Three-Dimensional Fundamental Solution for Transversely Isotropic Electro-Magneto-Thermo-Elastic Materials

Green's functions play an important role in electroelastic analyses of piezoelectric media. However, most works available on this topic are on the case of identical temperature. Based on the compact general solution of transversely isotropic piezothermoelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the three-dimensional Green's function for a steady point heat source in a two-phase piezothermoelectric media is presented by eight newly induced harmonic functions. Numerical results are given graphically by contours.

Three-dimensional Green’s functions for a steady point heat source in a functionally graded half-space and some related problems

Green's functions play an important role in electroelastic analyses of pyroelecric media. However, most works available on this topic are on case of identical temperature. Based on the 2D and 3D general solution of pyroelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the 2D and 3D Green's function for a steady line heat source and a steady point heat source in semi-infinite, infinite and two-phase pyroelectric plane and media is presented by newly induced harmonic functions with undetermined constants. The corresponding coupled field can be obtained by substituting these functions into the 2D and 3D general solution, and the undetermined constants can be obtained by boundary conditions, continuous conditions and equilibrium conditions.

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.

General solution and Green’s function for fluid-saturated infinite and semi-infinite orthotropic poro-thermoelastic materials

A point heat source on the surface of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material

Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions

The objective was to optimize the existing solution for the limit support pressure of a tunnel face. Firstly, based on the numerical simulation results, the existing three-dimensional analytical solution for pore water pressure distribution is expanded to a three-dimensional solution considering the pore water pressure distribution in the upper formation behind the tunnel face. Then, according to the results of physical model tests, a failure model considering the failure range in the upper formation behind the tunnel face is established, and the newly established three-dimensional solution for pore water pressure is introduced into the model, and then the limit effective support pressure of the tunnel face considering seepage is obtained by the method of soil–water joint calculation. Finally, the calculation results in this paper are compared with the experimental results, numerical simulation results and existing theoretical solutions. The major findings are as follows. The distribution of pore water pressure in the front and back strata above the tunnel face is basically symmetrical. The limit effective support pressure of the tunnel face will increase linearly with an increase in the hydraulic head difference between the tunnel face and the ground surface. The calculated results of the new limit equilibrium theory are obviously larger than those of the existing theory and numerical simulation and closer to the results of the physical model tests. Therefore, the new limit equilibrium model can better predict the limit effective support pressure of the tunnel face considering seepage and provide a reference for actual projects.

On the half-infinite crack problem in thermo-electro-elasticity

A quasi‐linear form of Richards' equation, which assumes exponential dependence of the hydraulic conductivity on the pressure head and depth, was used to analyze the flows from a point source and to a point sink in a cylindrically confined soil domain overlying a shallow groundwater table (WT). The evaporation flux at the surface was taken to be proportional to the matric flux potential (MFP). Analytical solutions for the time‐dependent and steady‐state problems were obtained by using integral transforms. The general solution for a point source in a finite‐length cylinder takes the form of double series, which contains the inverse of the finite Hankel transform in the radial direction coupled with the inverse of the generalized Fourier transform in the vertical direction. Numerical evaluation of this solution is straightforward but time consuming for both time‐dependent and steady‐state cases. An alternative solution for a steady point source that involves only the Hankel inverse transform was found to be more practicable. Steady flows from a surface point source toward a WT and from a WT toward a subsurface sink were analyzed by mapping the distributions of the pressure head and the MFP, together with the streamlines. The sink was simulated by a suction zone of prescribed radius whose strength was evaluated from the condition that MFP equals zero at a reference point at either the bottom or the top of the suction zone. Particular attention was paid to the choice of locations of the reference point for evaluating the sink strength, for which the solution for the sink remains physically relevant irrespective of whether or not the surface loses water by evaporation.

According to Green’s function for a half-infinite plane subjected to a concentrated force, a two-dimensional Green’s function is derived for an isotropic double-layer structure of equal thickness under a concentrated force applied above the surface. This derivation was based on the mirroring image method and Dirichlet’s principle of singularity. Initially, the characteristics of the equal-thickness double-layer structure were examined, leading to the formulation of the upper and lower surface and interface conditions of the computational structure. Subsequently, four series of harmonic functions were generated from ten harmonic functions using the general solutions to two harmonic functions. By substituting these four series functions into a two-dimensional general solution, eight recursive equations were derived. The equations were expressed through eight harmonic functions, which could be solved using the upper and lower surface conditions and interface connection conditions. The harmonic function under any force can be determined through Green’s function for an isotropic half-infinite plane structure, allowing for the computation of the displacement and stress fields of the isotropic double-layer structure with equal thickness. Finally, numerical examples were used to compare the interface calculation results of this study with the finite element method numerical results, demonstrating the validity of the algorithm. This method is computationally simple and converges quickly. Utilizing the principle of superposition, the stress fields and displacements of an isotropic finite element method equal-thickness double-layer structure can be solved for any directional force acting on the surface, providing a robust foundation for the engineering design and analysis of such problems.

In the boundary element method (BEM) for stress analysis, it is well known that thermal loads give rise to an additional volume integral in the primary form of the boundary integral equation (BIE). This volume integral needs to be further transformed to surface ones in order to retain the characteristic of the BEM as a boundary solution technique. In this study of the BEM for 3D thermoelasticity in general anisotropy, the fundamental solutions are expressed as Fourier series with coefficients calculated using an explicit-form Green’s function. In the exact volume-to-surface integral transformation associated with the term for the thermal effects in the BIE, a new kernel function is constructed. All formulations are implemented in an existing BEM code for 3D elastostatic analysis. Some numerical examples are presented to demonstrate the veracity of the formulations and the implementation, where the numerical results are compared with those obtained using the finite element method (FEM).

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

This paper deals with a hybrid displacement variational formulation of BEM for elastodynamics. The mathematical model for elastodynamics is obtained from the stationarity condition of a modified functional expressed in terms of displacements and tractions. These are assumed as independent on the boundary and in the domain. The boundary variables are expressed by using their nodal values while the domain displacement field is approximated with a linear combination of static fundamental solutions. The source point of every fundamental solution is located outside the domain. The problem of free vibrations is reduced to an algebraic eigenvalue problem, the solution of which is straightforward. The structural matrices result symmetric and positive definite and the domain integral, associated to the inertial term, can be transformed into a boundary one. Numerical results are in excellent agreement with existing ones.