Algorithmic Formulations of Evolutionary Anisotropic Plasticity Models Based on Non-Associated Flow Rule

Abstract In the present paper, orthotropic elasto-plastic constitutive formulations for sheet metal forming based on non-associated flow rule that assume distortion of yield function/plastic potential with ongoing deformation process are analyzed. The yield function/plastic potential are considered as two different functions with functional form as orthotropic quadratic Hill or non-quadratic Karafillis-Boyce stress function. Based on the principle of plastic work equivalence, anisotropy parameters of the utilized yield function/plastic potential are set as functions of the equivalent plastic strain. In the constitutive formulation, for this internal variable, evolution equation consistent with the same principle of plastic work equivalence is introduced. For DC06 sheet sample with reported significant variation of the incremental r-values with straining, predictions of the evolution of the yield stress and r-value directional dependences with straining obtained by the analyzed models are presented. The algorithmic formulations of the analyzed constitutive models are derived by application of the implicit return mapping algorithm. For the derived stress integration procedures the accuracy is investigated by calculating iso-error maps. The maps are compared according to the flow rule and involved orthotropic stress functions. It has been revealed that although there is a difference in maps configuration there is no prominent difference in error magnitudes.摘要：本文针对基于非关联流动法则（non-associated flow rule）的板材成形（sheet metal forming）正交各向异性弹塑性本构公式（orthotropic elasto-plastic constitutive formulations）展开分析，该模型假设屈服函数（yield function）/塑性势（plastic potential）随变形进程发生畸变。本文将屈服函数与塑性势视为两类不同的函数，其函数形式为正交各向异性二次Hill应力函数或非二次Karafillis-Boyce应力函数。基于塑性功等效原理，本文将所采用的屈服函数与塑性势的各向异性参数设定为等效塑性应变（equivalent plastic strain）的函数。在本构公式中，针对该内变量（internal variable），本文引入了与前述塑性功等效原理相一致的演化方程。针对DC06薄板试样（其增量r值随变形过程发生显著变化已有相关报道），本文给出了通过所分析模型得到的屈服应力演化以及r值随变形的各向异性依赖关系的预测结果。本文通过应用隐式回映算法（implicit return mapping algorithm）推导了所分析本构模型的算法公式。本文通过计算等误差图（iso-error maps）对所推导的应力积分过程的精度展开了研究，并依据流动法则与所涉及的正交各向异性应力函数对等误差图进行了对比分析。研究结果表明，尽管等误差图的构型存在差异，但误差幅值并无显著区别。