The temperature-dependent thermoelastic problem of elliptic inhomogeneity embedded in infinite matrix


Based on the pioneering works of Maxwell, ellipsoidal inhomogeneity embedded in continuum medium has been subsequently analyzed for potential applications in various fields. This analysis problem has been regarded as a classical problem in solids, and its solution has been shown to have important practical applications in the analysis of heterogeneous structures and composite materials. It is worth noting that for fields with ellipsoidal inhomogeneity, the elastic, thermal, magnetic and electric media governed by temperature-independent linear constitutive laws are generally known to be uniform. This property also applies to linear systems like piezoelectric. However, the analysis of ellipsoidal inhomogeneity for the temperature-dependent nonlinear medium is more complicated because the field distributions are not uniform in most cases. In addition, the coupling between different physical phenomena further complicities the nonlinear analysis. A good example of such a nonlinear problem is the temperature-dependent thermoelastic problem with nonlinear heat compatibility and conduction equations.

Currently, numerous studies have focused on developing nanostructured systems and hybrid composites. Nevertheless, the continuum analysis of the temperature-dependent thermoelastic problem is rarely studied despite its practical implications, especially in the development of temperature-dependent composites. This can be partly attributed to the lack of closed-form solutions for the thermoelastic problem of inhomogeneity in the temperature-dependent medium. Therefore, more and thorough studies are still needed to address this problem to enable an effective solution to nonlinear inhomogeneity problems in temperature-dependent media.

On this account, PhD candidate Kunkun Xie, Dr. Haopeng Song and Professor Cunfa Gao from Nanjing University of Aeronautics and Astronautics analyzed the temperature-dependent thermo-elastic problem of an elliptic inhomogeneity embedded in an infinite medium using the generalized complex variable method. Analytical analysis was used to obtain the temperature and thermoelastic fields. The influence of temperature on elastic modulus, thermal conductivity and thermal expansion coefficient were among the critical parameters that were fully accounted for in this study. The main aim of the study was to derive temperature-dependent composite solutions such that the elliptical inhomogeneity can be effectively used to estimate a variety of composite microstructure such as fibers and layered structure by varying the aspect ratio. Their research work is currently published in the International Journal of Engineering Science.

Results revealed that the temperature field distribution changes were more visible than that of the classical temperature-independent results despite the complicated nature of the thermoelastic field expressions. Both numerical and analytical results showed that whereas the thermal flux within the inhomogeneity is generally uniform, the thermal stress within the same inhomogeneity was a quadratic function with respect to the coordinates. Both σx and σy varied nonlinearity at the right inhomogeneity tip with respect to the remote thermal loads  , while σxy remained proportional to remoted thermal flux  . Furthermore, the existence of a specific value b/a responsible for maximizing the shear stress was confirmed based on the relationship between the normal and shear stress at the inhomogeneity tip.

In summary, the successful analysis of the thermoelastic problem of inhomogeneity in the temperature-dependent medium was reported. The analytical and numerical solutions of the thermoplastic fields were detailed. The thermal stress field was made up of two parts: (1) the thermal stress induced by the thermal expansion mismatch between the matrix and the inhomogeneity and (2) the nonlinear constitutive equation caused by the high dependency of the material parameter on the temperature. In a statement to Advances in Engineering, Corresponding author Dr. Haopeng Song explained that the study findings provide a powerful tool for effective analysis of the behaviors of temperature-independent composite for a wide range of applications.

About the author

KunKun Xie is a PhD candidate of solid mechanics in Nanjing university of aeronautics and astronautics(NUAA), China. He received his B.S degree in 2018, then he started his career of PhD under the supervision of Dr. Cunfa Gao and Dr. Haopeng Song. His major research interests are focus on theory of mathematical elasticity, joule thermal stress, heat transfer and temperature-dependent thermoelastic problem, and relative research have been published on some influential journals such as Mathematics and Mechanics of Solids, International Journal of Solids and Structures, International Journal of Engineering Science.

E-mail:[email protected]

About the author

Haopeng Song graduated from the College of Mechanical and Vehicle Engineering of Hunan University in 2010, worked as a visiting scholar at the University of Washington from 2015 to 2016, and is now an associate professor at the College of Aerospace Engineering of Nanjing University of Aeronautics and Astronautics. His main research interests include the analytical solution of elastic-electro-thermal coupling problem, the analytical solution of anisotropic thermoelastic problem, temperature dependent thermoelastic problem and unsteady thermoelastic problem, as well as the local analysis of thermal stress. At present, he has published more than 40 papers and has been invited to review manuscripts for many magazines.

E-mail:[email protected]

About the author

Cunfa Gao graduated from the Department of aircraft of Nanjing University of Aeronautics and Astronautics In 1998, and then engaged in post doctoral research in the Department of mechanics of Peking University. From 2000 to 2005, he successively engaged in scientific research at Dresden University of technology in Germany, Hong Kong University of science and technology and Shizuoka University in Japan. He is now a distinguished professor of Nanjing University of Aeronautics and Astronautics. He has long been engaged in the research of soft material, electromagnetic solid material and mechanical behavior and structural thermal stress. He served as editorial board member of J thermal stresses, guest editor of Acta mechanica special issue and editorial board member of Encyclopedia of thermal stresses. At present, he has published more than 100 papers and cited more than 1000 times.

E-mail:[email protected]


Xie, K., Song, H., & Gao, C. (2021). The temperature-dependent thermoelastic problem of an elliptic inhomogeneity embedded in an infinite matrixInternational Journal of Engineering Science, 166, 103523.

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