# Unified strength theory

The unified strength theory (UST). proposed by Yu Mao-Hong is a series of yield criteria (see yield surface) and failure criteria (see Material failure theory). It is a generalized classical strength theory which can be used to describe the yielding or failure of material begins when the combination of principal stresses reaches a critical value.

## Mathematical formulation

Mathematically, the formulation of UST is expressed in principal stress state as
$F={\sigma _{1}}-{\frac {\alpha }{1+b}}(b{\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$
$F'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{2}})-\alpha {\sigma _{3}}={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$
where ${\sigma _{1}},{\sigma _{2}},{\sigma _{3}}$ are three principal stresses, ${\sigma _{t}}$ is the uniaxial tensile strength and $\alpha$ is tension-compression strength ratio ($\alpha ={\sigma _{t}}/{\sigma _{c}}$). The unified yield criterion (UYC) is the simplification of UST when $\alpha =1$, i.e.
$f={\sigma _{1}}-{\frac {1}{1+b}}(b{\sigma _{2}}+{\sigma _{3}})={\sigma _{s}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {1}{2}}({\sigma _{1}}+{\sigma _{3}})$
$f'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{2}})-{\sigma _{3}}={\sigma _{s}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {1}{2}}({\sigma _{1}}+{\sigma _{3}})$

## Limit surfaces

The limit surfaces of the unified strength theory in principal stress space are usually a semi-infinite dodecahedron cone with unequal sides. The shape and size of the limiting dodecahedron cone depends on the parameter b and $\alpha$. The limit surfaces of UST and UYC are shown as follows.

## Derivation

Due to the relation (${\tau _{13}}={\tau _{12}}+{\tau _{23}}$), the principal stress state (${\sigma _{1}},{\sigma _{2}},{\sigma _{3}}$) may be converted to the twin-shear stress state (${\tau _{13}},{\tau _{12}};{\sigma _{13}},{\sigma _{12}}$) or (${\tau _{13}},{\tau _{23}};{\sigma _{13}},{\sigma _{23}}$). Twin-shear element models proposed by Mao-Hong Yu are used for representing the twin-shear stress state. Considering all the stress components of the twin-shear models and their different effects yields the unified strength theory as
$F={\tau _{13}}+b{\tau _{12}}+\beta ({\sigma _{13}}+b{\sigma _{12}})=C,{\text{ when }}{\tau _{12}}+\beta {\sigma _{12}}\geqslant {\tau _{23}}+\beta {\sigma _{23}}$
$F'={\tau _{13}}+b{\tau _{23}}+\beta ({\sigma _{13}}+b{\sigma _{23}})=C,{\text{ when }}{\tau _{12}}+\beta {\sigma _{12}}\leqslant {\tau _{23}}+\beta {\sigma _{23}}$
The relations among the stresses components and principal stresses read
${\tau _{13}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{3}}}\right)$
${\sigma _{13}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{3}}}\right)$
${\tau _{12}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{2}}}\right)$
${\sigma _{12}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{2}}}\right)$
${\tau _{23}}={\frac {1}{2}}\left({{\sigma _{2}}-{\sigma _{3}}}\right)$
${\sigma _{23}}={\frac {1}{2}}\left({{\sigma _{2}}+{\sigma _{3}}}\right)$
The $\beta$ and C should be obtained by uniaxial failure state
${\sigma _{1}}={\sigma _{t}},{\sigma _{2}}={\sigma _{3}}=0$
${\sigma _{1}}={\sigma _{2}}=0,{\sigma _{3}}=-{\sigma _{\text{c}}}$
By substituting Eqs.(4a), (4b) and (5a) into the Eq.(3a), and substituting Eqs.(4a), (4c) and (5b) into Eq.(3b), the $\beta$ and C are introduced as
$\beta ={\frac {{\sigma _{\text{c}}}-{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1-\alpha }{1+\alpha }}$
$C={\frac {1+b{\sigma _{\text{c}}}{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1+b}{1+\alpha }}{\sigma _{t}}$

## History

The development of the unified strength theory can be divided into three stages as follows. 1. Twin-shear yield criterion (UST with $\alpha =1$ and $b=1$)
$f={\sigma _{1}}-{\frac {1}{2}}({\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+{\sigma _{3}}}{2}}$
$f={\frac {1}{2}}({\sigma _{1}}+{\sigma _{2}})-{\sigma _{3}}={\sigma _{t}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+{\sigma _{3}}}{2}}$
2. Twin-shear strength theory (UST with $b=1$).
$F={\sigma _{1}}-{\frac {\alpha }{2}}({\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$
$F={\frac {1}{2}}({\sigma _{1}}+{\sigma _{2}}){\text{ - }}\alpha {\sigma _{3}}={\sigma _{t}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$
3. Unified strength theory.

## Applications

Unified strength theory has been used in Generalized Plasticity, Structural Plasticity, Computational Plasticity and many other fields