请教屈服面法向的偏导

lungo999

最近在看《连续体和结构的非线性有限元》，里面关于J2材料的径向返回算法有部分看不太懂，请教个弱弱的问题，自己写的J2材料的屈服面法向向量r对应力的偏导的表达式是否正确？

另外想请教书中的投射张量和单位法向矢量的表达式，不甚感激。

tonnyw

1# lungo999

I don't think your derivation is correct.

Let put the yield function the following way in tensorial form, we have
f = 1/2*(s_ij - alpha_ij)*(s_ij - alpha_ij) - 1/3*sigma_y^2
where sigma_y is the yield stress.
If you take the derivative of f over the stress tensor sigma_ij, you have
df/d(sigma_ij) = s_ji - alpha_ji This is the variable which you call n

For the radial return algorithm, you need to provide more information and make it more readable. The resolution is not good.

lungo999

2# tonnyw

谢谢回复。

对这种张量表达不甚清楚，我的问题就是来自于《连续体和结构的非线性有限元》那边经典的书。关于强化模型，书中没有特别的假设，就是说是J2材料。


版主给出的表达式为流动方向的另外一个表达形式，我的推导中的n给出的值除以系数后应该就是你说的n了。


就是想问下，上面推导的流动方向对于应力的偏导的表达式是否正确。还有附件的书中的关于投射张量的表达式如何推导不甚清楚，还请版主指教，固体力学基础不甚

tonnyw

I feel confused. I mentioned that your derivation is not correct. I just changed the format of the yield function by removing the square root.
The tensor S is the deviatoric one which is equal to sigma - sigma_m where sigma_m is the hydrostatic pressure. How about reading 王瑁成's book first?

lungo999

4# tonnyw

我也有点疑惑，版主你想说的是什么。很多书上给出的n的表达式都是我上面写的那样，我的问题的重点是n的偏导形式是不是对的，我在另外一本书中发现我的推导是对的，只是我不知道中间的推导中的投射张量的表达式。不过还是谢谢你的回复，希望下次回复别人可以用中文。

tonnyw

If you still think n = 3/(2*Y)*(S - a) is correct. Then the derivative of n over sigma is okay since S = sigma - sigma_m*I
where sigma_m is hydrostatic pressure and I is identity matrix.

lungo999

6# tonnyw

Thank you for your reply. I am glad to know that we have foreign experts in this bbs. I really appreciate.

By the way, you can find the above expression of n in many foreign books on material constitution.

tonnyw

7# lungo999

You really don't have to use English since I understand Chinese.

Do you think the denominator should be Y^2 instead of Y in the expression of n? I put some comments on your first post. Can you take a look?

lungo999

8# tonnyw

应该是那样的，很多书上都是这么写的，我也自己推过，这个公式在很多经典的书上都出现过，你可以看看庄茁翻译的《连续体和结构的非线性有限元》PAGE245。非常感谢版主的回复及管理员的关注。