A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

Abstract

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by

$$\begin{aligned} W_\mathrm {eH}(\varvec{F}) = \dfrac{\mu }{k}\, e^{\displaystyle k \left||\text{ dev }_n \log \varvec{U}\right||^2} + \dfrac{\kappa }{2 \hat{k}}\, e^{\displaystyle \hat{k} [\text{ tr } (\log \varvec{U})]^2 }, \end{aligned}$$

where \(\mu >0\) is the (infinitesimal) shear modulus, \(\kappa >0\) is the (infinitesimal) bulk modulus, k and \(\hat{k}\) are additional dimensionless material parameters, \(\varvec{U}=\sqrt{\varvec{F}^T\varvec{F}}\) is the right stretch tensor corresponding to the deformation gradient \(\varvec{F}\), \(\log \) denotes the principal matrix logarithm on the set of positive definite symmetric matrices, \(\text{ dev }_n \varvec{X} = \varvec{X}-\frac{\text{ tr } \varvec{X}}{n}\varvec{1}\) and \(||\varvec{X} ||= \sqrt{\text{ tr }\varvec{X}^T\varvec{X}}\) are the deviatoric part and the Frobenius matrix norm of an \(n\times n\)-matrix \(\varvec{X}\), respectively, and \(\text{ tr }\) denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging “eversion of elastic tubes” problem.

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Notes

  1. 1.

    Note that, although every objective and isotropic energy function can be expressed in terms of the (material) logarithmic strain tensor \(\log \varvec{U}\) (or the spatial strain tensor \(\log \varvec{V}\)), not every such energy can be expressed in terms of the logarithmic strain measures alone [33].

  2. 2.

    The invertibility of the mapping \(\varvec{B}\mapsto \varvec{\sigma }(\varvec{B})\) also holds for the volumetric-isochorically decoupled representation of the neo-Hooke and Mooney–Rivlin energies, which are in use for slightly compressible materials like rubber.

  3. 3.

    The first use of the definition \(\nu = -\frac{(\log V)_{22}}{(\log V)_{11}}\) is due to the famous German scientist Röntgen [43].

  4. 4.

    Truesdell remarks on the logarithmic strain that “[b]ecause of the difficulty of calculating the off-diagonal components of [\(\log \varvec{B}\)] in terms of the displacement gradient, Hencky’s theory is hard to use except in trivial cases”. [47, p. 202, (49.4)]

  5. 5.

    An everted rubber tube (shown in Fig. 15) was described by Truesdell as follows: “We see that the everted piece is a little longer than the other [identical, non-everted tube]. [...] With the naked eye we can see that the wall of the everted piece is a little thinner than it was originally. If we consider the part of the tube that lies a distance from the ends greater than one-fifth of the diameter, we can say that the everted piece, like the undeformed one, is very nearly a right-circular cylinder. We can idealize what we have seen by saying that an infinitely long, elastic, right-circular cylinder can be turned inside out so as to form another right-circular cylinder, having different radii”.

  6. 6.

    Note that for isotropic materials, the Kirchhoff stress can be obtained directly from the elastic energy as

    $$\begin{aligned} \varvec{\tau } = \frac{\partial W}{\partial \log \varvec{V}} = \frac{\partial W}{\partial \varvec{V}}\cdot \varvec{V}, \end{aligned}$$

    a formula first derived by Richter [42], see also [50].

  7. 7.

    The restriction of the three-dimensional exponentiated Hencky energy to planar strain is not polyconvex, while (56) is. The difference stems from the definition of the two-dimensional isochoric stretches \(\overline{\lambda }_i = (\det \varvec{U})^{-1/2}\,\lambda _i\).

  8. 8.

    The compressible hyperelastic models considered in [5] require the elastic energy potential W to be of the Valanis–Landel form [49]

    $$\begin{aligned} W(\varvec{F}) = \sum _{i=1}^3 w(\lambda _i) , \end{aligned}$$
    (66)

    with a function \(w:[0,\infty )\rightarrow \mathbb {R}\). In the past, energy functions of this type have been successfully applied in the incompressible case [19, 49], where they are in good agreement with experimental results. However, in the compressible case, an energy function of the form (66) without an additional volumetric energy term always implies zero lateral contraction for uniaxial stresses.

  9. 9.

    In the (unmodified) compressible case, the Abaqus updated Lagrangian implementation is not energy consistent, since the used Jaumann-rate of the Cauchy stress

    $$\begin{aligned} \overset{\triangle }{\varvec{\sigma }} = \dot{\varvec{\sigma }} + \varvec{\sigma }\cdot \varvec{W} - \varvec{W}\cdot \varvec{\sigma } \end{aligned}$$

    with \(\varvec{W}=\text{ skew }(\varvec{L}) = \frac{1}{2}(\varvec{L}-\varvec{L}^T)\) is not energy consistent with the Cauchy stress. The principle of virtual work must be implemented correctly for any choice of stress and objective stress-rate.

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Nedjar, B., Baaser, H., Martin, R.J. et al. A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes. Comput Mech 62, 635–654 (2018). https://doi.org/10.1007/s00466-017-1518-9

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Keywords

  • Exponentiated Hencky-logarithmic model
  • Spectral decomposition
  • Linearizations
  • Tangent moduli
  • Finite element method
  • Eversion of tubes