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 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.     

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).     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

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.     

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.     

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.     

Application of Krein’s theorem and bifurcation theory for stability analysis of a bladed rotor

This paper considers the stability and eigenvalue analyses for a bladed rotor which goes under cylindrical and conical whirling. The model consists of a group of flexible blades which are modeled by beams and rigid disk on the elastic bearings. The model is a Hamiltonian system which is perturbed by small dissipative forces. Krein’s theorem reveals that the forward whirling mode and the blade collective motion may cause instability when their frequencies cut themselves in the Campbell diagram. An unstable interaction between the blades and the conical whirling is discovered. The eigenmode and eigenvalue evolutions are determined on the stability boundary. The bifurcation analysis is performed by applying multiple scales method around the stability boundary. It is shown that the damping distribution between the blades and the bearings may shift the unstable mode.     

A generalization of the Coulomb’s friction law: from graphene to macroscale

At the nanoscale, differently to what happens at the macroscale, friction even without an applied normal pressure and spontaneous adhesion take place. In particular, the nanotribology between two layers of graphene, or other two-dimensional nanomaterials (even curved, such as nanotube walls), remains controversial. It is sufficient to say that friction between two graphene layers or nanotube walls is described in the current literature giving as “material property” a constant friction force or a constant friction shear strength, even if such views are obviously mutually exclusive. Is friction dominated by a strength, by a force or by an energy? Coupling elasticity and energy balance we solve this paradox deriving a generalization of the celebrated Coulomb’s friction law, reconciling the two current views. Molecular dynamics simulations on graphene are conducted to verify its validity at the nanoscale whereas statistical simulations confirm its validity even at the macroscale.     

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.     

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.     

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.     

Two-dimensional Eshelby’s problem for piezoelectric materials with a parabolic boundary

We use complex variable techniques to obtain analytic solutions of Eshelby’s problem consisting of an inclusion of arbitrary shape in an anisotropic piezoelectric plane with a parabolic boundary. The region of the physical plane below the parabola is mapped onto the lower half of the image plane. The problem is then more conveniently studied in the image plane rather than in the physical plane. The critical step in our approach lies in the construction of certain auxiliary functions in the image plane which allow for the technique of analytic continuation to be applied to an inclusion of arbitrary shape.     

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.     

Analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions: Duhamel’s theorem and separation of variables

An analytical resolution of the time-dependent one-dimensional heat conduction problem with time-dependent boundary conditions using the method of separation of variables and Duhamel’s theorem is presented. The two boundary conditions used are a time-dependent heat flux at one end and a varying temperature at the other end of the one-dimensional domain. It is put forth because the author found that the prescribed resolution method using separation of variables and Duhamel’s theorem presented in heat conduction textbooks is not directly applicable to problems with more than one time-dependent boundary condition. The analytical method presented in this paper makes use of one of the property of the heat conduction equation: the apparent linearity of the solutions. For that reason, in order to solve a problem with two time-dependent boundary conditions, the author first separates the initial problem into two independent but complementary problems, each with only one time-dependent boundary condition. Doing that, both simpler problems can be solved independently using a prescribed method that is known to work and the final solution can be obtained by joining the two independent solutions from the simpler separated problems. Every step of the resolution method is presented in this paper, along with a numerical validation of the final solution of three test case problems.     

The Navier–Stokes Equations Under a Unilateral Boundary Condition of Signorini’s Type

We propose a new outflow boundary condition, a unilateral condition of Signorini’s type, for the incompressible Navier–Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier–Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya’s class using the standard Galerkin’s method. For the proof of the existence of pressures, however, we offer a new method of analysis.     

Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals

This article presents in a closed form new influence functions of a unit point heat source on the displacements for three boundary value problems of thermoelasticity for a half-plane. We also obtain the corresponding new integral formulas of Green’s and Poisson’s types that directly determine the thermoelastic displacements and stresses in the form of integrals of the products of specified internal heat sources or prescribed boundary temperature and constructed already thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of three theorems. Based on these theorems and on derived early by author the general Green-type integral formula, we obtain in elementary functions new solutions to two particular boundary value problems of thermoelasticity for half-plane. The graphical presentation of the temperature and thermal stresses of one concrete boundary value problems of thermoelasticity for half-plane also is included. The proposed method of constructing thermoelastic Green’s functions and integral formulas is applicable not only for a half-plane, but also for many other two- and three-dimensional canonical domains of different orthogonal coordinate systems.