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.
Keywords
This is a preview of subscription content, log in via an institution.
Notes
- 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.
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
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)
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)
Babuska, I.: The finite element method with lagrangian multipliers. Numer. Math. 20(3), 179–192 (1973)
Babuska, I., Suri, M.: Locking effects in the finite element approximation of elasticity problems. Numer. Math. 62(1), 439–463 (1992)
Bathe, K.J.: Finite Element Procedures. Prentice Hall (2006)
Beirão Da Veiga, L., Brezzi, F., Marini, L.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)
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)
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)
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)
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)
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)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
Chapelle, D., Bathe, K.J.: The inf-sup test. Comput. Struct. 47, 537–545 (1993)
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
Flanagan, D., Belytschko, T.: A uniform strain hexahedron and quadrilateral with orthogonal hour-glass control. Int. J. Num. Methods Eng. 17, 679–706 (1981)
Hamila, N., Boisse, P.: Locking in simulation of composite reinforcement deformations. Analysis and treatment. Compos. Part A, 109–117 (2013)
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)
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)
Hughes, T.R.J.: The Finite Element Method. Prentice Hall, Englewood Cliffs, New Jersey (1987)
Korelc, J.: Automatic generation of finite-element code by simultaneous optimization of expressions. Theor. Comput. Sci. 187, 231–248 (1997)
Korelc, J.: Automatic generation of numerical codes with introduction to AceGen 4.0 symbolic code generator (2000). http://www.fgg.uni-lj.si/Symech
Korelc, J.: Computational Templates (2016). http://www.fgg.uni-lj.si/Symech
Korelc, J., Solinc, U., Wriggers, P.: An improved EAS brick element for finite deformation. Comput. Mech. 46, 641–659 (2010)
Korelc, J., Wriggers, P.: Automation of Finite Element Methods. Springer, Berlin (2016)
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)
Krysl, P.: Mean-strain eight-node hexahedron with stabilization by energy sampling. Int. J. Numer. Methods Eng. 103, 437–449 (2015)
Krysl, P.: Mean-strain 8-node hexahedron with optimized energy-sampling stabilization. Finite Elem. Anal. Des. 108, 41–53 (2016)
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)
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)
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)
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)
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)
Reese, S., Wriggers, P.: A new stabilization concept for finite elements in large deformation problems. Int. J. Numer. Methods Eng. 48, 79–110 (2000)
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)
Schröder, J.: Anisotropic polyconvex energies. In: Schröder, J. (ed.) Polyconvex Analysis, vol. 62, pp. 1–53. CISM, Springer, Wien (2009)
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)
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)
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)
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)
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)
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)
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)
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)
Wriggers, P.: Nonlinear Finite Elements. Springer, Berlin (2008)
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
Wriggers, P., Rust, W., Reddy, B.: A virtual element method for contact. Comput. Mech. 58, 1039–1050 (2016)
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)
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)
Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, vol. 2, 5th edn. Butterworth-Heinemann, Oxford, UK (2000)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-65463-8_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65462-1
Online ISBN: 978-3-319-65463-8
eBook Packages: EngineeringEngineering (R0)