Time-dependent Elasticity Research Articles

Forward Utility and Market Adjustments in Relative Investment-Consumption Games of Many Players

We study a portfolio management problem featuring many-player and mean-field competition, investment and consumption, and relative performance concerns under the forward performance processes (FPP) framework. We focus on agents using power (CRRA) type FPPs for their investment-consumption optimization problem an under common noise Merton model and we solve both the many-player and mean-field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a parameter we coin market consumption The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common behaviour. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle his risk-tolerance and of intertemporal (EIS) just like Epstein-Zin preferences under recursive utility framework and unlike the classical utility theory one. This, in turn, allows a finer analysis on the agent's consumption income and substitution regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent's perception of consumption in a non-trivial manner in addition to a time-dependent elasticity of conformity of the agent to the market's consumption intensity.

Mechanism behind Time Dependent Elasticity of Crumb Rubber-Nano-Asphalt Hybrids Using Discrete Relaxation Spectrum

Crumb rubber powder is a successfully used renewable material obtained from waste tire rubber, which has been incorporated into paving asphalt since the 1930s due to its good resistance to deformation and fatigue as well as its eco-friendly performance. In this study, carbon nanotubes and nano silica were incorporated into the terminal blend crumb rubber modified asphalt technology to remedy the issues of excessive desulfurization and degradation of ground tyre rubber with this technology. The mechanism behind the high temperature delayed elastic properties of the crumb rubber-nano-asphalt hybrids was experimentally investigated based on discrete relaxation spectrum. Development of the discrete relaxation spectra was accomplished by fitting on the 60°C storage modulus data tested by the dynamic shear rheometer using the generalized Maxwell model. Subsequently, the feasibility of characterizing delayed asphalt elasticity using main relaxation time was verified by test results from the 60°C creep and recovery test. Results indicated that the crumb rubber-nano-asphalt hybrids exhibited arrheodictic behavior and the asphalt elasticity was strengthened by two nano agents. Moreover, the elasticity reinforcement with carbon nanotubes was greater than with nano silica. Additionally, a good correlation was observed between the 60°C zero shear viscosity and main relaxation time, and greater 60°C zero shear viscosity was correlated to longer main relaxation times. Furthermore, longer main relaxation time of the asphalt was related to greater average recovery rate in the creep and recovery test. This research is expected to shed some light on the mechanism behind time-dependent elasticity of crumb rubber modified asphalt from the perspective of polymer physics.

Mathematical modelling of atrial and ventricular pressure–volume dynamics and their change with heart rate

This paper presents mathematical models that can simulate the cardiovascular system of a healthy sheep under normal resting conditions in which the heart rate changes significantly. The models include several new modelling features that are introduced progressively. The contraction of the cardiac chambers is modelled using a time-dependent muscle force with constant elasticity instead of time dependent elasticity. A new hypothesis about the mechanical contraction of the atria generates realistic pressure volume (PV) loops. The inter-ventricular interaction is modelled as well. Additionally, hysteresis is incorporated in the aortic valve to produce an end-systolic reverse (negative) flow. Most of the model parameter values are based on previous literature data while time periods of delay, atrial and ventricular contraction are derived using experimental data from 14 sheep. We provide new relationships between contraction time and delay as a function of heart period. The effects of different aspects of our modelling on the mean cardiac output, stroke volume, ejection time, ejection fraction and PV loops are studied. Model outputs are compared with published experimental results where possible, and are within a wide range of physiological observations.

Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencils

A new approach for the increase in the order of accuracy of high-order numerical techniques used for 2-D time-dependent elasticity (structural dynamics and wave propagation) has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding semi-discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of the linear finite elements with 9-point stencils is optimal and cannot be improved. However, we have calculated new 9-point stencils with the second order of accuracy that yield more accurate results than those obtained by the linear finite elements. We have also developed new 25-point stencils (similar to those for the quadratic finite and isogeometric elements) with the optimal sixth order of accuracy for the non-diagonal mass matrix and with the optimal fourth order of accuracy for the diagonal mass matrix. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for the high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by the high-order (up to the eighth order) finite elements with the non-diagonal mass matrix. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Damage of Solid Phase due to Explosion Effect Using Uzawa’s Algorithm

The paper deals with the impact of explosion on the damage development in the structure that forms a boundary of closed or semi-closed space – tunnel - the explosion in which is initiated. In the previous publications of the author the penalty formulation of the problem was used; now Uzawa’s algorithm is applied, which appears to be more promising algorithm accelerating the iterative process of damage development in the structure. Shock waves propagating in the airspace can be described by non-linear conservation equations starting with Euler’s equations which are used to be solved by forward differences of the system of pseudo-linear equations, using Arbitrary Lagrangian-Euler Method (ALE). Conservation equations are formulated for two phases: air and solid. In the solid phase, concrete lining, the time-dependent elasticity of free rectangular elements undergoing the damage, is solved by uniformly distributed boundary elements. The paper is focused on damage to the concrete lining due to impact of the explosion in the tunnel.

Study of temperature and pressure dependent elastic properties of porous ceramics

Resonant Ultrasound Spectroscopy (RUS) is a widely used experimental approach to investigate the elastic properties of solid materials at different temperature and pressure. In RUS, experimentally obtained vibrational resonance spectrum of a sample is used to calculate the full elastic tensor of the material. We report here a series of RUS measurements of novel porous ceramics material designed by LG which has applications in high temperature fuel cells. The elasticity and the acoustic damping of a porous material are influenced by its porosity and its saturation fluid. The time dependent elasticity obtained from RUS measurements exhibits a two-step process of stiffening with two distinct time constants. The stiffening-softening behavior during the heating-cooling cycles and the reduction of acoustic damping in the low-pressure conditions were observed depending on the synthesis history of the material.Resonant Ultrasound Spectroscopy (RUS) is a widely used experimental approach to investigate the elastic properties of solid materials at different temperature and pressure. In RUS, experimentally obtained vibrational resonance spectrum of a sample is used to calculate the full elastic tensor of the material. We report here a series of RUS measurements of novel porous ceramics material designed by LG which has applications in high temperature fuel cells. The elasticity and the acoustic damping of a porous material are influenced by its porosity and its saturation fluid. The time dependent elasticity obtained from RUS measurements exhibits a two-step process of stiffening with two distinct time constants. The stiffening-softening behavior during the heating-cooling cycles and the reduction of acoustic damping in the low-pressure conditions were observed depending on the synthesis history of the material.

Experiment study and FEM simulation on erythrocytes under linear stretching of optical micromanipulation

The elasticity of erythrocytes is an important criterion to evaluate the quality of blood. This paper presents a novel research on erythrocytes’ elasticity with the application of optical tweezers and the finite element method (FEM) during blood storage. In this work, the erythrocytes with different in vitro times were linearly stretched by trapping force using optical tweezers and the time dependent elasticity of erythrocytes was investigated. The experimental results indicate that the membrane shear moduli of erythrocytes increased with the increasing in vitro time, namely the elasticity was decreasing. Simultaneously, an erythrocyte shell model with two parameters (membrane thickness h and membrane shear modulus H) was built to simulate the linear stretching states of erythrocytes by the FEM, and the simulations conform to the results obtained in the experiment. The evolution process was found that the erythrocytes membrane thicknesses were decreasing. The analysis assumes that the partial proteins and lipid bilayer of erythrocyte membrane were decomposed during the in vitro preservation of blood, which results in thin thickness, weak bending resistance, and losing elasticity of erythrocyte membrane. This study implies that the FEM can be employed to investigate the inward mechanical property changes of erythrocyte in different environments, which also can be a guideline for studying the erythrocyte mechanical state suffered from different diseases.

Correlation between Viscoelastic Moduli and Atomic Rearrangements in Metallic Glasses.

Dynamical moduli, such as storage and loss moduli, characterize the viscoelasticity of materials (i.e., time-dependent elasticity) and convey important information about the relaxation processes of glasses and supercooled liquids. A fundamental question is what ultimately determines them in glassy materials. Here, for several model metallic glasses, we demonstrate that both the storage and loss moduli are uniquely determined by the most probable atomic nonaffine displacements, regardless of temperature or frequency. Moreover, the fast-moving atoms (which contribute to dynamical heterogeneity) do not contribute explicitly to the moduli. Our findings provide a physical basis for the origin of viscoelasticity in metallic glasses.

A Hybrid Model for Pricing and Hedging of Long Dated Bonds

Long dated fixed income securities play an important role in asset-liability management, in life insurance and in annuity businesses. This paper applies the benchmark approach, where the growth optimal portfolio (GOP) is employed as numeraire together with the real world probability measure for pricing and hedging of long dated bonds. It employs a time dependent constant elasticity of variance model for the discounted GOP and takes stochastic interest rate risk into account. This results in a hybrid framework that models the stochastic dynamics of the GOP and the short rate simultaneously. We estimate and compare a variety of continuous-time models for short-term interest rates using non-parametric kernel-based estimation. The hybrid models remain highly tractable and fit reasonably well the observed dynamics of proxies of the GOP and interest rates. Our results involve closed-form expressions for bond prices and hedge ratios. Across all models under consideration we find that the hybrid model with the 3/2 dynamics for the interest rate provides the best fit to the data with respect to lowest prices and least expensive hedges.

A Tractable Model for Indices Approximating the Growth Optimal Portfolio

The growth optimal portfolio (GOP) plays an important role in finance, where it serves as the numeraire portfolio, with respect to which contingent claims can be priced under the real world probability measure. This paper models the GOP using a time dependent constant elasticity of variance (TCEV) model. The TCEV model has high tractability for a range of derivative prices and fits well the dynamics of a global diversified world equity index. This is confirmed when pricing and hedging various derivatives using this index.

A tractable model for indices approximating the growth optimal portfolio

AbstractThe growth optimal portfolio (GOP) plays an important role in finance, where it serves as the numéraire portfolio, with respect to which contingent claims can be priced under the real world probability measure. This paper models the GOP using a time dependent constant elasticity of variance (TCEV) model. The TCEV model has high tractability for a range of derivative prices and fits well the dynamics of a global diversified world equity index. This is confirmed when pricing and hedging various derivatives using this index.

Viscoelasticity of stepped interfaces

Using molecular dynamics modeling, we show that interfaces in sputter deposited Cu-Nb superlattices exhibit time-dependent elasticity, i.e., viscoelasticity, under shear loading. In the high temperature and small strain rate limit, the interfacial shear modulus approaches a value proportional to the density of steps in the interface. It may therefore be possible to tailor the low-frequency shear moduli of interfaces by controlling their step densities.

A Hybrid Model for Equity Indices and Stochastic Interest Rates

The growth optimal portfolio (GOP) plays an important role in finance, in particular in derivative pricing, where it is employed as a num\'eraire portfolio, allowing to price contingent claims directly under the real world probability measure. This paper derives an extension of a time dependent constant elasticity of variance (TCEV) model which takes into account stochastic interest rate risk. This results in a hybrid framework that models the stochastic dynamics of the GOP and the short rate simultaneously. We estimate and compare a variety of continuous-time models for short-term interest rates using non-parametric kernel-based estimation. Taking interest rate dynamics into account, we show that the hybrid model remains highly tractable and fits well the observed dynamics of diversified equity indices and interest rates. Our results are important for pricing and hedging of various derivative products, allowing to derive closed-form solutions for standard derivatives. Across all models under consideration we find that the hybrid model with 3/2 dynamics for the interest rate provides the best fit to the data. It leads to the lowest prices and the least expensive hedges.

Numerical methods for solid mechanics on overlapping grids: Linear elasticity

This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids.

A flexible model for the pricing of perishable assets

We present a flexible and versatile model which addresses the problem of assigning optimal prices to assets whose value becomes zero after a fixed expiry date. (Such assets include the important example of seats on airline flights.) Our model is broad in scope, in particular encompassing the ability to deal with arrivals of customers in groups. It is highly adaptable and can be adjusted to deal with a very extensive set of circumstances.Our approach to the problem is based on elementary and intuitively appealing ideas. We model the arrival of customers (or groups of customers) according to an inhomogeneous Poisson process. We incorporate into the model time dependent price sensitivity (which may also be described as “time dependent elasticity of demand”). In this setting the solution to the asset pricing problem is achieved by setting up coupled systems of differential equations which are readily amenable to numerical solution via (for instance) a vectorised Runge–Kutta procedure. An attractive feature of our approach is that it unifies the treatment of discrete and continuous prices for the assets.

Toughening effect of cellulose nanowhiskers on polyvinyl acetate: Fracture toughness and viscoelastic analysis

AbstractThe toughening effect of cellulose nanowhiskers (CNWs) on modified polyvinyl acetate (PVAc) was analyzed with the help of morphology, relaxation, and creep behavior. The CNWs together with bound moisture at the matrix/whisker interfaces resulted in significant improvement in resistance to crack initiation and propagation. The magnitude of plastic deformation of the nanocomposites was higher than that of the neat PVAc. The relaxation temperature decreased, while the width of the damping peak increased with increasing CNW and moisture contents. The results from creep modeling showed that the instantaneous elastic modulus first increased and then decreased with the addition of CNWs, while the time‐dependent elasticity and viscosity decreased. The results suggested that the reinforcing effect of the CNWs was overwhelmed by the plasticizing effect of the bound moisture. Furthermore, low concentrations of CNWs significantly improved the fracture toughness of PVAc at the minor cost of strength, stiffness, and creep resistance. In this article, we present a novel approach to studying the toughening effect of CNWs on polymers using fracture tests and viscoelastic modeling. POLYM. COMPOS., 2011. © 2011 Society of Plastics Engineers

MEMORY EFFECT IN HOMOGENIZATION OF A VISCOELASTIC KELVIN–VOIGT MODEL WITH TIME-DEPENDENT COEFFICIENTS

This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin–Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.

Constitutive Relation of Yunjialing Anthracite Under Medium Strain Rate

Abstract By means of the split Hopkinson pressure bar (SHPB) testing system, this paper presents a dynamic constitutive relation of anthracite at a strain rate of e=5–85s−1. Generally, the dynamic stress-strain curve for this kind of anthracite under uni-axial compression has the following four stages: a non-linear loading stage, a plastic yielding stage, a strain-strengthening stage and an unloading breakage stage. Correspondingly, the initial elastic modulus Eb, the yielding strength σs and the ultimate strength σb increase along with an increasing strain rate. The time-dependent elasticity was identified when we analyzed the mechanical properties of anthracite. Based on characteristics of measured dynamic stress–strain curves and an analysis of existing rock dynamic constitutive models, as well as a preparatory simulation, a new visco-elastic damage model has been introduced in this paper. A linear spring is put parallel to two Maxwell units with different relaxation times to express two distinct plastic flows. The damage D is equal to [Eb–E(ɛi)]/Eb, where Eb is the beginning modulus and the E(ɛi) is the slope of a connected line between the origin point and any other point on a tested stress-strain curve. In the new constitutive model, one Maxwell unit with low relaxation time φ1 is used to describe the response of anthracite to a low strain rate, while the other, with a high relaxation time φ2 describes the response of anthracite to a high strain rate. Simulated stress-strain curves from the new model are consistent with the measured curves.

Sol – Gel Modelling Associated with the Rheology of Polymeric Precursors of Ceramic Materials

Abstract A general constitutive relation describing the change of viscoelastic behavior during the liquid - solid (sol - gel) transition which takes place in preceramic polymers is derived on the basis of Jeffrey’s 3-constants model with time dependent viscosities and elasticity. It is postulated that the sol - gel - transition can be analyzed analogous to the solutions of the Avrami equation used for modeling crystallization processes. Two different polymer systems used as precursor for the production of ceramic materials are investigated here: i) a mixture based on polysiloxane, alumatrane and isopropanol; ii) a non-oxidic carbodiimide gel based on the reaction of chlorosi-lanes with bis(trimethylsilyl)carbodiimide. Continuous measurements of the dynamic moduli versus reaction time, as well as creep tests at constant shear stress, evidenced both qualitative similarities and quantitative differences associated with the sol - gel transition of the two polymer systems. The shear rate and viscosity dependence of reaction time in creep tests, respectively the evolution of Lissajous figures associated with oscillatory experiments, are found to be consistent with the numerical simulations of the proposed constitutive relation.

Debris Effects on EHL Contact

A model was developed to study the effects of a rigid debris on elastohydrodynamic lubrication of rolling/sliding contacts. In order to achieve the objectives the time dependent Reynolds equation was modified to include the effects of an ellipsoidal shaped debris. The modified time dependent Reynolds and elasticity equations were simultaneously solved to determine the pressure and film thickness in EHL contacts. The debris force balance equation was solved to determine the debris velocity. The model was then used to obtain results for a variety of loads, speeds, and debris sizes. The results indicate that the debris has a significant effect on the pressure distribution and causes a dent on the rolling/sliding bounding surfaces. Depending on the size and location of the debris the pressure generated within the contact can be high enough to plastically deform the bounding surfaces. Debris smaller than the minimum film thickness do not enter the contact and only large and more spherical debris move toward the contact. [S0742-4787(11)00501-7]