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Discussion of “The general basis-free spin and its concise proof” by Meng and Chen, Acta Mech., https://doi.org/10.1007/s00707-022-03162-2

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Abstract

In the paper by Meng and Chen (Acta Mech 233:1307-1316, 2022), the polar (relative) and logarithmic spin tensors and the twirl tensors of Eulerian and Lagrangian triads are presented in the form of isotropic tensor functions of the stretch and stretching tensors. These tensor functions depend on the principal invariants and second-order polynomials of the stretch tensors and depend linearly on the stretching tensors. In the present paper, we show that these expressions are theoretically equivalent to the previously published basis-free expressions for these spin tensors represented by the same tensor functions, but in the form of dependencies on eigenvalues and their subordinate eigenprojections of the stretch tensors and linear dependencies on the stretching tensors. It is shown that the previously obtained expressions are preferred to those presented in the discussed paper both in the selection of spin tensors suitable for applications and in numerical implementations of expressions for these spin tensors.

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Notes

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

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

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

  4. 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 \(m=1\).

  5. Here we assume that the tensors \({\textbf{H}}\) and \({\textbf{h}}\) are a pair of objective counterparts of each other (i.e., \({\textbf{h}}={\textbf{R}} \cdot {\textbf{H}} \cdot {\textbf{R}}^T \quad \Leftrightarrow \quad {\textbf{H}}={\textbf{R}}^T \cdot {\textbf{h}} \cdot {\textbf{R}}\)). Hereinafter, the tensor \({\textbf{R}}\) is the (proper orthogonal) rotation tensor generated by the polar decomposition of the deformation gradient.

References

  1. Bertram, A.: Elasticity and Plasticity of Large Deformations, 4th edn. Springer, Cham (2021)

    Book  MATH  Google Scholar 

  2. Itskov, M.: Tensor Algebra and Tensor Analysis for Engineers (with Applications to Continuum Mechanics), 5th edn. Springer, Cham (2019)

    Book  MATH  Google Scholar 

  3. Luehr, C.P., Rubin, M.B.: The significance of projection operators in the spectral representation of symmetric second order tensors. Comput. Methods Appl. Mech. Eng. 84(3), 243–246 (1990). https://doi.org/10.1016/0045-7825(90)90078-Z

    Article  MathSciNet  MATH  Google Scholar 

  4. Korobeynikov, S.N.: Families of continuous spin tensors and applications in continuum mechanics. Acta Mech. 216(1), 301–332 (2011). https://doi.org/10.1007/s00707-010-0369-7

    Article  MATH  Google Scholar 

  5. Korobeynikov, S.N.: Objective tensor rates and applications in formulation of hyperelastic relations. J. Elast. 93(2), 105–140 (2008). https://doi.org/10.1007/s10659-008-9166-0

    Article  MathSciNet  MATH  Google Scholar 

  6. Korobeynikov, S.N.: Basis-free expressions for families of objective strain tensors, their rates, and conjugate stress tensors. Acta Mech. 229(3), 1061–1098 (2018). https://doi.org/10.1007/s00707-017-1972-7

    Article  MathSciNet  MATH  Google Scholar 

  7. Xiao, H., Bruhns, O.T., Meyers, A.: On objective corotational rates and their defining spin tensors. Int. J. Solids Struct. 35(30), 4001–4014 (1998). https://doi.org/10.1016/S0020-7683(97)00267-9

    Article  MathSciNet  MATH  Google Scholar 

  8. Xiao, H., Bruhns, O.T., Meyers, A.: Strain rates and material spins. J. Elast. 52(1), 1–41 (1998). https://doi.org/10.1023/A:1007570827614

    Article  MathSciNet  MATH  Google Scholar 

  9. Xiao, H., Bruhns, O.T., Meyers, A.: Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures. Arch. Mech. 50(6), 1015–1045 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Xiao, H., Bruhns, O.T., Meyers, A.: Direct relationship between the Lagrangean logarithmic strain and the Lagrangean stretching and the Lagrangean Kirchhoff stress. Mech. Res. Commun. 25(1), 59–67 (1998). https://doi.org/10.1016/S0093-6413(98)00007-X

    Article  MathSciNet  MATH  Google Scholar 

  11. Korobeynikov, S.N.: Analysis of Hooke-like isotropic hypoelasticity models in view of applications in FE formulations. Arch. Appl. Mech. 90(2), 313–338 (2020). https://doi.org/10.1007/s00419-019-01611-3

    Article  Google Scholar 

  12. Korobeynikov, S.N.: Family of continuous strain-consistent convective tensor rates and its application in Hooke-like isotropic hypoelasticity. J. Elast. 143(1), 147–185 (2021). https://doi.org/10.1007/s10659-020-09808-2

    Article  MathSciNet  MATH  Google Scholar 

  13. Scheidler, M.: Time rates of generalized strain tensors part I: component formulas. Mech. Mater. 11(3), 199–210 (1991). https://doi.org/10.1016/0167-6636(91)90002-H

    Article  Google Scholar 

  14. Bruhns, O.T., Xiao, H., Meyers, A.: New results for the spin of the Eulerian triad and the logarithmic spin and rate. Acta Mech. 155(1), 95–109 (2002). https://doi.org/10.1007/BF01170842

    Article  MATH  Google Scholar 

  15. Xiao, H., Bruhns, O.T., Meyers, A.: A natural generalization of hypoelasticity and Eulerian rate type formulation of hyperelasticity. J. Elast. 56(1), 59–93 (1999). https://doi.org/10.1023/A:1007677619913

    Article  MathSciNet  MATH  Google Scholar 

  16. Xiao, H., Bruhns, O.T., Meyers, A.: Existence and uniqueness of the integrable-exactly hypoelastic equation \(\overset{\circ }{\varvec {\tau }}{}^{\ast }=\lambda (\text{ tr }\,{\textbf{D} }){\textbf{I} }+2\mu {\textbf{D} }\) and its significance to finite inelasticity. Acta Mech. 138(1), 31–50 (1999). https://doi.org/10.1007/BF01179540

    Article  Google Scholar 

  17. Meng, C.Y., Chen, M.X.: The general basis-free spin and its concise proof. Acta Mech. 233(4), 1307–1316 (2022). https://doi.org/10.1007/s00707-022-03161-2

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The support from the Russian Federation Government (Grant No. P220-14.W03.31.0002) is gratefully acknowledged.

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  1. Lavrentyev Institute of Hydrodynamics SB RAS, Lavrentyev av., 15, Novosibirsk, 630090, Russia

    S. N. Korobeynikov

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  1. S. N. Korobeynikov
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Korobeynikov, S.N. Discussion of “The general basis-free spin and its concise proof” by Meng and Chen, Acta Mech., https://doi.org/10.1007/s00707-022-03162-2. Acta Mech 234, 825–829 (2023). https://doi.org/10.1007/s00707-022-03402-4

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