Measurement of Dominant Deformation Modes

I understand the sarcasam. I am dealing with a manager that does not understand nonlinear mechanics of materials but has experience in running automated FEA from a previous company. He refuses to believe that what his previous company did is incorrect. So I need to provide hard data to show him that yes our products show deformations other than uniaxial tension. He wants color plots not vectors. I believe that I may be fighting a losing battle.

As to the topic...

One idea I have is to use the principal streches of hte mateiral point compared to what the principle stretches should be for Uniaxial Tension, Biaxial Tension, and Planar Tension. I can calcualte a series of ratios if I assume that the max principla strecth to be the main strethc. The closer these coefficents are to 1.0 the more likely the mode of deformation falls into the category. Below are the ratios I came up with:

s1, s2, s3 = (max, mid, min) principal stretches

[U][B]Uniaxial Tension[/B][/U]

UT_N = s2 * sqrt(s1)

UT_M = s3 * sqrt(s1)

[U][B]Biaxial Tension[/B][/U]

EB_N = s1*s2

EB_M = s1^2 * s3

[U][B]Planar Tension[/B][/U]

PT_N = s2

PT_M = s1 * s3

I see your problem now, one I am quite familiar with!

Back to your problem, I am not so sure I understand your plan. If you compare principal stretches you are automatically comparing quantities defined in different coordinate systems. I mean, however complex the deformation, the computation of principal stretches implies locally rotating the coordinate system to an orientation such that the strain tensor is diagonal. The key aspect in uniaxial deformation is that such rotation is nihil everywhere, i.e. if you select your global orientation as the one for which the strain tensor of an arbitrary point is diagonal, then the strain tensor is diagonal in this system for all material points.

Let me clarify with an example. Imagine a bar of material undergoing uniaxial strain, apart from a small region undergoing localized shear (shear band, for example). Then you can choose the shear strain in such a way the principal strain is in magnitude constant! which if I understood you correctly, would invalidate your approach (your metric would suggest the deformation being uniaxial)

That us why i proposed to quantify the rotation needed to bring the strain tensor in an arbitrary point in the diagonal form. If your manager prefers colour plots, you can still:

1) choose a region of the product where the strain field in mainly uniaxial

2) Compute the principal strain orientation and consider this your reference orientation

3) For all other integration points, you compute the prinical stretches: the difference between their orientation and the orientation chosen at step 2) is then a scalar field you can use to obtain a colour plot.

As a variation of the procedure, you can also assign different weights to different material points, based on the principal stretch magnitude: in this way your metric will assign more importance to areas characterized by high strains.

Hope this is of some help, let us know how it goes!

PS: On second thought, why not try something simpler for our manager? Show him a plot (after performing a good selection of the global reference system, i.e. aligned as much as possible with the prevailing uniaxial strain field)) of the diagonal terms in the strain tensor (linked to normal strains), and the off-diagonal terms (linked to shear). In an uniaxial deformation the second plot should be zero everywhere!

Dear jrherron,

I think I know exactly what you need:
Reformulation of strain invariants at incompressibility
[url] http://link.springer.com/content/pdf/10.1007%2Fs00419-012-0652-2.pdf [/url]

In this paper there is an equation (23) of a strain mode exponent m:
m = 1.0 biaxial
m = 0.0 shear
m = -0.5 uniaxial

Mr. Inventor,

that is an interesting paper, but how do you put it at use? The deformation measure they introduced is local!

You could simulate a rubber part with a simple neo-hooke hyperelastic model.

After the simulation you can use the equation to determine the strain mode of the elements.

In Abaqus for example, you can use the create field output tool to calculate m for all elements.

Then you will see which strain modes the rubber part has, and you can determine which tests you need calibrate a suitable hyperelastic material model.