Unified strength theory

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]

## Contents

* 1 Mathematical Formulation
* 2 Limit surfaces of Unified Strength Theory
* 3 Derivation of Unified Strength Theory
* 4 History of Unified Strength Theory
* 5 Applications of the Unified Strength theory
* 6 References

## Mathematical Formulation

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

F=σ1−α1+b(bσ2+σ3)=σt, when σ2⩽σ1+ασ31+α

(1a)

F′=11+b(σ1+bσ3)−ασ3=σt, when σ2⩾σ1+ασ31+α

(1b)

where σ1,σ2,σ3 are three principal stresses, σtis the uniaxial tensile strength and α is tension-compression strength ratio (α=σt/σc). The unified yield criterion (UYC) is the simplification of UST when α=1, i.e.

f=σ1−11+b(bσ2+σ3)=σs, when σ2⩽12(σ1+σ3)

(2a)

f′=11+b(σ1+bσ2)−σ3=σs, when σ2⩾12(σ1+σ3)

(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 α. The limit surfaces of UST and UYC are shown as follows.

The limit surfaces of UST with α=0.6 The limit surfaces of UYC

## Derivation of Unified Strength Theory

Due to the relation (τ13=τ12+τ23), the principal stress state (σ1,σ2,σ3) may be converted to the twin-shear stress state (τ13,τ12;σ13,σ12) or (τ13,τ23;σ13,σ23). 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=τ13+bτ12+β(σ13+bσ12)=C, when τ12+βσ12⩾τ23+βσ23

(3a)

F′=τ13+bτ23+β(σ13+bσ23)=C, when τ12+βσ12⩽τ23+βσ23

(3b)

The relations among the stresses components and principal stresses read

τ13=12(σ1−σ3), σ13=12(σ1+σ3)

(4a)

τ12=12(σ1−σ2), σ12=12(σ1+σ2)

(4b)

τ23=12(σ2−σ3), σ23=12(σ2+σ3)

(4c)

The β and C should be obtained by uniaxial failure state

σ1=σt,σ2=σ3=0

(5a)

σ1=σ2=0,σ3=−σc

(5b)

By substituting Eqs.(4a), (4b) and (5a) into the Eq.(3a), and substituting Eqs.(4a), (4c) and (5b) into Eq.(3b), the β and C are introduced as

β=σc−σtσc+σt=1−α1+α, C=1+bσcσtσc+σt=1+b1+ασt

(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 α=1 and b=1)[8][9]

f=σ1−12(σ2+σ3)=σt, when σ2⩽σ1+σ32

(7a)

f=12(σ1+σ2)−σ3=σt, when σ2⩾σ1+σ32

(7b)

2\. Twin-shear strength theory (UST with b=1)[10].

F=σ1−α2(σ2+σ3)=σt, when σ2⩽σ1+ασ31+α

(8a)

F=12(σ1+σ2) - ασ3=σt, when σ2⩾σ1+ασ31+α

(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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