Mode II intersonic crack propagation in poroelastic media

A crack is steadily running in an elastic isotropic fluid-saturated porous solid at an intersonic constant speed c. The crack tip speeds of interest are bounded below by the slower between the slow longitudinal wave-speed and the shear wave-speed, and above by the fast longitudinal wave-speed. Biot’s theory of poroelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities n ∈ [0, 1] is investigated. Solids are obtained as the limit case n = 0, and the continuity of the energy release rate as the porosity vanishes is addressed. Three characteristic regions in the plane (n, c) are delineated, depending on the relative order of the body wave-speeds. Mode II loading conditions are considered, with a permeable crack surface. Cracks with and without process zones are envisaged. In each region, the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressure and velocity fields near the tip of the crack. For subsonic propagation, the asymptotic crack tip fields are known to be continuous in the body [Loret and Radi (2001) J Mech Phys Solids 49(5):995–1020]. In contrast, for intersonic crack propagation without a process zone, the asymptotic stress and pore pressure might display a discontinuity across two or four symmetric rays emanating from the moving crack tip. Under Mode II loading condition, the singularity exponent for energetically admissible tip speeds turns out to be weaker than 1/2, except at a special point and along special curves of the (n, c)-plane. The introduction of a finite length process zone is required so that 1. the energy release rate at the crack tip is strictly positive and finite; 2. the relative sliding of the crack surfaces has the same direction as the applied loading. The presence of the process zone is shown to wipe out possible first order discontinuities.     

Asymptotic normality of the Nadaraya–Watson semivariogram estimators

In this work, the Nadaraya–Watson semivariogram estimation is considered for both the isotropic and the anisotropic settings. Several properties of these estimators are analyzed and, particularly, their asymptotic normality is established in terms of unknown characteristics of the random process. The latter provides a theoretical procedure for construction of confidence intervals for the semivariogram via the normal quantiles, which in practice must be appropriately estimated. A numerical study is included to illustrate the performance of the Nadaraya–Watson estimation when used to obtain confidence intervals.      

Passivity-based sliding mode control for a polytopic stochastic differential inclusion system

Passivity-based sliding mode control for a polytopic stochastic differential inclusion (PSDI) system is considered. A control law is designed such that the reachability of sliding motion is guaranteed. Moreover, sufficient conditions for mean square asymptotic stability and passivity of sliding mode dynamics are obtained by linear matrix inequalities (LMIs). Finally, two examples are given to illustrate the effectiveness of the proposed method.     

Approximations to expected stopping times with applications to sequential estimation

In this paper we consider sharp asymptotic aproximations to renewal functions associated with general stochastic sequences. We establish a variant of Smith's renewal theorem with a bounded remainder term imposing moment conditions on the underlying stochastic sequence. This result is then applied to an asymptotic analysis of stopping times arising in sequential estimation problems. We show that the renewal theorem can be used for deriving approximations to the expected values of first passage times for random walks with dependent increments     

Sequential fixed width confidence intervals for regression parameters from censored data with a discrete covariate

Sequential fixed width interval estimation procedure is considered in a simple linear regression model where the independent variable (covariate) assumes only a finite number of values and the dependent variable (response) is randomly right censored. The censoring distribution may depend on the covariate values. The sequential procedure is shown to be consistent and efficient as the width of the confidence interval decreases to zero. The asymptotic distribution of the underlying stopping rale is obtained     

The performance of a sequential procedure for the estimation of the median survival time with specified accuracy

A bound on the order of convergence of the remainder term in the coverage probability of a fixed quantile of the survival distribution is obtained as the prescribed tolerance on the width of the interval shrinks to zero.     

Effect of Solubles on Disintegration of Distiller's Spent Grain Compacts During Superheated Steam Drying

Drying of densified spent grain in superheated steam (SS) may cause breakage and disintegration of the product in the initial stage of SS drying. The present work investigated the effect of solubles (0, 10, 30, 50, and 70% solubles) on the percentage change in length, diameter, volume; hardness; and the asymptotic modulus (E_A) of the cylindrical compacts during SS drying. An increase in dimensions along with a decrease in hardness and E_A of the compact was observed immediately after exposing it to SS. We observed a 47% increase in length and 12% increase in diameter for compacts having 0% solubles during the first five seconds of SS drying; however, when the soluble content increased to 70%, the percentage increase in length and diameter became 12 and 2%, respectively. Also, with an increase in the amount of solubles in the compact, a considerable increase in the hardness and E_A of the compact was observed. The study also aimed to find suitable explanatory variables for predicting the hardness and E_A of the compact using a stepwise forward regression method.     

Asymptotic stabilization of planar systems

Sufficient conditions for the existence of a C¹ asymptotically stabilizing feedback control for planar, affine systems are given. The methods are constructive and yield homogeneous feedback controls for homogeneous systems which allow the results to be extended to more general systems.     

Asymptotic modeling of the long-period oscillations of tribological parameters in the wear process of metals under heavy duty sliding conditions with application to structural health monitoring

Different aspects of the long-period oscillations of tribological parameters in the wear process of metals under constant heavy duty sliding conditions are theoretically investigated on the basis of the previously constructed mathematical model. The obtained asymptotic formula for the period of oscillations agrees with the quantitative predictions known in the literature. The period of oscillations is suggested as a diagnostic parameter in assessing the health of operating tribosystem. An approximate formula for the phase of oscillations of relaxation type in the Lotka-Volterra nonlinear mathematical model is derived by the relaxation oscillation method.     

Theory of initially twisted, composite, thin-walled beams

An asymptotically correct theory for initially twisted, thin-walled, composite beams has been constructed by the variational asymptotic method. The strain energy of the original, three-dimensional structure is first rigorously reduced to be a two-dimensional energy expressed in terms of shell strains. Then the two-dimensional strain energy is further reduced to be expressed in terms of the classical beam strain measures. The resulting theory is a classical beam model approximating the three-dimensional energy through the first-order of the initial twist. Consistent use of small parameters that are intrinsic to the problem allows a natural derivation for all thin-walled beams within a common framework, regardless of whether the section is open, closed, or strip-like. Several examples are studied using the present theory and the results are compared with a general cross-sectional analysis, VABS, and other published results.