# 塑性變形

## 數學詮釋

1934年，傑弗里·泰勒和其他兩名科學家幾乎同時發現物質的塑性形變可以用差排的理論解釋，這樣物質的形變應用一系列非線性的公式解釋。

## 屈服条件

### Tresca criterion

${\displaystyle \sigma _{1}-\sigma _{3}\geq \sigma _{0}}$

σ1為最高應力，σ3為最低應力，σ0為降伏強度。此公式可推導出一個表面，在表面內的狀態為彈性形變，在表面上的狀態為塑性形變，而不存在表面外的物質狀態。

### von Mises criterion

{\displaystyle {\begin{aligned}\sigma _{\mathrm {effective} }^{2}&={\tfrac {1}{2}}\left[\left(\sigma _{11}-\sigma _{22}\right)^{2}+\left(\sigma _{22}-\sigma _{33}\right)^{2}+\left(\sigma _{33}-\sigma _{11}\right)^{2}+6\left(\sigma _{12}^{2}+\sigma _{13}^{2}+\sigma _{23}^{2}\right)\right]\end{aligned}}}

## 參考

1. ^ M. Jirasek and Z. P. Bazant, 2002, Inelastic analysis of structures, John Wiley and Sons.
2. ^ W-F. Chen, 2008, Limit Analysis and Soil Plasticity, J. Ross Publishing
3. ^ M-H. Yu, G-W. Ma, H-F. Qiang, Y-Q. Zhang, 2006, Generalized Plasticity, Springer.
4. ^ W-F. Chen, 2007, Plasticity in Reinforced Concrete, J. Ross Publishing
5. ^ J. A. Ogden, 2000, Skeletal Injury in the Child, Springer.
6. ^ J-L. Leveque and P. Agache, ed., 1993, Aging skin:Properties and Functional Changes, Marcel Dekker.
7. ^ Gerolf Ziegenhain and Herbert M. Urbassek: Reversible Plasticity in fcc metals. In: Philosophical Magazine Letters. 89(11):717-723, 2009 DOI
8. ^ R. Hill, 1998, The Mathematical Theory of Plasticity, Oxford University Press.
9. ^ von Mises, R. (1913). Mechanik der Festen Korper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys., vol. 1, pp. 582–592.