ON COMPARATIVE GROWTH RELATIONSHIP OF ITERATED ENTIRE FUNCTIONS FROM THE VIEWPOINT OF SLOWLY CHANGING

A positive continuous function L= L(r) is called slowly if L(ar) ~ L(r) as r͢-∞   for every positive constant “a”. Lakshminarasimhan [14] introduced the idea of the functions of L-bounded index. Later Lahiri and Bhattacharjee [16] worked on the entire functions (i.e., functions analytic in the finite complex plane) of L-bounded index and of non uniform L-bounded index. The growth of an entire function f with respect to another entire function g is de.ned as the ratio of their maximum moduli for sufficiently large values of r. The same may be de.ned in terms of maximum terms as well as Nevanlinna’s characteristic functions of entire functions. In this paper we would like to investigate some comparative growth analysis of iterated entire functions (as de.ned by Lahiri and Banerjee [15]) on the basis of their maximum terms, maximum moduli and Nevanlinna.s characteristic functions and obtain some powerful results with a scope of further research in the concerned area

Experimental And Numerical Study Of Sonic Wave Propagation In Freezing Sand And Silt

Thesis (Ph.D.) University of Alaska Fairbanks, 2009A numerical model for delineating the temperature-velocity relationship of freezing porous media and soil is developed in Matlab based on Leclaire's Biot-type three-phase theory. Leclaire's theory gives lower sonic velocities than the experimental results because it does not take into consideration the effect of the solid-ice frame when water is freezing. To take the solid-ice effective frame into account, the average bulk and shear moduli estimation are modified with a proposed procedure. The modification gives higher P-wave and S-wave velocities that fit experimental data well. A comprehensive suite of physical and acoustic laboratory experiments are conducted on artificial sands, sand-clay mixtures and Fairbanks silts to investigate the temperature-velocity relationship during the freezing process and the effects of grain size and fine clay content. A Multi-channel ultrasonic scanning system (MUSS) is designed, installed and programmed for the experimental computerized ultrasonic tomography (CUST) study. The inward and outward freezing process and freezing front development in Fairbanks silt samples are observed using computerized ultrasonic tomography (CUST) in the laboratory. The experiments generate sonic wave velocity and temperature distribution during the freezing process. The freezing front is clearly identified in the CUST as a function of time and temperature. Comprehensive numerical finite element method (FEM) simulations, which account for the conduction in porous media, the latent heat effect and the nonlinear thermal properties of soil, are performed on the inward and outward freezing process of Fairbanks silt based on the experimental conditions. In conjunction with the temperature-velocity model developed in the study, sonic wave velocity tomograms are generated. The results are comparable with those obtained by CUST. The study indicates that CUST is an effective method for studying freezing processes and has potential for indirect measurement of unfrozen water content variations in the soil without interfering with the freezing process

Molecular Dynamic Simulation of Structures and Interfaces in Amorphous/Ordered Composites.

This thesis describes molecular dynamics simulation studies of the structure-property relationships of molecular network systems, including inorganic and organic bulk amorphous systems, as well as two different amorphous polymers at the interface with ordered substrates. A series of soda lime silicate glasses were simulated, with up to 50% total modification and varying ratios of sodium and calcium. The clustering of cations and second-neighbor connectivity affect vibrational modes and the compressibility vs. pressure behavior. Mean-field theory is unable to account for mixed modifier effects in soda lime silicates. The structure and tensile behavior of a dynamically reacted bulk epoxy network were studied, demonstrating an improved polymerization method for continuously monitoring properties as a function of network growth, including volumetric shrinkage and internal stresses. A bifunctional epoxy resin is reacted with two aliphatic amines at room temperature, comparing simulation size, amine functionality, and stoichiometry. The elastic properties change by only 1-2 GPa during the growth of the network within the achieved degree of conversion. Tensile strength increases by ~100 MPa. Systems with surplus amine hardener reach higher degrees of epoxide conversion, but lag in formation of an infinite network. As a simple model system for amorphous/ordered interfaces, a thin alkane film was placed onto a metallic substrate. The ordered substrate creates a layered polymer configuration within the adjacent 10 Å, as shown by density profiles, pair correlation functions, and monomer orientation statistics. This structural change also affects the mechanical properties, as the elastic moduli of nanoconfined alkane systems are higher than would be expected for a simple laminate composite, based on extrapolating from the bulk properties of the two materials. Lastly, epoxy/carbon laminate systems were investigated, comparing different epoxy layer thicknesses and amine functionality. The cure and shrinkage behavior mimic the bulk epoxy, though the percolation of an infinite cluster is delayed. Post-annealed structures show a nearly uniform decrease in both the elastic modulus and tensile strength. Local heterogeneity is important in predicting nanoscale mechanics for all systems investigated. Larger system size provides better accuracy in determining mechanical properties of simulated highly cross-linked network polymers.PHDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111417/1/kabeck_1.pd

A Finite Element Framework for Multiscale/Multiphysics Analysis of Structures with Complex Microstructures

This research work has contributed in various ways to help develop a better understanding of textile composites and materials with complex microstructures in general. An instrumental part of this work was the development of an object-oriented framework that made it convenient to perform multiscale/multiphysics analyses of advanced materials with complex microstructures such as textile composites. In addition to the studies conducted in this work, this framework lays the groundwork for continued research of these materials. This framework enabled a detailed multiscale stress analysis of a woven DCB specimen that revealed the effect of the complex microstructure on the stress and strain energy release rate distribution along the crack front. In addition to implementing an oxidation model, the framework was also used to implement strategies that expedited the simulation of oxidation in textile composites so that it would take only a few hours. The simulation showed that the tow architecture played a significant role in the oxidation behavior in textile composites. Finally, a coupled diffusion/oxidation and damage progression analysis was implemented that was used to study the mechanical behavior of textile composites under mechanical loading as well as oxidation. A parametric study was performed to determine the effect of material properties and the number of plies in the laminate on its mechanical behavior. The analyses indicated a significant effect of the tow architecture and other parameters on the damage progression in the laminates

Detection of Delaminations in Carbon Fiber Reinforced Polymers Embedded with Terfenol-D Particles Using Machine Learning

The characterization of the damage state of a system provides insight into its performance and safety during operation. In composite materials, specifically, fiber-reinforced polymers, delaminations form from the evolution of cracks in a matrix that leads to adhesion failure between adjacent laminae. Nondestructive evaluation (NDE) seeks to characterize the state of a material or system during non-operational times. A previously proposed NDE method employs embedded magnetostrictive particles between laminae of carbon fiber reinforced polymer (CFRP) for damage sensing. The phenomenon of magnetostriction couples the mechanical state of a material with its magnetic state so that a change in the local stress field alters its magnetic susceptibility. The change in magnetic susceptibility is measured using an induced sensing voltage. This work aims to provide a preliminary exploration of machine learning to predict the presence of a delamination using the embedded magnetostrictive particle NDE method with CFRP laminates. Machine learning algorithms\u27 ability to decipher and develop relationships among input features attracted its use since the visual examination of the experimental induced sensing voltage plots yielded inconsistent delamination predictions. This work investigated the feasibility of an analytical model based on the Euler-Bernoulli beam theory to generate data. This model utilized functional relationships to characterize the nonlinear behavior of the magnetostrictive material Terfenol-D. Fourier series relationships reduced the error in the characterization of the properties over the previously proposed functions. The previously proposed derived model failed to converge for the calculation of stress within the magnetostrictive material. Eight machine learning algorithms were employed using a Python script to classify the presence of a delamination within a unidirectional HexPly AS4/3501-6 CFRP embedded with Terfenol-D particles. The maximum accuracy achieved was approximately 80, whereas the average accuracy for all the models was just below 71%. The multi-layer perceptron (MLP) models, a neural network algorithm, produced the highest prediction accuracy for this two-class classification problem because of their ability to account for nonlinear relationships. A parametric study involving the architecture and activation function was necessary for the MLP models because of the range of obtained prediction accuracies. A direct relationship was observed between the number of hidden layers and the accuracy outliers. Despite the accuracy of the examined machine learning models\u27 being less than that of other NDE applications, this preliminary investigation demonstrated that machine learning could be paired with previously indiscernible experimental data to detect delaminations

Quantitative Ultrasonic Coda Wave (Diffuse Field) NDE of Carbon-Fiber Reinforced Polymer Plates

The increasing presence and applications of composite materials in aerospace structures precipitates the need for improved Nondestructive Evaluation (NDE) techniques to move from simple damage detection to damage diagnosis and structural prognosis. Structural Health Monitoring (SHM) with advanced ultrasonic (UT) inspection methods can potentially address these issues. Ultrasonic coda wave NDE is one of the advanced methods currently under investigation. Coda wave NDE has been applied to concrete and metallic specimens to assess damage with some success, but currently the method is not fully mature or ready to be applied for SHM. Additionally, the damage diagnosis capabilities and limitations of coda wave NDE applied to fibrous composite materials have not been widely addressed in literature. The central objective of this work, therefore, is to develop a quantitative foundation for the use of coda wave NDE for the inspection and evaluation of fibrous composite materials. Coda waves are defined as the superposition of late arriving wave modes that have been scattered or reflected multiple times. This results in long, complex signals where individual wave modes cannot be discriminated. One method of interpreting the changes in such signals caused by the introduction or growth of damage is to isolate and quantify the difference between baseline and damage signals. Several differential signal features are used in this work to quantify changes in the coda waves which can then be correlated to damage size and growth. Experimental results show that coda wave differential features are effective in detecting drilled through-holes as small as 0.4 mm in a 50x100x6 mm plate and discriminating between increasing hole diameter and increasing number of holes. The differential features are also shown to have an underlying basis function that is dependent on the hole volume and can be scaled by a material dependent coefficient to estimate the feature amplitude and size holes. The fundamental capabilities of the coda wave measurements, such as error, repeatability, and reproducibility, are also examined. Damage detection was found to be repeatable, reproducible, and relatively insensitive to noise. The measurements are found to be sensitive to thermal changes and absorbing boundaries. Several propagation models are also presented and discussed along with a brief analysis of coda wave signals and spectra

Viscoelasticity

This book contains a wealth of useful information on current research on viscoelasticity. By covering a broad variety of rheology, non-Newtonian fluid mechanics and viscoelasticity-related topics, this book is addressed to a wide spectrum of academic and applied researchers and scientists but it could also prove useful to industry specialists. The subject areas include, theory, simulations, biological materials and food products among others

Pressure and saturation estimation from PRM time-lapse seismic data for a compacting reservoir

Observed 4D effects are influenced by a combination of changes in both pressure and saturation in the reservoir. Decomposition of pressure and saturation changes is crucial to explain the different physical variables that have contributed to the 4D seismic responses. This thesis addresses the challenges of pressure and saturation decomposition from such time-lapse seismic data in a compacting chalk reservoir. The technique employed integrates reservoir engineering concepts and geophysical knowledge. The innovation in this methodology is the ability to capture the complicated water weakening behaviour of the chalk as a non-linear proxy model controlled by only three constants. Thus, changes in pressure and saturation are estimated via a Bayesian inversion by employing compaction curves derived from the laboratory, constraints from the simulation model predictions, time strain information and the observed fractional change in and . The approach is tested on both synthetic and field data from the Ekofisk field in the North Sea. The results are in good agreement with well production data, and help explain strong localized anomalies in both the Ekofisk and Tor formations. These results also suggest updates to the reservoir simulation model. The second part of the thesis focuses on the geomechanics of the overburden, and the opportunity to use time-lapse time-shifts to estimate pore pressure changes in the reservoir. To achieve this, a semi-analytical approach by Geertsma is used, which numerically integrates the displacements from a nucleus of strain. This model relates the overburden time-lapse time-shifts to reservoir pressure. The existing method by Hodgson (2009) is modified to estimate reservoir pressure change and also the average dilation factor or R-factor for both the reservoir and overburden. The R-factors can be quantified when prior constraints are available from a well history matched simulation model, and their uncertainty defined. The results indicate that the magnitude of R is a function of strain change polarity, and that this asymmetry is required to match the observed timeshifts. The recovered average R-factor is 16, using the permanent reservoir monitoring (PRM) data. The streamer data has recovered average R-factors in the range of 7.2 to 18.4. Despite the limiting assumptions of a homogeneous medium, the method is beneficial, as it treats arbitrary subsurface geometries, and, in contrast to the complex numerical approaches, it is simple to parameterise and computationally fast. Finally, the aim and objective of this research have been met predominantly by the use of PRM data. These applications could not have been achieved without such highly repeatable and short repeat period acquisitions. This points to the value in using these data in reservoir characterisation, inversion and history matching