## Abstract

Many engineering scenarios involve contact between beam structures or, eventually, self-contact. Specifically when dealing with a beam submitted to large torsion loads and considering large displacements and rotations, it is possible to occur self-contact. Beams with low bending stiffness loaded with large torsion can present a loop, followed by self-contact and sometimes a snarl formation. This work presents a numerical model and numerical procedures to solve such a kind of self-contact in beams under loop formation. Numerical examples are presented in the context of initially straight nitinol beams. Numerical tests showed that friction may influence loop and snarling formation. Numerical models could predict self-contact occurrence events with good agreement with experimental results already published in literature. Snarling patterns (when occurred) were also predicted correctly, but with delay when compared to experiments. This delay showed to be friction-dependent.

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## Notes

- 1.
Alternatively one could also write the virtual power theorem, leading to different equations. This was done for instance in [19].

- 2.
It was considered that the derivatives of matrices \({\varvec{\Psi } }^{\mathbf{i}+\mathbf{1}}\) and \({\varvec{\Upsilon }}^{\mathbf{i}+\mathbf{1}}\) have no important contributions to the tangent operator. The classical Newmark method was applied as described, considering the inertial and damping matrices. However, strictly speaking, the tangent operator should also contain Gateaux derivatives components from damping and inertia matrices, which were not considered. This does not imply in wrong solutions, once the weak form is complete and includes all contributions. The only possible implication would be a non-quadratic asymptotic convergence rate. However, good convergence rate was achieved in numerical examples here developed, even including large nonlinear behavior.

- 3.
The solution \((\upzeta _{1\mathrm{c}},{\upzeta }_{2\mathrm{c}})\) is not necessarily unique, once it depends on the nature of the curves. For straight non-parallel beams the solution is unique but, for generic spatial curves one can have multiple points obeying criterion for local minimum distance points. The present work assumes, as Zavarise and Wriggers [28], that beams are parameterized as straight lines for contact detection purposes, even after suffering bending. Clearly this is a geometric approximation, which should lead to good results in cases of meshes with small-length beam elements.

- 4.
This parameterization is used only in the contact evaluation of the here utilized model. It is completely different and independent from the parameterization used in structural model, presented in Sect. 2. For the simulations here presented, e.g., quadratic isoparametric interpolation was assumed for structural model, which is not consistent with the linear interpolation here assumed. This inconsistency, however, does not represent a drawback, once contact detection and structural interpolation are absolutely independent terms in problem weak form and can be with no problems be treated this way. Some tests were performed for frictionless contact considering higher order polynomials. However, increasing the interpolation degree would give a more precise contact detection, but does not ensure smoothness between beam elements, which could be achieved by Hermite interpolation using neighbor elements, such as done in [30]. This technique was not here applied and can be an object of future work.

- 5.
To shorten notation \(\mathbf{z}_{\mathbf{1}}^{\mathbf{i}+\mathbf{1}} ({\upzeta }_{1\hbox {c}} )\) and \(\mathbf{z}_{\mathbf{2}}^{\mathbf{i}+\mathbf{1}} ({\upzeta }_{2\hbox {c}} )\) are simply written by \(\mathbf{z}_{\mathbf{1}}^{\mathbf{i}+\mathbf{1}} \) and \(\mathbf{z}_{\mathbf{2}}^{\mathbf{i}+\mathbf{1}}\). All calculations involving gap function and its consistent linearization (leading to the tangent operator) for contact computation are evaluated using \({\upzeta }_{1\hbox {c}}\) and \({\upzeta }_{2\hbox {c}}\) values.

- 6.
- 7.
The parameters are related to the initial-touch position on both beams. These values have to be reset and re-set in case of losing and re-establishing contact. Furthermore, the actual beam has non-null cross section dimensions and, even small, strictly speaking will not lead to a line to line interaction, as implicitly modeled by the here used approach. Physically the contact interaction takes place on both beams external surface. Then, a more elaborated tangential gap function would have to consider the actual surface points of contact in both beams, mapping then on surfaces. This kinematics is absolutely more complicated than the proposed by Zavarise and Wriggers [28] approach. The real importance of such an effect, however, may take place only in problems with larger dimensions of cross section (when compared to beam length) or situations in which torsion and rolling may occur. An enhanced kinematic description involving ideas to be generalized to beam to beam contact is presented by Gay Neto et al. [37].

- 8.
- 9.
Pinball radius value, once enough for detecting contact properly, does not affect contact forces, gap calculations, etc. It is only a first detection tool to help in efficiency of contact detection.

- 10.
Initial conditions (displacement and velocity) of point “B” were considered to be null in the beginning of the third load-step. One can note that Fig. 8 imposed displacement is not a ramp loading, but a composition of a null derivative function in origin—actually a polynomial of second order, until the time achieves 1 s, followed by a constant slope function until the time 4.5 s is achieved. This imposed displacement series avoids a jerk in the very beginning of dynamics which could be done in a traditional ramp loading. This time-series showed smoother oscillation results along time than a traditional “ramp”. This was very useful, once dealing with a very low damping rate.

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## Acknowledgments

The authors acknowledge FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) for the support under the Grants 2012/09912-0 and 2014/11910-0. The second author acknowledges also the support by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) under the Grant 303091/2013-4.

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Gay Neto, A., Pimenta, P.M. & Wriggers, P. Self-contact modeling on beams experiencing loop formation.
*Comput Mech* **55, **193–208 (2015). https://doi.org/10.1007/s00466-014-1092-3

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### Keywords

- Contact
- Rod
- Instability
- Self-contact