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Discretization Methods for Solids Undergoing Finite Deformations

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Advanced Finite Element Technologies

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 566))

Abstract

Finite element methods for solving engineering problems are used since decades in industrial applications. This market is still growing and the underlying methodologies, formulations, and algorithms seem to be settled. But still there are open questions and problems when applying the finite element method to situations where finite strains occur. Another problem area is the incorporation of constraints into the formulations, such as incompressibility, contact, and directional constraints needed to formulate anisotropic material behavior. In this section, we present the basic continuum formulation and different discretization techniques that can be used to overcome the problems mentioned above. Additionally, a set of test problems is presented that can be applied to test new finite element formulations.

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Notes

  1. 1.

    Small letters are used for indices of vectors and tensors which are related to the basis \(\mathbf{e }_i\) of the current or spatial configuration. The quantities \(x_i\) are the spatial coordinates of X.

  2. 2.

    Note that this split represents physically a different strain energy function than (35).

  3. 3.

    The construction of such principle has advantages. One of them is that the development of efficient algorithms for the solution of the nonlinear equations can be based on optimization strategies.

  4. 4.

    This result corresponds to the variation \(\delta \mathbf{E }\), defined already (49). The partial derivative of W with respect to \(\mathbf{C }\) leads to the second Piola-Kirchhoff stress tensor \(\mathbf{S }\), see (30): \(\mathbf{S } = 2\,\partial W/\partial \mathbf{C }\). Hence Eq. (57) is equivalent to the weak form (50) for a hyperelastic material.

  5. 5.

    The user can overrule the automatic selection of the integration rule, but this is only necessary when special shape functions are used.

  6. 6.

    All data are provided as dimensionless constants, it is assumed that the dimensions match real physical data.

  7. 7.

    Here the variable p is the stress component related to the constraint, e.g., the stress in direction of \(\mathbf{a }\). It has to be scaled in order to yield the correct stress.

References

  • Altenbach, J., & Altenbach, H. (1994). Einführung in die Kontinuumsmechanik. Stuttgart: Teubner-Verlag.

    MATH  Google Scholar 

  • Arnold, D. N., Brezzi, F., & Douglas, J. (1984). Peers: A new mixed finite element for plane elasticity. Japan Journal of Applied Mathematics, 1, 347–367.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Becker, E., & Bürger, W. (1975). Kontinuumsmechanik. Stuttgart: B.G. Teubner.

    Book  MATH  Google Scholar 

  • Belytschko, T., Ong, J. S. J., Liu, W. K., & Kennedy, J. M. (1984). Hourglass control in linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 43, 251–276.

    Article  MATH  Google Scholar 

  • Braess, D. (1992). Finite elemente. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Brezzi, F., & Fortin, M. (1991). Mixed and hybrid finite element methods. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Chadwick, P. (1999). Continuum mechanics, Concise theory and problems. Mineola: Dover Publications.

    Google Scholar 

  • Chapelle, D., & Bathe, K. J. (1993). The inf-sup test. Computers and Structures, 47, 537–545.

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet, P. G. (1988). Mathematical elasticity I: Three-dimensional elasticity. Amsterdam: North-Holland.

    MATH  Google Scholar 

  • Cottrell, J. A., Hughes, T. J. R., & Bazilevs, Y. (2009). Isogeometric analysis: Toward integration of CAD and FEA. New York: Wiley.

    Book  Google Scholar 

  • Duffet, G., & Reddy, B. D. (1983). The analysis of incompressible hyperelastic bodies by the finite element method. Computer Methods in Applied Mechanics and Engineering, 41, 105–120.

    Article  MathSciNet  Google Scholar 

  • Eringen, A. (1967). Mechanics of Continua. New York: Wiley.

    MATH  Google Scholar 

  • Flory, P. (1961). Thermodynamic relations for high elastic materials. Transactions of the Faraday Society, 57, 829–838.

    Article  MathSciNet  Google Scholar 

  • Fraeijs de Veubeke, B. M. (1975). Stress function approach. In World Congress on the Finite Element Method in Structural Mechanics (pp. 1–51). Bournmouth.

    Google Scholar 

  • Häggblad, B., & Sundberg, J. A. (1983). Large strain solutions of rubber components. Computers and Structures, 17, 835–843.

    Article  Google Scholar 

  • Holzapfel, G. A. (2000). Nonlinear solid mechanics. Chichester: Wiley.

    MATH  Google Scholar 

  • Hughes, T. R. J. (1987). The finite element method. Englewood Cliffs: Prentice Hall.

    MATH  Google Scholar 

  • Korelc, J. (1997). Automatic generation of finite-element code by simultaneous optimization of expressions. Theoretical Computer Science, 187, 231–248.

    Article  MATH  Google Scholar 

  • Korelc, J. (2002). Multi-language and multi-environment generation of nonlinear finite element codes. Engineering with Computers, 18, 312–327.

    Article  Google Scholar 

  • Korelc, J. (2011). AceGen and AceFEM user manual.Technical report. University of Ljubljana. http://www.fgg.uni-lj.si/symech/.

  • Malvern, L. E. (1969). Introduction to the mechanics of a continuous medium. Englewood Cliffs: Prentice-Hall.

    MATH  Google Scholar 

  • Marsden, J. E., & Hughes, T. J. R. (1983). Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall.

    MATH  Google Scholar 

  • Mathematica. (2011). http://www.wolfram.com.

  • Oden, J. T., & Key, J. E. (1970). Numerical analysis of finite axisymmetrical deformations of incompressible elastic solids of revolution. International Journal of Solids and Structures, 6, 497–518.

    Article  MATH  Google Scholar 

  • Ogden, R. W. (1984). Non-linear elastic deformations. Chichester: Ellis Horwood and Wiley.

    MATH  Google Scholar 

  • Reese, S. (2005). On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity. Computer Methods in Applied Mechanics and Engineering, 194, 4685–4715.

    Article  MATH  Google Scholar 

  • Reese, S., & Wriggers, P. (2000). A new stabilization concept for finite elements in large deformation problems. International Journal for Numerical Methods in Engineering, 48, 79–110.

    Article  MATH  Google Scholar 

  • Simo, J. C., & Armero, F. (1992). Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 33, 1413–1449.

    Article  MathSciNet  MATH  Google Scholar 

  • Simo, J. C., Armero, F., & Taylor, R. L. (1993). Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Computer Methods in Applied Mechanics and Engineering, 110, 359–386.

    Article  MathSciNet  MATH  Google Scholar 

  • Simo, J. C., & Rifai, M. S. (1990). A class of assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29, 1595–1638.

    Article  MathSciNet  MATH  Google Scholar 

  • Simo, J. C., Taylor, R. L., & Pister, K. S. (1985). Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Computer Methods in Applied Mechanics and Engineering, 51, 177–208.

    Article  MathSciNet  MATH  Google Scholar 

  • Stenberg, R. (1988). A family of mixed finite elements for elasticity problems. Numerische Mathematik, 48, 513–538.

    Article  MathSciNet  MATH  Google Scholar 

  • Sussman, T., & Bathe, K. J. (1987). A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Computers and Structures, 26, 357–409.

    Article  MATH  Google Scholar 

  • Taylor, R. L. (2011). FEAP: A finite element analysis program. Technical report. http://www.ce.berkeley.edu/feap.

  • Truesdell, C., & Noll, W. (1965). The nonlinear field theories of mechanics. In S. Flügge (Ed.), Handbuch der Physik III/3. Berlin: Springer.

    Google Scholar 

  • Truesdell, C., & Toupin, R. (1960). The classical field theorie. Handbuch der Physik III/1. Berlin: Springer.

    Google Scholar 

  • Washizu, K. (1975). Variational methods in elasticity and plasticity (2nd ed.). Oxford: Pergamon Press.

    MATH  Google Scholar 

  • Wriggers, P. (2008). Nonlinear finite elements. Berlin: Springer.

    MATH  Google Scholar 

  • Zienkiewicz, O. C., & Taylor, R. L. (1989). The finite element method (4th ed., Vol. 1). London: McGraw Hill.

    Google Scholar 

  • Zienkiewicz, O. C., & Taylor, R. L. (2000). The finite element method (5th ed., Vol. 2). Oxford: Butterworth-Heinemann.

    Google Scholar 

  • Zienkiewicz, O. C., Taylor, R. L., & Too, J. M. (1971). Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3, 275–290.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Continuum Mechanics, Gottfried Wilhelm Leibniz Universität, Hannover, Germany

    Peter Wriggers

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  1. Peter Wriggers
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Correspondence to Peter Wriggers .

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Editors and Affiliations

  1. Institute of Mechanics, University of Duisburg-Essen, Essen, Germany

    Jörg Schröder

  2. University of Hannover, Hannover, Germany

    Peter Wriggers

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Wriggers, P. (2016). Discretization Methods for Solids Undergoing Finite Deformations. In: Schröder, J., Wriggers, P. (eds) Advanced Finite Element Technologies. CISM International Centre for Mechanical Sciences, vol 566. Springer, Cham. https://doi.org/10.1007/978-3-319-31925-4_2

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  • DOI: https://doi.org/10.1007/978-3-319-31925-4_2

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