Two-dimensional problem of thermoelectric materials with an elliptic hole or a rigid inclusion

Zhang, Aibing; Wang, Baolin; Ji, Wang; Du, Jianke

doi:10.1016/j.ijthermalsci.2017.03.020

Cited by 28 publications

(8 citation statements)

References 49 publications

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“…Introducing a new function

ψ false(z false) = χ^{'} false(z false)

and using the following superposition principle

\begin{matrix} right \begin{matrix} \begin{matrix}  \end{matrix} & 0true σ_{y} + σ_{x} = 4 \frac{\partial^{2} U}{\partial z \partial \bar{z}} = 4 \frac{\partial^{2} U_{p}}{\partial z \partial \bar{z}} + 4 \frac{\partial^{2} U_{h}}{\partial z \partial \bar{z}}, \end{matrix} \\ right σ_{y} - σ_{x} + 2 i τ_{x y} = 4 \frac{\partial^{2} U}{\partial z^{2}} = 4 \frac{\partial^{2} U_{p}}{\partial z^{2}} + 4 \frac{\partial^{2} U_{h}}{\partial z^{2}}, \end{matrix}

the components of the total stress can be expressed as Refs. []

\begin{matrix} right \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} σ_{y} + σ_{x} = \frac{μ λ_{v} δ}{2 κ} f^{'} false(z false) \bar{f^{'} false(z false)} + 2 [ϕ^{'} false(z false) + \bar{ϕ^{'} false(z false)}], \\ right σ_{y} - σ_{x} + 2 i τ_{x y} = \frac{μ λ_{v} δ}{2 κ} f^{''} false(z false) \bar{f (z)} + 2 [\bar{z} ϕ^{'' ...}] \end{matrix}

…”

Section: Basic Equationsmentioning

confidence: 99%

“…Additionally, the components of displacements and resultant forces

(F_{x}, F_{y})

on a certain directed curve s from any point A to B can be expressed by Refs. []

2 μ false(u_{x} + i u_{y} false) = β ϕ false(z false) - z \bar{ϕ^{'} false(z false)} - \bar{ψ (z)} - \frac{μ λ_{v} δ}{4 κ} \bar{f^{'} false(z false)} f false(z false) + 2 μ λ g false(z false),

i \int_{A}^{B} false(F_{x} + i F_{y} false) d s = {(|, [ϕ false(z false) + z \bar{ϕ^{'} false(z false)} + \bar{ψ (z)} + \frac{μ λ_{v} δ}{4 κ} \bar{f^{'} false(z false)} f false(z false)])}_{A}^{B},

where

g false(z false) = \int g^{'} (z) d z

and

β = \{\begin{matrix} ((), 3 - ν^{'}) / ((), 1 + ν^{'}) & false(plane stress false), \\ 3 - 4 ν^{'} & false(<...> \end{matrix}

…”

Section: Basic Equationsmentioning

confidence: 99%

“…F I G U R E 1 A bonded circular inhomogeneity subjected to a uniform electric current and energy flux at infinity Additionally, the components of displacements and resultant forces ( , ) on a certain directed curve from any point to can be expressed by Refs. [27,28]…”

Section: General Solutions Of Stress Fieldmentioning

confidence: 99%

“…However, to the best of our knowledge, study of the inhomogeneity problem in thermoelectric materials has yet not been reached due to the complicated nonlinear governing equations, only some investigations about the hole/crack problem or electrically insulated inclusion problem are published. [24][25][26][27][28][29] It is thus highly desirable if closed-form solutions for thermoelectric inhomogeneity can be derived.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Closed‐form solutions for a circular inhomogeneity in nonlinearly coupled thermoelectric materials

Yang²,

et al. 2019

Z Angew Math Mech

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Thermoelectricity is a class of material possessing the ability of interconverting heat and electricity. For the sake of high conversion efficiency, inclusions/fibers are usually introduced into thermoelectric materials. However, the discontinuity of material property and geometry brought by inclusions/fibers might result in severe stress concentration and thus greatly affect the mechanical properties of thermoelectric material in its engineering service. This paper introduces a nonlinear coupled thermal–electrical–mechanical formulation for the plane problem of a circular inhomogeneity embedded in a thermoelectric medium under a combined uniform electric current density and energy flux. A special attention is directed towards the induced thermal stress field which has often neglected in the literature. Using methods based on Cauchy integrals, the complex potentials of the electric field, temperature field and associated stress field are derived explicitly in both the inhomogeneity and the surrounding matrix. The analysis shows that the electrically and thermally induced stress has a linear relationship with the energy flux applied at infinity, but has a nonlinear relationship with the remote electric current density. Numerical simulations are also performed to examine how the inhomogeneity influences the electrically induced thermal stress concentration on the interface. The presented results can be directly used for reliability consideration in design and optimization of thermoelectric composites with circular fibers/inclusions.

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“…Introducing a new function

ψ false(z false) = χ^{'} false(z false)

and using the following superposition principle

\begin{matrix} right \begin{matrix} \begin{matrix}  \end{matrix} & 0true σ_{y} + σ_{x} = 4 \frac{\partial^{2} U}{\partial z \partial \bar{z}} = 4 \frac{\partial^{2} U_{p}}{\partial z \partial \bar{z}} + 4 \frac{\partial^{2} U_{h}}{\partial z \partial \bar{z}}, \end{matrix} \\ right σ_{y} - σ_{x} + 2 i τ_{x y} = 4 \frac{\partial^{2} U}{\partial z^{2}} = 4 \frac{\partial^{2} U_{p}}{\partial z^{2}} + 4 \frac{\partial^{2} U_{h}}{\partial z^{2}}, \end{matrix}

the components of the total stress can be expressed as Refs. []

\begin{matrix} right \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} σ_{y} + σ_{x} = \frac{μ λ_{v} δ}{2 κ} f^{'} false(z false) \bar{f^{'} false(z false)} + 2 [ϕ^{'} false(z false) + \bar{ϕ^{'} false(z false)}], \\ right σ_{y} - σ_{x} + 2 i τ_{x y} = \frac{μ λ_{v} δ}{2 κ} f^{''} false(z false) \bar{f (z)} + 2 [\bar{z} ϕ^{'' ...}] \end{matrix}

…”

Section: Basic Equationsmentioning

confidence: 99%

“…Additionally, the components of displacements and resultant forces

(F_{x}, F_{y})

on a certain directed curve s from any point A to B can be expressed by Refs. []

2 μ false(u_{x} + i u_{y} false) = β ϕ false(z false) - z \bar{ϕ^{'} false(z false)} - \bar{ψ (z)} - \frac{μ λ_{v} δ}{4 κ} \bar{f^{'} false(z false)} f false(z false) + 2 μ λ g false(z false),

i \int_{A}^{B} false(F_{x} + i F_{y} false) d s = {(|, [ϕ false(z false) + z \bar{ϕ^{'} false(z false)} + \bar{ψ (z)} + \frac{μ λ_{v} δ}{4 κ} \bar{f^{'} false(z false)} f false(z false)])}_{A}^{B},

where

g false(z false) = \int g^{'} (z) d z

and

β = \{\begin{matrix} ((), 3 - ν^{'}) / ((), 1 + ν^{'}) & false(plane stress false), \\ 3 - 4 ν^{'} & false(<...> \end{matrix}

…”

Section: Basic Equationsmentioning

confidence: 99%

Section: General Solutions Of Stress Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Closed‐form solutions for a circular inhomogeneity in nonlinearly coupled thermoelectric materials

Yang²,

et al. 2019

Z Angew Math Mech

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show abstract

“…However, discontinuous temperature distribution and severe stress concentration may be induced around the interface in the thermoelectric composites [11, 12]. The mismatch of thermoelastic properties between different components will cause serious interfacial stress concentration, for example, a flexible thermoelectric matrix containing a rigid inclusion [13]. It is worth pointing out that flexible thermoelectric materials often have low mechanical strength and fracture toughness [14, 15].…”

Section: Introductionmentioning

confidence: 99%

Analysis of a hollow fiber in thermoelectric materials considering interfacial thermal resistance

Xing

Gao

2021

Z Angew Math Mech

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Hollow fibers are commonly introduced into flexible thermoelectric materials for certain engineering applications. However, significant stress concentration may be caused in the vicinity of the interface between the fibers and thermoelectric matrix threatening the reliability of the composite structure. In this paper, we study the plane deformation problem of a rigid hollow fiber embedded in a thermoelectric material. By utilizing the complex variable method, the closed‐form solutions for electric, thermal and elastic fields in both of the fiber and the matrix are obtained. The effects of the thermal resistance and geometric parameters of the hollow fiber on the temperature and stress distributions are numerically examined. The results show that the temperature difference (amplitude of variation in temperature in the matrix side around the interface) and the stress field around the interface are proportion to the inner radius of the hollow fiber and the thermal resistance, respectively. Meanwhile, we find that the temperature and the stress distributions are more sensitive to the geometric parameters than to the thermal resistance. The results in this paper may serve as guidelines for design of fibers‐filled thermoelectric structures.

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A three-phase elliptical inhomogeneity in nonlinearly coupled thermoelectric materials

Wang

Schiavone

2022

Z. Angew. Math. Phys.

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Two-dimensional problem of thermoelectric materials with an elliptic hole or a rigid inclusion

Cited by 28 publications

References 49 publications

Closed‐form solutions for a circular inhomogeneity in nonlinearly coupled thermoelectric materials

Closed‐form solutions for a circular inhomogeneity in nonlinearly coupled thermoelectric materials

Analysis of a hollow fiber in thermoelectric materials considering interfacial thermal resistance

A three-phase elliptical inhomogeneity in nonlinearly coupled thermoelectric materials

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