Unified strength theory

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The unified strength theory (UST).[1][2][3][4] 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.[5][6][7]

## Mathematical formulation

[edit]

Mathematically, the formulation of UST is expressed in principal stress state as

F = σ 1 − α 1 + b ( b σ 2 + σ 3 ) = σ t , when σ 2 ⩽ σ 1 + α σ 3 1 + α {\displaystyle 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 ′ = 1 1 + b ( σ 1 + b σ 2 ) − α σ 3 = σ t , when σ 2 ⩾ σ 1 + α σ 3 1 + α {\displaystyle F'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{2}})-\alpha {\sigma _{3}}={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}}

(1b)

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.

f = σ 1 − 1 1 + b ( b σ 2 + σ 3 ) = σ s , when σ 2 ⩽ 1 2 ( σ 1 + σ 3 ) {\displaystyle 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 ′ = 1 1 + b ( σ 1 + b σ 2 ) − σ 3 = σ s , when σ 2 ⩾ 1 2 ( σ 1 + σ 3 ) {\displaystyle 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

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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.

The limit surfaces of UST with  α {\displaystyle \alpha } =0.6 The limit surfaces of UYC

## Derivation

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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.[1] Considering all the stress components of the twin-shear models and their different effects yields the unified strength theory as

F = τ 13 + b τ 12 + β ( σ 13 + b σ 12 ) = C , when τ 12 + β σ 12 ⩾ τ 23 + β σ 23 {\displaystyle 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 ′ = τ 13 + b τ 23 + β ( σ 13 + b σ 23 ) = C , when τ 12 + β σ 12 ⩽ τ 23 + β σ 23 {\displaystyle 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

τ 13 = 1 2 ( σ 1 − σ 3 ) {\displaystyle {\tau _{13}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{3}}}\right)} ,  σ 13 = 1 2 ( σ 1 + σ 3 ) {\displaystyle {\sigma _{13}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{3}}}\right)}

(4a)

τ 12 = 1 2 ( σ 1 − σ 2 ) {\displaystyle {\tau _{12}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{2}}}\right)} ,  σ 12 = 1 2 ( σ 1 + σ 2 ) {\displaystyle {\sigma _{12}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{2}}}\right)}

(4b)

τ 23 = 1 2 ( σ 2 − σ 3 ) {\displaystyle {\tau _{23}}={\frac {1}{2}}\left({{\sigma _{2}}-{\sigma _{3}}}\right)} ,  σ 23 = 1 2 ( σ 2 + σ 3 ) {\displaystyle {\sigma _{23}}={\frac {1}{2}}\left({{\sigma _{2}}+{\sigma _{3}}}\right)}

(4c)

The  β {\displaystyle \beta } and C should be obtained by uniaxial failure state

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

(5a)

σ 1 = σ 2 = 0 , σ 3 = − σ c {\displaystyle {\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  β {\displaystyle \beta } and C are introduced as

β = σ c − σ t σ c + σ t = 1 − α 1 + α {\displaystyle \beta ={\frac {{\sigma _{\text{c}}}-{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1-\alpha }{1+\alpha }}} ,  C = 1 + b σ c σ t σ c + σ t = 1 + b 1 + α σ t {\displaystyle 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

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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} )[8][9]

f = σ 1 − 1 2 ( σ 2 + σ 3 ) = σ t , when σ 2 ⩽ σ 1 + σ 3 2 {\displaystyle 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 = 1 2 ( σ 1 + σ 2 ) − σ 3 = σ t , when σ 2 ⩾ σ 1 + σ 3 2 {\displaystyle 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 {\displaystyle b=1} )[10].

F = σ 1 − α 2 ( σ 2 + σ 3 ) = σ t , when σ 2 ⩽ σ 1 + α σ 3 1 + α {\displaystyle 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 = 1 2 ( σ 1 + σ 2 ) - α σ 3 = σ t , when σ 2 ⩾ σ 1 + α σ 3 1 + α {\displaystyle 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[1].

## Applications

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Unified strength theory has been used in Generalized Plasticity,[11] Structural Plasticity,[12] Computational Plasticity[13] and many other fields[14][15]

## References

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1. ^ a b c Yu M. H., He L. N. (1991) A new model and theory on yield and failure of materials under the complex stress state. Mechanical Behaviour of Materials-6 (ICM-6). Jono M and Inoue T eds. Pergamon Press, Oxford, (3), pp. 841–846. https://doi.org/10.1016/B978-0-08-037890-9.50389-6
2. ^ Yu M. H. (2004) Unified Strength Theory and Its Applications. Springer: Berlin. ISBN 978-3-642-18943-2
3. ^ Zhao, G.-H.; Ed., (2006) Handbook of Engineering Mechanics, Rock Mechanics, Engineering Structures and Materials (in Chinese), China's Water Conservancy Resources and Hydropower Press, Beijing, pp. 20-21
4. ^ Yu M. H. (2018) Unified Strength Theory and Its Applications (second edition). Springer and Xi'an Jiaotong University Press, Springer and Xi'an. ISBN 978-981-10-6247-6
5. ^ Teodorescu, P.P. (Bucureşti). (2006). Review: Unified Strength Theory and its applications, Zentralblatt MATH Database 1931 – 2009, European Mathematical Society,Zbl 1059.74002, FIZ Karlsruhe & Springer-Verlag
6. ^ Altenbach, H., Bolchoun, A., Kolupaev, V.A. (2013). Phenomenological Yield and Failure Criteria, in Altenbach, H., Öchsner, A., eds., Plasticity of Pressure-Sensitive Materials, Serie ASM, Springer, Heidelberg, pp. 49-152.
7. ^ Kolupaev, V. A., Altenbach, H. (2010). Considerations on the Unified Strength Theory due to Mao-Hong Yu (in German: Einige Überlegungen zur Unified Strength Theory von Mao-Hong Yu), Forschung im Ingenieurwesen, 74(3), pp. 135-166.
8. ^ Yu M. H. (1961) Plastic potential and flow rules associated singular yield criterion. Res. Report of Xi'an Jiaotong University. Xi'an, China (in Chinese)
9. ^ Yu MH (1983) Twin shear stress yield criterion. International Journal of Mechanical Sciences, 25(1), pp. 71-74. https://doi.org/10.1016/0020-7403(83)90088-7
10. ^ Yu M. H., He L. N., Song L. Y. (1985) Twin shear stress theory and its generalization. Scientia Sinica (Sciences in China), English edn. Series A, 28(11), pp. 1174–1183.
11. ^ Yu M. H. et al., (2006) Generalized Plasticity. Springer: Berlin. ISBN 978-3-540-30433-3
12. ^ Yu M. H., Ma G. W., Li J. C. (2009) Structural Plasticity: Limit, Shakedown and Dynamic Plastic Analyses of Structures. ZJU Press and Springer: Hangzhou and Berlin. ISBN 978-3-540-88152-0
13. ^ Yu M. H., Li J. C. (2012) Computational Plasticity, Springer and ZJU Press: Berlin and Hangzhou. ISBN 978-3-642-24590-9
14. ^ Fan, S. C., Qiang, H. F. (2001). Normal high-velocity impaction concrete slabs-a simulation using the meshless SPH procedures. Computational Mechanics-New Frontiers for New Millennium, Valliappan S. and Khalili N. eds. Elsevier Science Ltd, pp. 1457-1462
15. ^ Guowei, M., Iwasaki, S., Miyamoto, Y. and Deto, H., 1998. Plastic limit analyses of circular plates with respect to unified yield criterion. International journal of mechanical sciences, 40(10), pp.963-976. https://doi.org/10.1016/S0020-7403(97)00140-9