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All sources NASA NTRS DOE OSTI

DOE OSTI2022

The contribution of Humboldt Glacier, northern Greenland, to sea-level rise through 2100 constrained by recent observations of speedup and retreat

Abstract. Humboldt Glacier, northern Greenland, has retreated and accelerated through the 21st century, raising concerns that it could be a significant contributor to future sea-level rise. We use a data-constrained ensemble of three-dimensional higher-order ice sheet model simulations to estimate the likely range of sea-level rise from the continued retreat of Humboldt Glacier. We first solve for basal traction using observed ice thickness, bed topography, and ice surface velocity from the year 2007 in a PDE-constrained (partial differential equation) optimization. Next, we impose calving rates to match mean observed retreat rates from winter 2007–2008 to winter 2017–2018 in a transient calibration of the exponent in the power-law basal friction relationship. We find that power-law exponents in the range of 1/7–1/5 – rather than the commonly used 1/3–1 – are necessary to reproduce the observed speedup over this period. We then tune an iceberg calving parameterization based on the von Mises stress yield criterion in another transient-calibration step to approximate both observed ice velocities and terminus position in 2017–2018. Finally, we use the range of basal friction relationship exponents and calving parameter values to generate the ensemble of model simulations from 2007–2100 under three climate forcing scenarios from CMIP5 (two RCP8.5 forcings, Representative Concentration Pathway) and CMIP6 (one SSP5-8.5 forcing, Shared Socioeconomic Pathway). Our simulations predict 5.2–8.7 mm of sea-level rise from Humboldt Glacier, significantly higher than a previous estimate (∼ 3.5 mm) and equivalent to a substantial fraction of the 40–140 mm predicted by ISMIP6 from the whole Greenland Ice Sheet. Our larger future sea-level rise prediction results from the transient calibration of our basal friction law to match the observed speedup, which requires a semi-plastic bed rheology. In many simulations, our model predicts the growth of a sizable ice shelf in the middle of the 21st century. Thus, atmospheric warming could lead to more retreat than predicted here if increased surface melt promotes hydrofracture of the ice shelf. Our data-constrained simulations of Humboldt Glacier underscore the sensitivity of model predictions of Greenland outlet glacier response to warming to choices of basal shear stress and iceberg calving parameterizations. Further, transient calibration of these parameterizations, which has not typically been performed, is necessary to reproduce observed behavior. Current estimates of future sea-level rise from the Greenland Ice Sheet could, therefore, contain significant biases.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Heterosynaptic plasticity in biomembrane memristors controlled by pH

Abstract In biology, heterosynaptic plasticity maintains homeostasis in synaptic inputs during associative learning and memory, and initiates long-term changes in synaptic strengths that nonspecifically modulate different synapse types. In bioinspired neuromorphic circuits, heterosynaptic plasticity may be used to extend the functionality of two-terminal, biomimetic memristors. In this article, we explore how changes in the pH of droplet interface bilayer aqueous solutions modulate the memristive responses of a lipid bilayer membrane in the pH range 4.97–7.40. Surprisingly, we did not find conclusive evidence for pH-dependent shifts in the voltage thresholds ( V* ) needed for alamethicin ion channel formation in the membrane. However, we did observe a clear modulation in the dynamics of pore formation with pH in time-dependent, pulsed voltage experiments. Moreover, at the same voltage, lowering the pH resulted in higher steady-state currents because of increased numbers of conductive peptide ion channels in the membrane. This was due to increased partitioning of alamethicin monomers into the membrane at pH 4.97, which is below the pKa (~5.3–5.7) of carboxylate groups on the glutamate residues of the peptide, making the monomers more hydrophobic. Neutralization of the negative charges on these residues, under acidic conditions, increased the concentration of peptide monomers in the membrane, shifting the equilibrium concentrations of peptide aggregate assemblies in the membrane to favor greater numbers of larger, increasingly more conductive pores. It also increased the relaxation time constants for pore formation and decay, and enhanced short-term facilitation and depression of the switching characteristics of the device. Modulating these thresholds globally and independently of alamethicin concentration and applied voltage will enable the assembly of neuromorphic computational circuitry with enhanced functionality. Impact statement We describe how to use pH as a modulatory “interneuron” that changes the voltage-dependent memristance of alamethicin ion channels in lipid bilayers by changing the structure and dynamical properties of the bilayer. Having the ability to independently control the threshold levels for pore conduction from voltage or ion channel concentration enables additional levels of programmability in a neuromorphic system. In this article, we note that barriers to conduction from membrane-bound ion channels can be lowered by reducing solution pH, resulting in higher currents, and enhanced short-term learning behavior in the form of paired-pulse facilitation. Tuning threshold values with environmental variables, such as pH, provide additional training and learning algorithms that can be used to elicit complex functionality within spiking neural networks. Graphical abstract

36 MATERIALS SCIENCE↗

DOE OSTI2021

In situ high-energy X-ray study of deformation mechanisms in additively manufactured 316L stainless steel

A key structural material planned to be used in the core of the Transformational Challenge Reactor is the additively manufactured 316L stainless steel (AM 316L SS). While the deformation of conventionally-manufactured 316 SS have been extensively studied, the deformation mechanism of AM 316L SS is less known. Here the room-temperature tensile behavior of AM 316L SS was studied in situ using high-energy X-ray diffraction and X-ray tomography. Diffraction yielded the lattice strains and the dislocation densi-ties of the AM 316L SS, which were compared to those of a conventionally processed 316H SS. It was found that while the conventional 316H SS had three distinct deformation stages, namely stage II ather-mal hardening, stage III dynamic recovery, and stage IV, the AM 316L experienced only the stages III and IV after an initial transitional period. The plastic instability stresses (PIS), i.e. the true ultimate tensile strength, were nearly the same in both materials. The AM 316L had very low dislocation barrier strength throughout the plastic deformation, which, in combination with the invariant PIS, explained its high dis-location storage capability and the attractive combination of strength and ductility. Tomography revealed morphological changes of the built-in pores in the AM 316L during deformation and fracture, and it was concluded that large pores close to the surface might have significant roles in the necking stage.

316 stainless steel↗

DOE OSTI2025

Probing Operando Electrochemical Strain Generation in α-NaFeO 2 Composite Cathodes during Cycling of Na-Ion Batteries

The transition metal oxide (TMO) cathodes in Na-ion batteries suffer from low-capacity retention. Chemo-mechanical instabilities lead to the deterioration of the electrochemical performance of TMO cathodes in Li-ion batteries. However, there is not much known about the chemo-mechanical instabilities in the TMO cathodes for Na-ion batteries. Understanding the governing forces behind the interplay between the electrochemical performance and mechanical stability in TMO cathodes is critical for the development of Na-ion batteries. Here, we synchronize the digital image correlation (DIC) technique with electrochemical analysis to capture the real-time deformation behavior of the α-NaFeO 2 cathodes during cycling. When the charge cutoff voltage is 3.6 V, the cathode experiences reversible deformations (except for the first cycle). There is negative strain (shrinkage) generation during Na extraction and positive strain (expansion) generation during the subsequent Na insertion. A detailed analysis of the potential-dependent strain rate evolution points out complicated phase transformations and nonequilibrium conditions in the α-NaFeO 2 cathodes during cycling. When the charge cutoff voltage was increased to 4.2 V, there was a rapid capacity loss and large plastic deformations in the α-NaFeO 2 cathodes. We provide an in-depth discussion about the possible mechanisms behind the chemo-mechanical instabilities in the α-NaFeO 2 . In conclusion, the correlation is critical to develop material-based strategies to mitigate instability mechanisms in TMO cathodes for Na-ion batteries.

Wable, Minal [University of Maryland Baltimore Cou↗

NASA NTRS2012

High-Temperature Shape Memory Polymers

physical conformation changes when exposed to an external stimulus, such as a change in temperature. Such materials have a permanent shape, but can be reshaped above a critical temperature and fixed into a temporary shape when cooled under stress to below the critical temperature. When reheated above the critical temperature (Tc, also sometimes called the triggering or switching temperature), the materials revert to the permanent shape. The current innovation involves a chemically treated (sulfonated, carboxylated, phosphonated, or other polar function group), high-temperature, semicrystalline thermoplastic poly(ether ether ketone) (Tg .140 C, Tm = 340 C) mix containing organometallic complexes (Zn++, Li+, or other metal, ammonium, or phosphonium salts), or high-temperature ionic liquids (e.g. hexafluorosilicate salt with 1-propyl-3- methyl imidazolium, Tm = 210 C) to form a network where dipolar or ionic interactions between the polymer and the low-molecular-weight or inorganic compound forms a complex that provides a physical crosslink. Hereafter, these compounds will be referred to as "additives". The polymer is semicrystalline, and the high-melt-point crystals provide a temporary crosslink that acts as a permanent crosslink just so long as the melting temperature is not exceeded. In this example case, the melting point is .340 C, and the shape memory critical temperature is between 150 and 250 C. PEEK is an engineering thermoplastic with a high Young fs modulus, nominally 3.6 GPa. An important aspect of the invention is the control of the PEEK functionalization (in this example, the sulfonation degree), and the thermal properties (i.e. melting point) of the additive, which determines the switching temperature. Because the compound is thermoplastic, it can be formed into the "permanent" shape by conventional plastics processing operations. In addition, the compound may be covalently cross - linked after forming the permanent shape by S-PEEK by applying ionizing radiation ( radiation, neutrons), or by chemical crosslinking to form a covalent permanent network. With respect to other shape memory polymers, this invention is novel in that it describes the use of a thermoplastic composition that can be thermally molded or solution-cast into complex "permanent" shapes, and then reheated or redissolved and recast from solution to prepare another shape. It is also unique in that the shape memory behavior is provided by a non-polymer additive.

Yoonessi, Mitra↗

DOE OSTI2021

Pseudoelasticity of SrNi2P2 Micropillar via Double Lattice Collapse and Expansion

The maximum recoverable strain of most crystalline solids is less than 1% because plastic deformation or fracture usually occurs at a small strain. In this work, we show that a SrNi 2 P 2 micropillar exhibits pseudoelasticity with a large maximum recoverable strain of ~14% under uniaxial compression via unique reversible structural transformation, double lattice collapse–expansion that is repeatable under cyclic loading. Its high yield strength (~3.8 ± 0.5 GPa) and large maximum recoverable strain bring out the ultrahigh modulus of resilience (~146 ± 19 MJ/m 3 ), a few orders of magnitude higher than that of most engineering materials. The double lattice collapse–expansion mechanism shows stress–strain behaviors similar to that of conventional shape-memory alloys, such as hysteresis and thermo-mechanical actuation, even though the structural changes involved are completely different. Our work suggests that the discovery of a new class of high-performance ThCr 2 Si 2 -structured materials will open new research opportunities in the field of pseudoelasticity.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2025

Influence of nanoscale interfaces on the dynamic deformation and spall failure of Cu–Fe alloy microstructures

Additively manufactured immiscible Cu–Fe alloys can exhibit a distribution of nanoscale interfaces due to the distribution of nanoscale clusters in the equilibrium and metastable phases. Molecular dynamics simulations investigate the role of such interfaces on the phase stability and transformation behavior during shock compression, as well as the mechanisms of damage nucleation during spall failure. The model multiphase Cu–Fe systems studied here comprise a distribution of Fe clusters in an FCC Cu matrix, as well as Cu clusters in a BCC Fe matrix. The length scales of the nanoscale clusters determine the energetics of the interfaces that can result in FCC/BCC (equilibrium) or FCC/FCC (metastable) interfaces in the Cu system matrix, and BCC/FCC (equilibrium) or BCC/BCC (metastable) interfaces in the Fe matrix. The MD simulations demonstrate that nanoscale metastable interface microstructures can induce plastic deformation and also stabilize phases or suppress phase transformations in metastable phase clusters. In contrast, equilibrium interfaces can influence phase transformation thresholds and serve as additional void nucleation sites during failure.

36 MATERIALS SCIENCE↗

NASA NTRS1985

Effects of High Mean Stress on High-cycle Fatigue Behavior of PWA 1480

PWA 1480 is a potential candidate material for use in the high-pressure fuel turbine blade of the space shuttle main engine. As an engine material it will be subjected to high-cycle fatigue loading superimposed on a high mean stress due to combined centrifugal and thermal loadings. The present paper describes the results obtained in an ongoing program at the Argonne National Laboratory, sponsored by NASA Lewis, to determine the effects of a high mean stress on the high-cycle fatigue behavior of this material. Straight-gauge high-cycle fatigue specimens, 0.2 inch in diameter and with the specimen axis in the 001 direction, were supplied by NASA Lewis. The nominal room temperature yield and ultimate strength of the material were 146 and 154 ksi, respectively. Each specimen was polished with 1-micron diamond paste prior to testing. However, the surface of each specimen contained many pores, some of which were as large as 50 micron. In the initial tests, specimens were subjected to axial-strain-controlled cycles. However, very little cyclic plasticity was observed.

Majumdar, S.↗

NASA NTRS1993

Nonlinear probabilistic finite element models of laminated composite shells

A probabilistic finite element analysis procedure for laminated composite shells has been developed. A total Lagrangian finite element formulation, employing a degenerated 3-D laminated composite shell with the full Green-Lagrange strains and first-order shear deformable kinematics, forms the modeling foundation. The first-order second-moment technique for probabilistic finite element analysis of random fields is employed and results are presented in the form of mean and variance of the structural response. The effects of material nonlinearity are included through the use of a rate-independent anisotropic plasticity formulation with the macroscopic point of view. Both ply-level and micromechanics-level random variables can be selected, the latter by means of the Aboudi micromechanics model. A number of sample problems are solved to verify the accuracy of the procedures developed and to quantify the variability of certain material type/structure combinations. Experimental data is compared in many cases, and the Monte Carlo simulation method is used to check the probabilistic results. In general, the procedure is quite effective in modeling the mean and variance response of the linear and nonlinear behavior of laminated composite shells.

Engelstad, S. P.↗

NASA NTRS1997

Fatigue-Life Prediction Methodology Using Small-Crack Theory

This paper reviews the capabilities of a plasticity-induced crack-closure model to predict fatigue lives of metallic materials using 'small-crack theory' for various materials and loading conditions. Crack-tip constraint factors, to account for three-dimensional state-of-stress effects, were selected to correlate large-crack growth rate data as a function of the effective-stress-intensity factor range (delta K(eff)) under constant-amplitude loading. Some modifications to the delta k(eff)-rate relations were needed in the near-threshold regime to fit measured small-crack growth rate behavior and fatigue endurance limits. The model was then used to calculate small- and large-crack growth rates, and to predict total fatigue lives, for notched and un-notched specimens made of two aluminum alloys and a steel under constant-amplitude and spectrum loading. Fatigue lives were calculated using the crack-growth relations and microstructural features like those that initiated cracks for the aluminum alloys and steel for edge-notched specimens. An equivalent-initial-flaw-size concept was used to calculate fatigue lives in other cases. Results from the tests and analyses agreed well.

Newmann, James C., Jr.↗

DOE OSTI2020

Scale bridging damage model for quasi-brittle metals informed with crack evolution statistics

Computationally efficient methods for bridging length scales, from highly resolved micro/meso-scale models that can explicitly model crack growth, to macro-scale continuum models that are more suitable for modeling large parts, have been of interest to researchers for decades. In this work, an improved brittle damage model is presented for the simulation of dynamic fracture in continuum scale quasi-brittle metal components. Crack evolution statistics, including the number, length, and orientation of individual cracks, are extracted from high-fidelity, finite discrete element method (FDEM) simulations and used to generate effective material moduli that reflect the material’s damaged state over time. This strategy allows for the retention of small-scale physical behaviors such as crack growth and coalescence in continuum scale hydrodynamic simulations. However, the high-fidelity simulations required to generate the crack statistics are computationally expensive. Thus, steps were taken to produce a flexible constitutive model to reduce the number of costly high-fidelity simulations needed to produce accurate results. A new stress based degradation criterion is introduced for the degradation of individual material zones. This allows for the development of a heterogeneous damage distribution within the bulk material. Then a flow stress model is added to the hydrodynamic simulation to account for plasticity in quasi-brittle materials. As a result, the effective moduli model can be applied to a larger range of materials. The effective moduli constitutive model is used to simulate beryllium flyer plate experiments. The results from the continuum scale simulations using statistics from a single high-fidelity simulation are found to be in excellent agreement with numerical and experimental velocity interferometer data. The same set of crack statistics are used to extrapolate the results of a higher rate flyer plate case using the effective moduli model. In conclusion, the extension of this model to higher rate cases shows promise for further reducing the number of costly high-fidelity simulations needed to generate crack statistics.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Hybrid Particle Method for Computational Shock Physics

A long-standing area of research for Eulerian shock wave physics codes has been the treatment of strength and damage for materials. Here we present a method that will aid in the analysis of strength and failure in shock physics applications where excessive diffusion of critical variables can occur and control the solution outcome. Eulerian methods excel for large deformation simulations in general but are inaccurate in capturing structural behavior. Lagrangian methods provide better structural response, but finite element meshes can become tangled. Therefore, a technique for merging Lagrangian and Eulerian treatments of material response, within a single numerical framework, was implemented in the Multiple Component computational shock physics hydrocode. The capability is a Lagrangian/Eulerian Particle Method (LEPM) that uses particles to interface a Lagrangian treatment of material strength with a more traditional Eulerian treatment of the Equation of State (EOS). Lagrangian numerical methods avoid the advection diffusion found in Eulerian methods, which typically strongly affects strength constitutive law internal variables, such as equivalent plastic strain, porosity and/or damage. The Lagrangian capability enhances existing capabilities and permits accurate predictions of high rate, large deformation and/or shock of mechanical structures.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Microstructural Evolution of Tantalum During Deformation and Subsequent Annealing

Microstructure-aware models are necessary to predict the behavior of material based on process knowledge or to extrapolate mechanical properties of materials to environmental conditions which are not easily reproduced in the laboratory, e.g., nuclear reactor environments. Elemental Ta provides a relatively simple BCC system in which to develop a microstructural understanding of deformation processes which can then be applied to more complicated BCC alloys. In situ neutron diffraction during compressive deformation and subsequent heat treatment have been used to monitor the evolution of microstructural features in Ta throughout simulated processing steps. Crystallographic texture and dislocation density are determined as a function of first plastic strain, then temperature. Lattice strains are determined and attributed to stresses at macroscopic, grain and dislocation length scales. The increase of the dislocation density through deformation and subsequent recovery during heat treatment is monitored through the changing diffraction line profile. Also, randomization of the texture is used as a signature of recrystallization. The recovery of dislocations through annihilation is not observed to depend on the initial dislocation density in the range studied here. In contrast, recrystallization is observed to depend strongly on the initially dislocation density.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Creep deformation and cavitation in an additively manufactured Al-8.6Cu-0.4Mn-0.9Zr (wt%) alloy

Creep deformation and cavitation were investigated at 300 ºC in both tension and compression for an additively manufactured Al-8.6Cu-0.5Mn-0.9Zr (wt%) alloy in the as-fabricated state and after various aging treatments (aging at 300 °C/200 h or 350 °C/24 h and overaging at 400 °C/200 h). Creep mechanisms at 300 °C were determined by relating the measured creep response to corresponding microstructural and X-ray computed tomography observations. In compression, alloys in the as-fabricated and two aging conditions exhibited similarly high creep resistance. Overaging (400 °C/200 h) led to substantial coarsening of intragranular θ-Al2Cu precipitates and an expected drop in their Orowan strengthening contribution. In tension, minimum strain rates comparable to those in compression were obtained at any given stress; however, upon accumulation of some plastic strain in the matrix, creep cavities started to form, leading to accelerated tertiary stage creep deformation and rupture. Cavitation occurred exclusively along melt pool boundaries due to locally enhanced diffusion enabled by (i) large grain-boundary area in adjacent fine-grained zones and (ii) localization of creep strain in nearby heat-affected zones. Although cavity growth was initially diffusion-controlled, its rate was determined by matrix creep rate, consistent with constrained cavity growth mechanisms. This study reveals how microstructural complexities induced by the additive manufacturing process affect the creep and cavitation behavior of Al-Cu-Mn-Zr alloys. The underlying creep and cavitation mechanisms uncovered in this study point to pathways that improve the high-temperature properties of additively manufactured alloys.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Reformulating a Gurson-based dynamic damage model and demonstrating improved predictive power and numerical robustness

The original Tepla (TEnsile PLAsticity) ductile damage model, based on the Gurson yield surface, has long been used to model damage evolution and material failure under dynamic loading. Unfortunately, Tepla suffered from mesh sensitivity, numerical instability, and limited predictive capability. Here, we theoretically reformulate Tepla to address these issues. We especially focus on the prediction of porosity, which is the key state variable used for modeling ductile damage, as compared to more easily measured surface velocities, which are at best an indirect measure of damage. Key model changes include separating the viscosity during volumetric void growth from underlying shear strength behavior and switching to an iterative bisection solver. The new Tepla is then calibrated on incipient spall experiments on half-hard copper and tantalum, which demonstrate its ability to simultaneously fit the model to recovered porosity distributions and measured surface velocities, a stringent test. Improved numerical behavior, such as greatly reduced mesh sensitivity, is also shown in those simulations. Finally, the new Tepla model is applied to several high-explosive loaded, sweeping wave experiments, showing the ability of the model to predict behavior on tests with significantly different loading conditions and histories than the calibration data.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Transport of Compressed Woody Biomass: Correlating Rheology and Microcompounder Measurements

Transport of milled solid biomass can be problematic in biomass conversion processes due to the material’s high cohesiveness, low bulk density, and poor flowability. Here, the goal is to investigate the flow behavior of compressed biomass under conditions similar to compression-screw feeders. To mimic the flow of woody biomass inside a compression-screw feeder, two lignocellulosic biomass feedstocks were fed to a co-rotating twin-screw microcompounder. Force and screw speed from the microcompounder were correlated to viscosity in a traditional rheometer, where low-density polyethylene served as a control. The deformation of compressed biomass in a lab-scale compounder is strongly dependent on screw speed and moisture content. Shear stress in woody biomass decreased with the shear rate, indicating negative plastic viscosity. Moreover, viscosity decreased one order of magnitude as screw speed increased from 10 to 90 rpm. Increasing moisture from 10 to 50% reduced the viscosity of the compressed biomass by 60%.

09 BIOMASS FUELS↗

DOE OSTI2025

Computational alchemy clarifies origins of alloy strengthening

Solid solution strengthening (SSS) is widely used to enhance mechanical properties of metals. Originally developed for dilute alloys, classical SSS theories are presently challenged by the rise of complex concentrated alloys (CCA) with nearly equiatomic compositions. Here, we propose and develop a method of “computational alchemy” in which interatomic interactions are modified to systematically vary two key physical parameters defining SSS - atomic size misfit and elastic stiffness misfit - over a maximally wide range of two misfits. The resulting alchemical alloys are subjected to massive (~10 8 atoms) molecular dynamics (MD) simulations reproducing full complexity of plastic strength response. At variance with prevailing views, stiffness misfit is observed to contribute to SSS on par if not more than size misfit. Furthermore, depending on exactly how two misfits are combined, they result in synergistic (amplification) or antagonistic (compensation) effect on alloy strengthening. Unlike real CCAs in which each component element comes with its own specific size and stiffness, our alchemical model alloys span the space of two misfits continuously revealing trends in alloy strengthening unrecognized so far. Our study demonstrates unique value of intentionally unrealistic models for gaining deep physical insights into material behaviors that are difficult to reveal otherwise.

36 MATERIALS SCIENCE↗

NASA NTRS1972

Slow and fast motion of cracks in inelastic solids. Part 1: Slow growth of cracks in a rate sensitive tresca solid. Part 2: Dynamic crack represented by the Dugdale model

An extension is proposed of the classical theory of fracture to viscoelastic and elastic-plastic materials in which the plasticity effects are confined to a narrow band encompassing the crack front. It is suggested that the Griffith-Irwin criterion of fracture, which requires that the energy release rate computed for a given boundary value problem equals the critical threshold, ought to be replaced by a differential equation governing the slow growth of a crack prior to the onset of rapid propagation. A new term which enters the equation of motion in the dissipative media is proportional to the energy lost within the end sections of the crack, and thus reflects the extent of inelastic behavior of a solid. A concept of apparent surface energy is introduced to account for the geometry dependent and the rate dependent phenomena which influence toughness of an inelastic solid. Three hypotheses regarding the condition for fracture in the subcritical range of load are compared. These are: (1) constant fracture energy (Cherepanov), (2) constant opening displacement at instability (Morozov) and (3) final stretch criterion (Wnuk).

Wnuk, M. P.↗