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Research Papers

Integrating the Pressure-Sensitive Nonassociative Plasticity by Exponential-Based Methods

[+] Author and Article Information
Mohammad Rezaiee-Pajand

Professor of Civil Engineering,
Ferdowsi University of Mashhad,
Mashhad, 91775-1111 Iran

Mehrdad Sharifian

Ph.D. Student of Structural Engineering

Mehrzad Sharifian

Ph.D. Structural Engineering
Ferdowsi University of Mashhad,
Mashhad, 91775-1111 Iran

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 4, 2012; final manuscript received January 18, 2013; published online May 13, 2013. Assoc. Editor: Marwan K. Khraisheh.

J. Eng. Mater. Technol 135(3), 031010 (May 13, 2013) (22 pages) Paper No: MATS-12-1131; doi: 10.1115/1.4024173 History: Received June 04, 2012; Revised January 18, 2013

A nonassociative plasticity model of Drucker–Prager yield surface coupled with a generalized nonlinear kinematic hardening is considered. Conforming to the plasticity model, two exponential-based methods, called fully explicit and semi-implicit, are recommended for integrating its constitutive equations. These techniques are proposed for the first time to solve nonlinear hardening materials. The integrations are thoroughly investigated by utilizing stress and strain-updating tests along with a boundary value problem in diverse grounds of accuracy, convergence rate, and efficiency. The results indicate that the fully explicit scheme is more accurate and efficient than the Euler's, but the same convergence rate as the classical integrations is also perceived. Having a quadratic convergence, the semi-implicit is noticeably the most accurate and efficient procedure to use for this plasticity model among the algorithms in question. Since the plasticity model is in a great consistency with discontinuously reinforced aluminum (DRA) composites, the suggested formulations can be utilized pragmatically. The tangent moduli of the proposed and Euler's strategies are derived and examined, as well, due to their vital role in achieving the asymptotic quadratic convergence rate of the Newton–Raphson solution in nonlinear finite-element analyses.

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References

Bridgman, P. W., 1947, “The Effect of Hydrostatic Pressure on the Fracture of Brittle Substances,” J. Appl. Phys., 18, pp. 246–258. [CrossRef]
Bridgman, P. W., 1952, Studies in Large Plastic Flow and Fracture With Special Emphasis on the Effect of Hydrostatic Pressure, McGraw-Hill, New York.
Spitzig, W. A., Sober, R. J., and Richmond, O., 1975, “Pressure Dependence of Yielding and Associated Volume Expansion in Tempered Martensite,” Acta Metall., 23, pp. 885–893. [CrossRef]
Spitzig, W. A., Sober, R. J., and Richmond, O., 1976, “The Effect of Hydrostatic Pressure on the Deformation Behavior of Maraging and HY-80 and Its Implication for Plasticity,” Metall. Trans. A, A7, pp. 457–463.
Spitzig, W. A., and Richmond, O., 1984, “The Effect of Pressure on the Flow Stress of Metals,” Acta Metall., 32, pp. 457–463. [CrossRef]
Wilson, C. D., 2002, “A Critical Reexamination of Classical Metal Plasticity,” ASME J. Appl. Mech., 69, pp. 63–68. [CrossRef]
Drucker, D. C., and Prager, W., 1952, “Soil Mechanics and Plastic Analysis or Limit Design,” Q. Appl. Math., 10, pp. 157–165.
Singh, A. P., Padmanabhan, K. A., Pandy, G. N., Murty, G. M. D., and Jha, S., 2000, “Strength Differential Effect in Four Commercial Steels,” J. Mater. Sci., 35, pp. 1379–1388. [CrossRef]
Altenbach, H., Stoychev, G. B., and Tushtev, K. N., 2001, “On Elastoplastic Deformation of Grey Cast Iron,” Int. J. Plast., 17, pp. 719–736. [CrossRef]
Chait, R., 1973, “The Strength Differential of Steel and Ti Alloys as Influenced by Test Temperature and Microstructure,” Scr. Metall., 7, pp. 351–363. [CrossRef]
Gil, C. M., Lissenden, C. J., and Lerch, B. A., 1999, “Yield of Inconel 718 by Axial-Torsional Loading at Temperatures Up to 649 °C,” J. Test. Eval., 27, pp. 327–336. [CrossRef]
Iyer, S. K., and Lissenden, C. J., 2000, “Initial Anisotropy of Inconel 718: Experiments and Mathematical Representation.” J. Eng. Mater. Technol., 122, pp. 321–326. [CrossRef]
Lewandowski, J. J., Wesseling, P., Prabhu, N. S., Larose, J., and Lerch, B. A., 2003, “Strength Differential Measurements in IN-718: Effects of Superimposed Pressure,” Metall. Mater. Trans. A, 34A, pp. 1736–1739. [CrossRef]
Lei, X., and Lissenden, C. J., 2007, “Pressure Sensitive Nonassociative Plasticity Model for DRA Composites,” ASME J. Eng. Mater. Technol., 129, pp. 255–264. [CrossRef]
Prager, W., 1956, “A New Method of Analyzing Stresses and Strains in Work Hardening Plastic Solids.” ASME J. Appl. Mech., 23, pp. 493–496.
Chakrabarty, J., 2000, Theory of Plasticity, 3rd ed. Elsevier Butterworth-Heinemann, Oxford, UK.
Bari, S., and Hassan, T., 2000, “Anatomy of Coupled Constitutive Models for Ratcheting Simulation,” Int. J. Plast., 16, pp. 381–409. [CrossRef]
Armstrong, P. J., and Frederick, C. O., 1966, “A Mathematical Representation of the Multiaxial Bauschinger Effect,” CEGB Report No. RD/B/N 731.
Chaboche, J. L., 1986, “Time-Independent Constitutive Theories for Cyclic Plasticity,” Int. J. Plast., 2, pp. 149–188. [CrossRef]
Chaboche, J. L., 1991, “On Some Modifications of Kinematic Hardening to Improve the Description of Ratcheting Effects,” Int. J. Plast., 7, pp. 661–678. [CrossRef]
Ohno, N., and Wang, J. D., 1993, “Kinematic Hardening Rules With Critical State of Dynamic Recovery—Part I: Formulations and Basic Features for Ratcheting Behavior,” Int. J. Plast., 9, pp. 375–390. [CrossRef]
Abdel-Karim, M., and Ohno, N., 2000, “Kinematic Hardening Model Suitable for Ratcheting With Steady-State,” Int. J. Plast., 16, pp. 225–240. [CrossRef]
Kang, G., 2004, “A Visco-Plastic Constitutive Model for Ratcheting of Cyclically Stable Materials and Its Finite Element Implementation,” Mech. Mater., 36, pp. 299–312. [CrossRef]
Chaboche, J. L., 2008, “A Review of Some Plasticity and Viscoplasticity Constitutive Theories,” Int. J. Plast., 24, pp. 1642–1693. [CrossRef]
Abdel-Karim, M., 2009, “Modified Kinematic Hardening Rules for Simulations of Ratcheting,” Int. J. Plast., 25, pp. 1560–1587. [CrossRef]
Rezaiee-Pajand, M., and Sinaie, S., 2009, “On the Calibration of the Chaboche Hardening Model and a Modified Hardening Rule for Uniaxial Ratcheting Prediction,” Int. J. Solids Struct., 46, pp. 3009–3017. [CrossRef]
Mroz, Z., 1967, “On the Description of Anisotropic Work Hardening,” J. Mech. Phys. Solids, 15, pp. 163–175. [CrossRef]
Dafalias, Y. F., and Popov, E. P., 1976, “Plastic Internal Variables Formalism of Cyclic Plasticity,” ASME J. Appl. Mech., 43, pp. 645–650. [CrossRef]
Tseng, N. T., and Lee, G. C., 1983, “Simple Plasticity Model of the Two-Surface Type,” ASCE J. Eng. Mech., 109, pp. 795–810. [CrossRef]
Krieg, R. D., and Krieg, D. B., 1977, “Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model,” ASME J. Pressure Vessel Technol., 99, pp. 510–515. [CrossRef]
Yoder, P. J., and Whirley, R. G., 1984, “On the Numerical Implementation of Elastoplastic Models,” ASME J. Appl. Mech., 51, pp. 283–288. [CrossRef]
Loret, B., and Prevost, J. H., 1986, “Accurate Numerical Solutions for Drucker-Prager Elastic-Plastic Models,” Comput. Methods Appl. Mech. Eng., 54, pp. 259–277. [CrossRef]
Ristinmaa, M., and Tryding, J., 1993, “Exact Integration of Constitutive Equations in Elastoplasticity,” Int. J. Numer. Methods Eng., 36, pp. 2525–2544. [CrossRef]
Wei, Z., Perić, D., and Owen, D. R. J., 1996, “Consistent Linearization for the Exact Stress Update of Prandtl–Reuss Non-Hardening Elastoplastic Models,” Int. J. Numer. Methods Eng., 39, pp. 1219–1235. [CrossRef]
Szabó, L., 2009, “A Semi-Analytical Integration Method for J2 Flow Theory of Plasticity With Linear Isotropic Hardening,” Comput. Methods Appl. Mech. Eng., 198, pp. 2151–2166. [CrossRef]
Rezaiee-Pajand, M., Sharifian, M., and Sharifian, M., 2011, “Accurate and Approximate Integrations of Drucker-Parger Plasticity With Linear Isotropic and Kinematic Hardening,” Eur. J. Mech. A/Solids, 30, pp. 345–361. [CrossRef]
Rezaiee-Pajand, M., and Sharifian, M., 2012, “A Novel Formulation for Integrating Nonlinear Kinematic Hardening Drucker-Prager's Yield Condition,” Eur. J. Mech. A/Solids, 31, pp. 163–178. [CrossRef]
Wilkins, M. L., 1964, “Calculation of Elastic-Plastic Flow,” Methods in Computational Physics, Vol. 3, Academic Press, New York.
Rice, J. R., and Tracey, D. M., 1973, “Computational Fracture Mechanics,” Numerical and Computer Methods in Structural Mechanics, S. J. Fenves, ed., Academic Press, New York, pp. 585–623.
Ortiz, M., and Popov, E. P., 1985, “Accuracy and Stability of Integration Algorithms for Elastoplastic Constitutive Relations,” Int. J. Numer. Methods Eng., 21, pp. 1561–1576. [CrossRef]
Kobayashi, M., and Ohno, N., 2002, “Implementation of Cyclic Plasticity Models Based on a General Form of Kinematic Hardening,” Int. J. Numer. Methods Eng., 53, pp. 2217–2238. [CrossRef]
Kobayashi, M., Mukai, M., Takahashi, H., Ohno, N., Kawakami, T., and Ishikawa, T., 2003, “Implicit Integration and Consistent Tangent Modulus of a Time-Dependent Non-Unified Constitutive Model,” Int. J. Numer. Methods Eng., 58, pp. 1523–1543. [CrossRef]
Kang, G., 2006, “Finite Element Implementation of Viscoplastic Constitutive Model With Strain-Range Dependent Cyclic Hardening,” Commun. Numer. Methods Eng., 22(2), pp. 137–153. [CrossRef]
Kan, Q. H., Kang, G. Z., and Zhang, J., 2007, “A Unified Visco-Plastic Constitutive Model for Uniaxial Time-Dependent Ratchetting and Its Finite Element Implementation,” Theor. Appl. Fract. Mech., 47, pp. 133–144. [CrossRef]
Coombs, W. M., Crouch, R. S., and Augarde, C. E., 2010, “Reuleaux Plasticity: Analytical Backward Euler Stress Integration and Consistent Tangent,” Comput. Methods Appl. Mech. Eng., 199, pp. 1733–1743. [CrossRef]
Hong, H. K., and Liu, C. S., 1999, “Internal Symmetry in Bilinear Elastoplasticity,” Int. J. Non-Linear Mech., 34, pp. 279–288. [CrossRef]
Hong, H. K., and Liu, C. S., 2000, “Internal Symmetry in the Constitutive Model of Perfect Elastoplasticity,” Int. J. Non-Linear Mech., 35, pp. 447–466. [CrossRef]
Hong, H. K., and Liu, C. S., 2000, “Lorentz Group on Minkowski Spacetime for Construction of the Two Basic Principles of Plasticity.” Int. J. Non-Linear Mech., 36, pp. 679–686. [CrossRef]
Liu, C. S., 2003, “Symmetry Groups and the Pseudo-Riemann Spacetimes for Mixed-Hardening Elastoplasticity,” Int. J. Solids Struct., 40, pp. 251–269. [CrossRef]
Liu, C. S., 2004, “A Consistent Numerical Scheme for the Von-Mises Mixed-Hardening Constitutive Equations.” Int. J. Plast., 20, pp. 663–704. [CrossRef]
Liu, C. S., 2004, “Internal Symmetry Groups for the Drucker-Prager Material Model of Plasticity and Numerical Integrating Methods.” Int. J. Solids Struct., 41, pp. 3771–3791. [CrossRef]
Auricchio, F., and Beirão da Veiga, L., 2003, “On a New Integration Scheme for Von-Mises Plasticity With Linear Hardening,” Int. J. Numer. Methods Eng., 56, pp. 1375–1396. [CrossRef]
Artioli, E., Auricchio, F., and Beirão da Veiga, L., 2005, “Integration Schemes for Von-Mises Plasticity Models Based on Exponential Maps: Numerical Investigations and Theoretical Considerations,” Int. J. Numer. Methods Eng., 64, pp. 1133–1165. [CrossRef]
Artioli, E., Auricchio, F., and Beirão da Veiga, L., 2006, “A Novel ‘Optimal’ Exponential-Based Integration Algorithm for Von-Mises Plasticity With Linear Hardening: Theoretical Analysis on Yield Consistency, Accuracy, Convergence and Numerical Investigations,” Int. J. Numer. Methods Eng., 4, pp. 449–498. [CrossRef]
Artioli, E., Auricchio, F., and Beirão da Veiga, L., 2007, “Second-Order Accurate Integration Algorithms for Von-Mises Plasticity With a Nonlinear Kinematic Hardening Mechanism,” Comput. Methods Appl. Mech. Eng., 196, pp. 1827–1846. [CrossRef]
Rezaiee-Pajand, M., and Nasirai, C., 2007, “Accurate Integration Scheme for Von-Mises Plasticity With Mixed-Hardening Based on Exponential Maps,” Eng. Comput., 24(6), pp. 608–635. [CrossRef]
Rezaiee-Pajand, M., and Nasirai, C., 2008, “On the Integration Schemes for Drucker-Prager's Elastoplastic Models Based on Exponential Maps.” Int. J. Numer. Methods Eng., 74, pp. 799–826. [CrossRef]
Rezaiee-Pajand, M., Nasirai, C., and Sharifian, M., 2010, “Application of Exponential-Based Methods in Integrating the Constitutive Equations With Multicomponent Kinematic Hardening,” ASCE J. Eng. Mech., 136(12), pp. 1502–1518. [CrossRef]
Rezaiee-Pajand, M., Nasirai, C., and Sharifian, M., 2011, “Integration of Nonlinear Mixed Hardening Models,” Multidiscip. Model. Mater. Struct.7(3), pp. 266–305. [CrossRef]
de Souza Neto, E. A., Perić, D., and Owen, D. R. J., 2008, Computational Methods for Plasticity: Theory and Applications, John Wiley and Sons, Ltd, New York.

Figures

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Fig. 1

Flow chart for the explicit exponential, EXF, and semi-implicit exponential, EXS, integrations

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Fig. 2

Treatment of the apex for associative model

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Fig. 3

Treatment of the apex for nonassociative model

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Fig. 6

Stress relative errors by FE, BE, and EXF for strain history 1

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Fig. 7

Stress relative errors by EXF and EXS for strain history 1

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Fig. 8

Stress relative errors by FE, BE, and EXF for strain history 2

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Fig. 9

Stress relative errors by EXF and EXS for strain history 2

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Fig. 10

Comparison of the accuracy of EXF and BE with EXS for strain history 1

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Fig. 11

Comparison of the accuracy of EXF and BE with EXS for strain history 2

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Fig. 12

Stress relative error by EXS in consecutive load-step sizes for strain history 1

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Fig. 13

Stress relative error by EXF in consecutive load-step sizes for strain history 1

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Fig. 14

Demonstration of the accuracy convergence rates of the integration schemes for strain history 1

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Fig. 17

Strain relative errors by FE, BE, and EXF for stress history 1

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Fig. 18

Strain relative errors by EXF and EXS for stress history 1

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Fig. 19

Strain relative errors by FE, BE, and EXF for stress history 2

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Fig. 20

Strain relative errors by EXF and EXS for stress history 2

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Fig. 21

The strip with an elliptical hole

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Fig. 22

The finite-element mesh

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Fig. 23

The history of the nonproportional loads

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Fig. 24

The elements involvement in plastic computations alongside their deformations

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