圆内旋轮线形状异质夹杂热弹性问题的解析解

论文研究具有圆内旋轮线型形状夹杂域的平面热弹性体在夹杂域内非均匀温度场作用下对弹性场所产生的影响，其中考虑的夹杂与基体的材料不同但是具有相同的剪切模量。借助黎曼映射理论，将平面光滑闭合曲线外部区域映射到单位圆外部区域，进而利用解析函数性质，结合柯西型积分与Faber多项式，求解得到夹杂域内外场势函数的显式解。通过得到的势函数求出内外场应力，并对应力分布进行分析，结果表明：一般形状异质夹杂时，内场应力值与有限元计算值相吻合；退化到椭圆同质夹杂时，与相关文献中的结果相同，但是更具一般性与实际可操作性。

Analytical Solution for the Thermaoelasticity Problem of an Inhomogeneous Inclusion with the Hypotrochoidal Shape in the Infinite Plane

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.