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Several time-stepping schemes for incompressibility problems are presented which can be solved directly for steady state or iteratively through the time domain. The difficulty of mixed interpolation is avoided by using these schemes. The schemes are applicable to problems of fluid and solid mechanics.
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.
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.
Films reinforced with woven fabrics are being considered for the development of a material suitable for long duration scientific balloons under a program managed by the National Aeronautics and Space Administration (NASA). Recently developed woven fabrics provide a relatively high strength to weight ratio compared to standard homogenous films. Woven fabrics also have better crack propagation resistance and rip stop capabilities when compared to homogenous lightweight, high strength polymeric films such as polyester and nylon. If joining is required, such as in the case of scientific balloons, woven fabrics have the advantage over polymeric thin films to utilize traditional textile methods as well as other techniques including hot sealing, adhesion, and ultrasonic means. Woven fabrics, however, lack the barrier properties required for helium filled scientific balloons, therefore lamination with homogenous films is required to provide the gas barrier capabilities required in these applications.
The results of an analytical and experimental study of the nonlinear response of thin, unstiffened, aluminum cylindrical shells with a long longitudinal crack are presented. The shells are analyzed with a nonlinear shell analysis code that accurately accounts for global and local structural response phenomena. Results are presented for internal pressure and for axial compression loads. The effect of initial crack length on the initiation of stable crack growth and unstable crack growth in typical shells subjected to internal pressure loads is predicted using geometrically nonlinear elastic-plastic finite element analyses and the crack-tip-opening angle (CTOA) fracture criterion. The results of these analyses and of the experiments indicate that the pressure required to initiate stable crack growth and unstable crack growth in a shell subjected to internal pressure loads decreases as the initial crack length increases. The effects of crack length on the prebuckling, buckling and postbuckling responses of typical shells subjected to axial compression loads are also described. For this loading condition, the crack length was not allowed to increase as the load was increased. The results of the analyses and of the experiments indicate that the initial buckling load and collapse load for a shell subjected to axial compression loads decrease as the initial crack length increases. Initial buckling causes general instability or collapse of a shell for shorter initial crack lengths. Initial buckling is a stable local response mode for longer initial crack lengths. This stable local buckling response is followed by a stable postbuckling response, which is followed by general or overall instability of the shell.
The results of an analytical and experimental study of the nonlinear response of thin, unstiffened, aluminum cylindrical shells with a long longitudinal crack are presented. The shells are analyzed with a nonlinear shell analysis code that accurately accounts for global and local structural response phenomena. Results are presented for internal pressure and for axial compression loads. The effect of initial crack length on the initiation of stable crack growth and unstable crack growth in typical shells subjected to internal pressure loads is predicted using geometrically nonlinear elastic-plastic finite element analyses and the crack-tip-opening angle (CTOA) fracture criterion. The results of these analyses and of the experiments indicate that the pressure required to initiate stable crack growth and unstable crack growth in a shell subjected to internal pressure loads decreases as the initial crack length increases. The effects of crack length on the prebuckling, buckling and postbuckling responses of typical shells subjected to axial compression loads are also described. For this loading condition, the crack length was not allowed to increase as the load was increased. The results of the analyses and of the experiments indicate that the initial buckling load and collapse load for a shell subjected to axial compression loads decrease as the initial crack length increases. Initial buckling causes general instability or collapse of a shell for shorter initial crack lengths. Initial buckling is a stable local response mode for longer initial crack lengths. This stable local buckling response is followed by a stable postbuckling response, which is followed by general or overall instability of the shell.
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A two-dimensional, elastic-plastic finite-element analysis was used with a critical crack-tip-opening angle (CTOA) fracture criterion to model stable crack growth in thin-sheet 2024-T3 aluminum alloy under monotonic loading after precracking at different cyclic stress levels. Tests were conducted on three types of specimens: middle-crack, three-hole-crack and blunt-notch tensile specimens. An experiment technique was developed to measure CTOA during crack growth initiation and stable tearing using a high-resolution video camera and recorder. Crack front shapes were also measured during initiation and stable tearing using a fatigue marker-load technique. Three-dimensional elastic-plastic finite-element analyses of these crack shapes for stationary cracks were conducted to study the crack-front opening displacements. Predicted load against crack extension on middle-crack tension specimens agreed well with test results even for large-scale plastic deformations. The analyses were able to predict the effects of specimen size and precracking stress history on stable tearing. Predicted load against load-line displacements agreed well with test results up to maximum load bu the analyses tended to overpredict displacements as crack grew beyond the maximum load under displacement-controlled conditions. During the initiation phase, the measured CTOA values were high but decreased and remained nearly constant after a small amount of stable tearing. The constant value of CTOA agree well with the calculated value from the finite-element analysis. The larger CTOA values measured at the sheet surface during the initiation phase may be associated with the crack tunneling observed in the tests. Three-dimensional analyses for nonstraight crack fronts predicted much higher displacements near the free surface than in the interior.
This paper uses normal mode analysis to examine topological changes in magnetic field lines at the magnetopause. For a modified Harris sheet, the normal modes include a symmetric solution (tearing mode) and an antisymmetric solution (twisting mode). Both modes form flux ropes. The tearing mode in the presence of shear flow is studied. By modeling the external region plasma with drift kinetic formalism, the growth rate of the tearing mode is found to be reduced by shear flow. On the other hand, the shear flow generates a Kelvin-Helmholtz instability which makes the magnetic field perturbation a twisting mode structure. When magnetopause parameters are used, it is difficult to induce a tearing mode in the presence of shear flow, though a shear flow twisting mode did grow under these conditions.
Transition to turbulence in magnetohydrodynamic (MHD) tearing jets has been invoked as a mechanism underlying some of the complex behavior observed in solar surges, including deceleration of the upflowing plasma and temporal correlations with types I and III radio bursts. In this paper we investigate a possible mechanism for this transition: three-dimensional secondary instabilities on two-dimensional saturated states. We find through linear analysis that these MHD configurations -- in particular, the tearing jet -- are secondarily unstable, with the dominant energy transfer from the one-dimensional field into the 3-dimensional fields. Using nonlinear simulations, we also investigate the system evolution after the secondary modes attain finite amplitude. When the tearing jet transitions to turbulence, the total kinetic energy drops rapidly corresponding to the deceleration of the jet. The electric field grows rapidly as the primary mode saturates and the three-dimensional secondary mode develops, and then decays quickly as the tearing jet becomes turbulent, providing a possible explanation for the finite duration of the associated meter-wave bursts. The electric field decays as the magnetic and velocity fields both decay. The system is dominated at late times by spanwise modes, which strongly resemble the magnetic field-aligned filamentary flows characteristic of many surges.
IMP 8 observations in the geomagnetic tail of energetic particles at hundreds of keV and of the magnetic field are examined for a substorm period from 1700 to 1800 UT on Mar. 3, 1976. For this interval, it is found that bursts of energetic protons detected at a downstream distance of about 37 earth radii in the midnight tail region are accompanied by east-west magnetic field perturbations indicative of Birkeland current sheets. The magnitude of the magnetic field perturbations range from about 1 to 12 nT, corresponding to current densities of 1 to 9 mA/m when integrated over the current sheet thickness. These values are comparable to previous observations of Birkeland currents in the near-earth tail (X greater than -20 earth radii) and with the auroral-arc-associated Birkeland current densities at low altitudes scaled out to the tail region. The favorable comparison suggests that the closure of small-scale Birkeland current systems, at least in these instances, lies tailward of about 40 earth radii. The observed Birkeland current system and the magnitude of current densities are found to be consistent with ion tearing instability in the tail. The inferred wavelength of the tearing mode along the tail axis is about 2 to 17 earth radii. In the context of the tearing model, the presence of Birkeland currents and the occasional isotropic distribution of energetic electrons (of more than 0.22 MeV) suggest that the energetic particle events in this study occur on closed magnetic field lines, even though strong tailward streaming dominates the particle anisotropy in these events.
Filaments and flares are prominent indicators of the magnetic fields of solar activity. These instability phenomena arise from the influence of weak transport effects (radiation and resistivity, respectively) on coronal magnetodynamics and energy flow. It has been shown that the filament and flare (tearing or reconnection) mechanisms are resistively coupled in sheared magnetic fields of the kind existing in active regions. The present paper expands this treatment to include the effects of compressibility and viscosity, which are most prominent at short wavelengths. The results show that compressibility affects the radiative mode, including a modest increase of its growth rate, and that viscosity modifies the tearing mode, partially through a decrease of its growth rate. A comprehensive discussion of the mode structures and flows is presented. The strongest effect found is a reversal, at very long wavelengths, of the radiative cooling of the resistive interior layer of the tearing mode, caused by compressional heating.
The reconnection rate in the presence of tearing mode turbulence is calculated. Tearing modes can produce turbulence in three-dimensional sheared magnetic fields. They produce a turbulent anomalous electron viscosity or hyperresistivity. Using quasi-linear and direct interaction approximations, the scaling of the hyperresistivity with magnetic perturbation strength and other parameters is estimated. This yields a reconnection rate, or inflow Mach number M, scaling as delta B, the ratio of rms magnetic fluctuations to mean magnetic field. This can greatly exceed the Sweet-Parker rate. The rate of heating of the current sheet is also large and can be many orders of magnitude larger than in a Sweet-Parker current sheet. Hyperresistivity produced by tearing mode turbulence may explain both fast reconnection and the heating of the solar corona.
A laboratory-scale batch of aluminum alloy sheeting samples in the 2XXX + Zr system has been produced by P/M techniques in the T8X temper and subjected to tensile and Kahn tear property tests in both the longitudinal and long-transverse directions. The results obtained were compared to those of a NASA study concerning the same alloy powders; it appears that laboratory production-scale sheet-sample properties are good to excellent predictors of pilot-scale process products' tensile and tear notch toughness properties. The tear resistance toughness of the laboratory samples was not predictive of the pilot-scale products, however.
The long-term nonlinear evolution of a tearing-mode-unstable sheared-field plasma in a turbulent environment is explored. Two different physical configurations are modeled, and a different computational system is used for each. Results of both sets of calculations show that magnetic tearing arises spontaneously provided that the initial turbulence energy level is below the natural saturation level of the tearing instability. The relationship between these results and those of previous calculations are briefly discussed, concluding that there are no significant unexplainable disagreements.
The possibility of producing rapid heating of a substantial coronal volume by current dissipation, driven by a tearing instability associated with a single unidirectional current system, is investigated. For the fast tearing mode, it is found that, for initial growth rates gamma(f) equal or less than 0.3/s, the electron heating is offset by convective losses, resulting in a very slow temperature rise, and hard X-ray emitting temperatures are never realized. For the larger growth rates, heating from 10 to the 7th to 10 to the 8th K can be achieved in a few seconds; however, in this regime, the maximum volume heated is some three to five orders of magnitude less than the volume of heated material that is inferred from hard X-ray emission measures. The results suggest that to achieve the required heating for the fast tearing mode, a more complicated geometry involving multiple small-scale, oppositely directed current channels may be necessary.