求助：关于“简单剪切时剪应力的问题” 的一个解释

guojunhang

如图，是简单剪切状态的一个图示，物体发生了剪切变形。
    虚线是变形前的状态，实线是变形后的状态，剪应力施加在物体的上表面。

    于是，施加力的面上有剪应力，问右边的面有没有剪应力？
    我的理解是：右边的面是自由表面，自由面上没有剪应力。
但是按照“剪应力互等定律”，这个面应该有剪应力？

于是我得出一个矛盾？

有还是没有？ 如图所示：

rock.li

剪应力互等定理的理解错误，建议你再好好看下这个知识点，然后再讨论。提示：一点处应力状态

hillyuan

哈哈。这个问题有意思！

我国知名学者陈至达院士在其大著《理性力学》第3.11节中以上图相同的例子证明了剪应力是非对称的。

该同学有院士的水平！

tonnyw

To have Shear stress reciprocal theorem, certain assumptions are needed. Maybe the attached paper is helpful.

guojunhang

谢谢各位的支持，我回去慢慢消化消化

hillyuan

tonnyw 发表于 2013-1-11 03:06
To have Shear stress reciprocal theorem, certain assumptions are needed. Maybe the attached paper is ...

Do you really consider the paper you suggest  worth reading? I read throught the paper and consider the authors are idiots in mechanics. They known something about strength of material and boldly apply those in elasticity, which they really understand nothing.

tonnyw

hillyuan 发表于 2013-1-11 11:06
Do you really consider the paper you suggest  worth reading? I read throught the paper and conside ...

Oh, I really didn't go through the whole paper. I cannot tell if it is worth reading.

rock.li

别用英文“扯淡”了，哪位出来讲解下？

tonnyw

我的理解是这样的，剪应力互等定理的成立条件是 1）微元体内没有体力矩，2)该定理的推导是在物理模型内以一点为圆心，半径无限小的邻域内，该邻域不能与边界接触。因此在边界上，不具备剪应力互等的条件。这和控制方程定义在开区间上有些类似。

tonnyw

hillyuan 发表于 2013-1-11 11:06
Do you really consider the paper you suggest  worth reading? I read throught the paper and conside ...

Indeed, I feel the paper is rather trashy after going through the paper. Originally I took a look at the abstract and simply felt it might be useful. Sorry for wasting your time.

hillyuan

tonnyw 发表于 2013-1-11 23:10
我的理解是这样的，剪应力互等定理的成立条件是 1）微元体内没有体力矩，2)该定理的推导是在物理模型内以一 ...

对于这个问题似乎不少人有疑问,如

http://forum.simwe.com/forum.php ... B%E4%BA%92%E7%AD%89

其中也包括某些院士. 对我来说,其基本概念的错误是明显的,但仔细想想要说清楚也不容易!

如果只要解释楼主的图,可以这样解释;在我们的模型中(三维欧氏空间)中,我们可以对一点加载,但我们不可能对一点的一个面加载,因为理想化的点不包含面!(上述连接的72楼有个较易理解的解释).因此你的见解2是错误的.

至于你的见解1,正确!但该假设在古典连续体力学中不成立.这并不难理解,如果我们定义力矩为力与距离的矢量积,对于尺寸为零的数学点来说它当然是一高阶无穷小.但如果我们定义的数学点为有向点(除去三个位移,还加上三个转动自由度,这已不是Riemman空间的问题了),这是可能的,这时应力不对称.这种模型最早由Cosserat(1896)提出,我们称之为nonclassic/nonlocal连续体.可参见,如Eringen(1968);Theory of micopolar elastiticy.

这是一个基本概念问题,建议花时间认真考虑一下.

tonnyw

hillyuan 发表于 2013-1-12 09:01
对于这个问题似乎不少人有疑问,如

http://forum.simwe.com/forum.php ... B%E4%BA%92%E7%AD%89

你还是没有说服我为什么见解2)有问题。我们使用连续体力学推导平衡方程： 应力张量的散度等于体力，就是建立在半径无穷小的单连域上的，该域与边界的交集为空。
剪应力互等定律也是基于该域上的力矩平衡得到的。我看不出有什么问题。

zsq-w

对于简单剪切时剪应力的问题”的问题，我觉得我对这个问题的理解与lz存在差异。

lz一楼的图形在力学教材中很常见，这个矩形（受剪之后变为平行四边形）应该是指物体内一个微元体。
这个工况下，右边面不是自由面，自然产生大小相等的剪应力。

hillyuan

tonnyw 发表于 2013-1-12 11:32
你还是没有说服我为什么见解2)有问题。我们使用连续体力学推导平衡方程： 应力张量的散度等于体力，就是 ...

1.  如果你要对一点施加载荷f,其表达式应为
     \sigma_(ij)*n_j=f_i
而不是\sigma_(ij)=f, which is just #1 doing.

2. 在古典连续体力学中,我们有三个运动学量(位移矢量)和三个动力学量(力矢量).这是一组对偶量,其点积是功.
   力矩是材料力学中使用的概念,它是一个积分量,一般用于板壳或梁.严格说古典连续体力学并不需要这个概念.当然如果我们知道板壳或梁断面的应力分布,作一积分计算即便可得到此量.

3. 在2下,我们不可能给一点施加力距.此时任意一点(不管它是否处于边界)的应力必是对称的,否则该点不平衡.

4. 如果我们扩展2,应力可以是不对称的.

hillyuan

zsq-w 发表于 2013-1-12 12:43
对于简单剪切时剪应力的问题”的问题，我觉得我对这个问题的理解与lz存在差异。

lz一楼的图形在力学教材中 ...

lz的意思是右边面是一边界面,是一自由表面.

tonnyw

hillyuan 发表于 2013-1-13 07:45
1.  如果你要对一点施加载荷f,其表达式应为
     \sigma_(ij)*n_j=f_i
而不是\sigma_(ij)=f, which is jus ...

Your description doesn't help me.

If we use the conservation of linear momentum, we get equilibrium and for statics it is divergence (stress tensor) = f

If we use the conservation of angular momentum, we get the shear stress reciprocal theorem if the body moment is zero.

For both cases, the derivation is based on a neighborhood of small radius within the model. It has nothing to do with boundary conditions.

For the model, we are talking about, we cannot pick a point on the boundary and then apply the conservation of angular momentum or linear momentum to get the equilibrium equation and shear stress reciprocal theorem.

As for \sigma_ij*n_j = f_i, it is just the traction on the surface whose normal vector is n_j. I don't see how it is related to what we are discussing.

hillyuan

tonnyw 发表于 2013-1-13 09:41
Your description doesn't help me.

If we use the conservation of linear momentum, we get equilibri ...

Well, my point 1 is just to explain where #1 is wrong, not relates to your question directly.

I seems like that you consider we cannot obtain the derivative of a tensor in boundary, doesn't you? If so, it is just a mathmetical question, not a mechanical one.

Do you think we cannot define a derivative at the boundary of close zone?

hillyuan

tonnyw 发表于 2013-1-14 05:33
First, I don't think my point 1 is wrong. Your argument is that the body moment doesn't exist if I u ...

1. In classical continuum mechanics, body moment is not allowed to exists.

2. To consider the case the body moment exists, you need the so-called cosserat or micoploar mechanics. Here you need additional degrees of freedom besides displacement.

3. You should have the equilibrium equation all around the body! Shear stress reciprocal theorem does hold on the boundary. Otherwise the boundary would explode!

tonnyw

First, I don't think my point 1 is wrong. Your argument is that the body moment doesn't exist if I understand you correctly. I believe there are cases where body moment exists. For instance, for magnetic material in strong magnetic field, we are going to have body moment. If we have body moment, the stress tensor is not symmetric any more.

Certainly you can take derivative of a tensor on boundary. But does it make sense to have such derivative? I guess the question should be: Do we have the equilibrium equation on the boundary? The answer is no. Does the shear stress reciprocal theorem still hold on the boundary? Negative.

For statics, we have Navier-Lame equation which is defined on the open domain excluding boundaries. Actually, all the governing equations are defined on the open domain.

tonnyw

hillyuan 发表于 2013-1-14 09:14
1. In classical continuum mechanics, body moment is not allowed to exists.

2. To consider the cas ...

1. In classical continuum mechanics, body moment is not allowed to exists.

2. To consider the case the body moment exists, you need the so-called cosserat or micoploar mechanics. Here you need additional degrees of freedom besides displacement.
>>Could you show us some of references on this? Among all the continuum mechanics and elasticity books I have read, body moment also exists.

3. You should have the equilibrium equation all around the body! Shear stress reciprocal theorem does hold on the boundary. Otherwise the boundary would explode!
>>Sorry I cannot convince you that the equilibrium equation such as Navier-Lame equation is defined on the open domain. If equilibrium equation holds on the closed domain, we would not have singularity issue.

As for boundary explosion, I don't see how it is possible. Can you give proof?