Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods
Introduction
Filamentary structures are found in many different systems and diverse applications e.g. biological systems such as DNA, proteins, bacterial fibers; engineering applications such as sub-oceanic cables; theory of liquid crystals and polymers [1], [2], [3]. The stability of filamentary structures is of intrinsic theoretical interest and has important practical implications for the application areas listed above. There are multiple approaches to stability analyses e.g. static bifurcation analysis, dynamic approaches, conjugate point methods, distinguished diagram methods etc. and the interested reader is referred to [4], [5], [6], [7], [8], [9] for further references.
We focus on the model problem of an intrinsically straight, twisted elastic rod, subject to clamped boundary conditions and a terminal end-load. We only consider a special class of Kirchhoff rods in this paper i.e. an inextensible, unshearable rod with circular cross-section [2]. In our previous work [10], we study the static stability of the natural straight state in terms of the second variation of a general quadratic strain energy, in three dimensions. We obtain explicit criteria for the static stability and the onset of instability, in terms of the twist and the tension and our aim, in this paper, is to quantify the dynamic implications of static stability criteria for Kirchhoff rods. The intricate relationship between static and dynamic stability is, in general, poorly understood (see [11] for examples of counter-intuitive behavior) and it is, therefore, of interest to study this relationship for these rods, where both computations can be carried out explicitly.
Our work builds on results reported by Caflisch and Maddocks in [12]. We work in a fully three-dimensional setting i.e. account for both in-plane and out-of-plane configurations and consider two different boundary-value problems: (i) clamped boundary conditions and (ii) clamped boundary conditions with isoperimetric constraints. Our first result is a novel characterization of static equilibria, given by local minimizers of the quadratic strain energy, in a time-independent setting. We use the Euler angle representation and only consider equilibria that avoid the polar singularities. This restriction excludes rod configurations which turn over themselves or rods with self-contact. Nevertheless, we can account for all equilibria within a large neighborhood of the unbuckled state, which comprises a large subset of physically admissible equilibria. We analyze the second variation of the quadratic strain energy [13], without appealing to associated eigenvalue problems, and obtain explicit stability criteria in terms of the Euler angles, which in turn yield explicit bounds for the curvature and torsion of the static equilibria. Our methods may be generalized to the quaternion representation of the rod configuration, which does not have any polar singularities but has an associated nonlinear constraint of unit length [14]. The ensuing analysis of the second variation of the strain energy would be more complicated in the quaternion representation but would offer more detailed information. The Euler angle representation suffices for the purposes of this paper, since our aim, in the first instance, is to illustrate the links between static and dynamic stability for a class of analytically tractable problems with transparent computations.
The dynamic evolution of the rod is governed by the Kirchhoff equations in three dimensions [2], [5]: a six-dimensional system of coupled nonlinear PDEs of second order in space and time, based on linear constitutive relations between moments and Darboux curvature. We do not address questions related to the existence and regularity of solutions of the Kirchhoff equations subject to different boundary conditions as we are mainly interested in the classification of static equilibria and the interplay between static and dynamic stability. The total energy, which is the sum of the kinetic and potential energy, is a conserved quantity [15], [2] and following the methods in [12], we establish the equivalence between the energy and a suitably defined dynamic norm in three dimensions. This equivalence allows us to prove the strong statement that local minimizers of the quadratic strain energy (under explicit hypothesis) are stable in the dynamic sense due to Liapounov. More precisely, we use the direct method due to Liapounov to prove the equivalence between static and dynamic stability (under explicit hypothesis) in a three-dimensional framework. This is a significant generalization of the powerful two-dimensional study carried out in [12]. We further extend our results to damped systems with an isotropic drag force proportional to the tangent-velocity; the resulting equations of motion are different to the traditional Kirchhoff equations in [2] and have additional damping terms. However, the total energy is a decreasing function of time and hence, can still be used as a Liapounov function. As a consequence, local minimizers of the strain energy retain dynamic stability in the presence of a suitably defined local drag force. These methods are generalized to arbitrary functionals, that are quadratic and strictly convex with respect to the derivatives of the model variables, wherein we derive explicit criteria that guarantee the equivalence between static and Liapounov stability.
We primarily focus on clamped boundary conditions in this paper and address the question of isoperimetric constraints at the end. In two dimensions, it is known that the classical buckling force for the constrained problem with isoperimetric constraints differs from the classical Euler buckling formula by a factor of 4 [16]. Previous work has demonstrated the onset of instability for terminal loads greater than an explicit critical value. In two dimensions, we use Wirtinger’s inequality to prove stability in the complementary regime. In three dimensions, we obtain explicit bounds for a critical tension in the presence of isoperimetric constraints, such that the unbuckled state is stable for all loads, . Our approach is novel and it is possible that such integral inequalities can be exploited to study post-buckling behavior too.
The paper is organized as follows. In Section 2, we review the general theory of Kirchhoff rods and elaborate on the concepts of static and dynamic stability. In Section 3, we use Hamilton’s principle to derive the equations of motion in terms of the Euler angle representation. In Section 4, we derive explicit static stability criteria and examine the relationship between static and dynamic stability for a clamped Kirchhoff rod and in Section 5, we generalize these results to damped systems. Finally, in Section 6, we consider the case of isoperimetric constraints and in Section 7, we present our conclusions and directions for future work.
Section snippets
Preliminaries
In this section, we recall the model problem in [10] and the concomitant concepts of static and dynamic Liapounov stability from [12], [7].
The Kirchhoff rod is initially aligned along the -axis of a Cartesian basis, , and is subject to an external force, , in the -direction along with a controlled end-rotation at a terminal end [10]. The end-rotation is characterized by a non-zero twist parameter, , throughout the paper. We work in the thin filament approximation and hence, all
Rod dynamics in Euler angles
The rod energy is the sum of the potential energy (defined in (6)) and the kinetic energy, where is a spin constant and is the rod mass-density (mass per unit length) [12], [15]. For simplicity and without loss of generality, we take . Following the methodology in [12] where the authors compute
Stability estimates for BVP I
Consider the BVP I in (3) with clamped boundary conditions. Our first result concerns the derivation of explicit stability criteria for an arbitrary solution of this boundary-value problem.
Damped systems
In this section, we consider a local drag force, , acting on the elastic rod, given by where is an inhomogeneous, positive damping coefficient. As a consequence, since i.e. the drag force is acting normal to the tangent vector of the rod. Our aim is to study dynamic stability of local minimizers of the potential energy in (6), in the presence of such local damping.
The resultant force vector,
Stability estimates for BVP II
The clamped boundary conditions in BVP I are the simplest choice of boundary conditions. Nevertheless, the boundary conditions and the isoperimetric constraints in BVP II are of practical interest since they naturally arise in classical experiments with controlled end-displacements. BVP II is technically harder than BVP I and the key difference arises from the inequalities, (12), (13) which, in turn, affects the corresponding stability estimates as shown below. Our first result reproduces a
Conclusions
We study the static and dynamic stability of arbitrary extremals of the general quadratic strain energy in (6), in a three-dimensional framework. In particular, we establish the equivalence between static stability and Liapounov stability under explicit conditions in Proposition 5. Our work heavily builds on the mathematical machinery in [12] and we generalize (to some extent) the two-dimensional results in [12] to three dimensions. As in [12], we use the direct method due to Liapounov, with
Acknowledgments
AM is supported by an EPSRC Career Acceleration Fellowship, EP/J001686/1, an OCCAM Visiting Fellowship and a Keble Research Fellowship, University of Oxford (till October 2012). AM would like to thank the Oxford Center for Collaborative Applied Mathematics for its hospitality over the months of August–October 2012, during which this work was completed. This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). AG
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