Forward and inverse problems in the mechanics of soft filaments

Abstract

Soft slender structures are ubiquitous in natural and artificial systems, in active and passive settings and across scales, from polymers and flagella, to snakes and space tethers. In this paper, we demonstrate the use of a simple and practical numerical implementation based on the Cosserat rod model to simulate the dynamics of filaments that can bend, twist, stretch and shear while interacting with complex environments via muscular activity, surface contact, friction and hydrodynamics. We validate our simulations by solving a number of forward problems involving the mechanics of passive filaments and comparing them with known analytical results, and extend them to study instabilities in stretched and twisted filaments that form solenoidal and plectonemic structures. We then study active filaments such as snakes and other slender organisms by solving inverse problems to identify optimal gaits for limbless locomotion on solid surfaces and in bulk liquids.

Instability-driven oscillations of elastic microfilaments

“…. , n, in the spirit of [36] and [37]. The unit tangent to edge i is defined as t i = i / i where i = i .…”

Topology, Geometry, and Mechanics of Strongly Stretched and Twisted Filaments: Solenoids, Plectonemes, and Artificial Muscle Fibers

“…where ρ is the material density,ω = vec(∂ t Q T Q) is the local angular velocity,k = vec(∂ s Q T Q) is the local strain vector (of curvatures and twist), S is the matrix of shearing and extensional rigidities, B is the matrix of bending and twisting rigidities, and f , c are the body force density and external couple density (see SI or [18] for details). Here the vectorā traced out byd ⊥ 1 (i.e., the projection ofd1 onto the normal-binormal plane) is shown in red while the curve associated with −d1 is shown in yellow (see Fig.…”

“…To follow the geometrically nonlinear deformations of the filament described by the equations above, we employ a recent simulation framework [18], wherein the filament is discretized in a set of n + 1 vertices {x i } n i=0 connected by edgesē i =x i+1 −x i , and a set of n frames {Q i } n−1 i=0 . The resulting discretized system of equations is integrated using an overdamped second order scheme, reducing the dynamical simulation to a quasi-static process, and accounting for self-contact forces (SI and [18] for details) while ignoring friction [?…”

“…To follow the geometrically nonlinear deformations of the filament described by the equations above, we employ a recent simulation framework [18], wherein the filament is discretized in a set of n + 1 vertices {x i } n i=0 connected by edgesē i =x i+1 −x i , and a set of n frames {Q i } n−1 i=0 . The resulting discretized system of equations is integrated using an overdamped second order scheme, reducing the dynamical simulation to a quasi-static process, and accounting for self-contact forces (SI and [18] for details) while ignoring friction [? ] To track the knot-like structures that form when the stretched and twisted filament can contact itself, we take advantage of the CFW theorem [22,23]: Link (Lk ) = Twist (Tw ) + Writhe (Wr ) Here, link is the oriented crossing number (or Gauss linking integral) of the centerline and auxiliary curveā(s) ( Fig.…”

Elastohydrodynamical instabilities of active filaments, arrays and carpets analyzed using slender body theory

“…We specify that for the straight filament n(s, t) = e 2 , p(s, t) = e 3 and t(s, t) = e 1 . The localized coordinate frame at s + ds is obtained by an infinitesimal rotation of the coordinate frame at s. The deformed state of the axis of the filament is then determined by [45,53]…”