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The chemical vapor deposition of polycrystalline diamond (PCD) films is typically done on substrates seeded with diamond nanoparticles. Specular laser reflectance and a continuous film model have been used to monitor the thickness of these films during their deposition. However, most seeds are isolated during the early stages of the deposition, which questions the utility of applying such a continuous film model for monitoring deposition before film formation. In this work, we present a model based on the Rayleigh theory of scattering for laser reflectance at substrates with growing nanodiamonds to capture the early stages of PCD deposition. The reflectance behavior predicted by our model differs from that of a continuous film, which is well-described by the continuous film model. This difference enlarges as the seed density used in our model decreases. We verify this trend experimentally by depositing diamond under identical conditions on substrates with various seed densities. A relation derived from our model is used to fit reflectance data from which seed densities are obtained that are proportional to those found with electron microscopy. We also show that relying on the continuous film model for describing the early stages of deposition can result in falsely deducing the existence of incubation, and that the continuous film model can be used safely beyond the early stages of deposition. Based on these findings, we delineate a robust method for obtaining growth rates and incubation periods from reflectance measurements. This work may also advance the general understanding of nanoparticle growth and formation., 32 pages, 9 figures, 1 table

Dilational materials, for which the angles between pairs of material fibers are preserved under deformations, are an important class of metamaterials. Although these materials are typically made by assembling discrete elemental building blocks in repeating patterns, continuum mechanics provides a powerful tool for exploring their macroscopic properties and response. We present an analysis of the constraint, the constitutive relation, and the equilibrium equations for homogeneous and isotropic dilational elastic material surfaces. We also describe the possibility of penalizing deviations from local area preservation to yield a framework for approximating isometric deformations of unstretchable elastic material surfaces.

We develop a constrained theory for constituent migration in bodies with microstructure described by a scalar phase field. The distinguishing features of the theory stem from a systematic treatment and characterization of the reactions needed to maintain the internal constraint given by the coincidence of the mass fraction and the phase field. We also develop boundary conditions for situations in which the interface between the body and its environment is structureless and cannot support constituent transport. In addition to yielding a new derivation of the Cahn–Hilliard equation, the theory affords an interpretation of that equation as a limiting variant of an Allen–Cahn type diffusion system arising from the unconstrained theory obtained by considering the mass fraction and the phase field as independent quantities. We corroborate that interpretation with three-dimensional numerical simulations of a recently proposed benchmark problem.

The difference between the differential geometric concept of an isometry between two given surfaces as purely mathematical objects and the kinematical concept of an isometric deformation of a single unstretchable material surface as a physical object is discussed. We clarify some misunderstandings that have been promoted in recent works concerning the mechanics of unstretchable material surfaces and we discuss this issue within the context of two specific examples. A revealing distinction between isometries and isometric deformations in two space dimensions is reviewed, and the use of rectifying developable surfaces to characterize the isometric deformation of rectangular material strips is analyzed.

Polymer-nanodiamond composites are excellent candidates for the fabrication of multifunctional hybrid materials. They integrate polymer flexibility and exceptional properties of nanodiamonds (NDs), such as biocompatibility, mechanical strength, color centers, and chemically-tailored surfaces. However, their development is hindered by the challenge of ensuring that NDs are homogeneously distributed in the composites. Here, we exploit colloidal coassembly between poly(isoprene-b-styrene-b-2-vinyl pyridine) (ISV) block copolymers (BCPs) and NDs to avoid ND self-agglomeration and direct ND spatial distribution. NDs were first air oxidized at 450 degrees C to obtain stable dispersions in dimethylacetamide (DMAc). By adding ISV into the dispersions, patchy hybrid micelles were formed due to H-bonds between NDs and ISV. The ISV-ND coassembly in DMAc was then used to fabricate nanocomposite films with a uniform sub-50 nm ND distribution, which has never been previously reported for an ND loading [Formula: see text] of more than 50 wt%. The films exhibit good transparency due to their well-defined nanostructures and smoothness and also exhibit an improved UV-absorption and hydrophilicity compared to neat ISV. More intriguingly, at a [Formula: see text] of 22 wt%, ISV and NDs coassemble into a network-like superstructure with well-aligned ND strings via a dialysis method. Transmission electron microscopy and dynamic light scattering measurements suggest a complex interplay between polymer-polymer, polymer-solvent, polymer-ND, ND-solvent, and ND-ND interactions during the formation of structures. Our work may provide an important foundation for the development of hierarchically ordered nanocomposites based on BCP-ND coassembly, which is beneficial for a wide spectrum of applications from biotechnology to quantum devices.

A prevalent strategy for synthesizing patchy nanoparticles is through the self-assembly of triblock terpolymers in selective solvents. Since the thermodynamic and kinetic factors that govern the morphology of the particles produced in this way are not fully understood, this strategy usually demands trial-and-error methodologies. We investigate the fundamental mechanisms that produce multiple types of patchy nanoparticles and identify the conditions needed to program the shapes of the nanoparticles and predict their assembly. Our findings demonstrate that particle morphology can be described in a generic fashion by accounting for the energetic balance between the conformation of the polymer coils and the formation of interfaces. This allows us to forecast the synthesis of patchy nanoparticles for systems with different triblock terpolymers and solvents. Since the shape, size, and distribution of the patches influence the growth of larger microscale structures, we construct a library of elemental nanoparticles, or building blocks, suitable for the study of hierarchically larger self-assembled aggregates and useful for streamlining the design of functional materials. Our results provide new insights into the intriguing mechanisms that determine the morphology of soft nanoscale objects, whether synthetic or naturally occurring.

A general framework is developed for finding the equations describing the equilibrium of an inextensional material surface with arbitrary flat reference shape that is deformed by applying tractions or moments to its edge. This is facilitated by using a representation of all isometric deformations of the material surface to convert the bending energy of the material surface to a line integral over the edge of the material surface. Euler–Lagrange equations are derived, leading to a complete and definitive set of equilibrium equations, which are a system of ordinary differential equations for the spatial directrix. Jump conditions that apply at points where the tangent and/or curvature of the edge may be discontinuous are also derived. As a simple but illustrative example, the deformation of a rectangular strip subject to various edge conditions is studied.

Kinematic chains made from links whose lengths and twist angles satisfy a certain proportionality rule are considered. A notion of exchange symmetry for the links of any such chain affords a new understanding of the mobility of well-known overconstrained mechanisms, exemplified by Bennett’s four-bar linkage, and provides a unifying framework for explaining the mobility of a recently discovered class of nontrivial underconstrained mechanisms. These findings allow for the design of mechanisms consisting of links that can be arbitrarily reordered without altering their number of internal degrees of freedom and, thus, of modular mechanical systems that can be assembled in different ways to serve different purposes. A nine-hinged linkage that can be assembled in ninety-four distinct ways, each evincing a single internal degree of freedom, is presented as an example of such a modular system.

Significance Linkages are the basic functional elements of any machine. Known established linkages with a single degree of freedom, which facilitates control, have so far consisted of six or fewer links. We introduce “Möbius kaleidocycles,” a class of single-degree of freedom ring linkages containing nontrivial linkages having less mobility than expected. Möbius kaleidocycles consist of arbitrarily many (but at least seven) identical hinge-joined links and may serve as building blocks in deployable structures, robotics, or chemistry. These linkages are chiral and have a nonorientable topology equivalent to 3π-twist Möbius bands. Other than technological promise, Möbius kaleidocycles pose a myriad of intriguing questions in mechanical engineering, physics, and various areas of mathematics, especially topology., Linkages are assemblies of rigid bodies connected through joints. They serve as the basis for force- and movement-managing devices ranging from ordinary pliers to high-precision robotic arms. Aside from planar mechanisms, like the well-known four-bar linkage, only a few linkages with a single internal degree of freedom—meaning that they can change shape in only one way and may thus be easily controlled—have been known to date. Here, we present “Möbius kaleidocycles,” a previously undiscovered class of single-internal degree of freedom ring linkages containing nontrivial examples of spatially underconstrained mechanisms. A Möbius kaleidocycle is made from seven or more identical links joined by revolute hinges. These links dictate a specific twist angle between neighboring hinges, and the hinge orientations induce a nonorientable topology equivalent to the topology of a 3π-twist Möbius band. Apart from having many technological applications, including perhaps the design of organic ring molecules with peculiar electronic properties, Möbius kaleidocycles raise fundamental questions about geometry, topology, and the limitations of mobility for closed loop linkages.

Recently, the rotational diffusivity of the coronavirus particle in suspension was calculated, from first principles, using general rigid bead-rod theory [M. A. Kanso, Phys. Fluids 32, 113101 (2020)]. We did so by beading the capsid and then also by replacing each of its bulbous spikes with a single bead. However, each coronavirus spike is a glycoprotein trimer, and each spike bulb is triangular. In this work, we replace each bulbous coronavirus spike with a bead triplet, where each bead of the triplet is charged identically. This paper, thus, explores the role of bulb triangularity on the rotational diffusivity, an effect not previously considered. We thus use energy minimization for the spreading of triangular bulbs over the spherical capsid. The latter both translates and twists the coronavirus spikes relative to one another, and we then next arrive at the rotational diffusivity of the coronavirus particle in suspension, from first principles. We learn that the triangularity of the coronavirus spike bulb decreases its rotational diffusivity. For a typical peplomer population of 74, bulb triangularity decreases the rotational diffusivity by 39 %.

Methane, oxygen and hydrogen in relative concentrations of 1:0.5:98.5 were employed to deposit diamond films by microwave plasma enhanced chemical vapor deposition on Si(001) substrates seeded with detonation nanodiamonds of density. The interaction between the nanodiamonds and the plasma was investigated as a function of the chamber pressure and the substrate temperature. At 40 Torr and 935 °C, 90% of the seeds were etched, and growth induction occurred with an impractically long time of 23.5 h. Varying the substrate temperature between 827 °C and 935 °C did not lead to faster growth induction. Reducing the chamber pressure to 30 Torr instead resulted in continuous films with induction time as short as 12 min. The induction time and the growth rate exhibited an Arrhenius dependence in the temperature range 769–884 °C with apparent activation energies of 3.7 eV/atom and 0.3 eV/atom, respectively. When at 30 Torr the concentration of the oxygen additive was raised to 1 vol%, the seeds were completely dissolved and no film deposition took place. However, film growth was possible with an oxygen admixture as high as 1.5 vol% on a diamond precursor layer of approximately 55 nm thickness, which was nucleated in an oxygen-free plasma.

Two-dimensional simulations are used to explore topological transitions that occur during the formation of films grown from grains that are seeded on substrates. This is done for a relatively large range of the initial value $\Phi_s$ of the grain surface fraction $\Phi$. The morphology of porous films is captured at the transition when grains connect to form a one-component network using newly developed raster-free algorithms that combine computational geometry and network theory. Further insight on the morphology of porous films and their suspended counterparts is obtained by studying the pore surface fraction $\Phi_p$, the pore over grain ratio, the pore area distribution, and the contribution of pores of certain chosen areas to $\Phi_p$. Pinhole survival is evaluated at the transition when film closure occurs using survival function estimates. The morphology of closed films ($\Phi = 1$) is also characterized and is quantified by measuring grain areas and perimeters. The majority of investigated quantities are found to depend sensitively on $\Phi_s$ and the long-time persistence of pinholes exhibits critical behavior as a function of $\Phi_s$. In addition to providing guidelines for designing effective processes for manufacturing thin films and suspended porous films with tailored properties, this work may advance the understanding of continuum percolation theory.

Using general rigid bead–rod theory, we explore the effect of twisting a macromolecule on its rheological properties in suspensions. We thus focus on macromolecules having the form of Möbius bands so that the number of twists can be incremented. We call these Möbius macromolecules. When represented in general rigid bead–rod theory, these macromolecules comprise beads whose centers all fall on a Möbius band. From first principles, we calculate the complex viscosity of twisted rings with zero to seven twists. We find that the zero-shear values of the viscosity and first normal stress coefficient increase with twisting. Furthermore, we find that the real part of the complex viscosity descends more rapidly, with frequency, with extent of twist. For the imaginary part of the complex viscosity, the more twisted, the higher the peak. For each part of the dimensionless complex viscosity and the first normal stress coefficient, the results fall on one of just three curves corresponding to zero, even, or odd numbers of twists. We also explore the effects of the length and the aspect ratio of twisted macromolecular suspensions. We close with a worked example for a suspension of triply twisted Möbius annulene.

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

In this letter, we demonstrate a relation between the boundary curvature $\kappa$ and the wrinkle wavelength $\lambda$ of a thin suspended film under boundary confinement. Experiments are done with nanocrystalline diamond films of thickness $t \approx 184$~nm grown on glass substrates. By removing portions of the substrate after growth, suspended films with circular boundaries of radius $R$ ranging from approximately 30 to 811 $\mu$m are made. Due to residual stresses, the portions of film attached to the substrate are of compressive prestrain $\epsilon_0 \approx 11 \times 10^{-4}$ and the suspended portions of film are azimuthally wrinkled at their boundary. We find that $\lambda$ monotonically decreases with $\kappa$ and present a model predicting that $\lambda \propto t^{1/2}(\epsilon_0 + \Delta R \kappa)^{-1/4}$, where $\Delta R$ denotes a penetration depth over which strain relaxes at a boundary. This relation is in agreement with our experiments and may be adapted to other systems such as plant leaves. Also, we establish a novel method for measuring residual compressive strain in thin films.

Meccanica : international journal of the Italian Association of Theoretical and Applied Mechanics, AIMETA 55(10), 1853-1868 (2020). doi:10.1007/s11012-020-01228-9, Published by Springer Science + Business Media B.V, Dordrecht [u.a.]

The study of material surfaces uses notions from classical differential geometry, such as the covariant gradient, the mean and Gaussian curvatures, and the Peterson–Mainardi–Codazzi and Gauss equations. These notions are traditionally introduced relative to local surface coordinates and involve Christoffel symbols. We proceed instead without recourse to coordinates using direct notation. After developing the formula for the covariant gradient relative to a surface metric, we derive versions of the Peterson–Mainardi–Codazzi and Gauss equations and Gauss’ Theorema Egregium relevant to a deformed material surface. We then apply our framework to kinematically constrained material surfaces. For material surfaces that can sustain only deformations that preserve either angles or lengths, we obtain explicit representations for the covariant gradient relative to the surface metric in terms of the surface gradient. We show also that a deformation of a material surface that preserves angles and areas must be length preserving and vice versa. Finally, we present an alternative derivation of the Peterson–Mainardi–Codazzi and Gauss equations for a deformed material surface subject to the provision that the surface metric derives from the metric for the ambient Euclidean space within which the surface is embedded. An Appendix involving coordinates is included to ease comparisons between our approach to covariant differentiation and associated derivations of the Peterson–Mainardi–Codazzi and Gauss equations and standard coordinate-based approaches.

The stored energy of an unstretchable material surface is assumed to depend only upon the curvature tensor. By control of its edge(s), the surface is deformed isometrically from its planar undistorted reference configuration into an equilibrium shape. That shape is to be determined from a suitably constrained variational problem as a state of relative minimal potential energy. We pose the variational problem as one of relative minimum potential energy in a spatial form, wherein the deformation of a flat, undistorted region D in E 2 to its distorted form S in E 3 is assumed specified. We then apply the principle that the first variation of the potential energy, expressed as a functional over S ∪ ∂ S , must vanish for all admissible variations that correspond to isometric deformations from the distorted configuration S and that also contain the essence of flatness that characterizes the reference configuration D , but is not covered by the single statement that the variation of S correspond to an isometric deformation. We emphasize the commonly overlooked condition that the spatial expression of the variational problem requires an additional variational constraint of zero Gaussian curvature to ensure that variations from S that are isometric deformations also contain the notion of flatness. In this context, it is particularly revealing to observe that the two constraints produce distinct, but essential and complementary, conditions on the first variation of S . The resulting first variation integral condition, together with the constraints, may be applied, for example, to the case of a flat, undistorted, rectangular strip D that is deformed isometrically into a closed ring S by connecting its short edges and specifying that its long edges are free of loading and, therefore, subject to zero traction and couple traction. The elementary example of a closed ring without twist as a state of relative minimum potential energy is discussed in detail, and the bending of the strip by opposing specific bending moments on its short edges is treated as a particular case. Finally, the constrained variational problem, with the introduction of appropriate constraint reactions as Lagrangian multipliers to account for the requirements that the deformation from D to S is isometric and that D is flat, is formulated in the spatial form, and the associated Euler–Lagrange equations are derived. We then solve the Euler–Lagrange equations for two representative problems in which a planar undistorted rectangular material strip is isometrically deformed by applied edge tractions and couple tractions (i.e., specific edge moments) into (i) a bent and twisted circular cylindrical helical state, and (ii) a state conformal with the surface of a right circular conical form.

In this work, we make a distinction between the differential geometric notion of an isometry relationship among two dimensional surfaces embedded in three-dimensional point space and the continuum mechanical notion of an isometric deformation of a two dimensional material surface. We illustrate the importance of separating the abstract theory of surfaces in differential geometry and their related differential geometric features from the physical notion of a material surface which is subject to a deformation from a given reference configuration. In differential geometry, while two surfaces may be isometric, the mapping between them that characterizes the isometry is simply a mapping between the points of the surfaces and not necessarily between corresponding material particles of a single deformed material surface. We review two equivalent characterizations of a smooth isometric deformation of a flat material surface into a curved surface, and emphasize the requirement that the referential directrix and rulings, and their deformed counterparts, must provide a basis for establishing a complete curvilinear coordinate covering of the material surface in both the reference and deformed states. Because this covering requirement has been overlooked in recent publications concerning the isometric bending of ribbons, we illustrate its importance in properly defining the deformation of a ribbon in the two examples of a flat rectangular material strip that is isometrically deformed into either (i) a portion of a circular cylindrical surface, or (ii) a portion of a circular conical surface. We then show how the accurate calculation of the bending energy in these two examples is influenced by this oversight. In example (i), the curvature along the generators of the deformed surface, generally helical in form, is constant. In this special circumstance overlooking the covering requirement, as has been done in the literature by integrating the specific bending energy, dependent only on the curvature, over a domain on the supporting circular cylindrical surface equal in area, though not equal in geometric form, to that of the deformed ribbon, gives the correct bending energy result. In example (ii), the curvature along a generator of the cone is not constant and the calculation of the bending energy is, indeed, compromised by this oversight. The historically important dimensional reductions that Sadowsky and Wunderlich introduced to study the bending energy and the equilibrium configurations of isometrically deformed rectangular strips have gained classical notoriety within the subject of elastic ribbons and Mobius bands. We show that the Sadowsky and Wunderlich functionals also overlook the covering requirement and that the exact bending energy is underestimated by these functionals, the Sadowsky functional being the lowest. We then show that the error in using these functionals can be great for a rectangular strip of given length $\ell$ and width $w$ , depending on the form of the isometric deformation and the size of the half-length-to-width ratio $w/2\ell$ . The Sadowsky functional is meant to apply to strips for which $w/2\ell$ is sufficiently small, in which case the covering requirement is of little consequence, and for such strips it yields an acceptable approximation of the actual bending energy. In such cases the Wunderlich functional shows only an incremental improvement over the Sadowsky calculation. While the Wunderlich functional is meant to apply accurately for all strips, without restricting the size of $w/2\ell$ , we show that in overlooking the covering requirement it greatly underestimates the actual bending energy for many isometrically deformed ribbons. In particular, we show relative errors between the exact, the Wunderlich, and the Sadowsky calculations of the bending energy as a function of $w/2\ell$ for the case of a rectangular strip which is isometrically deformed into a portion of a right circular conical surface, and we observe that the error in approximating the exact bending energy by the Wunderlich functional for reasonable ratios $w/2\ell$ is large and unacceptable. We then give an example of the isometric deformation of a rectangular strip whose Wunderlich functional predicts zero bending energy but for which the exact bending energy can be as large as one pleases. Finally, contrary to suggestions in the literature, we argue that Kirchhoff rod theory does not generally apply to the study of the isometric deformation of a thin rectangular strip because for this class of problems the through thickness dimension of the strip is assumed to be infinitesimal as compared to its width $w$ . For Kirchhoff rod theory to apply, these dimensions must be comparable.

We present a theory for coupled deformation and liquid permeation in polymer gels, allowing for large strains and rotations in conjunction with sharp interfaces separating regions of high and low polymer volume fraction. The theory is applied to pressure-driven liquid flow through a gel-filled rectangular channel, with the aim of investigating how the elastic and osmotic properties of the gel influence the resulting permeation process. Apart from streamwise flow in the direction of the applied pressure drop, the permeation process may involve transverse flow if the elastic response of the gel allows the normal stress acting on a plane parallel to the channel walls to be affected by the shear stress acting on the same plane. For certain elastic energy densities, including those of neo-Hookean and Mooney type, liquid permeation takes place exclusively in the direction of the applied pressure drop. Otherwise, spanwise permeation driven by the contribution to the normal stress distributed on planes parallel to the channel walls is also present. In contrast to the downstream flow, the transverse flow is transient and is accompanied by changes in composition leading to a nonuniform polymer distribution across the channel and thereby influencing the downstream flow profile. By analyzing the instantaneous response of the gel, we determine the driving force for the transverse flow and show that for strain-stiffening networks the transverse motion of the interstitial liquid is directed inward from the channel walls and that the polymer network motion spreads outward from the center of the channel toward the walls. The opposite occurs for strain-slackening networks. By analyzing the steady-state response, we demonstrate that the applied pressure gradient leads to nonuniform transverse distributions of polymer chains than can be continuous or discontinuous. Discontinuous distributions occur for an effective osmotic response involving attractive interactions and are accompanied by the formation of regions of high and low polymer volume fraction separated by sharp interfaces. Finally, we provide illustrative examples for a gel whose elastic and osmotic responses are modeled by the Gent elastic energy density and the Flory–Huggins mixing energy density, respectively.

We consider the implication of allowing the distortion rate in a fluid to be the sum of two smooth components, both incompatible in the sense that they are not gradients of vector fields. Whereas one of these components embodies a smooth distribution of slippage, the other describing the repair needed to ensure that the distortion rate is the gradient of the velocity field. Our considerations lead to a generalization of the Navier–Stokes equations for an incompressible fluid. A tensor field which characterizes incompatibility is introduced and a fundamental equation for its evolution is proposed. Properties of these equations are exemplified by revisiting the classical problem of pressure-driven flow in a plane, rectangular channel. For a sufficiently large instantaneously applied pressure drop, a novel action due to the diffusivity κ associated with the transport of incompatibility is illustrated in addition to the normal diffusive and dissipative affect of viscosity ν . The steady state of the velocity and incompatibility is governed by a system of two ordinary differential equations which are fully analyzed and the resulting fields are determined and discussed. The transient problem is solved numerically and the graphical results show how these fields are structured in time by diffusion and dissipation in the channel during the transition from rest to steady state. A precursor laminar flow persists until a particular time at which the wall shear stress becomes equal to an a priori given cut-off value that initiates incompatibility at the walls of the channel. This incompatibility diffuses inward and interacts synergistically with the viscous action present in the channel to flatten the velocity profile so as to take on a ‘plug flow-like’ appearance, and to redistribute the viscous dissipation and the dissipation due to incompatibility into boundary layers at the channel walls.

We perform a static analysis of a circular cylinder that forms a barrier between surfactant-laden and surfactant-free portions of a liquid–gas interface. In addition to determining the general implications of the balances for forces and torques, we quantify how the imbalance$\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{a}-\unicode[STIX]{x1D6FE}_{b}$between the uniform surface tension$\unicode[STIX]{x1D6FE}_{a}$of the surfactant-free portion of the interface and the uniform surface tension$\unicode[STIX]{x1D6FE}_{b}$of the surfactant-laden portion of the interface influences the load-bearing capacity of a hydrophobic cylinder. Moreover, we demonstrate that the difference between surface tensions on either side of a cylinder with a cross-section of arbitrary shape induces a horizontal force component$f^{h}$equal to$\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}$in magnitude, when measured per unit length of the cylinder. With an energetic argument, we show that this relation also applies to a rod-like barrier with cross-sections of variable shape. In addition, we apply our analysis to amphiphilic Janus cylinders and we discuss practical implications of our findings for Marangoni propulsion and surface pressure measurements.

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves., Revised version; 25 pages; 7 figures

We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset ${\mathcal{D}}$ of two-dimensional Euclidean point space $\mathbb{E}^{2}$ into a surface ${\mathcal{S}}$ in three-dimensional Euclidean point space $\mathbb{E}^{3}$ . To be isometric, such a deformation must preserve the length of every possible arc of material points on ${\mathcal{D}}$ . Characterizing the curves of zero principal curvature of ${\mathcal{S}}$ is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in $\mathbb{E}^{2}$ , based upon an a priori chosen pre-image form of the curves of zero principal curvature in ${\mathcal{D}}$ , and use that coordinate system to construct the most general isometric deformation of ${\mathcal{D}}$ to a smooth surface ${\mathcal{S}}$ . A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of ${\mathcal{S}}$ are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).

A general monotonicity formula for smooth constrained local extremizers of firstorder integral functionals subject to non-holonomic constraints is established. The result is then applied to recover some known monotonicity formulae and to discover some new monotonicity formulae of potential value.

We describe a method for constructing developable bands with N≥3 half twists. Each band is formed by threading a flat rectangular strip through a scaffold made from identical circular cylinders and smoothly connecting its short ends. The N cylinders in a scaffold are arranged with N-fold rotational symmetry. The number of half twists in a band is equal to the number N of cylinders in its scaffold and each band inherits the symmetry of its scaffold. Each scaffold admits a family of bands of the same length but variable width up to a maximum value determined by the features of the scaffold. Apart from orientable and nonorientable unknots, our method allows for the construction of bands with the topology of torus knots. We detail the geometric properties of the construction, discuss certain fundamental restrictions that must be met to ensure constructability, and calculate the elastic bending energy of each band. The rotational symmetry underlying the construction is essential for obtaining the presented bands, as the general non-symmetric problem is even more complex and has not yet been investigated. The bands and their corresponding scaffolds can be used as structural elements in practical applications, one of which we describe and analyze. The construction serves as a basis for a general framework for building a large variety of scaffolds and the corresponding unstretchable bands. Together, these assemblies can be used in architectural, interior, and machine design. They also open new avenues for the layout of conveyor belts in factories, airports, and other settings.

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies | k | ≤ κ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g = k .

A variational model is used to study the behavior of a flexible but inextensible loop spanned by a liquid film, with the objective of explaining the stability and buckling of flat circular configurations. Loops made from filaments with intrinsic curvature and/or intrinsic twist density are considered, but attention is restricted to filaments with circular cross sections and uniform mechanical properties. Loops made with intrinsic curvature but no intrinsic twist density exhibit in-plane and out-of-plane buckling modes corresponding to stable solution branches that bifurcate from the branch of flat circular solutions and out-of-plane buckling occurs at a lower value of the dimensionless surface tension of the liquid film than does in-plane buckling. Additionally, however, the destabilizing influence of the intrinsic curvature can be countered by increasing the torsional rigidity relative to the flexural rigidity. For a loop with both intrinsic curvature and intrinsic twist density, only one branch of stable solutions bifurcates from the flat circular solution branch, the in-plane and out-of-plane buckling modes are intertwined, and bifurcation occurs at a value of the dimensionless surface tension less than that governing the behavior of loops made from filaments that are intrinsically rectilinear. Moreover, increasing the torsional rigidity relative to the flexural rigidity has no or little stabilizing effect if the loop is either too short or too long and, in contrast to what occurs for loops with only intrinsic curvature, if the intrinsic twist density is sufficiently large then the destabilizing influence of the intrinsic curvature cannot be countered by increasing the torsional rigidity relative to the flexural rigidity, regardless of the length of the loop.

The classical procedure devised by Irving and Kirkwood in 1950 and completed slightly later by Noll produces counterparts of the basic balance laws of standard continuum mechanics starting from an ordinary Hamiltonian description of the dynamics of a system of material points. Post-1980 molecular dynamics simulations of the time evolution of such systems use extended Hamiltonians such as those introduced by Andersen, Nosé, and Parrinello and Rahman. The additional terms present in these extensions affect the statistical properties of the system so as to capture certain target phenomenologies that would otherwise be beyond reach. We here propose a physically consistent application of the Irving–Kirkwood–Noll procedure to the extended Hamiltonian systems of material points. Our procedure produces balance equations at the continuum level featuring non-standard terms because the presence of auxiliary degrees of freedom gives rise to additional fluxes and sources that influence the thermodynamic and transport properties of the continuum model. Being aware of the additional contributions may prove crucial when designing multiscale computational schemes in which information is exchanged between the atomistic and continuum levels.

We present a B-spline based conforming finite-element method of the streamfunction formulation of the stationary one-layer quasi-geostrophic equations for the study of the large scale wind-driven ocean circulation. The method encompasses standard simplifications of these equations, in particular the linear Stommel and Stommel–Munk models. A variational form of the method is developed and its consistency is established. In this formulation Dirichlet boundary conditions are enforced only weakly and stabilization is achieved via Nitsche’s method. Stability parameters are evaluated by solving a local generalized eigenvalue problem on the Dirichlet boundary and by monitoring the condition number of the resulting linear systems. Comparisons of the results from our simulations with previously published results and convergence studies lead us to conclude that our finite-element discretization is accurate.

A representation theorem for a smooth isometric mapping of a flat, connected domain \(\mathcal {D}\) in two-dimensional Euclidean point space \(\mathcal {E}^{2}\) into a surface \(\mathcal {S}\) in three-dimensional Euclidean point space \(\mathcal {E}^{3}\) is presented. The form of the mapping is shown to be necessary and sufficient to describe any such smooth isometry. Importantly, this work is not based upon the hypothesis that the mapped surface is ruled. In general, a mapping from a flat planar domain into a ruled surface is far from being isometric, and the property of being ruled is a partial consequence of our representation theorem.

Cox & Jones recently devised and studied an interesting variant of the classical Plateau problem, a variant in which a helical soap film is confined to a cylindrical tube with circular cross-section. Through experiments, numerics, and some analysis, they found that the length and (inner) radius of the tube strongly influence the equilibrium shape of the confined soap film. In this paper, an area minimization problem associated with determining the shape of the film is formulated and analyzed to determine which surfaces are local minima. The connection between a functional inequality and the associated eigenvalue problem plays an important role in the analysis. For helical films, a more detailed analysis is carried out and stability conditions consistent with the experimental and numerical results of Cox & Jones are obtained.

A phase-field model is used to capture the surfactant-driven formation of fracture patterns in particulate monolayers. The model is intended for the regime of closely-packed systems in which the mechanical response of the monolayer can be approximated as a linearly elastic solid. The model approximates the loss in tensile strength of the monolayer as the surfactant concentration increases through the evolution of a damage field. Initial-boundary value problems are constructed and spatially discretized with finite element approximations to the displacement and surfactant damage fields. A comparison between model-based simulations and existing experimental observations indicates a qualitative match in both the fracture patterns and temporal scaling of the fracture process. The importance of surface tension differences is quantified by means of a dimensionless parameter, revealing thresholds that separate different regimes of fracture. These findings are supported by newly performed experiments that validate the model and demonstrate the strong sensitivity of the fracture pattern to differences in surface tension., 10 pages, 11 figures, and 3 tables