Cylindrical System Of Vector Functions Research Articles

A novel method for calculating dislocation Green's functions and deformation in a transversely isotropic and layered elastic half-space

A novel and comprehensive method is proposed for calculating the dislocation Love numbers (DLNs), Green's functions (GFs), and the corresponding deformation in a transversely isotropic and layered elastic half-space. It is based on the newly introduced Fourier-Bessel series system of vector functions, along with the dual variable and position method. Two important features associated with this new system are: (1) it is much faster than the conventional cylindrical system of vector functions; (2) we can even pre-calculate the DLNs which are only possible in terms of this new system. This is due to the fact that the variables to be solved in the new system are functions of the simple discrete zero points of the Bessel functions, instead of the numerical integration of continuous Bessel functions between the neighboring zero points as in the conventional system. The introduced dual variable and position method is unconditionally stable as compared to the traditional propagator matrix method in dealing with layering. Exact asymptotic expressions of the DLNs for large wavenumber are further derived, which makes the Kummer's transformation applicable in accelerating the convergence of the corresponding GFs. For the reduced case of homogeneous and isotropic half-space, the present solutions amazingly reduce to the existing exact closed-form solutions. These new features are further seamlessly combined for calculating the deformation due to a finite dislocation (or a finite fault in geophysics) in the layered structure, which are demonstrated to be accurate and efficient.

Coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic and layered poroelastic half‐space

The coupled horizontal and rocking vibrations of a rigid circular disc resting on a transversely isotropic (TI) and layered poroelastic half-space is solved by a newly developed semi-analytical method for the first time. The poroelastic medium is modelled via Biot's poroelastodynamic theory. The Green's function corresponding to the horizontal uniform and vertical linearly changing patch loads on the surface of the layered half-space are first derived by virtue of the cylindrical system of vector functions and dual variable and position method. To solve the dynamic interaction, the influence function by the ring load is then introduced and calculated via the method of superposition. Together with the mixed boundary-value conditions, the dynamic horizontal, rocking, and coupling compliances are finally solved by a novel discretization scheme and an integral least-square approach. The developed fundamental solutions are verified by comparing with existing solutions in the purely elastic and poroelastic media. Numerical examples are presented to investigate the influence of poroelastic properties on the coupled vibration characteristics, which could be useful to the dynamic design of foundation on the saturated soil. It should be emphasized that the effect of the coupling compliance must be considered since its contribution can be as large as 26%, compared to the diagonal compliances.

Fundamental solution of general time-harmonic loading over a transversely isotropic, elastic and layered half-space: An efficient and accurate approach

Time-harmonic loading over layered elastic half-spaces has applications in various science and engineering fields. While various approaches have been proposed in solving the related boundary-value problems, in this paper, we propose a new approach, which is based on the novel Fourier-Bessel series system of vector functions and the dual variable and position method (DVP). While the DVP method was proposed recently and verified to be computationally stable and efficient, the Fourier-Bessel series system of vector functions is newly introduced. Similar to the cylindrical system of vector functions, the normal (dilatational) and shear (torsional) deformations (waves) can be separated and solved in terms of the LM- and N-types of the new vector function system. The new formulation is coded, and the corresponding algorithm/program is applied to a couple of cases. It is shown that, by comparing previous approaches, this new series system of vector functions is equally accurate, but much more computationally powerful. Since it is substantially time saving in calculation, it is hopeful that this new approach would have broad applications related to transient response and inverse problems in elastodynamics of layered systems.

Time-harmonic response of transversely isotropic and layered poroelastic half-spaces under general buried loads

A semi-analytical method is developed for solving the dynamic response of transversely isotropic, multilayered, and poroelastic half-spaces with different surface hydraulic conditions and subjected to time-harmonic vertical and horizontal loads buried in the layered half-space. The coupled governing equations of motion are presented in details in terms of the Biot's poroelastodynamic theory via the (u,p) formulation. The cylindrical system of vector functions is introduced to express the unknown primary quantities so that the coupled governing partial differential equations can be reduced and separated into two sets of first-order ordinary differential equations (i.e., the LM- and N-types). A recursive relation for the expansion coefficients among different layers is established by virtue of the stable and efficient dual variable and position method. Making use of the boundary and interface conditions, the fundamental solutions are obtained in terms of the vector-function system. The corresponding physical-domain solutions are then derived via an accurate semi-infinite integral algorithm. The developed fundamental solutions are carefully checked with existing solutions, and numerical examples are further presented to demonstrate the effect of material anisotropy, loading depth, material layering, and surface hydraulic condition on the dynamic response, which should be useful to design engineers. These solutions could be further served as benchmarks for future numerical methods.

Time-harmonic dislocations in a multilayered transversely isotropic magneto-electro-elastic half-space

Modeling layered systems with dislocations is very challenging; yet, it is important since most smart structures are made of multilayers to make best use of the combined effective property. As such, during the manufactures, defects, such as dislocations, could be introduced in the multilayers. In this article, we analytically find, for the first time, the response of three-dimensional multilayered magneto-electro-elastic systems due to time-harmonic dislocations. The dislocations are the most general, containing the elastic dislocations and discontinuity of the electric potential and/or magnetic potential over a circular region in any layer in the medium. The fully coupled partial differential equations of motion and the Gauss law for the magneto-electro-elastic materials are solved in terms of cylindrical system of vector functions, and the dual variable and position method is further introduced to treat the multilayers. Numerical examples are carried out based on the derived analytical solution to demonstrate the effects of the time-harmonic dislocations on the induced magneto-electro-elastic fields. This analytical solution is important in both electrodynamics and elastodynamics, with possible applications in material sciences and physics. The numerical results are useful in design process of smart devices made of magneto-electro-elastic solids applicable to other engineering fields like renewable energy.

Coupled horizontal and rocking vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces

A new semi-analytical method is developed for solving the coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic multilayered half-space with imperfect interfaces. By virtue of the dual variable and position method, the recursive relation among different layers is established so that the final expansion coefficients of the displacement and traction vectors corresponding to the surface loading can be expressed in terms of the cylindrical system of vector functions. The physical-domain response by the surface loading (i.e., the forward solution) is obtained by carrying out the semi-infinite line integral involved. To solve for the corresponding coupled interaction problem between the surface rigid disc and layered elastic medium, the loading disc area is discretized into annular load-rings containing basic horizontal constant and vertical linear loads with unknown weights. These weights are then determined by applying an integral least-square approach with a Lagrange multiplier. Finally, the coupled time-harmonic compliances are derived by virtue of the equilibrium of the applied load on the disc and the reaction force on the half-space side. Selected numerical examples on the horizontal, moment and coupling compliances are presented to study the effect of material layering, anisotropy, interface behavior as well as input frequency. The coupling compliance is investigated in detail for the first time as compared to the horizontal and moment ones. The fully coupled, semi-coupled and decoupled schemes are also discussed with numerical examples so that one can easily access the relative accuracy among them when an actual layered structure is under both horizontal and rocking vibrations. This is important since the coupling compliance could significantly contribute to the vibrational behavior of the layered system at certain frequency.

Concentrated sources on the interface of diffusive bimaterial media: The steady-state deformation

While the fundamental solutions of concentrated forces and dislocations in purely elastic media are available for various applications, solutions in the corresponding diffusive materials have seldom been derived due to the involved complexity. In this paper, in terms of the cylindrical system of vector functions and under the assumption of steady-state deformation, we derive analytically the elastic and diffusive fields induced by the concentrated forces and dislocations which are located on the interface of a transversely isotropic and diffusive bimaterial space. Solution of the concentrated diffusive source has been also derived. These solutions can further be reduced to the corresponding isotropic bimaterial case and the transversely isotropic full-space case. Based on the newly derived solutions, field quantities induced by different concentrated dislocation sources are presented numerically to demonstrate the effect of the diffusive coefficients, material heterogeneity, and dislocation types on the induced fields.

Vertical and torsional vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces

Rigid disc vibration over a layered half-space is essential to understand soil-structure interaction (SSI) between a foundation and the structure over it. In this article, we present a novel semi-analytical and very efficient method to calculate the SSI coefficients on the layered foundation due to both vertical and torsional vibrations. The material properties in each layer are transversely isotropic to take care of the possible different Young's moduli in different directions (vertical vs. horizontal). Furthermore, the interface between the layers is assumed to be general so that possible loose bonding between the layers (where the displacements and/or tractions are discontinuous across the interface) can be considered. The semi-analytical solution is based on the recently developed forward solution of the layered structure with imperfect interface under time-harmonic loadings within the circle on the surface. Since the present SSI problem is a mixed boundary-value problem, the method of superposition in terms of the powerful cylindrical system of vector functions combining with the integral least-square approach is proposed. To take care of multiple layers, the dual variable and position (DVP) is further introduced. The cylindrical system of vector functions has the advantage of separating the torsional vibration from the vertical vibration and yet expressing the two types of solutions uniformly. After validating the proposed semi-analytical method, numerical examples are presented to demonstrate the effect of material layering, interface imperfection, elastic anisotropy, and input frequency on the SSI coefficients, and on the surface displacement and stress variation. These new results should be also good benchmarks for future numerical methods.

Wave propagation in a multilayered magneto-electro-elastic half-space induced by external/internal circular time-harmonic mechanical loading

In this paper, a linearly magneto-electro-elastic (MEE) multilayered half-space which is dynamically loaded over a finite patch located either on the surface or internally within any layer is addressed. The completely coupled system of equations of motion for MEE is applied. The unknown primary field quantities consisting of displacements, electrical potential and magnetic potential are expressed in a cylindrical system of vector functions, where each of the vector functions is written in terms of the product of Bessel functions and Fourier series. In so doing, the components of displacement vector and traction tensor in the physical domain are expanded in terms of a set of expansion coefficients in the Hankel–Fourier transformed domain. In each field point the calculated expansion coefficients in the transformed domain are utilized to calculate the unknowns in the physical domain by integration. Different sets of numerical analyses are presented to investigate the effects of material anisotropy, material coupling, loading frequency and stacking sequences of layers on the responses due to internal/external loadings.

Static response of a layered magneto-electro-elastic half-space structure under circular surface loading

A cylindrical system of vector functions, the stiffness matrix method and the corresponding recursive algorithm are proposed to investigate the static response of transversely isotropic, layered magneto-electro-elastic (MEE) structures over a homogeneous half-space substrate subjected to circular surface loading. In terms of the system of vector functions, we expand the extended displacements and stresses, and deduce two sets of ordinary differential equations, which are related to the expansion coefficients. The solution to one of the two sets of these ordinary differential equations can be evaluated by using the stiffness matrix method and the corresponding recursive algorithm. These expansion coefficients are then integrated by adaptive Gaussian quadrature to obtain the displacements and stresses in the physical domain. Two types of surface loads, mechanical pressure and electric loading, are considered in the numerical examples. The calculated results show that the proposed technique is stable and effective in analyzing the layered half-space MEE structures under surface loading.

Effect of anisotropic base/interlayer on the mechanistic responses of layered pavements

Material anisotropy and thin interlayer are very common in layered pavement structures. Despite many numerical approaches in pavement analysis, very few theoretical methods are available in dealing with these two important issues. Thus, this study is focused on the effect of anisotropy and thin interlayer on the mechanistic responses of layered pavements. This is done using our analytical solutions based on the propagator matrix method in terms of the cylindrical system of vector functions. Our numerical results show that both anisotropy and thin interlayer could substantially contribute to the pavement responses.

Inverse calculation of elastic moduli in cross-anisotropic and layered pavements by system identification method

Material properties of cross-anisotropic (or transversely isotropic) elastic and layered systems including pavement structures are essential for the analysis of mechanical responses. Besides laboratory determination of these material properties, direct inversion using in situ input data is fundamental and more useful. In this paper, the system identification (SID) method with constraints is proposed to invert the elastic moduli in an anisotropic layered half space in general and in a layered pavement in particular. Since in the inverse calculation, the forward calculation is required, we have also presented briefly the forward calculation approach based on the cylindrical system of vector functions and the propagating matrix method. Our SID algorithm is then applied to three-layer and four-layer pavements with different numbers of cross-anisotropic layers, with the deflections at the surface of the layered pavement as inputs. Our numerical results demonstrate clearly that the proposed SID-based inverse method is accurate and efficient for a broad range of seed moduli.

Circular surface loading on a layered multiferroic half space

By introducing the cylindrical system of vector functions and the correspondingpropagating matrix, we present a semi-analytical solution for a layered multiferroic halfspace under a uniform vertical circular load on its surface. A two-layered system made ofBaTiO3 andCoFe2O4 is analyzed by the proposed method. The coupling feature among the elastic, electric, andmagnetic fields and the interplay between the adjacent layers are investigated. Inparticular, we find that the interfacial elastic, electric, and magnetic fields are very sensitiveto the thickness of the surface layer. Consequently, a critical thickness is found for eachfield quantity when it reaches its extreme value for varying thickness of the surface layer.This striking feature could be very useful as a theoretical reference for the optimal designof surface coatings.

Effect of temperature variation on pavement responses using 3D multilayered elastic analysis

The response of flexible pavement is largely influenced by the resilient modulus of the pavement profile. Different methods/approaches have been adopted in order to estimate or measure the resilient modulus of each layer assuming an average modulus within the layer. To account for the variation in the modulus of elasticity with depth within a layer, the layer should be divided into several sublayers and the modulus should be gradually varied between the layers. A powerful and innovative program has been developed utilizing the unique propagator matrix method and the cylindrical system of vector functions. Our new program can predict accurately and efficiently the response of flexible pavements of any number of layers/sublayers. Numerical results in this paper showed that, instead of assuming one response due to an average modulus, modulus variation with depth should be considered in any pavement analysis since it can capture the response envelopes of the pavement.

Efficient and accurate multi-layered elastostatic Green's functions via the bi-material Green's function

The three-dimensional elastostatic Green's functions for multi-layered half-spaces are expressed in terms of a cylindrical system of vector functions using the propagator matrix method. To obtain efficient and accurate results near the singularity at the interface between layers, the singularity extraction method is employed in the transform domain, based on the singular solutions for the bi-material full space expressed here in vector cylindrical function form. Numerical trials demonstrate that this approach improves on previous work, which relied on singularity extraction based on the well-known singular (Kelvin) solution for the homogenous full-space. A boundary element implementation illustrates the practical potential of this work to boundary value problems.

Responses of Piezoelectric, Transversely Isotropic, Functionally Graded, and Multilayered Half Spaces to Uniform Circular Surface Loadings

Abstract: In this paper, an analytical solution is presented to study the response of piezoelectric, transversely isotropic, functionally graded, and multilayered half spaces to uniform circular surface loadings (pressure or negative electric charge). The inhomogeneous material is exponentially graded in the vertical direction and can have multiple discrete layers. The propagator matrix method and cylindrical system of vector functions are used to first derive the solution in the transformed domain. In order to find the responses in the physicaldomain, which are expressed in one-dimensional infinite integrals of the Bessel function products, we introduced and adopted an adaptive Gauss quadrature. Two piezoelectric functionally graded half-space models are analyzed numerically: One is a functionally graded PZT4 half space, and the other a multilayered functionally graded half space with two different piezoelectric materials (PZT-4 and PZT-6B). The effect of different exponential factors of the functionally graded material on the field responses is clearly demonstrated. The difference of the responses between the two surface loading cases is also discussed via the numerical examples. The results should be particularly useful in the characterization of material properties using indentation tests, and could indirectly contribute to the design and manufacturing of piezoelectric functionally graded structures.

Static Green's functions in multilayered half spaces

This paper presents an efficient and accurate method for the calculation of static Green's functions in a multilayered transversely isotropic or isotropic half space. The cylindrical system of vector functions and the propagator matrix method are used to derive the Green's functions in the transformed domain. The well-known exponentially growing elements in the propagator matrix are fractionated out by propagating the matrix either upwards or downwards, depending upon the relative vertical location of the source and field points. The Green's functions in the physical domain are evaluated numerically by an adaptive Gauss quadrature with continued fraction expansions. Numerical examples are presented to show that very accurate Green's functions with relatively less Gauss quadrature points can be obtained. These examples also show clearly the effect of material layering and anisotropy on the displacement and stress fields.

Static response of a transversely isotropic and layered half-space to general dislocation sources

The static deformation problem of a transversely isotropic and layered half-space by general dislocation sources is solved. The solution is given in the Cartesian system and the cylindrical system of vector functions using multiplications of matrices. The source functions of general dislocations in a transversely isotropic medium are obtained in these two systems using a new approach. Explicit expressions for the surface displacements due to three- and two-dimensional dislocation sources are also obtained in terms of the cylindrical system and the Cartesian system of vector functions, respectively. It is shown that the present solution contains the solutions to the three- and two-dimensional source problems for the corresponding isotropic media, and therefore provides a unified solution for these two problems, which have long been studied separately. The formulation developed can be evaluated numerically to study the effects of anisotropy, as well as of Earth layering, on the static fields.