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Families of Hooke-like isotropic hyperelastic material models and their rate formulations

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Abstract

We introduce a new (HHE) family of Hooke-like isotropic hyperelastic material models associated with the Doyle–Ericksen family of strain tensors with Eulerian forms of constitutive relations (CRs). This family contains the well-known Hencky and Simo–Pister (modified neo-Hookean) isotropic hyperelastic material models. The main feature of the new family of material models is that the rate counterparts of CRs for material models from this family are simultaneously CRs for Hooke-like isotropic hypoelastic-type material models based on Eulerian continuous strain-consistent convective stress rates. Models from the new family extend the only previously known Hooke-like isotropic hypoelastic material model based on the corotational logarithmic stress rate with Hooke-like hyperelastic counterpart to an infinite number of such models based on non-corotational stress rates. In addition, we develop unified Eulerian forms of CRs and specific potential strain energies for the known families of Hill (HLIH) and Kellermann–Attard (K–A) Hooke-like isotropic hyperelastic material models. For all three families (HHE, HLIH, and K–A) of material models, new explicit basis-free (eigenprojection based) expressions for the fourth-order elasticity tensors with full symmetry are obtained. Expressions for the components of the Cauchy stress tensor versus axial stretch in the simple elongation problem are derived, and their plots for integer values of the parameter \(n=\pm 2,\pm 1, 0\) generating material models from the families considered are constructed. From an analysis of these plots, it is concluded that the Simo–Pister isotropic hyperelastic material model is the best model among those considered.

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Notes

  1. For brevity, these parameters \(\lambda \) and \(\mu \) will be referred to below as the Lamé parameters regardless of whether they are used in CRs for material models with infinitesimal or finite strains.

  2. A common practice to introduce elasticity laws into CRs for additive elasto-plasticity is to use CRs for Hooke-like isotropic hypoelasticity based on some corotational stress rate (for a more detailed analysis of these CRs, see Sect. 1 in [72]). In the early development of such formulations of CRs for elasto-plastic materials, researchers used mainly the Zaremba–Jaumann stress rate (see, e.g., [5, 10, 11, 84]). As noted in Sect. 1 in [72], one of the disadvantages of using this stress rate in CRs for additive elasto-plasticity is that in the absence of inelastic strain, CRs with this stress rate do not represent a rate form of CRs for any Cauchy/Green elastic material. It has been shown [123, 124] (see also [72]) that the only Hooke-like isotropic hypoelasicity material model based on corotational stress rate whose CRs are the rate form of CRs for Green elastic material is the model with CRs (12).

  3. The family of continuous strain-consistent convective stress rates was obtained using some constraints of continuity for stress rates introduced in [22].

  4. In [73], we have developed a method to determine whether some Hooke-like hypoelastic material model is the rate form of any hyperelastic material model. This method is simpler than the general method of Bernstein (cf., [15]) includes the following steps. The first step is to establish whether this hypoelasticity model passes the test for coaxiality of conjugate stress and strain tensors (i.e., whether the necessary condition of the Cauchy elasticity is satisfied). If the model passes this test, the second step is performed which is to try to explicitly determine CRs for the Cauchy elastic material, i.e., to derive the so-called response function \(\varvec{\phi }(\textbf{V})\) or its equivalent counterpart to check its physical consistency, e.g., to check the validity of the Baker–Ericksen (B–E) inequalities [110], which are sufficient conditions for this response function to represent CRs for the Cauchy elastic material [6, 7] (it is easy check that the B–E inequalities hold for CRs (58)). If such CRs are determined, the third step is performed which is to try to find the specific potential strain energy for the material model in question. If the specific potential energy is determined, it is concluded that the hypoelastic material model is indeed the rate form of some hyperelastic material model.

  5. Hereinafter, \(\mathcal {T}^2_{\text {sym}}\subset \mathcal {T}^2\) denotes the set of all symmetric second-order tensors.

  6. Hereinafter, fourth-order tensors with both major and twice minor symmetries will be called supersymmetric (cf., [61]) or fully symmetric (cf., [39]) tensors.

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

  8. 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 \(m=1\).

  9. Hill proposed the family of Lagrangian strain tensors \(\textbf{f}(\textbf{U})\) (cf., [55]); we use their Eulerian counterparts \(\textbf{f}(\textbf{V})\).

  10. This family is a subfamily of the Hill family of strain tensors. Some authors call this family of strain tensors the Seth–Hill family of strain tensors (see, e.g., [27]).

  11. Hereinafter, \(\mathcal {T}^2_{\text {skew}}\subset \mathcal {T}^2\) denotes the set of all skew-symmetric tensors.

  12. Since we assume that \(\textbf{x}(\textbf{X},t)\in C^2\) of t, it follows that \(\varvec{\ell },\textbf{d},\textbf{w},\varvec{\omega }^R\in C^0\) of t (cf., [102]).

  13. In [55] these CRs are given in Lagrangian form; in the present paper, we present alternative Eulerian expressions of CRs for models from this family.

  14. Expression (52) coincides with the expression for the tangent stiffness tensor for the Hooke-like hypoelastic isotropic material model based on the logarithmic stress rate (see Eqs. \((82)_2\), \((86)_1\), \((87)_1\), and (93) in [72]).

  15. The expression for \(\hat{W}_{\mu }^{(n)}\) was obtained in [73].

  16. In Eulerian version of CRs for the incompressible neo-Hookean isotropic hyperelastic material model, CRs have the form \(\varvec{\tau }=-p\textbf{I}+2\mu \textbf{e}^{(2)}\), where p is the hydrostatic pressure.

  17. Sometimes, the Hill stress rate is called the Biezeno–Hencky stress rate (cf., [62]).

  18. Note that CRs for additive elasto-plasticity also commonly use Eulerian tensors and these material models are implemented in FE codes using UL formulations of the SM equations (see, e.g., [4, 5, 10, 11, 81, 84]).

  19. Here \(\textbf{F}^{-T}\equiv (\textbf{F}^{-1})^T=(\textbf{F}^T)^{-1}\).

  20. The motion law (71) can only be used for compressible materials, and it is apparently difficult to implement in laboratory experiments. At the same time, this motion law makes it possible to highlight the dependence of stresses on volume changes for hyperelasticity models obtained in one way or another from hyperelasticity models for incompressible materials in the form of a generalization that takes into account material compressibility. In particular, in Sect. 7, we use the motion law (71) to evaluate the performance of some generalized (compressible) neo-Hookean material models generated by the classical incompressible neo-Hookean material model.

  21. For homogeneous deformation of a prismatic bar, equality (74) in fact reduces to the Considère (1888) condition for the instability of the deformation of a prismatic bar when the axial load on this bar reaches an extreme value (cf., [9, p. 166]).

  22. Nevertheless, it has been found [74] that HLIH material models based on the Itskov family of strain tensors (cf., [60, 61]) include such material models, in particular, the Pelzer isotropic hyperelastic material model, which can simulate engineering strains up to 150%.

  23. Hereinafter, the experimental data given in [63, 64, 108] will be referred to as gold standard experimental data.

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    S. N. Korobeynikov

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Korobeynikov, S.N. Families of Hooke-like isotropic hyperelastic material models and their rate formulations. Arch Appl Mech 93, 3863–3893 (2023). https://doi.org/10.1007/s00419-023-02466-5

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