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Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models

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Abstract

We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors (i.e., the Doyle–Ericksen, Curnier–Rakotomanana, Curnier–Zysset, Itskov, and Darijani–Naghdabadi families) and to the new family of SP strain tensors. In particular, these five scale functions include the scale function generating the Lagrangian and Eulerian Hencky strain tensors. We introduce the family of SPH models of isotropic hyperelastic materials (with Hill’s linear relations) which are generated by SP strain tensors and work-conjugate stress tensors based on Hill’s natural generalization of Hooke’s law. Five SPH models of isotropic hyperelastic materials are generated on the basis of chosen SP strain tensors and work-conjugate stress tensors. These models are tested by solving two problems with homogeneous strain and stress tensors fields: the simple elongation and simple shear problems. Analysis of these solutions shows that the solutions of both problems for the Hencky isotropic hyperelastic material model (one of the five generated SPH models of isotropic hyperelastic materials) are qualitatively different from the solutions for the remaining four material models. That is, the solutions using the Hencky isotropic hyperelastic material model are of yielding nature typical of inelastic deformation of metals whereas the solutions for the other four material models reproduce strain diagrams typical of rubber.

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Notes

  1. Hooke’s law using Cauchy stress and strain tensors cannot be directly used in large strains mechanics because of the non-objectivity of the Cauchy strain tensor [31].

  2. In the cited papers, an anomalous behavior of the solutions of problems using the St. Venant–Kirchhoff isotropic hyperelastic material model at large strains is noted.

  3. The rotated Kirchhoff stress tensor and the right logarithmic (Hencky) strain tensor are used in the Lagrangian version of the constitutive relations of the Hencky material model, and the Kirchhoff stress tensor and the left logarithmic (Hencky) strain tensor are used in the Eulerian version of the constitutive relations of this material model.

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

  5. Hereinafter, \(m\) is the eigenindex, \(\lambda_{i}>0\) are eigenvalues, and \(\mathbf{U}_{i}\) and \(\mathbf{V}_{i}\) (\(i=1,\ldots m\)) are subordinate eigenprojections (see, e.g., [11, 17, 34, 54, 69]) of the tensors \(\mathbf{U}\) and \(\mathbf{V}\), respectively.

  6. Some authors call this family of strain tensors the Seth–Hill family of strain tensors (see, e.g., [20]).

  7. In [20], this family is termed the rubber family.

  8. \(\mathbf{D}\in\mathcal{T}^{2}_{\text{sym}}\) is the Lagrangian rotated stretching or rotated strain rate tensor (see, e.g., [20]).

  9. \(\mathbf{d}\in\mathcal{T}^{2}_{\text{sym}}\) is the Eulerian stretching or strain rate tensor (see, e.g., [73]). The rotated stretching tensor \(\mathbf{D}\) is the Lagrangian counterpart of the stretching tensor \(\mathbf{d}\), i.e., \(\mathbf{D}=\mathbf{R}^{T}\cdot\mathbf{d} \cdot\mathbf{R}\).

  10. \(\bar{ \boldsymbol{\tau}}\in\mathcal{T}^{2}_{\text{sym}}\) is the Lagrangian rotated Kirchhoff (or Noll) stress tensor (see, e.g., [20]).

  11. \(\boldsymbol{\tau}\in\mathcal{T}^{2}_{\text{sym}}\) is the Eulerian Kirchhoff stress tensor (see, e.g., [41, 61]). The rotated Kirchhoff stress tensor \(\bar{\boldsymbol{\tau}}\) is the Lagrangian counterpart of the Kirchhoff stress tensor \(\boldsymbol{\tau}\), i.e., \(\bar{ \boldsymbol{\tau}}=\mathbf{R}^{T}\cdot\boldsymbol{\tau} \cdot \mathbf{R}\).

  12. Strain tensors in the pairs \((\mathbf{S}, \mathbf{E})\) and \((\mathbf{s},\mathbf{e})\) belong to the Hill family.

  13. Often the tensor \(\mathbf{P}^{T}\) is called the nominal stress tensor (see, e.g., [61]).

  14. Hereinafter, \(\sigma_{ij}\) (\(i,j=1,2,3\)) are the Cauchy stress tensor \(\boldsymbol{\sigma}\) components in the introduced coordinate system.

  15. In the literature on linear elasticity theory, the material parameters \(E\) and \(\nu\) are called Young’s modulus and Poisson’s ratio.

  16. The ranges of admissible values in (67) are obtained the from constitutive inequalities applied to the values of the parameters \(\lambda\) and \(\mu\) (\(\mu>0\), \(3\lambda+ 2\mu>0\), see, e.g., [6]) taking into account the equalities (56) and the standard assumption \(E>0\).

  17. The principal directions in the simple shear problem are determined, e.g., in [34].

  18. In the solution of this problem based on linear elasticity theory, the length of the axis of a rod of an isotropic material does not change.

  19. For homogeneous deformation of a prismatic bar, equality (85) 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., [6], p. 166).

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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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  1. Lavrentyev Institute of Hydrodynamics, Novosibirsk, 630090, Russia

    S. N. Korobeynikov

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

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Korobeynikov, S.N. Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models. J Elast 136, 159–187 (2019). https://doi.org/10.1007/s10659-018-9699-9

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