Showing papers on "Elasticity (economics) published in 2009"

The Elasticity of Taxable Income with Respect to Marginal Tax Rates: A Critical Review

Abstract: This paper critically surveys the large and growing literature estimating the elasticity of taxable income with respect to marginal tax rates (ETI) using tax return data First, we provide a theoretical framework showing under what assumptions this elasticity can be used as a sufficient statistic for efficiency and optimal tax analysis We discuss what other parameters should be estimated when the elasticity is not a sufficient statistic Second, we discuss conceptually the key issues that arise in the empirical estimation of the elasticity of taxable income using the example of the 1993 top individual income tax rate increase in the United States to illustrate those issues Third, we provide a critical discussion of most of the taxable income elasticities studies to date, both in the United States and abroad, in light of the theoretical and empirical framework we laid out Finally, we discuss avenues for future research

Nonlinear elasticity of monolayer graphene

Abstract: By combining continuum elasticity theory and tight-binding atomistic simulations, we work out the constitutive nonlinear stress-strain relation for graphene stretching elasticity and we calculate all the corresponding nonlinear elastic moduli. Present results represent a robust picture on elastic behavior and provide the proper interpretation of recent experiments. In particular, we discuss the physical meaning of the effective nonlinear elastic modulus there introduced and we predict its value in good agreement with available data. Finally, a hyperelastic softening behavior is observed and discussed, so determining the failure properties of graphene.

Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM

Abstract: Nonlocal elasticity theory is a popular growing technique for the mechanical analyses of MEMS and NEMS structures. The nonlocal parameter accounts for the small-size effects when dealing with nano-size structures such as single-walled carbon nanotubes (SWCNTs). In this article, nonlocal elasticity and Timoshenko beam theory are implemented to investigate the stability response of SWCNT embedded in an elastic medium. For the first time, both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction of the (SWCNT) with the surrounding elastic medium. A differential quadrature approach is utilized and numerical solutions for the critical buckling loads are obtained. Influences of nonlocal effects, Winkler modulus parameter, Pasternak shear modulus parameter and aspect ratio of the SWCNT on the critical buckling loads are analyzed and discussed. The present study illustrates that the critical buckling loads of SWCNT are strongly dependent on the nonlocal small-scale coefficients and on the stiffness of the surrounding medium.

Theory of Elasticity at the Nanoscale

Abstract: We have shown in a series of recent papers that the classical theory of elasticity can be extended to the nanoscale by supplementing the equations of elasticity for the bulk material with the generalized Young–Laplace equations of surface elasticity. This review article shows how this has been done in order to capture the often unusual mechanical and physical properties of nanostructured particulate and porous materials. It begins with a description of the generalized Young–Laplace equations. It then generalizes the classical Eshelby formalism for nano-inhomogeneities; the Eshelby tensor now depends on the size of the inhomogeneity and the location of the material point in it. Then the stress concentration factor of a spherical nanovoid is calculated, as well as the strain fields in quantum dots (QDs) with multi-shell structures and in alloyed QDs induced by the mismatch in the lattice constants of the atomic species. This is followed by a generalization of the micromechanical framework for determining the effective elastic properties and effective coefficients of thermal expansion of heterogeneous solids containing nano-inhomogeneities. It is shown, for example, that the elastic constants of nanochannel-array materials with a large surface area can be made to exceed those of the nonporous matrices through pore surface modification or coating. Finally, the scaling laws governing the properties of nanostructured materials are derived. The underlying cause of the size dependence of these properties at the nanoscale is the competition between surface and bulk energies. These laws provide a yardstick for checking the accuracy of experimentally measured or numerically computed properties of nanostructured materials over a broad size range and can thus help replace repeated and exhaustive testing by one or a few tests.

Local elasticity map and plasticity in a model Lennard-Jones glass.

Abstract: In this work we calculate the local elastic moduli in a weakly polydispersed two-dimensional Lennard-Jones glass undergoing a quasistatic shear deformation at zero temperature. The numerical method uses coarse-grained microscopic expressions for the strain, displacement, and stress fields. This method allows us to calculate the local elasticity tensor and to quantify the deviation from linear elasticity (local Hooke's law) at different coarse-graining scales. From the results a clear picture emerges of an amorphous material with strongly spatially heterogeneous elastic moduli that simultaneously satisfies Hooke's law at scales larger than a characteristic length scale of the order of five interatomic distances. At this scale, the glass appears as a composite material composed of a rigid scaffolding and of soft zones. Only recently calculated in nonhomogeneous materials, the local elastic structure plays a crucial role in the elastoplastic response of the amorphous material. For a small macroscopic shear strain, the structures associated with the nonaffine displacement field appear directly related to the spatial structure of the elastic moduli. Moreover, for a larger macroscopic shear strain we show that zones of low shear modulus concentrate most of the strain in the form of plastic rearrangements. The spatiotemporal evolution of this local elasticity map and its connection with long term dynamical heterogeneity as well as with the plasticity in the material is quantified. The possibility to use this local parameter as a predictor of subsequent local plastic activity is also discussed.

Link between surface elasticity and foam stability

Abstract: We have measured the surface dilational elastic moduli of bubbles immersed in water and soap bubbles in air. The short time response was obtained by submitting the bubbles to a rapid expansion after which the surface tension evolution was monitored, using either image analysis or pressure measurements. It was possible with this method to measure directly the Gibbs elasticity. The longer time response was obtained by submitting the bubbles to low frequency oscillations. Experiments were performed with solutions of non-ionic surfactants, C12E6, C12G2, their 1:1 mixture, Pluronic F-68 and 127 and the surface elastic moduli were compared with the stability of foams made with these surfactants. The foams evolve with time, first by Ostwald ripening, controlled by the low frequency elasticity, and then by bubbles coalescence, controlled by the high frequency elasticity.

On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework

Abstract: The mechanical testing of anisotropic nonlinearly elastic solids is a topic of considerable and increasing interest. The results of such testing are important, in particular, for the characterization of the material properties and the development of constitutive laws that can be used for predictive purposes. However, the literature on this topic in the context of soft tissue biomechanics, in particular, includes some papers that are misleading since they contain errors and false statements. Claims that planar biaxial testing can fully characterize the three-dimensional anisotropic elastic properties of soft tissues are incorrect. There is therefore a need to clarify the extent to which biaxial testing can be used for determining the elastic properties of these materials. In this paper this is explained on the basis of the equations of finite deformation transversely isotropic elasticity, and general planar anisotropic elasticity. It is shown that it is theoretically impossible to fully characterize the properties of anisotropic elastic materials using such tests unless some assumption is made that enables a suitable subclass of models to be preselected. Moreover, it is shown that certain assumptions underlying the analysis of planar biaxial tests are inconsistent with the classical linear theory of orthotropic elasticity. Possible sets of independent tests required for full material characterization are then enumerated.

Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory

Abstract: In the present work, nonlocal elasticity theory has been implemented to study the vibration response of single-layered graphene (SLGS) sheets. The nonlocal elasticity theory accounts for the small size effects when dealing with nanostructures. Influence of the surrounding elastic medium on the fundamental frequencies of the SLGS is investigated. Both Winkler-type and Pasternak-type models are employed to simulate the interaction of the graphene sheets with a surrounding elastic medium. On the basis of Hamilton’s principle governing differential equations for the aforementioned problems are derived. The nonlocal small scale coefficients get introduced into the nonlocal theory through the constitutive relations. Differential quadrature method is being employed and numerical solutions for the frequencies are obtained. Numerical results show that the fundamental frequencies of SLGS are strongly dependent on the small scale coefficients. Further, a nonlinear frequency response is observed for the SLGS with lar...

On the possible role of surface elasticity in emulsion stability.

Abstract: We have measured the short-time and long-time elastic responses to compression of various types of surfactant layers adsorbed at oil-water interfaces. We prepared reasonably monodisperse oil-in-water emulsions with the same surfactants and monitored the time evolution of the emulsion droplets' diameter. We used a broad variety of surfactants (cationic, nonionic, and small polymers) and alkanes with different chain lengths. The emulsion drop size evolution is first controlled by Ostwald ripening and later on by drop coalescence, the later step being quite short. The overall emulsion lifetime is therefore dominated by ripening and for a given oil appears well correlated with the low-frequency surface elasticity as expected (and not with the high-frequency one, which is expected to control coalescence). When the oil chain length is changed, the stability is related more to the oil solubility in water, which also controls ripening. The overall results demonstrate the great importance of surface elasticity in emulsion stability.

Laminin and biomimetic extracellular elasticity enhance functional differentiation in mammary epithelia - eScholarship

Abstract: In the mammary gland, epithelial cells are embedded in a ‘soft’ environment and become functionally differentiated in culture when exposed to a laminin‐rich extracellular matrix gel. Here, we define the processes by which mammary epithelial cells integrate biochemical and mechanical extracellular cues to maintain their differentiated phenotype. We used single cells cultured on top of gels in conditions permissive for β‐casein expression using atomic force microscopy to measure the elasticity of the cells and their underlying substrata. We found that maintenance of β‐casein expression required both laminin signalling and a ‘soft’ extracellular matrix, as is the case in normal tissues in vivo, and biomimetic intracellular elasticity, as is the case in primary mammary epithelial organoids. Conversely, two hallmarks of breast cancer development, stiffening of the extracellular matrix and loss of laminin signalling, led to the loss of β‐casein expression and non‐biomimetic intracellular elasticity. Our data indicate that tissue‐specific gene expression is controlled by both the tissues’ unique biochemical milieu and mechanical properties, processes involved in maintenance of tissue integrity and protection against tumorigenesis.

Nonlinear low-force elasticity of single-stranded DNA molecules.

Abstract: We reconcile single-molecule force-extension data with scaling theories of polymer elasticity: measurements of denatured single-stranded DNA show a regime where the extension grows as a nonlinear power law with force, in accordance with "tensile blob" models. Analysis of the salt dependence of this regime indicates that the polymer's Kuhn length is proportional to the Debye length; this contradicts the Odijk-Skolnick-Fixman theory, but agrees with other predictions. Finally, we identify a Theta condition of the polymer, and find that the wormlike chain model best describes the polymer's elasticity at this point.

Viscoelastic properties of oxide-coated liquid metals

Abstract: Many liquid metals exposed to air develop an oxide film on their outer surface. This film is sufficiently solid-like to provide mechanical stability to small liquid metal droplets, yet weak enough to allow the droplets to be malleable. These properties are useful in both micro-electronics and microfluidics; however, little is known about how to characterize them. Here we probe the elastic, yielding, and relaxation properties of oxide-coated gallium and eutectic gallium indium using a rheometer equipped with a parallel-plate geometry. By using parallel plates of different size, we show that surface stresses dominate bulk stresses. These experiments also demonstrate that the apparent elastic properties of the oxide film are highly sensitive to its strain history. Moreover, the apparent elasticity is sensitive to the stresses stored in the oxide skin. We probe these stresses and their time-dependence, with both torque and normal force measurements. We also characterize the time-dependence of the elasticity b...

Quantitative viscoelastic parameters measured by harmonic motion imaging

Abstract: Quantifying the mechanical properties of soft tissues remains a challenging objective in the field of elasticity imaging. In this work, we propose an ultrasound-based method for quantitatively estimating viscoelastic properties, using the amplitude-modulated harmonic motion imaging (HMI) technique. In HMI, an oscillating acoustic radiation force is generated inside the medium by using focused ultrasound and the resulting displacements are measured using an imaging transducer. The proposed approach is a two-step method that uses both the properties of the propagating shear wave and the phase shift between the applied stress and the measured strain in order to infer to the shear storage (G') and shear loss modulus (G''), which refer to the underlying tissue elasticity and viscosity, respectively. The proposed method was first evaluated on numerical phantoms generated by finite-element simulations, where a very good agreement was found between the input and the measured values of G' and G''. Experiments were then performed on three soft tissue-mimicking gel phantoms. HMI measurements were compared to rotational rheometry (dynamic mechanical analysis), and very good agreement was found at the only overlapping frequency (10 Hz) in the estimate of the shear storage modulus G' (14% relative error, averaged p-value of 0.34), whereas poorer agreement was found in G'' (55% relative error, averaged p-value of 0.0007), most likely due to the significantly lower values of G'' of the gel phantoms, posing thus a greater challenge in the sensitivity of the method. Nevertheless, this work proposes an original model-independent ultrasound-based elasticity imaging method that allows for direct, quantitative estimation of tissue viscoelastic properties, together with a validation against mechanical testing.

Nonlinear mesoscopic elasticity : the complex behaviour of granular media including rocks and soil

Abstract: Preface . Acknowledgements . 1 Introduction . 1.1 Systems . 1.2 Examples of Phenomena . 1.3 The Domain of Exploration . 1.4 Outline . References . 2 Microscopic/Macroscopic Formulation of the Traditional Theory of Linear and Nonlinear Elasticity . 2.1 Prefatory Remarks . 2.2 From Microscopic to Continuum . 2.3 Continuum Elasticity and Macroscopic Phenomenology . 2.4 Thermodynamics . 2.5 Energy Scales . References . 3 Traditional Theory of Nonlinear Elasticity, Results . 3.1 Quasistatic Response Linear and Nonlinear . 3.2 Dynamic Response Linear . 3.3 Quasistatic/Dynamic Response Nonlinear . 3.4 Dynamic Response Nonlinear . 3.5 Exotic Response Nonlinear . 3.6 Green Functions . References . 4 Mesoscopic Elastic Elements and Macroscopic Equations of State . 4.1 Background . 4.2 Elastic Elements . 4.3 Effective Medium Theory . 4.4 Equations of State Examples . References . 5 Auxiliary Fields . 5.1 Temperature . 5.2 Saturation . 5.3 The Conditioning Field, X . References . 6 Hysteretic Elastic Elements . 6.1 Finite Displacement Elastic Elements Quasistatic Response . 6.2 Finite Displacement Elastic Elements: Inversion . 6.3 Finite Displacement Elastic Elements: Dynamic Response . 6.4 Models with Hysteresis . 6.5 Summary . 6.6 Models with Hysteresis, Detail . References . 7 The Dynamics of Elastic Systems Fast and Slow . 7.1 Fast/Slow Linear Dynamics . 7.2 Fast Nonlinear Dynamics . 7.3 Auxiliary Fields and Slow Dynamics . 7.4 Summary . References . 8 Q and Issues of Data Modeling/Analysis . 8.1 Attenuation in Linear Elastic Systems . 8.2 Nonlinear Attenuation . 8.3 Why Measure Q ? 8.4 How to Measure Q. 8.5 Resonant Bar Revisited . References . 9 Elastic State Spectroscopies and Elastic State Tomographies . 9.1 Spectroscopies . 9.2 Tomographies, Linear, Inhomogeneous . 9.3 Tomographies, Nonlinear, Inhomogeneous . References . 10 Quasistatic Measurements . 10.1 Some Basic Observations . 10.2 Quasistatic Stress-Strain Data Hysteresis . 10.3 Coupling to Auxiliary Fields . 10.4 Inversion . References . 11 Dynamic Measurements . 11.1 Quasistatic-Dynamic . 11.2 Dynamic-Dynamic . 11.3 Examples of Systems . References . 12 Field Observations . 12.1 Active Probes . 12.2 Passive Probes . References . 13 Nonlinear Elasticity and Nondestructive Evaluation and Imaging . 13.1 Overview . 13.2 Historical Context . 13.3 Simple Conceptual Model of a Crack in an Otherwise Elastically Linear Solid . 13.4 Nonlinear Elastic Wave Spectroscopy in Nondestructive Evaluation (NEWS) . 13.5 Progressive Mechanical Damage Probed by NEWS Techniques . 13.6 Mechanical Damage Location and Imaging . 13.7 Other Methods for Extracting the Elastic Nonlinearity . 13.8 Summary . References . Color Plates . Index .

Elasticity of arrested short-ranged attractive colloids: homogeneous and heterogeneous glasses.

Abstract: We evaluate the elasticity of arrested short-ranged attractive colloids by combining an analytically solvable elastic model with a hierarchical arrest scheme. This new approach allows us to discriminate the microscopic (primary particle-level) from the mesoscopic (cluster-level) contribution to the macroscopic shear modulus. The results quantitatively predict experimental data in a wide range of volume fractions and indicate in which cases the relevant contribution is due to mesoscopic structures. On this basis we propose that different arrested states of short-ranged attractive colloids can be meaningfully distinguished as homogeneous or heterogeneous colloidal glasses in terms of the length scale which controls their elastic behavior.

A three‐dimensional C1 finite element for gradient elasticity

Abstract: In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C1 continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C1 finite elements and their perceived computational cost. In this context, the lack of three-dimensional C1 elements is of particular concern. In this paper we present a new C1 hexahedral element which, to the best of our knowledge, is the first three-dimensional C1 element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C1 elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good-quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.

A continuum description of nonlinear elasticity, slip and twinning, with application to sapphire

Abstract: A model is developed for elasticity, plasticity and twinning in anisotropic single crystals subjected to large deformations. Dislocation glide and deformation twinning are dissipative, while energy storage mechanisms associated with dislocation lines and twin boundaries are described via scalar internal state variables. Concepts from continuum crystal plasticity are invoked, with shearing rates on discrete glide and twinning systems modelled explicitly. The model describes aspects of thermomechanical behaviour of single crystals of alumina over a range of loading conditions. Resolved shear stresses necessary for glide or twin nucleation at low to moderate temperatures are estimated from nonlinear elastic calculations, theoretical considerations of Peierls barriers and stacking fault energies and observations from shock physics experiments. These estimates are combined with the existing data from high-temperature experiments to provide initial yield conditions spanning a wide range of temperatures. The model reflects hardening of glide and twin systems from dislocations accumulated during basal slip. Residual elastic volume changes, predicted from nonlinear elastic considerations and approximated dislocation line energies, are positive and proportional to the dislocation line density. While the model suggests that generation of very large dislocation densities could influence the pressure–volume response, volume increases from defects are predicted to be small in crystals deformed via basal glide on a single system.

Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model

Abstract: In this paper, flexural vibrations of cracked micro- and nanobeams are studied. The model is based on the theory of nonlocal elasticity applied to Euler–Bernouilli beams. The cracked-beam model is established using a proper modification of the classical cracked-beam theory consisting of dividing the cracked element into two segments connected by a rotational spring located at the cracked section. This model promotes a discontinuity in bending slope, which is proportional to the second derivative of the displacements. Frequency equations of cracked nanobeams with some typical boundary conditions are derived and the natural frequencies for different crack positions, crack lengths, and nonlocal length parameters are calculated. The results are compared with those corresponding to the classical local model, emphasizing the differences occurring when the nonlocal effects are significant.

Drop impact of yield-stress fluids

Abstract: The normal impact of a drop of yield-stress fluid on a flat rigid surface is investigated experimentally. Using different model fluids (polymer microgels, clay suspensions) and impacted surfaces (partially wettable, super-hydrophobic), we find a rich variety of impact regimes from irreversible viscoplastic coating to giant elastic spreading and recoil. A minimal model of inertial spreading, taking into account an elasto-viscoplastic rheology, allows explaining in a single framework the different regimes and scaling laws. In addition, semi-quantitative predictions for the spread factor are obtained when the measured rheological parameters of the fluid (elasticity, yield stress, viscosity) are injected into the model. Our study offers a means to probe the short-time rheology of yield-stress fluids and highlights the role of elasticity on the unsteady hydrodynamics of these complex fluids. Movies are available with the online version of the paper (go to journals.cambridge.org/flm).

Two finite element discretizations for gradient elasticity

Abstract: We present and compare two different methods for numerically solving boundary value problems of gradient elasticity. The first method is based on a finite-element discretization using the displacement formulation, where elements that guarantee continuity of strains (i.e., C1 interpolation) are needed. Two such elements are presented and shown to converge: a triangle with straight edges and an isoparametric quadrilateral. The second method is based on a finite-element discretization of Mindlin’s elasticity with microstructure, of which gradient elasticity is a special case. Two isoparametric elements are presented, a triangle and a quadrilateral, interpolating the displacement and microdeformation fields. It is shown that, using an appropriate selection of material parameters, they can provide approximate solutions to boundary value problems of gradient elasticity. Benchmark problems are solved using both methods, to assess their relative merits and shortcomings in terms of accuracy, simplicity and computa...

Velocity dispersion of guided waves propagating in a free gradient elastic plate: Application to cortical bone

Abstract: The classical linear theory of elasticity has been largely used for the ultrasonic characterization of bone. However, linear elasticity cannot adequately describe the mechanical behavior of materials with microstructure in which the stress state has to be defined in a non-local manner. In this study, the simplest form of gradient theory (Mindlin Form-II) is used to theoretically determine the velocity dispersion curves of guided modes propagating in isotropic bone-mimicking plates. Two additional terms are included in the constitutive equations representing the characteristic length in bone: (a) the gradient coefficient g, introduced in the strain energy, and (b) the micro-inertia term h, in the kinetic energy. The plate was assumed free of stresses and of double stresses. Two cases were studied for the characteristic length: h=10−4 m and h=10−5 m. For each case, three subcases for g were assumed, namely, g>h, g

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

Abstract: The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin–Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lame constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein–Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.

Non-entropic and reversible long-range deformation of an encapsulating bioelastomer

Abstract: Encapsulation is a widespread biological process particularly in the formation of protective egg cases of oviparous animals. The egg capsule wall of the channelled whelk Busycon canaliculum is an effective shock absorber with high reversible extensibility and a stiffness that changes significantly during extension. Here we show that post-stretch recovery in egg capsules is not driven by entropic forces as it is in rubber. Indeed, at fixed strain, force decreases linearly with increasing temperature, whereas in rubber elasticity the force increases. Instead, capsule wall recovery is associated with the internal energy arising from the facile and reversible structural α-helix -sheet transition of egg capsule proteins during extension. This behaviour is extraordinary in the magnitude of energy dissipated and speed of recovery and is reminiscent of strain-induced crystallization in some polymeric fibres and of superelastic deformations associated with diffusionless phase transitions in shape-memory alloys. Bioelastomers generally show elasticity similar to that of rubber, which originates from entropic forces linked to deformation. It is now shown that in the egg capsule of a large marine shell, the elasticity is instead based on a structural transition. The results could have a significant impact on engineering protective encapsulating systems inspired by natural elastomers.

Effects of encapsulation elasticity on the stability of an encapsulated microbubble.

Abstract: A model for gas transport from an encapsulated microbubble into the surrounding medium is developed and investigated incorporating the effects of encapsulation elasticity. Encapsulation elasticity stabilizes microbubbles against dissolution and explains the long shelf life of microbubble contrast agent. We consider air bubbles as well as bubbles containing perfluorocarbon gas. Analytical conditions between saturation level, surface tension and interfacial dilatational elasticity are determined for attaining non-zero equilibrium radius for these microbubbles. Numerical solution of the equation verifies the stability of the equilibrium radii. In an undersaturated medium all encapsulated bubbles dissolve. In a saturated medium, an encapsulated bubble is found to achieve a long-time stable radius when interfacial dilatational elasticity is larger than equilibrium surface tension. For bubbles with interfacial dilatational elasticity smaller than the equilibrium surface tension, stable bubble of non-zero radius can be achieved only when the saturation level is greater than a critical value. Even if they initially contain a gas other than air, bubbles that reach a stable radius finally become air bubbles. The model is applied to an octafluoropropane filled lipid-coated 2.5 μm bubble, which displayed a transient swelling due to air intake before reaching an equilibrium size. Effects of elasticity, shell permeability, initial mole fraction, initial radius and saturation level are investigated and discussed. Shell permeability and mole fraction do not affect the final equilibrium radius of the microbubble but affect the time scale and the transient dynamics. Similarly, the ratio of equilibrium radius to initial radius remains unaffected by the variation in initial radius.

Injective weak solutions in second-gradient nonlinear elasticity

Abstract: We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gateaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including "mixed" boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.

Bending elasticity of saturated and monounsaturated phospholipid membranes studied by the neutron spin echo technique

Abstract: We have used neutron spin echo (NSE) spectroscopy to study the effect of bilayer thickness and monounsaturation (existence of a single double bond on one of the aliphatic chains) on the physical properties of unilamellar vesicles. The bending elasticity of saturated and monounsaturated phospholipid bilayers made of phospholipids with alkyl chain length ranging from 14 to 20 carbons was investigated. The bending elasticity κc of phosphatidylcholines (PCs) in the liquid crystalline (Lα) phase ranges from 0.38 × 10−19 J for 1,2-dimyristoyl-sn-glycero-3-phosphocholine to 0.64 × 10−19 J for 1,2-dieicosenoyl-sn-glycero-3-phosphocholine. It was confirmed that, contrary to the strong effect on the main transition temperature, the monounsaturation has a limited influence on the bending elasticity of lipid bilayers. In addition, when the area modulus KA varies little with chain unsaturation or length, the elastic ratios (κc/KA)1/2 of saturated and monounsaturated phospholipid bilayers varies linearly with lipid hydrophobic thickness d which agrees well with the theory of ideal fluid membranes.