# Deformation gradient tensor from particle displacements in an inflating material

I don't have a mechanics background, but am trying very hard to read and understand how to approach this problem. I am struggling with understanding what's "correct" or the basic process of going from what I have to understanding the steps I need to take next.

Here is the system: An initially flat material clamped in a circular aperture is inflated in a bulge test. I cannot use any of the common spherical cap equations (I've done this already, but am trying to move to a better measurement). The shape that is formed is more of a mountain-like shape and not a hemisphere. I also don't want to assume pure bending with a plate approximation, as the deflection reaches about 4-5 times the thickness. I think it probably undergoes an initial bending stage and then transitions to stretching. It definitely goes beyond the elastic regime as the material eventually ruptures.

I have 3D images that give me a view of the shape, as well as the x,y,z coordinates (and hence displacement vectors) of hundreds of particles embedded in the material as it undergoes deformation.

My big questions are below (note that I don't mention what I "need" yet because I am not sure how this problem is commonly approached):

1) First, what is the general protocol for tackling a problem like this? Is there a normal flowchart that people work through when they see a problem like this?

2) Is there a way to get stress field if you figure out strain field first? How do you go from strain to stress?

3) With respect to figuring out the strain field: I have the displacement vectors for every particle throughout the experiment.

3a) Is it possible to get the deformation gradient F just using this collection of displacements? I don't know what function (if any) relates the coordinates of the point pre-deformation to the position post-deformation.

3b) Suppose I am able to get deformation gradient F. Just looking at F, all the information is in terms of the x, y, and z axes. But given that my shape goes from being a plate to a mountain, it doesn't seem like information in these three directions make a lot of sense in terms of "useful" information. Suppose I was curious about how much tranverse shear there is, or lateral shear, or just tension at different locations on the shape? But xz shear/xy shear doesn't mean anything if I have a mountain shape, right?

In that case, isn't it more useful to know the deformation gradient along a different axis oriented with respect to the important directions on the curved mountain shape? Like, suppose at any point in this mountain-like shape, I decide on 3 local orthogonal axes - one that is normal to the surface, one that is pointing "up" the mountain, and one that is tangent to the circular contour: and then I somehow got the deformation gradient with respect to these three axes at any point: Would this be meaningful in terms of answering my question of lateral/transverse shear on the shape vs tension?

Is this a stupid/wrong method?

I would be grateful for any guidance here -- even if I just know whether the direction I'm going in sounds right or would give meaningless information.

Thank you!

• Is the material anisotropic? – Solar Mike May 22 '20 at 6:56
• The composition varies across the material so... Somewhat, yes. – confusedstudent May 22 '20 at 17:21