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FATES size- and age-dependent mortality simulation output analysis scripts

This dataset contains python scripts to analyse outputs from the Functionally Assembled Terrestrial Ecosystem Simulator (FATES) and accompanies the paper “Needham, J.F., Chambers, J., Fisher, R., Knox, R., and Koven, C. D., Forest responses to simulated elevated CO2 under alternate hypotheses of size- and age-dependent mortality, 2020, Global Change Biology”. These scripts process FATES outputs from size- and age-dependent mortality simulations which were run to test the effect of different mechanisms of mortality on forest response to elevated CO2 (eCO2). Specifically, these scripts will process single plant functional type (PFT) simulations in which mortality is either a constant background rate, size-dependent or age-dependent. In each case, a simulation with constant woody NPP is compared to a simulation in which woody NPP increases by 25% to simulate the growth response of forests to eCO2. In addition, the data package contains scripts to analyse ensemble simulations with size- and age-dependent mortality and two PFTs, that were run to test the impact of different demographic rates on coexistence and the forest response to eCO2. Finally, the dataset includes scripts for processing simulations testing the sensitivity of results to allometry and to the recruitment scheme.

54 ENVIRONMENTAL SCIENCES↗

FATES size- and age-dependent mortality simulation outputs, 2019

This dataset contains outputs from the Functionally Assembled Terrestrial Ecosystem Simulator (FATES) and accompanies the paper “Needham, J.F., Chambers, J., Fisher, R., Knox, R., and Koven, C. D., Forest responses to simulated elevated CO2 under alternate hypotheses of size- and age-dependent mortality, 2020, Global Change Biology”. Data are unprocessed FATES outputs from size- and age-dependent mortality simulations which were run to test the effect of different mechanisms of mortality on forest response to elevated CO2 (eCO2). Specifically, this data package contains single plant functional type (PFT) simulations in which mortality is either a constant background rate, size-dependent or age-dependent. In each case, a simulation with constant woody NPP is compared to a simulation in which woody NPP increases by 25% to simulate the growth response of forests to eCO2. Ensemble simulations with size- and age-dependent mortality and two PFTs test the impact of different demographic rates on coexistence and the forest response to eCO2. Finally, the dataset includes simulations testing the sensitivity of results to allometry and to the recruitment scheme.

54 ENVIRONMENTAL SCIENCES↗

Impact of Pressure-Dependent Interfacial Tension and Contact Angle on Capillary Heterogeneity Trapping of CO2 in Storage Aquifers

Summary Carbon dioxide (CO2) capillary trapping increases the total amount of CO2 that can be effectively immobilized in storage aquifers. This trapping, manifesting itself as accumulated CO2 columns at a continuum scale, is because of capillary threshold effects that occur below low-permeability barriers. Considering that capillary pressure is dictated by heterogeneous pore throat size, the trapped CO2 column height and associated CO2 saturation will vary spatially within a storage aquifer. This variation will be influenced by two pressure-dependent interfacial parameters—CO2/brine interfacial tension (IFT) and CO2/brine/rock contact angle. Our objective is to understand how the pressure dependence of these two parameters affects the heterogeneity of capillary trapped CO2 at a continuum scale. Our conceptual model is a 1D two-zone system with the upper zone being a flow barrier (low permeability) and the lower zone being a flow path (high permeability). The inputs to this model include microfacies-dependent capillary pressure vs. saturation curves and permeability values. The input capillary pressure curves were collected in the literature that represents carbonate microfacies (e.g., dolograinstone) in a prevalent formation in the Permian Basin. We then used the Leverett j-function to scale the capillary pressure curve for the two zones that are assigned with the same or different microfacies. During scaling, we considered the influence of pressure on both the IFT and contact angle of CO2/brine/dolomite systems. We varied the zone permeability contrast ratio from 2 to 50. We then assumed capillary gravity equilibriums and calculated the CO2 saturation buildup corresponding to various trapped CO2 column heights. The CO2 saturation buildup is defined as the CO2 saturation in the lower layer minus that in the upper one. We found that the saturation buildup can be doubled when varying pressure in a storage aquifer, after considering pressure-dependent IFT and contact angles. Thus, assuming these two parameters to be constant across such aquifers would cause large errors in the quantification of capillary trapping of CO2. The whole study demonstrates the importance of considering pressure-dependent interfacial properties in predicting the vertical distribution of capillary trapped CO2. It has important implications in developing a better understanding of leakage risks and consequent storage safety.

Engineering↗

Progress Report on Alloy 617 Time Dependent Allowables

Time dependent allowable stresses are required in the ASME Boiler and Pressure Vessel Code for design of components in the temperature range where time dependent deformation (i.e., creep) is expected to become significant. There are time dependent allowable stresses in Section IID of the Code for use in the non-nuclear construction codes, however, there are additional criteria that must be considered in developing time dependent allowables for nuclear components. These criteria are specified in Section III NH. St is defined as the lesser of three quantities: 100% of the average stress required to obtain a total (elastic, plastic, primary and secondary creep) strain of 1%; 67% of the minimum stress to cause rupture; and 80% of the minimum stress to cause the initiation of tertiary creep. The values are reported for a range of temperatures and for time increments up to 100,000 hours. These values are determined from uniaxial creep tests, which involve the elevated temperature application of a constant load which is relatively small, resulting in deformation over a long time period prior to rupture. The stress which is the minimum resulting from these criteria is the time dependent allowable stress St. In this report data from a large number of creep and creep-rupture tests on Alloy 617 are analyzed using the ASME Section III NH criteria. Data which are used in the analysis are from the ongoing DOE sponsored high temperature materials program, form Korea Atomic Energy Institute through the Generation IV VHTR Materials Program and historical data from previous HTR research and vendor data generated in developing the alloy. It is found that the tertiary creep criterion determines St at highest temperatures, while the stress to cause 1% total strain controls at low temperatures. The ASME Section III Working Group on Allowable Stress Criteria has recommended that the uncertainties associated with determining the onset of tertiary creep and the lack of significant cavitation associated with early tertiary creep strain suggest that the tertiary creep criteria is not appropriate for this material. If the tertiary creep criterion is dropped from consideration, the stress to rupture criteria determines St at all but the lowest temperatures.

36 MATERIALS SCIENCE↗

Polynomial chaos expansions for dependent random variables

Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate PCE coefficients. In contrast, the application of PCE to models of dependent variables is much more challenging. Three approaches can be used to construct PCE of models of dependent variables. The first approach uses mapping methods where measure transformations, such as the Nataf and Rosenblatt transformation, can be used to map dependent random variables to independent ones; however we show that this can significantly degrade performance since the Jacobian of the map must be approximated. A second strategy is the class of dominating support methods. In these approaches a PCE is built using independent random variables whose distributional support dominates the support of the true dependent joint density; we provide evidence that this approach appears to produce approximations with suboptimal accuracy. A third approach, the novel method proposed here, uses Gram-Schmidt orthogonalization (GSO) to numerically compute orthonormal polynomials for the dependent random variables. This approach has been used successfully when solving differential equations using the intrusive stochastic Galerkin method, and in this paper we use GSO to build PCE using a non-intrusive stochastic collocation method. The stochastic collocation method treats the model as a black box and builds approximations of the input-output map from a set of samples. Building PCE from samples can introduce ill-conditioning which does not plague stochastic Galerkin methods. To mitigate this ill-conditioning we generate weighted Leja sequences, which are nested sample sets, to build accurate polynomial interpolants. We show that our proposed approach, GSO with weighted Leja sequences, produces PCE which are orders of magnitude more accurate than PCE constructed using mapping or dominating support methods.

97 MATHEMATICS AND COMPUTING↗

Freeze/Thaw-Induced Embolism: Probability of Critical Bubble Formation Depends on Speed of Ice Formation

Bubble formation in the conduits of woody plants sets a challenge for uninterrupted water transportation from the soil up to the canopy. Freezing and thawing of stems has been shown to increase the number of air-filled (embolized) conduits, especially in trees with large conduit diameters. Despite numerous experimental studies, the mechanisms leading to bubble formation during freezing have not been addressed theoretically. We used classical nucleation theory and fluid mechanics to show which mechanisms are most likely to be responsible for bubble formation during freezing and what parameters determine the likelihood of the process. Our results confirm the common assumption that bubble formation during freezing is most likely due to gas segregation by ice. If xylem conduit walls are not permeable to the salts expelled by ice during the freezing process, osmotic pressures high enough for air seeding could be created. The build-up rate of segregated solutes in front of the ice-water interface depends equally on conduit diameter and freezing velocity. Therefore, bubble formation probability depends on these variables. The dependence of bubble formation probability on freezing velocity means that the experimental results obtained for cavitation threshold conduit diameters during freeze/thaw cycles depend on the experimental setup; namely sample size and cooling rate. The velocity dependence also suggests that to avoid bubble formation during freezing trees should have narrow conduits where freezing is likely to be fast (e.g., branches or outermost layer of the xylem). Avoidance of bubble formation during freezing could thus be one piece of the explanation why xylem conduit size of temperate and boreal zone trees varies quite systematically.

59 BASIC BIOLOGICAL SCIENCES↗

Phytochrome-dependent coordinate control of distinct aspects of nuclear and plastid gene expression during anterograde signaling and photomorphogenesis

Light perception by photoreceptors impacts plastid transcription, development, and differentiation. This photoreceptor-dependent activity suggests a mechanism for photoregulation of gene expression in the nucleus and plastid that serves to coordinate expression of critical genes of these two organelles. This coordinate expression is required for proper stoichiometric accumulation of components needed for assembly of plastids, photosynthetic light-harvesting complexes and components such as phytochromes. Chloroplast-targeted sigma factors, which function together with the plastid-encoded RNA polymerase to regulate expression of plastid-encoded genes, and nuclear-encoded plastid development factors, such as GLK1 and GLK2, are targets of phytochrome regulation. Such phytochrome-dependent functions are hypothesized to allow light-dependent regulation, and feasibly tuning, of plastid components and function in response to changes in the external environment, which directly affects photosynthesis and the potential for light-induced damage. When the size and protein composition of the light-harvesting complexes are not tuned to the external environment, imbalances in electron transport can impact the cellular redox state and cause cellular damage. We show that phytochromes specifically regulate the expression of multiple factors that function to modulate plastid transcription and, thus, provide a paradigm for coordinate expression of the nuclear and plastid genomes in response to changes in external light conditions. As phytochromes respond to changes in the prevalent wavelengths of light and light intensity, we propose that specific phytochrome-dependent molecular mechanisms are used during light-dependent signaling between the nucleus and chloroplast during photomorphogenesis to coordinate chloroplast development with plant developmental stage and the external environment.

59 BASIC BIOLOGICAL SCIENCES↗

The Apparent Involvement of ANMEs in Mineral Dependent Methane Oxidation, as an Analog for Possible Martian Methanotrophy

On Earth, marine anaerobic methane oxidation (AOM) can be driven by the microbial reduction of sulfate, iron, and manganese. Here, we have further characterized marine sediment incubations to determine if the mineral dependent methane oxidation involves similar microorganisms to those found for sulfate-dependent methane oxidation. Through FISH and FISH-SIMS analyses using 13 C and 15 N labeled substrates, we find that the most active cells during manganese dependent AOM are primarily mixed and mixed-cluster aggregates of archaea and bacteria. Overall, our control experiment using sulfate showed two active bacterial clusters, two active shell aggregates, one active mixed aggregate, and an active archaeal sarcina, the last of which appeared to take up methane in the absence of a closely-associated bacterial partner. A single example of a shell aggregate appeared to be active in the manganese incubation, along with three mixed aggregates and an archaeal sarcina. These results suggest that the microorganisms (e.g., ANME-2) found active in the manganese-dependent incubations are likely capable of sulfate-dependent AOM. Similar metabolic flexibility for Martian methanotrophs would mean that the same microbial groups could inhabit a diverse set of Martian mineralogical crustal environments. The recently discovered seasonal Martian plumes of methane outgassing could be coupled to the reduction of abundant surface sulfates and extensive metal oxides, providing a feasible metabolism for present and past Mars. In an optimistic scenario Martian methanotrophy consumes much of the periodic methane released supporting on the order of 10,000 microbial cells per cm 2 of Martian surface. Alternatively, most of the methane released each year could be oxidized through an abiotic process requiring biological methane oxidation to be more limited. If under this scenario, 1% of this methane flux were oxidized by biology in surface soils or in subsurface aquifers (prior to release), a total of about 10 20 microbial cells could be supported through methanotrophy with the cells concentrated in regions of methane release.

58 GEOSCIENCES↗

Propagation of Solar Energetic Particles in Three-dimensional Interplanetary Magnetic Fields: Radial Dependence of Peak Intensities

A functional form I{sub max}(R)=kR{sup −α}, where R is the radial distance of a spacecraft, was usually used to model the radial dependence of peak intensities I{sub max}(R) of solar energetic particles (SEPs). In this work, the five-dimensional Fokker–Planck transport equation incorporating perpendicular diffusion is numerically solved to investigate the radial dependence of SEP peak intensities. We consider two different scenarios for the distribution of a spacecraft fleet: (1) along the radial direction line and (2) along the Parker magnetic field line. We find that the index α in the above expression varies in a wide range, primarily depending on the properties (e.g., location and coverage) of SEP sources and on the longitudinal and latitudinal separations between the sources and the magnetic foot points of the observers. Particularly, whether the magnetic foot point of the observer is located inside or outside the SEP source is a crucial factor determining the values of index α. A two-phase phenomenon is found in the radial dependence of peak intensities. The “position” of the break point (transition point/critical point) is determined by the magnetic connection status of the observers. This finding suggests that a very careful examination of the magnetic connection between the SEP source and each spacecraft should be taken in the observational studies. We obtain a lower limit of R{sup −1.7±0.1} for empirically modeling the radial dependence of SEP peak intensities. Our findings in this work can be used to explain the majority of the previous multispacecraft survey results, and especially to reconcile the different or conflicting empirical values of the index α in the literature.

79 ASTRONOMY AND ASTROPHYSICS↗

Beta-dependent Properties of Solar Wind Current Sheets at 1 au

We utilize plasma turbulence in interplanetary coronal mass ejections as a natural laboratory to analyze more than 4600 current sheets observed over a broad range of electron and proton beta values, 10 −2 ≲ β e ≲ 10 and 10 −3 ≲ β p ≲ 10. The analysis shows that the shear angle Δθ and the thickness λ CS of the current sheets depend on both electron and proton beta. Specifically, the dependencies on electron beta are Δθ ≈ 12.6° $β^{0.41}_{e}$ and, λ CS /λ p ≈ 1.9 $β^{0.26}_{e}$ where λ p is the local proton inertial length. We argue that the beta dependence of the shear angle is an intrinsic feature of solar wind turbulence arising from the natural correlation between turbulence intensity and plasma beta. In contrast, the beta dependence of current sheet scales is likely a general characteristic of plasma turbulence. According to recent theory, current sheets formed in turbulence are disrupted by the electron tearing instability once their thickness is smaller than a critical scale. We demonstrate that normalizing the thickness of current sheets by this critical scale eliminates the dependence on beta across the entire range of plasma beta. This observation not only supports the theory suggesting that current sheets mediate the transition from the inertial to the kinetic cascade through the electron tearing instability, but also demonstrates the plausibility of similar processes in astrophysical plasma. The presented results also indicate that the theory may be applicable beyond its formal limits, m e /m i ≪ β ≪ 1.

79 ASTRONOMY AND ASTROPHYSICS↗

Future changes in isoprene-epoxydiol-derived secondary organic aerosol (IEPOX SOA) under the Shared Socioeconomic Pathways: the importance of physicochemical dependency

Abstract. Secondary organic aerosol (SOA) is a dominant contributor of fine particulate matter in the atmosphere, but the complexity of SOA formation chemistry hinders the accurate representation of SOA in models. Volatility-based SOA parameterizations have been adopted in many recent chemistry modeling studies and have shown a reasonable performance compared to observations. However, assumptions made in these empirical parameterizations can lead to substantial errors when applied to future climatic conditions as they do not include the mechanistic understanding of processes but are rather fitted to laboratory studies of SOA formation. This is particularly the case for SOA derived from isoprene epoxydiols (IEPOX SOA), for which we have a higher level of understanding of the fundamental processes than is currently parameterized in most models. We predict future SOA concentrations using an explicit mechanism and compare the predictions with the empirical parameterization based on the volatility basis set (VBS) approach. We then use the Community Earth System Model 2 (CESM2.1.0) with detailed isoprene chemistry and reactive uptake processes for the middle and end of the 21st century under four Shared Socioeconomic Pathways (SSPs): SSP1–2.6, SSP2–4.5, SSP3–7.0, and SSP5–8.5. With the explicit chemical mechanism, we find that IEPOX SOA is predicted to increase on average under all future SSP scenarios but with some variability in the results depending on regions and the scenario chosen. Isoprene emissions are the main driver of IEPOX SOA changes in the future climate, but the IEPOX SOA yield from isoprene emissions also changes by up to 50 % depending on the SSP scenario, in particular due to different sulfur emissions. We conduct sensitivity simulations with and without CO2 inhibition of isoprene emissions that is highly uncertain, which results in factor of 2 differences in the predicted IEPOX SOA global burden, especially for the high-CO2 scenarios (SSP3–7.0 and SSP5–8.5). Aerosol pH also plays a critical role in the IEPOX SOA formation rate, requiring accurate calculation of aerosol pH in chemistry models. On the other hand, isoprene SOA calculated with the VBS scheme predicts a nearly constant SOA yield from isoprene emissions across all SSP scenarios; as a result, it mostly follows isoprene emissions regardless of region and scenario. This is because the VBS scheme does not consider heterogeneous chemistry; in other words, there is no dependency on aerosol properties. The discrepancy between the explicit mechanism and VBS parameterization in this study is likely to occur for other SOA components as well, which may also have dependencies that cannot be captured by VBS parameterizations. This study highlights the need for more explicit chemistry or for parameterizations that capture the dependence on key physicochemical drivers when predicting SOA concentrations for climate studies.

54 ENVIRONMENTAL SCIENCES↗

Heliocentric distance dependence of the interplanetary magnetic field

Numerous spacecraft measurements bearing on the heliocentric distance dependencies of both large- and small-scale properties of the interplanetary magnetic field (IMF) are assembled and compared. These data tend to indicate that the average of the radial field component varies as the inverse square of distance. However, the azimuthal component is rather strongly a function of time, being influenced by both the time-dependent solar wind speed and the evolution of the source field at the sun. Thus, unless the solar wind speed dependence is taken into account, individual sets of measurements by a single spacecraft give an azimuthal component gradient which is steeper than the inverse distance dependence predicted from the Parker spiral model. A least squares fit to the composite (five spacecraft) solar rotation average data set gives a result close to the inverse distance dependence. Preliminary Helios results suggest general consistency with the spiral model.

Behannon, K. W.↗

Position-dependent effective masses in semiconductor theory

A Hamiltonian possessing a position-dependent effective mass is sometimes employed to describe the motion of free carriers (electrons and holes) in semiconductors of nonuniform chemical composition. In the present investigation it is shown that Hamiltonians representative of carriers possessing position-dependent masses are not Galilean invariant, and that approximation schemes which lead to position-dependent mass equations are not unique. The concept of a position-dependent mass should, therefore, be abandoned. von Roos (1982) has shown how to circumvent the difficulties associated with position-dependent masses.

Von Roos, O.↗

On the penetration of a hot diapir through a strongly temperature-dependent viscosity medium

The ascent of a hot spherical body through a fluid with a strongly temperature-dependent viscosity has been studied using an axisymmetric finite element method. Numerical solutions range over Peclet numbers of 0.1 - 1000 from constant viscosity up to viscosity variations of 100,000. Both rigid and stress-free boundary conditions were applied at the surface of the sphere. The dependence of drag on viscosity variation was shown to have no dependence on the stress boundary condition except for a Stokes flow scaling factor. A Nusselt number parameterization based on the stress-free constant viscosity functional dependence on the Peclet number scaled by a parameter depending on the viscosity structure fits both stress-free and rigid boundary condition data above viscosity variations of 100. The temperature scale height was determined as a function of sphere radius. For the simple physical model studied in this paper pre-heating is required to reduce the ambient viscosity of the country rock to less than 10 to the 22nd sq cm/s in order for a 10 km diapir to penetrate a distance of several radii.

Daly, S. F.↗

Effects of a temperature-dependent rheology on large scale continental extension

The effects of a temperature-dependent rheology on large-scale continental extension are investigated using a thin viscous sheet model. A vertically-averaged rheology is used that is consistent with laboratory experiments on power-law creep of olivine and that depends exponentially on temperature. Results of the calculations depend principally on two parameters: the Peclet number, which describes the relative rates of advection and diffusion of heat, and a dimensionless activation energy, which controls the temperature dependence of the rheology. At short times following the beginning of extension, deformation occurs with negligible change in temperature, so that only small changes in lithospheric strength occur due to attenuation of the lithosphere. However, after a certain critical time interval, thermal diffusion lowers temperatures in the lithosphere, strongly increasing lithospheric strength and slowing the rate of extension. This critical time depends principally on the Peclet number and is short compared with the thermal time constant of the lithosphere. The strength changes cause the locus of high extensional strain rates to shift with time from regions of high strain to regions of low strain. Results of the calculations are compared with observations from the Aegean, where maximum extensional strains are found in the south, near Crete, but maximum present-day strain rates are largest about 300 km further north.

Sonder, Leslie J.↗

Effects of a temperature-dependent rheology on large-scale continental extension

The effects of a temperature-dependent rheology on large-scale continental extension are investigated using a thin viscous sheet model. A vertically-averaged rheology is used that is consistent with laboratory experiments on power-law creep of olivine and that depends exponentially on temperature. Results of the calculations depend principally on two parameters: the Peclet number, which describes the relative rates of advection and diffusion of heat, and a dimensionless activation energy, which controls the temperature dependence of the rheology. At short times following the beginning of extension, deformation occurs with negligible change in temperature, so that only small changes in lithospheric strength occur due to attenuation of the lithosphere. However, after a certain critical time interval, thermal diffusion lowers temperatures in the lithosphere, strongly increasing lithospheric strength and slowing the rate of extension. This critical time depends principally on the Peclet number and is short compared with the thermal time constant of the lithosphere. The strength changes cause the locus of high extensional strain rates to shift with time from regions of high strain to regions of low strain. Results of the calculations are compared with observations from the Aegean, where maximum extensional strains are found in the south, near Crete, but maximum present-day strain rates are largest about 300 km further north.

Sonder, Leslie J.↗

Dependability analysis of parallel systems using a simulation-based approach

The analysis of dependability in large, complex, parallel systems executing real applications or workloads is examined in this thesis. To effectively demonstrate the wide range of dependability problems that can be analyzed through simulation, the analysis of three case studies is presented. For each case, the organization of the simulation model used is outlined, and the results from simulated fault injection experiments are explained, showing the usefulness of this method in dependability modeling of large parallel systems. The simulation models are constructed using DEPEND and C++. Where possible, methods to increase dependability are derived from the experimental results. Another interesting facet of all three cases is the presence of some kind of workload of application executing in the simulation while faults are injected. This provides a completely new dimension to this type of study, not possible to model accurately with analytical approaches.

Sawyer, Darren Charles↗

Modeling time-dependent corrosion fatigue crack propagation in 7000 series aluminum alloys

Stress corrosion cracking and corrosion fatigue experiments were conducted with the susceptible S-L orientation of AA7075-T651, immersed in acidified and inhibited NaCl solution, to provide a basis for incorporating environmental effects into fatigue crack propagation life prediction codes such as NASA FLAGRO. This environment enhances da/dN by five to ten-fold compared to fatigue in moist air. Time-based crack growth rates from quasi-static load experiments are an order of magnitude too small for accurate linear superposition prediction of da/dN for loading frequencies above 0.001 Hz. Alternate methods of establishing da/dt, based on rising-load or ripple-load-enhanced crack tip strain rate, do not increase da/dt and do not improve linear superposition. Corrosion fatigue is characterized by two regimes of frequency dependence; da/dN is proportional to f(exp -1) below 0.001 Hz and to F(exp 0) to F(exp -0.1) for higher frequencies. Da/dN increases mildly both with increasing hold-time at K(sub max) and with increasing rise-time for a range of loading waveforms. The mild time-dependence is due to cycle-time-dependent corrosion fatigue growth. This behavior is identical for S-L nd L-T crack orientations. The frequency response of environmental fatigue in several 7000 series alloys is variable and depends on undefined compositional or microstructural variables. Speculative explanations are based on the effect of Mg on occluded crack chemistry and embritting hydrogen uptake, or on variable hydrogen diffusion in the crack tip process zone. Cracking in the 7075/NaCl system is adequately described for life prediction by linear superposition for prolonged load-cycle periods, and by a time-dependent upper bound relationship between da/dN and delta K for moderate loading times.

Mason, Mark E.↗