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 引用本文: 武守信, 魏吉瑞, 杨舒蔚. 基于能量等效原理的应变局部化分析:Ⅰ.一维解析解[J]. 力学学报, 2017, 49(3): 667-676.
Wu Shouxin, Wei Jirui, Yang Shuwei. ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE: Ⅰ. ONE-DIMENSIONAL ANALYTICAL SOLUTION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(3): 667-676.
 Citation: Wu Shouxin, Wei Jirui, Yang Shuwei. ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE: Ⅰ. ONE-DIMENSIONAL ANALYTICAL SOLUTION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(3): 667-676.

## ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE: Ⅰ. ONE-DIMENSIONAL ANALYTICAL SOLUTION

• 摘要: 基于热力学第一定律和非局部塑性理论，提出了一种求解应变局部化问题的非局部方法。对材料的每一点定义了局部和非局部两种状态空间，局部状态空间的内变量通过非局部权函数映射到非局部空间，成为非局部内变量。在应变软化过程中，局部状态空间中的塑性变形服从正交流动法则，材料的软化律在非局部状态空间中被引入。通过两个状态空间的塑性应变能耗散率的等效，得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布。应用本方法导出了一维应变局部化问题的解析解。解析解表明，应变局部化区域的尺寸只与材料内尺度有关；对于高斯型非局部权函数，局部化区域的尺寸大约是材料内尺度的6倍。一维算例表明，局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关，变形局部化区域的尺寸随着材料内尺度的减小而减小，同时塑性应变也随着材料内尺度的减小变得更加集中。当内尺度趋近于零时，应用本文方法得到的解与采用传统的局部塑性理论得到的解相同。

Abstract: Based on the first law of thermodynamics and the nonlocal plasticity theories, a new approach is proposed to solve the strain localization problems induced by strain softening. For each material point, two state spaces, local and nonlocal state spaces, are defined such that the local internal variable can be mapped, from the local state space by integral transformation with the nonlocal weighting function, into the nonlocal internal variable in the nonlocal state space. During strain softening, the plastic deformation follows the normal flow rule in the local state space and the softening law is introduced in the nonlocal state space. It is assumed that the strain softening is a global material behavior and the plastic energy dissipation within the entire material body is always positive. However, the balance of momentum is still satisfied locally. By equating the rates of the plastic energy dissipation in the two state spaces during strain softening, the localization zone and the plastic strain distribution become well-defined. Analytical solution for the one-dimensional strain localization is developed, and it is well shown that the plastic strain distribution and load-displacement curves are well-defined by the material properties, such as the softening modulus and internal length scale, as well as the geometry of the material body. For the Gaussian-type weighting functions the width of the localization zone is approximately six times the internal length scale. Numerical example demonstrates that the size of the localization zone decreases as the internal length scale is reduced, and the distribution of the plastic strain in the localization zone becomes more concentrated when the internal length scale becomes smaller. As the internal length scale approaches to zero, the solution reduces to the one predicted by the conventional local plasticity theory.

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