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Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring


For finite-strain plasticity with anisotropic yield functions and anisotropic hyperelasticity, we use the Kröner-Lee decomposition of the deformation gradient combined with a yield function written in terms of the Mandel stress. The source is here the right Cauchy-Green tensor provided by a FE discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential, which is compared with a classical backward-Euler method. The exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source, consistent with the approximation. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function. The constitutive system is solved by a damped Newton-Raphson algorithm for the plastic multiplier and the elastic right Cauchy-Green tensor \(\varvec{C}_{e}\). To ensure power-consistency, we make use of the elastic Mandel stress construction. Two numerical examples exhibit the comparative effectiveness of the Algorithm for very large elastic and plastic deformations. The elasto-plastic pinched cylinder makes use of as few as 2 steps for the total radius displacement of 300 mm and only 25 steps are required for the cup drawing problem.

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  1. 1.

    Betsch P, Stein E (1999) Numerical implementation of multiplicative elasto-plasticity into assumed strain elements with application to shells at large strains. Comp Method Appl Mech Eng 179:215–245

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bennett KC, Regueiro RA, Luscher DJ (2019) Anisotropic finite hyper-elastoplasticity of geomaterials with drucker-prager/cap type constitutive model formulation. Int J Plast 123:224–250

    Google Scholar 

  3. 3.

    Andelfinger U, Ramm E (1993) EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int J Numer Methods Eng 36:1311–1337

    MATH  Google Scholar 

  4. 4.

    Buchter N, Ramm E, Roehl D (1994) Three-dimensional extension of nonlinear shell formulation based on the enhanced assumed strain concept. Int J Numer Methods Eng 37:2551–3568

    MATH  Google Scholar 

  5. 5.

    Areias P, Mota Soares CA, Rabczuk T, Garçao J (2016) A finite-strain solid-shell using local Löwdin frames and least-squares strains. Comp Method Appl Mech Eng 311:112–133

    MATH  Google Scholar 

  6. 6.

    Eidel B, Gruttmann F (2003) Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation. Comput Mater Sci 28:732–742

    Google Scholar 

  7. 7.

    Mandel J (1973) Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int J Solids Struct 9:725–740

    MATH  Google Scholar 

  8. 8.

    Simo JC (1992) Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comp Method Appl Mech Eng 99:61–112

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Cuitiño A, Ortiz M (1992) A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics. Eng Comput 9:437–451

    Google Scholar 

  10. 10.

    de Souza Neto EA, Perić D, Owen DRJ (2008) Computational methods for plasticity: theory and applications. Wiley, West Sussex

    Google Scholar 

  11. 11.

    Shutov AV (2018) Efficient time stepping for the multiplicative Maxwell fluid including the Mooney-Rivlin hyperelasticity. Int J Numer Methods Eng 113(12):1851–1869

    MathSciNet  Google Scholar 

  12. 12.

    Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  13. 13.

    Simo JC, Hughes TJR (2000) Computational inelasticity, Corrected 2 edn. Springer, Berlin

    MATH  Google Scholar 

  14. 14.

    Areias P, Belytschko T (2006) Analysis of finite strain anisotropic elastoplastic fracture in thin plates and shells. J Aerospace Eng 19(4):259–270

    Google Scholar 

  15. 15.

    Ortiz M, Radovitzky RA, Repetto EA (2001) The computation of the exponential and logarithmic mappings and their first and second derivatives. Int J Numer Methods Eng 52:1431–1441

    MATH  Google Scholar 

  16. 16.

    Korelc J, Stupkiewicz S (2014) Closed-form matrix exponential and its application. Int J Numer Methods Eng 98:960–987

    MATH  Google Scholar 

  17. 17.

    Moler C, Van Loan C (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev 45(1):1–46

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Ward RC (1977) Numerical computation of the matrix exponential with accuracy estimate. SIAM J Numer Anal 14(4):600–610

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Higham NJ (2009) The scaling and squaring method for the matrix exponential revisited. SIAM Rev 51(4):747–764

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Cheng H-W, Yau SS-T (1997) More explicit formulas for the matrix exponential. Linear Algebra and its Applications, vol 262, pp 131–163

  21. 21.

    Lu J (2004) Exact expansions of arbitrary tensor functions \({\mathbf{F}}({\mathbf{A}})\). Int J Solids Struct 41:337–349

    MathSciNet  Google Scholar 

  22. 22.

    de Souza Neto EA (2001) The exact derivative of the exponential of an unsymmetric tensor. Comput Method Appl Mech Eng 190:2377–2383

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Fung TC (2004) Computation of the matrix exponential and its derivatives by scaling and squaring. Int J Numer Method Eng 59:1273–1286

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Sastre J, Ibánez J, Defez E, Ruiz P (2015) New scaling-squaring Taylor. SIAM J Sci Comput 37(1):A439–A455

    MATH  Google Scholar 

  25. 25.

    Baaser H (2004) The Padé-approximation for matrix exponentials applied to an integration algorithm preserving plastic incompressibility. Comput Mech 34:237–245

    MATH  Google Scholar 

  26. 26.

    Areias P, Matouš K (2008) Finite element formulation for modeling nonlinear viscoelastic elastomers. Comput Method Appl Mech Eng 197:4702–4717

    MATH  Google Scholar 

  27. 27.

    Al-Mohy AH, Higham NJ (2009) A new scaling and squaring algorithm for the matrix exponential. SIAM J Matrix Anal Appl 31(3):970–989

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Kröner E (1960) Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch Ration Mech Anal 4:273–334

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Lee EH, Liu DT (1967) Finite strain elastic-plastic theory particularly for plane wave analysis. J Appl Phys 38(1):19–27

    Google Scholar 

  30. 30.

    Lee EH (1969) Elasto-plastic deformation at finite strains. J Appl Mech-ASME 36:1–6

    MATH  Google Scholar 

  31. 31.

    Lubliner J (1990) Plasticity theory. Macmillan, Basingstoke

    MATH  Google Scholar 

  32. 32.

    Mandel J (1974) Foundations of continuum thermodynamics. Thermodynamics and plasticity. MacMillan, London, pp 283–304

    Google Scholar 

  33. 33.

    Gurtin ME (1981) An introduction to continuum mechanics, volume 158 of Mathematics in Science and Engineering. Academic Press, New York

  34. 34.

    Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Rational Mech Anal 13:167–178

  35. 35.

    Vladimirov I, Pietryga MP, Reese S (2010) Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming. Int J Plast 26:659–687

  36. 36.

    Wolfram Research Inc. (2007) Mathematica

  37. 37.

    Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18(4):312–327

    Google Scholar 

  38. 38.

    Bickart TA (1968) Matrix exponential: Approximation by truncated power series. Proc IEEE 56:372–373

    Google Scholar 

  39. 39.

    Petersen KB, Pedersen MS (2012) The matrix cookbook

  40. 40.

    Areias P (2021) Mandel-based plasticity with series exponential.

  41. 41.

    Murphy JG (2013) Transversely isotropic biological, soft tissue must be modelled using both anisotropic invariants. Eur J Mech A/Solids 42:90–96

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin

    MATH  Google Scholar 

  43. 43.

    Barlat F, Aretz H, Yoon JW, Karabin ME (2005) Linear transformation-based anisotropic yield functions. Int J Plast 21:1009–1039

    MATH  Google Scholar 

  44. 44.

    Yoshida F, Hamasaki H, Uemori T (2013) A user-friendly 3d yield function to describe anisotropy of steel sheets. Int J Plast 45:119–139

    Google Scholar 

  45. 45.

    Hill R (1948) A theory of yielding and plastic flow of anisotropic metals. Proc R Soc Lond 193:281–297

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Areias P Simplas. Portuguese Software Association (ASSOFT) registry number 2281/D/17

  47. 47.

    Areias P, Rabczuk T, Melo FJ, César de Sá JMA (2015) Coulomb frictional contact by explicit projection in the cone for finite displacement quasi-static problems. Comput Mech 55(1):57–72

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Wagner W, Klinkel S, Gruttmann F (2002) Elastic and plastic analysis of thin-walled structures using improved hexahedral elements. Comput Struct 80:857–869

    Google Scholar 

  49. 49.

    Comsa D-S, Banabic D (2007) Numerical simulation of sheet metal forming processes using a new yield criterion. Key Eng Mater 344:833–840

    Google Scholar 

  50. 50.

    Feng Z, Yoon S-Y, Choi J-H, Barrett TJ, Zecevic M, Barlat F, Knezevic M (2020) A comparative study between elasto-plastic self-consistent crystal plasticity and anisotropic yield function with distortional hardening formulations for sheet metal forming. Mech Mater 148:103422

    Google Scholar 

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The authors acknowledge the support of FCT, through IDMEC, under LAETA, project UIDB/50022/2020. The first author would like to thank the help from Professor Leonel Fernandes concerning ODE integrators.

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Correspondence to P. Areias.

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Areias, P., Rosa, P.A.R. & Rabczuk, T. Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring. Comput Mech 68, 391–404 (2021).

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  • Hyperelasto-plasticity
  • Anisotropy
  • Matrix exponential
  • Power series
  • Scaling and squaring
  • Mandel stress