求助：关于“简单剪切时剪应力的问题” 的一个解释

hillyuan

tonnyw 发表于 2013-1-14 10:27
1. In classical continuum mechanics, body moment is not allowed to exists.

2. To consider the cas ...

It seems that there are misundstanding between us and I would apologize about it because it would be arise from my vague word usage.

My word "moment" used above should be modified as  local angular momentum. I am sorry!

>>Could you show us some of references on this? Among all the continuum mechanics and elasticity books I have read, body moment also exists.

=> OK! If you mean local angular momentum exists, stress tensors should be unsymmetric. It's allright.

>>As for boundary explosion, I don't see how it is possible. Can you give proof?
=> What I want to say is that if equilibrium equation is not satified, the node would accelarate forever.

At last, I attached a file from Eringen: Machanics of continum, explaining the balance of local momenta and symmetricity of stress tensor. Maybe you may find the same contents in most testbook on continuum mechanics. But this books explicitly denotes that the body and surface couple stress not exist (two lines above equation 3.5.1), it is  why I post it here.

If it is possible, would you please tell me if there are something wrong, e.g. theorem 2 is wrong, or missing something, e.g.,　equation 3.5.7 should not be applied along the boundary and stress tensors should be unsymmetric there.

Thanks

tonnyw

hillyuan 发表于 2013-1-15 11:52
It seems that there are misundstanding between us and I would apologize about it because it would  ...

Thanks for pointing Eringen's book to me. Never had chance to read it before. Just took a look at it. It is great book.

Here are my comments:
1. The body moment per unit mass was referred as body couple per unit mass by Eringen as shown on page 100 equation (3.2.3)

2. Equation 3.5.7 has no problem. As you can see that Eringen got (3.5.7) and (3.5.8) from an arbitrary volume v (lower case) which are defined  in section 3.3. Here v is bounded by a closed surface s and v + s are contained in the body V. This clearly shows that the equilibrium equation and symmetric property of stress tensor are not defined on the boundary of the volume V.

hillyuan

tonnyw 发表于 2013-1-15 13:01
Thanks for pointing Eringen's book to me. Never had chance to read it before. Just took a look at  ...

1. The body moment per unit mass was referred as body couple per unit mass by Eringen as shown on page 100 equation (3.2.3)
=> I think you agree that body moment does not exist in nonpolar, by Eringen (classical in my above response), continuum mechanics. It's OK!

2. Equation 3.5.7 has no problem. As you can see that Eringen got (3.5.7) and (3.5.8) from an arbitrary volume v (lower case) which are defined  in section 3.3. Here v is bounded by a closed surface s and v + s are contained in the body V. This clearly shows that the equilibrium equation and symmetric property of stress tensor are not defined on the boundary of the volume V.
=> Is that clear? I think v+s denotes any region of body V, s contained by \Fai also. Otherwise, it should denotes theroem 1 and 2 don't satisfied along boundary explicitly and indicate what's the equilibrium condition along the boundary. Shouldn't we?
   I think following consideration would be help.
* We can imaginarily divide a body into two contaced bodies a and b. Then a/b and b/a apply external force to each other (e.g. Figure 13.1 of Gurtin: The mechanics and thermodynamics of continua). Because the interface is either interal and external surface. Stress along the interface boundary should symmetric.
  BY the way, I am still wondering how can we define the unsymmetric shear stress component along boundary if it is not symmetric in nonpolar or classical continuum mechanics.

Best regrads.

tonnyw

hillyuan 发表于 2013-1-15 14:25
1. The body moment per unit mass was referred as body couple per unit mass by Eringen as shown on  ...

1. Body moment just likes body force. Here Eringen assumes it is zero so he can have symmetric stress tensor.He just applied the continuum mechanics theory to the nonpolar material.

2. (3.5.7) and (3.5.8) is the results of local conservation of linear momentum and angular momentum. First as mentioned in Eringen's book, the local volume v is arbitrary one and it can be any small region contained in the body. But it has to be a small and open one closed by surface s. If you choose the point on the boundary of the body, the volume v is not open since there is no closed surface to contain it.

What you mentioned in your description is still about a point within the body. The point is just on the artificial boundary which Eringen created for the purpose of better understanding.

Let consider a simple 1D bar subject to pulling at one end and fixed on another.

Then we have the equilibrium equation like:
-E*A*d^2u/dx^2 = 0       0<x<L
Boundary condition:
u(0)=0, E*A*du/dx(L) = F

You can see that the equilibrium equation is defined on the open interval  (0, L).

You mentioned something about Chen Zhida. I am surprised that Chen had a fundamental misconception about the symmetry of stress tensor.

hillyuan

tonnyw 发表于 2013-1-15 22:53
1. Body moment just likes body force. Here Eringen assumes it is zero so he can have symmetric str ...

1. Body moment just likes body force. Here Eringen assumes it is zero so he can have symmetric stress tensor.He just applied the continuum mechanics theory to the nonpolar material.
=> Then we finally consent with each other about this point after all those discussion.
      When using micropolar theory, we need more dofs, more equlibrium equations, more material constants etc. It's another story.

Then we have the equilibrium equation like:
-E*A*d^2u/dx^2 = 0       0<x<L
Boundary condition:
u(0)=0, E*A*du/dx(L) = F
=> From "A first course in Finit elements"?. Maybe a spelling missing, an acceptable mistake. Could you find another one?

2. (3.5.7) and (3.5.8) is the results of local conservation of linear momentum and angular momentum. First as mentioned in Eringen's book, the local volume v is arbitrary one and it can be any small region contained in the body. But it has to be a small and open one closed by surface s. If you choose the point on the boundary of the body, the volume v is not open since there is no closed surface to contain it.
=> Do you consider "a small region of volume v bounded by a ..." a open one?
      An relavent question: When do volume->surface integral transfomation using Gauss's theroem, which is adopted in our deriavtion frequently, do you consider the volume is open?
   In fact, I don't remember any specific consideration is needed in such compat and dense closure. But I am not mathamatician  and cannnot say more about it anymore.

   At last, howerver, as a mechanical researcher, could you please write down the equlibrium equation along the boundary? It is important! Without it, the boundary point would accelarate forever. You may also need develop a constitutive equation for boundary material because the stress tensor may unsymmetric.
  On the other hand, the stress tensor on v+s is not smooth because there is a adrupt variation of stress tensor from symmetric to unsymmetric onto the boundary. You cannot obtain the deravitive of tensor here, at least in a common sense.
   You leave a hard work!

tonnyw

hillyuan 发表于 2013-1-16 11:01
1. Body moment just likes body force. Here Eringen assumes it is zero so he can have symmetric str ...

1. If we are dealing with polar materia where body couple or surface couple could happenl, we just have nonsymmetric stress tensor. The equilibrium equation is still the same one except the stress tensor is not symmetric. I don't see why we need more equlibrium equations. We might need more material constants.

2. The 1D equilibrium equation is just something I worked out for illustration. I don't know if the finite element book you refer to has the model. There is no typo. You can easily check all the governing equations for boundary value problems or initial boundary value problems and you will see that they all are defined on the open domain. Navier-Stokes equation, wave equation, heat conduction, Navier-Lame equation, etc.

2. For the small volume, it is open and bounded by the closed surface. This meets the divergence theorem requirements.

There is no equilibrium equation defined on the boundary. It doesn't exist. The boundary point would accelerated forever since you have internal force to balance it. For instance, for the 1D problem, on the boundary we have force F and we have the internal force E*A*du/dx(L) to balance it.

Usually when we derive the equilibrium equation, we choose a type of cube. If you choose a point on the boundary, it could be half of the cube buried in the model and another half has nothing. It doesn't make sense.

I have another example. Assume an elastic body consisting of two materials, at the material interface, the stresses are not symmetric at all.  We are completely fine. In this case, we cannot talk about shear stress reciprocal theorem for the material point at the material interface either.

We cannot take the derivative of stress tensor on the boundary because we don't know if the derivative exists or not.

hillyuan

tonnyw 发表于 2013-1-16 12:26
1. If we are dealing with polar materia where body couple or surface couple could happenl, we just ...

1. If we are dealing with polar materia where body couple or surface couple could happenl, we just have nonsymmetric stress tensor. The equilibrium equation is still the same one except the stress tensor is not symmetric. I don't see why we need more equlibrium equations. We might need more material constants.

=> Are you kidding? I think you must at least read some books like Eringen: Theory of micropolar elasticity, before you get such conclusion. If you consider it is too long, I have a article published many years ago

http://www.sciencedirect.com/science/article/pii/S0093641301001720
or
http://www.lib.kobe-u.ac.jp/repository/90000023.pdf

where a simple introducation is available.

I have no furthor comments about your 2,3 now. But if you found any example which obtains all the comoponent of the unsymmetric stress tensor. It is much grateful for you to tell me.

Much thanks.

tonnyw

hillyuan 发表于 2013-1-16 13:17
1. If we are dealing with polar materia where body couple or surface couple could happenl, we just ...

Here you are talking about the continuum mechanics which can catch the behavior of microstructure.

As mentioned in Eringen's book, he neglected the body couple (or body moment) to get the symmetric stress tensor for nonpolar materials. What I meant is that within the framework of his book ( I should have been more clear on this respect)  if he didn't neglect the body couple, then in the case of polar material, he would get the same form of governing equation and unsymmetric stress tensor. Just as the link illustrates
http://arxiv.org/ftp/arxiv/papers/1009/1009.3252.pdf

The continuum mechanics you talk about is another set. You may call it non-classical continuum mechanics and it doesn't conflict with my conclusion.  

With this, we can close the discussion. I am learning things. I appreciate this fruitful discussion.

hillyuan

tonnyw 发表于 2013-1-17 01:07
Here you are talking about the continuum mechanics which can catch the behavior of microstructure. ...

The paper you attached describes a standard micropolar theory.Equation 10, e.g., is equlibrium realtions with additional one to consider couple stress.

Do you know why we call it MICRO-polar, that's because it describes the effects of microstructure. All thoses theories , including strain gradient theory, micromorphic theory etc, could consider the effcts of microstrture. They  include a length scale of microstructure in their constitutive relationss. They can predict the size effect and avoid mesh dependent phenomena in instablity analysis. Therefore, they are also called nonlocal or non-classical. Please google nonlocal & micropolar before write down your words, such researchs is not the majority but there are a few.

tonnyw

hillyuan 发表于 2013-1-17 07:20
The paper you attached describes a standard micropolar theory.Equation 10, e.g., is equlibrium rea ...

As you see from the equation (3.2.7) in Eringen's book, if we negelct the body moment and surface couple, we will have symmetric stress tensor. If not, we will get equation (10b) as shown in my attached paper.

hillyuan

tonnyw 发表于 2013-1-17 09:32
As you see from the equation (3.2.7) in Eringen's book, if we negelct the body moment and surface  ...

I completely agree with you! If unsymmetric stress tensor exists, you needs equation (10b), which do not exists in nonpolar, or classical or local, continuum mechanics, to consider the existence of couple stress. By the way, this equation is just equation (1b-2) of my paper. The equlibrium equation of classic continuum mechanics is equation (10a), or (1b-1) in my paper.

That is why I say you need additional equlibrium equations. Furthernore, you need couple stress (first term in equation 10b), additional constitutive relation for couple stress etc also. That is why I say it is another story.

By the way, pay attention to the second term in equation (10b), which defines the no-syymetricity of stress tensor. Although the equation (10a) writen as the same form with nonpolar machanics, they are physically different.

tonnyw

I agree with you on the following aspects:
1. (10b) is another equilibrium equation based on the conservation of angular momentum.

2. l_i is the body moment per unit mass which causes the non-symmetry of stress tensor.

Question:
Why (10a) is physically different from the one for nonpolar material? It is still the equilibrium equation. Right? Of course, in this case, the stress tensor T is not symmetric.

hillyuan

tonnyw 发表于 2013-1-17 10:35
I agree with you on the following aspects:
1. (10b) is another equilibrium equation based on the con ...

It depends upon (10b) if stress is unsymmetric.