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Finite and Virtual Element Formulations for Large Strain Anisotropic Material with Inextensive Fibers

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Multiscale Modeling of Heterogeneous Structures

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 86))

Abstract

Anisotropic material with inextensive or nearly inextensible fibers introduce constraints in the mathematical formulations of the underlying differential equations from mechanics. This is always the case when fibers with high stiffness in a certain direction are present and a relatively weak matrix material is supporting these fibers. In numerical solution schemes like the finite element method or the virtual element method the presence of constraints—in this case associated to a possible fiber inextensibility compared to a matrix—lead to so called locking-phenomena. This can be overcome by special interpolation schemes as has been discussed extensively for volume constraints like incompressibility as well as contact constraints. For anisotropic material behaviour the most severe case is related to inextensible fibers. In this paper a mixed method is developed for finite elements and virtual elements that can handle anisotropic materials with inextensive and nearly inextensive fibers. For this purpose a classical ansatz, known from the modeling of volume constraint is adopted leading stable elements that can be used in the finite strain regime.

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Notes

  1. 1.

    It is well known that ill-conditioning can occur when a large penalty parameter \(C_c\) is selected. Thus in reality the penalty formulation is only able to approximately enforce the constraint condition (8).

  2. 2.

    In the linear case both conditions, while being different, yield a linear dependence on the components of the displacement gradient. Thus there the choice of using the same ansatz function for the pressure (incompressibility) and the fiber stress (anisotropy) is justified.

References

  1. Auricchio, F., de Veiga, L.B., Lovadina, C., Reali, A.: A stability study of some mixed finite elements for large deformation elasticity problems. Comput. Methods Appl. Mech. Eng. 194, 1075–1092 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auricchio, F., da Velga, L.B., Lovadina, C., Reali, A., Taylor, R.L., Wriggers, P.: Approximation of incompressible large deformation elastic problems: some unresolved issues. Comput. Mech. 52, 1153–1167 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuska, I.: The finite element method with lagrangian multipliers. Numer. Math. 20(3), 179–192 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuska, I., Suri, M.: Locking effects in the finite element approximation of elasticity problems. Numer. Math. 62(1), 439–463 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bathe, K.J.: Finite Element Procedures. Prentice Hall (2006)

    Google Scholar 

  6. Beirão Da Veiga, L., Brezzi, F., Marini, L.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)

    Google Scholar 

  7. Beirão Da Veiga, L., Lovadina, C., Mora, D.: A virtual element method for elastic and inelastic problems on polytope meshes. Comput. Methods Appl. Mech. Eng. 295, 327–346 (2015)

    Google Scholar 

  8. Belytschko, T., Bindeman, L.P.: Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems. Comput. Methods Appl. Mech. Eng. 88(3), 311–340 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Belytschko, T., Ong, J.S.J., Liu, W.K., Kennedy, J.M.: Hourglass control in linear and nonlinear problems. Comput. Methods Appl. Mech. Eng. 43, 251–276 (1984)

    Article  MATH  Google Scholar 

  10. Boerner, E., Loehnert, S., Wriggers, P.: A new finite element based on the theory of a cosserat point—extension to initially distorted elements for 2D plane strain. Int. J. Numer. Methods Eng. 71, 454–472 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. Revue francaise d’automatique, informatique, recherche operationnelle. Anal. Numer. 8(2), 129–151 (1974)

    Google Scholar 

  12. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  13. Chapelle, D., Bathe, K.J.: The inf-sup test. Comput. Struct. 47, 537–545 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chi, H., Beirão da Veiga, L., Paulino, G.: Some basic formulations of the virtual element method (VEM) for finite deformations. Comput. Methods Appl. Mech. Eng. (2016). https://doi.org/10.1016/j.cma.2016.12.020

  15. Flanagan, D., Belytschko, T.: A uniform strain hexahedron and quadrilateral with orthogonal hour-glass control. Int. J. Num. Methods Eng. 17, 679–706 (1981)

    Article  MATH  Google Scholar 

  16. Hamila, N., Boisse, P.: Locking in simulation of composite reinforcement deformations. Analysis and treatment. Compos. Part A, 109–117 (2013)

    Google Scholar 

  17. Helfenstein, J., Jabareen, M., Mazza, E., Govindjee, S.: On non-physical response in models for fiber-reinforced hyperelastic materials. Int. J. Solids Struct. 47(16), 2056–2061 (2010)

    Article  MATH  Google Scholar 

  18. Holzapfel, G., Gasser, T., Ogden, R.: A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elast. Phys. Sci. Solids 61(1–3), 1–48 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Hughes, T.R.J.: The Finite Element Method. Prentice Hall, Englewood Cliffs, New Jersey (1987)

    MATH  Google Scholar 

  20. Korelc, J.: Automatic generation of finite-element code by simultaneous optimization of expressions. Theor. Comput. Sci. 187, 231–248 (1997)

    Article  MATH  Google Scholar 

  21. Korelc, J.: Automatic generation of numerical codes with introduction to AceGen 4.0 symbolic code generator (2000). http://www.fgg.uni-lj.si/Symech

  22. Korelc, J.: Computational Templates (2016). http://www.fgg.uni-lj.si/Symech

  23. Korelc, J., Solinc, U., Wriggers, P.: An improved EAS brick element for finite deformation. Comput. Mech. 46, 641–659 (2010)

    Article  MATH  Google Scholar 

  24. Korelc, J., Wriggers, P.: Automation of Finite Element Methods. Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  25. Krysl, P.: Mean-strain eight-node hexahedron with optimized energy-sampling stabilization for large-strain deformation. Int. J. Num. Methods Eng. 103, 650–670 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Krysl, P.: Mean-strain eight-node hexahedron with stabilization by energy sampling. Int. J. Numer. Methods Eng. 103, 437–449 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Krysl, P.: Mean-strain 8-node hexahedron with optimized energy-sampling stabilization. Finite Elem. Anal. Des. 108, 41–53 (2016)

    Article  MathSciNet  Google Scholar 

  28. Loehnert, S., Boerner, E., Rubin, M., Wriggers, P.: Response of a nonlinear elastic general cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput. Mech. 36, 255–265 (2005)

    Article  MATH  Google Scholar 

  29. Mueller-Hoeppe, D.S., Loehnert, S., Wriggers, P.: A finite deformation brick element with inhomogeneous mode enhancement. Int. J. Numer. Methods Eng. 78, 1164–1187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nadler, B., Rubin, M.: A new 3-D finite element for nonlinear elasticity using the theory of a cosserat point. Int. J. Solids Struct. 40, 4585–4614 (2003)

    Article  MATH  Google Scholar 

  31. Reese, S.: On a consistent hourglass stabilization technique to treat large inelastic deformations and thermo-mechanical coupling in plane strain problems. Int. J. Numer. Methods Eng. 57, 1095–1127 (2003)

    Article  MATH  Google Scholar 

  32. Reese, S., Kuessner, M., Reddy, B.D.: A new stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Methods Eng. 44, 1617–1652 (1999)

    Article  Google Scholar 

  33. Reese, S., Wriggers, P.: A new stabilization concept for finite elements in large deformation problems. Int. J. Numer. Methods Eng. 48, 79–110 (2000)

    Article  MATH  Google Scholar 

  34. Sansour, C.: On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. Eur. J. Mech. A/Solids 27(1), 28–39 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Schröder, J.: Anisotropic polyconvex energies. In: Schröder, J. (ed.) Polyconvex Analysis, vol. 62, pp. 1–53. CISM, Springer, Wien (2009)

    Google Scholar 

  36. Schröder, J., Neff, P.: Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct. 40(2), 401–445 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schröder, J., Viebahn, N., Balzani, D., Wriggers, P.: A novel mixed finite element for finite anisotropic elasticity; the SKA-element simplified kinematics for anisotropy. Comput. Methods Appl. Mech. Eng. 310, 475–494 (2016)

    Article  MathSciNet  Google Scholar 

  38. Simo, J.C., Armero, F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 33, 1413–1449 (1992)

    Article  MATH  Google Scholar 

  39. Simo, J.C., Armero, F., Taylor, R.L.: Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput. Methods Appl. Mech. Eng. 110, 359–386 (1993)

    Article  MATH  Google Scholar 

  40. Simo, J.C., Rifai, M.S.: A class of assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29, 1595–1638 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. Simo, J.C., Taylor, R.L., Pister, K.S.: Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput. Methods Appl. Mech. Eng. 51, 177–208 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  42. ten Thjie, R.H.W., Akkerman, R.: Solutions to intra-ply shear locking in finite element analyses of fibre reinforced materials. Compos. Part A, 1167–1176 (2008)

    Google Scholar 

  43. Weiss, J.A., Maker, B.N., Govindjee, S.: Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput. Methods Appl. Mech. Eng. 135(1), 107–128 (1996)

    Article  MATH  Google Scholar 

  44. Wriggers, P.: Nonlinear Finite Elements. Springer, Berlin (2008)

    MATH  Google Scholar 

  45. Wriggers, P., Reddy, B., Rust, W., Hudobivnik, B.: Efficient virtual element formulations for compressible and incompressible finite deformations. Comput. Mech. (2017). https://doi.org/10.1007/s00466-017-1405-4

  46. Wriggers, P., Rust, W., Reddy, B.: A virtual element method for contact. Comput. Mech. 58, 1039–1050 (2016)

    Article  MathSciNet  Google Scholar 

  47. Wriggers, P., Schröder, J., Auricchio, F.: Finite element formulations for large strain anisotropic materials. Int. J. Adv. Model. Simul. Eng. Sci. 3(25), 1–18 (2016)

    Google Scholar 

  48. Zdunek, A., Rachowicz, W., Eriksson, T.: A novel computational formulation for nearly incompressible and nearly inextensible finite hyperelasticity. Comput. Methods Appl. Mech. Eng. 281, 220–249 (2014)

    Article  MathSciNet  Google Scholar 

  49. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, vol. 2, 5th edn. Butterworth-Heinemann, Oxford, UK (2000)

    Google Scholar 

  50. Zienkiewicz, O.C., Taylor, R.L., Too, J.M.: Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Methods Eng. 3, 275–290 (1971)

    Article  MATH  Google Scholar 

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Acknowledgements

The first and third author acknowledge the support of the “Deutsche Forschungsgemeinschaft” under contract of the Priority Program 1748 ‘Reliable simulation techniques in solid mechanics: Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project WR 19/50-1 and SCHR 570/23-1.

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Authors and Affiliations

  1. Leibniz Universität Hannover, Hannover, Germany

    P. Wriggers & B. Hudobivnik

  2. Universität Duisburg-Essen, Duisburg, Germany

    J. Schröder

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  1. P. Wriggers
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  2. B. Hudobivnik
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  3. J. Schröder
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Correspondence to P. Wriggers .

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Editors and Affiliations

  1. University of Zagreb, Zagreb, Croatia

    Jurica Sorić

  2. Institute of Continuum Mechanics, Leibniz Universität Hannover, Hannover, Germany

    Peter Wriggers

  3. LMT Cachan, Cachan, France

    Olivier Allix

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Wriggers, P., Hudobivnik, B., Schröder, J. (2018). Finite and Virtual Element Formulations for Large Strain Anisotropic Material with Inextensive Fibers. In: Sorić, J., Wriggers, P., Allix, O. (eds) Multiscale Modeling of Heterogeneous Structures. Lecture Notes in Applied and Computational Mechanics, vol 86. Springer, Cham. https://doi.org/10.1007/978-3-319-65463-8_11

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  • DOI: https://doi.org/10.1007/978-3-319-65463-8_11

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