arXiv subjects

Douglas P. Holmes

Publications and source records attributed to Douglas P. Holmes.

At least 19 recordsLinked to original sources

arXiv2024

The Collective Snapping of a Pair of Bumping Buckled Beams

When a pair of parallel buckling beams of unequal width make lateral contact under increasing compression, eventually either the thin or the thick beam will snap, leading to collective motion of the beam pair. Using experiments and FEM simulations, we find that the distance $D$ between the beams selects which beam snaps first, and that the critical distance $D^*$ scales linear with the combined width of the two beams. To understand this behavior, we show that the collective motion of the beams is governed by a pitchfork bifurcation that occurs at strains just below snapping. Specifically, we use a model of two coupled Bellini trusses to find a closed form expression for the location of this pitchfork bifurcation that captures the linear scaling of $D^*$ with beam width. Our work uncovers a novel elastic instability that combines buckling, snapping and contact nonlinearities. This instability underlies the packing of parallel confined beams, and can be leveraged in advanced metamaterials.

cond-mat.soft↗

arXiv2023

Wrinkling and developable cones in centrally confined sheets

Thin sheets respond to confinement by smoothly wrinkling, or by focusing stress into small, sharp regions. From engineering to biology, geology, textiles, and art, thin sheets are packed and confined in a wide variety of ways, and yet fundamental questions remain about how stresses focus and patterns form in these structures. Using experiments and molecular dynamics (MD) simulations, we probe the confinement response of circular sheets, flattened in their central region and quasi-statically drawn through a ring. Wrinkles develop in the outer, free region, then are replaced by a truncated cone, which forms in an abrupt transition to stress focusing. We explore how the force associated with this event, and the number of wrinkles, depend on geometry. Additional cones sequentially pattern the sheet, until axisymmetry is recovered in most geometries. The cone size is sensitive to in-plane geometry. We uncover a coarse-grained description of this geometric dependence, which diverges depending on the proximity to the asymptotic d-cone limit, where the clamp size approaches zero. This work contributes to the characterization of general confinement of thin sheets, while broadening the understanding of the d-cone, a fundamental element of stress focusing, as it appears in realistic settings.

cond-mat.soft↗

arXiv2023

Germain Curvature: The Case for Naming the Mean Curvature of a Surface after Sophie Germain

How do we characterize the shape of a surface? It is now well understood that the shape of a surface is determined by measuring how curved it is at each point. From these measurements, one can identify the directions of largest and smallest curvature, i.e. the principal curvatures, and construct two surface invariants by taking the average and the product of the principal curvatures. The product of the principal curvatures describes the intrinsic curvature of a surface, and has profound importance in differential geometry - evidenced by the Gauss's Theorema Egregium and the Gauss-Bonnet theorem. This curvature is commonly referred to as the Gaussian Curvature after Carl Friedrich Gauss, following his significant contributions to the emerging field of differential geometry in his 1828 work. The average, or mean curvature, is an extrinsic measure of the shape of a surface - that is, the shape must be embedded in a higher dimensional space to be measured. Beginning in 1811, and culminating in efforts in 1821 and 1826, the mathematician Sophie Germain identified the mean curvature as the appropriate measure for describing the shape of vibrating plates. Her hypothesis leads directly to the equations describing the behavior of thin, elastic plates. In letters to Gauss, she described her notion of a "sphere of mean curvature" that can be identified at each point on the surface. This contribution stimulated a period of rapid development in elasticity and geometry, and yet Germain has not yet received due credit for deducing the mathematical and physical significance of mean curvature. It is clear from the primary source evidence that this measure of mean curvature should bear the name of Sophie Germain.

math.HO↗

arXiv2022

Self-Ordering of Buckling, Bending, Bumping Beams

A collection of thin structures buckle, bend, and bump into each-other when confined. This contact can lead to the formation of patterns: hair will self-organize in curls; DNA strands will layer into cell nuclei; paper, when crumpled, will fold in on itself, forming a maze of interleaved sheets. This pattern formation changes how densely the structures can pack, as well as the mechanical properties of the system. How and when these patterns form, as well as the force required to pack these structures is not currently understood. Here we study the emergence of order in a canonical example of packing in slender-structures, i.e. a system of parallel growing elastic beams. Using experiments, simulations, and simple theory from statistical mechanics, we predict the amount of growth (or, equivalently, the amount of compression) of the beams that will guarantee a global system order, which depends only on the initial geometry of the system. Furthermore, we find that the compressive stiffness and stored bending energy of this meta-material is directly proportional to the number of beams that are geometrically frustrated at any given point. We expect these results to elucidate the mechanisms leading to pattern formation in these kinds of systems, and to provide a new mechanical meta-material, with a tunable resistance to compressive force.

cond-mat.soft↗

arXiv2021

Emergence of Structure in Columns of Grains and Elastic Loops

It is possible to build free-standing, load-bearing structures using only rocks and loops of elastic material. We investigate how these structures emerge, and find that the necessary maximum loop spacing (the critical spacing) is a function of the frictional properties of the grains and the elasticity of the confining material. We derive a model to understand both of these relationships, which depends on a simplification of the behavior of the grains at the edge of a structure. We find that higher friction leads to larger stable grain-grain and grain-loop contact angles resulting in a simple function for the frictional critical spacing, which depends linearly on friction to first order. On the other hand, a higher bending rigidity enables the loops to better contain the hydrostatic pressure of the grains, which we understand using a hydroelastic scale. These findings will illuminate the stabilization of dirt by plant roots, and potentially enable the construction of simple adhesion-less structures using only granular material and fiber.

cond-mat.soft↗

arXiv2021

Efficient snap-through of spherical caps by applying a localized curvature stimulus

In bistable actuators and other engineered devices, a homogeneous stimulus (e.g. mechanical, chemical, thermal, or magnetic) is often applied to an entire shell to initiate a snap-through instability. In this work, we demonstrate that restricting the active area to the shell boundary allows for a large reduction in its size, thereby decreasing the energy input required to actuate the shell. To do so, we combine theory with 1D finite element simulations of spherical caps with a non-homogeneous distribution of stimulus-responsive material. We rely on the effective curvature stimulus, i.e. the natural curvature induced by the non-mechanical stimulus, which ensures that our results are entirely stimulus-agnostic. To validate our numerics and demonstrate this generality, we also perform two sets of experiments, wherein we use residual swelling of bilayer silicone elastomers--a process that mimics differential growth--as well as a magneto-elastomer to induce curvatures that cause snap-through. Our results elucidate the underlying mechanics, offering an intuitive route to optimal design for efficient snap-through.

cond-mat.soft↗

arXiv2021

Delayed buckling of spherical shells due to viscoelastic knockdown of the critical load

We performed dynamic pressure buckling experiments on defect-seeded spherical shells made of a common silicone elastomer. Unlike in quasi-static experiments, shells buckled at ostensibly subcritical pressures (i.e. below the experimentally-determined load at which buckling occurs elastically), often following a significant time delay. While emphasizing the close connections to elastic shell buckling, we rely on viscoelasticity to explain our observations. In particular, we demonstrate that the lower critical load may be determined from the material properties, which is rationalized by a simple analogy to elastic spherical shell buckling. We then introduce a model centered on empirical quantities to show that viscoelastic creep deformation lowers the critical load in the same predictable, quantifiable way that a growing defect would in an elastic shell. This allows us to capture how both the critical deflection and the delay time depend on the applied pressure, material properties, and defect geometry. These quantities are straightforward to measure in experiments. Thus, our work not only provides intuition for viscoelastic behavior from an elastic shell buckling perspective, but also offers an accessible pathway to introduce tunable, time-controlled actuation to existing mechanical actuators, e.g. pneumatic grippers.

cond-mat.soft↗

arXiv2019

Packing Transitions in the Elastogranular Confinement of a Slender Loop

Confined thin structures are ubiquitous in nature. Spatial and length constraints have led to a number of novel packing strategies at both the micro-scale, as when DNA packages inside a capsid, and the macro-scale, seen in plant root development and the arrangement of the human intestinal tract. By varying the arc length of an elastic loop injected into an array of monodisperse, soft, spherical grains at varying initial number density, we investigate the resulting packing behaviors between a growing slender structure constrained by deformable boundaries. At low initial packing fractions, the elastic loop deforms as though it were hitting a flat surface by periodically folding into the array. Above a critical packing fraction $ϕ_c$, local re-orientations within the granular medium create an effectively curved surface leading to the emergence of a distinct circular packing morphology in the adjacent elastic structure. These results will bring new insight into the packing behavior of wires and thin sheets and will be relevant to modeling plant root morphogenesis, burrowing and locomotive strategies of vertebrates & invertebrates, and developing smart, steerable needles.

cond-mat.soft↗

arXiv2019

Elastogranularity in Binary Granular Mixtures

Frustration arises for a broad class of physical systems where confinement (geometric) or the presence of a perturbation (kinematic) prevents equilibration to a minimum energy state. By varying the diameter ratio and packing fraction in granular arrays surrounding a slowly elongating elastica, we characterize the resulting elastogranular interactions taking place in a transitional, amorphous medium. For low number density packings prepared with moderate to large bidispersity, we find the critical injected arclength to elicit jamming follows the same scaling law observed in monodisperse arrays. Beyond the jamming point, the elastica is seen to relax its bending energy within packings with progressively larger diameter ratios towards the shape expected when deforming within more fluid-like media.

cond-mat.soft↗

arXiv2018

Multistable Kirigami for Tunable Architected Materials

In nature, materials such as ferroelastics and multiferroics can switch their microstructure in response to external stimuli, and this reconfiguration causes a simultaneous modulation of its material properties. Rapid prototyping technologies have enabled origami and kirigami-inspired architected materials to provide a means for designing shape-shifting structures, and here we show how multistable structures inspired by kirigami provide novel design criteria for preparing mechanical metamaterials with tunable properties. By changing the geometry of kirigami unit cells, we obtain multistable kirigami lattice structures endowed with a bistable snap-through mechanism. We demonstrate the precise control of material stiffness, along with the ability to tune this property in situ by locally and reversibly switching the unit cell configurations. We anticipate these mechanical metamaterials will provide a platform to achieve in situ tunable electrical, optical, and mechanical properties for a variety of applications in multifunctional materials, two-dimensional materials, and soft robotics.

physics.app-ph↗

arXiv2018

Buckling of geometrically confined shells

We study the periodic buckling patterns that emerge when elastic shells are subjected to geometric confinement. Residual swelling provides access to range of shapes (saddles, rolled sheets, cylinders, and spherical sections) which vary in their extrinsic and intrinsic curvatures. Our experimental and numerical data show that when these structures are radially confined, a single geometric parameter -- the ratio of the total shell radius to the amount of unconstrained material -- predicts the number of lobes formed. We then generalize our model to account for a relaxation of the in-plane constraint by interpreting this parameter as an effective foundation stiffness. Experimentally, we show that reducing the transverse confinement of saddles causes the lobe number to decrease according to our effective stiffness model. Hence, one geometric parameter captures the wave number through a wide range of radial and transverse confinement, connecting the shell shape to the shape of the boundary that confines it. We expect these results to be relevant for an expanse of shell shapes, and thus apply to the design of shape--shifting materials and the swelling and growth of soft structures.

cond-mat.soft↗

arXiv2018

The nonlinear buckling behavior of a complete spherical shell under uniform external pressure and homogenous natural curvature

In this work, we consider the stability of a spherical shell under combined loading from a uniform external pressure and a homogenous natural curvature. Non-mechanical stimuli, such as one that tends to modify the rest curvature of an elastic body, are prevalent in a wide range of natural and engineered systems, and may occur due to thermal expansion, changes in pH, differential swelling, and differential growth. Here, we investigate how the presence of both an evolving natural curvature and an external pressure modifies the stability of a complete spherical shell. We show that due to a mechanical analogy between pressure and curvature, positive natural curvatures can severely destabilize a thin shell, while negative natural curvatures can strengthen the shell against buckling, providing the possibility to design shells that buckle at or above the theoretical limit for pressure alone, i.e. a knock-up factor. These results extend directly from the classical analysis of the stability of shells under pressure, and highlight the important role that non-mechanical stimuli can have on modifying the the membrane state of stress in a thin shell.

cond-mat.soft↗

arXiv2018

Elasticity and Stability of Shape Changing Structures

As we enter the age of designer matter - where objects can morph and change shape on command - what tools do we need to create shape-shifting structures? At the heart of an elastic deformation is the combination of dilation and distortion, or stretching and bending. The competition between the latter can cause elastic instabilities, and over the last fifteen years these instabilities have provided a multitude of ways to prescribe and control shape change. Buckling, wrinkling, folding, creasing, snapping have become mechanisms that when harmoniously combined enable mechanical metamaterials, self-folding origami, ultralight and ultrathin kirigami, and structures that appear to grow from one shape to another. In this review, I aim to connect the fundamentals of elastic instabilities to the advanced functionality currently found within mechanical metamaterials.

cond-mat.soft↗

arXiv2018

Static bistability of spherical caps

Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two `self-equilibrated' states, one natural and the other inside out (or `everted'). Though this is familiar from everyday life -- an umbrella is remarkably stable, yet a contact lens can be easily turned inside out -- the precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite element simulations to determine the threshold between bistability and monostability for shells of different solid angle. We compare these results with the prediction from shallow shell theory, showing that, when appropriately modified, this offers a very good account of bistability even for relatively deep shells. We then investigate the robustness of this bistability against pointwise indentation. We find that indentation provides a continuous route for transition between the two states for shells whose geometry makes them close to the threshold. However, for thinner shells, indentation leads to asymmetrical buckling before snap-through, while also making these shells more `robust' to snap-through. Our work sheds new light on the robustness of the `mirror buckling' symmetry of spherical shell caps.

cond-mat.soft↗

arXiv2018

Snapping of Bistable, Prestressed Cylindrical Shells

Bistable shells can reversibly change between two stable configurations with very little energetic input. Understanding what governs the shape and snap-through criteria of these structures is crucial for designing devices that utilize instability for functionality. Bistable cylindrical shells fabricated by stretching and bonding multiple layers of elastic plates will contain residual stress that will impact the shell's shape and the magnitude of stimulus necessary to induce snapping. Using the framework of non-Euclidean shell theory, we first predict the mean curvature of a nearly cylindrical shell formed by arbitrarily prestretching one layer of a bilayer plate with respect to another. Then, beginning with a residually stressed cylinder, we determine the amount of the stimuli needed to trigger the snapping between two configurations through a combination of numerical simulations and theory. We demonstrate the role of prestress on the snap-through criteria, and highlight the important role that the Gaussian curvature in the boundary layer of the shell plays in dictating shell stability.

cond-mat.soft↗

arXiv2018

Curvature-Induced Instabilities of Shells

Induced by proteins within the cell membrane or by differential growth, heating, or swelling, spontaneous curvatures can drastically affect the morphology of thin bodies and induce mechanical instabilities. Yet, the interaction of spontaneous curvature and geometric frustration in curved shells remains still poorly understood. Via a combination of precision experiments on elastomeric spherical bilayer shells, simulations, and theory, we show a spontaneous curvature-induced rotational symmetry-breaking as well as a snapping instability reminiscent of the Venus fly trap closure mechanism. The instabilities and their dependence on geometry are rationalized by reducing the spontaneous curvature to an effective mechanical load. This formulation reveals a combined pressurelike bulk term and a torquelike boundary term, allowing scaling predictions for the instabilities in excellent agreement with experiments and simulations. Moreover, the effective pressure analogy suggests a curvature-induced buckling in closed shells. We determine the critical buckling curvature via a linear stability analysis that accounts for the combination of residual membrane and bending stresses. The prominent role of geometry in our findings suggests the applicability of the results over a wide range of scales.

cond-mat.soft↗

arXiv2017

Kirigami Actuators

Thin elastic sheets bend easily and, if they are patterned with cuts, can deform in sophisticated ways. Here we show that carefully tuning the location and arrangement of cuts within thin sheets enables the design of mechanical actuators that scale down to atomically-thin 2D materials. We first show that by understanding the mechanics of a single, non-propagating crack in a sheet we can generate four fundamental forms of linear actuation: roll, pitch, yaw, and lift. Our analytical model shows that these deformations are only weakly dependent on thickness, which we confirm with experiments at centimeter scale objects and molecular dynamics simulations of graphene and MoS$_{2}$ nanoscale sheets. We show how the interactions between non-propagating cracks can enable either lift or rotation, and we use a combination of experiments, theory, continuum computational analysis, and molecular dynamics simulations to provide mechanistic insights into the geometric and topological design of kirigami actuators.

cond-mat.soft↗

arXiv2017

Elastogranular Mechanics: Buckling, Jamming, and Structure Formation

Confinement of a slender body into a granular array induces stress localization in the geometrically nonlinear structure, and jamming, reordering, and vertical dislocation of the surrounding granular medium. By varying the initial packing density of grains and the length of a confined elastica, we identify the critical length necessary to induce jamming, and demonstrate an intricate coupling between folds that localize along grain boundaries. Above the jamming threshold, the characteristic length of elastica deformation is shown to scale with the length over which force field fluctuations propagate in a jammed state, suggesting the ordering of the granular array governs the deformation of the slender structure. However, over confinement of the elastica will induce a form of stress relaxation in the granular medium by dislocating grains through two distinct mechanisms that depend on the geometry of the confined structure.

cond-mat.soft↗