Abstract
We present a rigorous treatment to the problem of a circular-arc crack in an infinite thermoelectric solid subjected to a combined electrical and thermal loading. Formulating the problem in terms of the complex potentials and reducing it to the Hilbert arc problem, the solutions of quantities in both thermoelectric field and the associated thermoelastic field are presented in a closed form based on the electrically permeable and thermally insulated crack model. The results show that the fields of heat flux, energy flux, and stress exhibit the traditional square-root singularity at tips of arc crack. The applied electric current and energy flux generate both mode I and mode II stress intensity factors (SIFs), which are dependent on the loading direction, electric conductivity, thermal conductivity, central angle of the crack, and thermoelastic isotropy. Electrically induced SIF is a quadratic function of applied electric current density, and thermally induced SIF is a linear function of the imposed total energy flux. In addition, the effects of the loading direction and half crack angle on the thermal SIFs are also presented in a graphic form. This is the first theoretical paper to study the effect of the crack shape on the fracture of fully coupled thermoelectric materials by a rigorous inference of physics and mathematics.
Similar content being viewed by others
References
Rowe, D.M.: Thermoelectrics Handbook: Macro to Nano. CRC Press, New York (2005)
Snyder, G.J., Toberer, E.S.: Complex thermoelectric materials. Nat. Mater. 7(2), 105–114 (2008)
Disalvo, F.J.: Thermoelectric cooling and power generation. Science 285(5428), 703–706 (1999)
Riffat, S.B., Ma, X.L.: Thermoelectrics: a review of present and potential applications. Appl. Therm. Eng. 23(8), 913–935 (2003)
Tritt, T.M., Subramanian, M.A.: Thermoelectric materials, phenomena, and applications: a bird’s eye view. Mater. Res. Soc. 31(3), 188–194 (2006)
Bell, L.E.: Cooling, heating, generating power, and recovering waste heat with thermoelectric systems. Science 321(5895), 1457–1461 (2008)
Hicks, L.D., Dresselhaus, M.S.: Effect of quantum-well structures on the thermoelectric figure of merit. Phys. Rev. B 47(19), 12727 (1993)
Venkatasubramanian, R., Siivola, E., Colpitts, T., et al.: Thin-film thermoelectric devices with high room-temperature figures of merit. Nature 413(6856), 597–602 (2001)
Poudel, B., Hao, Q., Ma, Y., et al.: High-thermoelectric performance of nanostructured bismuth antimony telluride bulk alloys. Science 320(5876), 634–638 (2008)
Hochbaum, A.I., Chen, R., Delgado, R.D., et al.: Enhanced thermoelectric performance of rough silicon nanowires. Nature 451(7175), 163–167 (2008)
Schmidt, R.D., Fan, X.F., Case, E.D., Sarac, P.B.: Mechanical properties of \(\text{ Mg }_{2}\text{ Si }\) thermoelectric materials with the addition of 0–4 vol% silicon carbide nanoparticles (SiCNP). J. Mater. Sci. 50(11), 4034–4046 (2015)
Zhao, D.G., Tian, C.W., Liu, Y.T., Zhan, C.W., Chen, L.D.: High temperature sublimation behavior of antimony in \(\text{ CoSb }_{3}\) thermoelectric material during thermal duration test. J. Alloys Compd. 509(6), 3166–3171 (2011)
Isoda, Y., Shinohara, Y., Imai, Y., Albart, N.I., Ohashi, O.: Thermal shock resistance and thermoelectric properties of boron doped iron disilicides. J. Jpn. Inst. Metall. 63(3), 391–396 (1999)
Schmidt, R.D., Case, E.D., Giles, J., Ni, J.E., Hogan, T.P.: Room-temperature mechanical properties and slow crack growth behavior of Mg\(_{2}\)Si thermoelectric materials. J. Electron. Mater. 41(6), 1210–1216 (2012)
Cui, J.L., Zhao, X.B., Zhao, W.M., Lu, Y.P.: Preparation, thermoelectric properties and interface analysis of n-type graded material \(\text{ FeSi }_{2}/\text{ Bi }_{2}\text{ Te }_{3}\). Mater. Sci. Eng. B Adv. 94(2), 223–228 (2002)
Zhao, D.G., Geng, H.R., Teng, X.Y.: Fabrication and reliability evaluation of \(\text{ CoSb }_{3}\)/W-Cu thermoelectric element. J. Alloys Compd. 517(7), 198–203 (2012)
Zhang, A.B., Wang, B.L.: Crack tip field in thermoelectric media. Theor. Appl. Fract. Mech. 66(16), 33–36 (2013)
Song, H.P., Gao, C.F., Li, J.Y.: Two-dimensional problem of a crack in thermoelectric materials. Therm. Stress 38(3), 325–337 (2015)
Zhang, A.B., Wang, B.L.: Temperature and electric potential fields of an interface crack in a layered thermoelectric or metal/thermoelectric material. Int. J. Therm. Sci. 104, 396–403 (2016)
Zhang, A.B., Wang, B.L.: Explicit solutions of an elliptic hole or a crack problem in thermoelectric materials. Eng. Fract. Mech. 151, 11–21 (2016)
Sevostianov, I., Kachanov, M.: On the elastic compliances of irregularly shaped cracks. Int. J. Fract. 114(3), 245–257 (2002)
Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Berlin (2013)
Sih, G.C., Paris, P.C., Erdogan, F.: Crack-tip, stress-intensity factors for plane extension and plate bending problems. J. Appl. Mech. 29(2), 306–312 (1962)
England, A.H.: Complex Variable Methods in Elasticity. Courier Corporation, New York (2003)
Ioakimidis, N.I., Theocaris, P.S.: A system of curvilinear cracks in an isotropic elastic half-plane. Int. J. Fract. 15(4), 299–309 (1979)
Datsyshin, A.P., Marchenko, G.P.: Interaction of curvilinear cracks with the boundary of an elastic half-plane. Mater. Sci. 20(5), 466–473 (1985)
Cheung, Y.K., Woo, C.W., Wang, Y.H.: Stress intensity factors for a circular arc crack by boundary collocation method. Eng. Fract. Mech. 34(4), 841–849 (1989)
Shiah, Y.C., Lin, Y.J.: An infinitely large plate weakened by a circular-arc crack subjected to partially distributed loads. J. Eng. Math. 49(1), 1–18 (2004)
Zhong, Z., Meguid, S.A.: Analysis of a circular arc-crack in piezoelectric materials. Int. J. Fract. 84(2), 143–158 (1997)
Zheng, M.M., Gao, C.F.: An arc-shaped crack in an electrostrictive material. Int. J. Eng. Sci. 48(9), 771–782 (2010)
Lin, C.B., Chen, S.C., Lee, J.L.: Explicit solutions of magnetoelastic fields in a soft ferromagnetic solid with curvilinear cracks. Eng. Fract. Mech. 76(12), 1846–1865 (2009)
Goupil, C., Seifert, W., Zabrocki, K., Muller, E., Snyder, G.J.: Thermodynamics of thermoelectric phenomena and applications. Entropy 13(8), 1481–1517 (2011)
Yang, Y., Ma, F.Y., Lei, C.H., Liu, Y.Y., Li, J.Y.: Nonlinear asymptotic homogenization and the effective behavior of layered thermoelectric composites. J. Mech. Phys. Solids 61(8), 1768–1783 (2013)
Liu, L.: A continuum theory of thermoelectric bodies and effective properties of thermoelectric composites. Int. J. Eng. Sci. 55(4), 35–53 (2012)
Zhang, A.B., Wang, B.L., Wang, J., Du, J.K., Xie, C.: Effect of cracking on the thermoelectric conversion efficiency of thermoelectric materials. J. Appl. Phys. 121(4), 045105 (2017)
Hao, F., Fang, D.N., Li, J.Y.: Thermoelectric transport in heterogeneous medium: the role of thermal boundary resistance. Eur. Phys. J. Appl. Phys. 58, 443–443 (2012)
Chao, C.K., Shen, M.H.: Thermal problem of curvilinear cracks in bonded dissimilar materials. J. Appl. Phys. 73(11), 7129–7137 (1993)
Chung, H.D., Beom, H.G., Choi, S.Y., Earmme, Y.Y.: Thermoelastic analysis of circular arc-shaped cracks. J. Therm. Stress. 21(2), 129–140 (1998)
Zhang, X.D., Du, Q.G., Jiang, X.Q., Gao, J.L.: Thermal stress and structural parameter selection of \(\text{ Bi }_{2}\text{ Te }_{3}\)-based thermoelectric modules. Chin. J. Power Sour. 35(4), 422–425 (2011)
Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw Hill, New York (1951)
Tokovyy, Y., Ma, C.C.: Analytical solutions to the 2D elasticity and thermoelasticity problems for inhomogeneous planes and half-planes. Arch. Appl. Mech. 79(5), 441–456 (2009)
Parkus, H.: Thermoelasticity. Springer, Berlin (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yu, C., Zou, D., Li, YH. et al. An arc-shaped crack in nonlinear fully coupled thermoelectric materials. Acta Mech 229, 1989–2008 (2018). https://doi.org/10.1007/s00707-017-2099-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-2099-6