Power-law Creep Research Articles

Nonlinear power-law creep of cell cortex: A minimal model

Experiments have revealed that biological cells exhibit a universal power-law rheology, but the underlying mechanisms remain elusive. Here, we present a minimal model to explain the power-law creep of cell cortex, which is abstracted as chains of crosslinkers with random binding energies. Using this model, we show that when both the load and chain length are small, the logarithm of both the strain and time scales with the fraction of unbound crosslinkers, leading to power-law creep with a constant exponent, as observed in many experiments. Increasing the load alters the latter relationship between time and unbinding fraction, and thus, increases the power-law exponent, explaining the stress-induced nonlinearity in some experiments. Increasing the chain length alters this relationship as well, and as a result, the exponent grows proportionally with the chain length, explaining the crosslinker-density-induced nonlinearity in other experiments. This work provides a mesoscopic explanation for the linear and nonlinear power-law creep of cell cortex and may serve as a basis for understanding the cytoskeletal mechanics.

Study on the investigation of creep behaviour in squeeze cast Ca-modified AZ91 magnesium alloy

ABSTRACT The AZ91 (Mg-Al) alloy is the most commonly used Mg-cast alloy with high specific strength and excellent die-castability; however, its poor creep performance limits its applications at elevated temperatures. In the present investigation, the effect of Ca addition on the microstructure development and creep performance of squeeze-cast AZ91 alloy recognised as AZX911 alloy has been investigated at 150°C–250°C temperature under 65 and 80 MPa stress. The addition of Ca in the AZ91 alloy results in the precipitation of the thermally stable Al2Ca intermetallic phase, which partially suppresses the precipitation of the thermally unstable β-Mg17Al12 phase and significantly improves the creep resistance of the AZ91 Mg alloy. The governing creep mechanism of AZX911 alloy is found to follow power law creep with n = 3.49 at 150°C that mainly results from dislocation climb. In contrast, a transition to power-law breakdown is observed at 175°C. The resistance to creep deformation of AZX911 further improves with heat treatment at 450°C as compared to the as-cast specimen of the same alloy due to the complete dissolution of the β-Mg17Al12 phase. The activation energy of phase dissolution, estimated using the Kissinger method, is found to be higher (923 kJ/mol) for the Al2Ca intermetallic phase as compared to that for β-Mg17Al12 phase (664 kJ/mol) in AZX911 alloy.

A novel Ta-contained TiAl alloy with excellent high temperature performance designed for powder hot isostatic pressing

TiAl alloys offer strong potential for replacing conventional nickel-base alloys in structural applications at high temperatures, which requires good high temperature performance. This study analyzes the creep performance and thermal exposure characteristics for a powder hot isostatic pressing (P-HIP) Ta-contained TiAl alloy. The microstructure characteristics, creep performance, and thermal exposure properties of the nearly lamellar (NL) alloy with a size of about 145 μm were investigated. The alloy remains 0.72 % fracture strain at room temperature after exposure at 750 ℃ for 1000 h. The fitting results of creep curves show that the creep stress index n is 15.82 at 750 ℃, indicating the power law creep mechanism. By reducing the interfacial energy, Ta can facilitate the formation of metastable structures and achieve refinement. The enrichment of Ta element at the lamellar edge, both observed after thermal exposure and creep, inhibits the growth of lamellae and hinders the expansion of cavities due to its low diffusion rate, improving the thermal stabilities and creep performance. The P-HIP Ta-contained TiAl alloy shows exciting high temperature performance.

Phase-field finite element modelling of creep crack growth in martensitic steels

Creep fracture presents a major concern for structural materials and critical components operating at elevated temperatures, thus requiring effective computational models. This study presents a phase-field framework for modelling creep crack growth and fracture behaviour of modern high-temperature materials such as Creep Strength Enhanced Ferritic (CSEF) martensitic steels. The model was formulated using the thermodynamic principles of the variational phase field theory of fracture and considering some physical aspects of creep fracture. Within the modelling framework, a dissipation potential dependent on creep damage is introduced to capture the effect of creep cavitation on the fracture energy and the creep crack growth resistance of the solid in a phenomenological manner. An elastoplastic power-law creep model is coupled to the phase-field formulations to account for the non-linear deformation processes ahead of the crack tip due to inelastic deformations, as well as their contribution to fracture at high temperatures. The capability of the proposed model is assessed against experimental data from compact tension (CT) creep tests conducted on P91 and P92 steels. Good agreement was obtained between the FE-predicted creep crack growth behaviour and the experimental measurements, showcasing the model’s feasibility. Further, numerical experiments were conducted using the proposed model to elucidate some key aspects influencing the fracture behaviour of martensitic steels. The proposed computational framework not only demonstrated good capability but was also able to offer improved mechanistic insights into the influence of material tendency to develop creep cavities on crack growth behaviour. This work contributes valuable insights into understanding the fracture process of CSEF steels at elevated temperatures and further demystifies the role of creep ductility.

Crystallographic orientation dependence in creep deformation of micron-thick silicon films for 3-D microstructures

In this study, the steady-state creep deformation properties of 5 μm-thick single crystal silicon (Si) films at elevated temperatures were investigated using punch creep-forming tests for the design of three-dimensional (3-D) microstructured MEMS. The relationship between the creep strain rate and stress of the Si films with {100}, {110}, and {111} planes at temperatures of 1223–1323 K was derived using finite element analysis following punch creep-forming tests, and a power-law creep constitutive equation for Si films was determined. The steady-state creep properties of the Si film samples varied clearly with crystallographic orientations. Among the three crystallographic orientations, the creep strain rate of the micron-thick Si films was the fastest for the {011} plane, followed by the {111} and {001} planes, in the applied stress range of 110 MPa to 220 MPa. This is consistent with the increasing order of magnitude of the thermal activation energy. The crystallographic orientation dependence of the steady-state creep properties of Si films is discussed based on the mobility of dislocation gliding per unit lattice and scanning electron microscope observations. The sample-size dependence of the creep strain rate of Si with the {001} plane was clearly observed between the micron- and millimeter-thick samples, which was attributed to the difference in the frequency factor of the power-law creep. However, the thermal-activation energy remained constant. This study successfully revealed the steady-state creep properties of Si films and provided useful engineering data for the design of 3-D Si-MEMS fabrication by the plastic processing of Si films.

Comparison study of creep constitutive laws in compaction of porous stainless steel 316 L

Modeling Powder Metallurgy Hot Isostatic Pressing (PM-HIP) is of paramount importance due to its critical role in modern manufacturing. Accurate modeling of PM-HIP and understanding the associated deformation behavior are essential for optimizing the process and advancing the design and production of high-performance materials in critical applications. To accurately simulate the deformation behavior during PM-HIP, this paper presents a comprehensive investigation of modeling PM-HIP processes using various creep models incorporating rate-dependent plasticity embedded into Finite Element Analysis (FEA). This study experimentally and computationally compares four distinct creep models: Kuhn McMeeking, power law creep, unified creep law, the rate-dependent creep that was modified by Abouaf et al. and with modification of Van Nguyen et al. for density dependency. The results showed the modified Abouaf's Model, uniquely designed to account for both primary and secondary creep, consistently exhibits the closest alignment with experimental outcomes compared to the other three creep models tested.

Research on the bearing creep characteristics and constitutive model of gangue filling body

The creep characteristics and potential deformation patterns of gangue backfill material are crucial in backfill mining operations. This study utilizes crushed gangue from the Gangue Yard in Fuxin City as the research material. An in-house designed, large-scale, triaxial gangue compaction test system was used. Triaxial compaction creep tests were conducted on gangue materials with varying particle size distributions. Analysis was performed based on different particle sizes, stresses, and confinement pressures. The study investigates the creep characteristics of the gangue under different conditions and explores the underlying causes. It reveals the relationship between the creep deformation of gangue materials and the passage of time. Mathematical methods are applied to develop a triaxial compaction creep power law model for gangue backfill materials. Finally, the creep results are fitted using an empirical formula approach.

The spatial and temporal characterization of type IV creep fracture in notched specimen of modified 9Cr–1Mo steel

In order to observe type IV fracture occurring in the HAZ fine-grained zone of a modified 9Cr–1Mo steel at high resolution, the creep damage process of a single notched specimen was continuously observed in three dimensions. The initiation, growth and coalescence behaviour of individual creep voids were clearly observed. This was combined with an analysis of the local stress state using the finite element method. Furthermore, all creep voids that could be observed in the final loading stage were traced backward in time to determine the loading stages at which the creep voids were initiated together with the growth behaviour of the individual creep voids. Significant creep void formation was observed a little further in from the bottom of the notch, where the maximum principal and hydrostatic stresses were greatest, while the stress triaxiality was greatest a little closer to the centre of the specimen than the maximum principal and hydrostatic stresses, where significant creep void growth was observed. It was found that the creep voids nucleated at an early loading stage dominated the creep failure of the specimen. The growth behaviour could be well approximated by the model that combined the diffusion law and power creep law with the constraint effect from neighbouring grains. Engineering parameters that describe the progress of type IV damage were also investigated.

The influence of viscous slab rheology on numerical models of subduction

Abstract. Numerical models of subduction commonly use diffusion and dislocation creep laws from laboratory deformation experiments to determine the rheology of the lithosphere. The specific implementation of these laws varies from study to study, and the impacts of this variation on model behavior have not been thoroughly explored. We run simplified 2D numerical models of free subduction in SULEC, with viscoplastic slabs following (1) a diffusion creep law, (2) a dislocation creep law, and (3) both simultaneously, as well as several variations of model 3 with reduced resistance to bending. We compare the results of these models to a model with a constant-viscosity slab to determine the impact of the implementation of different lithospheric flow laws on subduction dynamics. In creep-governed models, higher subduction velocity causes a longer effective slab length, increasing slab pull and asthenospheric drag, which, in turn, affect subduction velocity. Numerical and analogue models implementing constant-viscosity slabs lack this feedback but still capture morphological patterns observed in more complex models. Dislocation creep is the primary deformation mechanism throughout the subducting lithosphere in our models. However, both diffusion creep and dislocation creep predict very high viscosities in the cold core of the slab. At the trench, the effective viscosity is lowered by plastic failure, rendering effective slab thickness the primary control on bending resistance and subduction velocity. However, at depth, plastic failure is not active, and the viscosity cap is reached in significant portions of the slab. The resulting high slab stiffness causes the subducting plate to curl under itself at the mantle transition zone, affecting patterns in subduction velocity, slab dip, and trench migration over time. Peierls creep and localized grain size reduction likely limit the stress and viscosity in the cores of real slabs. Numerical models implementing only power-law creep and neglecting Peierls creep are likely to overestimate the stiffness of subducting lithosphere, which may impact model results in a variety of respects.

Thermal creep in a pre-hydrided Zr1 % Nb nuclear fuel cladding tube

The effect of hydrogen on the thermal creep behaviour of Zr1 %Nb alloy fuel cladding tubes for use in light water-cooled nuclear reactors was investigated. The tubes were hydrogen charged using an alternative method with a hydrogen content in the range 371–656 wppm H. Comparative constant-stress creep tests were carried out at 350 °C, under stress ranging from 150 to 225 MPa, on both non-hydrided and hydrided tubes. These creep tests were followed by a microstructural analysis of the specimens exposed to creep using SEM and TEM. It was found that these creep tests were conducted in the transition zone between the power-law and power-law breakdown creep regions. Hydrogen, either in solid solution (atomic hydrogen) or as precipitated zirconium hydrides, altered the creep behaviour of the hydrided tubes, and the impact of hydrogen depended on the hydrogen content CH. It was observed that under the creep loading conditions considered here with a hydrogen content CH < 450 wppm H, the presence of hydrogen resulted in an increase in the creep rate; however, a higher concentration of hydrogen caused a decrease in the creep rate. The difference in the effects of hydrogen on the creep rate was explained by the different roles of hydrogen in solid solution and in the form of zirconium hydride precipitates.

Creep-recovery deformation of 304 stainless-steel springs under low forces

The deformation behavior of materials at high temperatures determines the structural stability of mechanical structures under high-temperature service conditions. In this work, we prepare helical springs from 304 stainless-steel wires and study creep-recovery deformation of the helical springs under small forces at temperatures of 475–575 °C. In contrast to most methods reported in the literature, we use Kelvin representation of the Burgers model with a nonlinear dashpot in series connection to analyze the creep-recovery deformation of the helical springs and suggest the deformation mechanisms of the diffusion of interstitial atoms and dislocation generation/annihilation in transient creep and recovery of the helical springs under small forces. For the creep deformation, the stress exponent and activation energy for the plastic flow of the nonlinear dashpot are 2 and 57.8 kJ/mol, respectively, and the activation energy for the deformation flow of the linear dashpot is 41.8 kJ/mol. The nonlinear dashpot does not play any role in the recovery deformation, and the formation energy of defects for the recovery deformation of the helical springs is 51.4 kJ/mol. The approach used in this work provides a simple method to use a phenomenological model with a nonlinear dashpot to capture the power-law creep deformation of metallic materials.

Failure mechanisms at the Li anode/solid electrolyte interface during Li stripping

A precipitous increase in the resistance of the Li metal/solid electrolyte interface can occur during the stripping of Li from the electrode. This electrical failure has been typically attributed to the loss of contact associated with the growth of voids in the Li anode at the electrode/electrolyte interface. We first analyse the growth of voids at the electrode/electrolyte interface using a framework that couples the power-law creep deformation of the Li electrode and the flux of Li+ through a single-ion conductor solid electrolyte. We show that a modified Butler-Volmer kinetics where the local interfacial resistance decreases due to dislocations within the creeping Li predicts that voids indeed grow around interfacial sub-micron impurity particles. Consistent with observations that the increase in resistance of interface occurs earlier for thinner electrodes, we predict that the propensity of void growth increases with decreasing electrode thickness, and this is associated with the mechanical constraint imposed by the current collector. However, in contrast to the observations and rather counterintuitively, this analysis predicts that the cell voltage decreases with void growth. Consequently, we investigate an alternative mechanism of contact loss due to the deposition of insulating solute atoms within the Li electrode onto the interface. Predictions of the rising cell voltage using this analysis are in broad agreement with measurements. This leads us to hypothesize that although void growth occurs at the interface it is not the primary mechanism leading to the increase in interface resistance during stripping.

Activation entropy and enthalpy of creep in pure metals

Previously published results on the power law creep of a selection of fcc, bcc and hcp pure metals are discussed using a modified Stokes – Einstein equation originally describing Newtonian viscosity of spherical particles in a fluid. Activation energy is modified to include terms representing activation enthalpy, activation entropy and the effect of strain rate. It is similar to the Dorn equation for power law creep but with strain rate rather than stress as the independent variable. Curves of activation energy vs shear rate are linear rather than having a discontinuity as have those with stress as independent variable. All metals have linear curves of activation energy at zero rate vs temperature, the slope of which reflects the activation entropy and is similar for all metals. Intercepts which represent activation enthalpy are dependent on lattice structure.

Effect of damage accumulation on the asymptotic behavior of stresses ahead the crack tip

The subject of this study is the analysis of mechanical fields associated with a crack tip under creep conditions, taking into account the phenomenon of damage accumulation. The objective of the study is to perform finite element modeling, using the SIMULIA Abaqus software package, of uniaxial tension of a plate with a central horizontal crack under creep conditions, taking into account damage accumulation. For numerical simulation of creep, the Bailey-Norton power law is used. The power law of creep with the help of the user procedure UMAT (User Material) of the SIMULIA Abaqus package was supplemented with the Kachanov-Rabotnov kinetic equation of damage accumulation in a related formulation. In the calculation scheme of finite elements, the crack tip was modeled as a mathematical notch and as a notch with a finite radius of curvature. As a result of the calculations, the distributions of stresses, strains, and continuity under creep conditions were obtained, taking into account the accumulation of damage over time. Radial distributions of continuity, stresses, and strains are plotted over time at various distances from the crack tip. The subject of the study was the consideration of the asymptotic of the stress distribution. As a result of the study, it is shown that in the elastic region the asymptotic corresponds to the distribution under the elastic regime, and in the creep zone the asymptotics of Hutchinson, Rice and Rosengren (HRR-solution) is satisfied for different exponents nof the power law of creep.&#x0D; A comparison is made of the radial stress distributions in modeling without taking into account damage and in the case of taking into account damage accumulation. It is shown that the presence of damage significantly changes the asymptotics of the stress field in the vicinity of the crack tip.

Realistic long-term stress levels in a deep segmented tunnel lining, from hereditary mechanics-informed evaluation of strain measurements

Realistic estimation of stresses in segmented tunnel linings is a challenge tackled herein by means of a hybrid, i.e. computational-experimental, approach. Thermistor-equipped vibrating wire strain gauges were installed (i) into concrete samples undergoing one year-long uniaxial creep tests under the environmental conditions of the tunnel site, and (ii) into the tubbings making up the tunnel lining of the Koralm tunnel. The data obtained from the creep tests allow for calibration and validation of an integro-differential thermo-viscoelastic model. The creep function combines a power-law for short-term creep with a logarithmic law for long-term creep. The corresponding relaxation function is determined by means of Laplace-Carson transformation, inversion, and back-transformation. This is the basis for translating the circumferential strain histories measured in the tubbings of Ring 2013 in Koralm tunnel KAT3 into circumferential and longitudinal stress evolutions. They are mainly due to mechanical ground-shell interactions. The corresponding degree of utilization increases during the first four months after ring installation, and remains virtually constant thereafter. Stress fluctuations due to seasonal temperature variations play only a minor role. With regards to a long-term prognosis, it is very interesting to note that the strain measurements recorded in the tubbings, when plotted as function of the logarithm of time, follow bi-linear trends. These trends can be extrapolated to 150 years, the targeted service life of newly built tunnels in Austria. Throughout this period, the viscoelasticity-based estimates of the stresses in the vicinity of the strain sensors stay temporally constant, at some 40% of the strength of concrete.

Power law creep and delayed failure of gels and fibrous materials under stress.

Motivated by recent experiments studying the creep and breakup of a protein gel under stress, we introduce a simple mesoscopic model for the irreversible failure of gels and fibrous materials, and demonstrate it to capture much of the phenomenology seen experimentally. This includes a primary creep regime in which the shear rate decreases as a power law over several decades of time, a secondary crossover regime in which the shear rate attains a minimum, and a tertiary regime in which the shear rate increases dramatically up to a finite time singularity, signifying irreversible material failure. The model also captures a linear Monkman-Grant scaling of the failure time with the earlier time at which the shear rate attained its minimum, and a Basquin-like power law scaling of the failure time with imposed stress, as seen experimentally. The model furthermore predicts a slow accumulation of low levels of material damage during primary creep, followed by the growth of fractures leading to sudden material failure, as seen experimentally.

Viscoelastic response of gray matter and white matter brain tissues under creep and relaxation

Accurate measurement of the mechanical properties of brain tissue is of paramount importance for understanding its mechanics-biology relationship. Most published studies on brain viscoelasticity have been conducted using a single relaxation test, without validating the validity of linear viscoelasticity, which is insufficient to establish an accurate constitutive equation for brain tissue. We obtained the creep and relaxation profiles of fresh adult porcine white matter (N = 120) and gray matter (N = 56) under finite step-and-hold uniaxial compression, using a mechanical testing machine, with 16.67 mm/s loading rate and 80 s hold time. These curves were employed to determine viscoelastic properties and demonstrated an excellent fit with a concise power-law function. The average initial modulus for gray matter (GM) was 6.619 kPa, higher than that for white matter under transverse loading (WM-2D) at 5.579 kPa (p < 0.01), yet lower than that for white matter under axial loading (WM-1D) at 6.759 kPa (p = 0.0121). Notably, WM-2D exhibited the highest degree of fluidity (β = 0.216). Our findings reveal that gray matter behaves as a linear viscoelastic material with power-law creep compliance and relaxation modulus. Conversely, the creep and relaxation behavior of white matter deviates from the verification relationship derived from linear viscoelastic theory, indicating its nonlinearity. This fact underscores the inaccuracy of assuming a linear constitutive relationship to characterize the viscoelastic properties of white matter. By combining the power-law function with the experimentally obtained creep compliance and relaxation modulus, we offer a unique approach to determining the viscoelastic characteristics of brain tissue.

Monitoring early age elastic and viscoelastic properties of alkali-activated slag mortar by means of repeated minute-long loadings

This study investigates the development of elastic and creep properties of sodium hydroxide-activated blast furnace slag mortar, utilizing three different molarities of the activator solution and two solution-to-binder ratio and a reference OPC-based mixture, since the earliest age. The experimental phase involves a series of hourly-repeated 5-min long creep tests on the aging material. This approach enables continuous monitoring, facilitating the characterization of early-age elastic stiffness and creep properties. Linear regression is employed to calculate the tangent unloading (elastic) modulus and Poisson's ratio. To model short-term creep behaviour, a power-law creep function is utilized. The integration of these findings with calorimetry-derived evolutions of cumulative heat release establishes linear correlation between (compressive) strength and heat release, along with a power function relationship between unloading (elastic) modulus and heat release. An optimal alkali dosage (Na2O content) appears to be vital for long-term strength development. Additionally, the creep parameters, namely amplitude (A) and kinetic (K), demonstrate a gradual decrease, although they maintain values higher than those of the corresponding OPC mixture, as the heat release progresses.

The strength of a constrained lithium layer

A constrained compression test is developed to replicate the mechanical state of a lithium filament within a solid state battery. Lithium microspheres are compressed between parallel quartz plates into a pancake shape of thickness on the order of 15 µm. Full adhesion with no slip exists between the lithium and platens, and the attendant mechanical constraint implies that the average pressure on the pancake-shaped specimens increases with increasing aspect ratio of radius to height. In addition to mechanical constraint, a thickness-dependant size effect is observed whereby the apparent flow strength of the lithium increases from 0.7 MPa in the bulk to 2.0 MPa at a thickness of 15 µm. The lithium deforms in a power-law creeping manner at room temperature, and to simplify interpretation of the results, the relative velocity of the loading platens is adjusted to ensure that the true compressive strain rate is held fixed at 10−3 s−1. Additional measurements of lithium flow strength are obtained by subjecting the pancake-shaped specimens to simple shear. The size effect under shear loading is comparable to that in constrained compression. The observed size effect for lithium is consistent with that reported in the literature for lithium in indentation tests and in single pillar compression tests. Finally, the size effect of lithium in the power law creep regime is compared with that for rate-independent plasticity (for copper).

Power Law Creep Behavior Model of Third Generation Lead-Free Alloys Considering Isothermal Aging

Abstract In realistic applications, the solder joint is continually subjected to thermal-mechanical stress due to the difference in the coefficient of thermal expansion between the printed circuit board substrate and the electronic packaging components. Creep and fatigue processes were the most common causes of failure in electronic assemblies. Under isothermal aging, creep deformation becomes more prominent. The aged microstructure was recognized by intermetallic coarsening and the appearance of intergranular fracture generated by dynamic recrystallization in the bulk solder joint. In this study, the influence of Bi content on the creep behaviors of solder joints was investigated under various aging conditions. Three lead-free solder alloys, including SAC305, SAC-3Bi, and SAC-6Bi, are tested at room temperature. For each alloy, preliminary micro-indentation tests were conducted to define three stress levels for distinct aging conditions. After each test, displacement versus time data was gathered. A novel approach based on an empirical model was developed to systematically examine the development of the steady-state creep rate. A power dependency prediction model was developed to investigate the relationship between creep strain rate and stress levels. The steady-state creep rate of SAC305 is significantly higher than that of SAC-Bi alloys owing to the presence of bismuth (Bi) in the solid solution at room temperature. The creep properties showed less variation after 100 h of aging. SAC-Bi alloys showed less coarsening of the intermetallic compounds precipitates after aging than SAC305. In the SAC-Bi solder alloys, combinations of precipitate and solid solution hardening mechanisms were observed, while Ag3Sn particles were the dominant strengthening mechanism in the SAC305 alloy system.