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

$F={\sigma _{1}}-{\frac {\alpha }{1+b}}(b{\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$

(1a)

$F'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{3}})-\alpha {\sigma _{3}}={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$

(1b)

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.

$f={\sigma _{1}}-{\frac {1}{1+b}}(b{\sigma _{2}}+{\sigma _{3}})={\sigma _{s}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {1}{2}}({\sigma _{1}}+{\sigma _{3}})$

(2a)

$f'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{2}})-{\sigma _{3}}={\sigma _{s}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {1}{2}}({\sigma _{1}}+{\sigma _{3}})$

(2b)

Limit surfaces of Unified Strength Theory

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.

The limit surfaces of UST with $\alpha$=0.6

The limit surfaces of UYC

Derivation of Unified Strength Theory

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

$F={\tau _{13}}+b{\tau _{12}}+\beta ({\sigma _{13}}+b{\sigma _{12}})=C,{\text{ when }}{\tau _{12}}+\beta {\sigma _{12}}\geqslant {\tau _{23}}+\beta {\sigma _{23}}$

(3a)

$F'={\tau _{13}}+b{\tau _{23}}+\beta ({\sigma _{13}}+b{\sigma _{23}})=C,{\text{ when }}{\tau _{12}}+\beta {\sigma _{12}}\leqslant {\tau _{23}}+\beta {\sigma _{23}}$

(3b)

The relations among the stresses components and principal stresses read

${\tau _{13}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{3}}}\right)$, ${\sigma _{13}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{3}}}\right)$

(4a)

${\tau _{12}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{2}}}\right)$, ${\sigma _{12}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{2}}}\right)$

(4b)

${\tau _{23}}={\frac {1}{2}}\left({{\sigma _{2}}-{\sigma _{3}}}\right)$, ${\sigma _{23}}={\frac {1}{2}}\left({{\sigma _{2}}+{\sigma _{3}}}\right)$

(4c)

The $\beta$ and C should be obtained by uniaxial failure state

${\sigma _{1}}={\sigma _{t}},{\sigma _{2}}={\sigma _{3}}=0$

(5a)

${\sigma _{1}}={\sigma _{2}}=0,{\sigma _{3}}=-{\sigma _{\text{c}}}$

(5b)

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

$\beta ={\frac {{\sigma _{\text{c}}}-{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1-\alpha }{1+\alpha }}$, $C={\frac {1+b{\sigma _{\text{c}}}{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1+b}{1+\alpha }}{\sigma _{t}}$

(6)

History of Unified Strength Theory

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 $b=1$)

$f={\sigma _{1}}-{\frac {1}{2}}({\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+{\sigma _{3}}}{2}}$

(7a)

$f={\frac {1}{2}}({\sigma _{1}}+{\sigma _{2}})-{\sigma _{3}}={\sigma _{t}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+{\sigma _{3}}}{2}}$

(7b)

2. Twin-shear strength theory (UST with $b=1$).

$F={\sigma _{1}}-{\frac {\alpha }{2}}({\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$

(8a)

$F={\frac {1}{2}}({\sigma _{1}}+{\sigma _{2}}){\text{ - }}\alpha {\sigma _{3}}={\sigma _{t}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$

(8b)

3. Unified strength theory.

Applications of the Unified Strength theory

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