【UMAT6】中篇-Chaboche非线性随动硬化雅可比矩阵

大家好，我是九千CAE。

上一节讲解了 Chaboche 非线性随动硬化本构理论，书接上回，本节我们将介绍雅可比矩阵的获取。

Since

Then

Consider

Then the 2nd term is

Also consider

Then the 3rd term is

Substitute the 2nd and 3rd terms back, yielding

The termA can be rearranged as a similar as a elastic Jacobian matrix as

To derive termB and termC , we have to derive      and      first.

Note that in the previous formula,      

since      is a deviatoric tensor.

This formula gives one relation between      and     , the other relation can be derived from the yield function as

Combining above two relation and rearrange, one can obtain

The termB can be expressed as

Because     

Substitute back to termB

Finally, termC is

Substitute termA, B and C back and let

The right-hand side of the above formula gives the notation for the elasto-plastic Jacobian matrix. To fit the Voigt expressions of stress and strain, we need the 6x6 matrix form as below.

The partA is

The partB is 

(Note that the superscription 'def' is omitted since     ) 

The partC is