scholarly journals Indentation of a floating elastic sheet: geometry versus applied tension

2017 ◽  
Vol 473 (2206) ◽  
pp. 20170335 ◽  
Author(s):  
Finn Box ◽  
Dominic Vella ◽  
Robert W. Style ◽  
Jerome A. Neufeld
Keyword(s):  
Surface Tension ◽  
Hydrostatic Pressure ◽  
Indentation Depth ◽  
Liquid Bath ◽  
Tectonic Plates ◽  
Indentation Curve ◽  
Elastic Sheet ◽  
Applied Tension ◽  
Small Indentation ◽  
Theoretical Results

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.

Further theoretical analysis of concentration–pressure–flux waves in phloem transport systems

1978 ◽  
Vol 56 (8) ◽  
pp. 1086-1090 ◽  
Author(s):  
Jack M. Ferrier
Keyword(s):  
Hydrostatic Pressure ◽  
Water Flux ◽  
Sieve Tube ◽  
Phloem Transport ◽  
Time Dependent ◽  
Transport Systems ◽  
Flux Variation ◽  
Complex Interactions ◽  
Theoretical Results ◽  
Sugar Source

Theoretical results show that waves involving complex interactions between osmotic pressure, hydrostatic pressure, and fluxes of water and solute can occur in any phloem transport system surrounded by a semipermeable membrane. These results show that such waves can travel from sugar sink to sugar source as well as from sugar source to sugar sink. The time-dependent sugar concentration variation is shown to be caused largely by the time-dependent variation of the gradient of mass flow velocity in the sieve tube which is produced by the time-dependent variation of water flux across the membrane. This water flux variation is produced by a slight phase difference between osmotic and hydrostatic pressure variation. It is proposed that this phenomenon be called the concentration–pressure–flux (CPJ) wave.

Effects of Boundary Conditions and Initial Out-of-Roundness on the Strength of Thin-Walled Cylinders Subject to External Hydrostatic Pressure

1956 ◽  
Vol 23 (3) ◽  
pp. 351-358
Author(s):  
G. D. Galletly ◽  
R. Bart
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Semiempirical Methods ◽  
Test Results ◽  
External Hydrostatic Pressure ◽  
Simply Supported ◽  
Small Deflection ◽  
The Difference ◽  
Deflection Theory ◽  
Theoretical Results

Abstract Using classical small-deflection theory, an investigation was made of the effects of boundary conditions and initial out-of-roundness on the strength of cylinders subject to external hydrostatic pressure. The equations developed in this paper for initially out-of-round cylinders with clamped ends, and a slightly modified form of the equations previously derived by Bodner and Berks for simply supported ends, were applied to some actual test results obtained from nine steel cylinders which had been subjected to external hydrostatic pressure. Three semiempirical methods for determining the initial out-of-roundness of the cylinders also were investigated and these are described in the paper. The investigation indicates that if the initial out-of-roundness is determined in a manner similar to that suggested by Holt then the correlation between the experimental and theoretical results is quite good. The investigation also indicates that while the difference in collapse pressures for clamped-end and simply supported perfect cylinders may be quite considerable, this does not appear to be the case when initial out-of-roundnesses of a practical magnitude are considered.

Septal complexity in ammonoid cephalopods increased mechanical risk and limited depth

Paleobiology ◽  
1997 ◽  
Vol 23 (4) ◽  
pp. 470-481 ◽  
Author(s):  
Thomas L. Daniel ◽  
Brian S. Helmuth ◽  
W. Bruce Saunders ◽  
Peter D. Ward
Keyword(s):  
Surface Tension ◽  
Finite Element ◽  
Hydrostatic Pressure ◽  
Adaptive Response ◽  
Mechanical Stresses ◽  
Habitat Depth ◽  
Selective Pressures ◽  
Limited Depth ◽  
Buoyancy Control ◽  
Finite Element Stress

The evolution of septal complexity in fossil ammonoids has been widely regarded as an adaptive response to mechanical stresses imposed on the shell by hydrostatic pressure. Thus, septal (and hence sutural) complexity has been used as a proxy for depth: for a given amount of septal material greater complexity permitted greater habitat depth. We show that the ultimate septum is the weakest part of the chambered shell. Additionally, finite element stress analyses of a variety of septal geometries exposed to pressure stresses show that any departure from a hemispherical shape actually yields higher, not lower, stresses in the septal surface. Further analyses show, however, that an increase in complexity is consistent with selective pressures of predation and buoyancy control. Regardless of the mechanisms that drove the evolution of septal complexity, our results clearly reject the assertion that complexly sutured ammonoids were able to inhabit deeper water than did ammonoids with simpler septa. We suggest that while more complexly sutured ammonoids were limited to shallower habitats, the accompanying more complex septal topograhies enhanced buoyancy regulation (chamber emptying and refilling), through increased surface tension effects.

Nano-Indentation for Characterizing Mechanical Properties of Soft Materials

2013 ◽  
Author(s):  
A. Hossain ◽  
A. Mian
Keyword(s):  
Mechanical Properties ◽  
Indentation Depth ◽  
Contact Stiffness ◽  
Indentation Test ◽  
Soft Materials ◽  
Element Analysis ◽  
Fea Model ◽  
Computer Based ◽  
Loading And Unloading ◽  
Small Indentation

We have attempted to apply the computer-based finite element analysis (FEA) method to accurately measure the mechanical properties (e.g., hardness and elasticity) of a soft material by an indentation test. First, an axisymmetric model has been developed using commercially available FEA code ANSYS. The FEA model consisted of a thin Al-film resting on Si-substrate. A spherical indenter has been used to indent the Al-film, which traveled a predefined depth during the loading and unloading cycles. First, numerical simulations were conducted to get the force vs. displacement plot, which was later used to determine the modulus of elasticity and hardness of Al-film. The effects of substrate modulus and indentation depth were thoroughly investigated to determine the modulus and hardness of Al-film. The effect of friction, considered at the interface of indenter and Al-film, was found to offer minimum impact for relatively small indentation depth. The induced force on the Al-film by the indenter has been found to be higher with increasing indentation depth when friction was considered. However the contact stiffness, represented by the slope of the unloading curve, has been found almost the same with and without considering friction. The variation of substrate modulus has been found to be ineffective to capture the Al-film modulus for relatively small indentation depth. However for higher indentation depth, the substrate modulus has been found to offer profound effect to capture the film modulus. The hardness of the Al-film has also been found to be relatively unaffected with variation of substrate modulus. However, the hardness of the Al-film has been found to be higher with friction for relatively high indentation depth. Results obtained from this preliminary research are important to continue further investigation and to characterize the mechanical properties of other soft-materials, e.g., biofilms to minimize its detrimental effects and utilize its favorable aspects in industrial and biomedical applications.

scholarly journals The role of incoming waves in ice-edge dynamics

1991 ◽  
Vol 15 ◽  
pp. 96-100 ◽  
Author(s):  
Vernon A. Squire ◽  
Fox Colin
Keyword(s):  
Unit Length ◽  
Reflection And Transmission Coefficients ◽  
Radiation Stress ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Elastic Sheet ◽  
Driving Mechanisms ◽  
Ice Edge ◽  
Theoretical Results

The radiation stress at an ice edge due to incident sea and swell is reconsidered in the light of new theoretical results concerned with the calculation of the reflection and transmission coefficients at the edge of a thin elastic sheet. Both monochromatic seas and seas made up of a spectrum of periods are discussed, the latter invoking the Pierson-Moskowitz spectral model. It is found that the force per unit length due to radiation stress is comparable in magnitude with other driving mechanisms at the edge, i.e. with winds and currents.

scholarly journals The role of incoming waves in ice-edge dynamics

1991 ◽  
Vol 15 ◽  
pp. 96-100 ◽  
Author(s):  
Vernon A. Squire ◽  
Fox Colin
Keyword(s):  
Unit Length ◽  
Reflection And Transmission Coefficients ◽  
Radiation Stress ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Elastic Sheet ◽  
Driving Mechanisms ◽  
Ice Edge ◽  
Theoretical Results

The radiation stress at an ice edge due to incident sea and swell is reconsidered in the light of new theoretical results concerned with the calculation of the reflection and transmission coefficients at the edge of a thin elastic sheet. Both monochromatic seas and seas made up of a spectrum of periods are discussed, the latter invoking the Pierson-Moskowitz spectral model. It is found that the force per unit length due to radiation stress is comparable in magnitude with other driving mechanisms at the edge, i.e. with winds and currents.

Surface Tension Actuators Droplets in Microchannels

2005 ◽  
Author(s):  
Eric Baird ◽  
Kamran Mohseni
Keyword(s):  
Surface Tension ◽  
Contact Angle ◽  
Contact Angle Hysteresis ◽  
Unified Model ◽  
Analytic Model ◽  
Electrowetting On Dielectric ◽  
Fluid Velocities ◽  
Theoretical Results

A unified model is presented for the velocity of discrete droplets in microchannels actuated by surface tension modulation. Specific results are derived for the cases of electrowetting on dielectric (EWOD), dielectrophoresis (DEP), continuous electrowetting (CEW), and thermocapillary pumping (TCP). This treatment differs from previously published works by presenting one unified analytic model which is then simply applied to the specific cases of EWOD, CEW, DEP and TCP. In addition, the roles of equiliubrium contact angle and contact angle hysteresis are unambiguously described for each method. The model is shown to agree with experimental and theoretical results presented previously, predicting fluid velocities for a broad range of applications in digitized microfluidics.

Initiation of shape instabilities of free boundaries in planar Cauchy–Stefan problems

1993 ◽  
Vol 4 (4) ◽  
pp. 419-436 ◽  
Author(s):  
Qiang Zhu ◽  
Anthony Peirce ◽  
John Chadam
Keyword(s):  
Surface Tension ◽  
Large Time ◽  
Free Boundaries ◽  
Shape Stability ◽  
Stefan Problems ◽  
Moving Interfaces ◽  
Solidification Problem ◽  
Slow Moving ◽  
The Stability ◽  
Theoretical Results

The linearized shape stability of melting and solidifying fronts with surface tension is discussed in this paper by using asymptotic analysis. We show that the melting problem is always linearly stable regardless of the presence of surface tension, and that the solidification problem is linearly unstable without surface tension, but with surface tension it is linearly stable for those modes whose wave numbers lie outside a certain finite interval determined by the parameters of the problem. We also show that if the perturbed initial data is zero in the vicinity of the front, but otherwise quite general, it does not affect the stability. The present results complement those in Chadam & Ortoleva [4] which are only valid asymptotically for large time or equivalently for slow-moving interfaces. The theoretical results are verified numerically.

scholarly journals The effects of surface tension on the initial development of a free surface adjacent to an accelerated plate

2015 ◽  
Vol 776 ◽  
pp. 37-73 ◽  
Author(s):  
J. Uddin ◽  
D. J. Needham
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Local Structure ◽  
Contact Point ◽  
Initial Development ◽  
Early Stages ◽  
Inviscid Incompressible Fluid ◽  
Stationary Layer ◽  
Weak Surface ◽  
Theoretical Results

When a vertical rigid plate is uniformly accelerated horizontally from rest into an initially stationary layer of inviscid incompressible fluid, the free surface will undergo a deformation in the locality of the contact point. This deformation of the free surface will, in the early stages, cause a jet to rise up the plate. An understanding of the local structure of the free surface in the early stages of motion is vital in many situations, and has been developed in detail by King & Needham (J. Fluid Mech., vol. 268, 1994, pp. 89–101). In this work we consider the effects of introducing weak surface tension, characterized by the inverse Weber number $\mathscr{W}$, into the problem considered by King & Needham. Our approach is based upon matched asymptotic expansions as $\mathscr{W}\rightarrow 0$. It is found that four asymptotic regions are needed to describe the problem. The three largest regions have analytical solutions, whilst a numerical method based on finite differences is used to solve the time-dependent harmonic boundary value problem in the last region. Our results identify the local structure of the jet near the vicinity of the contact point, and we highlight a number of key features, including the height of this jet as well as its thickness and strength. We also present some preliminary experimental results that capture the spatial structure near the contact point, and we then show promising comparisons with the theoretical results obtained within this paper.

On oblique liquid curtains

2019 ◽  
Vol 876 ◽  
Author(s):  
Steven J. Weinstein ◽  
David S. Ross ◽  
Kenneth J. Ruschak ◽  
Nathaniel S. Barlow
Keyword(s):  
Surface Tension ◽  
Steady State ◽  
Characteristic Length ◽  
Volumetric Flow Rate ◽  
Liquid Sheet ◽  
Length Scale ◽  
Characteristic Length Scale ◽  
Cross Sectional ◽  
Average Speed ◽  
Theoretical Results

In a recent paper (J. Fluid Mech., vol. 861, 2019, pp. 328–348), Benilov derived equations governing a laminar liquid sheet (a curtain) that emanates from a slot whose centreline is inclined to the vertical. The equations are valid for slender sheets whose characteristic length scale in the direction of flow is much larger than its cross-sectional thickness. For a liquid that leaves a slot with average speed, $u_{0}$, volumetric flow rate per unit width, $q$, surface tension, $\unicode[STIX]{x1D70E}$, and density, $\unicode[STIX]{x1D70C}$, Benilov obtains parametric equations that predict steady-state curtain shapes that bend upwards against gravity provided $\unicode[STIX]{x1D70C}qu_{0}/2\unicode[STIX]{x1D70E}<1$. Benilov’s parametric equations are shown to be identical to those derived by Finnicum, Weinstein, and Ruschak (J. Fluid Mech., vol. 255, 1993, pp. 647–665). In the latter form, it is straightforward to deduce an alternative solution of Benilov’s equations where a curtain falls vertically regardless of the slot’s orientation. This solution is consistent with prior experimental and theoretical results that show that a liquid curtain can emerge from a slot at an angle different from that of the slot centreline.

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