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Micropolar hyperelasticity: constitutive model, consistent linearization and simulation of 3D scale effects

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Abstract

This study describes a computational framework for three-dimensional finite strain and finite curvature micropolar hyperelasticity. The model is based on the non-linear kinematic setting and features an appropriate hyperelastic material law which is derived within the thermodynamically consistent framework. The material tangent operator is obtained by consistent linearization. An implicit finite element method with a Newton-Raphson procedure is employed for the computation of the nodal displacements and rotations. A number of numerical examples is presented. The results demonstrate (i) that the methodology is capable of capturing 3D length scale effects in finite deformation, (ii) that it is robust and computationally efficient and (iii) that the proposed micropolar element tangent renders asymptotically quadratic convergence of the Newton-Raphson procedure. It is shown that the classical Neo-Hooke type material behaviour is recovered as a special case within the proposed micropolar setting.

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References

  1. Argyris J (1982) An excursion into large rotations. Comput Methods Appl Mech Eng 32(1): 85–155

    Article  MathSciNet  MATH  Google Scholar 

  2. Arslan H, Sture S (2008) Finite element simulation of localization in granular materials by micropolar continuum approach. Comput Geotech 35(4): 548–562

    Article  Google Scholar 

  3. Arzt E (1998) Size effects in materials due to microstructural and dimensional constraints: a comparative review. Acta Mater 46(16): 5611–5626

    Article  Google Scholar 

  4. Bauer S, Dettmer W, Perić D, Schäfer M (2011) Micropolar hyper-elastoplasticity: constitutive model, consistent linearization and simulation of 3D scale effects. Int J Numer Methods Eng. doi:10.1002/nme.4256

  5. Bauer S, Schäfer M (2009) Numerical aspects of finite rotation parameterization for non-linear micropolar finite element analysis. In: Ambrosio J (ed) 7th EUROMECH solid mechanics conference, pp 1–18. Lisbon

  6. Bauer S, Schäfer M, Grammenoudis P, Tsakmakis C (2010) Three-dimensional finite elements for large deformation micropolar elasticity. Comput Methods Appl Mech Eng 199(41–44): 2643–2654

    Article  MATH  Google Scholar 

  7. Begley M, Hutchinson J (1998) The mechanics of size-dependent indentation. J Mech Phys Solids 46(10): 2049–2068

    Article  MATH  Google Scholar 

  8. Betsch P (1998) On the parametrization of finite rotations in computational mechanics A classification of concepts with application to smooth shells. Comput Methods Appl Mech Eng 155(3–4): 273–305

    Article  MathSciNet  MATH  Google Scholar 

  9. de Borst R (1991) Simulation of strain localization: a reappraisal of the cosserat continuum. Eng Comput 8(4): 317–332

    Article  Google Scholar 

  10. de Borst R, Sluys L, Mühlhaus H, Pamin J (1993) Fundamental issues in finite element analyses of localization of deformation. Eng Comput 10(2): 99–121

    Article  Google Scholar 

  11. Cosserat E, Cosserat F (1909) Théorie de corps déformables. A. Herman et fils, Paris

    Google Scholar 

  12. de Souza Neto E, Perić D, Owen D (2008) Computational methods for plasticity: theory and applications. Wiley, Chichester

    Book  Google Scholar 

  13. Eringen A (1966) Linear theory of micropolar elasticity. J Math Mech 15(6): 909–923

    MathSciNet  MATH  Google Scholar 

  14. Eringen A (1999) Microcontinuum Field Theories, I: Foundations and Solids. Springer, New York

    Book  MATH  Google Scholar 

  15. Eringen A, Kafadar C (1976) Polar field theories. In: Eringen CA (eds) Continuum physics IV. Academic Press, New York, pp 2–75

    Google Scholar 

  16. Fleck N, Muller G, Ashby M, Hutchinson J (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42(2): 475–487

    Article  Google Scholar 

  17. Gauthier R, Jashman W (1975) A quest for micropolar elastic constants. J Appl Mech 42(2): 369–374

    Article  Google Scholar 

  18. Grammenoudis P, Tsakmakis C (2001) Hardening rules for finite deformation micropolar plasticity: Restrictions imposed by the second law of thermodynamics and the postulate of Il’iushin. Contin Mech Thermodyn 13(5): 325–363

    Article  MathSciNet  MATH  Google Scholar 

  19. Grammenoudis P, Tsakmakis C (2005) Finite element implementation of large deformation micropolar plasticity exhibiting isotropic and kinematic hardening effects. Int J Numer Methods Eng 62(12): 1691–1720

    Article  MATH  Google Scholar 

  20. Huang F, Liang K (1997) Boundary element analysis of stress concentration in micropolar elastic plate. Int J Numer Methods Eng 40(9): 1611–1622

    Article  Google Scholar 

  21. Huang F, Yan B, Yan J, Yang D (2000) Bending analysis of micropolar elastic beam using a 3-D finite element method. Int J Eng Sci 38(3): 275–286

    Article  MATH  Google Scholar 

  22. Hughes T, Pister K (1978) Consistent linearization in mechanics of solids and structures. Comput Struct 8: 391–397

    Article  MathSciNet  MATH  Google Scholar 

  23. Ibrahimbegović A (1997) On the choice of finite rotation parameters. Comput Methods Appl Mech Eng 149(1–4): 49–71

    Article  MATH  Google Scholar 

  24. Ibrahimbegović A, Koz̆ar I (1995) Non-linear Wilsons brick element for finite elastic deformations of three-dimensional solids. Commun Numer Methods Eng 11(8): 655–664

    Article  MATH  Google Scholar 

  25. Kaloni P, Ariman T (1967) Stress concentration effects in micropolar elasticity. Z Angew Math Phys ZAMP 18(1): 136–141

    Article  Google Scholar 

  26. Koz̆ar I, Ibrahimbegović A (1995) Finite element formulation of the finite rotation solid element. Finite Elem Anal Des 20(2): 101–126

    Article  MathSciNet  Google Scholar 

  27. Li L, Xie S (2004) Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in MEMS. Int J Mech Sci 46(11): 1571–1587

    Article  MATH  Google Scholar 

  28. Li X, Tang H (2005) A consistent return mapping algorithm for pressure-dependent elastoplastic cosserat continua and modelling of strain localisation. Comput Struct 83(1): 1–10

    Article  Google Scholar 

  29. Mindlin R (1963) Influence of couple-stresses on stress concentrations. Exp Mech 3(1): 1–7

    Article  MathSciNet  Google Scholar 

  30. Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11(1): 415–448

    Article  MathSciNet  MATH  Google Scholar 

  31. Mühlhaus H, Aifantis E (1991) A variational principle for gradient plasticity. Int J Solids Struct 28(7): 845–857

    Article  MATH  Google Scholar 

  32. Mühlhaus H, Vardoulakis I (1987) The thickness of shear bands in granular materials. Geotechnique 37(3): 271–283

    Article  Google Scholar 

  33. Perić D, Yu J, Owen D (1994) On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua. Int J Numer Methods Eng 37(8): 1351–1379

    Article  MATH  Google Scholar 

  34. Providas E, Kattis M (2002) Finite element method in plane cosserat elasticity. Comput Struct 80(27–30): 2059–2069

    Article  Google Scholar 

  35. Ramezani S, Naghdabadi R, Sohrabpour S (2008) Non-linear finite element implementation of micropolar hypo-elastic materials. Comput Methods Appl Mech Eng 197(49–50): 4149–4159

    Article  MATH  Google Scholar 

  36. Ramezani S, Naghdabadi R, Sohrabpour S (2009) Constitutive equations formicropolar hyper-elastic materials. Int J Solids Struct (14–15): 2765–2773

  37. Schäfer H (1962) Versuch einer elastizitätstheorie des zweidimensionalen ebenen Cosserat-Kontinuums. Miszellaneen der Angewandten Mechanik, pp 277–292

  38. Steinmann P (1994) A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int J Solids Struct 31(8): 1063–1084

    Article  MathSciNet  MATH  Google Scholar 

  39. Steinmann P (1995) Theory and numerics of ductile micropolar elastoplastic damage. Int J Numer Methods Eng 38(4): 583–606

    Article  MathSciNet  MATH  Google Scholar 

  40. Stölken J, Evans A (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46: 5109–5115

    Article  Google Scholar 

  41. Taylor R (2004) Feap Version 7.5 Programmer Manual. Department of Civil and Environmental Engineering/University of Berkeley, Berkeley

    Google Scholar 

  42. Wriggers P (2001) Nichtlineare finite-elemente-methoden. Springer, Berlin

    Book  Google Scholar 

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Authors and Affiliations

  1. Chair of Numerical Methods in Mechanical Engineering, Technische Universität Darmstadt, Dolivostrasse 15, 64293, Darmstadt, Germany

    S. Bauer & M. Schäfer

  2. Civil and Computational Engineering Research Centre, College of Engineering, Swansea University, Singleton Park, Swansea, SA2 8PP, UK

    W. G. Dettmer & D. Perić

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  1. S. Bauer
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  2. W. G. Dettmer
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  3. D. Perić
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  4. M. Schäfer
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Correspondence to S. Bauer.

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Bauer, S., Dettmer, W.G., Perić, D. et al. Micropolar hyperelasticity: constitutive model, consistent linearization and simulation of 3D scale effects. Comput Mech 50, 383–396 (2012). https://doi.org/10.1007/s00466-012-0679-9

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  • DOI: https://doi.org/10.1007/s00466-012-0679-9

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