Fracture of thermoelectric materials: An electrical and thermal strip saturation model
Introduction
Thermoelectric materials have the ability to directly convert heat into electricity and vice versa [1]. With continuous efforts in enhancing the thermoelectric materials’ performance in the past two decades [2], [3], [4], [5], the current maximum efficiency of a thermoelectric device reaches up to 16% [6]. Thus, thermoelectric materials have been broadly applied in waste heat recovery [7], cooling systems [8], self-powered electronic devices [9], etc. Thermoelectric materials are usually brittle solids, the defects like voids, cracks or holes are easy to be produced in the manufacturing processes or in-service conditions. The presence of these defects are reported to be the source of thermoelectric field concentration and stress concentration [10], reducing the material’s thermoelecric performance and even leading to the structural failure [11]. The widespread applications call for a better understanding of the mechanism of thermoelectric materials fracture.
Unlike the traditional thermoelastic or piezoelectric medium, the governing equations for the thermal and electrical transportation in thermoelectric materials are highly nonlinear, which makes the crack problem challenging to be analytically solved. By introducing an auxiliary function, Zhang and Wang [12] are able to decoupled the electric field from the non-linearly coupled thermoelectric system, and give the solution of a classical crack problem for thermoelectric materials. The solution framework is based on the complex variable method developed by Muskhelishvili [13], which is also popular in solving elastic and piezoelectric crack problems [14], [15], [16], [17], [18]. For example, Kuna [19] made a comprehensive review of using the complex method solving the piezoelectric crack problems. Wang [20] investigated the problem of a fiber inclusion in thermoelectric materials and derived the effective thermoelectric properties. In addition, the crack problem with different crack permeability is discussed by Song et al. [21], and Song and Song [22], they found that impermeable cracks lead to higher thermoelectric concentration at the crack tip. Wang and Wang [10] solved the inclined crack problem and found that the crack tip concentration is maximum (minimum) when the crack surface is vertical (parallel) to the current and heat flow direction. Wang et al., [23], [24] solved the inclusion problem and further discussed the influence of inclusion on effective thermoelectric properties.
Although significant progress regarding the fracture of thermoelectric materials has been made, there are still some unexplained discrepancies between theory and experiments. From the theoretical analysis, we find that the electric current and heat show singularity at the crack tip, which means the electric current and heat is infinite at the crack tip [12]. However, it is impossible that the electric current and heat flux can reach infinity under any circumstances since infinite electric current and heat flux cannot be sustained at the atomic level. In other words, the thermoelectric field concentration at the crack tip must be bounded in the view of experimental observation. An alternative theory to fill the discrepancy between theory and experiment is to consider the effects of electrical and thermal nonlinearity at the crack tip. Due to the impermeability of the crack, the electric current and heat will concentrate at the crack tip area as shown in Fig. 1(a). Furthermore, the concentrated heat flux at the crack tip may lead to thermal stress concentration and cause potential threat to the material strength.
The fracture is a multiscale physical process, in which we assume there is an intermediate length scale that the electrical and thermal field is yielded while the stress field remains singular as shown in Fig. 1(b). In the electrically and thermally yielded area, the thermal stresses still remain singular. Gao et al. [25] and Fulton and Gao [26] analyzed the isotropic piezoelectric fracture based on the electric nonlinearity model and give field predictions in broad agreement with experimental observations. This model is first proposed by Dugdale [27] in analyzing the elastic fracture, then extended by Gao et al. [25] to piezoelectric fracture. The full anisotropic piezoelectric fracture based on electric saturation model is later presented by Wang [28]. Other piezoelectric fracture problems based on the electrical saturation model can be found such as Ru [29], and Wang and Zhang [30], [31]. Generally, the piezoelectric fracture model considering electrical saturation gives better agreement with the experimental observation compared to the traditional linear fracture theory. This inspires us to present an electrical and thermal saturation model to give better agreement with the experiment observation for thermoelectric fracture.
In this paper, we propose a strip saturation model in which electrical current density and thermal flux have a saturation limit within the line segment extended from the crack tip. Since there are electric current and heat coupled in the thermoelectric transportation system, the thermal and electrical field should both show nonlinearity. The plan of this paper is as follows. Section 2 gives the basic thermoelectric equations and the fundamental solutions for the crack problem. Section 3 presents the classical solution of the thermoelectric crack problem and Section 4 gives the solution based on the strip saturation model. Some numerical results and discussions are given in Section 5 and the concluding remarks are presented in Section 6.
Section snippets
Description of the problem
Consider an infinity thermoelectric medium containing an inner crack (half-crack length is a) with an arbitrary inclination angle as shown in Fig. 2. Under the in-service conditions, there are thermoelectric loads (electric current and heat) applied at infinity along Y-direction. To simplify the solveing process, a global coordinate XOY and a local coordinate xOy are established, where
Basically, the crack problem is difficult to be directly solved in the global
The classical model
In this section, the classic model will be reviewed as a direct comparison to the saturation model. However, the solving process is more simplified due to the simplification to the nonlinear governing equations. To find the proper functions satisfying Eqs. (15), (16), we introduce a new analytical functionwhich has the following properties a) , at ; b) , at and . By using this function, the crack boundary conditions can be easily satisfied, and
The electrical and thermal saturation model
It is well-known that the field concentrations often occur at the crack tip, and the crack propagates along a strip ahead of the crack tip. Therefore, for the practicality and mathematical simplicity, we confine the saturation area to be a strip ahead of the crack tip to find the analytical solutions. We assume that the material is thermally and electrically yielded along a strip as shown in Fig. 4. The electric current and heat rise in the strip, but will never exceed a saturation limit
Numerical results and discussions
The numerical results in this section are based on the widely applied commercial thermoelectric material , whose material properties are given in Table 1 [23].
Concluding remarks
This paper proposes a thermal and electrical saturation model for the analysis of the crack problem in thermoelectric materials. The electric current and heat flux is assumed to be within a saturation limit along the strip at crack front. The analytical solutions of the thermoelectric field and thermal stress field are derived based on the complex variable method. The stress intensity factors are given for both the classical model and the saturation model. It is found that the thermoelectric
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by Research Innovation Fund of Shenzhen City of China (project No. JCYJ20170811160538023), the National Natural Science Foundation of China (project Nos. 11602072).
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