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At least 559 records · Page 31
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

Terrace width variations in complex Mercurian craters and the transient strength of cratered Mercurian and lunar crust

The effective cohesion of the cratered region during crater collapse is determined via the widths of slump terraces of complex craters. Terrace widths are measured for complex craters on Mercury; these generally increase outward toward the rim for a given crater, and the width of the outermost major terrace is generally an increasing function of crater diameter. The terrace widths on Mercury and a gravity-driven slump model are used to estimate the strength of the cratered region immediately after impact (about 1-2 MPa). A comparison with the previous study of lunar complex craters by Pearce and Melosh (1986) indicates that the transient strength of cratered Mercurian crust is no greater than that of the moon. The strength estimates vary only slightly with the geometric model used to restore the outermost major terrace to its precollapse configuration and are consistent with independent strength estimates from the simple-to-complex crater depth/diameter transition.

Leith, Andrew C.↗
NASA NTRS1993

Dust temperature distributions in star-forming condensations

The FIR spectra of the central IR condensations in the dense cores of molecular clouds AFGL 2591. B335, L1551, Mon R2, and Sgr B2 are reanalyzed here in terms of the distribution of dust mass as a function of temperature. FIR spectra of these objects can be characterized reasonably well by a given functional form. The general shapes of the dust temperature distributions of these objects are similar and closely resemble the theoretical computations of de Muizon and Rouan (1985) for a sample of 'hot centered' clouds with active star formation. Specifically, the model yields a 'cutoff' temperature below which essentially no dust is needed to interpret the dust emission spectra, and most of the dust mass is distributed in a broad temperature range of a few tens of degrees above the cutoff temperature. Mass, luminosity, average temperature, and column density are obtained, and it is found that the physical quantities differ considerably from source to source in a meaningful way.

Xie, Taoling↗
NASA NTRS1993

Turbulence modeling of free shear layers for high-performance aircraft

The High Performance Aircraft (HPA) Grand Challenge of the High Performance Computing and Communications (HPCC) program involves the computation of the flow over a high performance aircraft. A variety of free shear layers, including mixing layers over cavities, impinging jets, blown flaps, and exhaust plumes, may be encountered in such flowfields. Since these free shear layers are usually turbulent, appropriate turbulence models must be utilized in computations in order to accurately simulate these flow features. The HPCC program is relying heavily on parallel computers. A Navier-Stokes solver (POVERFLOW) utilizing the Baldwin-Lomax algebraic turbulence model was developed and tested on a 128-node Intel iPSC/860. Algebraic turbulence models run very fast, and give good results for many flowfields. For complex flowfields such as those mentioned above, however, they are often inadequate. It was therefore deemed that a two-equation turbulence model will be required for the HPA computations. The k-epsilon two-equation turbulence model was implemented on the Intel iPSC/860. Both the Chien low-Reynolds-number model and a generalized wall-function formulation were included.

Sondak, Douglas L.↗
NASA NTRS1994

Exercise detraining: Applicability to microgravity

Physical training exposes the various systems of the body to potent physiologic stimuli. These stimuli induce specific adaptations that enhance an individual's tolerance for the type of exercise encountered in training. The level of adaptation and the magnitude of improvement in exercise tolerance is proportional to the potency of the physical training stimuli. Likewise, our bodies are stimulated by gravity, which promotes adaptations of both the cardiovascular and skeletal muscles. Exposure to microgravity removes normal stimuli to these systems, and the body adapts to these reduced demands. In many respects the cessation of physical training in athletes and the transition from normal gravity to microgravity represent similar paradigms. Inherent to these situations is the concept of the reversibility of the adaptations induced by training or by exposure to normal gravity. The reversibility concept holds that when physical training is stopped (i.e., detraining) or reduced, or a person goes from normal gravity to microgravity, the bodily systems readjust in accordance with the diminished physiologic stimuli. The focus of this chapter is on the time course of loss of the adaptations to endurance training as well as on the possibility that certain adaptations persist, to some extent, when training is stopped. Because endurance exercise training generally improves cardiovascular function and promotes metabolic adaptations within the exercising skeletal musculature, the reversibility of these specific adaptations is considered. These observations have some applicability to the transition from normal to microgravity.

Coyle, Edward F.↗
NASA NTRS1994

Comparative transduction mechanisms of hair cells in the bullfrog utriculus. 1: Responses to intracellular current

Hair cells in the bullfrog sacculus are specifically adapted to sense small-amplitude, high-frequency linear accelerations. These hair cells display many properties that are undesirable or inappropriate for hair cells that must provide static gravity sensitivity. This study resulted in part due to an interest in seeing how the transduction mechanisms of hair cells in a gravity-sensing otolith endorgan would differ from those in the bullfrog sacculus. The bullfrog utriculus is an appropriate model for these studies, because its structure is representative of higher vertebrates in general and its function as a sensor of static gravity and dynamic linear acceleration is well known. Hair cells in the bullfrog utriculus, classifiable as Type 2 by cell body and synapse morphology, differ markedly in hair bundle morphology from those in the bullfrog sacculus. Moreover, the hair bundle morphologies of utricular hair cells, unlike those in the sacculus, differ in different membrane regions.

Baird, Richard A.↗
NASA NTRS1993

A new technique for deconvolution of data from instruments that make integral measurements, e.g. RIMS on DE-1

A general method for deconvolving an unknown function from integral measurements is described and applied to data from the instrument Retarding Ion Mass Spectrometer (RIMS) aboard the spacecraft Dynamics Explorer 1 (DE-1). The principal features of the method are: (1) it uses objective criteria based upon fundamental statistical principles, i.e. Bayesian statistics; (2) it provides for insertion of prior knowledge in a non-prejudicial, explicit manner through the choice of breakpoints that determine the bicubic spline expansion functions; (3) it prevents random fluctuations from controlling the fit to the data through the use of singular value decomposition and the elimination of small singular values; and (4) it guards agianst the introduction of spurious features into the result by including a penalty function and using the principle of generalized cross validation. Illustrative examples from RIMS data for H(+) and O(+) show that the method provides enhanced accuracy and detail in deconvolving the ion phase space density.

Perez, J. D.↗
NASA NTRS1997

Error Analysis System for Spacecraft Navigation Using the Global Positioning System (GPS)

The Flight Dynamics Division (FDD) at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) is currently developing improved space-navigation filtering algorithms to use the Global Positioning System (GPS) for autonomous real-time onboard orbit determination. In connection with a GPS technology demonstration on the Small Satellite Technology Initiative (SSTI)/Lewis spacecraft, FDD analysts and programmers have teamed with the GSFC Guidance, Navigation, and Control Branch to develop the GPS Enhanced Orbit Determination Experiment (GEODE) system. The GEODE system consists of a Kalman filter operating as a navigation tool for estimating the position, velocity, and additional states required to accurately navigate the orbiting Lewis spacecraft by using astrodynamic modeling and GPS measurements from the receiver. A parallel effort at the FDD is the development of a GPS Error Analysis System (GEAS) that will be used to analyze and improve navigation filtering algorithms during development phases and during in-flight calibration. For GEAS, the Kalman filter theory is extended to estimate the errors in position, velocity, and other error states of interest. The estimation of errors in physical variables at regular intervals will allow the time, cause, and effect of navigation system weaknesses to be identified. In addition, by modeling a sufficient set of navigation system errors, a system failure that causes an observed error anomaly can be traced and accounted for. The GEAS software is formulated using Object Oriented Design (OOD) techniques implemented in the C++ programming language on a Sun SPARC workstation. The Phase 1 of this effort is the development of a basic system to be used to evaluate navigation algorithms implemented in the GEODE system. This paper presents the GEAS mathematical methodology, systems and operations concepts, and software design and implementation. Results from the use of the basic system to evaluate navigation algorithms implemented on GEODE are also discussed. In addition, recommendations for generalization of GEAS functions and for new techniques to optimize the accuracy and control of the GPS autonomous onboard navigation are presented.

Truong, S. H.↗
NASA NTRS1997

Near-Optimal Re-Entry Trajectories for Reusable Launch Vehicles

A near-optimal guidance law for the descent trajectory for earth orbit re-entry of a fully reusable single-stage-to-orbit pure rocket launch vehicle is derived. A methodology is developed to investigate using both bank angle and altitude as control variables and selecting parameters that maximize various performance functions. The method is based on the energy-state model of the aircraft equations of motion. The major task of this paper is to obtain optimal re-entry trajectories under a variety of performance goals: minimum time, minimum surface temperature, minimum heating, and maximum heading change; four classes of trajectories were investigated: no banking, optimal left turn banking, optimal right turn banking, and optimal bank chattering. The cost function is in general a weighted sum of all performance goals. In particular, the trade-off between minimizing heat load into the vehicle and maximizing cross range distance is investigated. The results show that the optimization methodology can be used to derive a wide variety of near-optimal trajectories.

Chou, H.-C.↗
NASA NTRS2002

Rapid, specific determination of iodine and iodide by combined solid-phase extraction/diffuse reflectance spectroscopy

A new, rapid methodology for trace analysis using solid-phase extraction is described. The two-step methodology is based on the concentration of an analyte onto a membrane disk and on the determination by diffuse reflectance spectroscopy of the amount of analyte extracted on the disk surface. This method, which is adaptable to a wide range of analytes, has been used for monitoring ppm levels of iodine and iodide in spacecraft water. Iodine is used as a biocide in spacecraft water. For these determinations, a water sample is passed through a membrane disk by means of a 10-mL syringe that is attached to a disk holder assembly. The disk, which is a polystyrene-divinylbenzene composite, is impregnated with poly(vinylpyrrolidone) (PVP), which exhaustively concentrates iodine as a yellow iodine-PVP complex. The amount of concentrated iodine is then determined in only 2 s by using a hand-held diffuse reflectance spectrometer by comparing the result with a calibration curve based on the Kubelka-Munk function. The same general procedure can be used to determine iodide levels after its facile and exhaustive oxidation to iodine by peroxymonosulfate (i.e., Oxone reagent). For samples containing both analytes, a two-step procedure can be used in which the iodide concentration is calculated from the difference in iodine levels before and after treatment of the sample with peroxymonosulfate. With this methodology, iodine and iodide levels in the 0.1-5.0 ppm range can be determined with a total workup time of approximately 60 s with a RSD of approximately 6%.

Non-NASA Center↗
NASA NTRS1999

Transduction-like gene transfer in the methanogen Methanococcus voltae

Strain PS of Methanococcus voltae (a methanogenic, anaerobic archaebacterium) was shown to generate spontaneously 4.4-kbp chromosomal DNA fragments that are fully protected from DNase and that, upon contact with a cell, transform it genetically. This activity, here called VTA (voltae transfer agent), affects all markers tested: three different auxotrophies (histidine, purine, and cobalamin) and resistance to BES (2-bromoethanesulfonate, an inhibitor of methanogenesis). VTA was most effectively prepared by culture filtration. This process disrupted a fraction of the M. voltae cells (which have only an S-layer covering their cytoplasmic membrane). VTA was rapidly inactivated upon storage. VTA particles were present in cultures at concentrations of approximately two per cell. Gene transfer activity varied from a minimum of 2 x 10(-5) (BES resistance) to a maximum of 10(-3) (histidine independence) per donor cell. Very little VTA was found free in culture supernatants. The phenomenon is functionally similar to generalized transduction, but there is no evidence, for the time being, of intrinsically viral (i.e., containing a complete viral genome) particles. Consideration of VTA DNA size makes the existence of such viral particles unlikely. If they exist, they must be relatively few in number;perhaps they differ from VTA particles in size and other properties and thus escaped detection. Digestion of VTA DNA with the AluI restriction enzyme suggests that it is a random sample of the bacterial DNA, except for a 0.9-kbp sequence which is amplified relative to the rest of the bacterial chromosome. A VTA-sized DNA fraction was demonstrated in a few other isolates of M. voltae.

Methanococcus/genetics/growth & development/metabo↗
NASA NTRS2004

Picturing Data With Uncertainty

NASA is in the business of creating maps for scientific purposes to represent important biophysical or geophysical quantities over space and time. For example, maps of surface temperature over the globe tell scientists where and when the Earth is heating up; regional maps of the greenness of vegetation tell scientists where and when plants are photosynthesizing. There is always uncertainty associated with each value in any such map due to various factors. When uncertainty is fully modeled, instead of a single value at each map location, there is a distribution expressing a set of possible outcomes at each location. We consider such distribution data as multi-valued data since it consists of a collection of values about a single variable. Thus, a multi-valued data represents both the map and its uncertainty. We have been working on ways to visualize spatial multi-valued data sets effectively for fields with regularly spaced units or grid cells such as those in NASA's Earth science applications. A new way to display distributions at multiple grid locations is to project the distributions from an individual row, column or other user-selectable straight transect from the 2D domain. First at each grid cell in a given slice (row, column or transect), we compute a smooth density estimate from the underlying data. Such a density estimate for the probability density function (PDF) is generally more useful than a histogram, which is a classic density estimate. Then, the collection of PDFs along a given slice are presented vertically above the slice and form a wall. To minimize occlusion of intersecting slices, the corresponding walls are positioned at the far edges of the boundary. The PDF wall depicts the shapes of the distributions very dearly since peaks represent the modes (or bumps) in the PDFs. We've defined roughness as the number of peaks in the distribution. Roughness is another useful summary information for multimodal distributions. The uncertainty of the multi-valued data can also be interpreted by the number of peaks and the widths of the peaks as shown by the PDF walls.

Kao, David↗
NASA NTRS1997

Promethazine affects autonomic cardiovascular mechanisms minimally

Promethazine hydrochloride, Phenergan, is a phenothiazine derivative with antihistaminic (H1), sedative, antiemetic, anticholinergic, and antimotion sickness properties. These properties have made promethazine a candidate for use in environments such as microgravity, which provoke emesis and motion sickness. Recently, we evaluated carotid baroreceptor-cardiac reflex responses during two Space Shuttle missions 18 to 20 hr after the 50 mg intramuscular administration of promethazine. Because the effects of promethazine on autonomic cardiovascular mechanisms in general and baroreflex function in particular were not known, we were unable to exclude a possible influence of promethazine on our results. Our purpose was to determine the ground-based effects of promethazine on autonomic cardiovascular control. Because of promethazine's antihistaminic and anticholinergic properties, we expected that a 50-mg intramuscular injection of promethazine would affect sympathetically and vagally mediated cardiovascular mechanisms. Eight healthy young subjects, five men and three women, were studied at rest in recumbency. All reported drowsiness as a result of the promethazine injection; most also reported nervous excitation, dry mouth, and fatigue. Three subjects had significant reactions: two reported excessive anxiety and one reported dizziness. Measurements were performed immediately prior to injection and 3.1 +/- 0.1 and 19.5 +/- 0.4 hr postinjection. We found no significant effect of promethazine on resting mean R-R interval, arterial pressure, R-R interval power spectra, carotid baroreflex function, and venous plasma catecholamine levels.

Clinical Trial↗
NASA NTRS2006

Geology of the Gusec cratered plains from the Spirit rover transverse

The cratered plains of Gusev traversed by Spirit are generally low-relief rocky plains dominated by impact and eolian processes. Ubiquitous shallow, soil-filled, circular depressions, called hollows, are modified impact craters. Rocks are dark, fine-grained basalts, and the upper 10 m of the cratered plains appears to be an impact-generated regolith developed over intact basalt flows. Systematic field observations across the cratered plains identified vesicular clasts and rare scoria similar to original lava flow tops, consistent with an upper inflated surface of lava flows with adjacent collapse depressions. Crater and hollow morphometry are consistent with most being secondaries. The size frequency distribution of rocks >0.1 m diameter generally follows exponential functions similar to other landing sites for total rock abundances of 5-35%. Systematic clast counts show that areas with higher rock abundance and more large rocks have higher thermal inertia. Plains with lower thermal inertia have fewer rocks and substantially more pebbles that are well sorted and evenly spaced, similar to a desert pavement or lag. Eolian bed forms (ripples and wind tails) have coarse surface lags, and many are dust covered and thus likely inactive. Deflation of the surface _5-25 cm likely exposed two-toned rocks and elevated ventifacts and transported fines into craters creating the hollows. This observed redistribution yields extremely slow average erosion rates of _0.03 nm/yr and argues for very little long-term net change of the surface and a dry and desiccating environment similar to today's since the Hesperian (or _3 Ga).

Spirit↗
NASA NTRS2009

Improved Model of a Mercury Ring Damper

A short document discusses the general problem of mathematical modeling of the three-dimensional rotational dynamics of rigid bodies and of the use of Euler parameters to eliminate the singularities occasioned by the use of Euler angles in such modeling. The document goes on to characterize a Hamiltonian model, developed by the authors, that utilizes the Euler parameters and, hence, is suitable for use in computational simulations that involve arbitrary rotational motion. In this formulation unlike in prior Euler-parameter-based formulations, there are no algebraic constraints. This formulation includes a general potential energy function, incorporates a minimum set of momentum variables, and takes an explicit state-space form convenient for numerical implementation. Practical application of this formulation has been demonstrated by the development of a new and simplified model of the rotational motion of a rigid rotor to which is attached a partially filled mercury ring damper. Models like this one are used in guidance and control of spin-stabilized spacecraft and gyroscope-stabilized seekers in guided missiles.

Fahrenthold, Eric P.↗
NASA NTRS2010

Multiphysics Modeling for Dimensional Analysis of a Self-Heated Molten Regolith Electrolysis Reactor for Oxygen and Metals Production on the Moon and Mars

The technology of direct electrolysis of molten lunar regolith to produce oxygen and molten metal alloys has progressed greatly in the last few years. The development of long-lasting inert anodes and cathode designs as well as techniques for the removal of molten products from the reactor has been demonstrated. The containment of chemically aggressive oxide and metal melts is very difficult at the operating temperatures ca 1600 C. Containing the molten oxides in a regolith shell can solve this technical issue and can be achieved by designing a self-heating reactor in which the electrolytic currents generate enough Joule heat to create a molten bath. In a first phase, a thermal analysis model was built to study the formation of a melt of lunar basaltic regolith irradiated by a focused solar beam This mode of heating was selected because it relies on radiative heat transfer, which is the dominant mode of transfer of energy in melts at 1600 C. Knowing and setting the Gaussian-type heat flux from the concentrated solar beam and the phase and temperature dependent thermal properties, the model predicts the dimensions and temperature profile of the melt. A validation of the model is presented in this paper through the experimental formation of a spherical cap melt realized by others. The Orbitec/PSI experimental setup uses an 3.6-cm diameter concentrated solar beam to create a hemispheric melt in a bed of lunar regolith simulant contained in a large pot. Upon cooling, the dimensions of the vitrified melt are measured to validate the thermal model. In a second phase, the model is augmented by multiphysics components to compute the passage of electrical currents between electrodes inserted in the molten regolith. The current through the melt generates Joule heating due to the high resistivity of the medium and this energy is transferred into the melt by conduction, convection and primarily by radiation. The model faces challenges in two major areas, the change of phase as temperature increases, and the dominance of radiative heat flux as heat transfer mechanism within the melt the change of phase concerns the regolith itself which is present in states ranging from a fine grain regolith with low thermal conductivity and low density to a vitrified melt with much higher thermal conductivity, and higher density. As the regolith is heated, it starts to soften around 1300 C the melt iS very viscous and evolving gas bubbles out in thick, lava-like fashion. By 1600 C the regolith is completely melted and the viscosity is low The second challenge resides in the proper modeling of the radiative heat flux requiring the addition of the computing-demanding radiative-heat-transfer function to the general heat transfer equation. The model Includes temperature-dependent properties (density, thermal conductivity, heat capacity, and viscosity, and absorption coefficients) and solves the radiative heat flux equation assuming gray (fine grains) and semi-transparent (melt) media and using an absorption coefficient spectral found in the literature for terrestrial minerals similar in composition to those of lunar regolith simulant

Dominguez, Jesus A.↗
NASA NTRS2011

Michigan Orbital DEbris Survey Telescope Observations of the Geosynchronous Orbital Debris Environment. Observing Years: 2007-2009

NASA uses the Michigan Orbital DEbris Survey Telescope (MODEST), the University of Michigan's 0.61-m aperture Curtis-Schmidt telescope at the Cerro Tololo Inter-American Observatory in Chile, to help characterize the debris environment in geosynchronous orbit; this began in February 2001 and continues to the present day. Detected objects that are found to be on the U.S. Space Surveillance Network cataloged objects list are termed correlated targets (CTs), while those not found on the list are called uncorrelated targets (UCTs). This Johnson Space Center report provides details of observational and data-reduction processes for the entire MODEST dataset acquired in calendar years (CYs) 2007, 2008, and 2009. Specifically, this report describes the collection and analysis of 36 nights of data collected in CY 2007, 43 nights of data collected in CY 2008, and 43 nights of data collected in CY 2009. MODEST is equipped with a 2048 x 2048-pixel charged coupled device camera with a 1.3 by 1.3 deg field of view. This system is capable of detecting objects fainter than 18th magnitude (R filter) using a 5-s integration. This corresponds to a 20-cm diameter, 0.175-albedo object at 36,000 km altitude assuming a diffuse Lambertian phase function. The average number of detections each night over all 3 years was 26. The percentage of this number that represented the UCT population ranged from 34% to 18%, depending on the observing strategy and the field center location. Due to the short orbital arc over which observations are made, the eccentricity of the object s orbit is extremely difficult to measure accurately. Therefore, a circular orbit was assumed when calculating the orbital elements. A comparison of the measured inclination (INC), right ascension of ascending node (RAAN), and mean motion to the quantities for CTs from the U.S. Space Surveillance Network shows acceptable errors. This analysis lends credibility to the determination of the UCT orbital distributions. Figure 1 shows the size distribution of 3,143 objects detected in the data processed for CYs 2007, 2008, and 2009. The actual peak of the absolute magnitude distribution for the functional correlated targets is 10th magnitude, whereas the peak was 11th magnitude in 2002 2003 and 10th magnitude for 2004-2006. An absolute magnitude of 10.5 corresponds to objects with average diameters of 6.3 m, assuming an albedo of 0.175 and a diffuse Lambertian phase function. This result generally agrees with the known sizes of intact satellites. The absolute magnitude distribution for the UCTs is broad, but starts to roll off near 25 cm diameter or 17.5 magnitude. This roll off in the distribution reflects the detection capability of MODEST, not the true nature of the population. The true population is believed to continue at the same slope through fainter magnitudes based on comparisons with the LEO break-up law.

Abercromby, K. J.↗
NASA NTRS2012

Real-Time Distributed Embedded Oscillator Operating Frequency Monitoring

A document discusses the utilization of embedded clocks inside of operating network data links as an auxiliary clock source to satisfy local oscillator monitoring requirements. Modem network interfaces, typically serial network links, often contain embedded clocking information of very tight precision to recover data from the link. This embedded clocking data can be utilized by the receiving device to monitor the local oscillator for tolerance to required specifications, often important in high-integrity fault-tolerant applications. A device can utilize a received embedded clock to determine if the local or the remote device is out of tolerance by using a single link. The local device can determine if it is failing, assuming a single fault model, with two or more active links. Network fabric components, containing many operational links, can potentially determine faulty remote or local devices in the presence of multiple faults. Two methods of implementation are described. In one method, a recovered clock can be directly used to monitor the local clock as a direct replacement of an external local oscillator. This scheme is consistent with a general clock monitoring function whereby clock sources are clocking two counters and compared over a fixed interval of time. In another method, overflow/underflow conditions can be used to detect clock relationships for monitoring. These network interfaces often provide clock compensation circuitry to allow data to be transferred from the received (network) clock domain to the internal clock domain. This circuit could be modified to detect overflow/underflow conditions of the buffering required and report a fast or slow receive clock, respectively.

Pollock, Julie↗
NASA NTRS2016

Plasticity Tool for Predicting Shear Nonlinearity of Unidirectional Laminates Under Multiaxial Loading

This study implements a plasticity tool to predict the nonlinear shear behavior of unidirectional composite laminates under multiaxial loadings, with an intent to further develop the tool for use in composite progressive damage analysis. The steps for developing the plasticity tool include establishing a general quadratic yield function, deriving the incremental elasto-plastic stress-strain relations using the yield function with associated flow rule, and integrating the elasto-plastic stress-strain relations with a modified Euler method and a substepping scheme. Micromechanics analyses are performed to obtain normal and shear stress-strain curves that are used in determining the plasticity parameters of the yield function. By analyzing a micromechanics model, a virtual testing approach is used to replace costly experimental tests for obtaining stress-strain responses of composites under various loadings. The predicted elastic moduli and Poisson's ratios are in good agreement with experimental data. The substepping scheme for integrating the elasto-plastic stress-strain relations is suitable for working with displacement-based finite element codes. An illustration problem is solved to show that the plasticity tool can predict the nonlinear shear behavior for a unidirectional laminate subjected to multiaxial loadings.

Wang, John T.↗