Editing Plasticity (physics) (section)

== Yield criteria ==
[[File:Critere tresca von mises.svg|class=skin-invert-image|thumb|Comparison of Tresca criterion to Von Mises criterion]]
{{main|Yield (engineering)}}

If the stress exceeds a critical value, as was mentioned above, the material will undergo plastic, or irreversible, deformation. This critical stress can be tensile or compressive.  The Tresca and the [[von Mises yield criterion|von Mises]] criteria are commonly used to determine whether a material has yielded.  However, these criteria have proved inadequate for a large range of materials and several other yield criteria are also in widespread use.

===Tresca criterion===
The Tresca criterion is based on the notion that when a material fails, it does so in shear, which is a relatively good assumption when considering metals. Given the principal stress state, we can use [[Mohr's circle]] to solve for the maximum shear stresses our material will experience and conclude that the material will fail if

: <math>\sigma_1 - \sigma_3 \ge \sigma_0</math>

where ''σ''<sub>1</sub> is the maximum normal stress, ''σ''<sub>3</sub> is the minimum normal stress, and ''σ''<sub>0</sub> is the stress under which the material fails in uniaxial loading. A [[yield surface]] may be constructed, which provides a visual representation of this concept. Inside of the yield surface, deformation is elastic. On the surface, deformation is plastic. It is impossible for a material to have stress states outside its yield surface.

===Huber–von Mises criterion===
[[File:Yield surfaces.svg|class=skin-invert-image|thumb|right|The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder around the hydrostatic axis. Also shown is [[Henri Tresca|Tresca]]'s hexagonal yield surface.]]
{{main|Von Mises yield criterion}}

The Huber–von Mises criterion<ref>{{cite journal |last=von Mises |first=Richard |author-link=Richard von Mises |year=1913 |title=Mechanik der festen Körper im plastisch-deformablen Zustand |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen |series=Mathematisch-Physikalische Klasse |volume=1913 |issue=1 |pages=582–592 |url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002503697 }}</ref> is based on the Tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure. [[Tytus Maksymilian Huber|M. T. Huber]] was the first who proposed the criterion of shear energy.<ref>{{cite journal |last=Huber |first=Maksymilian Tytus |author-link=Tytus Maksymilian Huber |title=Właściwa praca odkształcenia jako miara wytezenia materiału |journal=Czasopismo Techniczne |location=Lwów |year=1904 |volume=22 }} Translated as {{cite journal |title=Specific Work of Strain as a Measure of Material Effort |journal=Archives of Mechanics |volume=56 |pages=173–190 |year=2004 |url=http://am.ippt.pan.pl/am/article/viewFile/v56p173/pdf }}</ref><ref>See {{cite book |first=Stephen P. |last=Timoshenko |author-link=Stephen Timoshenko |title=History of Strength of Materials |location=New York |publisher=McGraw-Hill |year=1953 |page=369 |url=https://books.google.com/books?id=tkScQmyhsb8C&pg=PA369 |isbn=9780486611877 }}</ref> Von Mises solves for an [[effective stress]] under uniaxial loading, subtracting out hydrostatic stresses, and states that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation.

: <math>\sigma_v^2 = \tfrac{1}{2}[(\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{11} - \sigma_{33})^2 + 6(\sigma_{23}^2 + \sigma_{31}^2 + \sigma_{12}^2)]</math>

Again, a visual representation of the yield surface may be constructed using the above equation, which takes the shape of an ellipse. Inside the surface, materials undergo elastic deformation. Reaching the surface means the material undergoes plastic deformations.