Engineering: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]
Mathematical Formulation
Mathematically, the formulation of UST is expressed in principal stress state as
(1a)
(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.
(2a)
(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.
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
(3a)
(3b)
The relations among the stresses components and principal stresses read
(4a)
(4b)
(4c)
The [math]\displaystyle{ \beta }[/math] and C should be obtained by uniaxial failure state
(5a)
(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
(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]
(7a)
(7b)
2. Twin-shear strength theory (UST with [math]\displaystyle{ b = 1 }[/math])[10].
(8a)
(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
- ↑ 1.0 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
- ↑ Yu M. H. (2004) Unified Strength Theory and Its Applications. Springer: Berlin. ISBN:978-3-642-18943-2
- ↑ 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
- ↑ 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
- ↑ 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
- ↑ 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.
- ↑ 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.
- ↑ 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)
- ↑ 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
- ↑ 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.
- ↑ Yu M. H. et al., (2006) Generalized Plasticity. Springer: Berlin. ISBN:978-3-540-30433-3
- ↑ 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
- ↑ Yu M. H., Li J. C. (2012) Computational Plasticity, Springer and ZJU Press: Berlin and Hangzhou. ISBN:978-3-642-24590-9
- ↑ 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
- ↑ 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
Original source: https://en.wikipedia.org/wiki/Unified strength theory.
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