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At least 685 records · Page 38
All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Droplet Deformation in an Extensional Flow: The Role of Surfactant Physical Chemistry

Surfactant-induced Marangoni effects strongly alter the stresses exerted along fluid particle interfaces. In low gravity processes, these stresses can dictate the system behavior. The dependence of Marangoni effects on surfactant physical chemistry is not understood, severely impacting our ability to predict and control fluid particle flows. A droplet in an extensional flow allows the controlled study of stretching and deforming interfaces. The deformations of the drop allow both Marangoni stresses, which resist tangential shear, and Marangoni elasticities, which resist surface dilatation, to develop. This flow presents an ideal model system for studying these effects. Prior surfactant-related work in this flow considered a linear dependence of the surface tension on the surface concentration, valid only at dilute surface concentrations, or a non-linear framework at concentrations sufficiently dilute that the linear approximation was valid. The linear framework becomes inadequate for several reasons. The finite dimensions of surfactant molecules must be taken into account with a model that includes surfaces saturation. Nonideal interactions between adsorbed surfactant molecules alter the partitioning of surfactant between the bulk and the interface, the dynamics of surfactant adsorptive/desorptive exchange, and the sensitivity of the surface tension to adsorbed surfactant. For example, cohesion between hydrocarbon chains favors strong adsorption. Cohesion also slows the rate of desorption from interfaces, and decreases the sensitivity of the surface tension to adsorbed surfactant. Strong cohesive interactions result in first order surface phase changes with a plateau in the surface tension vs surface concentration. Within this surface concentration range, the surface tension is decoupled from surface concentration gradients. We are engaged in the study of the role of surfactant physical chemistry in determining the Marangoni stresses on a drop in an extensional flow in a numerical and experimental program. Using surfactants whose dynamics and equilibrium behavior have been characterized in our laboratory, drop deformation will be studied in ground-based experiment. In an accompanying numerical study, predictive drop deformations will be determined based on the isotherm and equation of state determined in our laboratory. This work will improve our abilities to predict and control all fluid particle flows.

Stebe, Kathleen J.↗
DOE OSTI2021

Shear strength measurements and hydrostatic compression of rhenium diboride under high pressures

Shear strength measurements have been carried out on rhenium diboride, ReB 2 , to a pressure of 74 GPa using a Radial X-ray Diffraction (R-XRD) technique in a diamond anvil cell using platinum as an internal x-ray pressure standard. The R-XRD result has provided a unique insight into the deformation of hexagonal crystal lattice under non-hydrostatic compression and variation of shear strength with increasing pressure. From R-XRD data, we have estimated hydrostatic component of compression to determine an equation of state of rhenium diboride yielding a bulk modulus of K 0 = 366 ± 25 GPa with a pressure derivative $K^{'}_{0}$ = 4.3 ± 0.5 in good agreement with hydrostatic density functional theory calculations. Here, the average lower bound of shear strength (τ) from various diffraction planes was then calculated using the measured interplanar d-spacing (d m ) and hydrostatic component of d-spacing (d p ) to be shown to approach 6.7 ± 0.4 GPa at 70 GPa. Our results show that the anisotropic compression effects observed in ReB 2 under hydrostatic compression are correlated to electronic structure changes under compression as predicted by theoretical calculations.

36 MATERIALS SCIENCE↗
DOE OSTI2020

On the Cutting of Metals: A Mechanics Viewpoint

The mechanics of large-strain deformation in cutting of metals is discussed, primarily from viewpoint of recent developments in in situ analysis of plastic flow and microstructure characterization. It is shown that a broad range of deformation parameters can be accessed in chip formation—strains of 1–10, strain rates of 10–10 5 /s, and temperatures up to 0.7T m —and controlled. This range is far wider than achievable by any other single-stage, severe plastic deformation (SPD) process. The resulting extreme deformation conditions produce a rich variety of microstructures in the chip. Four principal types of chip formation—continuous, shear-localized, segmented, and mushroom-type—as elucidated first by Nakayama (1974, “The Formation of ‘Saw-Toothed Chip’ in Metal Cutting,” Proceedings of International Conference on Production Engineering, Tokyo, pp. 572–577) are utilized to emphasize the diverse plastic flow phenomena, especially unsteady deformation modes that prevail in cutting. These chip types are intimately connected with the underlying flow, each arising from a distinct mode and triggered by an instability phenomenon. The role of plastic flow instabilities such as shear banding, buckling, and fracture in mediating unsteady flow modes is expounded, along with consequences of the flow modes and chip types for the cutting. Sinuous flow is shown to be the reason why gummy (highly strain-hardening) metals, although relatively soft, are so difficult to cut. Synthesizing the various observations, a hypothesis is put forth that it is the stability of flow modes that determines the mechanics of cutting. Furthermore, this leads to a flow-stability phase diagram that could provide a framework for predicting chip types and process attributes.

42 ENGINEERING↗
DOE OSTI2022

Shear strain gradient in Cu/Nb nanolaminates: Strain accommodation and chemical mixing

Disentangling the intragranular and interfacial plasticity contribution to the overall strain accommodation is crucial to understanding the microstructural evolution and mass transport upon deformation in materials with the nanoscale feature size. In this study, we devise a new approach to tackle the issue by introducing shear strain gradients into Cu/Nb nanolaminates of different layer thicknesses with the shear perpendicular to the laminate interfaces. The measurement of the strain gradient and the resultant lattice disorientation enables a quantitative understanding of the intragranular and interfacial plasticity contribution. We found that intragranular slip entirely governs the deformation in the 300 nm-layer laminate and, unexpectedly, contributes ~80% of the total plasticity in the 30 nm-layer laminate. The high intragranular plasticity in the thin laminate is attributed to the large width of confined slip planes and their remnant potential for storing dislocations. In addition, substantial forced chemical mixing is observed in the top region of the 30 nm-layer laminate where the effective layer thickness is reduced below 8 nm. The transition of deformation mechanism from confined layer slip to dislocation transmission is largely responsible for the initiation of substantial mixing. Our method and findings shed light on the deformation mechanism and deformation-induced mass transport behavior in nanostructured materials.

36 MATERIALS SCIENCE↗
NASA NTRS1989

Shear testing of fiber reinforced metal matrix composites

This paper outlines the elements of a combined experimental/analytical methodology for accurate shear characterization of unidirectional composites in the linear and nonlinear range, with particular attention devoted to metal matrix composites. It is illustrated that consistent results can be obtained for a large class of composites from two commonly employed shear test methods currently in use by composites researchers when the influence of various factors that affect the determination of the actual shear response is properly analyzed. These factors include the effects of material anisotropy, specimen geometry, manner of load introduction, and test fixture design on the stress and deformation fields in the test section of off-axis and Iosipescu specimens. Common errors associated with the measurement of deformation fields and calculation of stress fields are discussed and quantified. Particular problems in the determination of the shear response of unidirectional boron/aluminum using the Iosipescu test are illustrated and discussed.

Pindera, Marek-Jerzy↗
NASA NTRS1991

New, unusual and nonclassical behavior of thin-walled composite structures

Examples are discussed which illustrated several new, unusual, and nonclassical effects that are present in thin-walled composite structures. Some of the unusual deformation characteristics occur naturally, such as the apparent increased flexibility of thin-walled beam structures due to bending-transverse shear and extension-transverse shear elastic couplings. Others are created intentionally, such as the exaggerated Poisson expansion in the tailored wing covers that produces elastic chordwise camber deformations.

Rehfield, Lawrence W.↗
DOE OSTI2026

Chemo-Mechanics of α-V 2 O 5 During Lithiation and Implications for Rechargeable Battery Cathodes

Chemo-mechanical degradation of layered oxide electrodes is strongly influenced by crystallographic anisotropy, local stress evolution, and ion insertion, yet the intrinsic mechanical response of layered materials remains incompletely understood. Indeed, most prior studies have focused on polycrystalline materials but single crystals enable direct observation of coupling between anisotropic ion diffusion and mechanical response. This study aims to determine how crystallographic anisotropy and lithiation affect deformation, fracture, and mechanical properties in single-crystal V 2 O 5 , and compares this behavior with polycrystalline counterparts. Polycrystalline V 2 O 5 thin films and single-crystal α-V 2 O 5 were studied using nanoindentation, scanning electron microscopy, focused ion beam cross-sectioning, and Raman spectroscopy. Single crystals were tested in pristine and chemically lithiated states, including experiments in which crystals were first plastically deformed via nanoindentation and subsequently lithiated. Polycrystalline films exhibited significantly higher hardness and elastic modulus than single crystals. Single crystals indented normal to the exposed (001) basal plane exhibited pronounced anisotropic deformation, including directional slip, crystallographically-guided cracking, anisotropic crack propagation, interlayer separation, and shear localization. Lithiation caused substantial softening, reduced hardness and modulus, and suppressed displacement bursts during nanoindentation, while previously indented regions showed crack formation and growth upon lithiation. Mechanical behavior of α-V 2 O 5 is strongly governed by crystallographic anisotropy and further altered by lithiation, with pre-existing deformation serving as a strong driver of fracture during ion insertion. These findings illuminate the coupling among ion insertion, deformation, and fracture in layered oxides and provide a basis for understanding and mitigating mechanical failure in electrochemical energy-storage materials.

Anisotropy↗
DOE OSTI2022

Comparison of cracking behavior of nanocrystalline Cu film on substrates of different plastic deformation mechanisms

Nanocrystalline (NC) metallic thin films on metal substrates usually undergo substantial cracking at applied strains of a few percent due to strain concentration at the film/substrate interfaces. In this work, we show that the cracking of a nanocrystalline Cu film can be significantly inhibited by a NiTi substrate. NC Cu films are deposited on 304 stainless steel, Kapton and NiTi substrates, in which plastic deformation occurs via dislocation slip, molecule chain disentanglement and collective lattice shear, respectively. NC Cu film on NiTi substrate starts to show observable cracks at an applied strain of 24%, in contrast to 6% for that on steel and 15% for that on Kapton. Further, the crack density in the film on NiTi substrate is two to three orders lower than the counterparts. The discrepancy is suggested to attribute to the reduced local strain concentration enabled by the uniform lattice shear and high strain hardening of the NiTi substrate.

36 MATERIALS SCIENCE↗
NASA NTRS1972

Theoretical study of corrugated plates: Shearing of a corrugated plate with curvilinear corrugations

A theoretical elastic analysis is presented for the in-plane shear of a corrugated plate with curvilinear corrugations and with discrete attachments between the ends of the corrugations and the surrounding members. The purpose of the analysis is to obtain information about the effective shear stiffness of the plate and the flexural strains that develop in it during the shearing process as a result of the attendant cross-sectional deformations. The crests and troughs of the corrugation cross section are assumed to be identical circular arcs. The following four kinds of discrete attachment at the corrugation ends are considered in the analysis: (1) point attachments in the troughs, (2) point attachments at the crests and in the troughs, (3) point attachments at mid-height, and (4) point attachments at mid-height, at the crests, and in the troughs.

Wu, L.↗
NASA NTRS1993

Strength and deformation properties of basaltic lava flows on planetary surfaces

Basaltic rocks are thought to constitute a volumetrically significant rock type on the Moon, Mercury, Mars, and Venus, in addition to the Earth. Spacecraft images of surfaces with known or suspected basaltic composition on these bodies, particularly on Venus, indicate that these rocks have been deformed in the brittle regime to form faults and perhaps dilatant cracks, in addition to folding and more distributed types of deformation. Predictions of brittle fracture or other types of deformation are made by comparing calculated stresses from a tectonic model to some criterion for rock strength. Common strength criteria used in the planetary science literature for near-surface deformation include a Griffith tensile-strength criterion for intact rock, a Mohr envelope for intact basalt, and a brittle strength envelope based on Byerlee's law of rock frictional resistance. However, planetary terrains of basaltic composition consist of much more than just intact basaltic rock. The aggregate basaltic material, termed the 'rock mass,' consists of both the intact rock and the associated fracture, faults, lithologic contacts, and other discontinuous surfaces. A basaltic rock mass is the relevant material for which strength properties must be defined and calculated model stresses must be compared to in order to more accurately predict brittle deformation. For example, the various strengths of a rock mass are less than that of intact material of the same composition. This means that tectonic models which compare stresses to intact failure strengths overestimate the stresses required for fracture and so underestimate the extent and magnitude of brittle deformation predicted in these models. On the other hand, rock mass shear strength can be greater than that predicted from Byerlee's law. The concept of rock mass strength is central to many engineering design studies in which calculated stresses are used to predict brittle fracture, and this experience indicates that brittle strength envelopes which assume properties for intact rock (Griffith parabolas) or sliding along a single, continuous surface (Byerlee's law) inadequately characterize the tensile, compressive, and shear strengths of rock masses. The criterion adopted here to relate stresses to rock mass fracture is based on a Griffith-type curve for tensile normal stress and a concave downward curve for compressive normal stress. It is the only available criterion that explicitly considers the weakening effects of discontinuities within the rock mass on the stress state required for fracture.

Schultz, Richard A.↗
NASA NTRS2000

Experiment Evaluation of Bifurcation in Sands

The basic principles of bifurcation analysis have been established by several investigators, however several issues remain unresolved, specifically how do stress level, grain size distribution, and boundary conditions affect general bifurcation phenomena in pressure sensitive and dilatant materials. General geometrical and kinematics conditions for moving surfaces of discontinuity was derived and applied to problems of instability of solids. In 1962, the theoretical framework of bifurcation by studying the acceleration waves in elasto-plastic (J2) solids were presented. Bifurcation analysis for more specific forms of constitutive behavior was examined by studying localization in pressure-sensitive, dilatant materials, however, analyses were restricted to plane deformation states only. Bifurcation analyses were presented and applied to predict shear band formations in sand under plane strain condition. The properties of discontinuous bifurcation solutions for elastic-plastic solids under axisymmetric and plane strain loading conditions were studied. The study focused on theory, but also references and comparisons to experiments were made. The current paper includes a presentation of a summary of bifurcation analyses for biaxial and triaxial (axisymmetric) loading conditions. The Coulomb model is implemented using incremental piecewise scheme to predict the constitutive relations and shear band inclination angles. Then, a comprehensive evaluation of bifurcation phenomena is presented based on data from triaxial experiments performed under microgravity conditions aboard the Space Shuttle under very low effective confining pressure (0.05 to 1.30 kPa), in which very high peak friction angles (47 to 75 degrees) and dilatancy angles (30 to 31 degrees) were measured. The evaluation will be extended to include biaxial experiments performed on the same material under low (10 kPa) and moderate (100 kPa) confining pressures. A comparison between the behavior under biaxial and triaxial loading conditions will be presented, and related issues concerning influence of confining pressure will be discussed.

Alshibi, Khalid A.↗
NASA NTRS2019

Extratropical Large-Scale Traveling Weather Systems in the Southern Hemisphere on Mars

From late-autumn through early-spring, the middle- and high-latitudes of both hemispheres of Mars and its predominantly carbon-dioxide atmosphere support mean equator-to-pole thermal contrasts, and then, support a strong mean westerly polar vortex. Observations from orbiting spacecraft indicate that this intense mean baroclinicity-barotropicity supports large-scale eastward traveling weather systems (i.e., transient, traveling synoptic-period waves, on the order of the Rossby deformation scale). On Earth, extratropical weather disturbances arise from wind-shear instabilities, and these are critical components of the terrestrial global circulation. So it is the case for Mars. Large-scale traveling weather systems on Mars serve as agents in the transport of heat, momentum and scalar and tracer quantities (e.g., atmospheric dust, watervapor, ice clouds, chemical species, etc). Such weather systems interact with other large-scale atmospheric circulation components, namely, quasi-stationary (i.e., forced Rossby) modes; global thermal tidal modes; and then, upon large-/continental- geographical scales, upslope/ down-slope flows amongst high relief, low relief, impact basins, and volcanic rises, and more. The character of Mars' traveling extratropical weather disturbances in its southern hemisphere during late winter through early spring is investigated using a high-resolution Mars global climate model (i.e., Mars GCM), and one from the Agency's Mars Climate Modeling Center (MCMC) based at the NASA Ames Research Center. The climate model includes several complex atmospheric physical packages. With such physics modules, our global climate simulations present comparatively well with observations of the planet's current water cycle (Haberle et al.,2019). The climate model is "forced" with an annual dust cycle (i.e., nudged based on MGS/TES observations). Compared to the northern-hemisphere counterparts, the southern synoptic-period weather disturbances and accompanying frontal waves have smaller meridional and zonal scales, and are less intense. Influences of the zonally asymmetric (i.e., east-west varying) topography on southern large-scale weather are investigated, in addition to large-scale up-slope/down-slope flows and the diurnal cycle. A southern storm zone in late winter and early spring presents in the western hemisphere via orographic influences from the Tharsis highlands, and the Argyre and Hellas impact basins. Geographically localized transient-wave activity diagnostics are constructed that illuminate dynamical differences amongst the simulations and these are presented.

Hollingsworth, Jeffery L.↗
NASA NTRSDate not supplied

Experimental Characterization of Non-Associative Plasticity Flow Rule Coefficients and Post-Peak Stress Degradation for the LS-DYNA MAT213 Model

This project is focused on developing an experimental framework for characterizing non-associative plasticity flow rule coefficients through coupon-scale tests for the LS-DYNA MAT213 model. The main objective is to characterize these coefficients based on the multi-scale (i.e., both microscopic and macroscopic) full-field measurement of the evolution of strain and stress fields. This paper focuses on presenting the experimental work on characterizing the full-scale stress-strain curves of T700/LM-PAEK composites under tension, compression, and shear loads. The experimental data set was intended to build a deformation sub-model in the MAT213 model for the material. The strain data were collected using both microscopic and macroscopic digital image correlation techniques. The microscopic technique was particularly useful for fracture cases under small strains. A preliminary simulation result obtained from the MAT213 model is also presented in the paper. The experimental framework herein will be extended to characterize post-peak stress degradation in the composite material and to develop a damage sub-model for the material. This project will contribute to developing a simulation tool based on the MAT213 model for simulating the rate-dependent impact damage in composites under multi-axial loading.

MAT213↗
NASA NTRS2012

Failure Analysis of T-38 Aircraft Burst Hydraulic Aileron Return Line

During maintenance troubleshooting for fluctuating hydraulic pressures, a technician found that a right hand aileron return line, on the flight hydraulic side, was ruptured (Fig. 1, 2). This tubing is part of the Hydraulic Flight Control Aileron Return Reducer to Aileron Manifold and is suspected to be original to the T-38 Talon trainer aircraft. Ailerons are small hinged sections on the outboard portion of a wing used to generate rolling motion thereby banking the aircraft. The ailerons work by changing the effective shape of the airfoil of the outer portion of the wing [1]. The drawing, Northrop P/N 3-43033-55 (6/1960), specifies that the line is made from 0.375 inch OD, aluminum 5052-0 tubing with a 0.049 inch wall thickness. WW-T-787 requires the tube shall be seamless and uniform in quality and temper [2]. The test pressure for this line is 3000 psi, and the operational pressure for this line is estimated to be between 45 psi and 1500 psi based on dynamic loading during flight. Examination of the fracture surface found evidence of arrest bands originating on the inner diameter (Fig 3). Ductile dimples are observed on the tube fractures (Fig. 4). The etched cross-section revealed thinning and work-hardening in the burst region (Fig. 5). The wall thickness just outside the work-hardened fracture region measured 0.035". Barlow's Formula: P = 2St/D, where P is burst pressure, S is allowable stress, t is wall thickness and D is the outer diameter of tube. Using the ultimate tensile strength of 28 ksi and a measured wall thickness of 0.035 inches at burst, P = 5.2 ksi (burst pressure). Using the yield of 13 ksi (YS) for aluminum 5052-0, plastic deformation will happen at P = 2.4 ksi suggesting plastic deformation occurred at a proof pressure of 3.0 ksi. Conclusion: The burst resulted from high stress, low-cycle fatigue. Evidence of arrest bands originating on the inner diameter. Fracture is predominately shear dimples, characteristic of high load ductile fractures (Fig 6). Section wall reduction in the burst region. Plastic deformation and thinning of the out-of-specification tube wall likely happened during the initial proof testing years ago. Metallography of tubing away from rupture site confirmed tubing was seamless. Based on the tube microstructure, it is likely that the initial wall thickness was about 30 % thinner than the requirement of 0.049 inches. Fracture initiated on the ID and progressed to the OD (shear lip). The tube is made of the correct material of 5052-0 aluminum as verified using Optical Emission Spectroscopy (Table 2). The tubing hardness tested 77 HV100 (77 HRE). This hardness is slightly above the requirement for 70 HRE maximum for aluminum 5052-0 in AMS 2658C [3].

Martinez, J. E.↗
NASA NTRS2000

One Century of Tectonic Deformation Along the Sumatran Fault from Triangulation and GPS Surveys

An analysis combining historical triangulation and recent Global Positioning System (GPS) survey measurements in West and North Sumatra, Indonesia, reveals a detailed slip history along the central part of the Sumatran Fault. The arc-parallel components of the combined velocity field are consistent with slip rates inferred from GPS data, ranging from 23 to 24 mm/yr. Between 1.0 S and 1.3 N the Sumatran Fault appears to be characterized by deep locking depths, on the order of 20 km, and the occurrence of large (M (sub w) approximately 7) earthquakes. The long-term (1883-1993) strains show simple right-lateral shear, with rates similar to GPS-measured, 1989-1993 strain rates. Coseismic deformation due to the 1892 Tapanuli and 1926 Padang Panjang earthquakes, estimated from triangulation measurements taken before and after the events, indicate that the main shocks were significantly larger than previously reported. The 1892 earthquake had a likely magnitude of M (sub w) approximately equal to 7.6, while the 1926 events appear to be comparable in size to the subsequent (M approximately 7) 1943 events, and an order of magnitude higher than previously reported.

Prawirodirdjo, L.↗
NASA NTRS1976

Higher-order effects of initial deformation on the vibrations of crystal plates

A system of plate equations for the thickness-shear and flexural vibrations superposed on large initial deflection due to bending is derived; in the stress-strain relations, the terms associated with the fourth-order elastic stiffness coefficients are retained. An explicit formula for the change in the fundamental cut-off thickness shear frequency is obtained, and the effects of the terms associated with the fourth-order constants appear to be significant for large gradients of the rotation angles.

Markenscoff, X.↗
NASA NTRS1999

Drop Breakup in Fixed Bed Flows as Model Stochastic Flow Fields

We examine drop breakup in a class of stochastic flow fields as a model for the flow through fixed fiber beds and to elucidate the general mechanisms whereby drops breakup in disordered, Lagrangian unsteady flows. Our study consists of two parallel streams of investigation. First, large scale numerical simulations of drop breakup in a class of anisotropic Gaussian fields will be presented. These fields are generated spectrally and have been shown in a previous publication to be exact representations of the flow in a dilute disordered bed of fibers if close interactions between the fibers and the drops are dynamically unimportant. In these simulations the drop shape is represented by second and third order small deformation theories which have been shown to be excellent for the prediction of drop breakup in steady strong flows. We show via these simulations that the mechanisms of drop breakup in these flows are quite different than in steady flows. The predominant mechanism of breakup appears to be very short lived twist breakups. Moreover, the occurrence of breakup events is poorly predicted by either the strength of the local flow in which the drop finds itself at breakup, or the degree of deformation that the drop achieves prior to breakup. It is suggested that a correlation function of both is necessary to be predictive of breakup events. In the second part of our research experiments are presented where the drop deformation and breakup in PDMS/polyisobutylene emulsions is considered. We consider very dilute emulsions such that coalescence is unimportant. The flows considered are simple shear and the flow through fixed fiber beds. Turbidity, small angle light scattering, dichroism and microscopy are used to interrogate the drop deformation process in both flows. It is demonstrated that breakup at very low capillary numbers occurs in both flows but larger drop deformation occurs in the fixed bed flow. Moreover, it is witnessed that breakup in the bed occurs continuously during flow and apparently with uniform probability through the bed length. The drop deformations witnessed in our experiments are larger than those predicted by the numerical simulations, and future plans to investigate these differences are discussed.

Shaqfeh, Eric S. G.↗
NASA NTRS1990

Strike-slip faulting, wrinkle ridges, and time variable stress states in the Coprates Region of Mars

The existence of strike-slip faults was recently documented in two locations on Mars. Two clear examples are reviewed located southeast of Valles Marineris and preliminary evidence is presented for more widespread strike-slip deformation elsewhere in Coprates. The first two examples show that strike-slip faulting occurred in a broad zone east of the Coprates Rise spanning approximately 400 km east-west by perhaps 1000 km north-south. The last example suggests that the growth of major wrinkle ridges throughout Coprates may have been influenced by horizontally directed shear stresses and that more than one generation of ridges was produced. Thus, 'compressional' deformation of ridged plains south of Valles Marineris was spatially heterogeneous and a temporal change in stress may have been involved.

Schultz, Richard A.↗