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

$F={\sigma }_1-\frac{\alpha }{1+b}(b{\sigma }_2+{\sigma }_3)={\sigma }_t,\text{ when }{\sigma }_2⩽\frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }$

(1a)

$F^{\prime }=\frac{1}{1+b}({\sigma }_1+b{\sigma }_3)-\alpha {\sigma }_3={\sigma }_t,\text{ when }{\sigma }_2⩾\frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }$

(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.

$f={\sigma }_1-\frac{1}{1+b}(b{\sigma }_2+{\sigma }_3)={\sigma }_s,\text{ when }{\sigma }_2⩽\frac{1}{2}({\sigma }_1+{\sigma }_3)$

(2a)

$f^{\prime }=\frac{1}{1+b}({\sigma }_1+b{\sigma }_2)-{\sigma }_3={\sigma }_s,\text{ when }{\sigma }_2⩾\frac{1}{2}({\sigma }_1+{\sigma }_3)$

(2b)

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.

File:UST limit surface wikipedia.jpg

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

File:UYC Plane.jpg

The limit surfaces of UYC

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

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

(3a)

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

(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

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 $b=1$)

$f={\sigma }_1-\frac{1}{2}({\sigma }_2+{\sigma }_3)={\sigma }_t,\text{ when }{\sigma }_2⩽\frac{{\sigma }_1+{\sigma }_3}{2}$

(7a)

$f=\frac{1}{2}({\sigma }_1+{\sigma }_2)-{\sigma }_3={\sigma }_t,\text{ when }{\sigma }_2⩾\frac{{\sigma }_1+{\sigma }_3}{2}$

(7b)

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

$F={\sigma }_1-\frac{\alpha }{2}({\sigma }_2+{\sigma }_3)={\sigma }_t,\text{ when }{\sigma }_2⩽\frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }$

(8a)

$F=\frac{1}{2}({\sigma }_1+{\sigma }_2)\text{ - }\alpha {\sigma }_3={\sigma }_t,\text{ when }{\sigma }_2⩾\frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }$

(8b)

3. Unified strength theory.

Applications

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