A PVDF Sensor for the In-situ Measurement of Stress Intensity Factors During Fatigue Crack Growth

AbstractSeveral analytical and numerical studies of inverse analysis are performed to verify the feasibility and accuracy of the proposed K-sensor. At first, the application to cracks in sheets under in-plane stresses is investigated and compared with the analytical solution for the GRIFFITH's crack under mixed mode. It was found that the convergence radius, where the electrodes have to be placed, must be smaller than half of the crack length, which is sufficient for real cracks of several millimeters. The obtained accuracy of crack tip location and (KI, KII)-factors is better than 1%. Second, the technique is applied to cracks in thin-walled plates of KIRCHHOFF type under bending and torsion moments. In this case, the plate intensity factors (k1, k2) are of interest. Again, the inverse identification procedure is studied by synthetic analytical and numerical solutions of simple crack configurations. Due to the assumptions of the KIRCHHOFF plate model, the sensors have to be placed outside a radius of 3 times plate thickness h. The obtained accuracy in position and intensity factors is quite sufficient as well. The practical realization of the K-factor sensor requires good electric signal measurement and amplification. Its experimental testing on components is ongoing work

Universality in the flooding of regular islands by chaotic states

We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.Comment: 11 pages, 12 figure

Correlations of chaotic eigenfunctions: a semiclassical analysis

We derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the Schr\"odinger equation. The system we considered is an energy-conserved Hamiltonian system possessing time-invariant symmetry. The energy smoothed autocorrelation function is expressed as a sum of three terms. The first one is analogous to Berry's conjecture, which is a Bessel function of the zeroth order. The second and the third terms are trace formulae made from special trajectories. The second term is found to be direction dependent in the case of spacing averaging, which agrees qualitatively with previous numerical observations in high-lying eigenstates of a chaotic billiard.Comment: Revtex, 13 pages, 1 postscript figur

Chaotic eigenfunctions in momentum space

We study eigenstates of chaotic billiards in the momentum representation and propose the radially integrated momentum distribution as useful measure to detect localization effects. For the momentum distribution, the radially integrated momentum distribution, and the angular integrated momentum distribution explicit formulae in terms of the normal derivative along the billiard boundary are derived. We present a detailed numerical study for the stadium and the cardioid billiard, which shows in several cases that the radially integrated momentum distribution is a good indicator of localized eigenstates, such as scars, or bouncing ball modes. We also find examples, where the localization is more strongly pronounced in position space than in momentum space, which we discuss in detail. Finally applications and generalizations are discussed.Comment: 30 pages. The figures are included in low resolution only. For a version with figures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp99-2.htm

About ergodicity in the family of limacon billiards

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure.Comment: 17 pages, 9 figure

Autocorrelation function of eigenstates in chaotic and mixed systems

We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation length. For localized states, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales.Comment: 30 pages, 12 figures. Some of the pictures are included in low resolution only. For a version with pictures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab

Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards

We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits GUE-like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunneling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.Comment: 18 pages, 8 figures, with 3 additional GIF animations available at http://chaos.fiz.uni-lj.si/~veble/boundary

Nano-wires with surface disorder: Giant localization lengths and dynamical tunneling in the presence of directed chaos

We investigate electron quantum transport through nano-wires with one-sided surface roughness in the presence of a perpendicular magnetic field. Exponentially diverging localization lengths are found in the quantum-to-classical crossover regime, controlled by tunneling between regular and chaotic regions of the underlying mixed classical phase space. We show that each regular mode possesses a well-defined mode-specific localization length. We present analytic estimates of these mode localization lengths which agree well with the numerical data. The coupling between regular and chaotic regions can be determined by varying the length of the wire leading to intricate structures in the transmission probabilities. We explain these structures quantitatively by dynamical tunneling in the presence of directed chaos.Comment: 15 pages, 12 figure

Fractional-Power-Law Level-Statistics due to Dynamical Tunneling

For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution P(s) over a wide range of small spacings s. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of P(s) which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant h.Comment: 4 pages, 2 figure