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Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring
Computational Mechanics ( IF 4.1 ) Pub Date : 2021-06-05 , DOI: 10.1007/s00466-021-02038-w
P. Areias , P. A. R. Rosa , T. Rabczuk

For finite-strain plasticity with anisotropic yield functions and anisotropic hyperelasticity, we use the Kröner-Lee decomposition of the deformation gradient combined with a yield function written in terms of the Mandel stress. The source is here the right Cauchy-Green tensor provided by a FE discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential, which is compared with a classical backward-Euler method. The exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source, consistent with the approximation. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function. The constitutive system is solved by a damped Newton-Raphson algorithm for the plastic multiplier and the elastic right Cauchy-Green tensor \(\varvec{C}_{e}\). To ensure power-consistency, we make use of the elastic Mandel stress construction. Two numerical examples exhibit the comparative effectiveness of the Algorithm for very large elastic and plastic deformations. The elasto-plastic pinched cylinder makes use of as few as 2 steps for the total radius displacement of 300 mm and only 25 steps are required for the cup drawing problem.



中文翻译:

通过幂级数和缩放/平方的指数近似的完全各向异性超弹塑性

对于具有各向异性屈服函数和各向异性超弹性的有限应变塑性,我们使用变形梯度的 Kröner-Lee 分解结合以曼德尔应力表示的屈服函数。此处的来源是由 FE 离散化提供的正确 Cauchy-Green 张量。对于流动定律的积分,我们采用矩阵指数的缩放/平方级数近似,将其与经典的后向欧拉方法进行比较。第二个 Piola-Kirchhoff 应力的精确雅可比是相对于该源确定的,与近似值一致。生成的系统是通过为每个屈服函数和超弹性应变能量密度函数生成符号源代码生成的。\(\varvec{C}_{e}\)。为了确保功率一致性,我们使用了弹性曼德尔应力结构。两个数值例子展示了该算法对于非常大的弹性和塑性变形的比较有效性。弹塑性挤压圆柱体只需 2 个步骤即可完成 300 毫米的总半径位移,而杯形拉伸问题仅需要 25 个步骤。

更新日期:2021-06-05
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