Discrete Elasto-Plastic Rods
Introduction
Rods are structures with length much larger than the width and thickness. Due to the slenderness of the geometry, this class of structures often undergo geometrically nonlinear deformation, as manifested in both natural (e.g. bacterial flagella) [1] and engineering (ropes, cables, and pipelines) [2] systems. The recent trend of exploiting large deformation in slender structures to design and fabricate metamaterials [3] indicates the need for computationally efficient simulation tools for slender structures. While an extensive amount of work has been done in modeling purely elastic rods in both mechanics [4] and computer graphics [5] communities, computational tools for rods with nonlinear constitutive relations, e.g. hyperelasticity [6], rate-dependent viscoelasticity [7], [8], and elasto-plasticity [9], are relatively scarce. A commonly used nonlinear constitutive relation is elasto-plasticity that manifests itself in both simple materials [10], [11], [12], [13] and advanced structures [14], [15], [16]. In this paper, we focus on modeling the elasto-plastic behavior in rod-like structures.
The first scholarly work on plasticity dates back to at least 1868, when Tresca [17] proposed an assumption that metals would flow when shear stress exceeds a threshold on basis of experimental observation. In rod-like structures, the deformation can be decomposed into three components — bending, twisting, and stretching. The latter mode is negligible since geometric slenderness makes stretching energetically expensive. However, bending and twisting often appear simultaneously to pose a coupled problem. Prior works on elasto-plastic deformation of rods typically consider only bending [18], [19] or twisting [20], [21], [22]. The few studies that consider combined bending and torsion restricted themselves to simplified material behavior, e.g. rigid-plastic [23], [24], [25], [26], [27], [28], elastic-perfectly plastic [29], [30], [31], and Ramberg–Osgood stress–strain law [32]. This motivates us to develop a simulation algorithm for elasto-plastic rods that can account for combined bending, twisting, and stretching for an infinite number of loading and unloading cycles.
With the development of computational capability in the past few decades, Finite Element Method (FEM) [33], [34], [35], [36] has become the preferred means for researchers and engineers in structural analysis involving elasto-plasticity. However, 3D simulations of slender rods undergoing bending and torsion with elasto-plastic constitutive law typically require volumetric mesh of small size. This can lead to computationally expensive simulations, especially when combining the nonlinearity from both geometric and material side. Recently, another type of numerical tools — Discrete Differential Geometry (DDG)-based methods [37] – are becoming increasingly popular in the computer graphics community for the simulation of thin elastic structures, e.g. hair and clothes, due to computational efficiency and robustness in handling geometric nonlinearity. The DDG-based approach starts with discretization of the smooth structure into a mass–spring-type system, while preserving the key geometric properties of actual physical objects [37]. Previous DDG-based methods have shown surprisingly successful performance in simulating slender structures, e.g. rods [5], [38], [39], [40], [41], ribbons [42], plates/shells [43], [44], [45], and gridshells [46]. On the other hand, previous DDG-based numerical frameworks of filamentary rods usually assume that the structure is in linear elastic regime. Even though a more general model – rate-dependent viscoelastic constitutive law – has been considered in a recent study [7], it cannot be directly used for the investigation of elasto-plastic behavior of rods. Elasto-plasticity requires discretization of rod cross section such that the boundary between the elastic and the plastic regimes can be tracked.
Here, we propose a numerical method – Discrete Elasto-Plastic Rods (DEPR) – that combines a geometrically nonlinear description of a rod following Discrete Elastic Rods (DER) [5], [38] method, with aa increment-based plasticity flow model [9] to simulate the elasto-plastic behavior of rod-like structures during large deflection. DER is based on the classical Kirchhoff rod model. The rod centerline is first discretized into a number of nodes and the cross-section of each node is later divided into multiple fiber-like elements. The kinematics of the centerline is formulated in a manner similar to the DER method and the strain tensor at any point in the solid body is assumed to be a function of the deformation (bending, twisting, stretching) of the centerline. Next, an implicit returning mapping algorithm is used to update the stress tensor of each element on the basis of the increment of strain tensor – computed from the deformation of the rod centerline – together with its current von Mises equivalent stress [47], [48]; The internal force vector, required by the equations of motion of rod system, is then computed from the increment of strain energies. In parallel with numerical investigation, we analytically obtain the force–displacement relations for some simple cases to quantitatively check the accuracy of our simulator. A good match between numerical simulation and analytical results in the linear phase indicates the correctness of the numerical scheme. While the analytical solutions do not hold for geometrically nonlinear deformation, the DDG-based simulation can robustly capture the nonlinearity from both geometric and material sides. Nonetheless, if the rod is undergoing pure bending (i.e. a beam) or torsion, several analytical and numerical methods discussed earlier in this section should be preferred due to low computational cost. DEPR is particularly useful when the slender rod is undergoing combined bending and twisting with repeated loading and unloading.
Our paper is organized as follows. In Section 2, we discuss the proposed DEPR simulator. Next, in Section 3, we conduct both numerical data and analytical results for several demonstrative examples. Finally, conclusive remarks and potential research avenues are presented in Section 4.
Section snippets
Methods
In this section, we introduce the numerical framework for simulating the mechanical response of elasto-plastic rods. We first discuss the kinematics of one dimensional rod-like structures in a discrete format, followed by a general rate-independent, increment-based nonlinear constitutive relation. Finally, we present the time marching scheme and the return mapping algorithm used in our simulator.
Results
In this section, we use several examples in increasing complexity to demonstrate our newly introduced increment-based discrete elasto-plastic rod model; also, analytical solutions are derived for some simple cases to show the accuracy of our numerical framework. Appendix D shows the convergence of the algorithm with the number of nodes and elements, using the second example discussed in this section.
Conclusion
We have developed a discrete numerical framework for the simulation of geometrically nonlinear deformation of one dimensional elasto-plastic structures. For this purpose, we first discretized the rod centerline into a number of nodes and edges, and formulated its geometrically nonlinear deformation following the well-established DER method. Next, the cross section of a rod at each node was meshed into multiple fiber-like elements, and the strain tensor at each fiber was related to the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We acknowledge support from the National Science Foundation (Award # IIS-1925360) and the Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles .
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X.L. and W.H. contributed equally to this work.