Cauchy Stress Components from Surface Force

[Lewis S]

Hi, I've been trying to figure out how to get Cauchy Stress Tensor components (~9) from a surface force for a while now. My background in this subject is not too deep, but I'm trying to build a renderer simulation in my free time.

I can get surface traction from a Stress Tensor:
t = ((1,2,3),(2,4,6),(3,6,1)) (1,0,0) = (1,2,3) where (1,0,0) is the normal (an example from a Dover book) but in numeric examples, the Stress Tensor is usually given, and not numerically derived.

I've been reading through a bunch of books (just picked one up today in the library) and attempting a bunch of multiplications, but I often end up with fewer (stress) components than 9 as a result and am going in circles. I think I'm missing something notational since I'm resulting to this time-intensive "guesswork."

I've also reached out to continuummechanics.org but I think it's less of a Q&A site.

Some sample numbers for an example: Force: (46.79,74.86,34.40) Normals: (0.4, 0.6, 0.693), (0.832,0.555,0), and (-0.385,-0.576,0.721)

 

[Orodruin]

You have not given enough information. You need to provide the force in each of the normal directions.

 

[Lewis S]

From what I've been reading, that makes sense. I see that sigma_ij = t_(ei) . e_j, (may have notated that incorrectly) in P. Chadwick's book so 3 forces may be necessary. If I were to simulate this, I think I would start in equilibrium, and impose an external surface force from a single direction. I could derive the negative force in the opposite direction, but maybe the other two planes are computed from 0? Or maybe I need to decompose this external force, I'll re-read through the literature..

In any case, I'm glad I'm trying to figure this out before implementing it, since maybe it's not feasible with my requirements. Thanks a ton for the insight & help also

 

[Orodruin]

I am sorry, but it is completely unclear what you are trying to accomplish. You started asking about finding the stress tensor given directions and forces and now you are suddenly talking about equilibrium.

 

[Lewis S]

OK, fair enough. To get the stress tensor, I'll need forces in each of the normal directions. With the problem I'm working on, I think I only have 1 force though. I'm basing the problem after "The material on side 2 exerts a force F on the material on side 1 through the surface S, due to interactions between the molecules of the material." P.C. Matthews (pg. 140). He then says they're related by F_i = P_ij * n_j * S where P_ij is the stress tensor.

Sorry for the lack of clarity, but I really do appreciate the attention here. I'll sleep on this problem for now..

 

[Lewis S]

To simplify my problem: I'm trying to model a system where a point receives a surface force. Eventually, I'll include more points and will try to translate this force to strain. I think initially, I'll need to represent the contact force as stress, hence the Cauchy Stress Tensor.

The numbers I gave in the first message are hypothetical numbers of what the force and normals of the point could be. I can provide any more info to clear up what I am trying to accomplish...

 

[Nidum]

I can provide any more info to clear up what I am trying to accomplish...

Please do . Start by explaining what you mean by the words ' where a point receives a surface force ' . Draw a picture .

 

[Chestermiller]

It seems to me the OP is trying to determine the state of stress (i.e., the stress tensor) within a body based only of the applied load distribution on its surface. This can't be done in general without solving the stress equilibrium equation in conjunction with the strain-displacement equations and Hooke's law. Of course, there are some very simple cases where it is possible, such as tensile loading of a rod. I don't know what the minimum requirements are to be able to do this, but, for certain, it requires that the stress distribution be homogeneous (and probably also that the system be statically determinate). In my judgment, the OP needs to do much more studying before being able to solve a typical complex loading problem.

 

[Lewis S]

Hi sorry for the delay. Think you hit the nail on the head. I was originally under the impression that the Stress Tensor must be solved before approaching the strain equations. I thought it was more of a parameter and was putting off understanding the strain equations so that I could focus on stress. But looking ahead, it seems there are some crucial pieces to the puzzle, for example:

Conservation of linear momentum: dT_ij/(dx_i) + pb_j = pf_j
Rate-of-deformation tensor: D_ij = 1/2(dv_i / dx_j + dv_j / dx_i)
Constitutive equation for a linear elastic solid: T_ij = C_ijrs * E_rs

There's a ton to learn next.

Also, the problem I was originally working on was trying to model a simple force f on a single point p. I was apply force via my computer's cursor or by other points in the scene and wanted either a plastic or elastic deformation to eventually result.

The screenshot above is from the renderer I'm building that shows an isolated point from other points in the scene.
The below screenshot shows many more points (millions) that form a continuous body.

I'll be studying and working on this for quite some time, but will try to remember to post back here if I figure it out.

Again, much appreciated

 

[Nidum]

You have a dense 2D array of fine angle cones with the cones arranged point upwards and with all the points at the same level so that all the points taken together approximate a plane surface .

You wish to model the deflection of a small sub area of this surface when it is subjected to a load .

Your ultimate aim is to be able to do any calculations in real time and produce an interactive display of results in graphic form .

The deflection of anyone cone with point loading is just a matter of relatively simple calculation . The only difficulty is that an actual zero diameter cone point and the material immediately below it would see infinite and near infinite stress levels even for very small loads . This means that the cone point would deflect permanently and in measured height by a relatively large amount for any case of loading . To deal with this problem you need to refine your concept of a point . You would in any case find all of this project easier to deal with if you used an assembly of cylinders rather than cones . We can discuss this separately if you wish .

The affected area of cones or cylinders will deflect under load in different ways depending on the shape of the load applicator and the line of action of the applied force . So any calculation has to consider load distribution .

(I think you can reasonably treat the load applicator as being infinitely hard so that it does not itself deflect) .

All do-able . With cylinders rather than cones and easily defined simple load cases actually not too difficult at all .

 

[Chestermiller]

Hi sorry for the delay. Think you hit the nail on the head. I was originally under the impression that the Stress Tensor must be solved before approaching the strain equations. I thought it was more of a parameter and was putting off understanding the strain equations so that I could focus on stress. But looking ahead, it seems there are some crucial pieces to the puzzle, for example:

Conservation of linear momentum: dT_ij/(dx_i) + pb_j = pf_j
Rate-of-deformation tensor: D_ij = 1/2(dv_i / dx_j + dv_j / dx_i)
Constitutive equation for a linear elastic solid: T_ij = C_ijrs * E_rs

There's a ton to learn next.

Also, the problem I was originally working on was trying to model a simple force f on a single point p. I was apply force via my computer's cursor or by other points in the scene and wanted either a plastic or elastic deformation to eventually result.

View attachment 212189

The screenshot above is from the renderer I'm building that shows an isolated point from other points in the scene.
The below screenshot shows many more points (millions) that form a continuous body.

View attachment 212190

I'll be studying and working on this for quite some time, but will try to remember to post back here if I figure it out.

Again, much appreciated

The rate of deformation tensor is for fluids. In your solids application, you should be using the strain tensor.

 

[Nidum]

It's actually quite easy if you limit deflection of the cones/cylinders to the one axis and have a load applicator of known shape such as a section of a spherical surface .

You do the calculation backwards . The number of points which the applicator contacts and the amount they are individually compressed can be worked out by simple geometry for any specified depth of impression by the applicator .

The compressions for each point being known the individual forces acting on them are known and so the total load coming from the applicator is known .

To reverse the calculation and determine the depth of impression and spread of points generated by a specific load on the applicator just requires some simple iteration .

 

[Lewis S]

Awesome guys, marking this as solved. Now just need to put in the time to get this done.

(As a side-note, yes this is just solids so rate of deformation tensor will be out of scope, noted. Fluid simulation steps are already worked out here http://meatfighter.com/fluiddynamics/GPU_Gems_Chapter_38.pdf thankfully).