The elastic fields of an infinite plane with an inhomogeneous inclusion due to the non-uniform distribution of temperature on the inclusion is studied, where the inclusion is shaped by the hypocycloid curve and has different properties from those of the matrix, except for the same elastic shear modulus. This is called the inhomogeneous thermoelastic problem. By means of the knowledge of complex variable functions, the closed-form solution for the general case of inhonmogeneous thermoelastic problems is firstly solved. Then through the Riemann conformal mapping, the exterior of the inclusion is mapped onto the exterior of the unit circle. Furthermore, in virtue of the features of analytical functions, and combining the Cauchy-type integrals with the Faber polynomials, we obtain the explicit analytical formulae of the K-M potentials inside and outside the inclusion with the region of inclusion affected by the temperature of a polynomial distribution. The stress fields are calculated from the potentials and illustrated in the cases for different polynomial distributions of temperature. It is found that the internal stress field is in good agreement with the finite element results, and the same as the reported solutions in the literature when the inclusion is elliptical. On top of that, the new formulae are of more generality and applicability.

Pipes conveying fluid are widely applied in heat exchanger systems, nuclear power plants, chemical process plants, marine risers, etc. However, the excessive piping vibration can cause leaks, fatigue failures and noises. Thus, investigations on the vibration suppression of pipes are of theoretical and practical significance. In this study, we construct a theoretical model to investigate the nonlinear dynamics of a simply-supported pipe conveying pulsating fluid equipped with a nonlinear energy sink (NES). By taking the deflection-dependent axial force into consideration, the nonlinear governing equations of the system are obtained. Based on the Galerkin method and the Runge-Kutta algorithm, the resulting equations are discretized and solved. Numerical results for the nonlinear dynamical responses of the pipes with and without NES are presented. It is found that pipe vibration can be effectively suppressed by the NES. Comparing with the pipe without NES under the same condition, the stability and nonlinear vibration characteristics of the pipe are greatly affected when the NES is attached. The effects of NES parameters on the stability and vibration response of the system are elaborately addressed. Numerical results show that an increase in the nonlinear (cubic) stiffness k, dissipation s or mass ratio e can improve the suppression of pipe vibration; and the improvement in the suppression of pipe vibration by increasing dissipation s is more significant than those by increasing other NES parameters. It shows that the best mounting position for the NES to reduce pipe vibration is at the midpoint of the pipe. In addition, it is found that an increase in dissipation s can shrink the unstable region in the frequency domain, while other NES parameters have little effects on the instability. Therefore, dissipation s is the most effective parameter for the nonlinear energy sink to control the vibration of pipes conveying fluid.