arXiv subjects

L. Mahadevan

Publications and source records attributed to L. Mahadevan.

At least 73 records · Page 4Linked to original sources

arXiv2021

Geometric control of topological dynamics in a singing saw

The common handsaw can be converted into a bowed musical instrument capable of producing exquisitely sustained notes when its blade is appropriately bent. Acoustic modes localized at an inflection point are known to underlie the saw's sonorous quality, yet the origin of localization has remained mysterious. Here we uncover a topological basis for the existence of localized modes, that relies on and is protected by spatial curvature. By combining experimental demonstrations, theory and computation, we show how spatial variations in blade curvature control the localization of these trapped states, allowing the saw to function as a geometrically tunable high quality oscillator. Our work establishes an unexpected connection between the dynamics of thin shells and topological insulators, and offers a robust principle to design high quality resonators across scales, from macroscopic instruments to nanoscale devices, simply through geometry.

cond-mat.soft↗

arXiv2021

Geometrical dynamics of edge-driven surface growth

Accretion of mineralized thin wall-like structures via localized growth along their edges is observed in a range of physical and biological systems ranging from molluscan and brachiopod shells to carbonate-silica composite precipitates. To understand the shape of these mineralized structures, we develop a mathematical framework that treats the thin-walled shells as a smooth surface left in the wake of the growth front that can be described as an evolving space curve. Our theory then takes an explicit geometric form for the prescription of the velocity of the growth front curve, along with some compatibility relations and a closure equation related to the nature of surface curling. The result is a set of equations for the geometrical dynamics of a curve that leaves behind a compatible surface. Solutions of these equations capture a range of geometric precipitate patterns seen in abiotic and biotic forms across scales. In addition to providing a framework for the growth and form of these thin-walled morphologies, our theory suggests a new class of dynamical systems involving moving space curves that are compatible with non-Euclidean embeddings of surfaces.

cond-mat.soft↗

arXiv2021

Wallpaper group kirigami

Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms and simple planar tilings. Here we complement these approaches by directly linking kirigami patterns to the symmetry associated with the set of seventeen repeating patterns that fully characterize the space of periodic tilings of the plane. We start by showing how to construct deployable kirigami patterns using any of the wallpaper groups, and then design symmetry-preserving cut patterns to achieve arbitrary size changes via deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles and connect these results to the underlying kirigami-based lattice structures. All together, our work provides a systematic approach for creating a broad range of kirigami-based deployable structures with any prescribed size and symmetry properties.

cond-mat.soft↗

arXiv2021

Active nematic defects and epithelial morphogenesis

Inspired by recent experiments that highlight the role of nematic defects in the morphogenesis of epithelial tissues, we develop a minimal framework to study the dynamics of an active curved surface driven by its nematic texture. Allowing the surface to evolve via relaxational dynamics leads to a theory linking nematic defect dynamics, cellular division rates and Gaussian curvature. Regions of large positive (negative) curvature and positive (negative) growth are colocalized with the presence of positive (negative) defects. Applying this framework to the dynamics of cultured murine neural progenitor cells (NPCs) in an ex-vivo setting, we find that cells accumulate at positive defects and are depleted at negative defects. In contrast, applying this to the dynamics of a basal marine invertebrate \emph{Hydra} in an in-vivo setting, we show that activity stabilizes a bound $+1$ defect state by creating an incipient tentacle, while a bound $+1$ defect state surrounded by two $-1/2$ defects can create a stationary ring configuration of tentacles, consistent with observations.

cond-mat.soft↗

arXiv2021

Geometry, Analysis and Morphogenesis: Problems and Prospects

The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discus open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.

math.GM↗

arXiv2021

Geodesics and isometric immersions in kirigami

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.

math.DG↗

arXiv2021

Defect-mediated dynamics of coherent structures in active nematics

Active fluids, such as cytoskeletal filaments, bacterial colonies and epithelial cell layers, exhibit distinctive orientational coherence, often characterized by nematic order and topological defects. By contrast, little is known about positional coherence -- i.e., how a hidden dynamic skeleton organizes the underlying chaotic motion -- despite this being one of their most prominent and experimentally accessible features. Using a combination of dynamical systems theory, experiments on two-dimensional mixtures of microtubules and kinesin and hydrodynamic simulations, we characterize positional coherence in active nematics. These coherent structures can be identified in the framework of Lagrangian dynamics as moving attractors and repellers, which orchestrate complex motion. To understand the interaction of positional and orientational coherence on the dynamics of defects, we then analysed observations and simulations and see that +1/2 defects move and deform the attractors, thus functioning as control centers for collective motion. Additionally, we find that regions around isolated +1/2 defects undergo high bending and low stretching/shearing deformations, consistent with the local stress distribution. The stress is minimum at the defect, while high differential stress along the defect orientation induces folding. Our work offers a new perspective to describe self-organization in active fluids, with potential applications to multicellular systems.

cond-mat.soft↗

arXiv2021

An additive algorithm for origami design

Inspired by the allure of additive fabrication, we pose the problem of origami design from a new perspective: how can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve this problem in two steps: by first identifying the geometric conditions for the compatible completion of two separate folds into a single developable four-fold vertex, and then showing how this foundation allows us to grow a geometrically compatible front at the boundary of a given folded seed. This yields a complete marching, or additive, algorithm for the inverse design of the complete space of developable quad origami patterns that can be folded from flat sheets. We illustrate the flexibility of our approach by growing ordered, disordered, straight and curved folded origami and fitting surfaces of given curvature with folded approximants. Overall, our simple shift in perspective from a global search to a local rule has the potential to transform origami-based meta-structure design.

cond-mat.soft↗

arXiv2021

Combing a double helix

Combing hair involves brushing away the topological tangles in a collective curl. Using a combination of experiment and computation, we study this problem that naturally links topology, geometry and mechanics. Observations show that the dominant interactions in hair are those of a two-body nature, corresponding to a braided homochiral double helix. Using this minimal model, we study the detangling of an elastic double helix via a single stiff tine that moves along it, leaving two untangled filaments in its wake. Our results quantify how the forces of detangling are correlated with the magnitude and spatial extent of the link density, a topological quantity, that propagates ahead of the tine. This in turn provides a measure of the maximum characteristic length of a single combing stroke, and thus the trade-offs between comfort, efficiency and speed of combing in the many-body problem on a head of hair.

cond-mat.soft↗

arXiv2021

Instability-induced patterning of a jelling jet

When a thin stream of aqueous sodium alginate is extruded into a reacting calcium chloride bath, it polymerizes into a soft elastic tube that spontaneously forms helical coils due to the ambient fluid drag. We quantify the onset of this drag-induced instability and its nonlinear evolution using experiments, and explain the results using a combination of scaling, theory and simulations. By co-extruding a second (internal) liquid within the aqueous sodium alginate jet and varying the rates of co-extrusion of the two liquids, as well as the diameter of the jet, we show that we can tune the local composition of the composite filament and the nature of the ensuing instabilities to create soft filaments of variable relative buoyancy, shape and mechanical properties. All together, by harnessing the fundamental varicose (jetting) and sinuous (buckling) instabilities associated with the extrusion of a jelling filament, we show that it is possible to print complex three-dimensional filamentous structures in the ambient fluid.

cond-mat.soft↗

arXiv2021

Static adhesion hysteresis in elastic structures

Adhesive interactions between elastic structures such as graphene sheets, carbon nanotubes, and microtubules have been shown to exhibit hysteresis due to irrecoverable energy loss associated with bond breakage, even in static (rate-independent) experiments. To understand this phenomenon, we start with a minimal theory for the peeling of a thin sheet from a substrate, coupling the local event of bond breaking to the nonlocal elastic relaxation of the sheet and show that this can drive static adhesion hysteresis over a bonding/debonding cycle. Using this model we quantify hysteresis in terms of the adhesion and elasticity parameters of the system. This allows us to derive a scaling relation that preserves hysteresis at different levels of granularity while resolving a seeming paradox of lattice trapping in the continuum limit of a discrete fracture process. Finally, to verify our theory, we use new experiments to demonstrate and measure adhesion hysteresis in bundled microtubules.

physics.bio-ph↗

arXiv2020

Compact reconfigurable kirigami

Kirigami involves cutting a flat, thin sheet that allows it to morph from a closed, compact configuration into an open deployed structure via coordinated rotations of the internal tiles. By recognizing and generalizing the geometric constraints that enable this art form, we propose a design framework for compact reconfigurable kirigami patterns, which can morph from a closed and compact configuration into a deployed state conforming to any prescribed target shape, and subsequently be contracted into a different closed and compact configuration. We further establish a condition for producing kirigami patterns which are reconfigurable and rigid deployable allowing us to connect the compact states via a zero-energy family of deployed states. All together, our inverse design framework lays out a new path for the creation of shape-morphing material structures.

cond-mat.soft↗

arXiv2020

Models of benthic bipedalism

Walking is a common bipedal and quadrupedal gait and is often associated with terrestrial and aquatic organisms. Inspired by recent evidence of the neural underpinnings of primitive aquatic walking in the little skate Leucoraja erinacea, we introduce a theoretical model of aquatic walking that reveals robust and efficient gaits with modest requirements for body morphology and control. The model predicts undulatory behavior of the system body with a regular foot placement pattern which is also observed in the animal, and additionally predicts the existence of gait bistability between two states, one with a large energetic cost for locomotion and another associated with almost no energetic cost. We show that these can be discovered using a simple reinforcement learning scheme. To test these theoretical frameworks, we built a bipedal robot and show that its behaviors are similar to those of our minimal model: its gait is also periodic and exhibits bistability, with a low efficiency gait separated from a high efficiency gait by a "jump" transition. Overall, our study highlights the physical constraints on the evolution of walking and provides a guide for the design of efficient biomimetic robots.

q-bio.QM↗

arXiv2020

Optimal policies for mitigating pandemic costs

Several non-pharmaceutical interventions have been proposed to control the spread of the COVID-19 pandemic. On the large scale, these empirical solutions, often associated with extended and complete lockdowns, attempt to minimize the costs associated with mortality, economic losses and social factors, while being subject to constraints such as finite hospital capacity. Here we pose the question of how to mitigate pandemic costs subject to constraints by adopting the language of optimal control theory. This allows us to determine top-down policies for the nature and dynamics of social contact rates given an age-structured model for the dynamics of the disease. Depending on the relative weights allocated to life and socioeconomic losses, we see that the optimal strategies range from long-term social-distancing only for the most vulnerable, to partial lockdown to ensure not over-running hospitals, to alternating-shifts with significant reduction in life and/or socioeconomic losses. Crucially, commonly used strategies that involve long periods of broad lockdown are almost never optimal, as they are highly unstable to reopening and entail high socioeconomic costs. Using parameter estimates from data available for Germany and the USA, we quantify these policies and use sensitivity analysis in the relevant model parameters and initial conditions to determine the range of robustness of our policies. Finally we also discuss how bottom-up behavioral changes can also change the dynamics of the pandemic and show how this in tandem with top-down control policies can mitigate pandemic costs even more effectively.

physics.soc-ph↗

arXiv2020

Flow-driven branching in a frangible porous medium

Channel formation and branching is widely seen in physical systems where movement of fluid through a porous structure causes the spatiotemporal evolution of the medium in response to the flow, in turn causing flow pathways to evolve. We provide a simple theoretical framework that embodies this feedback mechanism in a multi-phase model for flow through a fragile porous medium with a dynamic permeability. Numerical simulations of the model show the emergence of branched networks whose topology is determined by the geometry of external flow forcing. This allows us to delineate the conditions under which splitting and/or coalescing branched network formation is favored, with potential implications for both understanding and controlling branching in soft frangible media.

physics.flu-dyn↗

arXiv2020

Self-excited motions of volatile drops on swellable sheets

When a volatile droplet is deposited on a floating swellable sheet, it becomes asymmetric, lobed and mobile. We describe and quantify this phenomena that involves nonequilibrium swelling, evaporation and motion, working together to realize a self-excitable spatially extended oscillator. Solvent penetration causes the film to swell locally and eventually buckle, changing its shape and the drop responds by moving. Simultaneously, solvent evaporation from the swollen film causes it to regain its shape once the droplet has moved away. The process repeats and leads to complex pulsatile spinning and/or sliding movements. We use a one-dimensional experiment to highlight the slow swelling of and evaporation from the film and the fast motion of the drop, a characteristic of excitable systems. Finally, we provide a phase diagram for droplet excitability as a function of drop size and film thickness and scaling laws for the motion of the droplet.

cond-mat.soft↗

arXiv2020

Control of connectivity and rigidity in prismatic assemblies

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

cond-mat.soft↗

arXiv2020

Coordinated Crawling via Reinforcement Learning

Rectilinear crawling locomotion is a primitive and common mode of locomotion in slender, soft-bodied animals. It requires coordinated contractions that propagate along a body that interacts frictionally with its environment. We propose a simple approach to understand how these coordinations arise in a neuromechanical model of a segmented, soft-bodied crawler via an iterative process that might have both biological antecedents and technological relevance. Using a simple reinforcement learning algorithm, we show that an initial all-to-all neural coupling converges to a simple nearest-neighbor neural wiring that allows the crawler to move forward using a localized wave of contraction that is qualitatively similar to what is observed in D. melanogaster larvae and used in many biomimetic solutions. The resulting solution is a function of how we weight gait regularization in the reward, with a tradeoff between speed and robustness to proprioceptive noise. Overall, our results, which embed the brain-body-environment triad in a learning scheme, has relevance for soft robotics while shedding light on the evolution and development of locomotion.

physics.bio-ph↗