caoer 发表于 2011-7-14 10:48:21

hyperelastic shell element算法


由于壳单元采用了多自由度,除displacement 以外又增加了director。
那么如何建立其 strain invarient 呢?
难道是将两个分别基于displacement和director的deformation gradient叠加起来求
最终壳单元的deformation gradient?从而推导出green strain,并得出I1, I2, I3吗?


北鹰南飞 发表于 2011-7-14 11:06:33

本帖最后由 北鹰南飞 于 2011-7-14 11:11 编辑

1# caoer

caoer 发表于 2011-7-14 11:27:02

-- 是的,但是一种特殊的本构,非线性本构

-- 壳和solid是不一样的,壳如何计算deformaiton gradient呢?

-- 同意,壳多了几个自由度,不一定是3个,也可能是2个。普通钢材不是超弹物质,泊松比一般低于0.4。弹性属性也是属于线性变化(塑性阶段除外)。

hillyuan 发表于 2011-7-19 11:59:15

When using degenerated shell element (see, e.g. Zienkiewicz & Taylor; FEM for solid and structural mechanics, chapter 15: shells as a special case of three-dimensional analysis ), you can get displacement gradient just as that of solid elements. In this case, there is coupling between gradient from displacement and director change, not only 两个分别基于displacement和director的deformation gradient叠加起来求最终壳单元的deformation gradient.

The only difference between solid and shell elements is that for shell elements, stress(33), normal stress along shell normal is zero. You should modify your constutive relation (not just hyperelastic relation) to satisfy this condition. And, keep stress(33)=0 when doing stress update calculation.

caoer 发表于 2011-7-20 08:28:00

本帖最后由 caoer 于 2011-7-19 19:34 编辑

interesting statement.
so you meant the deformation graident F = epi_{3x3} + 1_{3x3}, where the epi{3x3} is converted from the (15.11)? The director variables are not involved to the hyperelastic model? I may agree with you about this if deformation is small.

hillyuan 发表于 2011-7-20 09:19:15

I am not sure what you mean! But

1. The director variables enter Eq.(15.11) as it is obtained from Eq.(15.4), where director exists.

2. There is no strain(33) in Eq.(15.11), because stress(33)=0.0, the energy term of stress(33)*strain(33) doesn't exist.
    You cannot obtain strain(33) directly but from supposition stress(33)=0.0

3. Such supposition (stress(33)=0.0) is characteristic of shell elements, it is irrelevent with all other things.

caoer 发表于 2011-7-20 22:17:26

1. this equation makes sense, which also answer my questions, thanks.
2&3. I agree, the transverse normal stress/strain is zero if thickness is constant.
This set of equation looks good and doable. Thanks!

hillyuan 发表于 2011-7-21 08:22:04

Thickness is not constant and strain(33) is not zero. In some casees, such as stamping processes, considering the variation of shell thickness is quite important.

tonnyw 发表于 2011-7-21 10:02:58


由于壳单元采用了多自由度,除displacement 以外又增加了 ...
caoer 发表于 2011-7-14 10:48

I thought this problem is already well addressed in Bathe's papers, such as

Bathe mainly focused on degenerated 3D shell elements.

mxlzhenzhu 发表于 2013-8-31 18:50:22

本帖最后由 mxlzhenzhu 于 2013-8-31 18:51 编辑

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