In-plane Elasticity Research Articles

Mechanics of Curved Plasma Membrane Vesicles: Resting Shapes, Membrane Curvature, and In-Plane Shear Elasticity

Highly curved cell membrane structures, such as plasmalemmal vesicles (caveolae) and clathrin-coated pits, facilitate many cell functions, including the clustering of membrane receptors and transport of specific extracellular macromolecules by endothelial cells. These structures are subject to large mechanical deformations when the plasma membrane is stretched and subject to a change of its curvature. To enhance our understanding of plasmalemmal vesicles we need to improve the understanding of the mechanics in regions of high membrane curvatures. We examine here, theoretically, the shapes of plasmalemmal vesicles assuming that they consist of three membrane domains: an inner domain with high curvature, an outer domain with moderate curvature, and an outermost flat domain, all in the unstressed state. We assume the membrane properties are the same in these domains with membrane bending elasticity as well as in-plane shear elasticity. Special emphasis is placed on the effects of membrane curvature and in-plane shear elasticity on the mechanics of vesicle during unfolding by application of membrane tension. The vesicle shapes were computed by minimization of bending and in-plane shear strain energy. Mechanically stable vesicles were identified with characteristic membrane necks. Upon stretch of the membrane, the vesicle necks disappeared relatively abruptly leading to membrane shapes that consist of curved indentations. While the resting shape of vesicles is predominantly affected by the membrane spontaneous curvatures, the membrane shear elasticity (for a range of values recorded in the red cell membrane) makes a significant contribution as the vesicle is subject to stretch and unfolding. The membrane tension required to unfold the vesicle is sensitive with respect to its shape, especially as the vesicle becomes fully unfolded and approaches a relative flat shape.

New interaction formulae for beam-columns in Eurocode 3: The French–Belgian approach

This paper presents one of the two new proposals for beam-column interaction formulae, which will be made available in the forthcoming EN version of Eurocode 3. It is based on second-order in-plane elasticity, and has been derived in order to exhibit an understandable physical background, as well as accuracy and consistency with the other resistance and stability formulae of the code. The formulae ensure all the required continuities: between the cross-section classes, from plasticity to elasticity when slenderness and axial force are increasing, and continuity between all the individual stability member checks and cross-section verifications. The paper shows how the basic format may be extended to cover a wider range of cases, i.e. spatial elastic–plastic behaviour with possible lateral–torsional buckling. Through an extensive comparison with the results of FEM numerical simulations of about 15 000 beam-columns, the proposal is found to be safe and accurate.

323 血管内皮細胞内チャネルの形成メカニクス

The existence of a transendothelial channel, which is considered to be involved in specific of macromolecules, is reported. In this study, the generation mechanism of vascular endothelial chained vesicles and the transendothelial channel is theoretically investigated based on minimization of bending and in-plane shear strain energy of the membrane. A system of nonlinear differential equations is derived according to the variational principle and is reduced to a two point boundary value problem. The equations are solved through the Shooting method along with Newton-Raphson and Runge-Kutta method. The computed shape suggests that the effect of in-plane shear elasticity and outer cytoplasmic membrane are significant and these should be taken into account in numerical simulation.

Lipid phase separation in phospholipid bilayers and monolayers modeling the plasma membrane

It is postulated that biological membrane lipids are heterogeneously distributed into lipid microdomains. Recent evidence indicates that docosahexaenoic acid-containing phospholipids may be involved in biologically important lipid phase separations. Here we investigate the elastic and thermal properties of a model plasma membrane composed of egg sphingomyelin (SM), cholesterol and 1-stearoyl-2-docosahexaenoyl- sn-glycerophosphoethanolamine (SDPE). Two techniques are employed, pressure–area isotherms on monolayers to examine condensation and interfacial elasticity behavior, and differential scanning calorimetry (DSC) on bilayers to evaluate phase separations. Significant levels of condensation are observed for mixtures of SM and cholesterol. Surface elasticity measurements indicate that cholesterol decreases and SDPE increases the in-plane elasticity of SM monolayers. At X SDPE≥0.15 in SM, a more horizontal region emerges in the pressure–area isotherms indicating ‘squeeze out’ of SDPE from the monolayers. Addition of cholesterol to equimolar amounts of SM and SDPE further increases the amount of ‘squeeze out’, supporting the concept of phase separation into a cholesterol- and SM-rich liquid ordered phase and a SDPE-rich liquid disordered phase. This conclusion is corroborated by DSC studies where as little as X Chol=0.0025 induces a phase separation between the two lipids.

Bioengineering. Large Deformation Mechanics of Plasma Membrane Chained Vesicles in Cells.

The clathrin-coated pits, vesicles and chained vesicles on the inner surface of the plasma membrane facilitate the cell to transport specific extracellular macromolecules. This cellular process is strongly involved with large mechanical deformations of the plasma membrane accompanied by changes in membrane curvature. The assembly of the clathrin coat is thought to provide curvature into the membrane. Hence, effects of in-plane shear elasticity due to these coat structure may be significant on the vesicular mechanics. In this study, large deformation mechanics of plasma membrane chained vesicles in cells have been formulated based on minimization of bending and in-plane shear strain energy of the membrane. Effects of outer surrounding cytoplasmic flat membrane upon mechanically stable shapes of the vesicles were revealed, while effects of in-plane shear elasticity were partly discussed.

Alternative solution methods for crack problems in plane anisotropic elasticity, with examples

Two-dimensional crack problems of homogeneous, anisotropic, linear elasticity are solved using the Riemann–Hilbert method. To this end, the Riemann–Hilbert problem of line-discontinuity is formulated for anisotropic plane problems and the necessary parameters and functions are identified. For illustration, the method is applied to obtain the complete stress field and the stress intensity factors for a crack in an infinite anisotropic plate which is loaded on a part of one of its faces. Then, the well-established method of continuously distributed edge-dislocations is considered and illustrated via some example problems; e.g., an infinite anisotropic plate under uniform farfield loads containing: 1. a closed frictional crack and a pair of arbitrarily-located single edge-dislocations, and 2. an infinite row of equally-spaced parallel open cracks. The illustrative examples reveal that the first method offers an effective solution technique for problems where unbalanced tractions are applied on crack surfaces, whereas for problems with self-equilibrating loads applied on the crack faces, the second method is generally well suited. In addition, the method of resultant forces along the crack is discussed and its formulation in terms of the dislocation density functions and also the crack-opening displacements (which is new) is presented. The solutions to some of the example problems are provided in some detail, and for others, just the key formulae (e.g., stress functions and stress intensity factors) are calculated and analyzed. In brief, this paper presents the generalization of the Riemann–Hilbert method from isotropic to anisotropic in-plane elasticity problems, and also provides a collection of certain basic two-dimensional anisotropic crack problems; some of the results here are also new.

AN ANALYTICAL MODEL FOR COMPOSITE BOX-BEAMS

A direct analytical beam formulation is developed for predicting the effective elastic stiffnesses and corresponding load deformation behavior of tailored box beams. Deformation of the beam is described by extension, bending, torsion, transverse shearing, and torsion related out-of-plane warping. The present analysis is developed without much dependence on thin-walled beam theory. Formulation of the analysis is in terms of a two-dimensional cross-section, instead of an infinitely thin contour enable stresses, strains, elastic properties to vary through the thickness of the beam walls in a manner consistent to the true laminated structure nature of the composite structure. This analytical model is very useful in studying non-classical phenomenon such as warping and in-plane elasticity. Governing differential equations for the extension, bending, torsion and shearing of the beam are derived using Newtonian approach. Validation studies are carried out for composite designs with no coupling, designs With varying degrees of extension-torsion and bending shear coupling, as well as • tdesigns with varying degrees of torsion-bending and extension shear coupling. The analysis is evaluated by correlation with existing experimental results.

Acyl Chain-Length Asymmetry Alters the Interfacial Elastic Interactions of Phosphatidylcholines

Phosphatidylcholines (PCs) with stearoyl (18:0) sn-1 chains and variable-length, saturated sn-2 acyl chains were synthesized and investigated using a Langmuir-type film balance. Surface pressure was monitored as a function of lipid molecular area at various constant temperatures between 10°C and 30°C. Over this temperature range, 18:0–10:0 PC displayed only liquid-expanded behavior. In contrast, di-14:0 PC displayed liquid-expanded behavior at 24°C and 30°C, but two-dimensional phase transitions were evident at 20°C, 15°C, and 10°C. The average molecular area of 18:0–10:0 PC was larger than that of liquid-expanded di-14:0 PC at equivalent surface pressures, and the shapes of their liquid expanded isotherms were somewhat dissimilar. Analysis of the elastic moduli of area compressibility ( C s −1) as a function of molecular area revealed shallower slopes in the semilog plots of 18:0–10:0 PC compared to di-14:0 PC. At membrane-like surface pressures (e.g., 30 mN/m), 18:0–10:0 PC was 20–25% more elastic (in an in-plane sense) than di-14:0 PC. Other PCs with varying degrees of chain-length asymmetry (18:0–8:0 PC, 18:0–12:0 PC, 18:0–14:0 PC, 18:0–16:0 PC) were also investigated to determine whether the higher in-plane elasticity of fluid-phase 18:0–10:0 PC is a common feature of PCs with asymmetrical chain lengths. Two-dimensional phase transitions in 18:0–14:0 PC and 18:0–16:0 PC prevented meaningful comparison with other fluid-phase PCs at 30 mN/m. However, the C s −1 values for fluid-phase 18:0–8:0 PC and 18:0–12:0 PC were similar to that of 18:0–10:0 PC (85–90 mN/m). These values showed chain-length asymmetrical PCs to have 20–25% greater in-plane elasticity than fluid-phase PCs with mono- or diunsaturated acyl chains.

Phosphatidylcholine acyl unsaturation modulates the decrease in interfacial elasticity induced by cholesterol

The effect of cholesterol on the interfacial elastic packing interactions of various molecular species of phosphatidylcholines (PCs) has been investigated by using a Langmuir-type film balance and analyzing the elastic area compressibility moduli (Cs(-1)) as a function of average cross-sectional molecular area. Emphasis was on the high surface pressure regions (pi > or = 30 mN/m) which are thought to mimic biomembrane conditions. Increasing levels of cholesterol generally caused the in-plane elasticity of the mixed monolayers to decrease. Yet, the magnitude of the cholesterol-induced changes was markedly dependent upon PC hydrocarbon structure. Among PC species with a saturated sn-1 chain but different sn-2 chain cis unsaturation levels [e.g., myristate (14:0), oleate (18:1delta9(c), linoleate (18:2delta9,12(c), arachidonate (20:4delta5,8,11,14(c), or docosahexenoate (22:6delta4,7,10,13,16,19(c)], the in-plane elasticity moduli of PC species with higher sn-2 unsaturation levels were less affected by high cholesterol mol fractions (e.g., >30 mol %) than were the more saturated PC species. The largest cholesterol-induced decreases in the in-plane elasticity were observed when both chains of PC were saturated (e.g., di-14:0 PC). When both acyl chains were identically unsaturated, the resulting PCs were 20-25% more elastic in the presence of cholesterol than when their sn-1 chains were long and saturated (e.g., palmitate). The mixing of cholesterol with PC was found to diminish the in-plane elasticity of the films beyond what was predicted from the additive behavior of the individual lipid components apportioned by mole and area fraction. Deviations from additivity were greatest for di-14:0 PC and were least for diarachidonoyl PC and didocosahexenoyl PC. In contrast to Cs(-1) analyses, sterol-induced area condensations were relatively unresponsive to subtle structural differences in the PCs at high surface pressures. Cs(-1) versus average area plots also indicated the presence of cholesterol concentration-dependent, low-pressure (<14 mN/m) phase boundaries that became more prominent as PC acyl chain unsaturation increased. Hence, area condensations measured at low surface pressures often do not accurately portray which lipid structural features are important in the lipid-sterol interactions that occur at high membrane-like surface pressures.

The interfacial elastic packing interactions of galactosylceramides, sphingomyelins, and phosphatidylcholines

The interfacial elastic packing interactions of different galactosylceramides (GalCers), sphingomyelins (SMs), and phosphatidylcholines (PC) were compared by determining their elastic area compressibility moduli (Cs-1) as a function of lateral packing pressure (pi) in a Langmuir-type film balance. To assess the relative contributions of the lipid headgroups as well as those of the ceramide and diacylglycerol hydrocarbon regions, we synthesized various GalCer and SM species with identical, homogeneous acyl residues and compared their behavior to that of PCs possessing similar hydrocarbon structures. For PCs, this meant that the sn-1 acyl chain was long and saturated (e.g., palmitate) and the sn-2 chain composition was varied to match that of GalCer or SM. When at equivalent pi and in either the chain-disordered (liquid-expanded) or chain-ordered (liquid-condensed) state, GalCer films were less elastic than either SM or PC films. When lipid headgroups were identical (SM and PC), Cs-1 values (at equivalent pi) for chain-disordered SMs, but not chain-ordered SMs, were 25-30% higher than those of PCs. Typical values for fluid phase (liquid-expanded) GalCer at 30 mN/m and 24 degrees C were 158 (+/- 7) mN/m, whereas those of SM were 135 (+/- 7) mN/m and those of PC were 123 (+/- 2) mN/m. Pressure-induced transitions to chain-ordered states (liquid-condensed) resulted in significant increases (two- to fourfold) in the "in-plane" compressibility for all three lipid types. Typical Cs-1 values for chain-ordered GalCers at 30 mN/m and 24 degrees C were between 610 and 650 mN/m, whereas those of SM and of PC were very similar and were between 265 and 300 mN/m. Under fluid phase conditions, the pi-Cs-1 behavior for each lipid type was insensitive to whether the acyl chain was saturated or monounsaturated. Measurement of the Cs-1 values also provided an effective way to evaluate the two-dimensional phase transition region of SMs, GalCers, and PCs. Modest heterogeneity in the acyl composition led to transitional broadening. Our findings provide useful information regarding the in-plane elasticity of lipids that are difficult to investigate by alternative methods, i.e., micropipette aspiration technique. The results also provide insight into the stability of sphingolipid-enriched, membrane microdomains that are thought to play a role in the sorting and trafficking of proteins containing glycosylphosphatidylinositol anchors with cells.

Aeroelastic response, loads, and stability of a composite rotor in forward flight

The aeroelastic response, blade and hub loads, and shaft-fixed aeroelastic stability are investigated for a helicopter with elastically tailored composite rotor blades. A finite element analysis including nonclassical effects such as transverse shear, torsion related warping, and in-plane elasticity is integrated with the University of Margland Advanced Rotorcraft Code. The analysis is correlated against both experimental data and detailed finite element results. Correlation of rotating natural frequencies of coupled composite box-beams is generally within 5-10%

Aeroelastic Response, Loads, and Stability of a Composite Rotor in Forward Flight

The aeroelastic response, blade and hub loads, and shaft-fixed aeroelastic stability are investigated for a helicopter with elastically tailored composite rotor blades. A finite element analysis including nonclassical effects such as transverse shear, torsion related warping, and in-plane elasticity is integrated with the University of Margland Advanced Rotorcraft Code. The analysis is correlated against both experimental data and detailed finite element results. Correlation of rotating natural frequencies of coupled composite box-beams is generally within 5-10%

The Mechanics of Axially Symmetric Liposomes

Hotani has filmed morphological transformations in unilamellar liposomes, starting from a spherical shape, when the interior volume decreases steadily. Hotani's liposomes showed no evidence of general thermal fluctuations. We use a finite-deformation theory of axisymmetric, quasi-static thin shells to analyze theoretically bifurcations and changes of shape in liposomes under decreasing volume. The main structural action in a lipid bilayer is generally agreed to be its elastic resistance to bending, and it is usual to regard surface deformation as being like that of a two-dimensional liquid. We find, however, that some in-plane shear elasticity is also needed in order to produce the observed post-bifurcation behavior. Such an elasticity would be difficult to measure directly.

A cylindrical shell with an arbitrarily oriented crack

In this paper the general problem of a shallow shell with constant curvatures is considered. It is assumed that the shell contains an arbitrarily oriented through crack and the material is specially orthotropic. The nonsymmetric problem is solved for arbitrary self-equilibrating crack surface tractions, which, added to an appropriate solution for an uncracked shell, would give the result for a cracked shell under most general loading conditions. The problem is reduced to a system of five singular integral equations in a set of unknown functions representing relative displacements and rotations on the crack surfaces. The stress state around the crack tip is asymptotically analyzed and it is shown that the results are identical to those obtained from the two-dimensional in-plane and anti-plane elasticity solutions. The numerical results are given for a cylindrical shell containing an arbitrarily oriented through crack. Some sample results showing the effect of the Poisson's ratio and the material orthotropy are also presented.