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Applied Mathematics and Mechanics (English Edition) ›› 2015, Vol. 36 ›› Issue (10): 1359-1370.doi: https://doi.org/10.1007/s10483-015-1988-9

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Relations between cubic equation, stress tensor decomposition, and von Mises yield criterion

Haoyuan GUO1, Liyuan ZHANG2, YajunYIN1, Yongxin GAO3   

  1. 1. Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China;
    2. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China;
    3. Department of Mathematics, Tianjin University, Tianjin 300072, China
  • 收稿日期:2014-10-08 修回日期:2015-03-04 出版日期:2015-10-01 发布日期:2015-10-01
  • 通讯作者: Liyuan ZHANG E-mail:zhangly@ustb.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Nos. 11072125 and 11272175), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20130002110044), and the China Postdoctoral Science Foundation (No. 2015M570035)

Relations between cubic equation, stress tensor decomposition, and von Mises yield criterion

Haoyuan GUO1, Liyuan ZHANG2, YajunYIN1, Yongxin GAO3   

  1. 1. Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China;
    2. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China;
    3. Department of Mathematics, Tianjin University, Tianjin 300072, China
  • Received:2014-10-08 Revised:2015-03-04 Online:2015-10-01 Published:2015-10-01
  • Contact: Liyuan ZHANG E-mail:zhangly@ustb.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Nos. 11072125 and 11272175), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20130002110044), and the China Postdoctoral Science Foundation (No. 2015M570035)

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摘要： Inspired by Cardano's method for solving cubic scalar equations, the additive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new viewpoint. This decomposition simplifies the cubic tensor equation, decouples the spherical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.

关键词: von Mises yield criterion, cubic tensor equation, Caylay-Hamilton theorem, decomposition of spherical/deviatoric tensor (DSDT), Cardano's method

Abstract: Inspired by Cardano's method for solving cubic scalar equations, the additive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new viewpoint. This decomposition simplifies the cubic tensor equation, decouples the spherical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.

Key words: Caylay-Hamilton theorem, decomposition of spherical/deviatoric tensor (DSDT), von Mises yield criterion, cubic tensor equation, Cardano's method

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