In this work, an elastic-damage evolution analysis is carried out for a cylinder under torsion made of a material obeying a gradient damage model with softening. Both semi-analytical and asymptotic approaches are developed to analyze the elastic, axisymmetric and bifurcation stages. We show the existence of a fundamental branch where the damage field is asymmetric and localized within a finite thickness from the boundary. By minimizing a generalized Rayleigh quotient, the bifurcation time and modes are obtained as a function of the length scale \(\epsilon =\ell /R\) involving a material internal length and the cylinder radius. We will then focus on these size effects by assuming that \(\epsilon \) is a small parameter in an asymptotic setting. After justification, specific spatial and temporal rescaled variables are introduced for the boundary layer problem. It is shown that the axisymmetric damage evolution and the bifurcation are governed by two universal functions independent of the length scale. The simulation results obtained by the semi-analytical approach are formally justified by the asymptotic methods.

在这项工作中，对一种圆柱体进行了弹性损伤演化分析，该圆柱体是由一种材料制成的，该材料在服从梯度损伤模型的同时具有软化特性。半解析和渐近方法都可以用来分析弹性，轴对称和分叉阶段。我们显示了基本分支的存在，在该分支中，损伤场是不对称的，并且位于边界的有限厚度内。通过最小化广义瑞利商，获得分叉时间和众数，这是涉及材料内部长度和圆柱半径的长度比例\（\ε= \ ell / R \）的函数。然后，我们将假定\（\ epsilon \）专注于这些大小效应是渐近设置中的一个小参数。证明合理之后，针对边界层问题引入特定的空间和时间重新定标变量。结果表明，轴对称损伤的演化和分叉受两个独立于长度尺度的通用函数支配。通过渐近方法正式证明了通过半分析方法获得的仿真结果是正确的。