Thermoelastostatic equilibrium of a spatial quadrant: Green’s function and solution in integrals

In this study, a new Green??s function and a new Green-type integral formula for a 3D boundary value problem (BVP) in thermoelastostatics for a quarter-space are derived in closed form. On the boundary half-planes, twice mixed homogeneous mechanical boundary conditions are given. One boundary half-plane is free of loadings and the normal displacements and the tangential stresses are zero on the other one. The thermoelastic displacements are subjected by a heat source applied in the inner points of the quarter-space and by mixed non-homogeneous boundary heat conditions. On one of the boundary half-plane, the temperature is prescribed and the heat flux is given on the other one. When the thermoelastic Green??s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by ??-Dirac??s function. All results are obtained in elementary functions that are formulated in a special theorem. As a particular case, when one of the boundary half-plane of the quarter-space is placed at infinity, we obtain the respective results for half-space. Exact solutions in elementary functions for two particular BVPs for a thermoelastic quarter-space and their graphical presentations are included. They demonstrate how to apply the obtained Green-type integral formula as well as the derived influence functions of an inner unit point body force on volume dilatation to solve particular BVPs of thermoelasticity. In addition, advantages of the obtained results and possibilities of the proposed method to derive new Green??s functions and new Green-type integral formulae not for quarter-space only, but also for any canonical Cartesian domain are also discussed.     

Three-dimensional Green’s functions for transversely isotropic thermoelastic bimaterials

Green’s functions for transversely isotropic thermoelastic biomaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoelastic material in terms of harmonic functions and introduce six new harmonic functions. The three-dimensional Green’s function having a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of temperature and stress distributions at the interface. The temperature contours are normal to the interface for the isotropic material but not for the orthotropic one. The normal stress contours are parallel to the interface at the boundary in the isotropic region only and shear failure is most likely at the heat source due to the highly degenerated direction of shear stress contours.     

Static and Dynamic Green’s Functions in Peridynamics

We derive the static and dynamic Green’s functions for one-, two- and three-dimensional infinite domains within the formalism of peridynamics, making use of Fourier transforms and Laplace transforms. Noting that the one-dimensional and three-dimensional cases have been previously studied by other researchers, in this paper, we develop a method to obtain convergent solutions from the divergent integrals, so that the Green’s functions can be uniformly expressed as conventional solutions plus Dirac functions, and convergent nonlocal integrals. Thus, the Green’s functions for the two-dimensional domain are newly obtained, and those for the one and three dimensions are expressed in forms different from the previous expressions in the literature. We also prove that the peridynamic Green’s functions always degenerate into the corresponding classical counterparts of linear elasticity as the nonlocal length tends to zero. The static solutions for a single point load and the dynamic solutions for a time-dependent point load are analyzed. It is analytically shown that for static loading, the nonlocal effect is limited to the neighborhood of the loading point, and the displacement field far away from the loading point approaches the classical solution. For dynamic loading, due to peridynamic nonlinear dispersion relations, the propagation of waves given by the peridynamic solutions is dispersive. The Green’s functions may be used to solve other more complicated problems, and applied to systems that have long-range interactions between material points.     

On Green’s function for a bimaterial elastic half-plane

The problem of a point force acting in a composite, two-dimensional, isotropic elastic half-plane is considered. An exact solution is obtained, using Mellin transforms and the Melan solution for a point force in a homogeneous half-plane.     

Green’s functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper, we obtain Green’s functions of two-dimensional (2D) piezoelectric quasicrystal (PQC) in half-space and bimaterials. Based on the elastic theory of QCs, the Stroh formalism is used to derive the general solutions of displacements and stresses. Then, we obtain the analytical solutions of half-space and bimaterial Green’s functions. Besides, the interfacial Green’s function for bimaterials is also obtained in the analytical form. Before numerical studies, a comparative study is carried out to validate the present solutions. Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force, the line dislocation, the line charge, and the phason line force. As a result, the coupling effect among the phonon field, the phason field, and the electric field is prominent, and the butterfly-shaped contours are characteristic in 2D PQCs. In addition, the changes of material parameters cause variations in physical quantities to a certain degree.     

Green’s functions of a surface-stiffened transversely isotropic half-space

Green’s functions of a transversely isotropic half-space overlaid by a thin coating layer are analytically obtained. The surface coating is modeled by a Kirchhoff thin plate perfectly bonded to the half-space. With the aid of superposition technique and employing appropriate displacement potential functions, the Green’s functions are expressed in two parts; (i) a closed-form part corresponding to the transversely isotropic half-space with surface kinematic constraints, and (ii) a numerically evaluated part reflecting the interaction between the half-space and the plate in the form of semi-infinite integrals. Some limiting cases of the problem such as surface-stiffened isotropic half-space, Boussinesq and Cerruti loadings, and extremely flexible and rigid plates are also studied. For the classical Cerruti problem in transversely isotropic materials, the effects of incompressibility are highlighted. Numerical results are provided to show the effects of material anisotropy, relative stiffness factor, and load buried depth. The obtained Green’s functions play a key role in treating further mixed-boundary-value problems in surface stiffened transversely isotropic half-spaces.     

High-order derivatives of Green’s functions in magneto-electro-elastic materials

Three-dimensional Green’s functions and their arbitrary order derivatives in general anisotropic magneto-electro-elastic materials are derived by using Fourier transform. They are analytical solutions expressed in line integral forms, and can be evaluated by a standard numerical integration method. With this method, we can obtain results with high accuracy. Besides, a numerical finite difference method is also given to evaluate the second-order derivatives quickly. When setting the appropriate material coefficients to zero, the piezoelectric, piezomagnetic, and purely anisotropic elastic Green’s functions and their derivatives can all be obtained from the current solutions.     

Green’s functions for a loaded rolling tyre

A new formulation to determine the unit impulse response (Green’s) functions of a loaded rotating tyre in the vehicle-fixed (Eulerian) reference frame for tyre/road noise predictions is presented. The proposed formulation makes use of the set of eigenfrequencies and eigenmodes for the statically loaded tyre obtained from a finite element (FE) model of the tyre. A closed-form expression for the Green’s functions of a rotating tyre in the Eulerian reference system as a function of the eigenfrequencies and eigenmodes of the statically loaded tyre is found. Non-linear effects during loading are accounted for in the FE model, while the frequency shift due to the rotational velocity is included in the calculation of the Green’s functions. In the literature on tyre/road noise these functions are generally used to determine the tyre response during tyre/road contact calculations. The presented formulation opens the possibility to solve the contact problem directly in the Eulerian reference frame and to include local tyre softening due to non-linear effects while keeping the computational advantage of describing the tyre dynamics as a set of impulse response functions. The advantage of obtaining the Green’s functions in the Eulerian reference system is that only the Green’s functions corresponding to the potential contact zone need to be determined, which significantly reduces the computational cost of solving the tyre/road contact and since the mesh is fixed in space, a finer mesh can be used for the potential contact zone, improving the accuracy of the contact force calculations. Although these effects might be less pronounced if a more accurate tyre model is used, it is found that using the Green’s functions of the loaded tyre in a contact force calculation leads to smaller forces than in the unloaded case, lower frequencies are present in the response and they decrease faster as the rotational velocity increases.     

Dynamic Green’s functions of a two-phase saturated medium subjected to a concentrated force

The Green’s functions of a two-phase saturated medium subjected to a concentrated force are known to play an important role in seismology, earthquake engineering, soil dynamics, geophysics, and dynamic foundation theory. This paper presents a physical method for obtaining the dynamic Green’s functions of a two-phase saturated medium for materials considered to be isotropic and for low frequencies. First, the pore-fluid pressure in a two-phase saturated medium is divided into two parts: flow pressure and deformation pressure. Next, based on the compatibility condition of Biot’s equation and the property of the δ-function, the problem of coupled_fast and slow dilational waves is solved using the decomposition condition of the potential dilation field. The Green’s function for a concentrated force is then obtained by solving Biot’s complex modular equations, and their physical characteristics are discussed. The behavior of Green’s functions for the solid and fluid phases of a δ-impulsive force is investigated, from which the Green’s functions for a unit Heaviside force are also obtained by time integration. Finally, the present Green’s functions for a unit Heaviside force are compared with those obtained by a purely mathematical method; the two differ in form, but the numerical results are identical. The physical meaning of the expressions of Green’s functions obtained in this paper is evident. Therefore, the results may benefit future research on the dynamic responses of a two-phase saturated medium.     

Green’s function for a prestressed thermoelastic half-space with an inhomogeneous coating

A mathematical model is developed for an inhomogeneous thermoelastic prestressed half-space consisting of a stack of homogeneous or functionally graded layers rigidly attached to a homogeneous base. Each component of the inhomogeneous medium is subjected to initial mechanical stresses and temperature. Successive linearization of the constitutive relations of the nonlinear mechanics of a thermoelastic medium is performed using the theory of superposition of small deformations on finite deformations with the inhomogeneity of the medium taken into account. Integral formulas are derived to explore dynamic processes in inhomogeneous prestressed thermoelastic media.     

Decoupling coefficients of dilatational wave for Biot’s dynamic equation and its Green’s functions in frequency domain

Green’s functions for Biot’s dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering, rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green’s functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term “decoupling coefficient” for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green’s functions. The correctness of the solution is demonstrated by numerically comparing the current solution with Cheng’s previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green’s functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method (BEM) and other applications.     

Interaction between a moving two-mass oscillator and an infinite homogeneous structure: Green’s functions method

This paper deals with the interaction between a vehicle and a slab track using the model of moving wheel for both frequency and time-domain. The vehicle is reduced to a moving two-mass oscillator and the slab track is considered as an infinite structure consisting of elastically supported double Euler–Bernoulli beams. In order to perform the time-domain analysis, a semi-analytical method based on the outstanding properties of the time-domain Green’s functions of the slab track has been developed. The method allows the computing of the non-linear wheel/rail contact (the contact loss and the non-linear contact stiffness). The vehicle/track interaction due to the polygonal wheel and the corrugated rail has been investigated and the running velocity and non-linear wheel/rail contact influences have been pointed out.     

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.     

Derivatives of the three-dimensional Green’s functions for anisotropic materials

A concise formulation is presented for the derivatives of Green’s functions of three-dimensional generally anisotropic elastic materials. Direct calculation for derivatives of the Green’s function on the Cartesian coordinate system is a common practice, which, however, usually leads to a complicated course. In this paper the Green’s function derived by Ting and Lee [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. The Quarterly Journal of Mechanics and Applied Mathematics 50 (3) 407–426] is extended to obtain the derivatives. Using a spherical coordinate system, the Green’s function can be shown as the composition of two independent functions, one depends only on the radial distance of the field point to the origin and the other is in spherical angles. The method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly. With the application of Cauchy residue theorem, the contour integral can be evaluated in terms of the Stroh eigenvalues of a sextic equation. For the degenerate case, evaluation of residues at multiple poles is also given. Applications of the present result are made to examine the Green’s functions and stress components for isotropic and transversely isotropic materials. The results are in exact agreement with existing solutions.     

Green’s functions of an exponentially graded transversely isotropic half-space

By virtue of a complete set of displacement potential functions and Hankel transform, the analytical expressions of Green’s function of an exponentially graded elastic transversely isotropic half-space is presented. The given solution is analytically in exact agreement with the existing solution for a homogeneous transversely isotropic half-space. Employing a robust asymptotic decomposition technique, the Green’s function is decomposed to the closed-form Green’s function corresponding to the homogeneous transversely isotropic half-space and grading term with strong decaying integrands. This representation is very useful for numerical methods which are based on boundary-integral formulations such as boundary-element method since the numerically evaluated part is not responsible for the singularity. The high accuracy of the proposed numerical scheme is confirmed by some numerical examples.     

Time-harmonic Green’s functions for anisotropic magnetoelectroelasticity

Two-dimensional (2-D) and three-dimensional (3-D) time-harmonic Green’s functions for linear magnetoelectroelastic solids are derived in this paper by means of Radon-transform. Displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time-harmonic point force, point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2-D case and surface integrals over a unit sphere in 3-D case. This dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms. The singular terms coincide with the Green’s functions for the static problem and may be further reduced to closed form expressions. The proposed Green’s functions can be used in the corresponding boundary element method (BEM) formulation.     

Three-dimensional dynamic Green’s functions for a multilayered transversely isotropic half-space

With the aid of a method of displacement potentials, an efficient and accurate analytical derivation of the three-dimensional dynamic Green’s functions for a transversely isotropic multilayered half-space is presented. Constituted by proper algebraic factorizations, a set of generalized transmission–reflection matrices and internal source fields that are free of any numerically unstable exponential terms are proposed for effective computations of the potential solution. Three-dimensional point-load Green’s functions for stresses and displacements are given, for the first time, in the complex-plane line-integral representations. The present formulations and solutions are analytically in exact agreement with the existing solutions given by Pak and Guzina (2002) for the isotropic case. For the numerical computation of the integrals, a robust and effective methodology which gives the necessary account of the presence of singularities including branch points and poles on the path of integration is laid out. A comparison with the existing numerical solutions for multilayered isotropic half-space is made to confirm the accuracy of the numerical solutions.     

Green’s functions for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials

Summary The anti-plane deformation problem of an interfacial debounding crack between a circular piezoelectric inclusion and a piezoelectric matrix is investigated by means of the complex variables method. For a line load applied within the matrix or inside the inclusion, Greens functions are presented for the complex potentials, intensity factors and electric fields on the crack faces, respectively, in closed and explicit form. The solutions are valid for both permeable and impermeable crack models. It is shown that, in the general case of permeable cracks, the electric field singularity is always proportional to the stress singularity.The first author (C.F.Gao) would like to express his gratitude for the support of the Alexander von Humboldt Foundation (Germany).     

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.