Three-dimensional General Solution Research Articles

Three-Dimensional General Solutions of Orthorhombic Quasicrystals With Constraints

Abstract This study employs the Lur'e operator method to derive generalized solutions for orthorhombic quasicrystals, incorporating anisotropy factors as constraints. The solutions derived contain the Lekhnitskii–Hu–Nowacki and Elliott–Lodge solutions as special cases. The corresponding fundamental solutions or Green's functions within the infinite space are also derived, offering a comprehensive characterization of quasicrystal anisotropy. It is noteworthy that Green's functions in orthorhombic quasicrystals can be simplified to those in hexagonal quasicrystals or conventional orthorhombic crystals, with possible broad engineering applications.

Three-dimensional quasi-static general solution for isotropic chemoelastic materials and their application

Based on potential theory, the three-dimensional quasi-static general solution for isotropic chemoelastic materials is presented in this work. Through the three-dimensional general solution, the Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived. This can serve as theoretical guidance for future engineering practices. Four functions constitute the expressions of the general solution that satisfy the harmonic functions and the quasi-static transport equation, respectively. The Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived by combining the general solution with the chemical balance boundary conditions at infinity. It can be expressed in terms of the error function and elementary functions. Finally, the numerical results are provided, as shown in the contours. These results can be used to analyze the variation law in the coupling fields of isotropic chemoelastic materials. The corresponding analysis can provide a theoretical basis for elucidating the mechanism of the chemoelastic coupling problem in further work.

Three-dimensional Green's functions for fluid and isotropic thermoelastic solid two-phase materials under heat loading

Thermal exchange between liquid and solid is a common problem for various engineering structures. The thermal stresses caused by temperature change have great influence on the engineering structures. A three dimensional model for a coupling thermal and mechanical field was established to investigate thermal stress under heat loading. In this study, the three dimensional Green's functions for fluid and isotropic thermoelastic two-phase materials with internal heat source are obtained and used to study the thermal stress problems between solid and liquid. By virtue of the compact three-dimensional general solutions expressed in harmonic functions, four newly introduced harmonic functions are constructed, and all components of coupled field are expressed in terms of elementary functions which are very convenient to use. Numerical results are given graphically by contours. The corresponding analysis will contribute to obtain stress characteristics of the fluid-solid thermal coupling field.

Three-dimensional exact solutions of transversely isotropic coated structures under tilted circular flat punch contact

Because of the large stress gradient in the coated structure and the more and more thin thickness of coating, it is too hard to guarantee the calculation accuracy by the commercial software in the design of the coated structure. In this case, a research work for an accurate and efficient calculation method for the coated structure is necessary and will contribute to the accurate design and quality promotion of the coated structure. In this paper, the three-dimensional solutions of transversely isotropic coated structure under tilted circular flat punch contact are derived based on the three-dimensional general solutions of the transversely isotropic material. The solutions are obtained in two cases including the smooth contact and the frictional contact. The numerical results show that this method has the good convergence, high accuracy, high efficiency and high stability. Based on this method, the optimal thickness of the coating and the interfacial stress between the coating and the substrate can be obtained accurately. The corresponding analysis will contribute to the strengthening design of the coated structure.

Three-dimensional exact solutions of homogeneous transversely isotropic coated structures under spherical contact

Because of the large stress gradient in the coated structure and the more and more thin thickness of coating, it is too hard to guarantee the calculation accuracy by the commercial software in the design of the coated structure. In this case, a research work for an accurate and efficient calculation method for the coated structure is necessary and will contribute to the accurate design and quality promotion of the coated structure. In this paper, the three-dimensional solutions of homogeneous transversely isotropic coated structure under spherical contact loadings are derived based on the three-dimensional general solutions of the transversely isotropic material. The solutions are obtained in two cases including the smooth contact and the frictional contact. The numerical results show that this method has the good convergence, high accuracy, high efficiency and high stability. Based on this method, the optimal thickness of the coating and the interfacial stress between the coating and the substrate can be obtained accurately. Because the interfacial stress can result to the debonding failure of coated structures, a method for predicting interface security state with evaluation criteria for interfacial failure is proposed. The corresponding analysis will contribute to the strengthening design of the coated structure.

Three-dimensional Green’s functions for a fluid and pyroelectric two-phase material

The three-dimensional Green’s functions for a steady-state point heat source in the interior of a fluid and pyroelectric two-phase material are presented in this paper. By virtue of the three-dimensional general solutions which are expressed in harmonic functions, six newly introduced harmonic functions with undetermined constants are constructed. Then, all the pyroelectric components in the fluid and pyroelectric two-phase material can be derived by substituting these harmonic functions into the corresponding general solutions. And the undetermined constants can be obtained by the interface compatibility conditions and the mechanical, electric and thermal equilibrium conditions. Numerical results are given graphically by contours.

Green's Function for Transversely Isotropic Thermoelastic Diffusion Bimaterials

The aim of the present article is to study the Green's function in transversely isotropic thermoelastic diffusion bimaterial. With this objective, first the three-dimensional general solution in transversely isotropic thermoelastic diffusion bimaterial is derived. On the basis of general solution, Green's function, with a concentrated heat source in steady state, is completely solved using harmonic functions. The components of displacement, stress, temperature distribution, and mass concentration are expressed in terms of elementary functions. The resulting quantities are computed numerically and illustrated graphically. A particular case of three-dimensional Green function in transversely isotropic thermoelastic bimaterial has been deduced from the present investigation.

Three-dimensional thermo-elastic general solutions of one-dimensional hexagonal quasi-crystal and fundamental solutions

This Letter presents three-dimensional general solutions for static problems in thermo-elasticity of one-dimensional hexagonal quasi-crystals. Two displacement potentials are introduced to simplify the equilibrium equations in terms of displacement and temperature. Rigorous operator theory and generalized Almansiʼs theorem are applied to derive the general solutions in terms of five quasi-harmonic functions. To show the significance of the general solutions, a semi-infinite space and an infinite space, both of which are subjected to a heat source, are considered. In these two cases, closed form fundamental phonon–phason-elastic fields are expressed by elementary functions, which play an important role in numerical simulations.

Exact Analysis of Simply Supported Functionally Graded Piezothermoelectric Plates

This paper presents an exact analysis of a functionally graded piezothermo-electric rectangular plate that is simply supported, electrically grounded and isothermal on its four lateral edges. The governing equations are established for an orthotropic functionally graded piezothermoelectric plate under an assumption that the mechanical, electrical, and thermal properties of the material have the same exponential dependence on the thickness-coordinate. An exact three-dimensional general solution in the form of double Fourier series is derived for arbitrary distributions of combined mechanical, electrical, and thermal loadings at the top and bottom surfaces of the plate. Numerical results are presented for three special cases of uniformly distributed loads at the top and bottom surfaces of the plate, and the effect of truncation of the series on the accuracy of the solution is discussed.

On the equilibrium of piezoelectric bodies of revolution

By means of the three-dimensional general solution in displacement functions (weighted harmonic functions) for piezoelectric materials, the general solution of stress components and electric displacements expressed by the displacement functions is derived by use of the constitutive relation and the equilibrium equations. Based on this general solution, a series of problems is solved by the trial-and-error method, including circular plate (or cylinder), annular plate (or hollow cylinder), cone and hollow cone. These problems are circular plates and cylinders under uniform radial or axial tension and electric displacements as well as pure bending, simply-supported circular plates subjected to uniformly distributed loads, rotating disks and circular shafts, cones or hollow cones subjected to concentrated forces plus charge and concentrated force couple at their apex, etc. Analytical solutions to various problems are obtained. When the cone apex angle 2 α equals π, the solutions for the cases of concentrated forces plus point charges and torsion reduce to the simple and practical solutions of the half-space problem.

Wave-induced soil response in a nearly saturated sea-bed of finite thickness

Consolidation and storage equations are used to develop a closed-form solution for the wave-induced pore pressure, soil displacements and effective stresses in an elastic sea-bed subject to a system of two intersecting waves. A homogeneous soil matrix of finite thickness in isotropic and saturated conditions only is considered. The three-dimensional general solutions so developed are readily reducible to the conditions for soil of infinite thickness, and also for the limiting cases of two-dimensional progressive and standing waves for soil of finite thickness, for which no explicit solutions have previously been available. Verification of this solution with results of two-dimensional solutions available from a semi-analytical method, a boundary-layer approximation and a numerical model is carried out. Validation is performed by comparison with experimental results. The effects on wave-induced pore pressure of sea-bed thick- ness, shear modulus of soil and grain size are discussed. Les équations de stockage et de consolidation sont à la base de la solution approchée qui a été développée et qui permet de calculer la pression interstitielle, les déplacements et les contraintes effectives régnant dans un banc marin, élastique, soumis à deux régimes de vagues sécants. On considère que le sol est homogène, isotrope, saturé et d'épaisseur fine. Les solutions générales tridimensionnelles mises en oeuvre sont facilement transposables à un sol d'épaisseur infinie ou à un sol d'épaisseur finie soumis à un système bidimensionnel de vagues progressives puis stationnaires, cas pour lequel il n'y avait, jusqu'à présent, aucune solution explicite. Cette solution a été vérifée à l'aide des différentes solutions bidimensionnelles calculées à l'aide d'une méthode semi-analytique, d'une approximation couche-limite et d'un modèle numérique. Elle est validée par les résultats expérimentaux. On étudie également l'influence de l'paisseur du banc, du module de cisaillement du sol et de la taille des grains sur la pression interstitielle créée par les mouvements marins.