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
where σ 1 , σ 2 , σ 3 {\displaystyle {\sigma _{1}},{\sigma _{2}},{\sigma _{3}}} are three principal stresses, σ t {\displaystyle {\sigma _{t}}} is the uniaxial tensile strength and α {\displaystyle \alpha } is tension-compression strength ratio ( α = σ t / σ c {\displaystyle \alpha ={\sigma _{t}}/{\sigma _{c}}} ). The unified yield criterion (UYC) is the simplification of UST when α = 1 {\displaystyle \alpha =1} , i.e.
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 α {\displaystyle \alpha } . The limit surfaces of UST and UYC are shown as follows.
Derivation Due to the relation ( τ 13 = τ 12 + τ 23 {\displaystyle {\tau _{13}}={\tau _{12}}+{\tau _{23}}} ), the principal stress state ( σ 1 , σ 2 , σ 3 {\displaystyle {\sigma _{1}},{\sigma _{2}},{\sigma _{3}}} ) may be converted to the twin-shear stress state ( τ 13 , τ 12 ; σ 13 , σ 12 {\displaystyle {\tau _{13}},{\tau _{12}};{\sigma _{13}},{\sigma _{12}}} ) or ( τ 13 , τ 23 ; σ 13 , σ 23 {\displaystyle {\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
The relations among the stresses components and principal stresses read
The β {\displaystyle \beta } and C should be obtained by uniaxial failure state
By substituting Eqs.(4a), (4b) and (5a) into the Eq.(3a), and substituting Eqs.(4a), (4c) and (5b) into Eq.(3b), the β {\displaystyle \beta } and C are introduced as
History The development of the unified strength theory can be divided into three stages as follows. 1. Twin-shear yield criterion (UST with α = 1 {\displaystyle \alpha =1} and b = 1 {\displaystyle b=1} )
2. Twin-shear strength theory (UST with b = 1 {\displaystyle b=1} ).
3. Unified strength theory.
Applications Unified strength theory has been used in Generalized Plasticity, Structural Plasticity, Computational Plasticity and many other fields
References