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How to supply a visualization for the displacement gradient tensor

Hi all, 

[Warning: The writing is long, as is usually the case with my posts :)]

It all began with a paper that I proposed for an upcoming conference in India. The extended abstract got accepted, of course, but my work is still in progress, and today I am not sure if I can meet the deadline. So, I may perhaps withdraw it, and then submit a longer version of it to a journal, later.

Anyway, here is a gist of the idea behind the paper. I am building a very small pedagogical software called "toyDNS." DNS stands for Displacement, straiN, and streSs, and the order of the letters in the acronymn emphasizes what I (now) believe is the correct hierarchical order for the three concepts. Anyway, let's keep the hierarchical order aside and look into what the software does---which I guess could be more interesting.

The sofware is very very small and simple. It begins by showing the user a regular 2D grid (i.e. squares). The user distorts the grid using the mouse (somewhat similar to the action of an image-warping software). The software then, immediately (in real time, without using menus etc.) computes and shows the following fields in the adjacent windows: (i) the displacement vector field, (ii) the displacement gradient tensor field, (iii) the rotation field, (iv) the strain field, (v) and the stress field. The software assumes plane-stress, linear elasticity, and uses static configuration data for material properties like nu and E. The software also shows the boundary tractions (forces) that would be required to maintain the displacement field that the user has specified.

Basically, the idea is that the beginning undergraduate student encountering the mechanics of materials for the first time, gets to see the importance of the rotation field (which is usually not emphasized in textbooks or courses), and thereby is able to directly appreciate the reason why an arbitrary displacement field does uniquely determines the corresponding stress fields but why the converse is not true---why an arbitrary stress/strain field cannot uniquely determine a corresponding displacement field. To illustrate this point (call it the compatibility issue if you wish) is the whole rationale behind this toy software.

Now, when it comes to visualizing the fields, I can always use arrows for showing the vector fields of displacements and forces. For strains and stresses, I can use Lame's ellipse (in 2D). In fact, since the strain and stress fields are symmetric, in 2D, they each have only 3 components, which means that the symmetric tensor object as a whole can directly map onto an RGB (or HLS) color-space, and so, I can also show a single, full-color field plot for the strain (or stress) field.

Ok. So far, so good.

The problem is with the displacement gradient tensor (DG for short here). Since the displacement field is arbitrary, there is no symmetry to the DG tensor. Hence, even in 2D, there are 4 independent components to it---i.e. one component too many than what can be accomodated in the three-component color-space. So, a direct depiction of the tensor object taken as a whole is not possible, and something else has to be done. So, I thought of the following idea.

First, the notation. Assume that the DG tensor is being described thus:

DG11 DG12
DG21 DG22


du/dx du/dy
dv/dx dv/dy

where DGij are the components of the DG tensor, u and v are the x- and y-components of the displacement field, and the d's represent the partial differentation. (Also imagine as if the square brackets of the matrix notation are placed around the components listing above.)

Consider that DGij can be taken to represent a component of a vector that refers to the i-th face and j-th direction. Understanding this scheme is easier to do for the stress tensor. For the stress tensor, Sij is the component of the traction vector acting across the i-the face and pointing in the j-th direction. For instance, in fig. 2.3 here:, T^{e_1} is the vector acting across the face normal to the 1-axis.

Even if the DG tensor is not symmetric, the basic idea would still apply, wouldn't it?

Thus, each row in the DG tensor represents a vector: the first row is a vector acting on the face normal to the x-axis, and the second is another vector (which, for DG, is completely indpendent of the first) acting on the face normal to the y-axis. For 2D, subsitute "line" in place of "face."

If I now show these two vectors, they would completely describe the DG tensor. This representation would be somewhat similar to the "cross-bars" visualization commonly used in engineering software for the stress tensor, wherein the tensor field is shown using periodically cross-bars---very convenient if the grid is regular and uniform and has square elements.

Notice a salient difference, however. Since the DG tensor is asymmetric, the two vectors will not in general lie at right-angles to each other. The latter is the case only with the symmetric tensors such as the strain and stress tensors.

My question is this: Do you see any issues with this kind of visualization for the DG tensor? Is there any loss of generality by following this scheme of visualization? I mean, I read some literature on visualization of asymmetric tensors, and noticed that they sometimes worry about the eigenvalues being complex, not real. I think that complex eigenvalues would not be a consideration for the above kind of depiction of the DG tensor---the rotation part will be separately shown in a separate window anyway. But, still, I wanted to have the generality aspect cross-checked. Hence this post. Am I missing something? assuming too much? What are the other things, if any, that I need to consider? Also: Would you be "intuitively" comfortable with this scheme? Can you think of or suggest any alternatives?

Comments are welcome.




You can use du/dx = F - I = RU - I where R is the rotation (which can be represented by an axial vector) and U is the stretch which can be represented by an ellipsoid.  See for more ideas in terms of eigenvalues of unsymmetric matrices (p. 48).

-- Biswajit


0. I have something urgent at work here coming up, so this is just a "quick" note. I will write a more elaborate reply this evening or tomorrow.

Anyway, let me first note that I have caught a mistake---an inconsistency---in my own description above. See if you can catch it. Hint: It makes the visualization for DG that I thought of, inconsistent with the visualization of the strain/stress tensors as cross-bars oriented along principal axes. And, the inconsistency implies that the visualization for DG, at least in the form stated above, would not be so "intuitive"ly useful.

1. Biswajit, I downloaded and saw Brannon's Mohrs_Circle document. I began at p. 44, and got a bit stumped around eq. 3.1 and 3.2, because she uses a notation that is not quite "standard" (Dieter, Shames, Beer and Johnson, Sadd...), transposing, as she does, rows and columns around. Therefore, I need to work carefully through her notes, supplying component definitions in "my" terms at each step.

On another note, I also am not sure the point you wish to make with the equation you give above. Could you please elaborate on the idea?

2. Another point. I have also now noticed Brannon's papers that advocate the use of Mohr's circle in tensor visualizations. I would like to go through these ideas in detail (and so give me some time to come back on those).

Yet, I can state right away that I would like to stick by my contentions that Mohr's circle is not at all intuitive and that visualization along the lines of Lame's ellipse are far superior. To add (Hindi) "masala": Towards this end, I should be happy to take on the entire Sandia National Labs!

However, of course, I wouldn't mind supplying Mohr's circles as an additional visualization (not the primary one).

More, as I said, later. Sorry for the inconsistency (to be pointed out in detail in a couple of days).

- - - - -

wallstedt's picture

Ajit, I think that anything that can be done to better understand and visualize tensors is a great idea; I also echo Biswajit's pointer to the work of Rebecca Brannon.  There was another project done at Utah by way of visualization of tensors: some computer science people put together a routine to visualize the deformation and rotation of tensors which can be seen at the link below.

WaiChing Sun's picture

Hi AJit,

   I think this paper (see URL below) may be helpful. 



Digital Object Identifier: 10.1109/VISUAL.2004.80 




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