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At least 271 records · Page 15

Elastic Properties in Tension and Shear of High Strength Nonferrous Metals and Stainless Steel - Effect of Previous Deformation and Heat Treatment

A resume is given of an investigation of the influence of plastic deformation and of annealing temperature on the tensile and shear elastic properties of high strength nonferrous metals and stainless steels in the form of rods and tubes. The data were obtained from earlier technical reports and notes, and from unpublished work in this investigation. There are also included data obtained from published and unpublished work performed on an independent investigation. The rod materials, namely, nickel, monel, inconel, copper, 13:2 Cr-Ni steel, and 18:8 Cr-Ni steel, were tested in tension; 18:8 Cr-Ni steel tubes were tested in shear, and nickel, monel, aluminum-monel, and Inconel tubes were tested in both tension and shear. There are first described experiments on the relationship between hysteresis and creep, as obtained with repeated cyclic stressing of annealed stainless steel specimens over a constant load range. These tests, which preceded the measurements of elastic properties, assisted in devising the loading time schedule used in such measurements. From corrected stress-set curves are derived the five proof stresses used as indices of elastic or yield strength. From corrected stress-strain curves are derived the secant modulus and its variation with stress. The relationship between the forms of the stress-set and stress-strain curves and the values of the properties derived is discussed. Curves of variation of proof stress and modulus with prior extension, as obtained with single rod specimens, consist in wavelike basic curves with superposed oscillations due to differences of rest interval and extension spacing; the effects of these differences are studied. Oscillations of proof stress and modulus are generally opposite in manner. The use of a series of tubular specimens corresponding to different amounts of prior extension of cold reduction gave curves almost devoid of oscillation since the effects of variation of rest interval and extension spacing were removed. Comparison is also obtained between the variation of the several properties, as measured in tension and in shear. The rise of proof stress with extension is studied, and the work-hardening rates of the various metals evaluated. The ratio between the tensile and shear proof stresses for the various annealed and cold-worked tubular metals is likewise calculated. The influence of annealing or tempering temperature on the proof stresses and moduli for the cold-worked metals and for air-hardened 13:2 Cr-Ni steel is investigated. An improvement of elastic strength generally is obtained, without important loss of yield strength, by annealing at suitable temperature. The variation of the proof stress and modulus of elasticity with plastic deformation or annealing temperature is explained in terms of the relative dominance of three important factors: namely, (a) internal stress, (b) lattice-expansion or work-hardening, and (c) crystal reorientation. Effective values of Poisson's ratio were computed from tensile and shear moduli obtained on tubular specimens. The variation of Poisson's ratio with plastic deformation and annealing temperature is explained in terms of the degree of anisotropy produced by changes of (a) internal stress and (b) crystal orientation.

METALS - ELASTICITY↗

Control of micromachined deformable mirrors

A micromachined deformable mirror with pixelated electrostatic actuators is proposed. The paper begins with a physical description of the proposed mirror. Then a mathematical model in the form of a nonlinear partial differential equation describing the mirror surface deformations is derived. This model is used to derive the required voltages for the actuators to achieve a specified static deformation of the mirror surface. This is followed by the derivation of a static nonlinear feedback controller for achieving noninteracting actuation. Then the structure for a complete control system for wavefront correction is proposed. The paper concludes with a discussion of the physical implementation of the proposed control system.

Agronin, M. L.↗

Patterns of brittle deformation under extension on Venus

The development of fractures at regular length scales is a widespread feature of Venusian tectonics. Models of lithospheric deformation under extension based on non-Newtonian viscous flow and brittle-plastic flow develop localized failure at preferred wavelengths that depend on lithospheric thickness and stratification. The characteristic wavelengths seen in rift zones and tessera can therefore provide constraints on crustal and thermal structure. Analytic solutions were obtained for growth rates in infinitesimal perturbations imposed on a one-dimensional, layered rheology. Brittle layers were approximated by perfectly-plastic, uniform strength, overlying ductile layers exhibiting thermally-activated power-law creep. This study investigates the formation of faults under finite amounts of extension, employing a finite-element approach. Our model incorporates non-linear viscous rheology and a Coulomb failure envelope. An initial perturbation in crustal thickness gives rise to necking instabilities. A small amount of velocity weakening serves to localize deformation into planar regions of high strain rate. Such planes are analogous to normal faults seen in terrestrial rift zones. These 'faults' evolve to low angle under finite extension. Fault spacing, orientation and location, and the depth to the brittle-ductile transition, depend in a complex way on lateral variations in crustal thickness. In general, we find that multiple wavelengths of deformation can arise from the interaction of crustal and mantle lithosphere.

Neumann, G. A.↗

Global, quasi-steady dynamics of the distant solar wind 2: Deformation of the heliospheric current sheet

A three-dimensional (3-D) magnetohydrodynamic (MHD) numerical model is used to trace global deformations of the heliospheric current sheet (HCS) caused by large-scale dynamical interactions associated with corotating solar wind flows. Configurations incorporating the tilted-dipole geometry are investigated out past 30 AU for a variety of dipole tilt angles, alpha. Inclusion of the full, 3-D interplanetary dynamics allows north-south displacements and the east-west warping of the HCS by advective corotational effects to be accurately assessed for the first time. It is found that large-scale spatial correlations between velocity and density imposed at the coronal source (i.e., the geometric arrangement whereby regions of high-speed low-density material lying adjacent to areas of slow, dense flow interact obliquely under the influence of solar rotation) result in a distinctive pattern of deformation of the HCS. For an alpha = 30 deg tilted-dipole example, it is shown that typical zonal variations in radial velocity lead to significant folding of the HCS within about 5 AU of the Sun. By 10 AU, additional sharp bends appear near the latitudinal extremes of the HCS surface, where it is overtaken by shock fronts driven by 3-D corotating interaction regions (CIRs). Moreover, the model suggests that inside about 20 AU, major plasma structures are systematically organized, about the HCS, such that the greatest concentrations of material and magnetic field (the centroids of the 3-D CIR structures) are coincident with the folded crests of the HCS (near heliographic latitudes lambda = +/- alpha). Thus, in these circumstances many of the more interesting dynamical features inconveniently lie well away from the heliographic equator. At larger heliocentric distances, where neighboring CIRs begin to interact strongly, the warping of the HCS abates dramatically and the association between folds in the HCS and major field and density concentrations is weakened and ultimately breaks down. On the basis of the model calculations, quasi-steady deformations and related phenomena should figure prominently in the interplanetary medium, except during those periods when the magnetic dipole and solar spin axes are nearly aligned (alpha is approximately less than 10 deg). Even then, they may be of significance if there are any substantial local warps or kinks in the streamer belt.

Pizzo, V. J.↗

Viscosity estimates for the crust and upper mantle from patterns of lacustrine shoreline deformation in the Eastern Great Basin

The deformed shorelines of Lake Bonneville constitute a classic source of information on lithospheric elastic thickness and upper mantle viscosity. We describe and apply a new model to a recently augmented data set. New data better constrain both the complex spatio-temporal pattern of the lake load and the crustal deformation response to that load. The history of lake level fluctuations has been significantly refined and somewhat modified. This is due to both more radiocarbon dates from within the Bonneville basin and to an improved calibration of the radiocarbon timescale itself. The data which constrain the crustal deformation pattern consist of ages and shoreline elevations from several hundred points which sample three major levels of Lake Bonneville and corresponding elevations from the high stands of three smaller lakes situated to the west of Lake Bonneville. The geometry of the Earth model incorporates an arbitrary number of layers overlying a half-space, and the rheology of each level can accommodate an arbitrary number of Maxwell viscoelastic elements in parallel. The inverse modeling comprises three complementary approaches: for the simplest configurations, we performed a direct search of the parameter space and delineated the irregular boundary of the subspace of acceptable models. For more complex configurations, we constrained the elastic parameters to their seismically determined values and then solved for viscosity versus depth profiles by either expressing the log(viscosity) versus log(depth) profile as a series of specially constructed orhtogonal polynomials, or by allowing each of 8-10 layers (plus the half-space) to have an independently determined viscosity. We found that the data do not strongly support (nor can they conclusively exclude) a more complex rheology than simple Maxwell viscoelasticity. The orthogonal polynomial solution exhibits an essentially monotonic decrease in viscosity with depth.

Bills, Bruce G.↗

A spectral formalism for computing three-dimensional deformations due to surface loads. 2: Present-day glacial isostatic adjustment

Using a spherically symmetric, self-gravitating, linear viscoelastic Earth model, we predict present-day three-dimensional surface deformation rates and baseline evolutions arising as a consequence of the late Pleistocene glacial cycles. In general, we use realistic models for the space-time geometry of the final late Pleistocene deglaciation event and incorporate a gravitationally self-consistent ocean meltwater redistribution. The predictions of horizontal velocity presented differ significantly, in both their amplitude and their spatial variation, from those presented in earlier analysis of others which adopted simplified models of both the late Pleistocene ice history and the Earth rheology. An important characteristic of our predicted velocity fields is that the melting of the Laurentide ice sheet over Canada is capable of contributing appreciably to the adjustment in Europe. The sensitivity of the predictions to variations in mantle rheology is investigated by considering a number of different Earth models, and by computing appropriate Frechet kernels. These calculations suggest that the sensitivity of the deformations to the Earth's rheology is significant and strongly dependent on the location of the site relative to the ancient ice sheet. The effects on the predictions of three-dimensional deformation rates of altering the ice history or adopting approximate models for the ocean meltwater redistribution have also been considered and found to be important (the former especially so). Finally, for a suite of Earth models we provide predictions of the velocity of a number of baselines in North America and Europe. We find that, in general, both radial and tangential motions contribute significantly to baseline length changes, and that these contributions are a strong function of the Earth model. We have, furthermore, found a set of Earth models which, together with the ICE-3G deglaciation chronology, produce predictions of baseline length changes that are consistent with very long baseline interferometry measurements of baselines within Europe.

Mitrovica, J. X.↗

Geodetic measurement of deformation in the Ventura basin region, southern California

We have measured the deformation in the Ventura basin region, southern California, with Global Positioning System (GPS) measurements carried out over 4.6 years between 1987 and 1992. The deformation within our network is spatically variable on scales of tens of kilometers, with strain rates reaching 0.6 +/- 1 micro-rad/yr in the east-central basin. Blocklike rotations are observed south and northwest of the basin where the maximum shear strain rates are an order of magnitude lower (0.06 +/- 1 micro-rad/yr to the south). We also observed clockwise rotations of 1 deg - 7 deg/m.y. Shear strain rates determined by comparing angle changes from historical triangulation spanning several decades and GPS measurements give consistent, though less precise, results. The geodetic rates of shortening across the basin and Western Transverse Ranges are lower than those estimated from geological observations, but the patterns of deformation from the two methods agree qualitatively.

Donnellan, Andrea↗

Inelastic deformation mechanisms in a transverse MMC lamina under compression

An investigation was undertaken to study the inelastic deformation mechanisms in (90)(sub 8) Ti 15-3/SCS-6 lamina subjected to pure compression. Both mechanical behavior and microstructural evaluation were undertaken at room temperature, 538 and 650 C. Results indicate that mechanical response and deformation characteristics are significantly different in monotonic tension and compression. The inelastic deformation mechanisms in compression are controlled by radial fiber fracture, matrix plasticity and fiber-matrix debonding. The radial fiber fracture is a new damage mode observed for metal-matrix composites (MMC).

Newaz, Golam M.↗

Late Cenozoic flexural deformation of the middle U.S. Atlantic passive margin

Despite the century-long recognition of regional epeirogeny along the middle Atlantic passive margin, relatively few studies have focused on understanding postrift uplift mechanisms. Here, we demonstrate that epeirogenic uplift of the central Appalachian Piedmont and subsidence of the Salisbury Embayment represent first-order, flexural isostatic processes driven by continental denudation and offshore deposition. Our results show that regional epeirogenic processes, present on all Atlantic-type passive margins, are best resolved by specific stratigraphic and geomorphic relationships, rather than topography. A simple one-dimensional geodynamic model, constrained by well-dated Baltimore Canyon trough, Coastal Plain, and lower Susquehanna River (piedmont) stratigraphy, simulates flexural deforamtion of the U.S. Atlantic margin. The model represents the passive margin lithosphree as a uniformly thick elastic plate, without horizontal compressive stresses, that deforms flexurally under the stress of strike-averaged, vertically applied line loads. Model results illustrate a complex interaction among margin stratigraphy and geomorphology, the isostatic repsonse to denudational and depositional processes, and the modulating influence of exogenic forces such as eustasy. The current elevation, with respect to modern sea level, of fluvial terraces and correlateive Coastal Plain deposits or unconformities is successfully predicted through the synthesis of paleotopography, eustatic change, and margin flexure. Results suggest that the middle U.S. Atlantic margin landward of East Coast Magnetic Anomaly is underlain by lithoshpere with an average elastic thickness of 40 km (flexural rigidity, D = 4 X 10(exp 23) N m), the margin experience an average, long-term denudation rate of approximately 10m/m.y., and the Piedmont has been flexurally upwaped between 35 and 130 meters in the last 15 m.y. Long term isostatic continental uplift resulting rom denudation and basin subsidence resulting rom sediment loading are accomodated primately by a convex-up flexural hinge, physiographically represented by the Fall Zone. Our results elucidate an inherent danger in using topography alone to constrain late-stage passive margin deformation mechanisms. Only through careful synthesis of field stratigraphic and geomorphic elements such as fluvial terraces, Coastal Plain deposits, and offshore stratigraphy can age control be extended from the offshore depositional setting to the erosionally dominated continent. This sudy demonstrates that despite a relatively subdued topography, the middle U.S. Atlantic margin experiences progressive flexural isostatic deformation similar to that proposed for high-relief margins characterized by great escarpments. Thus margin topographic diversity remains a function of other factors, such as lithospheric composition and/or structure, supracrustal stratigraphy and structure, degree of drainage integration, drainage divide migration and climate.

Pazzaglia, Frank J.↗

Temperature Dependent Cyclic Deformation Mechanisms in Haynes 188 Superalloy

The cyclic deformation behavior of a wrought cobalt-base superalloy, Haynes 188, has been investigated over a range of temperatures between 25 and 1000 C under isothermal and in-phase thermomechanical fatigue (TMF) conditions. Constant mechanical strain rates (epsilon-dot) of 10(exp -3)/s and 10(exp -4)/s were examined with a fully reversed strain range of 0.8%. Particular attention was given to the effects of dynamic strain aging (DSA) on the stress-strain response and low cycle fatigue life. A correlation between cyclic deformation behavior and microstructural substructure was made through detailed transmission electron microscopy. Although DSA was found to occur over a wide temperature range between approximately 300 and 750 C the microstructural characteristics and the deformation mechanisms responsible for DSA varied considerably and were dependent upon temperature. In general, the operation of DSA processes led to a maximum of the cyclic stress amplitude at 650 C and was accompanied by pronounced planar slip, relatively high dislocation density, and the generation of stacking faults. DSA was evidenced through a combination of phenomena, including serrated yielding, an inverse dependence of the maximum cyclic hardening with epsilon-dot, and an instantaneous inverse epsilon-dot sensitivity verified by specialized epsilon-dot -change tests. The TMF cyclic hardening behavior of the alloy appeared to be dictated by the substructural changes occuring at the maximum temperature in the TMF cycle.

Rao, K. Bhanu Sankara↗

Large Deformation Dynamic Bending of Composite Beams

Studies were conducted on the large deformation response of composite beams subjected to a dynamic axial load. The beams were loaded with a moderate eccentricity to promote bending. The study was primarily experimental but some finite element results were obtained. Both the deformation and the failure of the beams were of interest. The static response of the beams was also studied to determine potential differences between the static and dynamic failure. Twelve different laminate types were tested. The beams tested were 23 in. by 2 in. and generally 30 plies thick. The beams were loaded dynamically with a gravity-driven impactor traveling at 19.6 ft/sec and quasi-static tests were conducted on identical beams in a displacement controlled manner. For laminates of practical interest, the failure modes under static and dynamic loadings were identical. Failure in most of the laminate types occurred in a single event involving 40% to 50% of the plies. However, failure in laminates with 300 or 150 off-axis plies occurred in several events. All laminates exhibited bimodular elastic properties. The compressive flexural moduli in some laminates was measured to be 1/2 the tensile flexural modulus. No simple relationship could be found among the measured ultimate failure strains of the different laminate types. Using empirically determined flexural properties, a finite element analysis was reasonably accurate in predicting the static and dynamic deformation response.

Derian, E. J.↗

Deformation of Fluid Column by Action of Axial Vibration and Some Aspects of High-Rate Thermocapillary Convection

The deformation of the fluid column by an action of a low-frequency vibration is considered. It is shown that behavior of the free fluid surface depends on the frequency of applied vibration and its amplitude. In the area of very low frequencies when fluid has time to comment on travel of bounding solid walls limiting column, the harmonical oscillations of free surface with given frequency are observed. With increase of vibration frequency the steady-state relief on free fluid surface is formed. If the amplitude of vibration is very small and the frequency corresponding to the first peak in the vibration spectrum on the Mir orbital station, the deformation of free surface tends to zero. Fluid flow induced thermocapillary effect on deformed free surface is more unstable as in the case of smooth cylindrical surface. It was shown that width of heating zone affects very essentially the flow pattern and transition to oscillatory regime of thermocapillary convection.

Feonychev, Alexander I.↗

Ice in Channels and Ice-Rock Mixtures in Valleys on Mars: Did They Slide on Deformable Rubble Like Antarctic Ice Streams?

Recent studies of ice streams in Antarctica reveal a mechanism of basal motion that may apply to channels and valleys on Mars. The mechanism is sliding of the ice on deformable water-saturated till under high pore pressures. It has been suggested by Lucchitta that ice was present in outflow channels on Mars and gave them their distinctive morphology. This ice may have slid like Antarctic ice streams but on rubbly weathering products rather than till. However, to generate water under high pore pressures, elevated heatflow is needed to melt the base of the ice. Either volcanism or higher heatflow more than 2 b.y. ago could have raised the basal temperature. Regarding valley networks, higher heatflow 3 b.y. ago could have allowed sliding of ice-saturated overburden at a few hundred meters depth. If the original, pristine valleys were somewhat deeper than they are now, they could have formed by the same mechanism. Recent sounding of the seafloor in front of the Ross Ice Shelf in Antarctica reveals large persistent patterns of longitudinal megaflutes and drumlinoid forms, which bear remarkable resemblance to longitudinal grooves and highly elongated streamlined islands found on the floors of martian outflow channels. The flutes are interpreted to have formed at the base of ice streams during the last glacial advance. Additional similarities of Antarctic ice streams with martian outflow channels are apparent. Antarctic ice streams are 30 to 80 km wide and hundreds of kilometers long. Martian outflow channels have similar dimensions. Ice stream beds are below sea level. Carr determined that most common floor elevations of martian outflow channels lie below martian datum, which may have been close to or below past martian sea levels. The Antarctic ice stream bed gradient is flat and locally may go uphill, and surface slopes are exceptionally. Martian channels also have floor gradients that are shallow or go uphill locally and have low surface gradients. The depth to the bed in ice streams is 1 to 1.5 km. At bankful stage, the depth of the fluid in outflow channels was 1 to 2 km, according to the height of bordering scarps. The similarity between Antarctic ice streams and martian outflow channels suggests that ice may have flowed through and shaped the outflow channels, and that perhaps the mechanism of motion of Antarctic ice streams also operated in outflow channels. In addition, sliding on deformable rubble may explain the formation of small valley networks. The large Siple Coast Antarctic ice streams are thought to slide over longitudinally grooved, deforming till, where much of the movement is within the till. The till is saturated with water at high pore pressures that nearly supports all of the weight of the ice. The small differential between overburden pressure and pore pressure at the bed is more important than the volume of water, but water needs to be supplied to the till interface. For pore pressures to remain high, the ice streams have to act as a seal that blocks the flow of water through them, and the rock underneath has to be of low permeability to prevent the water from draining away.

Lucchitta, B. K.↗

An Improved Computational Technique for Calculating Electromagnetic Forces and Power Absorptions Generated in Spherical and Deformed Body in Levitation Melting Devices

An improved computational technique for calculating the electromagnetic force field, the power absorption and the deformation of an electromagnetically levitated metal sample is described. The technique is based on the volume integral method, but represents a substantial refinement; the coordinate transformation employed allows the efficient treatment of a broad class of rotationally symmetrical bodies. Computed results are presented to represent the behavior of levitation melted metal samples in a multi-coil, multi-frequency levitation unit to be used in microgravity experiments. The theoretical predictions are compared with both analytical solutions and with the results or previous computational efforts for the spherical samples and the agreement has been very good. The treatment of problems involving deformed surfaces and actually predicting the deformed shape of the specimens breaks new ground and should be the major usefulness of the proposed method.

Zong, Jin-Ho↗

Model Deformation Measurements at NASA Langley Research Center

Only recently have large amounts of model deformation data been acquired in NASA wind tunnels. This acquisition of model deformation data was made possible by the development of an automated video photogrammetric system to measure the changes in wing twist and bending under aerodynamic load. The measurement technique is based upon a single view photogrammetric determination of two dimensional coordinates of wing targets with a fixed third dimensional coordinate, namely the spanwise location. A major consideration in the development of the measurement system was that use of the technique must not appreciably reduce wind tunnel productivity. The measurement technique has been used successfully for a number of tests at four large production wind tunnels at NASA and a dedicated system is nearing completion for a fifth facility. These facilities are the National Transonic Facility, the Transonic Dynamics Tunnel, and the Unitary Plan Wind Tunnel at NASA Langley, and the 12-FT Pressure Tunnel at NASA Ames. A dedicated system for the Langley 16-Foot Transonic Tunnel is scheduled to be used for the first time for a test in September. The advantages, limitations, and strategy of the technique as currently used in NASA wind tunnels are presented. Model deformation data are presented which illustrate the value of these measurements. Plans for further enhancements to the technique are presented.

Burner, A. W.↗

High Strain Rate Deformation Modeling of a Polymer Matrix Composite: Composite Micromechanical Model - Part 2

Recently applications have exposed polymer matrix composite materials to very high strain rate loading conditions, requiring an ability to understand and predict the material behavior under these extreme conditions. In this second paper of a two part report, a three-dimensional composite micromechanical model is described which allows for the analysis of the rate dependent, nonlinear deformation response of a polymer matrix composite. Strain rate dependent inelastic constitutive equations utilized to model the deformation response of a polymer are implemented within the micromechanics method. The deformation response of two representative laminated carbon fiber reinforced composite materials with varying fiber orientation has been predicted using the described technique. The predicted results compare favorably to both experimental values and the response predicted by the Generalized Method of Cells, a well-established micromechanics analysis method.

Goldberg, Robert K.↗

The Effects of Variable Mass and Geometry, Pretwist, Shear Deformation and Rotatory Inertia on the Resonant Frequencies of Intact Long Bones: A Finite Element Model Analysis

The influence of pretwist, nonuniformities in mass and flexural stiffness, rotatory inertia and shear deformation on the natural frequencies of intact bones is evaluated by means of a linear elastic, finite-element model which has been programmed for solution on the digital computer. Theoretical results are compared to the results on the forced vibration of intact canine radii obtained experimentally by Thompson. Surprisingly, inclusion of fairly large pretwist angles (from -14 to 12 deg for one specimen) had little affect on the first three frequencies of transverse vibration in either the cranial or lateral directions. Inclusion of shear deformation reduced the third-mode frequency in the stiffest (lateral) direction by about six percent, otherwise shear deformation played a minor role in determining natural frequencies. Similarly. rotatory inertia had negligible influence up to the third natural frequency. The predominant influence on the first three natural frequencies of transverse vibration could be attributed to the variations in mass and flexural stiffness along the length of the test specimens. Different effective moduli of elasticity are required to yield correct absolute values for the frequencies which correspond to experimental findings. thus implying the presence of some inhomogeneities in material properties around the bone cross-section and/or along its length.

Young, Donald R.↗

Assessment of Localized Deformations in Sand Using X-Ray Computed Tomography

The internal fabric and localized deformation patterns of triaxial sand specimens were investigated using Computed Tomography (CT). Three displacement-controlled, conventional, drained axisymmetric (triaxial) experiments were conducted on dry Ottawa sand specimens at very low effective confining stresses (0.05, 0.52, and 1.30 kPa) in a microgravity environment aboard the Space Shuttle during the NASA STS-79 mission. CT scanning was p'erformed on these flight specimens, as well as on an uncompressed specimen and a specimen tested in a terrestrial laboratory at 1.30 kPa effective confining stress. CT demonstrated high accuracy in detecting specimen inhomogeneity and localization patterns. Formation of deformation patterns is dependent on the effective confining stress and gravity. Multiple symmetrical radial shear bands were observed in the specimens tested in a microgravity environment. In the axial direction, two major conical surfaces were developed. Nonsymmetrical spatial deformation was observed in the 1-G specimen. Analysis tools were developed to quantify the spatial density change. Void ratio variation within and outside the shear bands was calculated and discussed.

Alshibli, Khalid A↗