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 $F = {\sigma _1} - \frac{\alpha }(b{\sigma _2} + {\sigma _3}) = {\sigma _t},\, {\text{ when }}{\sigma _2} \leqslant \frac$|undefined(1a) $F' = \frac{1}({\sigma _1} + b{\sigma _3}) - \alpha {\sigma _3} = {\sigma _t},\, {\text{ when }} {\sigma _2} \geqslant \frac$|undefined(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. $f = {\sigma _1} - \frac{1}(b{\sigma _2} + {\sigma _3}) = {\sigma _s},{\text{ when }} {\sigma _2} \leqslant \frac{1}{2}({\sigma _1} + {\sigma _3})$|undefined(2a) $f' = \frac{1}({\sigma _1} + b{\sigma _2}) - {\sigma _3} = {\sigma _s},{\text{ when }} {\sigma _2} \geqslant \frac{1}{2}({\sigma _1} + {\sigma _3})$|undefined(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.

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 $F = {\tau _{13}} + b{\tau _{12}} + \beta ({\sigma _{13}} + b{\sigma _{12}}) = C, {\text{ when }}  {\tau _{12}} + \beta {\sigma _{12}} \geqslant {\tau _{23}} + \beta {\sigma _{23}}$|undefined(3a) $F' = {\tau _{13}} + b{\tau _{23}} + \beta ({\sigma _{13}} + b{\sigma _{23}}) = C, {\text{ when }}{\tau _{12}} + \beta {\sigma _{12}} \leqslant {\tau _{23}} + \beta {\sigma _{23}}$|undefined(3b) The relations among the stresses components and principal stresses read ${\tau _{13}} = \frac{1}{2}\left( {{\sigma _1} - {\sigma _3}} \right)$, ${\sigma _{13}} = \frac{1}{2}\left( {{\sigma _1} + {\sigma _3}} \right)$|undefined(4a) ${\tau _{12}} = \frac{1}{2}\left( {{\sigma _1} - {\sigma _2}} \right)$, ${\sigma _{12}} = \frac{1}{2}\left( {{\sigma _1} + {\sigma _2}} \right)$|undefined(4b) ${\tau _{23}} = \frac{1}{2}\left( {{\sigma _2} - {\sigma _3}} \right)$, ${\sigma _{23}} = \frac{1}{2}\left( {{\sigma _2} + {\sigma _3}} \right)$|undefined(4c) The $$\beta $$ and C should be obtained by uniaxial failure state ${\sigma _1} = {\sigma _t},{\sigma _2} = {\sigma _3} = 0$(5a) ${\sigma _1} = {\sigma _2} = 0,{\sigma _3} = - {\sigma _{\text{c}}}$|undefined(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 $\beta = \frac = \frac$, $C = \frac = \frac{\sigma _t}$(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 $$ b  = 1$$) $f = {\sigma _1} - \frac{1}{2}({\sigma _2} + {\sigma _3}) = {\sigma _t},{\text{ when }} {\sigma _2} \leqslant \frac{2}$|undefined(7a) $f = \frac{1}{2}({\sigma _1} + {\sigma _2}) - {\sigma _3} = {\sigma _t},{\text{ when }} {\sigma _2} \geqslant \frac{2}$|undefined(7b) 2. Twin-shear strength theory (UST with $$ b = 1$$). $F = {\sigma _1} - \frac{\alpha }{2}({\sigma _2} + {\sigma _3}) = {\sigma _t}, {\text{ when }}{\sigma _2} \leqslant \frac$|undefined(8a) $F = \frac{1}{2}({\sigma _1} + {\sigma _2}){\text{ - }}\alpha {\sigma _3} = {\sigma _t},{\text{ when }} {\sigma _2} \geqslant \frac$|undefined(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