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Probabilistic elasto-plasticity: solution and verification in 1D

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Abstract

In this paper, a solution is presented for evolution of probability density function (PDF) of elastic–plastic stress–strain relationship for material models with uncertain parameters. Developments in this paper are based on already derived general formulation presented in the companion paper. The solution presented is then specialized to a specific Drucker–Prager elastic–plastic material model. Three numerical problems are used to illustrate the developed solution. The stress–strain response (1D) is given as a PDF of stress as a function of strain. The presentation of the stress–strain response through the PDF differs significantly from the traditional presentation of such results, which are represented by a single, unique curve in stress–strain space. In addition to that the numerical solutions are verified against closed form solutions where available (elastic). In cases where the closed form solution does not exist (elastic–plastic), Monte Carlo simulations are used for verification.

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Notes

  1. The yield parameter α is an internal variable and is a function of the friction angle ϕ given by \(\alpha = 2 \,{\rm sin}\phi/(\sqrt3(3-{\rm sin} \phi))\) (e.g., Chen and Han [1]).

  2. The plastic slope α′ is a rate of change of friction angle governing linear hardening.

References

  1. Chen WF, Han DJ (1988) Plasticity for structural engineers. Springer, Heidelberg

  2. Gardiner CW (2004) Handbook of stochastic methods for physics, chemistry and the natural science. Complexity, 3rd edn. Springer, Heidelberg

    Google Scholar 

  3. Jeremić B, Sett K, Kavvas ML (2007) Probabilistic elasto-plasticity: formulation in 1D. Acta Geotech doi:10.1007/s11440-007-0036-x

  4. Kavvas ML (2003) Nonlinear hydrologic processes: conservation equations for determining their means and probability distributions. J Hydrol Eng 8(2):44–53

    Article  Google Scholar 

  5. Langtangen H (1991) A general numerical solution method for Fokker-Planck equations with application to structural reliability. Probab Eng Mech 6(1):33–48

    Article  Google Scholar 

  6. Montgomery DC, Runger GC (2003) Applied statistics and probability for engineers, 3rd edn. Wiley, New York, 10158 pp

  7. Oberkampf WL, Trucano TG, Hirsch C (2002) Verification, validation and predictive capability in computational engineering and physics. In: Proceedings of the foundations for verification and validation on the 21st century workshop. Johns Hopkins University/Applied Physics Laboratory, Laurel, pp 1–74

  8. Risken H (1989) The Fokker–Planck equation: methods of solution applications, 2nd edn. Springer, Heidelberg

    Google Scholar 

  9. Rose C, Smith MD (2002) Mathematical statistics with mathematica. Springer texts in statistics. Springer, New York

    Google Scholar 

  10. Van Kampen NG (1976) Stochastic differential equations. Phys Rep 24:171–228

    Article  Google Scholar 

  11. Wolfram S (1991) Mathematica: a system for doing mathematics by computer, 2nd edn. Addison-Wesley, Redwood City

    Google Scholar 

  12. Wolfram Research Inc (2003) Mathematica Version 5.0. Wolfram Research, Champaign

    Google Scholar 

Download references

Acknowledgment

The work presented in this paper was supported in part by a number of Agencies listed below: Civil, Mechanical and Manufacturing Innovation program, Directorate of Engineering of the National Science Foundation, under Award NSF–CMMI–0600766 (cognizant program director Dr. Richard Fragaszy); Civil and Mechanical System program, Directorate of Engineering of the National Science Foundation, under Award NSF–CMS–0337811 (cognizant program director Dr. Steve McCabe); Earthquake Engineering Research Centers Program of the National Science Foundation under Award Number NSF–EEC–9701568 (cognizant program director Dr. Joy Pauschke); California Department of Transportation (Caltrans) under Award #59A0433 (cognizant program director Dr. Saad El-Azazy)

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of California, Davis, CA, 95616, USA

    Kallol Sett, Boris Jeremić & M. Levent Kavvas

Authors
  1. Kallol Sett
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  2. Boris Jeremić
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  3. M. Levent Kavvas
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Correspondence to Boris Jeremić.

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Sett, K., Jeremić, B. & Kavvas, M.L. Probabilistic elasto-plasticity: solution and verification in 1D. Acta Geotech. 2, 211–220 (2007). https://doi.org/10.1007/s11440-007-0037-9

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  • DOI: https://doi.org/10.1007/s11440-007-0037-9

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