基于能量等效原理的应变局部化分析:Ⅱ.有限元解法

以非局部塑性理论为基础,应用状态空间理论,通过局部和非局部两个状态空间的塑性能量耗散率等效原理,提出了一种求解应变局部化问题的新方法,以得到与网格无关的数值解.针对二维问题的屈服函数和流动法则导出了求解非局部内变量的一般方程,并提出了在有限元环境中求解应变局部化问题的应力更新算法.为了验证所提出的方法,对1个一维拉杆和3个二维平面应变加载试件进行了有限元分析.数值结果表明,塑性应变的分布和载荷-位移曲线都随着网格的变小而稳定地收敛,应变局部化区域的尺寸只与材料内尺度有关,而对有限元网格的大小不敏感.对于一维问题,当有限元网格尺寸减小时,数值解收敛于解析解.对于二维剪切带局部化问题,数值解随着网格尺寸的减小而稳定地向唯一解收敛.当网格尺寸减小时,剪切带的宽度和方向基本上没有变化.而且得到的塑性应变分布和网格变形是平滑的.这说明,所提方法可以克服经典连续介质力学模型导致的网格相关性问题,从而获得具有物理意义的客观解.此模型只需要单元之间的位移插值函数具有C~0连续性,因而容易在现有的有限元程序中实现而无需对程序作大的修改.

ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE: Ⅱ. FINITE ELEMENT SOLUTION

Founded on the nonlocal plasticity and the state space theories, a new approach is proposed to find the mesh-independent solution of the strain localization problems by equating the rates of plastic energy dissipation in the local and nonlocal state spaces. Following the previous paper by the authors, general formulas are developed for the solution of the nonlocal internal variables in the two- and more than two-dimensional problems. A stress updating algorithm is proposed to integrate the rate form constitutive equations in the finite element context. To verify the proposed approach, a one-dimensional model problem and three two-dimensional plane strain problems are solved numerically by the finite element method. Numerical results show that the plastic strain distributions and the load-displacement curves stably converge with refinement of the finite element mesh. The size of the localization zone depends only on the internal length scale and is insensitive to the mesh size. For the one-dimensional problem, numerical solutions converge to the analytical ones. For the two-dimensional problems, although no analytical solutions are available, the numerical solutions converge toward the unique ones. The width and the inclination are almost not changed as the mesh size is reduced. Also, the distribution of the plastic strains and the deformation patterns are smooth in the entire domain. A slope stability problem and a plane strain test of a coal specimen are also solved numerically to demonstrate the robustness of the proposed approach. It is well shown that the proposed approach can overcome the drawbacks of the classical continuum theory and lead to physically meaningful, mesh-independent solution of strain softening problems. Because only C0 continuity is needed between element boundaries, the proposed approach is easy to be incorporated into the existing finite element code without substantial modification.