2维各向异性弹性力学中的Stroh公式和某些不变量

THE STROH FORMALISM AND CERTAIN INVARIANCES IN TWO-DIMESIONAL ANISOTROPIC ELASTICITY

本文简要综述了各向异性弹性体2维变形的Stroh公式。该公式数学上优美而应用上有力。它的最近进展使我们将已有的复数形式的解转变成实数形式,进而可去解决一些迄今尚未解出的问题。结果,发现了许多有趣的而在物理上难以预料的现象。例如,对于界面包含x_3轴的各向异性双介质体,如果线力和线位错都沿x_3轴,那末包含x_3轴的任意径向平面上的面拉力将不随径向平面的选择而变化。而且,在任何固定点上的应力与界面的方位无关。要揭示的其他一些不变性质是:双介质材料中的虚拟力,界面裂纹上的应力奇异,界面裂纹表面上的振荡,等等;另外,在具有一个有限裂纹,或一个椭圆孔,或一个刚性椭圆核的无穷大均匀介质中的某些应力和位移也是不变量。这些不变量的大部分尚未有它们的物理解释。在实数形式的解中,Barnett-Lothe张量常常出现。在绕x_3轴转动时,这些张量的分量的某些组合是不变的。这些组合将在本文中推导并列出。最后我们给出了某些各向异性弹性材料,它们的Barnett-Lothe张量是不变量,即这些张量的所有分量在绕x_3轴转动时是不变的。

The Stroh formalism for two-dimensional deformations of an aniso-tropic elastic body is reviewed briefly. The formalism is mathematically elegant and technically powerful. Recent advances in the formalism allow us to simplify existing complex form solutions to real form ones and to solve heretofore unsolved problems. As a result many interesting and physically unexpected phenomena have been discovered. For instance, for the anisotropic bimaterial with the interface containing the x3-axis subject to a line force and a line dislocation along the x3-axis, the surface traction on any radial plane containing the x3-axis is invariant with respect to the radial planes. Moreover the stresses at any fixed point are independent of the location of the interface.Other invariance properties discovered include the image force in the bi-materials, the stress singularities at the interface crack, the oscillation of the interface crack surface, and certain invariant stresses and displacements in the homogeneous infinite medium with the presence of a finite crack, an elliptic hole, or a rigid elliptic inclusion. Many of these invariance properties defy a physical interpretation. In the real form solutions, the Barnett-Lothe tensors appear frequently. Certain combinations of the components of these tensors are invariant under rotations about the x3-axis. They arc derived and presented here.Finally we identify some anisotropic elastic materials for which the Barnett-Lothe tensors are invariant, i.e., all components of the tensors are invariant under rotations about the x3-axis.