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A copper wire (42 cm x 0.25 cm diameter) twisted to failure did not change its dimensions significantly. How can I visualize or quantify the cold work done (tensile at the exterior, compressive in the interior)? Does it still have a regular crystalline structure, or is it just a bunch of dislocations held together by copper atom chains?

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The stress state of the wire under torsion is neither tensile nor compressive, but pure shear. The shear strain is zero along the axis of twist, i.e. the central axis of the wire, and increases as you move radially away from this axis. Specifically, the shear strain at a point is given by the distance between the point and the axis of twist, multiplied by the twist per unit length of wire.

You're correct in that the number of dislocations increases as you cold-work the material. However, by definition, you need a crystal to have dislocations, so it's incorrect to assume that the material no longer has a crystalline structure. During work-hardening (i.e. plastic deformation), dislocations are created at a density on the order of the twist per unit length divided by the Burgers vector of a dislocation.

For more information, refer to section 2 of Fleck, N. A., Muller, G. M., Ashby, M. F., & Hutchinson, J. W. (1994). Strain gradient plasticity: Theory and experiment. Acta Metallurgica et Materialia, 42(2), 475–487. doi:10.1016/0956-7151(94)90502-9.

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The deformation of the copper wire was extreme: 89% elongation calculated for the outer fiber from the number of turns to fracture. The diameter did not change, but some length increase occurred (~2%). It was brittle, and snapped when bent a little. I don’t think it had lots of dislocations in a primarily crystalline structure; I think it had random packing to some extent, on the outside, anyway.

Copper has FCC structure (first photo, 6 x 6 x 6 atoms, from WebElements). The second photo is a random-packed 11 x 11 x 11 cluster. FCC and HCC structures have the densest possible packing of spheres: 74%; irregular packing generally comes in at about 64%. Jammed sphere packing from a diluted or tunneled FCC crystal can be as light as 49.365% packed, with over 50% of void volume (third picture). https://arxiv.org/pdf/0707.4263.pdf.

Picture 1

Picture 2

Picture 3

Irregular packing is reminiscent of liquid or amorphous (glassy) metals, and might be a fair description of the wire at its surface, but probably not so at the center, on the axis of twist.

Thank you @Chris and @blacksmith37! Now I have a better idea of what I was looking for by building on your answers.

So now a more refined question is, what would be a good way to describe and quantify the packing from the axis to the surface of the wire? Measuring density would give only an average. Or how could we measure the change in crystallinity across the diameter of such a wire? Is there an easy way to polish and etch (at an amateur level) across the diameter?

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