functionally graded materials (FGMs) with the presence of defects, such as dislocations and cracks, is of great scientific interest and practical importance. Existing methods for dislocation and crack problems are typically for homogenous materials or inhomogeneous materials with special kinds of variations in material properties, for instance variable shear modulus according to power-law or exponential functions and constant Poisson's ratio. Applicability of these methods to the practical problems with general inhomogeneity is questionable because variations of the real material properties are arbitrary and cannot be well approximated by special functions. To fill this gap, we present here a unified mathematical treatment of interfacial edge dislocation problems in three-dimensional FGMs. The basic idea is to use a multilayered system consisting of large number of dissimilar sublayers to approximate the arbitrary inhomogeneity in the FGMs. A novel solution approach is developed for calculating elastic field in the multilayered elastic medium due to interfacial edge dislocations of arbitrary shape and distribution profile. Comparing to the existing methods for multilayered elasticity, the present treatment is featured with high computational efficiency, accuracy, stability regardless the total number of sublayers. On the other hand, it is also mathematically consistent in that existing closed-form solutions for edge dislocations at bi-materials interface and homogeneous full space can be fully recovered. Moreover, it can accurately and clearly identify and isolate the singularities in the solutions, which enables the high-precision calculation of elastic field at the vicinity of the source plane. Numerical studies are also conducted on three examples associated with interfacial edge dislocation in multilayered coating and internal edge dislocation in bonded material with graded interfacial zone to show the applicability of the present treatment. It is shown that the present treatment can well address the dislocation problems in FGMs with arbitrary material inhomogeneity.

Multilayered Elastic Medium under Axisymmetric Loading and Surface Energy

This paper presents a solution scheme for analysis of a multilayered elastic medium under axisymmetric loading and surface energy effects by adopting Gurtin-Murdoch surface elasticity theory. Love’s strain function and Hankel integral transform are employed to derive the general solutions, and the obtained solutions are employed in the determination of the stiffness matrix for each layer. The global stiffness equation of a multi-layered system is assembled by considering the continuity of traction and displacements at each layer interface. The numerical solutions to the global equation yield displacements and stresses at the interfaces of the layered medium under axisymmetric loading. The accuracy of the proposed solution scheme is verified by comparing with existing solutions. Selected numerical results are presented to demonstrate a significant influence of surface energy on elastic fields of a multilayered elastic medium under axisymmetric loading.

Analytical D’Alembert Series Solution for Multi-Layered One-Dimensional Elastic Wave Propagation with the Use of General Dirichlet Series

A general initial-boundary value problem of one-dimensional transient wave propagation in a multi-layered elastic medium due to arbitrary boundary or interface excitations (either prescribed tractions or displacements) is considered. Laplace transformation technique is utilised and the Laplace transform inversion is facilitated via an unconventional method, where the expansion of complex-valued functions in the Laplace domain in the form of general Dirichlet series is used. The final solutions are presented in the form of finite series involving forward and backward travelling wave functions of the d’Alembert type for a finite time interval. This elegant method of Laplace transform inversion used for the special class of problems at hand eliminates the need for finding singularities of the complex-valued functions in the Laplace domain and it does not need utilising the tedious calculations of the more conventional methods which use complex integration on the Bromwich contour and the techniques of residue calculus. Justification for the solutions is then considered. Some illustrations of the exact solutions as time-histories of stress or displacement of different points in the medium due to excitations of arbitrary form or of impulsive nature are presented to further investigate and interpret the mathematical solutions. It is shown via illustrations that the one-dimensional wave motions in multi-layered elastic media are generally of complicated forms and are affected significantly by the changes in the geometrical and mechanical properties of the layers as well as the nature of the excitation functions. The method presented here can readily be extended for three-dimensional problems. It is also particularly useful in seismology and earthquake engineering since the exact time-histories of response in a multi-layered medium due to arbitrary excitations can be obtained as finite sums.

Reflecting and scattering of acoustic wave from sea ices

In order to build an efficient underwater acoustic sensor network in the Arctic Ocean environment, transmission characteristics of under-ice acoustic channels need comprehensive understanding. The reflecting and scattering of acoustic waves from sea ices have great influences on under-ice acoustic channels. Both topology and structure of sea surface ices are very complex and variable. The physical dimension, acoustic property and interface roughness of sea ices depend not only on local environment, but also on climate and formation time. Therefore, it is of great significance to develop a model of reflecting and scattering of acoustic waves from sea ices for investigating the sound propagation in the under-ice environment. Assuming that sea ices are a multi-layered elastic solid medium and the ice-water interface is rough and satisfies the boundary condition of perturbation, we develop a system of linear equations to solve the coherent reflection coefficient of the incident sound wave from water to sea ice. The coherent reflection coefficient is a function of the frequency of sound wave and incident grazing angle, and is numerically evaluated. The influences of ice thickness and ice-water interface roughness on the coherent reflection coefficient are analyzed. Furthermore, the method of calculating scattering coefficient by using the power spectrum density of the scattering field is introduced. The scattering coefficient as a function of the scattering grazing angle is numerically evaluated. The influences of ice thickness and ice-water interface roughness on scattering coefficient are analyzed. The results show that both the coherent reflection coefficient and the scattering coefficient are dependent on the frequency of acoustic wave, ice thickness and grazing angle. The coherent reflection coefficient is close to 1.0 and the scattering coefficient is less than 0.01 when incident grazing angle is less than 15°. In addition, the frequency of acoustic wave and ice thickness have weak influences on them. However, the frequency of acoustic wave and ice thickness have significant influences on the coherent reflection coefficient and the scattering coefficient when the incident grazing angle is big, say, greater than 30°. In general, the thicker the ice is, the smaller the coherent reflection coefficient and the scattering coefficient are. The coherent reflection coefficient is less than 0.18 when the ice thickness is greater than 10.0 m and the frequency of acoustic waves is greater than 2 kHz. The ice-water interface roughness has great influences on both the coherent reflection coefficient and the scattering coefficient. The rougher of the ice-water interface is, the smaller the coherent reflection coefficient is, and the bigger the scattering coefficient is.

Axisymmetric coupled thermo-mechanical response of multilayered elastic medium

Most thermal stress analyses are based on the uncoupled theory of thermoelasticity, which considers that the determination of the temperature field is independent from the stress and displacement fields. This paper derives an extended precise integration solution for the coupled thermoelastic problems with the integral transformation techniques. From the axisymmetric fundamental governing equations, an ordinary differential matrix equation is achieved by means of a Laplace–Hankel transform. An extended precise integration method is introduced to solve the above matrix equation in the transformed domain, and the actual solution is acquired by a numerical inverse transformation. Finally, some numerical examples are given to demonstrate the accuracy of the proposed theory and to illustrate the influence of the coupling of thermoelasticity and layered characteristics on the thermo-mechanical response.

Elastodynamic Inversion of Multilayered Media via Surface Waves—Part I: Methodologies

A rigorous theoretical foundation for solving elastodynamic inverse problem of multilayered media under an impulse load is established in this paper. The inversion is built upon the forward dynamic analysis of multilayered elastic media using transfer matrix approach, with which displacement continuity is assumed at the interfaces of upper and lower adjacent layers. Formulations for inverse analysis are derived in both the time domain and the complex frequency domain. Least square estimates and nonlinear optimization algorithms are used to implement parameter identification. The proposed theory and formulae can be utilized to develop a computer software for nondestructive evaluation of laminated civil and aerospace structure (highway and airport pavements, bridge decks, soil foundations, aircraft wing, etc.), exploration and dynamic source detection and identification, and petroleum exploration in geophysics.

Analysis of Stress and Displacement Field in Flexible Layered Pavement Based on Boundary Element Method

AbstractIn order to study the stress and the displacement of the flexible multi-layered elastic pavement under various loads distributed in arbitrary shape, the boundary element method for the flexible layered pavement was presented using the fundamental solutions multi-layered solids for the analysis. With combination of the generalized Kelvin fundamental solution and the boundary element method, the boundary integral equation was established for the boundary value problem of the flexible layered pavement to explain the singular integral appearing in the kernel function. The method, which does not involve integral of layered material’s interface, can be used to solve the stress and the displacement of the multi-layered elastic media under various loads distributed in arbitrary shape. The analytical result shows that the present method is of higher precision and can be used for analysis of the flexible pavement which mechanical parameters change along the depth because of using the generalized Kelvin fund...

A continuum-based model for analysis of laterally loaded piles in layered soils

An analysis is developed to calculate the response of laterally loaded piles in multilayered elastic media. The displacement fields in the analysis are taken to be the products of independent functions that vary in the vertical, radial and circumferential directions. The governing differential equations for the pile deflections in different soil layers are obtained using the principle of minimum potential energy. Solutions for pile deflection are obtained analytically, whereas those for soil displacements are obtained using the one-dimensional finite difference method. The input parameters needed for the analysis are the pile geometry, the soil profile, and the elastic constants of the soil and pile. The method produces results with accuracy comparable with that of a three-dimensional finite element analysis but requires much less computation time. The analysis can be extended to account for soil non-linearity.

Elastic analysis of axially loaded single pile in multilayered soils

Based on the vertical point solution of multilayered elastic medium, this paper presents a method for the elastic analysis of axially loaded single pile in layered soils. The method is easily incorporated into a computational scheme for accurate numerical analysis and prediction of the pile behavior under loading. The results of the method are compared well with the existing solutions in the literature. A parametric study for a six-layered elastic soil is further presented to show the effects of layered soils on the behavior of a single pile.

Boundary element analysis of axially loaded piles embedded in a multi-layered soil

In a field, piles are likely installed in a multi-layered soil. Analysis of axially loaded piles in a multi-layered soil is complicated and deserves more attention. A boundary element method is used in this study to analyze an axially loaded single pile in a multi-layered soil using the solution for vertical and horizontal axisymmetric ring loads in a multi-layered elastic medium. Good and reasonable agreement is obtained between the proposed and published solutions for a single pile in a homogenous soil, a finite soil, and a Gibson soil. The proposed solution is also used to evaluate an axially loaded single pile in a multi-layered (8 layers) soil.

A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems

We present a novel multigrid (MG) procedure for the efficient solution of the large non-symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi-layered elastic medium. The governing equations involve a highly non-linear coupled system of integro-partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N3). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub-sets of the finest grid are used to treat mixed variable problems associated with ‘pinch points.’ Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss–Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.

A Study on Zigzag Dispersion Curves in Rayleigh Wave Exploration

AbstractThe guided waves in multilayered elastic media are studied. The mechanism of zigzag dispersion curves in the Rayleigh Wave Exploration is analyzed. It is proved that the zigzag dispersion curve can not be obtained by a single guided mode. The relation of the excitation intensity of each mode to the formation parameters is studied. We obtained many results about zigzag dispersion curves that were not available in previous literatures. The effects of the position and thickness of the low‐velocity layer and other parameters on the zigzag dispersion curves of Rayleigh wave are also analyzed. Finally, comparative analyses between the practical data and theoretical results are carried out. Theoretical results are consistent to the practical data.