Spectral isolation of naturally reductive metrics on simple Lie groups

We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.Comment: 19 pages, new title and abstract, revised introduction, new result demonstrating that any collection of isospectral compact symmetric spaces must be finite, to appear Math Z. (published online Dec. 2009

Orientation effects on the measurement and analysis of critical CTOA in an aluminum alloy sheet

Fracture tests were conducted on 76.2mm wide, 2.3mm thick middle crack tension (M(T)) specimens machined from 2024-T3 aluminum sheet. The specimens were tested on the T-L orientation and comparisons were made to similar tests conducted in the L-T orientation. Measurement of critical crack tip opening angle (CTOA), applied stress, and crack front shape were made as a function of crack extension. A two-dimensional, elastic-plastic finite element analysis was used to simulate the fracture behavior for both orientations. The results indicate that the T-L orientation had a 10 percent lower stress at fracture than similar tests conducted in the L-T orientation. Correspondingly, the critical CTOA in the T-L tests reached a constant value of 4.7 degrees after 2-3mm of crack extension and the L-T tests reached a value of 6 degrees. The fracture surfaces of the T-L specimens were observed to remain flat, while those of the L-T specimens transitioned to a 45 degree slant fracture after about 2-3mm of crack extension. The tunneling behavior of the two orientations also differed; the T-L specimens reached a deeply tunneled stabilized crack front shape while, the L-T specimens were observed to have only a small amount of tunneling once the crack began to grow on the 45 degree slant. The two-dimensional, elastic-plastic finite element analysis was able to simulate the fracture behavior for both the T-L and L-T orientations

A Mixed-Mode I/II Fracture Criterion and Its Application in Crack Growth Predictions

A crack tip opening displacement (CTOD)-based, mixed mode fracture criterion is developed for predicting the onset and direction of crack growth. The criterion postulates that crack growth occurs in either the Mode I or Mode II direction, depending on whether the maximum in either the opening or the shear component of CTOD, measured at a specified distance behind the crack tip, attains a critical value. For crack growth direction prediction, the proposed CTOD criterion is shown to be equivalent to seven commonly used crack growth criteria under linearly elastic and asymptotic conditions. Under elastic-plastic conditions the CTOD criterion's prediction of the dependence of the crack growth direction on the crack-up mode mixity is in excellent agreement with the Arcan test results. Furthermore, the CTOD criterion correctly predicts the existence of a crack growth transition from mode I to mode II as the mode mixity approaches the mode II loading condition. The proposed CTOD criterion has been implemented in finite element crack growth simulation codes Z1P2DL and FRANC2DL to predict the crack growth paths in (a) a modified Arcan test specimen and fixture made of AL 2024-T34 and (b) a double cantilever beam (DCB) specimen made of AL 7050. A series of crack growth simulations have been carried out for the crack growth tests in the Arcan and DCB specimens and the results further demonstrate the applicability of the mixed mode CTOD fracture criterion crack growth predictions and residual strength analyses for airframe materials

Influence of crack history on the stable tearing behavior of a thin-sheet material with multiple cracks

Fracture tests were conducted on 2.3mm thick, 305mm wide sheets of 2024-T3 aluminum alloy with from one to five collinear cracks. The cracks were introduced (crack history) into the specimens by three methods: saw cutting, fatigue precracking at a low stress range, and fatigue precracking at a high stress range. For the single crack tests, the initial crack history influenced the stress required for the onset of stable crack growth and the first 10mm of crack growth. The effect on failure stress was about 4 percent or less. For the multiple crack tests, the initial crack history was shown to cause differences of more than 20 percent in the link-up stress and 13 percent in failure stress. An elastic-plastic finite element analysis employing the CTOA fracture criterion was used to predict the fracture behavior of the single and multiple crack tests. The numerical predictions were within 7 percent of the observed link-up and failure stress in all the tests

Stable tearing behavior of a thin-sheet material with multiple cracks

Fracture tests were conducted on 2.3mm thick, 305mm wide sheets of 2024-T3 aluminum alloy with 1-5 collinear cracks. The cracks were introduced (crack history) into the specimens by three methods: (1) saw cutting; (2) fatigue precracking at a low stress range; and (3) fatigue precracking at a high stress range. For the single crack tests, the initial crack history influenced the stress required for the onset of stable crack growth and the first 10mm of crack growth. The effect on failure stress was about 4 percent or less. For the multiple crack tests, the initial crack history was shown to cause differences of more than 20 percent in the link-up stress and 13 percent in failure stress. An elastic-plastic finite element analysis employing the Crack Tip Opening Angle (CTOA) fracture criterion was used to predict the fracture behavior of the single and multiple crack tests. The numerical predictions were within 7 percent of the observed link-up and failure stress in all the tests

Measurement and analysis of critical CTOA for an aluminum alloy sheet

The stable tearing behavior of thin sheets of 2024-T3 aluminum alloy was investigated for middle crack tension, M(T), and compact tension, C(T), specimens. The surface crack-tip opening angle (CTOA), applied loads, crack extension, and local displacements were measured. A critical CTOA fracture criterion was incorporated into a two-dimensional, elastic plastic finite element code and used to simulate the experimental fracture behavior. The CTOA measurements and observations of the fracture surfaces have shown that large values for surface CTOA were observed for small crack extensions (less than the sheet thickness); substantial tunneling of the crack was associated with small crack extensions; crack tunneling in the M(T) specimen was less than that observed in the C(T) configuration; for larger crack extensions, the measured CTOA values were determined to be approximately 6 degrees for both the M(T) and C(T) configuration; and for larger crack extensions, crack tunneling remained constant. The two-dimensional finite element predictions of fracture behavior assumed a constant critical CTOA value of 6 degrees and accounted for local crack tip constraint with a plane strain core of elements ahead of the crack tip. The plane strain core extended 5 mm above the crack plane. The simulations were within +/- 4 percent of the maximum applied load for the C(T) tests within 2 percent for the M(T) tests

XANES and Mg isotopic analyses of spinels in Ca-Al-rich inclusions: Evidence for formation under oxidizing conditions

Ti valence measurements in MgAl_2O_4 spinel from calcium-aluminum-rich inclusions (CAIs) by X-ray absorption near-edge structure (XANES) spectroscopy show that many spinels have predominantly tetravalent Ti, regardless of host phases. The average spinel in Allende type B1 inclusion TS34 has 87% Ti^(+4). Most spinels in fluffy type A (FTA) inclusions also have high Ti valence. In contrast, the rims of some spinels in TS34 and spinel grain cores in two Vigarano type B inclusions have larger amounts of trivalent titanium. Spinels from TS34 have approximately equal amounts of divalent and trivalent vanadium. Based on experiments conducted on CAI-like compositions over a range of redox conditions, both clinopyroxene and spinel should be Ti^(+3)-rich if they equilibrated with CAI liquids under near-solar oxygen fugacities. In igneous inclusions, the seeming paradox of high-valence spinels coexisting with low-valence clinopyroxene can be explained either by transient oxidizing conditions accompanying low-pressure evaporation or by equilibration of spinel with relict Ti^(+4)-rich phases (e.g., perovskite) prior to or during melting. Ion probe analyses of large spinel grains in TS34 show that they are enriched in heavy Mg, with an average Δ^(25)Mg of 4.25 ± 0.028‰, consistent with formation of the spinel from an evaporating liquid. Δ^(25)Mg shows small, but significant, variation, both within individual spinels and between spinel and adjacent melilite hosts. The Δ^(25)Mg data are most simply explained by the low-pressure evaporation model, but this model has difficulty explaining the high Ti^(+4) concentrations in spinel

Speckle from phase ordering systems

The statistical properties of coherent radiation scattered from phase-ordering materials are studied in detail using large-scale computer simulations and analytic arguments. Specifically, we consider a two-dimensional model with a nonconserved, scalar order parameter (Model A), quenched through an order-disorder transition into the two-phase regime. For such systems it is well established that the standard scaling hypothesis applies, consequently the average scattering intensity at wavevector _k and time t' is proportional to a scaling function which depends only on a rescaled time, t ~ |_k|^2 t'. We find that the simulated intensities are exponentially distributed, with the time-dependent average well approximated using a scaling function due to Ohta, Jasnow, and Kawasaki. Considering fluctuations around the average behavior, we find that the covariance of the scattering intensity for a single wavevector at two different times is proportional to a scaling function with natural variables mt = |t_1 - t_2| and pt = (t_1 + t_2)/2. In the asymptotic large-pt limit this scaling function depends only on z = mt / pt^(1/2). For small values of z, the scaling function is quadratic, corresponding to highly persistent behavior of the intensity fluctuations. We empirically establish a connection between the intensity covariance and the two-time, two-point correlation function of the order parameter. This connection allows sensitive testing, either experimental or numerical, of existing theories for two-time correlations in systems undergoing order-disorder phase transitions. Comparison between theory and our numerical results requires no adjustable parameters.Comment: 18 pgs RevTeX, to appear in PR