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 Template:CenterTemplate:Right
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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. Template:CenterTemplate:Right
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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.

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

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 Template:CenterTemplate:Right
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The relations among the stresses components and principal stresses read Template:CenterTemplate:Right
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The $\beta$ and C should be obtained by uniaxial failure state
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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
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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 $b=1$)
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2. Twin-shear strength theory (UST with $b=1$).
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3. Unified strength theory.

Applications

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