J Ashwin, Gendelman Oleg, Procaccia Itamar, Shor Carmel
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Aug;88(2):022310. doi: 10.1103/PhysRevE.88.022310. Epub 2013 Aug 20.
It is well known experimentally that well-quenched amorphous solids exhibit a plastic instability in the form of a catastrophic shear localization at a well-defined value of the external strain. The instability may develop to a shear band that in some cases is followed by a fracture. It is also known that the values of the yield strain (and yield stress), as well as the direction of the shear band with respect to the principal stress axis, vary considerably with variations in the external loading conditions. In this paper we present a microscopic theory of these phenomena for two-dimensional athermal amorphous solids that are strained quasistatically. We present analytic formulas for the yield strains for different loading conditions, as well as for the angles of the shear bands. We explain that the external loading conditions determine the eigenvalues of the quadrupolar Eshelby inclusions which model the nonaffine displacement field. These inclusions model elementary plastic events and determine both the yield strain and the direction of the shear band. We show that the angles of the shear bands with respect to the principal stress axis are limited theoretically between 30° and 60°. Available experimental data conform to this prediction.
实验上众所周知，充分淬火的非晶态固体在外部应变达到明确值时，会以灾难性剪切局部化的形式表现出塑性不稳定性。这种不稳定性可能发展为剪切带，在某些情况下随后会发生断裂。还已知屈服应变（和屈服应力）的值，以及剪切带相对于主应力轴的方向，会随着外部加载条件的变化而有很大差异。在本文中，我们针对二维无热非晶态固体在准静态应变下的这些现象提出了一种微观理论。我们给出了不同加载条件下屈服应变以及剪切带角度的解析公式。我们解释说，外部加载条件决定了模拟非仿射位移场的四极埃舍尔比夹杂的本征值。这些夹杂模拟基本塑性事件，并决定屈服应变和剪切带的方向。我们表明，剪切带相对于主应力轴的角度理论上限制在30°至60°之间。现有的实验数据符合这一预测。
