Improvement of return mapping algorithm in GTN model by reducing the number of equations to be solved

GTNモデルのリターンマッピングアルゴリズムの方程式の削減による改良

真野 晃宏 ; 今井 隆太*; 宮本 裕平*; 勝山 仁哉  ; Li, Y. 

Mano, Akihiro; Imai, Ryuta*; Miyamoto, Yuhei*; Katsuyama, Jinya; Li, Y.

GTNモデルに基づく有限要素解析では、塑性変形に伴うボイドの発生・成長・合体を考慮して材料の降伏曲面を更新しながら塑性変形挙動が評価される。一般的に、非線形なひずみと応力の関係はリターンマッピングアルゴリズムにより計算されるが、陰解法による解析を行う場合、複数の未知変数を含む連立方程式を解く必要があると同時に、反復計算を行う必要がある。そのため、GTNモデルのように方程式の数が多い場合、収束解を得るのが困難となる場合がある。このような背景から、汎用有限要素解析ソフトウェアAbaqusでは、GTNモデルに基づく解析を陰解法により行うことができない。そこで、本研究では、未知変数間の相関関係の存在を示す陰関数定理を適用することにより、リターンマッピングアルゴリズムにおいて解くべき方程式の数を3つに削減し、GTNモデルに基づく陰解法による解析を安定的に実施することを可能とした。また、改良手法を用いて、大規模塑性変形を伴う配管の4点曲げ破壊試験を対象に再現解析を実施し、負荷変位と反力の関係を良好に再現できたことを確認した。

Finite element analysis (FEA) based on Gurson-Tvergaard-Needleman (GTN) model is utilized for the simulation of the non-linear plastic behavior of a material. In the analysis, the variation of the size of yield surface based on the void nucleation, growth and coalescence in a material is considered. When the analysis is conducted by an implicit method, the stress values are generally updated with a time increment considering the size of yield surface by using the so-called return mapping algorithm. This algorithm requires solving simultaneous nonlinear equations related to a constitutive model. Thus, their solutions are usually obtained by an iterative calculation scheme. However, when the number of equations is large as with the GTN model, it becomes difficult to obtain the converged solutions in the iterative calculation scheme. In the commercial FEA software Abaqus version 2018, the analysis based on the GTN model considering the void coalescence cannot be performed by the implicit method, in which the converged solutions are difficult to obtain. In this research, the number of equations in return mapping algorithm of the GTN model was reduced by applying the implicit function theorem and the convergence of the analysis was improved. Based on the return mapping algorithm improved regarding the number of equations, the analysis based on GTN model was conducted by the implicit method in Abaqus taking the 4-point pipe bending test as an example. As the result, it was confirmed that the variation in load against displacement in the test could be simulated.