# Lévy-Mises Equations

Once the yield criterion is satisfied, we can no longer expect to use the equations of elasticity. We must develop a theory to predict plastic strains from the imposed stresses.

When a body is subjected to stresses of sufficient magnitude, it will plastically deform (or fracture). The nature of the stresses depend on the particular forces applied to the body and, often, the same resulting deformation may be achieved by applying forces in different ways. For instance, a ductile metallic rod may be extended (elongated) a given amount either by a single force along its axis (i.e. a tensile stress) or by the combined action of several forces acting in different directions (i.e. multi-axial loading). A simple example of the latter multi-axial loading situation to obtain the same extension in the metallic rod as that obtained in pure tension is to apply a reduced tensile stress while simultaneously compressing the rod along its length. Under such multi-axial loading, the behaviour of ductile metallic materials can be described by the *Lévy-Mises equations*, which relate the principal components of strain increments during plastic deformation to the principal applied stresses.

In general, there will be both plastic (non-recoverable) and elastic (recoverable) strains.

However, to a first approximation, we can ignore the elastic strain assuming that the plastic strains will dominate in a deformation processing situation. We can therefore treat the material as a rigid-plastic, i.e. a material which is perfectly rigid prior to yielding and perfectly plastic afterwards.

Since plasticity is a form of flow, we can relate the strain rate, \({{{\rm{d}}\varepsilon } \over {{\rm{dt}}}}\) to stress σ.

Plastic flow is similar to fluid flow, except that any rate of flow (strain rate) can occur for the same yield stress.

From symmetry we can show that in an isotropic body, the principal axes of stress and strain rate coincide, i.e. it goes the way you push it.

With respect to principal axes \(\frac{{{{\dot \varepsilon }_1}}}{{{{\sigma '}_1}}} = \frac{{{{\dot \varepsilon }_2}}}{{{{\sigma '}_2}}} = \frac{{{{\dot \varepsilon }_3}}}{{{{\sigma '}_3}}}\),

where \({\dot \varepsilon _i} = \) \(\frac{{{\rm{d}}\varepsilon }}{{{\rm{dt}}}}\)\({\rm{ (}}i = 1,3)\) , the normal strain rate parallel to i^{th} axis.

= deviatoric component of normal stress parallel to the i^{th} axis, and

\[{\sigma '_1} = {\sigma _1} - \frac{1}{3}\left( {{\sigma _1} + {\sigma _2} + {\sigma _3}} \right)\]

If we consider small intervals of time δt, and call the resultant changes in strain δε_{1}, δε_{2}, δε_{3}, it follows that,

\[\frac{{\delta {\varepsilon _1}}}{{{\sigma _1} - \frac{1}{2}\left( {{\sigma _2} + {\sigma _3}} \right)}} = \frac{{\delta {\varepsilon _2}}}{{{\sigma _2} - \frac{1}{2}\left( {{\sigma _3} + {\sigma _1}} \right)}} = \frac{{\delta {\varepsilon _3}}}{{{\sigma _3} - \frac{1}{2}\left( {{\sigma _1} + {\sigma _2}} \right)}}\] the **Lévy-Mises equations**.

As \(\frac{1}{3}\left( {{\sigma _1} + {\sigma _2} + {\sigma _3}} \right)\) is an invariant of the stress tensor, it also turns out that these equations apply even if stresses and strains are not referred to principal axes, so

\[\frac{{\delta {\varepsilon _{11}}}}{{{\sigma _{11}} - \frac{1}{2}\left( {{\sigma _{22}} + {\sigma _{33}}} \right)}} = \frac{{\delta {\varepsilon _{22}}}}{{{\sigma _{22}} - \frac{1}{2}\left( {{\sigma _{33}} + {\sigma _{11}}} \right)}} = \frac{{\delta {\varepsilon _{33}}}}{{{\sigma _{33}} - \frac{1}{2}\left( {{\sigma _{11}} + {\sigma _{22}}} \right)}}\]

for a general stress tensor and plastic strain increments δε_{11}, δε_{22} and δε_{33}.

The above Lévy-Mises equations describe precisely the relationships between the normal stresses (arising from any general applied stress situation with respect to a particular set of orthogonal axes) and the resulting normal plastic strains (deformation) of a body referred to the same set of orthogonal axes. In many situations, the precise stresses are not known accurately and so more empirical approaches can be very helpful in describing the deformation of a body when subjected to applied forces. A number of these approaches are considered in this TLP. However, several require further constraints, in particular the need to work in two dimensions and this introduces the concepts of **plane stress** and **plane strain**.