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DOE OSTI2024

Measurement of the Static Nonlinear Third-Order Elastic Moduli of Rocks: Problems and Applicability

The third-order elastic (TOE) model has been used to describe the widely observed nonlinear mechanical behaviors of earth materials. In addition to linear elastic constants ( λ , μ ), three nonlinear elastic moduli ( A , B , C ) are required for isotropic rocks. Contrary to previous research on dynamic TOE moduli, this study followed the protocol to measure strain and stress under uniaxial and hydrostatic compressive tests statically, which were later used to invert for the full set of TOE moduli for four standard rock types with differing pore structures of a porous oolitic limestone, quartz-rich sandstones, and a dense crystalline basalt. The applicability of the TOE model to characterize nonlinearity depends on the fulfillment of path-independence and small-strain assumptions. Using the measured static TOE moduli, the finite element model demonstrates that the stress in the vicinity of the wellbore is more amplified than the stress in the linear elastic case, which leads to a wider zone of rock failure around the wellbore. Due to the long-standing discrepancy between static and dynamic moduli, the rarely reported full set of static TOE moduli is necessary and will benefit future research in understanding the effect of rock nonlinearity on geophysical and geomechanical applications, such as long-term safe storage of CO 2 and generating process of geohazards.

58 GEOSCIENCES↗

DOE OSTI2025

10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes

We have developed the Optimal Local Truncation Error Method (OLTEM) with 10-th order of accuracy on unfitted Cartesian meshes for a system of 3-D elasticity equations with smooth irregular interfaces. 5 x 5 x 5 = 125-point stencils (similar to those for quadratic finite elements) for elastic heterogeneous materials are used for OLTEM. There are no unknowns at the interface points between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal 10-th order of accuracy for OLTEM on unfitted Cartesian meshes, i.e., the increase by 7 orders in accuracy compared to quadratic finite elements on conformal meshes. A new post-processing procedure provides the 9-th order of accuracy for stresses in the 3-D case. Similar to basic computations it uses OLTEM with the 125-point stencils, the interface conditions and the elasticity equations. It was shown that the use of the elasticity equations for post-processing improves the accuracy of 0.1% stresses by 6 orders compared to post-processing without the use of PDEs. At an accuracy of for stresses, OLTEM with the new post-processing procedure reduces the number of degrees of freedom by 360 - 8000 times compared to quadratic finite elements with similar stencils. OLTEM with the 125-point stencils yields even more accurate results than high-order finite elements with much wider stencils. OLTEM provides accurate numerical results for compressible and nearly incompressible materials.

elasticity equations↗

DOE OSTI2026

Elastic-wave sensitivity-guided adaptive seismic survey design for cost-effective monitoring of geological carbon storage

Effective seismic monitoring is essential for verifying CO₂ containment, detecting potential leakage, and optimizing operational decisions in geologic carbon storage. Here, this study presents a time-adaptive, elastic-wave sensitivity-guided framework for designing cost-effective seismic monitoring layouts for tracking CO₂ plume migration. The method is based on elastic-wave sensitivity analysis, which quantifies how variations in subsurface properties impact seismic wavefields. Two complementary design strategies are developed: one based on selecting a fixed number of seismic sources (Method A), and the other based on selecting source–receiver pairs contributing to a fixed fraction of cumulative elastic-wave sensitivity energy (Method B). The optimization workflow to identify source–receiver configurations with the highest detection potential is demonstrated using a hypothetical GCS scenario at the Kimberlina site in California using simulations of elastic-wave sensitivity data at multiple post-injection timesteps. Results show that both strategies adapt to evolving plume geometries and wavefield sensitivities, with Method B offering broader spatial coverage and Method A ensuring simpler deployment. This framework enables site-specific, cost-effective, and risk-informed seismic survey designs, enhancing the ability to monitor CO₂ migration over time in evolving geological environments

58 GEOSCIENCES↗

DOE OSTI2024

High-temperature structure, elasticity, and thermal expansion of ε-ZrH 1.8

Zirconium hydride is a promising candidate material for nuclear microreactor applications as a solid-state moderator component, owing to its favorable neutronics properties and good thermal stability over other metal hydrides. Here, in the present work, the crystal structure, thermal expansion, and elastic properties of the hydrogen-rich ε phase hydride were measured at elevated temperatures in the range 300–900 K. Samples were prepared by direct hydriding Zircaloy-4 metal – a nuclear-grade zirconium alloy. Room-temperature lattice parameters agree well with those reported from literature for unalloyed zirconium hydride and fall within an observed quadratic H-content dependence. The coefficients of thermal expansion, determined from lattice expansion and dilatometry, agree well within our work but were about 30 % lower than those reported by others for unalloyed hydrides. Density functional theory-based molecular dynamics simulations were used to compare with thermal expansion and elasticity measurements. Results showed lattice parameter temperature dependence and slope of thermal expansion align with those from measurements. Based on diffraction scans at select temperatures, ε phase remained stable in air up to at least 770 K. Likewise, dilatometry showed smooth thermal expansion up to the thermal decomposition temperature around 950 K. The precise decomposition temperature was not determined via diffraction due to sparse scanning. The complete elastic property measurements were gathered for ε-phase Ziracloy-4 hydride for the first time. Young's modulus was lower compared to the metal and δ hydride phases. High-temperature elasticity measurements were limited to <350 K due to acoustic dissipation effects.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Electrostatically driven pattern formation in mixed charged–neutral multicomponent elastic membranes

Multicomponent crystalline and amorphous elastic shells exhibit heterogeneous surface patterns that provide distinctive functionalities in cellular environments. Such patterning typically arises from the competition between short-range attractive and long-range repulsive interactions in membranes. Here, we demonstrate that the intrinsic competition between electrostatic repulsion and elastic deformation is sufficient to drive spontaneous surface patterning in elastic shells, requiring no additional attractive interactions. Using numerical simulations, we demonstrate pattern formation in mechanically homogeneous membranes with heterogeneous surface charge composition across different topologies, including spheres, discs, and flat periodic membranes. We also examine patterns in crystalline and amorphous shells of coassembled charged and neutral components with different bending rigidities. At low charge fraction, discrete charged surface domains form. At intermediate charge fraction, the competition between electrostatics and elasticity leads to elongated domains (rods) of the charged component, which results in lamellar patterns at nearly equal fraction of the charged and neutral components. At high charge fraction, nanodomains of the neutral component form. Amorphous shells exhibit similar progressions but with disordered structures rather than ordered lamellar patterns. These pattern morphologies are observed in both the closed shells and flat membranes. As salt concentration increases, all patterns coarsen due to the screening of electrostatic interactions.

charge heterogeneity↗

DOE OSTI2026

Nonmonotonic-potential description of polarization effects, fusion, and nuclear rainbows in elastic scattering of 6 Li + 12 C at 4.5–600 MeV

The experimental differential cross-section (CS) and analyzing power (AP) data of the 6 Li + 12 C elastic scattering over a wide laboratory energy scale (4.5MeV≤𝐸 lab ≤600MeV) are analyzed within the framework of the optical model (OPM) using nonmonotonic (NM) nucleus–nucleus potentials. The real part of the NM potentials is derived from the Pauli-embodied energy density-functional (EDF) formalism with the sudden approximation. The real part of the noncentral spin-orbit and tensor terms, as well as the imaginary parts, are treated phenomenologically. The effect of the radius of sensitivity on the CS and AP data is found to be more important at lower energies. The diffractive and refractive scattering with Airy structures in the whole angular region of the elastic scattering across the studied energy range is successfully described within the OPM using the NM 6 Li + 12 C potential. The near- and far-side (N and F) decomposition of the total elastic-scattering amplitudes has also been studied using our NM potentials. The evolution of the Airy minima in the angular distributions, coupled with the fitting of the AP data, provides an accurate description of Airy minima of different orders. The OPM calculations with the NM potentials describe exceptionally well the CS, vector analyzing power (VAP), and tensor analyzing power data at 𝐸 lab =9.0,19.24,20,30, and 50MeV. In agreement with our past successful descriptions of CS and the opposite signs of the VAP data for the 6 Li and 7 Li elastic scattering using NM potentials in OPM, the present results appear to provide a better fit, so far, than those obtained from the coupled-channels method. The fusion cross sections of 6 Li + 12 C have been predicted in the energy range (4.5MeV≤𝐸 lab ≤20MeV), fitting the experimental data well in the range 𝐸 lab =2.97–11.77MeV. The EDF potential without any energy dependence and renormalization is also found to describe satisfactorily the experimental CS and AP data at energies up to several hundreds of MeV.

6 ≤ A ≤ 19↗

DOE OSTI2025

Anomalous elastic softening in ferroelectric hafnia under pressure

his study employs first-principles density-functional theory (DFT) calculations to explore the elastic and mechanical properties of ferroelectric hafnia (HfO 2 ) in its polar orthorhombic 𝑃⁢𝑐⁢𝑎⁢2 1 phase under varying hydrostatic pressure conditions up to 30 GPa. Utilizing a plane-wave basis set and Perdew-Burke-Ernzerhof generalized-gradient approximation for solids in our DFT calculations, we investigate both pure and yttrium-substituted HfO 2 . Our findings reveal an anomalous reduction in the 𝐶 33 component of the elastic tensor with increasing pressure, which becomes significant above 15 GPa and signals a potential pressure-driven structural phase transition at higher pressure. The analysis of atomic displacements under pressure sheds light on the unusual mechanical behavior and phase stability of this material. Additionally, we observe a transition from an indirect band gap to a direct band gap with increasing pressure, which could have significant implications for optical applications. Here, the effects of yttrium substitution on the mechanical and electronic properties are further examined, revealing that yttrium substitution softens the elastic response of this material and reduces the electronic band gap. These results enhance our understanding of elastic and mechanical responses of ferroelectric hafnia and its potential for applications in microelectronics, piezoelectric devices, and nonvolatile ferroelectric random-access memories. Further experimental validation is recommended to confirm our predictions and explore the practical implications of the observed phase transitions and electronic behavior of the ferroelectric hafnia under high-pressure conditions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRSDate not supplied

Predicted Performance Effects of Blade Elasticity on Testing of Rotors for Mars

The Rotor Optimization for the Advancement of Mars Exploration project (ROAMX) has designed and manufactured a set of rotor blades optimized for hover on Mars. These blades will soon be tested for hover performance in the NASA Ames Research Center Planetary Aeolian Laboratory, a large vacuum chamber capable of rotor testing at reduced pressures. To match the tip Mach numbers during this testing with the tip Mach numbers that would be experienced in the Martian atmosphere, the rotor will be spun faster in the vacuum chamber than it would be spun on Mars. While this accomplishes the goal of simulating the tip Mach numbers of Mars operations, the elevated RPM may have undesired consequences for performance, due to elastic deflections of the blades. The purpose of this paper is to use CAMRAD II, a rotorcraft comprehensive analysis code, to predict the effect on performance of blade elasticity at elevated RPM in the vacuum chamber environment. Results are presented for different thrust conditions and different rotor speeds, and comparisons are made between rigid and elastic blade predictions. Overall, the CAMRAD II analysis suggests that blade elasticity does influence rotor performance in the operating conditions of interest, although the effect is moderate enough to conclude that the PAL testing will be a good simulation of Mars operation.

Blade elasticity↗

NASA NTRS2023

Unconservatism of Linear-Elastic Fracture Mechanics (LEFM) Analysis Post Autofrettage

The scope of the NASA Engineering and Safety Center (NESC) Unconservatism of Linear-Elastic Fracture Mechanics (LEFM) Analysis Post Autofrettage assessment was to evaluate the influence of the elastic-plastic autofrettage cycle on elastic cycles post autofrettage and generate data to support damage tolerance life analysis verification approach. The decision to include an elastic-plastic autofrettage cycle is informed by other factors beyond damage tolerance, including manufacturing and fatigue crack initiation. The results of the NESC assessment are contained in this report.

Linear-Elastic Fracture Mechanics↗

DOE OSTI2024

Effects of irradiation damage on the elastic properties and hardness of complex transition metal diborides

Multi-principal component transition metal (TM) diborides represent a class of high-entropy ceramics (HECs) that have received considerable interest in recent years owing to their promising properties for extreme environment applications that include thermal/ environmental barriers, hypersonic vehicles, turbine engines, and next-generation nuclear reactors. While the addition of chemical disorder through the random distribution of TM elements on the cation sublattice has offered opportunities to tailor elastic stiffness and hardness, the effects of irradiationinduced structural damage on the physical properties of these complex materials have remained largely unexplored. To this end, changes in the hardness and elastic moduli of a high-entropy TM diboride (Hf0.2Nb0.2Ta0.2Ti0.2Zr0.2)B2 and three of its quaternary subsets following irradiation with 10 MeV gold (Au) ions to fluences of up to 6 × 10^15 Au cm-2 are investigated at the micrometer and sub-micrometer length-scales via the dispersion of laser-generated surface acoustic waves (SAW) and nanoindentation, respectively. The nanoindentation measurements show that the TM diborides exhibit an initial increase in hardness following irradiation with energetic Au ions, with a subsequent decrease in hardness following further irradiation. One quaternary composition, (Hf1/3Ta1/3Ti1/3)B2, exhibits a notable exception to the trend and continues to exhibit an increase in hardness with ion irradiation fluence. Although differences in the absolute values of the effective elastic moduli obtained from the measured SAW dispersion and nanoindentation are observed (and attributed to microstructural variations at the measurement length-scale), both techniques yield similar trends in the form of an initial reduction and subsequent saturation in the elastic modulus with increasing ion irradiation fluence. The quaternary TM diboride (Hf1/3Ta1/3Ti1/3)B2 again exhibits a departure from this trend. The high-entropy TM diboride (Hf0.2Nb0.2Ta0.2Ti0.2Zr0.2)B2 exhibits the greatest recovery in hardness and modulus when irradiated to high ion fluences following initial changes at low fluence, indicating superior resistance to radiation-induced damage over its quaternary counterparts. Opportunities for designing HECs with superior hardness and modulus for enhanced radiation resistance (compared to their single constituent counterparts) by tailoring chemical disorder and bond character in the lattice are discussed.

36 - MATERIALS SCIENCE↗

DOE OSTI2025

Elastic cross section is entanglement entropy

We present universal relations between entanglement entropy, which quantifies the quantum correlation between subsystems, and the cross section, which is the primary observable for high-energy particle scattering, by employing a careful formulation of wave packets for the incoming particles. For 2-to-2 elastic scattering with no initial entanglement and subdividing the system along particle labels, we show that both the Rényi and Tsallis entropies in the final states are directly proportional to the elastic cross section in units of the transverse size for the initial wave packets, which is then interpreted as the elastic scattering probability. The relations do not depend on the underlying dynamics of the quantum field theory and are valid to all orders in coupling strengths. Furthermore, computing quantum correlations between momentum and nonkinematic data leads to entanglement entropies expressed as various semi-inclusive elastic cross sections. Our result gives rise to a novel “area law” for entanglement entropy in a two-body system. Published by the American Physical Society 2025

Low, Ian (ORCID:0000000275709597)↗

NASA NTRS1971

On the anisotropic elastic properties of hydroxyapatite.

Experimental measurements of the isotropic elastic moduli on polycrystalline specimens of hydroxyapatite and fluorapatite are compared with elastic constants measured directly from single crystals of fluorapatite in order to derive a set of pseudo single crystal elastic constants for hydroxyapatite. The stiffness coefficients thus derived are given. The anisotropic and isotropic elastic properties are then computed and compared with similar properties derived from experimental observations of the anisotropic behavior of bone.

Katz, J. L.↗

NASA NTRS1972

The elastic energy and character of quakes in solid stars and planets

The quadrupolar mechanical energy of a rotating axially symmetric solid planet (with or without a liquid interior) is calculated using methods previously developed for neutron stars in which an elastic reference tensor is introduced to describe the build-up of elastic energy in the star. The basic parameters of the theory (the gravitational energy A and elastic energy B) depend upon the internal structure of the planet and may be calculated from specific planetary models. Explicit expressions are obtained for the Love numbers, and for the planetary wobble frequency. The theory provides a simple relationship between changes in shape or axis of figure of the planet and elastic energy release. The theory is extended to describe the Earth by taking into account isostasy, triaxiality and the observed lithospheric configuration.

Pines, D.↗

NASA NTRS1972

Lattice model calculation of elastic and thermodynamic properties at high pressure and temperature

The elastic constants and the entire frequency spectrum were calculated up to high pressure for the alkali halides in the NaCl lattice, based on an assumed functional form of the inter-atomic potential. The quasiharmonic approximation is used to calculate the vibrational contribution to the pressure and the elastic constants at arbitrary temperature. By explicitly accounting for the effect of thermal and zero point motion, the adjustable parameters in the potential are determined to a high degree of accuracy from the elastic constants and their pressure derivatives measured at zero pressure. The calculated Gruneisen parameter, the elastic constants and their pressure derivatives are in good agreement with experimental results up to about 600 K. The model predicts that for some alkali halides the Grunesen parameter may decrease monotonically with pressure, while for others it may increase with pressure, after an initial decrease.

Demarest, H. H., Jr.↗

NASA NTRS1974

Fore and aft elastic response characteristics of 34 x 9.9, type 7, 14 ply-rating aircraft tires of bias-ply, bias-belted, and radial-belted design

An investigation was conducted to determine the fore-and-aft elastic response characteristics of 34 x 9.9, type VII, 14 ply-rating aircraft tires of bias-ply, bias-belted, and radial-belted design. The investigation consisted of static and rolling tests on dry concrete pavements at the Langley aircraft landing loads and traction facility; statistical techniques which related the measured tire elastic characteristics to variations in the vertical load, inflation pressure, braking force and/or tire vertical deflection; and a semiempirical analysis which related the tire elastic behavior to measured wheel slippage during steady-state braking. The bias-belted tire developed the largest spring constant value for most loading conditions; the radial-belted tire, the smallest. The elastic response of the tire free periphery to static braking included both tread stretch and carcass torsional wind-up about the axle for the bias-ply and bias-belted tires and carcass wind-up alone for the radial-belted tire.

Tanner, J. A.↗

NASA NTRS1973

Elastic properties of granular materials under uniaxial compaction cycles

Data on andesitic and basaltic sands are presented showing compressional sound velocity, density, and creep as functions of uniaxial loading through several compaction cycles. Maximum pressures over which acoustic measurements were made were in the range from 600 to 700 bars. The dynamic elastic modulus varies with pressure in a manner analogous to that of a static elastic modulus defined by small pressure perturbations on a typical compaction cycle. After several compaction cycles, two compressional elastic moduli apparently exist at low pressure (thus two modes of compressional wave propagation through the samples are indicated). The elastic moduli observations are briefly discussed in terms of a general expression for compressibility.

Warren, N.↗

NASA NTRS1977

Strength and rigidity of the elastic lunar lithosphere and implications for present-day mantle convection in the moon

Lunar gravity data are used to estimate the thickness of the elastically defined lunar lithosphere, and the load-bearing capability of lithosphere with respect to lunar mascons is analyzed by representing the elastic lithosphere as a plane elastic plate subject to spatially concentrated loads from above and buoyant support from below. Gravity perturbations are computed and compared with observations, the spectrum of lunar gravity perturbations is derived, and effects of curvature are considered. The magnitude of the elastic stresses associated with a lithospheric flexure model maintaining the irregularities of the lunar gravity field is shown to be quite similar to that of the lithospheric stresses found on earth. It is concluded that associated differences in gravity, heat flow, and lithospheric thickness all combine to make the lunar lithosphere much more rigid and stronger than that of earth. Implications for present-day lunar convection are discussed.

Kuckes, A. F.↗

NASA NTRS1978

Martian lithospheric thickness from elastic flexure theory

The thickness of the elastic lithosphere in the Tharsis region of Mars is estimated from effects due to the surface load of Olympus Mons. Deformation (vertical displacement) and stress are calculated using elastic flexure theory for a range of possible lithospheric thicknesses (T), modeling the lithosphere as a thin elastic shell and the interior as a Newtonian fluid. For T below 150 km, displacement and stress rise rapidly with decreasing thickness. For T near 100 km, deformation of the region surrounding the volcano would be clearly visible in the topography, and resulting tensional stresses exceeding 5 kbar should produce observable fracturing at the surface. In contrast, for T near 200 km deformation is minimal and the tensional stress, being less than a kilobar, would not result in extensive fracturing. Since significant deformation and fracturing are not observed, it is concluded that the Martian elastic lithosphere is at least 150 km in thickness. Seismic, tectonic, and gravity observations all suggest a thick Martian lithosphere as well.

Thurber, C. H.↗