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All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

Issues and Strategies in Solving Multidisciplinary Optimization Problems

Optimization research at NASA Glenn Research Center has addressed the design of structures, aircraft and airbreathing propulsion engines. The accumulated multidisciplinary design activity is collected under a testbed entitled COMETBOARDS. Several issues were encountered during the solution of the problems. Four issues and the strategies adapted for their resolution are discussed. This is followed by a discussion on analytical methods that is limited to structural design application. An optimization process can lead to an inefficient local solution. This deficiency was encountered during design of an engine component. The limitation was overcome through an augmentation of animation into optimization. Optimum solutions obtained were infeasible for aircraft and airbreathing propulsion engine problems. Alleviation of this deficiency required a cascading of multiple algorithms. Profile optimization of a beam produced an irregular shape. Engineering intuition restored the regular shape for the beam. The solution obtained for a cylindrical shell by a subproblem strategy converged to a design that can be difficult to manufacture. Resolution of this issue remains a challenge. The issues and resolutions are illustrated through a set of problems: Design of an engine component, Synthesis of a subsonic aircraft, Operation optimization of a supersonic engine, Design of a wave-rotor-topping device, Profile optimization of a cantilever beam, and Design of a cylindrical shell. This chapter provides a cursory account of the issues. Cited references provide detailed discussion on the topics. Design of a structure can also be generated by traditional method and the stochastic design concept. Merits and limitations of the three methods (traditional method, optimization method and stochastic concept) are illustrated. In the traditional method, the constraints are manipulated to obtain the design and weight is back calculated. In design optimization, the weight of a structure becomes the merit function with constraints imposed on failure modes and an optimization algorithm is used to generate the solution. Stochastic design concept accounts for uncertainties in loads, material properties, and other parameters and solution is obtained by solving a design optimization problem for a specified reliability. Acceptable solutions can be produced by all the three methods. The variation in the weight calculated by the methods was found to be modest. Some variation was noticed in designs calculated by the methods. The variation may be attributed to structural indeterminacy. It is prudent to develop design by all three methods prior to its fabrication. The traditional design method can be improved when the simplified sensitivities of the behavior constraint is used. Such sensitivity can reduce design calculations and may have a potential to unify the traditional and optimization methods. Weight versus reliability traced out an inverted-S-shaped graph. The center of the graph corresponded to mean valued design. A heavy design with weight approaching infinity could be produced for a near-zero rate of failure. Weight can be reduced to a small value for a most failure-prone design. Probabilistic modeling of load and material properties remained a challenge.

Patnaik, Surya↗
DOE OSTI2021

Molecular dynamics simulations of inelastic x-ray scattering from shocked copper

By taking the spatial and temporal Fourier transforms of the coordinates of the atoms in molecular dynamics simulations conducted using an embedded-atom-method potential, we calculate the inelastic scattering of x rays from copper single crystals shocked along [001] to pressures of up to 70 GPa. Above the Hugoniot elastic limit, we find that the copious stacking faults generated at the shock front introduce strong quasi-elastic scattering (QES) that competes with the inelastic scattering signal, which remains discernible within the first Brillouin zone; for specific directions in reciprocal space outside the first zone, the QES dominates the inelastic signal overwhelmingly. The synthetic scattering spectra we generate from our Fourier transforms suggest that energy resolutions of order 10 meV would be required to distinguish inelastic from quasi-elastic scattering within the first Brillouin zone of shock-loaded copper. We further note that high-resolution inelastic scattering also affords the possibility of directly measuring particle velocities via the Doppler shift. These simulations are of relevance to future planned inelastic scattering experiments at x-ray Free Electron Laser facilities.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

A continuum damage approach to spallation and the role of microinertia

Spall failure is of interest due to its prevalence in high strain rate problems in which the spallation is driven by the interaction of release waves. In this article, a porosity-based damage model that includes microinertial effects is used to examine spall failure. The model is successfully calibrated to plate impact-driven experiments and then used to evaluate experimental conditions producing more extreme strain rate conditions, such as those in laser-driven experiments. The incorporation of microinertia allows us to better understand the increase in apparent macroscopic spall strength seen at high strain rates. Correspondingly, we conclude that the incorporation of microinertial effects improves the model’s predictive capabilities. Microinertial effects result in more severe local tensile stresses that affect the damage evolution, and microinertia can play a significant role in the approach to the material’s ideal strength at extreme loading rates. Additionally, a preliminary parametric study is also carried out to investigate the role of microstructural aspects such as nucleation volume fraction and initial pore radius. One counter-intuitive result from the microinertial effects is that, for a given nucleation site volume fraction, having larger initial pore nucleation sites can lead to an improved macroscopic spall strength.

36 MATERIALS SCIENCE↗
NASA NTRS1996

Noninvasive Determination of Bone Mechanical Properties using Vibration Response: A Refined Model and Validation in vivo

Accurate non-invasive mechanical measurement of long bones is made difficult by the masking effect of surrounding soft tissues. Mechanical Response Tissue Analysis (MRTA) offers a method for separating the effects of the soft tissue and bone; however, a direct validation has been lacking. A theoretical analysis of wave propagation through the compressed tissue revealed a strong mass effect dependent on the relative accelerations of the probe and bone. The previous mathematical model of the bone and overlying tissue system was reconfigured to incorporate the theoretical finding. This newer model (six-parameter) was used to interpret results using MRTA to determine bone cross-sectional bending stiffness, EI(sub MRTA). The relationship between EI(MRTA) and theoretical EI values for padded aluminum rods was R(exp 2) = 0.999. A biological validation followed using monkey tibias. Each bone was tested in vivo with the MRTA instrument. Postmortem, the same tibias were excised and tested to failure in three-point bending to determine EI(sub 3-PT) and maximum load. Diaphyseal Bone Mineral Density (BMD) measurements were also made. The relationship between E(sub 3-PT) and in vivo EI(sub MRTA) using the six-parameter model is strong (R(exp 2) = 0.947) and better than that using the older model (R(exp 2) = 0.645). EI(MRTA) and BMD are also highly correlated (R(exp 2) = 0.853). MRTA measurements in vivo and BMD ex vivo are both good predictors of scaled maximum strength (R(exp 2) = 0.915 and R(exp 2) = 0.894, respectively). This is the first biological validation of a non-invasive mechanical measurement of bone by comparison to actual values. The MRTA technique has potential clinical value for assessing long-bone mechanical properties.

Roberts, S. G.↗
NASA NTRS2002

Fatigue Crack Measurement in Composite Materials by Ultrasonic Methods

The nondestructive detection of intra-ply microcracking in unlined pressure vessels fabricated from composite materials is critical to ensuring mission success. Microcracking in composite structures due to combined fatigue and cryogenic thermal loading can be very troublesome to detect in-service and when it begins to link through the thickness can cause leakage and failure of the structure. These leaks may lead to loss of pressure/propellant, increased risk of explosion and possible cryo-pumping. The work presented herein develops a method and an instrument to locate and measure intraply fatigue cracking through the thickness of laminated composite material by means of correlation with ultrasonic resonance. Resonant ultrasound spectroscopy provides measurements which are, sensitive to both the microscopic and macroscopic properties of an object. Elastic moduli, acoustic attenuation, and geometry can all be probed. The approach is based on the premise of half-wavelength resonance. The method injects a broadband ultrasonic wave into the test structure using a swept frequency technique. This method provides dramatically increased energy input into the test article, as compared to conventional spike pulsed ultrasonics. This relative energy increase improves the ability to measure finer details in the materials character, such as micro-cracking and porosity. As the micro-crack density increases, more interactions occur with the higher frequency (small wavelength) components of the signal train causing the spectrum to shift toward lower frequencies. Preliminary experiments have verified a measurable effect on the resonance spectrum of the ultrasonic data to detect microcracking. Methods involving self organizing neural networks and other clustering algorithms show that the resonance ultrasound signatures from composites vary with the degree of microcracking and can be separated and identified.

Walker, James L.↗
NASA NTRS1996

Noninvasive Determination of Bone Mechanical Properties Using Vibration Response: A Refined Model and Validation in vivo

Accurate non-invasive mechanical measurement of long bones is made difficult by the masking effect of surrounding soft tissues. Mechanical response tissue analysis (MRTA) offers a method for separating the effects of the soft tissue and bone; however, a direct validation has been lacking. A theoretical analysis of wave propagation through the compressed tissue revealed a strong mass effect dependent on the relative accelerations of the probe and bone. The previous mathematical model of the bone and overlying tissue system was reconfigured to incorporate the theoretical finding. This newer model (six-parameter) was used to interpret results using MRTA to determine bone cross-sectional bending stiffness, EI(sub MRTA). The relationship between EI(sub MRTA) and theoretical EI values for padded aluminum rods was R(sup 2) = 0.999. A biological validation followed using monkey tibias. Each bone was tested in vivo with the MRTA instrument. Postmortem, the same tibias were excised and tested to failure in three-point bending to determine EI(sub 3-PT) and maximum load. Diaphyseal bone mineral density (BMD) measurements were also made. The relationship between EI(sub 3-PT) and in vivo EI(sub MRTA) using the six-parameter model is strong (R(sup 2) = 0.947) and better than that using the older model (R(sup 2) = 0.645). EI(sub MRTA) and BMD are also highly correlated (R(sup 2) = 0.853). MRTA measurements in vivo and BMD ex vivo are both good predictors of scaled maximum strength (R(sup 2) = 0.915 and R(sup 2) = 0.894, respectively). This is the first biological validation of a non- invasive mechanical measurement of bone by comparison to actual values. The MRTA technique has potential clinical value for assessing long-bone mechanical properties.

Roberts, S. G.↗
DOE OSTI2024

Passive Islanding Detection of Inverter-Based Resources in a Noisy Environment

Islanding occurs when a load is energized solely by local generators and can result in frequency and voltage instability, changes in current, and poor power quality. Poor power quality can interrupt industrial operations, damage sensitive electrical equipment, and induce outages upon the resynchronization of the island with the grid. This study proposes an islanding detection method employing a Duffing oscillator to analyze voltage fluctuations at the point of common coupling (PCC) under a high-noise environment. Unlike existing methods, which overlook the noise effect, this paper mitigates noise impact on islanding detection. Power system noise in PCC measurements arises from switching transients, harmonics, grounding issues, voltage sags and swells, electromagnetic interference, and power quality issues that affect islanding detection. Transient events like lightning-induced traveling waves to the PCC can also introduce noise levels exceeding the voltage amplitude by more than seven times, thus disturbing conventional detection techniques. The noise interferes with measurements and increases the nondetection zone (NDZ), causing failed or delayed islanding detection. The Duffing oscillator nonlinear dynamics enable detection capabilities at a high noise level. The proposed method is designed to detect the PCC voltage fluctuations based on the IEEE standard 1547 through the Duffing oscillator. For the voltages beyond the threshold, the Duffing oscillator phase trajectory changes from periodic to chaotic mode and sends an islanded operation command to the inverter. The proposed islanding detection method distinguishes switching transients and faults from an islanded operation. Experimental validation of the method is conducted using a 3.6 kW PV setup.

Energy & Fuels↗
NASA NTRS2003

Advanced Computational Aeroacoustics Methods for Fan Noise Prediction

Direct computation of fan noise is presently not possible. One of the major difficulties is the geometrical complexity of the problem. In the case of fan noise, the blade geometry is critical to the loading on the blade and hence the intensity of the radiated noise. The precise geometry must be incorporated into the computation. In computational fluid dynamics (CFD), there are two general ways to handle problems with complex geometry. One way is to use unstructured grids. The other is to use body fitted overset grids. In the overset grid method, accurate data transfer is of utmost importance. For acoustic computation, it is not clear that the currently used data transfer methods are sufficiently accurate as not to contaminate the very small amplitude acoustic disturbances. In CFD, low order schemes are, invariably, used in conjunction with unstructured grids. However, low order schemes are known to be numerically dispersive and dissipative. dissipative errors are extremely undesirable for acoustic wave problems. The objective of this project is to develop a high order unstructured grid Dispersion-Relation-Preserving (DRP) scheme. would minimize numerical dispersion and dissipation errors. contains the results of the funded portion of the project. scheme on an unstructured grid has been developed. constructed in the wave number space. The characteristics of the scheme can be improved by the inclusion of additional constraints. Stability of the scheme has been investigated. Stability can be improved by adopting the upwinding strategy.

Envia, Edmane↗
NASA NTRS2004

Foreign Object Damage Resistance of Uncoated and EBC Coated SiC/SiC Composites

SiC/SiC composites fabricated by the melt infiltration method are candidate materials for the turbine components such as combustor liners, nozzle vanes and blades because of their high temperature strength, and thermal conductivity. To avoid surface recession in a combustion environment containing moisture, these materials also require an environmental barrier coating (EBC). Under thermal loading condition alone, the EBC coated SiC/SiC components show microstructural and strength stability after -1 5000 hr exposure to 12OO0C in combustion environment. However, the stability of SiC/SiC composites under impact conditions is not fully understood. In this study, both uncoated and EBC coated SiC/SiC composite specimens were impact tested by steel-ball projectiles at room temperature and at 1300 C in air. After impact, the specimens were analyzed under a scanning electron microscope to assess coating damage, and then tensile and bend tested at room temperature to determine their residual strength. The extent of coating and substrate damage with increasing projectile velocity was imaged and analyzed using nondestructive evaluation (NDE) methods such as micro focus x-ray radiography, tomography, and thermal wave imaging. The mechanical property results of impacted specimens are correlated with the NDE results and the scanning electron microscopy (SEM) observations. Influence of projectile velocity on impact damage of the coating/substrate, and the implication coating damage on durability of the composite will be discussed.

Bhatt, R. T.↗
NASA NTRS2009

Oxygen Ion Heat Rate within Alfvenic Turbulence in the Cusp

The role that the cleft/cusp has in ionosphere-magnetosphere coupling makes it a dynamic and important region. It is directly exposed to the solar wind, making it possible for the entry of electromagnetic energy and precipitating electrons and ions from dayside reconnection and other dayside events. It is also a significant source of ionospheric plasma, contributing largely to the mass loading of the magnetosphere with large fluxes of outflowing ions. Crossing the cusp/cleft near 5100 km, the Polar instruments observe the common correlation of downward Poynting flux, ion energization, soft electron precipitation, broadband extremely low-frequency (BB-ELF) emissions, and density depletions. The dominant power in the BB-ELF emissions is now identified to be from spatially broad, low frequency Alfv nic structures. For a cusp crossing, we determine using the Electric Field Investigation (EFI), that the electric and magnetic field fluctuations are Alfv nic and the electric field gradients satisfy the inequality for stochastic acceleration. With all the Polar 1996 horizontal crossings of the cusp, we determine the O+ heating rate using the Thermal Ion Dynamics Experiment (TIDE) and Plasma Wave Investigation (PWI). We then compare this heating rate to other heating rates assuming the electric field gradient criteria exceeds the limit for stochastic acceleration for the remaining crossings. The comparison suggests that a stochastic acceleration mechanism is operational and the heating is controlled by the transverse spatial scale of the Alfvenic waves.

Coffey, Victoria N.↗
DOE OSTI2020

Direct observations of shock-induced melting in a porous solid using time-resolved x-ray diffraction

Here in this work we provide direct evidence of shock-induced melting and associated kinetics in a porous solid (aluminum powder) using time-resolved x-ray diffraction. Unambiguous evidence of melting in 50% porous aluminum (Al) powder samples, shocked to peak pressures between ~13 - 19 GPa, was provided by the broadening of the Debye-Scherrer ring corresponding to the (111) peak. Shocked Al powder did not melt completely in any of our experiments within the durations of measurement. Incomplete (partial) melting of the powder, even after several hundreds of nanoseconds of shock loading, provides insights into thermal transport with Al powder particles under high-pressure dynamic loading. Such insights are quite valuable for developing well-constrained melting models and thermodynamic equations of state for porous Al and other porous solids relevant to planetary and materials science.

36 MATERIALS SCIENCE↗
DOE OSTI2021

The achievement of the T e,div feedback control by CD 4 seeding on EAST

A multi-function divertor feedback control system has been built on Experimental Advanced Superconducting Tokamak (EAST) to treat the divertor heat load issue. With the real-time data of the Langmuir probes and the impurity seeding, the divertor electron temperature (T e,div ) is well controlled to achieve the partial detachment phase. The first trial of the CD 4 seeding for the T e,div reduction has been achieved on EAST long-pulse discharge. In the seeding phase, the T e,div was maintained close to 5 eV, and the surface temperature of the target plate (T surface,div ) had a reduction of about 150° C. The plasma stored energy had a reduction in the control phase, so it is necessary to find a way to keep the good plasma confinement in the next step. The CD 4 injection also mitigated the low hybrid wave coupling rate in some degree. The big volume of the CD 4 injection lifted the Greenwald density fraction from ~0.4 to ~0.7, which made the SOL into high recycling state. As a result, most of the injected carbon particles were in the high ionized state, and with the lower of the T e,div , the tungsten line emission was suppressed obviously.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2023

Ultrasonic Inspection During Autoclave Cure of Reflowable-Interface Composite Joints

In structural bonds, the interface between adherend and adhesive is nearly two-dimensional making the interface susceptible to minute quantities of contamination, which can cause weak bonds. Regulatory organizations such as the Federal Aviation Administration (FAA) often require redundant load paths in secondary-bonded primary-structures due to uncertainty in bond performance. To address this issue, the NASA Convergent Aeronautics Solutions (CAS): Adhesive Free Bonding of Composites (AERoBOND) project is investigating reformulated aerospace epoxy-matrix resins to enable reflow and diffusion of the resin at the joint interface during a secondary bonding and cure process that can eliminate the material discontinuity at the interface. Implementing in-situ process monitoring enables assessing the bond quality during processing rather than waiting for post-fabrication mechanical testing to be completed. In this paper, an in-situ ultrasonic inspection system is used to monitor the joining of three composite laminates assembled using the AERoBOND technique. For each panel, the amplitude of the wave reflection at the joint was measured throughout the cure cycle. The results indicate the timing and extent of reflow and cure of the epoxy resin at the joint. Factors indicating end of cure were ascertained and a metric was developed to qualitatively predict acceptable mode-II fracture toughness based on the ultrasonic amplitude at the joint line during cure. In addition, since the inspection system scans an area of the joint, high-resolution localized results can be obtained across the joint.

Tyler B Hudson↗
DOE OSTI2020

Catalina Repower Feasibility Study: NREL Phases I & II Summary Report

Engineers at the National Renewable Energy Laboratory (NREL) supported Southern California Edison (SCE) and the United States Environmental Protection Agency (EPA) by conducting technical and economic analyses for energy systems at Santa Catalina (Catalina) Island, which is located 22 miles off the coast of Long Beach, California. This effort was part of a broader Repower Catalina Feasibility Study that was also supported by NV5, an engineering consulting firm and project partner to NREL for this analysis. This document describes NREL’s techno-economic modeling and optimization analysis for the first two phases of this project which focus on supply-side generation and energy storage options for Catalina. SCE’s goal for this analysis is to determine a strategy for electricity generation on Catalina Island that results in lower energy costs, improved energy resiliency, and reduced air emissions. EPA goals for this effort are to reduce emissions of air pollution and encourage renewable energy development on contaminated and formerly contaminated lands when such development is aligned with the community’s vision for the site. Currently, an on-island SCE power plant serves the Catalina Island electrical load with 6 reciprocating diesel generators totaling 9.4 MW; 23 propane-fueled microturbines totaling 1.5 MW; and a 1-MW, 7.2-MWh sodium sulfur battery energy storage system (BESS). In 2017, the electricity consumption on the island was 29.1 GWh, with an average load of 3.3 MW and peak load of approximately 5.5 MW. Considering new environmental standards on diesel generator emissions from California’s South Coast Air Quality Management District, a 60% renewable energy target for 2030 laid out in California’s Senate Bill 100, SCE’s Clean Power Electrification Pathway, and the characteristics of the island’s existing diesel generators, SCE is seeking to evaluate the technical and economic implications of different energy technology options to determine a path forward. Phases I and II of the Repower Catalina Feasibility Study, summarized in this document, evaluated the following: Interconnection with the mainland via an undersea cable; On-island fossil fuel generation, including diesel, propane, and/or liquified natural gas (LNG); On-island renewable energy (RE) technologies, including solar photovoltaics (PV), wind turbines, and wave energy devices; BESS to support the above generation technologies; Initial analysis of the potential impacts of implementing energy efficiency measures. Results indicate strong techno-economic potential for a mix of on-island diesel and/or propane generators, solar PV, BESS, and energy efficiency measures to help SCE and Catalina achieve their goals compliant with California’s emissions and clean energy standards while minimizing electricity life cycle costs (LCC) over the 30-year analysis period. This document summarizes the considerations and findings of Phases I and II, focusing on high-level takeaways from Phase I and more detailed results from Phase II, and discusses a potential path forward for Phase III.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2023

Additive Manufacturing for Powering the Blue Economy Applications: A Tidal Turbine Blade Case Study: Preprint

As the marine renewable energy industry continues to expand, innovation in the manufacturing space must grow accordingly to reduce costs and ensure the economic feasibility of new technologies. Additive manufacturing, more commonly known as 3D printing, provides an alternative for rapid prototyping of marine hydrokinetic technologies, particularly supporting Powering the Blue Economy initiatives of the U.S. Department of Energy Water Power Technologies Office. This study explores the application of additive manufacturing in the development of marine hydrokinetic structures, focusing on material and printing method selection, design, and analysis of a 3D-printed spar for an axial-flow tidal turbine blade. Corrosion-resistant metals were deemed ideal due to the loads and harsh marine environment the blade would experience. Laser metal deposition methods were determined to be the most effective and scalable for the considered scale. The designed spar adapts its geometry to the blade - a feature uniquely suited to additive manufacturing - and is intended to serve as the blade's primary structural component. A finite element model was used to study stresses and deformations under loading conditions. The spar was manufactured using 316L stainless steel through direct energy deposition, and defects were assessed and recorded. Future efforts will include mechanical testing of the spar. This research establishes a benchmark process for using additive manufacturing in developing marine hydrokinetic structures, paving the way for future optimization and techno-economic analysis.

additive manufacturing↗
NASA NTRS1977

Unsteady flow in a supersonic cascade with strong in-passage shocks

Linearized theory is used to study the unsteady flow in a supersonic cascade with in-passage shock waves. We use the Wiener-Hopf technique to obtain a closed-form analytical solution for the supersonic region. To obtain a solution for the rotational flow in the subsonic region we must solve an infinite set of linear algebraic equations. The analysis shows that it is possible to correlate quantitatively the oscillatory shock motion with the Kutta condition at the trailing edges of the blades. This feature allows us to account for the effect of shock motion on the stability of the cascade. Unlike the theory for a completely supersonic flow, the present study predicts the occurrence of supersonic bending flutter. It therefore provides a possible explanation for the bending flutter that has recently been detected in aircraft-engine compressors at higher blade loadings.

Goldstein, M. E.↗
NASA NTRS1988

Radial Alfven velocity profiles in the magnetosphere and their relation to ULF wave field-line resonances

The radial Alfven velocity, V(A), in the magnetosphere was investigated using magnetic field and plasma data from the ISEE-1 spacecraft. The harmonic frequency structure calculated from these data was compared with the observed resonant wave structure in the frequency range 5-100 mHz and over the region L = 3-9. It was found that the frequency of these resonances depend on the A(V), which is a function of the magnetic field strength and plasma density. The toroidal mode resonance frequency structure was modeled for the first four harmonics, with excellent agreement between the modeled and observed harmonics for a passs which does not show a plasmapause gradient. In the presence of a steep plasmapause gradient, the model was found to predict a correspondingly steep increase in the resonance frequencies at the plasmapause, by about 5 mHz for the first harmonic and by 20 mHz for the fourth harmonic. Special consideration is given to the effect of heavy ion mass loading on the modeling.

Fraser, B. J.↗
NASA NTRS1994

Crystalline structure and symmetry dependence of acoustic nonlinearity parameters

A quantitative measure of elastic wave nonlinearity in crystals is provided by the acoustic nonlinearity parameters. The nonlinearity parameters are defined for arbitrary propagation modes for solids of arbitrary crystalline symmetry and are determined along the pure mode propagation directions for 33 crystals of cubic symmetry from data reported in the literature. The magnitudes of the nonlinearity parameters are found to exhibit a strong dependence on the crystalline structure and symmetries associated with the modal direction in the solid. Calculations based on the Born-Mayer potential for crystals having a dominant repulsive contribution to the elastic constants from the interatomic pair potential suggest that the origin of the structure dependence is associated with the shape rather than the strength of the potential. Considerations based on variations in crystal symmetry during loading along pure mode propagation directions of face-centered-cubic solids provide a qualitative explanation for the dependence of the acoustic nonlinearity parameters on modal direction.

Cantrell, John H.↗