arXiv subjects

J. A. Hanna

Publications and source records attributed to J. A. Hanna.

At least 37 records · Page 2Linked to original sources

arXiv2020

Anisotropic swelling of anisotropic elastic panels

While isotropic in-plane swelling problems for thin elastic sheets have been studied extensively in recent years, many shape-programmable materials, including nematic solids and 3D-printed structures, are anisotropic, as are most industrial sheet materials. In this theoretical work, we consider central swelling and shrinkage of plates of aspect ratio and material properties relevant to the manufacture of engineered wood composite panels in which both in-plane swelling and material stiffness are highly orthotropic, leading to multiple separations in energy scales. With transverse swelling in the soft direction, and gradients in the stiff direction, the warped plates adopt two distinct types of configurations, axisymmetric and twisted, which we illustrate with toy models. We employ a two-parameter family of isometries to embed the metric programmed by the swelling, thus reducing the problem to one of minimizing bending energy alone. A simple argument is seen to closely predict averaged axisymmetric curvatures. While purely cylindrical shapes are unobtainable by pure in-plane swelling, they can be closely approximated in a highly anisotropic system. However, anisotropy can favor twisting, and breaks a degenerate soft deformation mode associated with minimal surfaces in isotropic materials. Bifurcations from axisymmetric to twisted shapes can be induced by anisotropy or by certain attributes of a central shrinkage profile. Finally, we note how our findings indicate practical limitations on the diagnosis of moisture inhomogeneities in manufactured panels by observation of warped conformations, due to the sensitivity of the qualitative response to specifics of the profile.

cond-mat.soft↗

arXiv2020

An integrable family of torqued, damped, rigid rotors

Expositions of the Euler equations for the rotation of a rigid body often invoke the idea of a specially damped system whose energy dissipates while its angular momentum magnitude is conserved in the body frame. An attempt to explicitly construct such a damping function leads to a more general, but still integrable, system of cubic equations whose trajectories are confined to nested sets of quadric surfaces in angular momentum space. For some choices of parameters, the lines of fixed points along both the largest and smallest moment of inertia axes can be simultaneously attracting. Limiting cases are those that conserve either the energy or the magnitude of the angular momentum. Parallels with rod mechanics, micromagnetics, and particles with effective mass are briefly discussed.

physics.class-ph↗

arXiv2020

Exact and approximate mechanisms for pure bending of sheets

Direct measurement of the moment-curvature response of sheets or wires up to high curvatures can aid in modeling creasing, pleating, and other forming operations. We consider theoretical issues related to the geometry of pure bending. We present a linkage design that, for homogeneous deformation of the sample, results in an exact pure bending state up to arbitrarily high curvature, by combining a cochleoidal trajectory with an angle-doubling mechanism. The full mechanism is single degree of freedom and position controlled, a desirable feature for measurement of soft materials. We also present an optimal approximation to the cochleoid using a more easily implemented circular trajectory, and compare this with an existing commercial system for fabric testing. We quantify the error of the approximate test by calculating the deformation of an Euler \emph{elastica}, a structure with linear moment-curvature response. While the circular mechanism can approximate the exact boundary conditions quite well up to moderately large curvatures, the resulting curvature of the test sample still deviates significantly from homogeneous response. We also briefly discuss expectations of localization behavior in pure bending tests.

cond-mat.soft↗

arXiv2019

Some observations on variational elasticity and its application to plates and membranes

Energies and equilibrium equations for thin elastic plates are discussed, with emphasis on several issues pertinent to recent approaches in soft condensed matter. Consequences of choice of basis, choice of invariant strain measures, and of approximating material energies as purely geometric in nature, are detailed. Ambiguities in the definitions of energies based on small-strain expansions, and in a typical informal process of dimensional reduction, are noted. A simple example serves to demonstrate that a commonly used bending energy has undesirable features, and it is suggested that a new theory based on Biot strains be developed. A compact form of the variation of a plate energy is presented. Throughout, the divergence form of equations is emphasized. An appendix relates the naive approach adopted in the main text with standard quantities in continuum mechanics.

cond-mat.soft↗

arXiv2018

Contrasting bending energies from bulk elastic theories

The choice of elastic energies for thin plates and shells is an unsettled issue with consequences for much recent modeling of soft matter. Through consideration of simple deformations of a thin body in the plane, we demonstrate that four bulk isotropic quadratic elastic theories have fundamentally different predictions with regard to bending behavior. At finite thickness, these qualitative effects persist near the limit of mid-surface isometry, and not all theories predict an isometric ground state. We discuss how certain kinematic measures that arose in early studies of rod mechanics lead to coherent definitions of stretching and bending, and promote the adoption of these quantities in the development of a covariant theory based on stretches rather than metrics.

cond-mat.soft↗

arXiv2018

On the planar elastica, stress, and material stress

We revisit the classical problem of the planar Euler \emph{elastica} with applied forces and moments, and present a classification of the shapes in terms of tangentially conserved quantities associated with spatial and material symmetries. We compare commonly used director, variational, and dynamical systems representations, and present several illustrative physical examples. We remark that an approach that employs only the shape equation for the tangential angle obscures physical information about the tension in the body.

physics.class-ph↗

arXiv2018

Partial constraint singularities in elastic rods

We present a unified classical treatment of partially constrained elastic rods. Partial constraints often entail singularities in both shapes and reactions. Our approach encompasses both sleeve and adhesion problems, and provides simple and unambiguous derivations of counterintuitive results in the literature. Relationships between reaction forces and moments, geometry, and adhesion energies follow from the balance of energy during quasistatic motion. We also relate our approach to the balance of material momentum and the concept of a driving traction. The theory is generalizable and can be applied to a wide array of contact, adhesion, gripping, and locomotion problems.

physics.class-ph↗

arXiv2018

The saturation bifurcation in coupled oscillators

We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies.

nlin.PS↗

arXiv2018

Impact-induced acceleration by obstacles

We explore a surprising phenomenon in which an obstruction accelerates, rather than decelerates, a moving flexible object. It has been claimed that the right kind of discrete chain falling onto a table falls \emph{faster} than a free-falling body. We confirm and quantify this effect, reveal its complicated dependence on angle of incidence, and identify multiple operative mechanisms. Prior theories for direct impact onto flat surfaces, which involve a single constitutive parameter, match our data well if we account for a characteristic delay length that must impinge before the onset of excess acceleration. Our measurements provide a robust determination of this parameter. This supports the possibility of modeling such discrete structures as continuous bodies with a complicated constitutive law of impact that includes angle of incidence as an input.

physics.class-ph↗

arXiv2018

Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

Motivated by observations of snap-through phenomena in buckled elastic strips subject to clamping and lateral end translations, we experimentally explore the multi-stability and bifurcations of thin bands of various widths and compare these results with numerical continuation of a perfectly anisotropic Kirchhoff rod. Our choice of boundary conditions is not easily satisfied by the anisotropic structures, forcing a cooperation between bending and twisting deformations. We find that, despite clear physical differences between rods and strips, a naive Kirchhoff model works surprisingly well as an organizing framework for the experimental observations. In the context of this model, we observe that anisotropy creates new states and alters the connectivity between existing states. Our results are a preliminary look at relatively unstudied boundary conditions for rods and strips that may arise in a variety of engineering applications, and may guide the avoidance of jump phenomena in such settings. We also briefly comment on the limitations of current strip models.

cond-mat.soft↗

arXiv2017

Pick-up and impact of flexible bodies

Picking up, laying down, colliding, rolling, and peeling are partial-contact interactions involving moving discontinuities. We examine the balances of momentum and energy across a moving discontinuity in a string, with allowance for injection or dissipation by singular supplies. We split the energy dissipation according to its invariance properties, discuss analogies with systems of particles and connections with the literature on shocks and phase transition fronts in various bodies, and derive a compatibility relation between supplies of momentum and translation-invariant energy. For a moving contact discontinuity between a string and a smooth rigid plane in the presence of gravity, we find a surprising asymmetry between the processes of picking up and laying down, such that steady-state kinks in geometry and associated jumps in tension are not admissible during pick-up. This prediction is consistent with experimental observations. We briefly discuss related problems including the falling folded chain, peeling of an adhesive tape, and the "chain fountain". Our approach is applicable to the study of impact and locomotion, and to systems such as moored floating structures and some musical instruments that feature vibrating string and cable elements interacting with a surface.

physics.class-ph↗

arXiv2016

Catenaries in viscous fluid

This work explores a simple model of a slender, flexible structure in a uniform flow, providing analytical solutions for the translating, axially flowing equilibria of strings subjected to a uniform body force and drag forces linear in the velocities. The classical catenaries are extended to a five-parameter family of curves. A sixth parameter affects the tension in the curves. Generic configurations are planar, represented by a single first order equation for the tangential angle. The effects of varying parameters on representative shapes, orbits in angle-curvature space, and stress distributions are shown. As limiting cases, the solutions include configurations corresponding to "lariat chains" and the towing, reeling, and sedimentation of flexible cables in a highly viscous fluid. Regions of parameter space corresponding to infinitely long, semi-infinite, and finite length curves are delineated. Almost all curves subtend an angle less than $π$ radians, but curious special cases with doubled or infinite range occur on the borders between regions. Separate transitions in the tension behavior, and counterintuitive results regarding finite towing tensions for infinitely long cables, are presented. Several physically inspired boundary value problems are solved and discussed.

physics.flu-dyn↗

arXiv2015

A conserved quantity in thin body dynamics

Thin, solid bodies with metric symmetries admit a restricted form of reparameterization invariance. Their dynamical equilibria include motions with both rigid and flowing aspects. On such configurations, a quantity is conserved along the intrinsic coordinate corresponding to the symmetry. As an example of its utility, this conserved quantity is combined with linear and angular momentum currents to construct solutions for the equilibria of a rotating, flowing string, for which it is akin to Bernoulli's constant.

physics.class-ph↗

arXiv2015

Jump conditions for strings and sheets from an action principle

I present conditions for compatibility of velocities, conservation of mass, and balance of momentum and energy across moving discontinuities in inextensible strings and sheets of uniform mass density. The balances are derived from an action with a time-dependent, non-material boundary, and reduce to matching of material boundary conditions if the discontinuity is stationary with respect to the body. I first consider a point discontinuity in a string and a line discontinuity in a sheet, in the context of classical inertial motion in three Euclidean dimensions. I briefly comment on line discontinuities terminating in point discontinuities in a sheet, discontinuous line discontinuities in a sheet, and an approach to dynamic fracture that treats a crack tip in a sheet as a time-dependent boundary point. I provide two examples of general solutions for conservative sheet motions near a line discontinuity. The approach also enables treatment of actions depending on higher derivatives of position; I thus derive balances for an \emph{elastica} which are applicable to moving contact problems.

physics.class-ph↗

arXiv2013

Dipoles in thin sheets

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.

cond-mat.soft↗

arXiv2013

At the end of a moving string

One cannot pull an open, curved string along itself. This fact is clearly reflected in the unwrapping motion of a string or chain as it is dragged around an object, and implies strong consequences for slender structures in passive locomotion, whether industrial cables and sheets or the ribbons of rhythmic gymnastics. We address a basic problem in the dynamics of flexible bodies, namely the solution of the string equations with a free boundary. This system is the backbone of many fluid-structure interactions, and also a model problem for thin structures where geometric nonlinearities cannot be ignored. We consider planar dynamics under the restriction that the spatially-dependent stress profile in the string is time-independent, which results in a conservation law form for the equations. We find a new exact solution whose range of validity is time-dependent, limited to greater than a distance scaling as $t^{\frac4/3}$ from the free end. The remainder of the distance is covered by splicing to another exact solution for a straight string, which introduces an error into the combined solution. The splicing also implies a $t^{-\frac2/3}$ singularity in acceleration, apparently corresponding to a whipping motion of the vanishing straight segment.

physics.flu-dyn↗

arXiv2013

Rotating strings

Analytical expressions are provided for the configurations of an inextensible, flexible, twistable inertial string rotating rigidly about a fixed axis. Solutions with trivial radial dependence are helices of arbitrary radius and pitch. Non-helical solutions are governed by a cubic equation whose roots delimit permissible values of the squared radial coordinate. Only curves coplanar with the axis of rotation make contact with it.

physics.class-ph↗

arXiv2012

Unwrapping Chains

A loop of chain can move along its own tangents, maintaining a steady shape. An open-ended chain undergoing a nontrivial motion must change its shape. One consequence is that chains pulled around objects will fail to follow the contours of the objects, unwrapping themselves instead. This short note accompanies a fluid dynamics video submission (83068) to the APS DFD Gallery of Fluid Motion 2012.

physics.flu-dyn↗