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Erschienen in:
Functional Analysis, Boundary Value Problems and Finite Elements Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

Discretization Methods for Solids Undergoing Finite Deformations

verfasst von : Peter Wriggers

Erschienen in: Advanced Finite Element Technologies

Verlag: Springer International Publishing

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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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Vorheriges Kapitel Functional Analysis, Boundary Value Problems and Finite Elements
Nächstes Kapitel Three-Field Mixed Finite Element Methods in Elasticity
Fußnoten
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
Note that this split represents physically a different strain energy function than (35).
 
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
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
The user can overrule the automatic selection of the integration rule, but this is only necessary when special shape functions are used.
 
6
All data are provided as dimensionless constants, it is assumed that the dimensions match real physical data.
 
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.
 
Literatur
Altenbach, J., & Altenbach, H. (1994). Einführung in die Kontinuumsmechanik. Stuttgart: Teubner-Verlag.MATH
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.MathSciNetCrossRefMATH
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.MathSciNetCrossRefMATH
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.CrossRefMATH
Brezzi, F., & Fortin, M. (1991). Mixed and hybrid finite element methods. Berlin: Springer.CrossRefMATH
Chadwick, P. (1999). Continuum mechanics, Concise theory and problems. Mineola: Dover Publications.
Ciarlet, P. G. (1988). Mathematical elasticity I: Three-dimensional elasticity. Amsterdam: North-Holland.MATH
Cottrell, J. A., Hughes, T. J. R., & Bazilevs, Y. (2009). Isogeometric analysis: Toward integration of CAD and FEA. New York: Wiley.CrossRef
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.MathSciNetCrossRef
Eringen, A. (1967). Mechanics of Continua. New York: Wiley.MATH
Flory, P. (1961). Thermodynamic relations for high elastic materials. Transactions of the Faraday Society, 57, 829–838.MathSciNetCrossRef
Fraeijs de Veubeke, B. M. (1975). Stress function approach. In World Congress on the Finite Element Method in Structural Mechanics (pp. 1–51). Bournmouth.
Häggblad, B., & Sundberg, J. A. (1983). Large strain solutions of rubber components. Computers and Structures, 17, 835–843.CrossRef
Holzapfel, G. A. (2000). Nonlinear solid mechanics. Chichester: Wiley.MATH
Hughes, T. R. J. (1987). The finite element method. Englewood Cliffs: Prentice Hall.MATH
Korelc, J. (1997). Automatic generation of finite-element code by simultaneous optimization of expressions. Theoretical Computer Science, 187, 231–248.CrossRefMATH
Korelc, J. (2002). Multi-language and multi-environment generation of nonlinear finite element codes. Engineering with Computers, 18, 312–327.CrossRef
Malvern, L. E. (1969). Introduction to the mechanics of a continuous medium. Englewood Cliffs: Prentice-Hall.MATH
Marsden, J. E., & Hughes, T. J. R. (1983). Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall.MATH
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.CrossRefMATH
Ogden, R. W. (1984). Non-linear elastic deformations. Chichester: Ellis Horwood and Wiley.MATH
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.CrossRefMATH
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.CrossRefMATH
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.MathSciNetCrossRefMATH
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.MathSciNetCrossRefMATH
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.MathSciNetCrossRefMATH
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.MathSciNetCrossRefMATH
Sussman, T., & Bathe, K. J. (1987). A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Computers and Structures, 26, 357–409.CrossRefMATH
Truesdell, C., & Noll, W. (1965). The nonlinear field theories of mechanics. In S. Flügge (Ed.), Handbuch der Physik III/3. Berlin: Springer.
Truesdell, C., & Toupin, R. (1960). The classical field theorie. Handbuch der Physik III/1. Berlin: Springer.
Washizu, K. (1975). Variational methods in elasticity and plasticity (2nd ed.). Oxford: Pergamon Press.MATH
Wriggers, P. (2008). Nonlinear finite elements. Berlin: Springer.MATH
Zienkiewicz, O. C., & Taylor, R. L. (1989). The finite element method (4th ed., Vol. 1). London: McGraw Hill.
Zienkiewicz, O. C., & Taylor, R. L. (2000). The finite element method (5th ed., Vol. 2). Oxford: Butterworth-Heinemann.
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.CrossRefMATH
Metadaten
Titel
Discretization Methods for Solids Undergoing Finite Deformations
verfasst von
Peter Wriggers
Copyright-Jahr
2016
Buch
Advanced Finite Element Technologies

Print ISBN: 978-3-319-31923-0

Electronic ISBN: 978-3-319-31925-4

Copyright-Jahr: 2016

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

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