免映射數值積分法於混合加載異向性非線性硬化彈塑性模型之發展及其在金屬材料和生物力學中的應用

Abstract

混合控制問題，為彈塑性模式在部分應力與部分應變已知，求解對應（其餘）部分的應力和應變響應之數學問題，如平面應力問題就是一個常見的混合控制問題。實際上，大多數「應變控制」實驗，都屬於混合控制問題：例如單軸應變控制試驗、軸扭應變控制試驗和雙軸應變控制試驗。處理混合控制問題時，基於純應力或純應變控制的塑性數值積分法是無法滿足混合控制的已知給定條件，因此需要額外的處理方法。在本研究中，我們針對異向性材料彈塑性模型，引入了Armstrong-Frederick和Chaboche硬軟化規則，探討在混合控制條件下，彈塑性模型中微分方程組和降伏面之間的內在對稱性，並透過利用李代數(Lie algebra)和李群(Lie group)的特性來更新未知應力與應變，而所獲的應力可自動滿足降伏條件，不需進一步迭代，也不需額外的處理方法來滿足混合控制的已知給定條件。此數值積分方法，我們稱為免映射(Return-free)法。再者，針對依據李代數和李群理論發展出的無需預測/修正的數值積分方法，我們進行了誤差分析，確認免映射法在不同初始條件下的精度。接著，我們利用免映射法研究了異性向材料的收縮比和塑性應變比，也與模擬密質骨與鬆質骨在不同加載路徑的力學行為。最後，本研究發展出免映射法在商用有限元素軟體ABAQUS 的UMAT，以利未來應用免映射法在複雜幾何外型物體的應力分析。

The mixed control problem refers to the mathematical problem of solving for the corresponding (remaining) stresses and strains in an elasto-plastic model when some stresses and strains are known, such as the plane stress problem, which is a common mixed control problem. In fact, most "strain-controlled" experiments belong to the mixed control problem category, e.g., uniaxial strain-controlled tests, torsion strain-controlled tests, and biaxial strain-controlled tests. When dealing with mixed control problems, plastic numerical integration methods based on pure stress or pure strain control cannot satisfy the known given conditions of mixed control, thus requiring additional treatment methods. In this study, we introduced the Armstrong-Frederick and Chaboche hardening/softening rules for anisotropic elasto-plastic material models and explored the intrinsic symmetry between the differential equation system and yield surface in the elasto-plastic model under mixed control conditions. By utilizing the properties of Lie algebra and Lie groups, we updated the unknown stresses and strains, where the obtained stresses automatically satisfy the yield condition without further iterations or additional treatment methods to meet the known given conditions of mixed control. We refer to this numerical integration method as the return-free method. Furthermore, for the predictor/corrector-free numerical integration method developed based on the Lie algebra and the Lie group theory, we performed error analysis to confirm the accuracy of the return-free method under different initial conditions. We then used the return-free method to study the contraction ratios and plastic strain ratios of anisotropic materials, as well as the mechanical behavior of cortical and trabecular bones under different loading paths. Finally, we developed a user material subroutine (UMAT) for the return-free method in the commercial finite element software ABAQUS to facilitate future applications of the return-free method in stress analysis of complex geometric bodies.