I'm trying to come up with a way to describe deformation on a body at each point on that body. My problem can be expressed and solved in two dimensions. Given two triangles, one being the initial shape, the other being the initial shape transformed arbitrarily :

Deformation of a triangle

It's evident by looking at this that points along the left edge are now more distant to each other, points along the right edge are closer, points along the bottom edge have the same distance as before, and any point inside that triangle is blend of those three states. Is there any engineering standard to describe this, for any position on the body? I've looked at stress and strain, but I'm not sure if either describe what I want. In the end, I'm trying to find a value (or possibly two? Maybe a vector?) that describe, for a tiny area, if that area is compressed or stretched after the transformation.

PS.: I'm not a mechanical engineer if that wasn't already obvious. :)


  • $\begingroup$ See en.wikipedia.org/wiki/Simple_shear. The Wikipedia page talks about a rectangle not a triangle, but that makes no difference. $\endgroup$
    – alephzero
    Jun 4, 2017 at 22:13
  • $\begingroup$ @alephzero thanks. I've clarified that the transformation is arbitrary. However, just describing the transformation is not enough - I want to know, for every point, if that point is closer or farther to it's neighbors than before. $\endgroup$
    – subb
    Jun 4, 2017 at 22:21
  • $\begingroup$ I suggest you research basic finite element modelling techniques, which frequently deal with triangular elements. Part of the modelling is precisely a description of the change from the original to the deformed configuration at any point in the surface. $\endgroup$
    – Wasabi
    Jun 5, 2017 at 2:53
  • $\begingroup$ @subb you are changing your original question "a standard description" as you now want to know "if that point is closer or further from its neighbours than before". As well as looking up FEM as suggested you should look at tension and compression. But first, you need to make sure you understand your own question. $\endgroup$
    – Solar Mike
    Jun 5, 2017 at 4:49
  • $\begingroup$ " I've looked at stress and strain" strain is precisely the quantity you are seeking. $\endgroup$
    – agentp
    Jun 5, 2017 at 20:35

1 Answer 1


Displacement field - a vector field, that assigns a displacement vector to each infinitesimal particle of the body over time.

The object, in this case will be usually defined by a density field (scalar field) with zero density being "outside the object."

This description works on all kinds of matter - gas, fluid mechanics, and non-rigid solids, their behavior changing with the kind of interaction between neighboring infinitesimal particles (stress/strain, pressure, viscosity etc.) which all interact with 2nd derivative of position over time (=acceleration) resulting in different equations of motion/displacement, but the underlying description of object changing remains a displacement field of one kind of another.


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