A multiple scale theory is developed for the prediction of damage evolution in heterogeneous viscoelastic media. Asymptotic expansions of the field variables are used to derive a global scale viscoelastic constitutive equation that includes the effects of local scale damage. Damage, in the form discrete cracks, is allowed to grow according to a micromechanically-based viscoelastic traction-displacement law. Finite element formulations have been developed for both the global and local scale problems. These formulations have been implemented into a two-scale computational model Numerical results are given for several example problems in order to demonstrate the effectiveness of the technique.

In this master thesis, it is proposed to solve in the large thedifferential equationZ2(d2y/dz2) + z (dy/dz) (b0+b1zm) + (c0+c1zm)y = 0Here, m is an arbitrary positive integer, the variable z is complex as are the constants bi, ci (i = 0, 1). It is also assumed that the roots of the indicial equation about the regular singular point z=0 are such that their difference is incongruent to zero modulo m.

The first chapter comprises a survey of valuations on totally ordered structures, developing notation and properties. A contraction map is induced by the exponential map on the value group $G$ of an ordered exponential field $K$ with respect to the natural valuation $v_{G}$. By studying the algebraic properties of Abelian groups with contractions, the theory of these groups is shown to be model complete, complete, decidable and to admit elimination of quantifiers. Hardy fields provide an example of non-archimedean exponential fields and of differential fields and therefore, they play a very important role in our research.<p>In accordance with Rosenlicht we define asymptotic couples and then give a short exposition of some basic facts about asymptotic couples. The theory $T_{P}$ of closed asymptotic triples, as defined in Section 2.4, is shown to be complete, decidable and to have elimination of quantifiers. This theory, as well as the theory $T$ of closed $H$-asymptotic couples do not have the independence property. The main result of the second chapter is that there is a formal connection between asymptotic couples of $H$-type and contraction groups.<p>A given valuation of a differential field of characteristic zero is a differential valuation if an analogue of l'Hospital's rule holds. We present in the third chapter, a survey of the most important properties of a differential valuation. The theorem of M. Rosenlicht regarding the construction of a differential field with given value group is given with a detailed proof. There exists a Hardy field, whose value group is a given asymptotic couple of Hardy type, of finite rank. We also investigate the problem of asymptotic integration.

The first chapter comprises a survey of valuations on totally ordered structures, developing notation and properties. A contraction map is induced by the exponential map on the value group $G$ of an ordered exponential field $K$ with respect to the natural valuation $v_{G}$. By studying the algebraic properties of Abelian groups with contractions, the theory of these groups is shown to be model complete, complete, decidable and to admit elimination of quantifiers. Hardy fields provide an example of non-archimedean exponential fields and of differential fields and therefore, they play a very important role in our research.<p>In accordance with Rosenlicht we define asymptotic couples and then give a short exposition of some basic facts about asymptotic couples. The theory $T_{P}$ of closed asymptotic triples, as defined in Section 2.4, is shown to be complete, decidable and to have elimination of quantifiers. This theory, as well as the theory $T$ of closed $H$-asymptotic couples do not have the independence property. The main result of the second chapter is that there is a formal connection between asymptotic couples of $H$-type and contraction groups.<p>A given valuation of a differential field of characteristic zero is a differential valuation if an analogue of l'Hospital's rule holds. We present in the third chapter, a survey of the most important properties of a differential valuation. The theorem of M. Rosenlicht regarding the construction of a differential field with given value group is given with a detailed proof. There exists a Hardy field, whose value group is a given asymptotic couple of Hardy type, of finite rank. We also investigate the problem of asymptotic integration.

Fretting fatigue occurs when there is a small amplitude oscillatory movement between two contacting surfaces while the bodies are undergoing fatigue loading. Fretting fatigue conditions can substantially reduce the fatigue life of a component. Many engineering components such as Ti-6Al-4V gas turbine engine disks in military aircraft commonly experience fretting fatigue conditions that can potentially lead to catastrophic failure of critical components. The aim of this study is to characterize the behavior of Ti-6Al-4V under fretting fatigue conditions. Experiments are performed to analyze the influence of stress amplitude, stress ratio, and contact geometry. The effect of surface treatments such as low plasticity burnishing on the fretting fatigue life is also explored. The experimental results are being used to validate a proposed crack nucleation life prediction model. The proposed model utilizes a crack nucleation parameter H that is based on the strength of the singular stress field at the contact boundary. An advantage of this singular parameter is that neither a coefficient of friction nor the location of the stick/slip boundary needs to be determined. These two parameters are often difficult to define with certainty a priori. H is also independent of geometry making it well suited for use as a design parameter for designing structural joints and other fitted connections between components.

Three investigations were conducted on the nature of the radiance field in clear ocean water. It is important to understand the sunlight intensity below the sea surface because this leads to an understanding of how ocean creatures navigate in shallow and deep water. The nature of the radiance field is also gives an understanding of the living environment for ocean animals. Hydrolight 4.1, a simulation software developed by Curtis D. Mobley, was used to calculate the spectral radiance in clear ocean water for multiple wavelengths from the surface down through the asymptotic region. The first study found, as expected, that Raman scattering has little effect on wavelengths of light that are less than 500 nm. The effect of Raman scattering increases with increasing wavelength, and with increasing depth. The second study found the region of the water column where the radiance field is asymptotic. The third investigation found the effect of changing the mean square slope, or variance of the water-wave slope distribution. This effect is greatest near the surface and for a more truncated mean square slope integral. There are three peaks in percent difference to the ideal case, near the surface, one in the solar beam and the others near the critical angle of water.

In this paper, we derive the asymptotic distribution of the Augmented Dickey-Fuller t Test statistics, t_{ADF}, against a generalized fractional integrated process (for example: ARFIMA(p,1+d,q) ,|d|<1/2,and p, q be positive integer) by using the propositions of Lee and Shie (2003). Then we discuss why the power decreases with the increasing lags in the same and large enough sample size T when d is unequal to 0. We also get that the estimator of the disturbance's variance, S^2, has slightly increasing bias with increasing k. Finally, we support the conclusion by the Monte Carlo experiments.

A multiple scale theory is developed for the prediction of damage evolution in heterogeneous viscoelastic media. Asymptotic expansions of the field variables are used to derive a global scale viscoelastic constitutive equation that includes the effects of local scale damage. Damage, in the form discrete cracks, is allowed to grow according to a micromechanically-based viscoelastic traction-displacement law. Finite element formulations have been developed for both the global and local scale problems. These formulations have been implemented into a two-scale computational model Numerical results are given for several example problems in order to demonstrate the effectiveness of the technique.

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Under consideration are two-point boundary value problems for a system of second order differential equations which contains a small parameter multiplying the highest dereivatives. We prove the existence of solutions exhibiting left and right boundary layers by constructing upper and lower solutions of the system. The behavior of the solutions as the parameter tends to zero is also established. Of special interest is the existence of a compound boundary layer (i.e., one involving two scales) at the left endpoint of the interval.