A Trying to understand couple stress in continua

[Klaus3]

In books that deal with continuum modelling of materials that include the presence of internal rotations/torques in the microstructure, the torque is defined as follows:

$$ \tau = \int ((r \times t + m) dA) + \int ( (r \times f + l) dV )$$

From this definition, the stress tensor and couple stress tensor are derived.

This is how it is defined in these papers
Eringen, A.C.: Theory of micropolar elasticity(1968) [Specifically Eq 14.6]
Toupin, Elastic Materials with Couple-stresses (1964)

Where

$t$ is the traction vector
$m$is the couple stress
$f$ is the body force
$l$ is the body couple
$r$ is the position vector

The addition of the couple terms are justified by mentioning they represent internal torques in the material. What i don't understand are two things:

1. Aren't these internal torques already captured by the traction vector term? I imagine you can have a continuous distribution of t on a surface that starts on one direction and continuously reduces, reaching 0 and then flipping to the other direction.
2. Since the couple stress is a flat addition of torque, how do you differentiate it from the traction torque? Isn't there a chance of double counting? What mathematically distinguishes the $t$ function from the $m$ function?

Thanks.

 

[Lnewqban]

$t$ is the traction vector
$m$is the couple stress
$f$ is the body force
$l$ is the body couple

Could you represent those in a diagram, the way in which are normally represented in those documents?

 

[Klaus3]

I am not aware of any force diagram that includes the couples in that case, and have no idea how would they appear, the texts only lay down the equations. $t$ are represented by contact forces which you have plenty of examples here https://en.wikipedia.org/wiki/Cauchy_stress_tensor and $f$ are body fields that act on the whole body, like gravitational forces.

For the couples, no idea. Thats mostly why i can't really understand it, my visualization of torques are all force related, so the couples that i imagine would simply look like torques of surface and body forces that cancel each other, but they are described as different things.

 

[A.T.]

The addition of the couple terms are justified by mentioning they represent internal torques in the material. What i don't understand are two things:

1. Aren't these internal torques already captured by the traction vector term? I imagine you can have a continuous distribution of t on a surface that starts on one direction and continuously reduces, reaching 0 and then flipping to the other direction.

Just guessing here: Is there a confusion about 'internal ...' vs. 'body ...'. Body forces/torques are external forces/torques, applied from outside the body to every point of the body.

2. Since the couple stress is a flat addition of torque, how do you differentiate it from the traction torque? Isn't there a chance of double counting? What mathematically distinguishes the $t$ function from the $m$ function?

Again guessing: When you consider the contact force distribution on an infitiesmally small region around a point on the surface, you can combine those forces into a pure torque m around that point and a force t acting effectively at that point, which creates a torque r x t around some point of the body. So the total torque around some point of the body, from the surface loading at some point at the surface is the sum of those torques: r x t + m. The r vector goes from the point of the body to some point on the surface.

The second integral is the same thing for the external body forces/torques applied directly to points thoughout the volume of the body.

 

[Andy Resnick]

In books that deal with continuum modelling of materials that include the presence of internal rotations/torques in the microstructure, the torque is defined as follows:

$$ \tau = \int ((r \times t + m) dA) + \int ( (r \times f + l) dV )$$

From this definition, the stress tensor and couple stress tensor are derived.

Looking at my reference ("The classical field theories", Truesdell and Toupin), The fields 'l' and 'm' are added to remove the restriction that all torques arise from forces (the contact load). Often (the 'non-polar' case), l = 0 and m = 0 .

I couldn't find any examples where l and/or m are non-zero, maybe they are used in active media where forces and couples are generated internally?

Edit: looking at another reference ("Continuum Mechanics and Thermodynamics", Tadmor, Miller, Elliot), it is also mentioned that those two terms appear in "multipolar" theories, for example magnetic materials in a magnetic field or polarized materials in an electric field. They suggest the references "Continuum Mechanics" by Jaunzemis and "Introduction to the mechanics of a continuous medium" by Malvern for more details about multipolar theories.

 

[Klaus3]

Just guessing here: Is there a confusion about 'internal ...' vs. 'body ...'. Body forces/torques are external forces/torques, applied from outside the body to every point of the body.

They are external to a particular system you choose, you can choose either the entire body, as such the forces would be fully external or to a particular volume inside the entire body, where the forces would be both internal and external.

Again guessing: When you consider the contact force distribution on an infitiesmally small region around a point on the surface, you can combine those forces into a pure torque m around that point and a force t acting effectively at that point, which creates a torque r x t around some point of the body. So the total torque around some point of the body, from the surface loading at some point at the surface is the sum of those torques: r x t + m. The r vector goes from the point of the body to some point on the surface.

The second integral is the same thing for the external body forces/torques applied directly to points thoughout the volume of the body.

Isn't this still a decomposition of the contact force still? It's still derived from the same function $r \times t$. The key factor here is that neither of the couples appear in the momentum balance, if they are derived from $t$ then they should, which is not the case.

Looking at my reference ("The classical field theories", Truesdell and Toupin), The fields 'l' and 'm' are added to remove the restriction that all torques arise from forces (the contact load). Often (the 'non-polar' case), l = 0 and m = 0 .

I got the reference and this statement is strange, how can torques arise from anything that is not a force? Are there torques that arise out of pure couples? I'm unaware.

I couldn't find any examples where l and/or m are non-zero, maybe they are used in active media where forces and couples are generated internally?

Edit: looking at another reference ("Continuum Mechanics and Thermodynamics", Tadmor, Miller, Elliot), it is also mentioned that those two terms appear in "multipolar" theories, for example magnetic materials in a magnetic field or polarized materials in an electric field. They suggest the references "Continuum Mechanics" by Jaunzemis and "Introduction to the mechanics of a continuous medium" by Malvern for more details about multipolar theories.

Just talking about electromagnetic effects, they would be included already in the body force term, no? As they are body field forces just like gravity.

Going through multiple references gives me the idea that these inclusions do not come from "purely continuous" models, but instead from models where the discrete nature of the microstructure is modelled, and when "averaged" they result in these additional terms, and those terms are dependent of microscopic kinematic properties (say, a microscopic velocity/rotation). If you think of a continuum as a collection of infinite particle points, there is inherently a disregard for the microstructure.

 

[Andy Resnick]

I got the reference and this statement is strange, how can torques arise from anything that is not a force? Are there torques that arise out of pure couples? I'm unaware.

Just talking about electromagnetic effects, they would be included already in the body force term, no? As they are body field forces just like gravity.

Going through multiple references gives me the idea that these inclusions do not come from "purely continuous" models, but instead from models where the discrete nature of the microstructure is modelled, and when "averaged" they result in these additional terms, and those terms are dependent of microscopic kinematic properties (say, a microscopic velocity/rotation). If you think of a continuum as a collection of infinite particle points, there is inherently a disregard for the microstructure.

Not sure what to say- just forwarding what I hoped was useful information.