Implicit Integration of the Unified Yield Criterion in the Principal Stress Space
Publication: Journal of Engineering Mechanics
Volume 145, Issue 7
Abstract
An implicit numerical integration algorithm is presented for the unified yield criterion, which could encompass most other yield criteria. The modification matrix, which is used to convert the continuum tangent modular matrix into the consistent tangent modular matrix, is derived for the return to planes, lines, and the apex of the unified yield criterion with multisurface plasticity with discontinuities. The stress update and consistent tangent modular matrix are first derived in closed form in the principal stress space, and then they are transformed back into the general stress space by coordinate transformation. Three numerical examples are used to demonstrate the effectiveness of the presented algorithm. The correctness of the developed algorithm is validated by the analytical solution and ABAQUS solution with the built-in Mohr-Coulomb model. The developed algorithm is also demonstrated to be least twice more efficient than the ABAQUS built-in algorithm. The presented algorithm for the unified yield criterion can facilitate the understanding of the effect the intermediate principal stress. With the increase in , the force versus deflection curve at the midspan increases for the beam under three-point bending, and the critical radius of the elastoplastic interface decreases (i.e., the plastic zone becomes small) for the circular tunnel under hydrostatic pressure.
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Acknowledgments
This work was supported by the University of Technology Sydney (UTS), Australia through a Chancellor’s Postdoctoral Research Fellowship and by Australian Research Council (ARC) through a Discovery Project (DP160104661). The first author is grateful to Associate Professor Johan Clausen of Department of Civil Engineering, Aalborg University, for fruitful discussion and kindly providing his codes. The third author is a recipient of UTS KTP Visiting Fellowship.
References
ABAQUS. 2014. Abaqus theory guide. Version 6.14. Providence, RI: Dassault Systemes Simulia Corp.
Abbo, A., A. Lyamin, S. Sloan, and J. Hambleton. 2011. “A C2 continuous approximation to the Mohr-Coulomb yield surface.” Int. J. Solids Struct. 48 (21): 3001–3010. https://doi.org/10.1016/j.ijsolstr.2011.06.021.
Abbo, A., and S. Sloan. 1995. “A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion.” Comput. Struct. 54 (3): 427–441. https://doi.org/10.1016/0045-7949(94)00339-5.
Bićanić, N. 1997. “Detection of multiple active yield conditions for Mohr-Coulomb elasto-plasticity.” Comput. Struct. 62 (1): 51–61. https://doi.org/10.1016/S0045-7949(96)00267-2.
Bigoni, D., and F. Laudiero. 1989. “The quasi-static finite cavity expansion in a non-standard elasto-plastic medium.” Int. J. Mech. Sci. 31 (11–12): 825–837. https://doi.org/10.1016/0020-7403(89)90027-1.
Clausen, J., L. Damkilde, and L. Andersen. 2006. “Efficient return algorithms for associated plasticity with multiple yield planes.” Int. J. Num. Methods Eng. 66 (6): 1036–1059. https://doi.org/10.1002/nme.1595.
Clausen, J., L. Damkilde, and L. Andersen. 2007. “An efficient return algorithm for non-associated plasticity with linear yield criteria in principal stress space.” Comput. Struct. 85 (23): 1795–1807. https://doi.org/10.1016/j.compstruc.2007.04.002.
Clausen, J., L. Damkilde, and L. V. Andersen. 2015. “Robust and efficient handling of yield surface discontinuities in elasto-plastic finite element calculations.” Eng. Comput. 32 (6): 1722–1752. https://doi.org/10.1108/EC-01-2014-0008.
Cook, R. D. 2007. Concepts and applications of finite element analysis. New York: Wiley.
Coombs, W., R. Crouch, and C. Heaney. 2013. “Observations on Mohr-Coulomb plasticity under plane strain.” J. Eng. Mech. 139 (9): 1218–1228. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000568.
Crisfield, M. 1987. “Plasticity computations using the Mohr-Coulomb yield criterion.” Eng. Comput. 4 (4): 300–308. https://doi.org/10.1108/eb023708.
Crisfield, M. A., J. J. Remmers, and C. V. Verhoosel. 2012. Nonlinear finite element analysis of solids and structures. New York: Wiley.
De Borst, R. 1987. “Integration of plasticity equations for singular yield functions.” Comput. Struct. 26 (5): 823–829. https://doi.org/10.1016/0045-7949(87)90032-0.
de Souza Neto, E. A., D. Peric, and D. R. Owen. 2011. Computational methods for plasticity: Theory and applications. New York: Wiley.
Galic, M., P. Marovic, and Z. Nikolic. 2011. “Modified Mohr-Coulomb-Rankine material model for concrete.” Eng. Comput. 28 (7): 853–887. https://doi.org/10.1108/02644401111165112.
Kachanov, L. M. 1971. Foundations of the theory of plasticity. Amsterdam, Netherlands: North-Holland.
Karaoulanis, F. E. 2013. “Implicit numerical integration of nonsmooth multisurface yield criteria in the principal stress space.” Arch. Comput. Methods Eng. 20 (3): 263–308. https://doi.org/10.1007/s11831-013-9087-3.
Koiter, W. T. 1953. “Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface.” Q. Appl. Math. 11 (3): 350–354. https://doi.org/10.1090/qam/59769.
Larsson, R., and K. Runesson. 1996. “Implicit integration and consistent linearization for yield criteria of the Mohr-Coulomb type.” Mech. Cohesive-Frict. Mater. 1 (4): 367–383. https://doi.org/10.1002/(SICI)1099-1484(199610)1:4%3C367::AID-CFM19%3E3.0.CO;2-F.
Lee, Y.-K., S. Pietruszczak, and B.-H. Choi. 2012. “Failure criteria for rocks based on smooth approximations to Mohr-Coulomb and Hoek-Brown failure functions.” Int. J. Rock Mech. Min. Sci. 56 (Dec): 146–160. https://doi.org/10.1016/j.ijrmms.2012.07.032.
Lin, C., and Y. M. Li. 2015. “A return mapping algorithm for unified strength theory model.” Int. J. Num. Methods Eng. 104 (8): 749–766. https://doi.org/10.1002/nme.4937.
Nayak, G., and O. Zienkiewicz. 1972. “Elasto-plastic stress analysis. A generalization for various constitutive relations including strain softening.” Int. J. Num. Methods Eng. 5 (1): 113–135. https://doi.org/10.1002/nme.1620050111.
Perić, D., and E. de Souza Neto. 1999. “A new computational model for Tresca plasticity at finite strains with an optimal parametrization in the principal space.” Comput. Methods Appl. Mech. Eng. 171 (3–4): 463–489. https://doi.org/10.1016/S0045-7825(98)00221-7.
Simo, J., J. Kennedy, and S. Govindjee. 1988. “Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms.” Int. J. Num. Methods Eng. 26 (10): 2161–2185. https://doi.org/10.1002/nme.1620261003.
Simo, J. C., and R. L. Taylor. 1985. “Consistent tangent operators for rate-independent elastoplasticity.” Comput. Methods Appl. Mech. Eng. 48 (1): 101–118. https://doi.org/10.1016/0045-7825(85)90070-2.
Taiebat, H. A., and J. P. Carter. 2008. “Flow rule effects in the Tresca model.” Comput. Geotech. 35 (3): 500–503. https://doi.org/10.1016/j.compgeo.2007.06.012.
Wille, K., S. El-Tawil, and A. E. Naaman. 2014. “Properties of strain hardening ultra high performance fiber reinforced concrete (UHP-FRC) under direct tensile loading.” Cem. Concr. Compos. 48 (Apr): 53–66. https://doi.org/10.1016/j.cemconcomp.2013.12.015.
Wu, C., Li, J., and Y. Su. 2018. Development of ultra-high performance concrete against blasts: From materials to structures. Cambridge, UK: Woodhead Publishing.
Xu, S.-Q., and M.-H. Yu. 2006. “The effect of the intermediate principal stress on the ground response of circular openings in rock mass.” Rock Mech. Rock Eng. 39 (2): 169–181. https://doi.org/10.1007/s00603-005-0064-5.
Yu, M. H. 2011. Unified strength theory and its applications. Berlin: Springer.
Yu, M.-H., L. He, and L. Song. 1985. “Twin shear stress theory and its generalization.” Sci. China Ser. A-Math. Phys. Astron. Technol. Sci. 28 (11): 1174–1183.
Yu, M.-H., and J.-C. Li. 2012. Computational plasticity: With emphasis on the application of the unified strength theory. Berlin: Springer.
Yu, M.-H., S.-Y. Yang, S. Fan, and G.-W. Ma. 1999. “Unified elasto-plastic associated and non-associated constitutive model and its engineering applications.” Comput. Struct. 71 (6): 627–636. https://doi.org/10.1016/S0045-7949(98)00306-X.
Zhang, C., J. Wang, and J. Zhao. 2010. “Unified solutions for stresses and displacements around circular tunnels using the unified strength theory.” Sci. China Technol. Sci. 53 (6): 1694–1699. https://doi.org/10.1007/s11431-010-3224-0.
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©2019 American Society of Civil Engineers.
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Received: Mar 4, 2018
Accepted: Nov 16, 2018
Published online: Apr 25, 2019
Published in print: Jul 1, 2019
Discussion open until: Sep 25, 2019
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