Ni Luqun, Markenscoff Xanthippi
Department of Mechanical and Aerospace Engineering , University of California , San Diego, La Jolla, CA 92093-0411, USA.
Proc Math Phys Eng Sci. 2016 Jul;472(2191):20160256. doi: 10.1098/rspa.2016.0256.
The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.
具有变换应变的著名埃舍尔比夹杂的动态推广是从零维开始的（亚声速）自相似膨胀椭球形夹杂。Ni和Markenscoff（即将发表。（doi:10.1016/j.jmps.2016.02.025））最近基于拉东变换得到了偏微分方程组的解，而本文提出了一种替代方法。在自相似膨胀运动中，恒定约束应变的埃舍尔比性质在膨胀椭球的内部区域有效，在该区域粒子速度为零（空洞）。动态埃舍尔比张量以积分形式得到。由此，通过一个极限过程得到静态埃舍尔比张量，当轴的膨胀速度趋于零且时间趋于无穷大时，而它们的乘积等于静态轴的长度。这使得埃舍尔比问题成为其动态推广的极限。
