Abstract
Hill’s linear isotropic hyperelastic material models based on the one-parameter (\(r\)) Itskov family of strain tensors (including the Hencky, Pelzer, and Mooney strain tensors generating the H, P, and M material models, respectively) were used to obtain exact solutions of the simple torsion problem for circular cross-section rods from the H, P, and M materials. The “exact” solutions available in the literature for the problem of generalized torsion of cylindrical rods with free edges in the axial direction were analyzed. The objectives of the present study are to develop and implement new formulations of these material models in the commercial MSC.Marc nonlinear FE software and to verify these formulations using the above-mentioned exact solutions. Computer simulations of the simple and generalized torsion of cylindrical specimens were carried out using three material models (H, P, and M) and the standard Mooney–Rivlin model. The results of computer simulations of the resultant moment and the resultant axial force (in the problem of simple torsion) or axial elongation (in the problem of generalized torsion) were compared with exact solutions. For the simple torsion problem, the solutions obtained by the two methods are similar, but for the problem of generalized torsion, these solutions are similar only for sufficiently small values of the torsion parameter. We explain the discrepancy for sufficiently large values of the torsion parameter by the fact that the so-called “exact” solutions cease to be exact because of the assumptions made by other authors in obtaining these solutions. We assume that for all values of the torsion parameter, our numerical solutions are close to the true exact solutions. Computer simulations showed that the Pelzer material model is similar in performance to the Mooney–Rivlin model.
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Notes
A similar generalization of Hooke’s law can be performed for any pair of conjugate Eulerian stress and strain tensors \(\mathbf{s}\) and \(\mathbf{e}\) (cf. Korobeynikov 2019).
In our classification, this material model is more briefly called the H material model.
Since material models from the HLIH family based on the Itskov family of strain tensors belong to the family of SPH isotropic hyperelastic material models (cf. Korobeynikov 2019), below the family of these material models will be called for brevity the family of SPHr isotropic hyperelastic material models (cf. Korobeynikov et al. 2022).
Hereinafter, \(\mathcal{T}^{2}_{\text{sym}}\subset \mathcal{T}^{2}\) denotes the set of all symmetric second-order tensors.
The number \(m\) (\(1\leq m\leq 3\)) will be called the eigenindex.
Some authors call this family of strain tensors the Seth–Hill family of strain tensors (see, e.g., Curnier and Rakotomanana 1991).
In Curnier and Rakotomanana (1991), this family is termed the rubber family.
In Curnier and Zysset (2006), this family is termed the metric family.
See Appendix A in Korobeynikov et al. (2022) for details. 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 \(m=1\).
See Korobeynikov 2019, Eq. \((79)_{4}\).
This dependence was obtained with retention of terms up to the second order in the torsion parameter \(\bar{\gamma}\).
In particular, these HLIH materials include St. Venant–Kirchhoff material (with \(m=2\)).
We introduce the parameters \(E\) and \(\nu \) for isotropic hyperelastic materials with Hill’s linear relations, which are related to the Lamé parameters \(\lambda \) and \(\mu \) by the equalities
$$\lambda =\frac{E \nu}{(1+\nu )(1-2\nu )},\quad \mu = \frac{E}{2(1+\nu )}. $$In the literature on linear elasticity theory, the material parameters \(E\) and \(\nu \) are called Young’s modulus and Poisson’s ratio.
To integrate the equations of quasistatic motion, we use an adaptive time step with prescribed values \(\epsilon _{U}=\epsilon _{F}=0.01\) (the quantities \(\epsilon _{U}\) and \(\epsilon _{F}\) are the relative tolerances for the displacements and internal forces when using the Newton–Raphson iterative procedure).
The relative differences \(d_{M}\) and \(d_{N}\) are simultaneously the relative errors for the solutions of the same problem only for the M-R model. For the remaining (H, P, and M) material models, the quantities \(d_{M}\) and \(d_{N}\) correspond to the relative differences in the resultant moment and compressive force obtained by solving different problems (see Remark 5).
The literature “exact” solutions (37) and (41) of the problem of generalized torsion in the form of the dependence \(\varepsilon (\bar{\gamma})\) were actually obtained under some assumptions on the relative smallness of the torsion parameter \(\bar{\gamma}\); therefore solutions obtained by computer simulation (shown by dotted curves in Fig. 12) without any assumptions on the relative smallness of this parameter can be considered close to the present “exact” solutions.
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The support from Russian Federation government (Grant No. P220-14.W03.31.0002) is gratefully acknowledged.
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S.N.K. wrote the main manuscript text, T.A.R. got exact solutions and performed FE simulations, and A.Yu. Larickin performed FE simulations and prepared all figures. All authors reviewed the manuscript.
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Korobeynikov, S.N., Larichkin, A.Y. & Rotanova, T.A. Simulating cylinder torsion using Hill’s linear isotropic hyperelastic material models. Mech Time-Depend Mater (2023). https://doi.org/10.1007/s11043-023-09592-1
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DOI: https://doi.org/10.1007/s11043-023-09592-1