An arc-shaped crack in nonlinear fully coupled thermoelectric materials

Yu, Chuanbin; Li, Yu‐Hao; Yang, Haibing; Gao, Cun‐Fa

doi:10.1007/s00707-017-2099-6

Cited by 24 publications

(15 citation statements)

References 35 publications

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“…In the following analysis, the assumption of temperature‐independent elastic constants is adopted. By combining the equilibrium equations, compatibility equations, thermoelastic stress‐strain relationship, one can easily obtain the governing equations in terms of Airy stress function U as

\nabla^{4} U + E λ \nabla^{2} T = 0

. Thus, one has from Equation that

\nabla^{4} U = - E λ \nabla^{2} T = \frac{2 μ λ_{v} δ}{κ} f^{''} false(z false) \bar{f^{''} false(z false)},

where μ is the shear modulus and

\begin{matrix} left E = ({, \begin{matrix} E^{'}, \\ E^{'} / false(1 - {ν^{'}}^{2} false), \end{matrix}) & left λ = ({, \begin{matrix} λ^{'}, \\ false(1 + ν^{'} false) λ^{'}, \end{matrix}) & left λ_{v} = ({, \begin{matrix} false(1 + ν^{'} false) λ^{'}, \\ (1 + ν^{'}) λ^{'} / false(1 - ν^{'} false), \end{matrix}) & left \begin{matrix} \begin{matrix} plane & stress \end{matrix} \\ \begin{matrix} plane & strain \end{matrix} \end{matrix} \end{matrix}

with

E^{'}

ν^{'}

and

λ^{'}

being the Young's modulus, Poisson's ratio and thermal expansion coefficient, respectively.…”

Section: Basic Equationsmentioning

confidence: 99%

“…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%

“…By combining the equilibrium equations, compatibility equations, thermoelastic stressstrain relationship, one can easily obtain the governing equations in terms of Airy stress function as ∇ 4 + ∇ 2 = 0. [28] Thus, one has from Equation (9) that…”

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%

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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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\nabla^{4} U + E λ \nabla^{2} T = 0

. Thus, one has from Equation that

\nabla^{4} U = - E λ \nabla^{2} T = \frac{2 μ λ_{v} δ}{κ} f^{''} false(z false) \bar{f^{''} false(z false)},

where μ is the shear modulus and

\begin{matrix} left E = ({, \begin{matrix} E^{'}, \\ E^{'} / false(1 - {ν^{'}}^{2} false), \end{matrix}) & left λ = ({, \begin{matrix} λ^{'}, \\ false(1 + ν^{'} false) λ^{'}, \end{matrix}) & left λ_{v} = ({, \begin{matrix} false(1 + ν^{'} false) λ^{'}, \\ (1 + ν^{'}) λ^{'} / false(1 - ν^{'} false), \end{matrix}) & left \begin{matrix} \begin{matrix} plane & stress \end{matrix} \\ \begin{matrix} plane & strain \end{matrix} \end{matrix} \end{matrix}

with

E^{'}

ν^{'}

and

λ^{'}

being the Young's modulus, Poisson's ratio and thermal expansion coefficient, respectively.…”

Section: Basic Equationsmentioning

confidence: 99%

“…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%

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

Yang²,

et al. 2019

Z Angew Math Mech

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An Eshelby inclusion in a nonlinearly coupled thermoelectric bi‐material and tri‐material

Wang

Schiavone

2023

Z Angew Math Mech

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We first derive an analytic solution to Eshelby's problem concerning a circular inclusion within one of two perfectly bonded nonlinearly coupled thermoelectric half‐planes. The inclusion undergoes a prescribed uniform electric current‐free thermoelectric potential gradient and a prescribed uniform energy flux‐free temperature gradient. Closed‐form expressions for the six analytic functions characterizing the thermoelectric fields of electric current density and energy flux in all three phases of the composite are obtained primarily with the aid of analytic continuation. A general method is then proposed for the analytic solution of Eshelby's problem of an inclusion of arbitrary shape in a nonlinearly coupled thermoelectric bi‐material. Finally, we extend our ideas to the case of a tri‐material by developing an analytic solution to the thermoelectric problem associated with a circular Eshelby inclusion in a nonlinearly coupled thermoelectric tri‐material composed of two semi‐infinite thermoelectric media bonded together through an intermediate thermoelectric layer of finite thickness.

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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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An arc-shaped crack in nonlinear fully coupled thermoelectric materials

Cited by 24 publications

References 35 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

An Eshelby inclusion in a nonlinearly coupled thermoelectric bi‐material and tri‐material

A three-phase elliptical inhomogeneity in nonlinearly coupled thermoelectric materials

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