Showing papers on "Elasticity (economics) published in 2005"

Theory of Elasticity

Abstract: Stress Tensor- Deformation of a Continuum- The Constitutive Laws of the Linear Theory of Elasticity- Governing Relationships in the Linear Theory of Elasticity- Three-dimensional Problems in the Theory of Elasticity- Saint-Venant's Problem- The Plane Problem of the Theory of Elasticity- Constitutive Laws for Nonlinear Elastic Bodies- Problems and Methods of the Nonlinear Theory of Elasticity

Size-dependent elasticity of nanowires: Nonlinear effects

Abstract: We employ a molecular statics approach based on embedded-atom-method interatomic potentials to study the elasticity of copper nanowires along [001], [110], and [111] crystallographic directions. Self-consistent comparison with the bulk response clearly shows that the overall nanowire elasticity is primarily due to nonlinear response of the nanowire core. While the surface-stress-induced surface elasticity modifies the behavior for ultrathin nanowires, their contribution is always considerably smaller than that due to nonlinear elasticity of the nanowire core. More importantly, for all three orientations, the surface is softer than an equivalently strained bulk, and the overall nanowire softening or stiffening is determined by orientation-dependent core elasticity.

The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries

Abstract: The non-Newtonian flow of dilute aqueous polyethylene oxide (PEO) solutions through micro-fabricated planar abrupt contraction-expansions is investigated. The small lengthscales and high deformation rates in the contraction throat lead to significant extensional flow effects even with dilute polymer solutions having time constants on the order of milliseconds. By considering the definition of the elasticity number, El = Wi/Re, we show that the lengthscale of the geometry is key to the generation of strong viscoelastic effects, such that the same flow behaviour cannot be reproduced using the equivalent macro-scale geometry using the same fluid. We observe significant vortex growth upstream of the contraction plane, which is accompanied by an increase of more than 200% in the dimensionless extra pressure drop across the contraction. Streak photography and video-microscopy using epifluorescent particles shows that the flow ultimately becomes unstable and three-dimensional. The moderate Reynolds numbers (0.44 ≤ Re ≤ 64) associated with these high Weissenberg number (0 ≤ Wi ≤ 548) micro-fluidic flows results in the exploration of new regions of the Re-Wi parameter space in which the effects of both elasticity and inertia can be observed. Understanding such interactions will be increasingly important in micro-fluidic applications involving complex fluids and can best be interpreted in terms of the elasticity number, El = Wi/Re, which is independent of the flow kinematics and depends only on the fluid rheology and the characteristic size of the device.

Friction enhances elasticity in granular solids

Abstract: For years, engineers have used elastic and plastic models to describe the properties of granular solids, such as sand piles and grains in silos. However, there are theoretical and experimental results that challenge this approach. Specifically, it has been claimed that stress in granular solids propagates in a manner described by wave-like (hyperbolic) equations, rather than the elliptic equations of static elasticity. Here we report numerical simulations of the response of a two-dimensional granular slab to an external load, revealing that both approaches are valid--albeit on different length scales. For small systems that can be considered mesoscopic on the scale of the grains, a hyperbolic-like, strongly anisotropic response is expected. However, in large systems (those typically considered by engineers), the response is closer to that predicted by traditional isotropic elasticity models. Static friction, often ignored in simple models, plays a key role: it increases the elastic range and renders the response more isotropic, even beyond this range.

Radial elasticity of multiwalled carbon nanotubes.

Abstract: We report an experimental and a theoretical study of the radial elasticity of multiwalled carbon nanotubes as a function of external radius. We use atomic force microscopy and apply small indentation amplitudes in order to stay in the linear elasticity regime. The number of layers for a given tube radius is inferred from transmission electron microscopy, revealing constant ratios of external to internal radii. This enables a comparison with molecular dynamics results, which also shed some light onto the applicability of Hertz theory in this context. Using this theory, we find a radial Young modulus strongly decreasing with increasing radius and reaching an asymptotic value of 30 � 10 GPa.

A model for the dynamics of gas bubbles in soft tissue

Abstract: Understanding the behavior of cavitation bubbles driven by ultrasonic fields is an important problem in biomedical acoustics. Keller-Miksis equation, which can account for the large amplitude oscillations of bubbles, is rederived in this paper and combined with a viscoelastic model to account for the strain-stress relation. The viscoelastic model used in this study is the Voigt model. It is shown that only the viscous damping term in the original equation needs to be modified to account for the effect of elasticity. With experiment determined viscoelastic properties, the effects of elasticity on bubble oscillations are studied. Specifically, the inertial cavitation thresholds are determined using R(max)/R(0), and subharmonic signals from the emission of an oscillating bubble are estimated. The results show that the presence of the elasticity increases the threshold pressure for a bubble to oscillate inertially, and subharmonic signals may only be detectable in certain ranges of radius and pressure amplitude. These results should be easy to verify experimentally, and they may also be useful in cavitation detection and bubble-enhanced imaging.

Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity

Abstract: The aim of this paper is to study the elastic stress and strain fields of dislocations and disclinations in the framework of Mindlin's gradient elasticity. We consider simple but rigorous versions of Mindlin's first gradient elasticity with one material length (gradient coefficient). Using the stress function method, we find modified stress functions for all six types of Volterra defects (dislocations and disclinations) situated in an isotropic and infinitely extended medium. By means of these stress functions, we obtain exact analytical solutions for the stress and strain fields of dislocations and disclinations. An advantage of these solutions for the elastic strain and stress is that they have no singularities at the defect line. They are finite and have maxima or minima in the defect core region. The stresses and strains are either zero or have a finite maximum value at the defect line. The maximum value of stresses may serve as a measure of the critical stress level when fracture and failure may occur. Thus, both the stress and elastic strain singularities are removed in such a simple gradient theory. In addition, we give the relation to the nonlocal stresses in Eringen's nonlocal elasticity for the nonsingular stresses.

Nonaffine correlations in random elastic media

Abstract: Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements from a uniform reference state grow linearly with position , and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements as the difference between and affine displacements, and we investigate the nonaffinity correlation function and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants (none of which are infinite), by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that scales as where the amplitude is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short- and long-range correlations in random elastic moduli.

Elastic moduli of soils dependent on pressure: a hyperelastic formulation

Abstract: The elastic behaviour of granular materials is non-linear, in that the small-strain tangent stiffness depends on the stress level. The elastic moduli typically vary as power functions of the mean stress. Simple models of this nonlinearity can result in behaviour that violates the laws of thermodynamics. To guarantee that an elasticity model is thermodynamically acceptable it must be possible to derive the elastic behaviour from a free energy potential (or alternatively from a complementary energy potential). In this paper elasticity models are derived that allow for variation of elastic moduli as power functions of mean stress, while guaranteeing thermodynamic acceptability. The important issue of the dependence of secant stiffness on strain amplitude (a phenomenon related to dissipation processes in the soil) is acknowledged but not addressed here.

Mechanical elasticity of single and double clamped silicon nanobeams fabricated by the vapor-liquid-solid method

Abstract: Atomic force microscopy has been used to characterize the mechanical elasticity of Si nanowires synthesized by the vapor-liquid-solid method. The nanowires are horizontally grown between the two facing Si(111) sidewalls of microtrenches prefabricated on a Si(110) substrate, resulting in suspended single and double clamped nanowire-in-trench structures. The deflection of the nanowires is induced and measured by the controlled application of normal forces with the microscope tip. The observed reversibility of the nanowire deflections and the agreement between the measured deflection profiles and the theoretical behavior of single and double clamped elastic beams demonstrate the overall beamlike mechanical behavior and the mechanical rigidity of the clamping ends of the nanowire-in-trench structures. These results demonstrate the potential of the nanowire-in-trench fabrication approach for the integration of VLS grown nanostructures into functional nanomechanical devices.

Silicone dielectric elastomer actuators: Finite-elasticity model of actuation

Abstract: A theory for the actuation strain of a silicone dielectric elastomer actuator with a simple geometry is developed. The stress is a function of two variables, the strain and the applied voltage. A lamination type stress–strain function was employed, due to the nature of the electrodes. All parameters are obtained from physical measurements. Then, measurements of the blocking force are shown to correspond to what is expected from the Maxwell stress. Actuation strain measurements are performed and compared with the developed theory. The developed model has no free parameters, yet it exhibits the features of the actuation strain: the optimum load is reproduced correctly (within the chosen experiment resolution of 5 g). The discrepancy on the actuation strain between the measurement results and the model vary between 15% and 37% at the optimum, which are ascribed to inaccuracies in dimension measurements of the actuator and the inherent crudeness in the assumption that the stress and strain fields are constant throughout the actuator. We conclude that even for high strains, the actuation of dielectric elastomer actuators is well described by only considering the elasticity of the material and the Maxwell stress due to the applied electric field.

Optimization of electromechanical coupling for a thin-film PZT membrane: II. Experiment

Abstract: In this two-part paper, the optimization of the electromechanical coupling coefficient for thin-film piezoelectric devices is investigated both analytically and experimentally. The electromechanical coupling coefficient is crucial to the performance of piezoelectric energy conversion devices. A membrane-type geometry is chosen for the study. In part I a one-dimensional model is developed for a membrane composed of two layers, a passive elastic material and a piezoelectric material. The lumped-parameter model is then used to explore the effect of design and process parameters, such as residual stress, substrate thickness, piezoelectric thickness and electrode coverage, on the electromechanical coupling coefficient. The model shows that the residual stress has the most substantial effect on the electromechanical coupling coefficient. For a given substrate material and thickness an optimum piezoelectric thickness can be found to achieve the maximum coupling coefficient. The substrate stiffness affects the magnitude of the maximum coupling coefficient that can be obtained. Electrode coverage was found to be important to electromechanical coupling. The model predicts an optimum electrode coverage of 42% of the membrane area. The model developed in part I formed the basis for the parameters studied experimentally in part II.

Elasticity and critical bending moment of model colloidal aggregates.

Abstract: The bending mechanics of singly bonded colloidal aggregates are measured using laser tweezers. We find that the colloidal bonds are capable of supporting significant torques, providing a direct measurement of the tangential interactions between particles. A critical bending moment marks the limit of linear bending elasticity, past which small-scale rearrangements occur. These mechanical properties underlie the rheology and dynamics of colloidal gels formed by diffusion-limited cluster aggregation, and give critical insight into the contact interactions between Brownian particles.

Effect of chemistry on the stability and elasticity of the perovskite and post‐perovskite phases in the MgSiO3‐FeSiO3‐Al2O3 system and implications for the lowermost mantle

Abstract: [1] Using first-principles calculations we predict the effects of composition on the perovskite to post-perovskite phase transition. The transition is predicted at 107 GPa for pure MgSiO3. The addition of Al2O3 slightly increases this transition pressure, and the addition of Fe2+ considerably reduces it; the FeSiO3 end-member term is stable in the post-perovskite modification with respect to perovskite at all pressures. We also determine the static equations of state, densities, elasticity and seismic wave velocities. At the transition Vp increases slightly, and Vs increases significantly, consistent with the seismic observations for D″. The addition of both Fe2+ and Al2O3 decrease the seismic wave velocities.

A Mixed Finite Element Method for Elasticity in Three Dimensions

Abstract: We describe a stable mixed finite element method for linear elasticity in three dimensions.

Elasticity of planar fiber networks

Abstract: A micromechanics model is proposed for the elasticity of planar fiber networks (FNs). The FN is created by random deposition of linearly elastic straight rods within a region. The rods are bonded rigidly at contacts. Under external in-plane loading, the FN deformation consists of fiber bending, elongation, and contraction. An effective constitutive relation for fiber network is developed by averaging the strain energy dissipated by all possible fiber deformations in all directions. Numerical calculations are performed to analyze the effects of fiber aspect ratio and fiber concentration on the effective stiffness of the planar random FN. Finite element analysis (FEA) is performed and compared with the theoretical predictions of the effective FN moduli at several fiber concentrations. FEA results are in good agreement with theoretical predictions. The present model can be used for the prediction of mechanical properties, scaling analysis, and optimization of fiber assemblies.

On dislocations in a special class of generalized elasticity

Abstract: In this paper we consider and compare special classes of static theories of gradient elasticity, nonlocal elasticity, gradient micropolar elasticity and nonlocal micropolar elasticity with only one gradient coefficient. Equilibrium equations are presented but higher-order boundary conditions are not of concern here, since they are not required for the problems considered. The connection between gradient theory and nonlocal theory is discussed for elasticity as well as for micropolar elasticity. Nonsingular solutions for the elastic fields of screw and edge dislocations are given. Both the elastic deformation (distortion, strain, bend-twist) and the force and couple stress tensors do not possess any singularity unlike ‘classical’ theories. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Chemorheological response of elastomers at elevated temperatures: Experiments and simulations

Abstract: An experimental study and a method for simulating the constitutive response of elastomers at temperatures in the chemorheological range (90–150 °C for natural rubber) are presented. A comprehensive set of uniaxial experiments for a variety of prescribed temperature histories is performed on natural rubber specimens that exhibit finite elasticity, entropic stiffening with temperature, viscoelasticity, scission, and oxygen diffusion/reaction effects. The simulation approach is based on a multi-network framework for finite elasticity, isothermal incompressibility, thermal expansion, and temperature-induced degradation. The model extends previous work to account for kinetics of scission for arbitrary time-varying temperature histories and incorporates the effects of viscoelastic relaxation and diffusion-limited oxidative scission. The model is calibrated to experiments performed on a commercially-available filled natural rubber material, and numerical simulations are compared favorably to experiments for a variety of temperature histories.

Rectangular mixed finite elements for elasticity

Abstract: We present a family of stable rectangular mixed finite elements for plane elasticity. Each member of the family consists of a space of piecewise polynomials discretizing the space of symmetric tensor fields in which the stress field is sought, and another to discretize the space of vector fields in which the displacement is sought. These may be viewed as analogues in the case of rectangular meshes of mixed finite elements recently proposed for triangular meshes. As for the triangular case the elements are closely related to a discrete version of the elasticity differential complex.

Three-dimensional problems in the theory of elasticity

Abstract: The displacement of the “point of observation” M in an unbounded elas- tic medium subjected to a concentrated force P applied at the “point of source” Q is determined by means of the Kelvin-Somigliana formula, eq. (3.5.9) of Chapter 4, $$ u(M,Q) = \hat U(M,Q) \cdot P. $$ (1.1.1)

Formation, manipulation, and elasticity measurement of a nanometric column of water molecules.

Abstract: Nanometer-sized columns of condensed water molecules are formed by an atomic-resolution force microscope operated in ambient conditions. An unusual stepwise decrease of the force gradient associated with the ultrathin water bridge in the tip-substrate gap is observed during its stretch, exhibiting regularity in step heights (approximately 0.5 N/m) and plateau lengths (approximately 1 nm). Such "quantized" elasticity is indicative of an atomic-scale stick slip at the tip-water interface. A thermodynamic-instability-induced rupture of the water meniscus (5 nm long and 2.6 nm wide) is also found. This work opens a high-resolution study of the structure and interface dynamics of a nanometric aqueous column.

Directionally controlled expandable device and methods for use

Abstract: Systems and methods for directionally controlling expansion of an expandable device are described. One such device includes an expandable body comprising a first wall portion and a second wall portion. The first wall portion comprises a high elasticity material. The second wall portion comprises a material having an elasticity lower than the elasticity of the first wall portion. When the body is expanded, expansion of the second wall portion is constrained more than expansion of the first wall portion. Expansion of the body is directed outwardly from the high elasticity first wall portion. Such a device is useful for providing cavities in interior body regions.

Effect of cross-sectional design on the modulus of elasticity and toughness of fiber-reinforced composite materials

Abstract: Statement of problem Many current fabrication protocols for dental fiber-reinforced composites use hand lay-up techniques and technician design input. Little information exists regarding how the manipulation of the cross-sectional design of a prosthesis might affect the modulus of elasticity and toughness. Purpose The aim of this study was to determine the effect of simple and complex cross-sectional designs on the modulus of elasticity and toughness of fiber-reinforced composite used for dental prostheses. Material and methods Two particulate composites (BelleGlass HP and Targis) were reinforced with ultra-high–molecular-weight polyethylene fiber ribbon (Connect), woven E-glass fibers (Vectris Frame), or unidirectional R-glass fibers (Vectris Pontic). A range of fiber positions, orientations, or geometries were incorporated into the rhombic specimens (2 × 2 × 25 mm 3 ) to achieve simple and complex experimental cross-sectional designs. The control specimen did not contain fiber reinforcement. Specimens (n=6) were stored 1 week in distilled water at 37°C prior to 3-point load testing to determine the modulus of elasticity (GPa) and toughness (MPa). The data within each main fiber group were subjected to 1-way analysis of variance and a Tukey post hoc test (α=.05). Cross-sections of randomly selected test specimens (n=2) were made for scanning electron microscope (SEM) analysis of the fiber distribution. Results The mean modulus of elasticity varied from 8.7 ± 2.0 GPa (Targis control) to 21.6 ± 1.4 GPa (2 unidirectional glass fiber reinforcements, 1 each at the tension side and the compression side). Mean toughness varied from 0.07 ± 0.02 MPa (unidirectional glass fiber positioned at the compression side) as the lowest mean, to 4.53 ± 0.89 MPa (unidirectional glass fiber positioned at the tension side) as the highest. Significant differences were identified between specimen groups in each main category (all groups P P =.003). SEM micrographs showed fiber distribution in the cross section of test specimens to correspond with the intended fiber geometry. Conclusion The modulus of elasticity of the composite specimens increased when 1 or more glass fiber groups were located at the compression side of the specimen. Toughness was most effectively increased when 1 or more fiber groups were located at the tension side of the specimen.

Displacement measurement method and apparatus, strain measurement method and apparatus, elasticity and visco-elasticity constants measurement apparatus, and the elasticity and visco-elasticity constants measurement apparatus-based treatment apparatus

Abstract: The present invention provides elastic constant and visco elastic constant measurement apparatus etc. for measuring in the ROI in living tissues elastic constants such as shear modulus, Poisson's ratio, Lame constants, etc., visco elastic constants such as visco shear modulus, visco Poisson's ratio, visco Lame constants, etc. and density even if there exist another mechanical sources and uncontrollable mechanical sources in the object. The elastic constant and visco elastic constant measurement apparatus is equipped with storage of deformation data measured in the ROI, a calculation of elastic and visco elastic constants by calculating the shear modulus etc. at arbitrary point in the ROI from measured strain tensor data etc., wherein the calculation of the elastic and visco elastic constants numerically determines elastic constants etc. from the first order partial differential equations relating elastic constants etc. and strain tensor etc.

Poisson ratio effects and critical values in spherical and cylindrical hertzian contacts

Abstract: 1. ABSTRACT This work determines the location of the greatest elastic di stress in spherical and cylindrical Hertzian contacts based upon the distortion energy and the maximum shear stress theories. The ratios between the maximum pressure, the von Mises stress, and the maximum shear stress ar e determined and fitted by empirical formulations for a wide range of the Poisson ratio, which represents materia l compressibility. Some similarities exist between cylindrical and spherical contacts, where for many metallic m aterials the maximum von Mises or shear stresses emerge beneath the surface. However, in cylindrical contact i f ny of the materials is excessively compressible then the maximum von Mises stress appears at the surface. The cor responding Poisson ratios are found. The critical forces that cause yielding onset, and the corresponding interferenc es a d radius or half-width contact are derived along with the maximum stored strain energy. It is shown that the distressing stresses decrease as Poisson’s ratio increases (i.e., as the material approaches incompressibility) . The results obtained herein are then used to calibrate FEA meshes intended for cases that do not have closed-form solutions.

Finite-amplitude Rayleigh-Bénard convection and pattern selection for viscoelastic fluids

Abstract: The influence of inertia and elasticity on the onset and stability of Rayleigh–Benard thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The Galerkin projection method is used to obtain the departure from the conduction state. The solution is capable of displaying complex dynamical behaviour for viscoelastic fluids in the elastic and inertio-elastic ranges, which correspond to . The amplitude of motion is found to be little influenced by fluid elasticity or retardation time, especially as the Rayleigh number increases. However, the range of stability of the stationary thermal convection narrows considerably for viscoelastic fluids. In this case, oscillatory thermal convection is favoured. The onset and the stability of other steady convective patterns, namely hexagons and squares, are studied in the inertio-elastic range by using an amplitude equation approach. The range of stability of each pattern is examined, simultaneously allowing the validation of the two-dimensional picture.

Mixed Spectral Finite Elements for the Linear Elasticity System in Unbounded Domains

Abstract: In this paper, we present a mixed formulation of a spectral element approximation of the linear elasticity system. After studying the main features of this approach, we construct perfectly matched layers (PMLs) for modeling unbounded domains. Then, algorithmic issues are discussed and numerical results are given.

In vitro time-dependent response of periodontal ligament to mechanical loading

Abstract: This study examined the time-dependent response of bovine periodontal ligament (PDL). Applying linear viscoelastic theory, the objective was 1) to examine the linearity of the PDL's response in ter...