Exponential Energy Function Research Articles

Evaluation of the Use of the Yeoh and Mooney-Rivlin Functions as Strain Energy Density Functions for the Ground Substance Material of the Annulus Fibrosus

Due to the importance of the intervertebral disc in the mechanical behavior of the human spine, special attention has been paid to it during the development of finite element models of the human spine. The mechanical behavior of the intervertebral disc is nonlinear, heterogeneous, and anisotropic and, due to the low permeability, is usually represented as a hyperelastic model. The intervertebral disc is composed of the nucleus pulposus, the endplates, and the annulus fibrosus. The annulus fibrosus is modeled as a hyperelastic matrix reinforced with several fiber families, and researchers have used different strain energy density functions to represent it. This paper presents a comparative study between the strain energy density functions most frequently used to represent the mechanical behavior of the annulus fibrosus: the Yeoh and Mooney-Rivlin functions. A finite element model of the annulus fibrosus of the L4-L5 segment under the action of three independent and orthogonal moments of 8 N-m was used, employing Abaqus software. A structured mesh with eight divisions along the height and the radial direction of annulus fibrosus and tetrahedron elements for the endplates were used, and an exponential energy function was employed to represent the mechanical behavior of the fibers. A total of 16 families were used; the fiber orientation varied with the radial coordinate from 25° on the outer boundary to 46° on the inner boundary, measuring it with respect to the transverse plane. The mechanical constants were taken from the reported literature. The range of motion was obtained by finite element analysis using different values of the mechanical constants and these results were compared with the reported experimental data. It was found that the Yeoh function showed a better fit to the experimental range of motion than the Mooney-Rivlin function, especially in the nonlinear region.

Elastic constants influence on the L4-L5-S1 annuli fibrosus behavior, a probabilistic finite element analysis.

A probabilistic finite element (FE) analysis of the L4-L5 and L5-S1 human annulus fibrosus (AF) was conducted to obtain a better understanding of the biomechanics of the AF and to quantify its influence on the range of motion (ROM) of the L4-L5 and L5-S1 segments. The FE models were composed of the AF and the upper and lower endplates. The AF was represented as a continuous material composed of a hyperelastic isotropic Yeoh matrix reinforced with two families of fibers described with an exponential energy function. The caudal endplate was fully restricted and 8 Nm pure moment was applied to the cranial endplate in flexion, extension, lateral flexion and axial rotation. The mechanical constants were determined randomly based on a normal distribution and average values reported. Results of the 576 models show that the ROM was more sensitive to the initial stiffness of the fibers rather than to the stiffening coefficient represented in the exponential function. The ROM was more sensitive to the input variables in extension, flexion, axial rotation and lateral bending. The analysis showed an increased probability for the L5-S1 ROM to be higher in flexion, extension and axial rotation, and smaller in lateral flexion, with respect to the L4-L5 ROM. An equation was proposed to obtain the ROM as a function of the elastic constants of the fibers and it may be used to facilitate the calibration process of the human spine segments and to understand the influence of each elastic constant on the ROM.

FAST MAGNETIC RECONNECTION AND PARTICLE ACCELERATION IN RELATIVISTIC LOW-DENSITY ELECTRON-POSITRON PLASMAS WITHOUT GUIDE FIELD

Magnetic reconnection and particle acceleration in relativistic Harris sheets in low-density electron-positron plasmas with no guide field have been studied by means of two-dimensional particle-in-cell simulations. Reconnection rates are of the order of one when the background density in a Harris sheet is of the order of 1% of the density in the current sheet, which is consistent with previous results in the non-relativistic regime. It has been demonstrated that the increase of the Lorentz factors of accelerated particles significantly enhances the collisionless resistivity needed to sustain a large reconnection electric field. It is shown analytically and numerically that the energy spectrum of accelerated particles near the X-line is the product of a power law and an exponential function of energy, γ–1/4exp (– aγ1/2), where γ is the Lorentz factor and a is a constant. However, in the low-density regime, while the most energetic particles are produced near X-lines, many more particles are energized within magnetic islands. Particles are energized in contracting islands by multiple reflection, but the mechanism is different from Fermi acceleration in magnetic islands for magnetized particles in the presence of a guide field. In magnetic islands, strong core fields are generated and plasma beta values are reduced. As a consequence, the fire-hose instability condition is not satisfied in most of the island region, and island contraction and particle acceleration can continue. In island coalescence, reconnection between two islands can accelerate some particles, however, many particles are decelerated and cooled, which is contrary to what has been discussed in the literature on particle acceleration due to reconnection in non-relativistic hydrogen plasmas.

Fung's model of arterial wall enhanced with a failure description.

One of the seminal contributions of Y.C. Fung to biomechanics of soft tissue is the introduction of the models of arterial deformation based on the exponential stored energy functions, which are successfully used in various applications. The Fung energy functions, however, explain behavior of intact arteries and do not include a description of arterial failure. The latter is done in the present work where Fung's model is enhanced with a failure description. The description is based on the introduction of a limiter for the stored energy - the average energy of chemical bonds, which can be interpreted as a failure constant characterizing the material 'toughness'. The limiting failure energy controls materials softening, which indicates the onset of failure. We demonstrate the efficiency of the enhanced Fung formulation on a problem of the arterial inflation under internal pressure. We show, particularly, that residual stresses delay the onset of failure. The considered softening hyperelasticity approach is an alternative to the simplistic pointwise failure criteria of strength of materials on the one hand and the sophisticated approach of damage mechanics involving internal variables on the other hand.

Remodeling of strain energy function of common bile duct post obstruction

Biliary duct obstruction is an important clinical condition that affects millions of people worldwide. We have previously shown that the common bile duct (CBD) undergoes significant growth and remodelling post obstruction. The mechanical stress-strain relation is expected to change due to growth and remodeling in response to obstruction and hence pressure-overload. The objective of the present study was to characterize the material properties of the CBD of the sham group and at 3 hours, 12 hours, 2 days, 8 days and 32 days (n=5 in each group) after obstruction. The Fung's exponential strain energy function was used to relate stress to strain of the normal and remodeled ducts. The Marquardt-Levenberg and genetic algorithm methods were used for the determination of material constants for the exponential strain energy function. The elastic constants obtained by the two methods did not differ (p>0.254). We obtained stable and convergent material constants at every time point. The material constant that dictates the nonlinear rate of change in the circumferential stress with respect to strain increased significantly after obstruction acutely in the first 12 hrs., decreased at 2 days and finally increased montonically during the remaining 30 days. The acute initial increase in stiffness was simply due to the elastic response. The chronic remodeling process results in an initial circumferential softening followed by stiffening of the CBD. This study has important clinical significance for patients with CBD obstruction and for endoscopists and surgeons who treat the condition.

Modeling of a plasma processing machine for semiconductor wafer etching using energy-functions-based neural networks

The complex processing of plasma etching and deposition is highly nonlinear and its modeling is intractable by analytical basic-principles techniques. Neural network approaches have shown initial success for specific plasma processes in extracting implicit relations/models based on input-output measurements. The resulting modeling techniques naturally depend on the neural structure, the adopted learning algorithms, and the specific plasma process and machine. We describe a plasma processing machine designed and in operation at Michigan State University, East Lansing, which has been equipped with select sensing devices. The machine exhibits a hysteresic nonlinearity in the desirable processing modes of operation. The experimental data characterize a testbed plasma etching process using Argon gas with control inputs including incident microwave power, pressure, and cavity size. The internal states and the outputs include reflected power, electric field, and ion density. We employ several tailored networks with novel learning algorithms derived from functions that include the polynomial and the exponential energy functions. It is shown that the learning algorithms enable fast and satisfactory convergence of parameters (weights and biases) in several scenarios of modeling and generalizing the input-state-output relations of the plasma process.

Evidence for weak electron-multiphonon coupling in a-Si:H and alloys from transient photocurrent analyses

The temperature dependence of the dispersion parameter α(T) has been analysed by transient secondary photocurrent measurements in a-Si:H, a-SiGe:H and a-SiC:H. It has turned out, that weak-electron multiphonon coupling (WEMPC) determines the thermalization of electrons into localized tail states and that the trap probability b t is an exponential function of energy with a characteristic energy kT P = 40 meV in the range T<300K. Increasing temperature (T > 300K) increases kT P too. In comparison to the conventional multiple trapping model (b t =const.) the movement of the demarcation energy E d (t) is dependent on kT P and is considerably slowed down.

Breakup of helium nuclei in high-temperature plasmas

The rate of breakup of helium nuclei by particle-induced reactions is computed. It is shown that the rate is determined by the endothermic reaction p+ 4He→ 3He+d, becoming effective at kT≳few MeV; the characteristic breakup temperature has a logarithmic dependence on density and available time. The rate is computed from experimental data on the reaction cross section, extrapolating to lower energies near threshold with insertion of Coulomb-barrier penetration factors. The Coulommb factor for the outgoing particles, being essentially an exponential function of energy, results in a “Gamow peak” for the reaction rate and, effectively, a shift of the threshold energy from 18 MeV to about 24 MeV. Competing particle-induced breakup reactions are considered, in particular,the process p+ 4He→ 3He+p+n. It is found that the energy dependence of the cross section ( σ 3) for this reaction can be understood in terms that ( σ 2) of the former reaction (→ 3He+d) from phase space considerations. The magnitude of the calculated σ 2/σ 3 also agrees well with experimental data, being determined by the 2- and 3-body phase space factors and the short-range deuteron wave function u d(0). A brief discussion is given of the possibility of the occurence of helium breakup in astrophysical plasmas. The phenomenon might be seen in matter ejected from the vicinity of black holes, for example.

Background of coincidence type beta-ray scintillation spectrometer

The performance of the low background β-ray scintillation spectrometer using a coincidence method with a G.M. counter is described. The background spectrum of the coincidence spectrometer is correlated theoretically to the background spectrum of the scintillation counter. If the latter spectrum is represented by a simple exponential function of energy E, B 0 e − μE, the former spectrum is also similar in shape and its counting rate is reduced by a factor of ϵκS/4 μV, where ϵ, V, S and 1/κ are the intrinsic efficiency of the G.M. tube, the volume of the scintillator, the total surface area of the G.M. tube, and the energy loss of electron per unit path length in the scintillator respectively. The results of calculation were examined experimentally with three detectors of different shapes and sizes.