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An Eshelby inclusion of arbitrary shape in a nonlinearly coupled thermoelectric material

ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK 2023年第2期74卷 1-10页

作者： Wang, Xu Schiavone, Peter East China Univ Sci & Technol Sch Mech & Power Engn 130 Meilong Rd Shanghai 200237 Peoples R China Univ Alberta Donadeo Innovat Ctr Engn 10 203 Dept Mech Engn Edmonton AB ABT6G 1H9 Canada

In this paper, we present a general method, based on the techniques of analytic continuation and conformal mapping, for the analytic solution of Eshelby's problem concerned with a two-dimensional inclusion of arbitrary shape in an infinite homogeneous and isotropic nonlinearly coupled thermoelectric plane. The inclusion is subjected to a prescribed uniform electric current-free thermoelectric potential gradient and a uniform energy flux-free temperature gradient. The corresponding boundary value problem is studied in both the physical and image planes. The closed-form general solution is found to be exact provided that the associated mapping function contains only a finite number of terms. Elementary expressions for the internal electric current density and energy flux in the physical plane are obtained. Examples of elliptical, hypotrochoidal and rectangular inclusions are presented to demonstrate the solution method. Interestingly, in this case Eshelby's uniformity property is found invalid for an elliptical inclusion.

关键词： Nonlinear thermoelectricity Eshelby's problem complex variable method Conformal mapping Analytic continuation

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A nonlinearly coupled thermoelectric material with a parabolic inhomogeneity

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JOURNAL OF THERMAL STRESSES 2023年第5期46卷 399-414页

作者： Wang, Xu Schiavone, Peter East China Univ Sci & Technol Sch Mech & Power Engn Shanghai Peoples R China Univ Alberta Dept Mech Engn Edmonton AB Canada East China Univ Sci & Technol Sch Mech & Power Engn 130 Meilong Rd Shanghai 200237 Peoples R China Univ Alberta Dept Mech Engn 10 203 Donadeo Innovat Ctr Engn Edmonton AB T6G 1H9 Canada

We use complex variable methods to study the two-dimensional thermoelectric and thermoelastic problems associated with a nonlinearly coupled thermoelectric parabolic inhomogeneity embedded in an infinite nonlinearly coupled thermoelectric matrix subjected to uniform remote electric current density and uniform remote energy flux. Exact closed-form solutions to the problems are derived primarily with the aid of an identity established on the parabolic interface. Our analysis indicates that the electrically and thermally induced stresses exhibit quadratic distributions inside the parabolic inhomogeneity.

关键词： Closed-form solution complex variable method nonlinear thermoelectricity parabolic inhomogeneity thermoelastic field thermoelectric field

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An elliptical inhomogeneity under nonuniform heat flux

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MATHEMATICS AND MECHANICS OF SOLIDS 2023年第8期28卷 1790-1797页

作者： Wang, Xu Schiavone, Peter East China Univ Sci & Technol Sch Mech & Power Engn 130 Meilong Rd Shanghaxi 200237 Peoples R China Univ Alberta Dept Mech Engn 10-203 Donadeo Innovat Ctr Engn Edmonton AB T6G 1H9 Canada

We use complex variable techniques to study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity perfectly bonded to an infinite matrix subjected to a nonuniform heat flux at infinity. Specifically, the nonuniform remote heat flux takes the form of a linear distribution. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. Explicit closed-form expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

关键词： Elliptical inhomogeneity nonuniform heat flux complex variable method

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An analytical approximate solution for stress and displacement around a shallow circular tunnel excavated under concave slope topography

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APPLIED MATHEMATICAL MODELLING 2023年第1期124卷 445-487页

作者： Cai, Yongxiang Huang, Houxu Cai, Guojun Cai, Yi Yan, Chao Shen, Huazhang Anhui Jianzhu Univ Sch Civil Engn Hefei Peoples R China Anhui Jianzhu Univ Anhui Inst Intelligent Underground Detect Technol Hefei Peoples R China

For shallow buried tunnels under slope, overloading on the ground surface and unloading of excavation both significantly influence the stability of the tunnel surrounding rock and slope. This study proposes an analytical approximate solution for the stress and displacement around shallow buried circular tunnel under slope boundary. In derivation process, the original problem is divided into two models: Model A is a slope plane without any holes subjected to gravity and surcharge load along the ground surface, and Model B is a tunnel contained in a slope plane loaded by released stresses along the tunnel boundary. Furthermore, Model B is decomposed into another two models: Model B1 represents a slope without holes subjected to virtual surface force along the ground surface, and Model B2 is an infinite domain containing a tunnel loaded by another virtual surface force along its boundary. Two sets of virtual surface forces are then achieved by an iterative calculation. The solution, which strictly meets all stress boundary and compatibility conditions of the original problem, are ultimately solved by superposing the solution of Models A, B1 and B2, respectively obtained by the method proposed by the authors (Models A, B1) and complex variable theory (Model B2). The convergence of iteration process and good agreement between the derived and numerical results under the same assumptions verify the correctness of the proposed solution. Furthermore, a parametric investigation is performed to investigate the influence of surcharge load position, slope topography and tunnel position on the stress and displacement around the tunnel and near the ground. The proposed solution is of great value for the conceptual understanding of mechanical behavior of shallow tunneling in slope terrain and for verifying the numerical models. Moreover, the solution is useful for the preliminary design of tunnels at shallow depth under slope topography.

关键词： Shallow tunnel Analytical approximate solution complex variable method Surcharge load

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Uniformity of anti-plane stresses within a three-phase nonlinear elastic inhomogeneity of arbitrary shape

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ARCHIVE OF APPLIED MECHANICS 2023年第11期93卷 4261-4272页

We first investigate the anti-plane shear deformation of a three-phase nonlinear elastic elliptical inhomogeneity under uniform remote anti-plane stresses. In the three-phase composite, the p-Laplacian nonlinear elastic elliptical inhomogeneity is bonded to the infinite linear elastic matrix via a linear elastic interphase layer resulting in two confocal elliptical interfaces. It is proved that the internal stress field inside the nonlinear elliptical inhomogeneity is nevertheless unconditionally uniform. Secondly, we examine the uniformity of internal anti-plane stresses within a three-phase nonlinear elastic non-elliptical inhomogeneity. In this case, the p-Laplacian nonlinear elastic non-elliptical inhomogeneity is again bonded to the infinite linear elastic matrix through a linear elastic interphase layer. The internal uniform stress field is achieved via the solution of two coupled nonlinear equations. Detailed numerical results validate our theoretical analysis.

关键词： Three-phase inhomogeneity p-Laplacian Anti-plane elasticity Uniform stress field complex variable method

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A DOUBLE COATED CIRCULAR ELASTIC INHOMOGENEITY WITH INTERNAL UNIFORM DEVIATORIC STRESSES

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JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES 2023年第5期18卷 741-749页

作者： Wang, Xu Schiavone, Peter East China Univ Sci & Technol Sch Mech & Power Engn Shanghai Peoples R China Univ Alberta Dept Mech Engn Edmonton AB Canada

We show that the internal deviatoric stresses within a double coated circular elastic inhomogeneity can remain uniform when the infinite elastic matrix is subjected to a uniform remote deviatoric load. The internal uniform stress field remains valid when the relative thickness of the outer coating is determined through the solution of a cubic equation for given material parameters and the relative thickness of the inner coating. The explicit expression for the internal uniform deviatoric stress field within the circular inhomogeneity is presented and demonstrated graphically. We also accomplish the design of a harmonic double coated circular inhomogeneity with internal uniform stresses. In this second design, both the relative thickness of the outer coating and the ratio of the shear modulus of the matrix to that of the outer coating are determined by solving two coupled nonlinear equations numerically.

关键词： double coated circular inhomogeneity deviatoric load uniform stress field harmonic inhomogeneity complex variable method

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Dynamic response of a lined tunnel to plane SH waves in a bi-material space

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MECCANICA 2023年第7期58卷 1427-1441页

作者： Ri, Song-Chol Wang, Shuhong Jin, Hak-Son Northeastern Univ Sch Resource & Civil Engn Shenyang 110819 Peoples R China Kim Chaek Univ Technol Dept Mech Sci Pyongyang North Korea Northeastern Univ Key Lab Minist Educ Safe Min Deep Met Mines Shenyang 110819 Peoples R China Kim Il Sung Univ Fac Energy Sci Pyongyang North Korea

In this paper, an analytical solution for the scattering of plane SH waves by a lined tunnel buried in an inhomogeneous medium is presented via the complex variable method. The problem model is a bi-material with a homogeneous medium containing a circular cavity and an inhomogeneous one embedding a lined tunnel. For the convenience of calculation, the physical domain including straight and circular interfaces is transformed into the image plane with four concentric circles by introducing the conformal mapping functions. The boundary conditions are applied to find the unknown coefficients of the wave functions. Then, the factors affecting the dynamic response to SH wave are investigated through the calculation results for the dynamic stress concentration factor (DSCF) at several boundary surfaces.

关键词： SH wave Inhomogeneous medium complex variable method Conformal mapping technique

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Antiplane scattering of SH waves by a shallow lined tunnel in a horizontal exponentially inhomogeneous half-space

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ARCHIVE OF APPLIED MECHANICS 2023年第3期93卷 1107-1122页

作者： Ri, Song-Chol Wang, Shuhong Jin, Hak-Son Jang, Paek-San Northeastern Univ Sch Resource & Civil Engn Shenyang 110819 Peoples R China Kim Chaek Univ Technol Dept Mech Sci Pyongyang North Korea Northeastern Univ Key Lab Minist Educ Safe Min Deep Met Mines Shenyang 110819 Peoples R China Kim Il Sung Univ Fac Energy Sci Pyongyang North Korea Kim Chaek Univ Technol Inst Nanophys Engn Pyongyang North Korea

In this study, based on the plane elastic complex variable theory, employing image technique and conformal mapping technique, an analytical solution for antiplane scattering of SH waves by a lined tunnel in an exponentially graded half-space is derived, and the dynamic stress concentration factor (DSCF) around the tunnel is investigated. The medium is a bimaterial consisting of a semi-infinite homogeneous space and an exponentially inhomogeneous half-space with a lined circular tunnel. The governing equation is normalized into a Helmholtz equation with constant coefficients in complex coordinates based on the plane complex variable theory. The conformal mapping technique is used to convert the physical plane with two half-spaces including a lined tunnel into an image region consisting of three concentric circles. By applying the boundary conditions and the principle of orthogonality of trigonometric functions, a series of infinite algebraic equation are constructed and the unknown coefficients for the scattered wave functions are calculated. The numerical calculations are performed by considering the several parameters of medium and various conditions, and then, the influences of medium parameters on the dynamic response of the tunnel are analyzed based on the calculation results.

关键词： SH wave Scattering Lined tunnel complex variable method Conformal mapping technique

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Thermoelectric field of an eccentrically coated circular nonlinear thermoelectric inhomogeneity

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INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 2023年 148卷

We study the nonlinearly coupled thermoelectric problem associated with an eccentrically coated circular inhomogeneity embedded in an infinite thermoelectric matrix subjected to uniform remote electric current density and uniform remote energy flux. Using conformal mapping and analytic continuation to enforce the continuity conditions across the two perfect circular interfaces, we reduce the original problem to the determination of only two analytic functions. Once these two analytic functions are determined via satisfaction of the singular behaviors of the two original analytic functions defined in the image of the domain occupied by the matrix, all of the six original analytic functions in the three phases characterizing the thermoelectric fields are obtained.

关键词： Eccentric coating Circular inhomogeneity Thermoelectric material Nonlinear coupling complex variable method Analytic solution

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A partially debonded circular inhomogeneity in nonlinear thermoelectricity

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CONTINUUM MECHANICS AND THERMODYNAMICS 2023年第1期35卷 267-278页

We study the two-dimensional thermoelectric problem associated with a circular inhomogeneity partially bonded to an infinite matrix subjected to uniform remote electric current density and energy flux. Both the inhomogeneity and the matrix are composed of nonlinearly coupled thermoelectric materials. The four analytic functions characterizing the thermoelectric fields in the two-phase composite are derived rigorously, in closed-form, by solving two Riemann-Hilbert problems with discontinuous coefficients. We obtain elementary expressions for the normal electric current density and normal energy flux along the bonded portion of the circular interface as well as the thermoelectric potential and temperature jumps across the remaining debonded section.

关键词： Thermoelectric material Nonlinear thermoelectricity Circular inhomogeneity Interface crack complex variable method Riemann-Hilbert problem

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