Modeling the Bauschinger effect is usually accomplished by introducing a second-order back-stress or directional hardening tensor. The objective of this paper is to propose a simpler scalar model of the Bauschinger effect based on a scalar directional hardening parameter that is determined by integration of an evolution equation. The behavior of this scalar model is compared to a tensorial model for a number of load cases. Strongly objective numerical algorithms are developed for integrating the evolution equations for both the tensorial and scalar models. Also, a consistent tangent is developed for both models. Obviously, the numerical implementation of the scalar model is significantly less complicated than for the tensorial model. Examples show that the tensorial and scalar models predict the same results for cyclic proportional triaxial extension and triaxial compression loadings. In contrast, the tensorial model predicts a Bauschinger effect for cyclic proportional pure torsion loading which is not predicted by the scalar model. More complicated examples with nonproportional loading paths and inhomogeneous deformations indicate that, relative to the tensorial model, the scalar model accounts for directional hardening fairly well and the simplicity of the model makes it an attractive option to add to isotropic hardening models.

通常，通过引入二阶背应力或定向硬化张量来完成对鲍辛格效应的建模。本文的目的是基于由演化方程积分确定的标量定向硬化参数，提出一种更简单的包辛格效应标量模型。对于许多情况，将此标量模型的行为与张量模型进行比较。开发了用于整合张量和标量模型的演化方程的强目标数值算法。同样，为这两个模型开发了一致的切线。显然，与张量模型相比，标量模型的数值实现要简单得多。实例表明，张量和标量模型对于循环比例三轴扩展和三轴压缩载荷预测了相同的结果。相反，张量模型预测标量模型未预测的周期性比例纯扭力载荷的鲍辛格效应。具有非比例加载路径和不均匀变形的更复杂示例表明，相对于张量模型，标量模型可以很好地说明方向硬化，并且模型的简单性使其成为添加到各向同性硬化模型的有吸引力的选择。