彈塑性組成方程式由常微分方程組 (ordinary differential equations) 以及代數等式與 不等式 (algebra equations and inequalities) 所組成。為了得到塑性週期時正確的應力值， 所有代數方程和不等式必須在整個塑性週期內同時滿足，亦即在更新應力值時必須滿足應 力點確實落在降伏面上的限制。本研究首先引用 Liu et al. [26] 之研究探討異向性彈塑性 的內在對稱性引入新變數並建立增廣力空間，由李群變換 (Lie group transformation) 建 構免映射法 (return-free 數值積分方法)，在不需特別修正結果下更新應力之餘亦滿足降伏 條件。再者我們探討並展現異向性彈塑性模式數值模擬時的力學行為，其中我們著重觀察 一致性誤差 (consistency error)，亦即數值積分法在模擬塑性階段中計算並更新應力態至 降伏面上精確與否的指標、平均誤差來探討加載路徑對一致性誤差的影響，最後由前者觀 察到的極值以進行等誤差 (iso-error) 的分析。在徑向加載的範例中與其他數值積分法進行 一致性誤差等高線圖的比較，其最終結果與其他誤差分析下免映射法展現了預期內的優良 精度。然而彈塑數值模式僅考慮的應變控制不能完整的反應材料受力與變形行為，為此文 中假設應力與應變在現實中都擁有未知的項目、重新安排組成方程以求得耦合之應力與應 變，故稱應力應變混合控制模型。最後我們針對異向性材料骨皮質進行單軸加載、單軸循 環加載與五種雙軸向試驗路徑來模擬並討論循環加載硬化/軟化等機械行為。

An elastoplastic model for anisotropic-pressure-sensitive materials is proposed and its numerical integration is established in the present study. An anisotropic yield sur- face, a nonlinear isotropic hardening, and a nonlinear kinematic hardening are consid- ered. To simulate the behavior of the anisotropic-pressure-sensitive material exactly, the return-free integration which was initialized by Liu et al. [26], which automati- cally updates the stress on the yield surface during plastic phase is developed. The performance of the return-free integration was examined to demonstrate its consis- tency errors, average errors, and iso-errors. The influence of non-zero initial condition of stress, pre-straining path, the loading paths on the consistency error is conducted. The convergence analysis of average error is investigated and the iso-error maps is established. All error analysis reveal that the return-free integration for the model of anisotropic material with the anisotropic yield surface and the nonlinear isotropic- kinematic-mixed hardening rule is stable, acceptable in performance, and reliable. In three-dimensional elastoplasticity of strain driven case, the strain components were assume to known entirely. Which is in equilibrium to the boundaries were fixed or determined. However, in experiment or simulation the boundaries were free. There- fore, we assumed that the components of both stress and strain exist unknown parts, hence the constitutive equations are rearranged to obtain the coupled stress and strain formulation, i.e. the stress-strain mixed control formulation. The demonstration for the anisotropic material included the cyclic hardening/softening behavior and stress evolution under bi-axial strain path.

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