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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946March, 2018Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[An upper bound on the particle-laden dependency of shear stresses at solid-fluid interfaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170332?rss=1
In modern advanced manufacturing processes, such as three-dimensional printing of electronics, fine-scale particles are added to a base fluid yielding a modified fluid. For example, in three-dimensional printing, particle-functionalized inks are created by adding particles to freely flowing solvents forming a mixture, which is then deposited onto a surface, which upon curing yields desirable solid properties, such as thermal conductivity, electrical permittivity and magnetic permeability. However, wear at solid–fluid interfaces within the machinery walls that deliver such particle-laden fluids is typically attributed to the fluid-induced shear stresses, which increase with the volume fraction of added particles. The objective of this work is to develop a rigorous strict upper bound for the tolerable volume fraction of particles that can be added, while remaining below a given stress threshold at a fluid–solid interface. To illustrate the bound’s utility, the expression is applied to a series of classical flow regimes.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0332hwp:master-id:royprsa;rspa.2017.03322018-03-21Research articles47422112017033220170332<![CDATA[Continuum modelling of segregating tridisperse granular chute flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170384?rss=1
Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength _{ij} between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, _{ij}, Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0384hwp:master-id:royprsa;rspa.2017.03842018-03-14Research articles47422112017038420170384<![CDATA[A hydrostatic model of the Wirtz pump]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170533?rss=1
The Wirtz pump is not only an excellent example of alternative technology, using as it does the kinetic energy of a stream to raise a proportion of its water, but its mathematical modelling also poses several intriguing problems. We give some history of the Wirtz pump and describe its operation. Taking a novel dynamical systems approach, we then derive a discrete mathematical model in the form of a mapping that describes its hydrostatic behaviour. Our model enables us to explain several aspects of the behaviour of the pump as well as to design one that gives approximately maximal, and maximally constant, output pressure.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0533hwp:master-id:royprsa;rspa.2017.05332018-03-21Research articles47422112017053320170533<![CDATA[Resilience of riverbed vegetation to uprooting by flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170547?rss=1
Riverine ecosystem biodiversity is largely maintained by ecogeomorphic processes including vegetation renewal via uprooting and recovery times to flow disturbances. Plant roots thus heavily contribute to engineering resilience to perturbation of such ecosystems. We show that vegetation uprooting by flow occurs as a fatigue-like mechanism, which statistically requires a given exposure time to imposed riverbed flow erosion rates before the plant collapses. We formulate a physically based stochastic model for the actual plant rooting depth and the time-to-uprooting, which allows us to define plant resilience to uprooting for generic time-dependent flow erosion dynamics. This theory shows that plant resilience to uprooting depends on the time-to-uprooting and that root mechanical anchoring acts as a process memory stored within the plant–soil system. The model is validated against measured data of time-to-uprooting of Avena sativa seedlings with various root lengths under different flow conditions. This allows for assessing the natural variance of the uprooting-by-flow process and to compute the prediction entropy, which quantifies the relative importance of the deterministic and the random components affecting the process.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0547hwp:master-id:royprsa;rspa.2017.05472018-03-14Research articles47422112017054720170547<![CDATA[Vibration of carbon nanotubes with defects: order reduction methods]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170555?rss=1
Order reduction methods are widely used to reduce computational effort when calculating the impact of defects on the vibrational properties of nearly periodic structures in engineering applications, such as a gas-turbine bladed disc. However, despite obvious similarities these techniques have not yet been adapted for use in analysing atomic structures with inevitable defects. Two order reduction techniques, modal domain analysis and modified modal domain analysis, are successfully used in this paper to examine the changes in vibrational frequencies, mode shapes and mode localization caused by defects in carbon nanotubes. The defects considered are isotope defects and Stone–Wales defects, though the methods described can be extended to other defects.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0555hwp:master-id:royprsa;rspa.2017.05552018-03-14Research articles47422112017055520170555<![CDATA[Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170589?rss=1
A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surface waves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0589hwp:master-id:royprsa;rspa.2017.05892018-03-21Research articles47422112017058920170589<![CDATA[Effects of geometric nonlinearity in an adhered microbeam for measuring the work of adhesion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170594?rss=1
Design against adhesion in microelectromechanical devices is predicated on the ability to quantify this phenomenon in microsystems. Previous research related the work of adhesion for an adhered microbeam to the beam's unadhered length, and as such, interferometric techniques were developed to measure that length. We propose a new vibration-based technique that can be easily implemented with existing atomic force microscopy tools or similar metrology systems. To make such a technique feasible, we analysed a model of the adhered microbeam using the nonlinear beam theory put forth by Woinowsky–Krieger. We found a new relation between the work of adhesion and the unadhered length; this relation is more accurate than the one by Mastrangelo & Hsu (Mastrangelo & Hsu 1993 J. Microelectromech. S., 2, 44–55. (doi:10.1109/84.232594)) which is commonly used. Then, we derived a closed-form approximate relationship between the microbeam's natural frequency and its unadhered length. Results obtained from this analytical formulation are in good agreement with numerical results from three-dimensional nonlinear finite-element analysis.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0594hwp:master-id:royprsa;rspa.2017.05942018-03-07Research articles47422112017059420170594<![CDATA[On the theory of drainage area for regular and non-regular points]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170693?rss=1
The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area was proposed (Gallant & Hutchinson. 2011 Water Resour. Res.47, W05535. (doi:10.1029/2009WR008540)), with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here, we show that such a differential equation can be derived from a continuity equation (Chen et al. 2014 Geomorphology219, 68–86. (doi:10.1016/j.geomorph.2014.04.037)) and extend the theory to critical and singular points both by applying Gauss’s theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines and watershed divides, can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0693hwp:master-id:royprsa;rspa.2017.06932018-03-14Research articles47422112017069320170693<![CDATA[Small nanoparticles, surface geometry and contact forces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170723?rss=1
In this molecular dynamics study, we examine the local surface geometric effects of the normal impact force between two approximately spherical nanoparticles that collide in a vacuum. Three types of surface geometries—(i) crystal facets, (ii) sharp edges, and (iii) amorphous surfaces of small nanoparticles with radii R<10 nm—are considered. The impact forces are compared with their macroscopic counterparts described by nonlinear contact forces based on Hertz contact mechanics. In our simulations, edge and amorphous surface contacts with weak surface energy reveal that the average impact forces are in excellent agreement with the Hertz contact force. On the other hand, facet collisions show a linearly increasing force with increasing compression. Our results suggest that the nearly spherical nanoparticles are likely to enable some nonlinear dynamic phenomena, such as breathers and solitary waves observed in granular materials, both originating from the nonlinear contact force.
]]>2018-03-21T01:44:55-07:00info:doi/10.1098/rspa.2017.0723hwp:master-id:royprsa;rspa.2017.07232018-03-21Research articles47422112017072320170723<![CDATA[Magic angles for fibrous incompressible elastic materials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170728?rss=1
In the analysis of the mechanical behaviour of fibre-reinforced incompressible elastic bodies, there is a special angle of orientation of the fibres which leads to a particular mechanical response. This angle has been called a ‘magic angle’ due to its appearance as if by magic in many different aspects of the mechanics of fibrous solids including several examples in biology. It occurs most commonly not only in structural elements composed of circular cylindrical tubes or cylinders reinforced by helically wound fibres but also in flat thin sheets reinforced by fibres in the plane. The occurrence of such a special angle was classically demonstrated using a simple purely geometric analysis in the context of a lattice composed of a single family of helically wound inextensible fibres. Recently, the magic angle concept has been discussed in the framework of nonlinear hyperelasticity for anisotropic materials with detailed constitutive modelling. Our purpose here is to describe some other contexts in which the magic angle occurs starting from earlier work in a special theory of linear elasticity for inextensible fibres and proceeding to relatively accessible models of hyperelasticity. We discuss the role of the magic angle in the quasi-isotropic mechanical response of fibre-reinforced composites as well as the implications for material instability.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0728hwp:master-id:royprsa;rspa.2017.07282018-03-21Research articles47422112017072820170728<![CDATA[A canonical form of the equation of motion of linear dynamical systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170809?rss=1
The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0809hwp:master-id:royprsa;rspa.2017.08092018-03-07Research articles47422112017080920170809<![CDATA[Post-buckling of a pressured biopolymer spherical shell with the mode interaction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170834?rss=1
Imperfection sensitivity is essential for mechanical behaviour of biopolymer shells characterized by high geometric heterogeneity. The present work studies initial post-buckling and imperfection sensitivity of a pressured biopolymer spherical shell based on non-axisymmetric buckling modes and associated mode interaction. Our results indicate that for biopolymer spherical shells with moderate radius-to-thickness ratio (say, less than 30) and smaller effective bending thickness (say, less than 0.2 times average shell thickness), the imperfection sensitivity predicted based on the axisymmetric mode without the mode interaction is close to the present results based on non-axisymmetric modes with the mode interaction with a small (typically, less than 10%) relative errors. However, for biopolymer spherical shells with larger effective bending thickness, the maximum load an imperfect shell can sustain predicted by the present non-axisymmetric analysis can be significantly (typically, around 30%) lower than those predicted based on the axisymmetric mode without the mode interaction. In such cases, a more accurate non-axisymmetric analysis with the mode interaction, as given in the present work, is required for imperfection sensitivity of pressured buckling of biopolymer spherical shells. Finally, the implications of the present study to two specific types of biopolymer spherical shells (viral capsids and ultrasound contrast agents) are discussed.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0834hwp:master-id:royprsa;rspa.2017.08342018-03-07Research articles47422112017083420170834<![CDATA[Manipulating nanoparticle transport within blood flow through external forces: an exemplar of mechanics in nanomedicine]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170845?rss=1
A large number of nanoparticles (NPs) have been raised for diverse biomedical applications and some of them have shown great potential in treatment and imaging of diseases. Design of NPs is essential for delivery efficacy due to a number of biophysical barriers, which prevents the circulation of NPs in vascular flow and their accumulation at tumour sites. The physiochemical properties of NPs, so-called ‘4S’ parameters, such as size, shape, stiffness and surface functionalization, play crucial roles in their life journey to be delivered to tumour sites. NPs can be modified in various ways to extend their blood circulation time and avoid their clearance by phagocytosis, and efficiently diffuse into tumour cells. However, it is difficult to overcome these barriers simultaneously by a simple combination of ‘4S’ parameters for NPs. At this moment, external triggerings are necessary to guide the movement of NPs, which include light, ultrasound, magnetic field, electrical field and chemical interaction. The delivery system can be constructed to be sensitive to these external stimuli which can reduce the non-specific toxicity and improve the efficacy of the drug-delivery system. From a mechanics point of view, we discuss how different forces play their roles in the margination of NPs in blood flow and tumour microvasculature.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0845hwp:master-id:royprsa;rspa.2017.08452018-03-21Review articles47422112017084520170845<![CDATA[Stochastic isotropic hyperelastic materials: constitutive calibration and model selection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170858?rss=1
Biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains, owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests. For these materials, although hyperelastic models calibrated to mean data are useful, stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications. We combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress–strain function or the nonlinear shear modulus, which is a function of the deformation, estimated from experimental tests. These quantities can take on different values, corresponding to possible outcomes of the experiments. As multiple models can be derived that adequately represent the observed phenomena, we apply Occam’s razor by providing an explicit criterion for model selection based on Bayesian statistics. We then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0858hwp:master-id:royprsa;rspa.2017.08582018-03-14Research articles47422112017085820170858<![CDATA[Reviewers in 2017]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20180091?rss=1
2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2018.0091hwp:master-id:royprsa;rspa.2018.00912018-03-07Editorial47422112018009120180091<![CDATA[Correction to 'The ductile/brittle transition provides the critical test for materials failure theory]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20180114?rss=1
2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2018.0114hwp:master-id:royprsa;rspa.2018.01142018-03-21Correction47422112018011420180114