Dynamic Green’s Functions for an Anisotropic Multilayered Poroelastic Half-Space

The dynamic responses of an anisotropic multilayered poroelastic half-space to a point load or a fluid source are studied based on Stroh formalism and Fourier transforms. Taking the boundary conditions and the continuity of the materials into consideration, the three-dimensional Green’s functions of generalized concentrated forces (force and fluid source) applied at the free surface, interface and in the interior of a layer are derived in the Fourier transformed domain, respectively. The actual solutions in the frequency domain can further be acquired by inverting the Fourier transform. Finally, numerical examples are carried out to verify the presented theory and discuss the Green’s fields due to three cases of a concentrated force or a fluid source applied at three different locations for an anisotropic multilayered poroelastic half-space.

     

Image singularities of Green’s functions for isotropic elastic bimaterials subjected to concentrated forces and dislocations

Green’s functions for isotropic materials in the two-dimensional problem for elastic bimaterials with perfectly bonded interface are reexamined in the present study. Although the Green’s function for an isotropic elastic bimaterial subjected to a line force or a line dislocation has been discussed by many authors, the physical meaning and the structure of the solution are not clear. In this investigation, the Green’s function for an elastic bimaterial is shown to consist of eight Green’s functions for a homogeneous infinite plane. One of the novel features is that Green’s functions for bimaterials can be expressed directly by knowing Green’s functions for the infinite plane. If the applied load is located in material 1, the solution for the half-plane of material 1 is constructed with the help of five Green’s functions corresponding to the infinite plane. However, the solution for the half-plane of material 2 only consists of three Green’s functions for the infinite plane. One of the five Green’s functions of material 1 and all the three Green’s functions of material 2 have their singularities located in the half-plane where the load is applied, and the other four image singularities of material 1 are located outside the half-plane at the same distance from the interface as that of the applied load. The nature and magnitude of the image singularities for both materials are presented explicitly from the principle of superposition, and classified according to different loads. It is known that for the problem of anisotropic bimaterials subjected to concentrated forces and dislocations, the image singularities are simply concentrated forces and dislocations with the stress singularity of order O(1/r). However, higher orders (O(1/r²) and O(1/r³)) of stress singularities are found to exist in this study for isotropic bimaterials. The highest order of the stress singularity is O(1/r³) for the image singularities of material 1, and is O(1/r²) for material 2. Using the present solution, Green’s functions associated with the problems of elastic half-plane with free and rigidly fixed boundaries, for homogeneous isotropic elastic solid, are obtained as special cases.     

Three-dimensional Green’s functions for composite laminates

The three-dimensional Green’s functions due to a point force in composite laminates are solved by using generalized Stroh formalism and two-dimensional Fourier transforms. Each layer of the composite is generally anisotropic and linearly elastic. The interfaces between different layers are parallel to the top and bottom surfaces of the composite and are perfectly bonded. The Green’s functions of point forces applied at the free surface, interface, and in the interior of a layer are derived in the Fourier transformed domain respectively. The surfaces are imposed by a proportional spring-type boundary condition. The spring-type condition may be reduced to traction-free, displacement-fixed, and mirror-symmetric conditions. Numerical examples are given to demonstrate the validity and elegance of the present formulation of three-dimensional point-force Green’s functions for composite laminates.     

半无限弹性空间域内点加振格林函数的计算

本文给出了满足全部自由面边界条件的半无限弹性空间域内点加振的Ｇｒｅｅｎ函数，利用变形的Ｈａｎｋｅｌ函数，在复数域内进行无限积分的有限化，从而使Ｇｒｅｅｎ函数的计算变得比较简单和方便。     

Green’s functions for plane problem under various boundary conditions and applications

Green’s functions of a point dislocation as well as a concentrated force for the plane problem of an infinite plane containing an arbitrarily shaped hole under stress, displacement, and mixed boundary conditions are stated. The Green’s functions are obtained in closed forms by using the complex stress function method along with the rational mapping function technique, which makes it possible to deal with relatively arbitrary configurations. The stress functions for these problems consist of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. These Green’s functions can be derived without carrying out any integration. The applications of the Green’s functions are demonstrated in studying the interaction of debonding and cracking from an inclusion with a line crack in an infinite plane subjected to remote uniform tension. The Green’s functions should have many other potential applications such as in boundary element method analysis. The boundary integral equations can be simplified by using the Green’s functions as the kernels.     

Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces

By virtue of two systems of vector functions and the propagator matrix method, Green’s functions for transversely isotropic, piezoelectric functionally graded (exponentially graded in the vertical direction), and multilayered half spaces are derived. It is observed that the homogeneous solution and propagator matrices for each functionally graded layer in the transformed domain are independent of the choice of the two systems of vector functions. For a point force and point charge density applied at any location of the functionally graded half space, Green’s functions are expressed in terms of one-dimensional infinite integrals. To carry out the numerical integral involving Bessel functions, an adaptive Gauss quadrature approach is introduced and modified. The piezoelectric functionally graded Green’s functions include those in the corresponding elastic functionally graded media as special results with the latter being also unavailable in the literature. Two piezoelectric functionally graded half-space models are analyzed numerically: one is a functionally graded PZT-4 half space, and the other a coated functionally graded PZT-4 layer over a homogeneous BaTiO₃ half space. The effects of different exponential factors on Green’s function components are clearly demonstrated, which could be useful in the design and manufacturing of piezoelectric functionally graded structures.     

Anti-plane shear Green’s functions for an isotropic elastic half-space with a material surface

This paper presents analytical Green’s function solutions for an isotropic elastic half-space subject to anti-plane shear deformation. The boundary of the half-space is modeled as a material surface, for which the Gurtin–Murdoch theory for surface elasticity is employed. By using Fourier cosine transform, analytical solutions for a point force applied both in the interior or on the boundary of the half-space are derived in terms of two particular integrals. Through simple numerical examples, it is shown that the surface elasticity has an important influence on the elastic field in the half-space. The present Green’s functions can be used in boundary element method analysis of more complicated problems.     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

On generalized Kelvin solutions in a multilayered elastic medium

This paper presents fundamental singular solutions for the generalized Kelvin problems of a multilayered elastic medium of infinite extent subjected to concentrated body force vectors. Classical integral transforms and a backward transfer matrix method are utilized in the analytical formulation of solutions in both Cartesian and cylindrical coordinates. The solution in the transform domain has no functions of exponential growth and is invariant with respect to the applied forces. The convergence of the solutions in the physical domain is rigorously and analytically verified. The solutions satisfy all required constraints including the basic equations and the interfacial conditions as well as the boundary conditions. In particular, singular terms of the generalized Kelvin solutions associated with the point and ring types of concentrated body force vectors are obtained in exact closed-forms via an asymptotic analysis. Numerical results presented in the paper illustrate that numerical evaluation of the solutions can be easily achieved with very high accuracy and efficiency and that the layering material inhomogeneity has a significant effect on the elastic field.The Canadian Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Superposing scheme for the three-dimensional Green’s functions of an anisotropic half-space

This paper presents a method of superposition for the half-space Green’s functions of a generally anisotropic material subjected to an interior point loading. The mathematical concept is based on the addition of a complementary term to the Green’s function in an anisotropic infinite domain. With the two-dimensional Fourier transformation, the complementary term is derived by solving the generalized Stroh eigenrelation and satisfying the boundary conditions on the free surface with the use of Green’s functions in the full-space case. The inverse Fourier transform leads to the contour integrals, which can be evaluated with the application of Cauchy residue theorem. Application of the present results is made to obtain analytical expression for the orthotropic materials which were not reported previously. The closed-form solutions for the transversely isotropic and isotropic materials derived directly from the solutions as being a special case are also given in this paper.     

Boundary element elastic stress analysis of 3D generally anisotropic solids using fundamental solutions based on Fourier series

The authors have very recently proposed an efficient, accurate alternative scheme to numerically evaluate etc. Green’s function, U(x), and its derivatives for three-dimensional, general anisotropic elasticity. These quantities are necessary items in the formulation of the boundary element method (BEM). The scheme is based on the double Fourier series representation of the explicit, exact, algebraic solution derived by Ting and Lee (1997) [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostic Green’s function for general anisotropic linear elastic solid. Q. J. Mech. Appl. Math. 50, 407–426] expressed in terms of Stroh’s eigenvalues. By taking advantage of some its characteristics, the formulations in this double Fourier series approach are revised and several of the analytical expressions are re-arranged in the present study. This results in significantly fewer terms to be summed in the series thereby improving further the efficiency for evaluating the Green’s function and its derivatives. These revised Fourier series representations of U(x) and its derivatives are employed in a BEM formulation for three-dimensional linear elastostatics. Some numerical examples are presented to demonstrate the veracity of the implementation and its applicability to the elastic stress analysis of generally anisotropic solids. The results are compared with known solutions in the literature where possible, and with those obtained using the commercial finite element code ANSYS. Excellent agreement is obtained in all cases.     

Comprehensive form of solution for Lamb＇s dynamic problem expressed by Green＇s functions＊

By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld＇s identity as well as Green＇s functions of Stokes＇ solution pertaining the conventional elastic dynamic equation, the results of Green＇s function in an infinite space of an axisymmetric coordinate are shown in this paper. After employing a supplementary influence field and the boundary conditions in the free surface of a senti-space, the authors obtain the solutions of Green＇s function for Lamb＇s dynamic problem. Besides, the vertical displacement uzz and the radial displacement urz can match Lamb＇s previous results, and the solutions of the linear expansion source u~r and the linear torsional source uee are also given in the paper. The authors reveal that Green＇s function of Stokes＇ solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb＇s problem for various situations. It may benefit the investigation of deepening and development of Lamb＇s problems and solution for pertinent dynamic problems conveniently.     

State space solution to 3D multilayered elastic soils based on order reduction method

Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.     

Steady and transient Green’s functions for anisotropic conduction in an exponentially graded solid

The problem of the determination of Green’s function in conduction for a rectilinearly anisotropic solid with an exponential grading along a certain direction is studied. Domains of an unbounded space and a half-space, either three-dimensional or two-dimensional, are considered. Along the boundary of the domain, homogeneous boundary conditions of the first and second kinds are imposed. We find interestingly that, under this specific type of grading, the Green’s functions permit an algebraic decomposition, which will in turn greatly simplify the formulation. The method of Fourier transform is employed for the Green’s function for a half-space or a half-plane. Although the derivation process is quite tedious, we show analytically that the inverse transform can be found exactly and their resulting expressions are surprisingly neat and compact. In addition, both steady-state and transient-state field solutions are considered. By taking Laplace transform with respect to the time variable, we show that the mathematical frameworks for the steady-state and transient-state Green’s functions are entirely analogous. Thereby, the transient-state Green’s function is readily obtained by taking Laplace inverse transform back to the time domain. These derived fundamental solutions will serve as benchmark results for modeling some inhomogeneous materials. In the absence of grading term, we have verified analytically that our solutions agree exactly with previously known Green’s functions for homogeneous media.     

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

Theory of images in plane-layered axisymmetric elastostatics

A fundamental theorem which automatically transforms the solution due to the presence of an arbitrary axisymmetric singularity in an unbounded homogeneous isotropic elastic solid into the corresponding solution for two perfectly bonded isotropic semi-infinite elastic solids is systematically applied in a stepwise fashion to obtain the complete image system when an arbitrary axisymmetric singularity is operative in or near a thick elastic layer which separates two other dissimilar isotropic semi-infinite elastic solids. The solution is closed and well structured. As an illustration, the distant effect in the interface layer produced by an influencing normal point force is luminously revealed to be two-dimensional, consisting of a combination at the origin of a bending hot spot and an infinite line of centres of dilatation. We conclude the paper with a complete theory of images for a free elastic layer under the influence of both axisymmetric and asymmetric singularities. We find that if the influencing displacement field is the gradient of a harmonic function, then the calculation of the induced elastic field reduces to the operation of differentiation only.     

Cracked elastic layer under compressive mechanical loads

We consider boundary value problem in which an elastic layer containing a finite length crack is under compressive loading. The crack is parallel to the layer surfaces and the contact between crack surfaces are either frictionless or with adhesive friction or Coulomb friction.Based on fourier integral transformation techniques the solution of the formulated problems is reduced to the solution of a singular integral equation, then, using Chebyshev’s orthogonal polynomials, to an infinite system of linear algebraic equations. The regularity of these equations is established. The expressions for stress and displacement components in the elastic layer are presented. Based on the developed analytical algorithm, extensive numerical investigations have been conducted.The results of these investigations are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mode II type singularity.     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract

Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.     

A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function

Based on Biot’s theory, the dynamic 2.5-D Green’s function for a saturated porous medium is obtained using the Fourier transform and the potential decomposition methods. The 2.5-D Green’s function corresponds to the solutions for the following two problems: the point force applied to the solid skeleton, and the dilatation source applied within the pore fluid. By performing the Fourier transform on the governing equations for the 3-D Green’s function, the governing differential equations for the two parts of the 2.5-D Green’s function are established and then solved to obtain the dynamic 2.5-D Green’s function. The derived 2.5-D Green’s function for saturated porous media is verified through comparison with the existing solution for 2.5-D Green’s function for the elastodynamic case and the closed-form 3-D Green’s function for saturated porous media. It is further demonstrated that a simple form 2-D Green’s function for saturated porous media can be been obtained using the potential decomposition method.