Engineering: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

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

[math]\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 }} }[/math]

(1a)

[math]\displaystyle{ F' = \frac{1}{{1 + b}}({\sigma _1} + b{\sigma _3}) - \alpha {\sigma _3} = {\sigma _t},\, {\text{ when }} {\sigma _2} \geqslant \frac{{{\sigma _1} + \alpha {\sigma _3}}}{{1 + \alpha }} }[/math]

(1b)

where [math]\displaystyle{ {\sigma _1},{\sigma _2},{\sigma _3} }[/math] are three principal stresses, [math]\displaystyle{ {\sigma _{t}} }[/math]is the uniaxial tensile strength and [math]\displaystyle{ \alpha }[/math] is tension-compression strength ratio ([math]\displaystyle{ \alpha = {\sigma _t}/{\sigma _c} }[/math]). The unified yield criterion (UYC) is the simplification of UST when [math]\displaystyle{ \alpha = 1 }[/math], i.e.

[math]\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}) }[/math]

(2a)

[math]\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}) }[/math]

(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 [math]\displaystyle{ \alpha }[/math]. The limit surfaces of UST and UYC are shown as follows.

The limit surfaces of UST with [math]\displaystyle{ \alpha }[/math]=0.6
The limit surfaces of UYC

Derivation of Unified Strength Theory

Due to the relation ([math]\displaystyle{ {\tau _{13}} = {\tau _{12}} + {\tau _{23}} }[/math]), the principal stress state ([math]\displaystyle{ {\sigma _1},{\sigma _2},{\sigma _3} }[/math]) may be converted to the twin-shear stress state ([math]\displaystyle{ {\tau _{13}},{\tau _{12}};{\sigma _{13}},{\sigma _{12}} }[/math]) or ([math]\displaystyle{ {\tau _{13}},{\tau _{23}};{\sigma _{13}},{\sigma _{23}} }[/math]). 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

[math]\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}} }[/math]

(3a)

[math]\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}} }[/math]

(3b)

The relations among the stresses components and principal stresses read

[math]\displaystyle{ {\tau _{13}} = \frac{1}{2}\left( {{\sigma _1} - {\sigma _3}} \right) }[/math], [math]\displaystyle{ {\sigma _{13}} = \frac{1}{2}\left( {{\sigma _1} + {\sigma _3}} \right) }[/math]

(4a)

[math]\displaystyle{ {\tau _{12}} = \frac{1}{2}\left( {{\sigma _1} - {\sigma _2}} \right) }[/math], [math]\displaystyle{ {\sigma _{12}} = \frac{1}{2}\left( {{\sigma _1} + {\sigma _2}} \right) }[/math]

(4b)

[math]\displaystyle{ {\tau _{23}} = \frac{1}{2}\left( {{\sigma _2} - {\sigma _3}} \right) }[/math], [math]\displaystyle{ {\sigma _{23}} = \frac{1}{2}\left( {{\sigma _2} + {\sigma _3}} \right) }[/math]

(4c)

The [math]\displaystyle{ \beta }[/math] and C should be obtained by uniaxial failure state

[math]\displaystyle{ {\sigma _1} = {\sigma _t},{\sigma _2} = {\sigma _3} = 0 }[/math]

(5a)

[math]\displaystyle{ {\sigma _1} = {\sigma _2} = 0,{\sigma _3} = - {\sigma _{\text{c}}} }[/math]

(5b)

By substituting Eqs.(4a), (4b) and (5a) into the Eq.(3a), and substituting Eqs.(4a), (4c) and (5b) into Eq.(3b), the [math]\displaystyle{ \beta }[/math] and C are introduced as

[math]\displaystyle{ \beta = \frac{{{\sigma _{\text{c}}} - {\sigma _{\text{t}}}}}{{{\sigma _{\text{c}}} + {\sigma _{\text{t}}}}} = \frac{{1 - \alpha }}{{1 + \alpha }} }[/math], [math]\displaystyle{ C = \frac{{1 + b{\sigma _{\text{c}}}{\sigma _{\text{t}}}}}{{{\sigma _{\text{c}}} + {\sigma _{\text{t}}}}} = \frac{{1 + b}}{{1 + \alpha }}{\sigma _t} }[/math]

(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 [math]\displaystyle{ \alpha = 1 }[/math] and [math]\displaystyle{ b = 1 }[/math])[8][9]

[math]\displaystyle{ f = {\sigma _1} - \frac{1}{2}({\sigma _2} + {\sigma _3}) = {\sigma _t},{\text{ when }} {\sigma _2} \leqslant \frac{{{\sigma _1} + {\sigma _3}}}{2} }[/math]

(7a)

[math]\displaystyle{ f = \frac{1}{2}({\sigma _1} + {\sigma _2}) - {\sigma _3} = {\sigma _t},{\text{ when }} {\sigma _2} \geqslant \frac{{{\sigma _1} + {\sigma _3}}}{2} }[/math]

(7b)

2. Twin-shear strength theory (UST with [math]\displaystyle{ b = 1 }[/math])[10].

[math]\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 }} }[/math]

(8a)

[math]\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 }} }[/math]

(8b)

3. Unified strength theory[1].

Applications of the Unified Strength theory

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

References

  1. ↑ 1.0 1.1 1.2 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



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