Editing Plasticity (physics) (section)

== Mathematical descriptions ==
=== Deformation theory ===
[[File:stress-strain1.svg|class=skin-invert-image|thumb|right|An idealized uniaxial [[stress-strain curve]] showing elastic and plastic deformation regimes for the deformation theory of plasticity]]

There are several mathematical descriptions of plasticity.<ref name=Hill>{{cite book |first=Rodney |last=Hill |author-link=Rodney Hill |year=1998 |title=The Mathematical Theory of Plasticity |publisher=Oxford University Press |isbn=0-19-850367-9 }}</ref> One is deformation theory (see e.g. [[Hooke's law]]) where the [[Cauchy stress tensor]] (of order d-1 in d dimensions) is a function of the strain tensor. Although this description is accurate when a small part of matter is subjected to increasing loading (such as strain loading), this theory cannot account for irreversibility.

Ductile materials can sustain large plastic deformations without [[fracture]].  However, even ductile metals will fracture when the [[strain (materials science)|strain]] becomes large enough—this is as a result of [[work hardening]] of the material, which causes it to become [[brittle]].  [[Heat treatment]] such as [[annealing (metallurgy)|annealing]] can restore the [[ductility]] of a worked piece, so that shaping can continue.

=== Flow plasticity theory ===
{{main|Flow plasticity theory}}
In 1934, [[Egon Orowan]], [[Michael Polanyi]] and [[Geoffrey Ingram Taylor]], roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of [[dislocations]].  The mathematical theory of plasticity, [[flow plasticity theory]], uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation.