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Continuum Elements

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Abstract

After a short introduction in the main targets of element development for continuum elements this chapter will consider different approaches ranging from standard up to special formulations.

Keywords

  • Deformation Gradient
  • Strain Energy Function
  • Hexahedral Element
  • Follower Load
  • Enhanced Assumed Strain

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  • DOI: 10.1007/978-3-319-39005-5_6
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Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5
Fig. 6.6
Fig. 6.7

Notes

  1. 1.

    The importance of this point will diminish with the increasing computing power, but at the moment it is still of concern.

  2. 2.

    In general, no potential is associated with the pressure load. However under certain boundary conditions the total assembled tangent matrix can be symmetrical, e.g. for internal pressure in a closed solid. An in depth discussion can be found in e.g. Sewell (1967) and Schweizerhof (1982).

  3. 3.

    In the elastic case the condition is \(J_F=\det {{\varvec{F}}} = 1\) and for plastic flow the condition \(J_p = \det {{\varvec{F}}}_p=1\) holds.

  4. 4.

    For general nonlinear applications there exists no formulation of the BB-condition. However it can be applied the to the tangent spaces that belong to a given state of deformation and pressure, as e.g. provided in (6.31).

  5. 5.

    Here the Lame constant \(\lambda \) has to be neglected since incompressibility is directly enforced by the Lagrange multiplier term \(p\,(\,J_F-1\,)\) in (6.36).

  6. 6.

    The three-dimensional version of this element can be developed exactly along the lines of the two-dimensional formulation. Only the number of shape quadratic functions for the displacement field changes to 10 for the tetrahedral element and thee mixed interpolations for the pressure are linear with values at the vertices of the tetrahedra.

  7. 7.

    The order of \(\alpha \) and p is here not arbitrary since the condensation is based on the solution of a \(2 \times 2\) equation system with a zero on the diagonal related to p. A standard solver is used for the condensation, hence the dilatation variable \(\alpha \) has to be first.

  8. 8.

    It is also possible to formulate the Hu-Washizu principle in other work conjugate variables. Examples are the 2nd Piola–Kirchhoff stress tensor and the Green–Lagrangian strain tensor \(\varvec{E}\) or the application of the Biot stress tensor \(\varvec{T}_B\) together with the right stretch tensor \(\varvec{U}\). From the viewpoint of continuum mechanics these formulations are equivalent. However due to the fact that the strain measures \({{\varvec{F}}}\), \(\varvec{E}\) and \(\varvec{U}\) are different, their enhancement will lead to different finite element approximations and discretizations.

  9. 9.

    This representation deviates from the form advocated in Simo and Armero (1992) in such way that \(\bar{\varvec{H}}\) was introduced as a gradient and hence could be interpolated without using \({{\varvec{F}}}_0\).

  10. 10.

    Due to the fact that the data are related to a nearly incompressible material with a Poisson ratio of \(\nu = 0.4983\) the mixed O2/P1 element converges to a slightly different solution since the incompressibility is enforced exactly in O2/P1.

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Korelc, J., Wriggers, P. (2016). Continuum Elements. In: Automation of Finite Element Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-39005-5_6

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