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Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity

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Abstract

This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lamé parameter \(\lambda =0\) and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

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Notes

  1. In the present paper, we show that, actually, the Hooke-like isotropic hypoelastic model based on the lower Oldroyd stress rate also has a Cauchy/Green elastic counterpart.

  2. Using a similar criterion, Kojic and Bathe [43] have shown that the Hooke-like isotropic hypoelastic model based on the Zaremba–Jaumann stress rate is not a Cauchy/Green elastic model.

  3. Hereinafter, the superscript \(T\) denotes the transposition of a tensor.

  4. The number \(m\) (\(1\leq m \leq 3\)) will be called the eigenindex.

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

  6. Tensors \(\mathbf{X},\,\mathbf{Y}\in \mathcal{T}^{2}_{\textup{sym}}\) will be called orthogonal if the equality \(\mathbf{X}:\mathbf{Y}=0\) is satisfied (cf., [1], p. 3588).

  7. 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 \(\boldsymbol{\Psi }\) such that \(\boldsymbol{\Psi }\cdot \boldsymbol{\Psi }^{T}=\mathbf{I}\) and \(\det \boldsymbol{\Psi }=1\)).

  8. 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., [12]).

  9. Since we assume that \(\mathbf{x}(\mathbf{X},t)\in C^{2}\) of \(t\), it follows that \(\mathbf{l},\mathbf{d},\mathbf{w},\boldsymbol{\omega }^{R}\in C^{0}\) of \(t\) (cf., [65]).

  10. The fact that the polar spin tensor \(\boldsymbol{\omega }^{R}\) is independent of dilatancy is also noted in [14, 66].

  11. The proof of this fact follows similar lines to the proof of Theorem 9.1 in [45] on the objectivity of corotational rates of the form (20).

  12. Restrictions of the form \(1<\lambda _{i}\) will be removed below (see Theorem 4.2), where the standard ranges \(\lambda _{i}\) (\(0<\lambda _{i}<\infty \)) will be used.

  13. Note that as \(\lambda _{j}\rightarrow \lambda _{i}\), the expression for the lower line of Eq. (30) is transformed to the expression for the upper line of this equation.

  14. The fact that the spin tensors \(\boldsymbol{\Omega }^{\log }\) and \(\boldsymbol{\omega }^{\log }\) belong to the family of continuous material spin tensors was shown in [45].

  15. Classical definitions of the upper and lower convective tensor rates \(\mathbf{h}^{\sharp }\equiv \dot{\hat{h}}^{ij}\hat{\mathbf{e}}_{i} \otimes \hat{\mathbf{e}}_{j}\) and \(\mathbf{h}^{\flat }\equiv \dot{\hat{h}}_{ij}\hat{\mathbf{e}}^{i} \otimes \hat{\mathbf{e}}^{j}\) of the Eulerian tensor \(\mathbf{h}=\hat{h}^{ij}\hat{\mathbf{e}}_{i}\otimes \hat{\mathbf{e}}_{j}= \hat{h}_{ij}\hat{\mathbf{e}}^{i}\otimes \hat{\mathbf{e}}^{j}\) (\(\hat{\mathbf{e}}_{i}\)/\(\hat{\mathbf{e}}^{i}\) are covariant/contravariant or tangent/cotangent convective bases, \(i,j=1,2,3\); the summation is performed over repeated indices from 1 to 3) are given by Oldroyd (cf., [58], see also Appendix in [59]). For the modern definition of the tensor rates \(\mathbf{h}^{\sharp }\) and \(\mathbf{h}^{\flat }\) and their Lagrangian counterparts \(\mathbf{H}^{\sharp }\) and \(\mathbf{H}^{\flat }\) as the Lie time derivatives of the tensors \(\mathbf{h}\) and \(\mathbf{H}\), see, e.g., [37, 44, 52].

  16. \(\bar{\boldsymbol{\tau }}^{\log }\) and \(\boldsymbol{\tau }^{\log }\) are the Lagrangian and Eulerian logarithmic stress rates associated with the spin tensors \(\boldsymbol{\Omega }=\boldsymbol{\Omega }^{\log }\) and \(\boldsymbol{\omega }=\boldsymbol{\omega }^{\log }\) determined from expressions (58).

  17. See [48] (Sect. 1) for a discussion of formulations of classical/generalized hypoelasticity.

  18. The fact that CRs (97) correspond only to Cauchy (and not Green) elastic material is shown, e.g., in [64].

  19. This conclusion is consistent with conclusions from many previous studies (see, e.g., [64, 67]).

  20. Expressions for the components of the Cauchy stress tensor in the simple shear problem for a Hencky isotropic hyperelastic material sample were also obtained and given in [3, 11, 47, 56].

  21. The “∘” sign between maps denotes their composition.

  22. Because the tangent spaces \(T_{P}\check{\beta}\) and \(T_{P}\hat{\beta}\) and the cotangent spaces \(T_{P}^{*}\hat{\beta}\) and \(T_{P}^{*}\check{\beta}\) coinside with the Euclidean vector space \(\mathbb{V}\) (see Definition 2.1 of configuration in [44]), it follows that \(\mathbf{F}\in \mathbb{V}\otimes\mathbb{V}\), and the tensor \(\mathbf{F}^{T}\) can be identified with a cotangent mapping.

  23. Hereinafter, ℂ is the fourth-order tangent stiffness tensor.

  24. The issue of continuity of corotational tensor rates in eigenvalue coalescence is considered in [45]. In particular, it has been shown in the cited work that the Gurtin–Spear corotational tensor rates associated with the twirl tensors of principal triads [30] do not pass the continuity test; therefore, Hooke-like hypoelasticity models based on the Gurtin–Spear corotational stress rates are not recommended for applications.

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Acknowledgements

The supports from the Russian Foundation for Basic Research (Grant No. 18-08-00358) and grant No. P220-14.W03.31.0002 from the Government of the Russian Federation 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. Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity. J Elast 143, 147–185 (2021). https://doi.org/10.1007/s10659-020-09808-2

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