Eulerian Formulation of Spatially Constrained Elastic Rods

Abstract

Slender elastic rods are ubiquitous in nature and technology. For a vast majority of applications, the rod deflection is restricted by an external constraint and a significant part of the elastic body is in contact with a stiff constraining surface. The research work presented in this doctoral dissertation formulates a computational model for the solution of elastic rods constrained inside or around frictionless tube-like surfaces. The segmentation strategy adopted to cope with this complex class of problems consists in sequencing the global problem into, comparatively simpler, elementary problems either in continuous contact with the constraint or contact-free between their extremities. Within the conventional Lagrangian formulation of elastic rods, this approach is however associated with two major drawbacks. First, the boundary conditions specifying the locations of the rod centerline at both extremities of each elementary problem lead to the establishment of isoperimetric constraints, i.e., integral constraints on the unknown length of the rod. Second, the assessment of the unilateral contact condition requires, in principle, the comparison of two curves parametrized by distinct curvilinear coordinates, viz. the rod centerline and the constraint axis. Both conspire to burden the computations associated with the method. To streamline the solution along the elementary problems and rationalize the assessment of the unilateral contact condition, the rod governing equations are reformulated within the Eulerian framework of the constraint. The methodical exploration of both types of elementary problems leads to specific formulations of the rod governing equations that stress the profound connection between the mechanics of the rod and the geometry of the constraint surface. The proposed Eulerian reformulation, which restates the rod local equilibrium in terms of the curvilinear coordinate associated with the constraint axis, describes the rod deformed configuration by means of either its relative position with respect to the constraint axis (contact-free segments) or its angular position on the constraint surface (continuous contacts.) This formulation circumvents both drawbacks that afflict the conventional Lagrangian approach associated with the segmentation strategy. As the a priori unknown domain, viz. the rod length, is substituted for the known constraint axis, the free boundary problem and the associated isoperimetric constraints are converted into a classical two-point boundary value problem. Additionally, the description of the rod deflection by means of its eccentricity with respect to the constraint axis trivializes the assessment of the unilateral contact condition. Along continuous contacts, this formulation expresses the strain variables, measuring the rod change of shape, in terms of the geometric invariants of the constraint surface, and emphasizes the influence of the constraint local geometry on the reaction pressure. Formalizing the segmentation strategy, a computational model that exploits the Eulerian formulation of the rod governing equations is devised. To solve the quasi-static deflection of elastic rods constrained inside or around a tube-like surface, this computational model identifies the number of contacts, their nature (either discrete or continuous), and the rod configuration at the connections that satisfies the unilateral contact condition and preserves the rod integrity along the sequence of elementary problems.