arXiv subjects

L. Mahadevan

Publications and source records attributed to L. Mahadevan.

At least 127 records · Page 7Linked to original sources

arXiv2014

Forced tearing of ductile and brittle thin sheets

Tearing a thin sheet by forcing a rigid object through it leads to complex crack morphologies; a single oscillatory crack arises when a tool is driven laterally through a brittle sheet, while two diverging cracks and a series of concertina-like folds forms when a tool is forced laterally through a ductile sheet. On the other hand, forcing an object perpendicularly through the sheet leads to radial petal-like tears in both ductile and brittle materials. To understand these different regimes we use a combination of experiments, simulations and simple theories. In particular, we describe the transition from brittle oscillatory tearing via a single crack to ductile concertina tearing with two tears by deriving laws that describe the crack paths and wavelength of the concertina folds and provide a simple phase diagram for the morphologies in terms of the material properties of the sheet and the relative size of the tool.

cond-mat.mtrl-sci↗

arXiv2014

Exactly isochoric deformations of soft solids

Many materials of contemporary interest, such as gels, biological tissues and elastomers, are easily deformed but essentially incompressible. Traditional linear theory of elasticity implements incompressibility only to first order and thus permits some volume changes, which become problematically large even at very small strains. Using a mixed coordinate transformation originally due to Gauss, we enforce the constraint of isochoric deformations exactly to develop a linear theory with perfect volume conservation that remains valid until strains become geometrically large. We demonstrate the utility of this approach by calculating the response of an infinite soft isochoric solid to a point force that leads to a nonlinear generalization of the Kelvin solution. Our approach naturally generalizes to a range of problems involving deformations of soft solids and interfaces in 2 dimensional and axisymmetric geometries, which we exemplify by determining the solution to a distributed load that mimics muscular contraction within the bulk of a soft solid.

cond-mat.soft↗

arXiv2014

2D and Axisymmetric Incompressible Elastic Green's Functions

We compile a list of 2-D and axisymmetric Green's functions for isotropic full and half spaces, to complement our letter "Linear elasticity of incompressible solids". We also extend the isotropic exactly incompressible linear theory from our letter to include isotropic neo-Hookean solids subject to a large pre-strain, and present Green's functions in these cases. The Green's function for a pre-strained half-space reproduces the Biot instability.

cond-mat.soft↗

arXiv2014

The dynamics of sperm cooperation in a competitive environment

Sperm cooperation has evolved in a variety of taxa and is often considered a response to sperm competition, yet the benefit of this form of collective movement remains unclear. Here we use fine-scale imaging and a minimal mathematical model to study sperm aggregation in the rodent genus $Peromyscus$. We demonstrate that as the number of sperm cells in an aggregate increase, the group moves with more persistent linearity but without increasing speed; this benefit, however, is offset in larger aggregates as the geometry of the group forces sperm to swim against one another. The result is a non-monotonic relationship between aggregate size and average velocity with both a theoretically predicted and empirically observed optimum of 6-7 sperm/aggregate. To understand the role of sexual selection in driving these sperm group dynamics, we compared two sister-species with divergent mating systems and find that sperm of $P.\,maniculatus$ (highly promiscuous), which have evolved under intense competition, form optimal-sized aggregates more often than sperm of $P.\,polionotus$ (strictly monogamous), which lack competition. Our combined mathematical and experimental study of coordinated sperm movement reveals the importance of geometry, motion and group size on sperm velocity and suggests how these physical variables interact with evolutionary selective pressures to regulate cooperation in competitive environments.

q-bio.CB↗

arXiv2014

Surface sulci in squeezed soft solids

The squeezing of soft solids, the constrained growth of biological tissues, and the swelling of soft elastic solids such as gels can generate large compressive stresses at their surfaces. This causes the otherwise smooth surface of such a solid to becomes unstable when its stress exceeds a critical value. Previous analyses of the surface instability have assumed two-dimensional plane-strain conditions, but in experiments isotropic stresses often lead to complex three-dimensional sulcification patterns. Here we show how such diverse morphologies arise by numerically modeling the lateral compression of a rigidly clamped elastic layer. For incompressible solids, close to the instability threshold, sulci appear as I-shaped lines aligned orthogonally with their neighbors; at higher compressions they are Y-shaped and prefer a hexagonal arrangement. In contrast, highly compressible solids when squeezed show only one sulcified phase characterized by a hexagonal sulcus network.

cond-mat.soft↗

arXiv2014

Synchronization in a stochastic Hebbian network of phase oscillators

We investigate a generalized Kuramoto phase-oscillator model with Hebb-like couplings that evolve according to a stochastic differential equation on various topologies. Numerical simulations show that even with identical oscillators, there is a regime in the nearest-neighbor coupling topologies and a complex network topology where oscillators move between an in phase and anti-phase state. Phase diagrams show the transition probabilities as a function of the noise strength and rate of evolution of network coupling. A minimal theoretical model allows us to understand these transitions.

cond-mat.stat-mech↗

arXiv2014

Feedback induced phase transitions in active porous media

Flow through passive porous media is typically described in terms of a linear theory relating current fluxes and driving forces, in the presence of a prescribed heterogeneous permeability. However, many porous systems such as glacial drainage networks, erosional river bed networks, vascular networks, social insect swarms and animal architectures such as termite mounds are continuously remodeled by the flow and thence modify the flow, i.e. they are active. Here we consider a minimal model for an active porous medium where flow and resistance are coupled to each other. Using numerical simulations, we show that this results in both channelization and wall-building transitions depending on the form of the feedback. A continuum model allows us to understand the qualitative features of the resulting phase diagram, and suggests ways to realize complex architectures using simple rules in engineered systems.

cond-mat.stat-mech↗

arXiv2014

How a Blister Heals

We use experiments to study the dynamics of the healing of a blister, a localized bump in a thin elastic layer that is adhered to a soft substrate everywhere except at the bump. We create a blister by gently placing a glass cover slip on a PDMS substrate. The pressure jump across the elastic layer drives fluid flow through micro-channels that form at the interface between the layer and the substrate; these channels coalesce at discrete locations as the blister heals and eventually disappear at a lower critical radius. The spacing of the channel follows a simple scaling law that can be theoretically justified, and the kinetics of healing is rate limited by fluid flow, but with a non-trivial dependence on the substrate thickness that likely arises due to channelization. Our study is relevant to a variety of soft adhesion scenarios.

cond-mat.soft↗

arXiv2014

Models for elastic shells with incompatible strains

The three-dimensional shapes of thin lamina such as leaves, flowers, feathers, wings etc, are driven by the differential strain induced by the relative growth. The growth takes place through variations in the Riemannian metric, given on the thin sheet as a function of location in the central plane and also across its thickness. The shape is then a consequence of elastic energy minimization on the frustrated geometrical object. Here we provide a rigorous derivation of the asymptotic theories for shapes of residually strained thin lamina with nontrivial curvatures, i.e. growing elastic shells in both the weakly and strongly curved regimes, generalizing earlier results for the growth of nominally flat plates. The different theories are distinguished by the scaling of the mid-surface curvature relative to the inverse thickness and growth strain, and also allow us to generalize the classical Föppl-von Kármán energy to theories of prestrained shallow shells.

math.AP↗

arXiv2013

Collective thermoregulation in bee clusters

Swarming is an essential part of honeybee behaviour, wherein thousands of bees cling onto each other to form a dense cluster that may be exposed to the environment for several days. This cluster has the ability to maintain its core temperature actively without a central controller, and raises the question of how this is achieved. We suggest that the swarm cluster is akin to an active porous structure whose functional requirement is to adjust to outside conditions by varying its porosity to control its core temperature. Using a continuum model that takes the form of a set of advection-diffusion equations for heat transfer in a mobile porous medium, we show that the equalization of an effective "behavioural pressure", which propagates information about the ambient temperature through variations in density, leads to effective thermoregulation. Our model extends and generalizes previous models by focusing the question of mechanism on the form and role of the behavioural pressure, and allows us to explain the vertical asymmetry of the cluster (as a consequence of buoyancy driven flows), the ability of the cluster to overpack at low ambient temperatures without breaking up at high ambient temperatures, and the relative insensitivity to large variations in the ambient temperature. Finally, our theory makes testable hypotheses for how the cluster bee density should respond to externally imposed temperature inhomogeneities, and suggests strategies for biomimetic thermoregulation.

q-bio.PE↗

arXiv2013

Statistical Mechanics of Multistability in Microscopic Shells

Unlike macroscopic multistable mechanical systems such as snap bracelets or elastic shells that must be physically manipulated into various conformations, microscopic systems can undergo spontaneous conformation switching between multistable states due to thermal fluctuations. Here we investigate the statistical mechanics of shape transitions in small elastic elliptical plates and shells driven by noise. By assuming that the effects of edges are small, which we justify exactly for plates and shells with a lenticular section, we decompose the shapes into a few geometric modes whose dynamics are easy to follow. We use Monte Carlo simulations to characterize the shape transitions between conformation minima as a function of noise strength, and corroborate our results using a Fokker-Planck formalism to study the stationary distribution and the mean first passage time problem. Our results are applicable to objects such as such as graphene flakes or protein β-sheets, where fluctuations, geometry and finite size effects are important.

cond-mat.stat-mech↗

arXiv2013

Digital instability of a confined elastic meniscus

Thin soft elastic layers serving as joints between relatively rigid bodies may function as sealants, thermal, electrical, or mechanical insulators, bearings, or adhesives. When such a joint is stressed, even though perfect adhesion is maintained, the exposed free meniscus in the thin elastic layer becomes unstable, leading to the formation of spatially periodic digits of air that invade the elastic layer, reminiscent of viscous fingering in a thin fluid layer. How- ever, the elastic instability is reversible and rate-independent, dis- appearing when the joint is unstressed. We use theory, experiments, and numerical simulations to show that the transition to the digital state is sudden (first-order), the wavelength and amplitude of the fingers are proportional to the thickness of the elastic layer, and the required separation to trigger the instability is inversely proportional to the in-plane dimension of the layer. Our study reveals the energetic origin of this instability and has implications for the strength of polymeric adhesives; it also suggests a method for patterning thin films reversibly with any arrangement of localized fingers in a digital elastic memory, which we confirm experimentally.

cond-mat.soft↗

arXiv2013

Elastic Platonic Shells

On microscopic scales, the crystallinity of flexible tethered or cross linked membranes determines their mechanical response. We show that by controlling the type, number and distribution of defects on a spherical elastic shell, it is possible to direct the morphology of these structures. Our numerical simulations show that by deflating a crystalline shell with defects, we can create elastic shell analogs of the classical Platonic solids. These morphologies arise via a sharp buckling transition from the sphere which is strongly hysteretic in loading-unloading. We construct a minimal Landau theory for the transition using quadratic and cubic invariants of the spherical harmonic modes. Our approach suggests methods to engineer shape into soft spherical shells using a frozen defect topology.

cond-mat.soft↗

arXiv2013

Swarming, swirling and stasis in sequestered bristle-bots

The collective ability of organisms to move coherently in space and time is ubiquitous in any group of autonomous agents that can move and sense each other and the environment. Here we investigate the origin of collective motion and its loss using macroscopic self-propelled Bristle-Bots, simple automata made from a toothbrush and powered by an onboard cell phone vibrator-motor, that can sense each other through shape-dependent local interactions, and can also sense the environment non-locally via the effects of confinement and substrate topography. We show that when Bristle-Bots are confined to a limited arena with a soft boundary, increasing the density drives a transition from a disordered and uncoordinated motion to organized collective motion either as a swirling cluster or a collective dynamical stasis. This transition is regulated by a single parameter, the relative magnitude of spinning and walking in a single automaton. We explain this using quantitative experiments and simulations that emphasize the role of the agent shape, environment, and confinement via boundaries. Our study shows how the behavioral repertoire of these physically interacting automatons controlled by one parameter translates into the mechanical intelligence of swarms.

cond-mat.soft↗

arXiv2013

Increased Network Interdependency Leads to Aging

Although species longevity is subject to a diverse range of selective forces, the mortality curves of a wide variety of organisms are rather similar. We argue that aging and its universal characteristics may have evolved by means of a gradual increase in the systemic interdependence between a large collection of biochemical or mechanical components. Modeling the organism as a dependency network which we create using a constructive evolutionary process, we age it by allowing nodes to be broken or repaired according to a probabilistic algorithm that accounts for random failures/repairs and dependencies. Our simulations show that the network slowly accumulates damage and then catastrophically collapses. We use our simulations to fit experimental data for the time dependent mortality rates of a variety of multicellular organisms and even complex machines such as automobiles. Our study suggests that aging is an emergent finite-size effect in networks with dynamical dependencies and that the qualitative and quantitative features of aging are not sensitively dependent on the details of system structure.

q-bio.PE↗

arXiv2012

Geometric Mechanics of Periodic Pleated Origami

Origami is the archetype of a structural material with unusual mechanical properties that arise almost exclusively from the geometry of its constituent folds and forms the basis for mechanical metamaterials with an extreme deformation response. Here we consider a simple periodically folded structure Miura-ori, which is composed of identical unit cells of mountain and valley folds with four-coordinated ridges, defined completely by 2 angles and 2 lengths. We use the geometrical properties of a Miura-ori plate to characterize its elastic response to planar and non-planar piece- wise isometric deformations and calculate the two-dimensional stretching and bending response of a Miura-ori sheet, and show that the in-plane and out-of-plane Poisson's ratios are equal in magnitude, but opposite in sign. Our geometric approach also allows us to solve the inverse design problem of determining the geometric parameters that achieve the optimal geometric and mechanical response of such structures.

physics.class-ph↗

arXiv2012

Flagellar dynamics of a connected chain of active, Brownian particles

Eukaryotic flagella are active structures with a complex architecture of microtubules, motor proteins and elastic links. They are capable of whiplike motions driven by motors sliding along filaments that are themselves constrained at an end. Here, we show that active, self-propelled particles that are connected together to form a single chain that is anchored at one end can produce the graceful beating motions of flagella. We use a combination of numerical simulations, scaling analysis and mean field continuum elastic theory to demarcate the phase diagram for this type of oscillatory motion as a function of the filament length, passive elasticity, propulsion force and longitudinal persistence of propulsion directions. Depending on the nature of the anchoring, we show that filament either undergoes flagella-like beating or assumes a steadily rotating coiled conformation. Our system is simpler than its biological inspiration, and thus could be experimentally realized using a variety of self-propelled particles.

cond-mat.soft↗

arXiv2012

Scale and Nature of Sulcification Patterns

Sulci are surface folds commonly seen in strained soft elastomers and form via a strongly subsubcriticalcritical, yet scale-free instability. Treating the threshold for nonlinear instability as a nonlinear critical point, we explain the nature of sulcus patterns in terms of the scale and translation symmetries which are broken by the formation of an isolated, small sulcus. Our perturbative theory and simulations show that sulcus formation in a thick, compressed slab can arise either as a supercritical or as a weakly subcritical bifurcation relative to this nonlinear critical point, depending on the boundary conditions. An infinite number of competing, equilibrium patterns simultaneously emerge at this critical point, but the one selected has the lowest energy. We give a simple, physical explanation for the formation of these sulcification patterns using an analogy to a solid-solid phase transition with a finite energy of transformation.

cond-mat.soft↗