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 function for T-stress of semi-infinite plane crack

Green’s function for the T-stress near a crack tip is addressed with an analytic function method for a semi-infinite crack lying in an elastical, isotropic, and infinite plate. The cracked plate is loaded by a single inclined concentrated force at an interior point. The complex potentials are obtained based on a superposition principle, which provide the solutions to the plane problems of elasticity. The regular parts of the potentials are extracted in an asymptotic analysis. Based on the regular parts, Gre...     

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.     

Analytical solution for squeeze film damping of MEMS perforated circular plates using Green’s function

Squeezed air film between two closely spaced vibrating microstructures is the important source of energy dissipation and has profound effects on the dynamics of microelectromechanical systems (MEMS). Perforations in the design are one of the methods to model these damping effects. The literature reveals that the analytical modeling of squeeze film damping of perforated circular microplates is less explored; however, these microplates are also an imperative part of the numerous MEMS devices. Here, we derive an analytical model of transverse and rocking motions of a perforated circular microplate. A modified Reynolds equation that incorporates compressibility and rarefaction effects is utilized in the analysis. Pressure distribution under the vibrating microplate is derived by using Green’s function and also derived by finite element method (FEM) to visualize the pressure distribution under perforated and non-perforated areas of the microplate. The analytical damping results are validated with previous renowned analytical models and also with the FEM results. The outcomes confirm the potential of the present analytical model to accurately predict the squeeze film damping parameters.     

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.     

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 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.     

New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.     

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.     

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.     

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.     

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.     

On Saint-Venant’s Problem with Concentrated Loads

The classical Saint-Venant problem is to find a solution of the traction problem of elastostatics in a finite cylinder ?? loaded over its bases. We prove that the problem has a unique solution for equilibrated surface forces $\hat{ \boldsymbol { s}}\in W^{-1,q}(\partial\Omega)$ , with q??(2?? ₀,+??) for some positive ? ₀ depending on ??. Hence $\hat{ \boldsymbol { s}}$ can model force acting on ???, concentrated on sets of zero Lebesgue surface measure of ???. Moreover, if $\hat{ \boldsymbol { s}}$ is equilibrated on each basis, we give a simple proof of the Toupin estimate expressing Saint-Venant??s principle.     

On a class of method for solving problems with random boundary notches and/or cracks

As we know,problems with boundary imperfections(notches or cracks)are the more importantone in practical fracture analysis.Frequently,these imperfections would appear in the boundaries ofthe bodies as a randomly distributed group,and under the loading circumstances would grow up to beunstable cracks which induces catastrophic fracture of the bodies.For right evaluation the fracturebehavior of the bodies with such boundary imperfections,it demands mathematical solutions for pro-blems with random boundary notches and/or cracks.     

On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach

In this paper, the self-adjointness of Eringen’s nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen’s model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever’s fundamental vibration frequency with respect to increasing Eringen’s small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen’s nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.     

Artificial boundary conditions for Euler–Bernoulli beam equation

In a semi-discretized Euler-Bernoulli beam equa- tion, the non-nearest neighboring interaction and large span of temporal scales for wave propagations pose challenges to the effectiveness and stability for artificial boundary treat- ments. With the discrete equation regarded as an atomic lattice with a three-atom potential, two accurate artificial boundary conditions are first derived here. Reflection co- efficient and numerical tests illustrate the capability of the proposed methods. In particular, the time history treatment gives an exact boundary condition, yet with sensitivity to nu- merical implementations. The ALEX （almost EXact） bound- ary condition is numerically more effective.