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Analysis of Hooke-like isotropic hypoelasticity models in view of applications in FE formulations

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Abstract

This paper presents an analysis of the constitutive relations of Hooke-like isotropic hypoelastic material models in Lagrangian and Eulerian forms generated using corotational stress rates with associated spin tensors from the family of material spin tensors. Explicit expressions were obtained for the Lagrangian and Eulerian tangent stiffness tensors for the hypoelastic materials considered. The main result of this study is a proof that these fourth-order tensors have full symmetry only for material models generated using two corotational stress rates: the Zaremba–Jaumann and the logarithmic ones. In the latter case, the Hooke-like isotropic hypoelastic material is simultaneously the Hencky isotropic hyperelastic material. For the material models considered, basis-free expressions for the material and spatial tangent stiffness tensors are obtained that can be implemented in FE codes. In particular, new basis-free expressions are derived for the tangent stiffness (elasticity) tensors for the Hencky isotropic hyperelastic material model.

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Notes

  1. The history of the derivation of this type of constitutive relations is described in [81, 83]. In these papers, the role of Zaremba (1903) and Jaumann (1911) as pioneers in this area of continuum mechanics is highlighted.

  2. Nevertheless, it is possible to construct scenarios of thermomechanical processes that give the current stress distribution in the Earth’s crust, as was done, e.g., in a simulation study [2, 50] of geophysical processes using the commercial MSC.Marc code.

  3. Hill [40] defines elasto-plastic materials as materials for which some objective rate of the Cauchy stress tensor \(\varvec{\sigma }^\nabla \) is linked to the stretching tensor \({\mathbf {d}}\) by a first-order homogeneous relation, but the coefficients of this relation also implicitly depend on the tensor \({\mathbf {d}}\).

  4. In some papers (cf., [76]), the Gurtin–Spear corotational rate is referred to as the Sowerby–Chu corotational rate [75] one.

  5. Hereinafter, tensors having both major and twice minor symmetries will be called supersymmetric (cf., [44]) or fully symmetric (cf., [26]) tensors.

  6. The number m (\(1\le m\le 3\)) will be called the eigenindex.

  7. Hereinafter, the notation \(\sum _{i\ne j=1}^{m}\) denotes the summation over \(i,j=1,\ldots , m\) and \(i\ne j\) and this summation is assumed to vanish when \(m=1\).

  8. Hereinafter, the subset \({\mathcal {T}}^{2\,+}_\text {orth}\subset {\mathcal {T}}^2\) denotes the set of all proper orthogonal second-order tensors (i.e., the tensors \(\varvec{\varPsi }\) such that \(\varvec{\varPsi }\cdot \varvec{\varPsi }^T={\mathbf {I}}\) and \(\det \varvec{\varPsi }=1\)).

  9. Hereinafter, we assume that all the tensors \({\mathbf {H}}\in {\mathcal {T}}^2\) are sufficiently smooth functions of a monotonically increasing parameter t (time), and we define the material time derivative (material rate) of the tensor \({\mathbf {H}}\): \(\dot{{\mathbf {H}}}\equiv \partial {\mathbf {H}}/\partial t\).

  10. The tensors \(\varvec{\varOmega }^L,\,\varvec{\omega }^E\in {\mathcal {T}}^2_{\text {skew}}\) are the twirl tensors of the Lagrangian and Eulerian triads, respectively.

  11. Hereinafter, the tensor \({\mathbb {O}}\) is the zero fourth-order tensor.

  12. In most of the studies cited, hypoelasticity relations are written in Eulerian form using the Cauchy stress tensor \(\varvec{\sigma }\), rather than the Kirchhoff stress tensor \(\varvec{\tau }\), to determine corotational stress rates. However, in the simple shear problem, \(J=1\), whence \(\varvec{\sigma }=\varvec{\tau }\), so that for all hypoelasticity models in the simple shear problem, the constitutive relations of Hooke-like isotropic hypoelastic material models based on corotational rates have form (30).

  13. The oscillating behavior of the Cauchy stress tensor components for this material model was first noted by Prager [69].

  14. The Green–Naghdi corotational rate of the Eulerian tensor \({\mathbf {h}}\in {\mathcal {T}}^2\) is defined as \({\mathbf {h}}^{GN} \equiv \dot{{\mathbf {h}}} - \varvec{\omega }^R \cdot {\mathbf {h}} + {\mathbf {h}} \cdot \varvec{\omega }^R\).

  15. The more general statement holds: For the isotropic Cauchy elastic material, tensors in pairs \((\bar{\varvec{\tau }},{\mathbf {U}})\) and \((\varvec{\tau },{\mathbf {V}})\) are coaxial (cf., [63]).

  16. In particular, hypoelastic materials do not depend on natural time (cf., [40]).

  17. The last statement can be generalized: The Cauchy stress tensor \(\varvec{\sigma }\) and any Eulerian strain tensor \({\mathbf {e}}\) from the Hill family are work-conjugate not in the classical sense due to the equality \({\mathbf {e}}^{\varDelta }={\mathbf {d}}\), where \({\mathbf {e}}^{\varDelta }\) is some convective rate of this tensor which is a corotational rate only if \({\mathbf {e}}={\mathbf {e}}^{(0)}\) and this corotational rate is logarithmic (cf., [16]).

  18. Sometimes, the Hill stress rate is called the Biezeno–Hencky stress rate (cf., [45]).

  19. More precisely, the tensors used in (30)\(_2\) have Eulerian objectivity, which is only considered in [83]. Starting from the book by Ogden [63], objective tensors include Lagrangian tensors, along with Eulerian ones.

  20. This statement contradicts the statement (see [60]) of the equivalence of hypoelasticity formulations based on any corotational rate, including the Gurtin–Spear one.

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Acknowledgements

The supports from the Russian Foundation for Basic Research (Grant No. 18-08-00358) and Grant from Russian Federation Government No. P220-14.W03.31.0002 are gratefully acknowledged. The author thanks the anonymous reviewers whose comments and suggestions helped in revising the manuscript.

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Korobeynikov, S.N. Analysis of Hooke-like isotropic hypoelasticity models in view of applications in FE formulations. Arch Appl Mech 90, 313–338 (2020). https://doi.org/10.1007/s00419-019-01611-3

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