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

F = σ_{1} - \frac{α}{1 + b} (b σ_{2} + σ_{3}) = σ_{t}, when σ_{2} ⩽ \frac{σ_{1} + α σ_{3}}{1 + α}

(1a)

F^{'} = \frac{1}{1 + b} (σ_{1} + b σ_{3}) - α σ_{3} = σ_{t}, when σ_{2} ⩾ \frac{σ_{1} + α σ_{3}}{1 + α}

(1b)

where $σ_{1}, σ_{2}, σ_{3}$ are three principal stresses, $σ_{t}$ is the uniaxial tensile strength and $α$ is tension-compression strength ratio ( $α = σ_{t} / σ_{c}$ ). The unified yield criterion (UYC) is the simplification of UST when $α = 1$ , i.e.

f = σ_{1} - \frac{1}{1 + b} (b σ_{2} + σ_{3}) = σ_{s}, when σ_{2} ⩽ \frac{1}{2} (σ_{1} + σ_{3})

(2a)

f^{'} = \frac{1}{1 + b} (σ_{1} + b σ_{2}) - σ_{3} = σ_{s}, when σ_{2} ⩾ \frac{1}{2} (σ_{1} + σ_{3})

(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.

Derivation of Unified Strength Theory

Due to the relation ( $τ_{13} = τ_{12} + τ_{23}$ ), the principal stress state ( $σ_{1}, σ_{2}, σ_{3}$ ) may be converted to the twin-shear stress state ( $τ_{13}, τ_{12}; σ_{13}, σ_{12}$ ) or ( $τ_{13}, τ_{23}; σ_{13}, σ_{23}$ ). 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}

(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}

(3b)

The relations among the stresses components and principal stresses read

τ_{13} = \frac{1}{2} (σ_{1} - σ_{3})

,

σ_{13} = \frac{1}{2} (σ_{1} + σ_{3})

(4a)

τ_{12} = \frac{1}{2} (σ_{1} - σ_{2})

,

σ_{12} = \frac{1}{2} (σ_{1} + σ_{2})

(4b)

τ_{23} = \frac{1}{2} (σ_{2} - σ_{3})

,

σ_{23} = \frac{1}{2} (σ_{2} + σ_{3})

(4c)

The $β$ and C should be obtained by uniaxial failure state

σ_{1} = σ_{t}, σ_{2} = σ_{3} = 0

(5a)

σ_{1} = σ_{2} = 0, σ_{3} = - σ_{c}

(5b)

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

β = \frac{σ_{c} - σ_{t}}{σ_{c} + σ_{t}} = \frac{1 - α}{1 + α}

,

C = \frac{1 + b σ_{c} σ_{t}}{σ_{c} + σ_{t}} = \frac{1 + b}{1 + α} σ_{t}

(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 = σ_{1} - \frac{1}{2} (σ_{2} + σ_{3}) = σ_{t}, when σ_{2} ⩽ \frac{σ_{1} + σ_{3}}{2}

(7a)

f = \frac{1}{2} (σ_{1} + σ_{2}) - σ_{3} = σ_{t}, when σ_{2} ⩾ \frac{σ_{1} + σ_{3}}{2}

(7b)

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

F = σ_{1} - \frac{α}{2} (σ_{2} + σ_{3}) = σ_{t}, when σ_{2} ⩽ \frac{σ_{1} + α σ_{3}}{1 + α}

(8a)

F = \frac{1}{2} (σ_{1} + σ_{2}) - α σ_{3} = σ_{t}, when σ_{2} ⩾ \frac{σ_{1} + α σ_{3}}{1 + 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

↑ ^1.0 ^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

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[Yu1991-1] 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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