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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946October, 2017Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Extended lubrication theory: improved estimates of flow in channels with variable geometry]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170234?rss=1
Lubrication theory is broadly applicable to the flow characterization of thin fluid films and the motion of particles near surfaces. We offer an extension to lubrication theory by starting with Stokes equations and considering higher-order terms in a systematic perturbation expansion to describe the fluid flow in a channel with features of a modest aspect ratio. Experimental results qualitatively confirm the higher-order analytical solutions, while numerical results are in very good agreement with the higher-order analytical results. We show that the extended lubrication theory is a robust tool for an accurate estimate of pressure drop in channels with shape changes on the order of the channel height, accounting for both smooth and sharp changes in geometry.
]]>2017-10-04T00:05:17-07:00info:doi/10.1098/rspa.2017.0234hwp:master-id:royprsa;rspa.2017.02342017-10-04Research articles47322062017023420170234<![CDATA[Modelling wave-induced sea ice break-up in the marginal ice zone]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170258?rss=1
A model of ice floe break-up under ocean wave forcing in the marginal ice zone (MIZ) is proposed to investigate how floe size distribution (FSD) evolves under repeated wave break-up events. A three-dimensional linear model of ocean wave scattering by a finite array of compliant circular ice floes is coupled to a flexural failure model, which breaks a floe into two floes provided the two-dimensional stress field satisfies a break-up criterion. A closed-feedback loop algorithm is devised, which (i) solves the wave-scattering problem for a given FSD under time-harmonic plane wave forcing, (ii) computes the stress field in all the floes, (iii) fractures the floes satisfying the break-up criterion, and (iv) generates an updated FSD, initializing the geometry for the next iteration of the loop. The FSD after 50 break-up events is unimodal and near normal, or bimodal, suggesting waves alone do not govern the power law observed in some field studies. Multiple scattering is found to enhance break-up for long waves and thin ice, but to reduce break-up for short waves and thick ice. A break-up front marches forward in the latter regime, as wave-induced fracture weakens the ice cover, allowing waves to travel deeper into the MIZ.
]]>2017-10-04T00:05:17-07:00info:doi/10.1098/rspa.2017.0258hwp:master-id:royprsa;rspa.2017.02582017-10-04Research articles47322062017025820170258<![CDATA[Indentation of a floating elastic sheet: geometry versus applied tension]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170335?rss=1
The localized loading of an elastic sheet floating on a liquid bath occurs at scales from a frog sitting on a lily pad to a volcano supported by the Earth’s tectonic plates. The load is supported by a combination of the stresses within the sheet (which may include applied tensions from, for example, surface tension) and the hydrostatic pressure in the liquid. At the same time, the sheet deforms, and may wrinkle, because of the load. We study this problem in terms of the (relatively weak) applied tension and the indentation depth. For small indentation depths, we find that the force–indentation curve is linear with a stiffness that we characterize in terms of the applied tension and bending stiffness of the sheet. At larger indentations, the force–indentation curve becomes nonlinear and the sheet is subject to a wrinkling instability. We study this wrinkling instability close to the buckling threshold and calculate both the number of wrinkles at onset and the indentation depth at onset, comparing our theoretical results with experiments. Finally, we contrast our results with those previously reported for very thin, highly bendable membranes.
]]>2017-10-11T00:28:31-07:00info:doi/10.1098/rspa.2017.0335hwp:master-id:royprsa;rspa.2017.03352017-10-11Research articles47322062017033520170335<![CDATA[Nonlinear mechanics of non-rigid origami: an efficient computational approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170348?rss=1
Origami-inspired designs possess attractive applications to science and engineering (e.g. deployable, self-assembling, adaptable systems). The special geometric arrangement of panels and creases gives rise to unique mechanical properties of origami, such as reconfigurability, making origami designs well suited for tunable structures. Although often being ignored, origami structures exhibit additional soft modes beyond rigid folding due to the flexibility of thin sheets that further influence their behaviour. Actual behaviour of origami structures usually involves significant geometric nonlinearity, which amplifies the influence of additional soft modes. To investigate the nonlinear mechanics of origami structures with deformable panels, we present a structural engineering approach for simulating the nonlinear response of non-rigid origami structures. In this paper, we propose a fully nonlinear, displacement-based implicit formulation for performing static/quasi-static analyses of non-rigid origami structures based on ‘bar-and-hinge’ models. The formulation itself leads to an efficient and robust numerical implementation. Agreement between real models and numerical simulations demonstrates the ability of the proposed approach to capture key features of origami behaviour.
]]>2017-10-11T00:28:31-07:00info:doi/10.1098/rspa.2017.0348hwp:master-id:royprsa;rspa.2017.03482017-10-11Research articles47322062017034820170348<![CDATA[Kinematic dynamos in spheroidal geometries]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170432?rss=1
The kinematic dynamo problem is solved numerically for a spheroidal conducting fluid of possibly large aspect ratio with an insulating exterior. The solution method uses solenoidal representations of the magnetic field and the velocity by spheroidal toroidal and poloidal fields in a non-orthogonal coordinate system. Scaling of coordinates and fields to a spherical geometry leads to a modified form of the kinematic dynamo problem with a geometric anisotropic diffusion and an anisotropic current-free condition in the exterior, which is solved explicitly. The scaling allows the use of well-developed spherical harmonic techniques in angle. Dynamo solutions are found for three axisymmetric flows in oblate spheroids with semi-axis ratios 1≤a/c≤25. For larger aspect ratios strong magnetic fields may occur in any region of the spheroid, depending on the flow, but the external fields for all three flows are weak and concentrated near the axis or periphery of the spheroid.
]]>2017-10-04T00:05:17-07:00info:doi/10.1098/rspa.2017.0432hwp:master-id:royprsa;rspa.2017.04322017-10-04Research articles47322062017043220170432<![CDATA[Twistor theory at fifty: from contour integrals to twistor strings]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170530?rss=1
We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space–time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold—the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics—anti-self-duality equations on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang–Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose’s proposal for a role of gravity in quantum collapse of a wave function.
]]>2017-10-11T00:28:31-07:00info:doi/10.1098/rspa.2017.0530hwp:master-id:royprsa;rspa.2017.05302017-10-11Review articles47322062017053020170530<![CDATA[A bioinspired study on the compressive resistance of helicoidal fibre structures]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170538?rss=1
Helicoidal fibre structures are widely observed in natural materials. In this paper, an integrated experimental and analytical approach was used to investigate the compressive resistance of helicoidal fibre structures. First, helicoidal fibre-reinforced composites were created using three-dimensionally printed helicoids and polymeric matrices, including plain, ring-reinforced and helix-reinforced helicoids. Then, load–displacement curves under monotonic compression tests were collected to measure the compressive strengths of helicoidal fibre composites. Fractographic characterization was performed using an X-ray microtomographer and scanning electron microscope, through which crack propagations in helicoidal structures were illustrated. Finally, mathematical modelling was performed to reveal the essential fibre architectures in the compressive resistance of helicoidal fibre structures. This work reveals that fibre–matrix ratios, helix pitch angles and interlayer rotary angles are critical to the compressive resistance of helicoidal structures.
]]>2017-10-12T02:18:05-07:00info:doi/10.1098/rspa.2017.0538hwp:master-id:royprsa;rspa.2017.05382017-10-12Research articles47322062017053820170538