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J. A. Hanna

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

At least 19 recordsLinked to original sources
arXiv2026

Some explorations of Kirchhoff dynamics

The classical problem of a rigid body in an ideal fluid is a six-dimensional dynamical system with three conserved quantities. Despite its long history and its importance as a reduced model of many fluid-structure problems, basic questions remain, including the shapes and connectivities of the three-dimensional submanifolds of solutions, and the nature of chaotic motions on these. We explore the problem with a particular ellipsoidal body, beginning with integrable motions--- steady linear translation and rotation and periodic planar tumbling and fluttering. The stability of these states depends on the relative magnitudes of linear and angular momentum or energy. Linear stability, including Floquet, analysis of lower-dimensional perturbed systems is consistent with direct integration of the full system. We document instabilities leading to a variety of regular and chaotic motions, including flipping and twirling, whose trajectories appear to follow paths near connections between the integrable states. Such indirect observation of these connections provides insight into the structure underlying the rich dynamics of this simple system. For the submanifold where linear and angular momentum are perpendicular, we further examine how solutions fit together to fill momentum space in different zones of stability of the integrable states, and thereby uncover additional bifurcations in connectivity and what appear to be new integrable solutions.

physics.class-ph↗
arXiv2026

A derivation of viscous thin film flow equations on curved surfaces

General equations are derived for slow viscous thin fluid film flows on curved surfaces through an extension of Leal's pedagogical approach, which leaves the characteristic velocity scale unspecified and employs a direct through-thickness integration of the continuity equation. The derivation neglects inertia, and includes gravitational, capillary, and Marangoni effects, the latter coupling the thickness dynamics to free-surface transport of a dilute, non-diffusing surfactant. The resulting general expression incorporates the leading order terms of each type, as well as additional terms that become leading order for nongeneric cases. A few examples are briefly presented and literature comparisons made. The importance of gradients in curvature is emphasized, and it is suggested that nondimensionalization of geometric features might lead to further useful generalizations. This relatively simple formulation is intended as a starting point for exploring interactions between geometry, gravity, and surface tension.

physics.flu-dyn↗
arXiv2026

A sequence of elastic patterns in a sheared bent sheet

We document a sequence of bifurcations and elastic patterns in sheared bent sheets of intermediate aspect ratio. The sheets undergo inversion of curvature through the passage of localized features, often in S-shaped pairs. Nested force-displacement hysteresis loops provide experimental evidence for snaking. Several mechanisms for coarsening and refinement of the patterns are observed, including splitting, merging, and escape through open boundaries. While most forces, including that required for full snap-through, scale with the length of the sheet, the initial drop in force upon pattern nucleation decreases rapidly with length.

cond-mat.soft↗
arXiv2026

Closing a catenary loop: the lariat chain, the string shooter, and the heavy elastica

We review, critique, and extend results related to the problem of closed loop shape equilibria of a string shooter, a type of catenary consisting of steady, axially moving configurations of an inertial, inextensible, perfectly flexible string in the presence of gravity and drag forces. We relate to similar problems, including the lariat (no gravity), chain fountain (not closed), and heavy \emph{elastica} (bending stiffness). We focus on the difficulty inherent to continuing a catenary through a vertical orientation, necessary to close a loop, which difficulty changes in nature as the system undergoes bifurcations with increasing drag. We construct solutions by implementing available analytical results, and numerically generate additional solutions with added bending stiffness. We briefly discuss global balances of linear, angular, and pseudo-momentum for this system.

physics.class-ph↗
arXiv2026

Buckling mediated by mobile localized elastic excitations

Experiments reveal that structural transitions in thin sheets are mediated by the passage of transient and stable mobile localized elastic excitations. These ``crumples'' or ``d-cones'' nucleate, propagate, interact, annihilate, and escape. Much of the dynamics occurs on millisecond time scales. Nucleation sites correspond to regions where generators of the ideal unstretched surface converge. Additional stable intermediate states illustrate two forms of quasistatic inter-crumple interaction through ridges or valleys. These interactions create pairs from which extended patterns may be constructed in larger specimens. The onset of localized transient deformation with increasing sheet size is correlated with a characteristic stable crumple size, whose measured scaling with thickness is consistent with prior theory and experiment for localized elastic features in thin sheets. We offer a new theoretical justification of this scaling.

cond-mat.soft↗
arXiv2025

A ruled narrow elastic strip model with corrected energy

We present a new one-dimensional model for elastic strips based on a nondevelopable ruled surface. An auxiliary field regularizes the Sadowsky narrow-strip model to allow nonzero twist with vanishing curvature. The energy exhibits the scalings derived by Freddi and co-workers, and for a certain choice of parameter, convexifies the Sadowsky energy without patching. We present the kinematics and energetics of the model, and employ a variational approach featuring a rotation tensor to derive equilibrium equations. We perform a regular perturbation expansion to study the model behavior close to inflection points. When the energy is convex, curvature and moment are continuous at inflection points, while the auxiliary function suffers a jump, leading to a discontinuity in the ruled embedding for any finite width.

cond-mat.soft↗
arXiv2025

Shear stresses in fluid and solid membranes with bending elasticity

Comparison of a few simple models of fluid and solid membranes illustrates how shear stresses can arise from a bending energy through a coupling between curvature and surface stresses, a feature incidental to the fluid or solid nature of the material. In particular, it is shown how a fluid-like Helfrich bending energy contributes shear stresses, while a related solid-like energy, the correct continuum limit of a Seung-Nelson discrete bending term, produces a stress tensor with purely isotropic tangential part. A distinction is noted between the resistance to tangential flows on the surface and the ability to support tangential stresses. The tangential projection of the divergence of the stress, or the pseudomomentum balance, is viewed in the light of some statements and observations in the literature. Different types of constraints, invariance properties, and shape equations are briefly discussed. The state of a patch of unidirectionally-bent material is connected with prior analyses of the pulling of membrane tethers.

cond-mat.soft↗
arXiv2025

Energies for elastic plates and shells from quadratic-stretch elasticity

We derive stretching and bending energies for isotropic elastic plates and shells. Through the dimensional reduction of a bulk elastic energy quadratic in Biot strains, we obtain two-dimensional bending energies quadratic in bending measures featuring a bilinear coupling of stretches and geometric curvatures. For plates, the bending measure is invariant under spatial dilations and naturally extends primitive bending strains for straight rods. For shells or naturally-curved rods, the measure is not dilation invariant, and contrasts with previous \emph{ad hoc} postulated forms. The corresponding field equations and boundary conditions feature moments linear in the bending measures, and a decoupling of stretching and bending such that application of a pure moment results in isometric deformation of a unique neutral surface, primitive behaviors in agreement with classical linear response but not displayed by commonly used analytical models. We briefly comment on relations between our energies, those derived from a neo-Hookean bulk energy, and a commonly used discrete model for flat membranes. Although the derivation requires consideration of stretch and rotation fields, the resulting energy and field equations can be expressed entirely in terms of metric and curvature components of deformed and reference surfaces.

cond-mat.soft↗
arXiv2024

Assorted remarks on bending measures and energies for plates and shells, and their invariance properties

In this note, we address several issues, including some raised in recent works and commentary, related to bending measures and energies for plates and shells, and certain of their invariance properties. We discuss overlaps and distinctions in results arising from two different definitions of stretching, correct an error and citation oversights in our prior work, reiterate some of the early history of dilation-invariant bending measures, and provide additional brief observations regarding the relative size of energetic terms and the symmetrization of bending measures. A particular point of emphasis is the distinction between dilation-invariant measures and a recently introduced non-dilation-invariant measure for shells and curved rods. In the course of this discussion, we provide a simpler presentation of the elementary, but much neglected, fact that the through-thickness derivative of tangential stretch of material near the mid-surface of a thin body is the product of the mid-surface stretch and change in curvature.

cond-mat.soft↗
arXiv2023

Anomalous curvature evolution and geometric regularization of energy focusing in the snapping dynamics of a flexible body

We examine the focusing of kinetic energy and the amplification of various quantities during the snapping motion of the free end of a flexible structure. This brief but violent event appears to be a regularized finite-time singularity, with remarkably large spikes in velocity, acceleration, and tension easily induced by generic initial and boundary conditions. A numerical scheme for the inextensible string equations is validated against available experimental data for a falling chain and further employed to explore the phenomenon. We determine that the discretization of the equations, equivalent to the physically discrete problem of a chain, does not provide the regularizing length scale, which in the absence of other physical effects must then arise from the geometry of the problem. An analytical solution for a geometrically singular limit, a falling perfectly-folded string, accounts surprisingly well for the scalings of several quantities in the numerics, but can only indirectly suggest a behavior for the curvature, one which seems to explain prior experimental data but does not correspond to the evolution of the curvature peak in our system, which instead displays a newly observed anomalously slow scaling. A simple model, incorporating only knowledge of the initial conditions along with the anomalous and singular-limit scalings, provides reasonable estimates for the amplifications of relevant quantities. This is a first step to predict and harness arbitrarily large energy focusing in structures, with a practical limit set only by length scales present in the discrete mechanical system or the initial conditions.

physics.class-ph↗
arXiv2023

Avoiding localization instabilities in rotary pleating

Rotary pleating is a widely used process for making filters out of nonwoven fabric sheets. This involves indirect elastic-plastic bending of pre-weakened creases by continuously injecting material into an accordion-shaped pack. This step can fail through a localization instability that creates a kink in a pleat facet instead of in the desired crease location. In the present work, we consider the effects of geometric and material parameters on the rotary pleating process. We formulate the process as a multi-point variable-arc-length boundary value problem for planar inextensible rods, with hinge connections. Both the facets (rods) and creases (hinges) obey nonlinear moment-curvature or moment-angle constitutive laws. Some unexpected aspects of the sleeve boundary condition at the point of material injection, common to many continuous sheet processes, are noted. The process, modeled as quasistatic, features multiple equilibria which we explore by numerical continuation. The presence of, presumably stable, kinked equilibria is taken as a conservative sign of potential pleating failure. Failure may also occur due to localization at the injection point. We may thus obtain "pleatability surfaces" that separate the parameter space into regions where mechanical pleating will succeed or fail. Successful pleating depends primarily on the distance between the injection point and the pleated pack. Other factors, such as the crease stiffness and strength relative to that of the facets, also have an influence. Our approach can be adapted to study other pleating and forming processes, the deployment and collapse of folded structures, or multi-stability in compliant structures.

cond-mat.soft↗
arXiv2022

Pseudomomentum: origins and consequences

The balance of pseudomomentum is discussed and applied to simple elasticity, ideal fluids, and the mechanics of inextensible rods and sheets. A general framework is presented in which the simultaneous variation of an action with respect to position, time, and material labels yields bulk balance laws and jump conditions for momentum, energy, and pseudomomentum. The example of simple elasticity of space-filling solids is treated at length. The pseudomomentum balance in ideal fluids is shown to imply conservation of vorticity, circulation, and helicity, and a mathematical similarity is noted between the evaluation of circulation along a material loop and the J-integral of fracture mechanics. Integration of the pseudomomentum balance, making use of a prescription for singular sources derived by analogy with the continuous form of the balance, directly provides the propulsive force driving passive reconfiguration or locomotion of confined, inhomogeneous elastic rods. The conserved angular momentum and pseudomomentum are identified in the classification of conical sheets with rotational inertia or bending energy.

physics.class-ph↗
arXiv2022

Momentum and pseudomomentum in a shallow water equation

A basic shallow water system with variable topography is analyzed from the point of view of a Lagrangian derivation of momentum, energy, and pseudomomentum balances. A two-dimensional action and associated momentum equation are derived. The latter is further manipulated to derive additional equations for energy and pseudomomentum. This revealed structure emphasizes broken symmetries in space and a reference configuration, and preserved symmetry in time.

physics.flu-dyn↗
arXiv2022

Dilation-invariant bending of elastic plates, and broken symmetry in shells

We propose bending energies for isotropic elastic plates and shells. For a plate, we define and employ a surface tensor that symmetrically couples stretch and curvature such that any elastic energy density constructed from its invariants is invariant under spatial dilations. This kinematic measure and its corresponding isotropic quadratic energy resolve outstanding issues in thin structure elasticity, including the natural extension of primitive bending strains for straight rods to plates, the assurance of a moment linear in the bending measure, and the avoidance of induced mid-plane strains in response to pure moments as found in some commonly used analytical plate models. Our analysis also reveals that some other commonly used numerical models have the right invariance properties, although they lack full generality at quadratic order in stretch. We further extend our result to naturally-curved rods and shells, for which the pure stretching of a curved rest configuration breaks dilation invariance; the new shell bending measure we provide contrasts with previous \emph{ad hoc} postulated forms. The concept that unifies these theories is not dilation invariance, but rather through-thickness uniformity of strain as a definition of pure stretching deformations. Our results provide a clean basis for simple models of low-dimensional elastic systems, and should enable more accurate analytical probing of the structure of singularities in sheets and membranes.

cond-mat.soft↗
arXiv2021

Exterior dissipation, proportional decay, and integrals of motion

Given a dynamical system with $m$ independent conserved quantities, we construct a multi-parameter family of new systems in which these quantities evolve monotonically and proportionally, and are replaced by $m-1$ conserved linear combinations of themselves, with any of the original quantities as limiting cases. The modification of the dynamics employs an exterior product of gradients of the original quantities, and often evolves the system towards asymptotic linear dependence of these gradients in a nontrivial state. The process both generalizes and provides additional structure to existing techniques for selective dissipation in the literature on fluids and plasmas, nonequilibrium thermodynamics, and nonlinear controls. It may be iterated or adapted to obtain any reduction in the degree of integrability. It may enable discovery of extremal states, limit cycles, or solitons, and the construction of new integrable systems from superintegrable systems. We briefly illustrate the approach by its application to the cyclic three-body Toda lattice, driven from an aperiodic orbit towards a limit cycle.

physics.class-ph↗
arXiv2021

Quadratic-stretch elasticity

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.

cond-mat.soft↗
arXiv2021

Cutting holes in bistable folds

A folded disk is bistable, as it can be popped through to an inverted state with elastic energy localized in a small, highly-deformed region on the fold. Cutting out this singularity relaxes the surrounding material and leads to a loss of bistability when the hole dimensions reach a critical size. These dimensions are strongly anisotropic and feature a surprising re-entrant behavior, such that removal of additional material can re-stabilize the inverted state. A model of the surface as a wide annular developable strip is found to capture the qualitative observations in experiments and simulations. These phenomena are consequential to the mechanics and design of crumpled elastic sheets, developable surfaces, origami and kirigami, and other deployable and compliant structures.

cond-mat.soft↗