Inelastic Strain Rate Research Articles

Development of a Flow Rule Based on a Unified Plasticity Model for 13Cr-4Ni Low-Carbon Martensitic Stainless Steel Subject to Post-Weld Heat Treatment

This study aims to develop a flow rule for evaluating the relaxation and redistribution of residual stresses during the post-weld heat treatment (PWHT) of hydroelectric runners made from low-carbon martensitic stainless steel (13Cr-4Ni composition). During the PWHT, austenite reforms in the filler metal and surrounding areas of the base metal near welded joints. The evolving inelastic strain rate with reformed austenite led to defining two distinct flow rules in the pure martensitic (α′) and austenitic (γ) phases. A linear rule of mixture was then applied to assess global effective stress based on the inelastic strain rate and current austenite fraction during the PWHT. A unified constitutive model incorporating drag stress and back stress, evolving with creep and plastic deformation mechanisms during the PWHT, described the stress–strain behavior. To validate this analysis, a third flow rule was determined in the 18% tempered austenitic microstructure, compared with the rule of mixture’s effective stress contribution from each phase on the inelastic strain rate. Isothermal constant strain rate tests in stabilized crystalline microstructures evaluated constants specific to their respective flow rules. This study demonstrates the stability of reformed austenite at elevated temperatures during slow cooling and its significant influence on the mechanical properties of 13Cr-4Ni steels. The effectiveness of estimating yield stress using the rule of mixture based on individual phase behaviors is also confirmed.

Creep‐fatigue deformation characteristics and life prediction model of Inconel 718 superalloy under hybrid stress–strain‐controlled mode

AbstractIn this paper, the creep‐fatigue responses of the nickel‐based superalloy Inconel 718 have been investigated by conducting hybrid stress–strain‐controlled creep‐fatigue interaction (HCFI) loading experiments at 650°C with different dwell stresses and dwell periods. Deformation responses demonstrate that stress dwell leads to complex cyclic responses, such as an increase in softening rate and changes in inelastic strain rate. Then, a novel creep equivalent stress determination method is proposed based on stress relaxed characteristics. After modifying the continuum damage mechanics (CDM) model with this method, accurate prediction results of the creep‐fatigue cyclic life under strain‐controlled creep‐fatigue loading and HCFI loading can be obtained. Finally, the damage evolutions of Inconel 718 under HCFI loading are analyzed by CDM model. The results of damage evolution indicate that the trend of damage accumulation under different test conditions is consistent with the trend of inelastic strain in the material.

Creep-Fatigue Life Evaluation for Grade 91 Steels with Various Origins and Service Histories

Grade 91 steel is widely used in the boilers and piping of thermal power plants. There has been significant research interest in understanding the variations in creep characteristics among different heats of this steel for effective plant management. In recent years, thermal power plants have been subjected to frequent load changes and startup/shutdown to adjust power supply and demand and stabilize frequencies. These operational shifts have raised concerns regarding the potential for creep-fatigue damage in high-temperature components. Therefore, this research focuses on creep-fatigue properties of Grade 91 steel and their predictability. Tensile, creep, strain-controlled fatigue, and strain-controlled creep-fatigue tests were performed on six Grade 91 steels with different heats and/or histories, and the characteristics in each test were compared. As a result, the variations in creep-fatigue life among the materials were found to be correlated with the difference in creep characteristics and stress level during stress relaxation. Furthermore, the study involved a comparative assessment of the predictive performance of creep-fatigue life using five different approaches: time fraction, classical ductility exhaustion, modified ductility exhaustion, energy-based, and hybrid approaches. Among these approaches, the hybrid approach, based on inelastic strain energy density at fracture formulated as a function of inelastic strain rate, exhibited the most accurate predictive performance.

The viscoplastic constitutive model of TC4 alloy under high temperature

TC4 alloy is widely used in the high temperature environment. This paper proposed an ununified viscoplastic constitutive model according to the uniaxial creep experiment and high temperature tensile experiment of TC4 alloy. The inelastic strain rate was decomposed into two parts: creep strain rate and plastic strain rate. The creep strain was calculated by multiaxial ductility exhaustion creep damage law, and the plastic strain was get from the yield function containing the damage and the kinematic hardening effect. The implicit stress integration algorithm and the consistent tangent modulus of this model were derived. The material constants of the proposed model were determined by using genetic algorithm. The proposed constitutive model was compiled in a UMAT subroutine of finite element software Abaqus and 3D eight nodes isoparametric brick element was employed to simulate the creep and the tensile behavior of TC4 alloy. Form the result, the proposed constitutive model can predict the three stages of creep curves and the softening stage of tensile curves accurately.

Multi-scale modelling of creep cavity nucleation and growth in polycrystalline Type 316 stainless steel

ABSTRACT A common creep damage mode in Type 316 stainless steel under high-temperature power plant conditions is intergranular cavitation. A review of the literature has confirmed that cavitation in Type 316 is controlled by nucleation, which is not fully understood. In order to provide further insights into the physics of this process, existing strain-based empirical and stress-based (classical nucleation theory) nucleation models are modified in this study by considering experimentally-observed features of cavity nucleation in Type 316. The models are employed locally within a newly-developed crystal plasticity finite element (CPFE)-interface element framework. Modelling results suggest that the strain-based model as a function of local inelastic strain rate does not explain the physical nature of the nucleation process as observed experimentally. By contrast, the modified classical nucleation theory is able to capture features of the observed macroscopic failure response and the distribution of cavities in the microstructure. A number of missing features are identified in the mechanistic model, which need to be incorporated in future unified cavity nucleation theories. These findings highlight key aspects of the nucleation process, which need to be examined experimentally.

Effects of dwell time on the isothermal and thermomechanical fatigue behavior of 316LN stainless steel

Isothermal fatigue (IF) and thermomechanical fatigue (TMF) tests with and without tensile dwell time at various strain amplitudes were conducted on 316LN stainless steel. The results showed that the inelastic relaxation strain rate is more appropriate to characterize creep damage than the amount of stress relaxation. The introduction of tensile strain hold imparted a more deleterious effect on the TMF than the IF performance based on a statistical analysis of the number of secondary cracks and their size distribution, which was primarily attributed to the more severe oxidation-assisted cracking at the interface between the oxide scale and substrate. Most importantly, the lowest fatigue life occurred in the TMF test with tensile dwell time at the low strain amplitude of 0.4% rather than in the creep-fatigue test at the isothermal maximum temperature, indicating that it was not always conservative to predict the life of engineering components based on the IF data.

Damage ratio strength criterion for cast iron

According to the assumption of small strains and total strain decomposition of cast iron, the expressions of the inelastic principal strain rate and relative energy dissipation rate of cast iron are derived, and the parameters of the tensile and compressive damage ratio of cast iron in the inelastic phase are proposed. Based on the least energy consumption principle, combined with the above expressions and parameters, the damage ratio strength criterion for cast iron under biaxial stress states is established. The damage ratio variable expression of the cast iron damage ratio strength criterion considering the effect of Lode angle is proposed. Furthermore, the values of the damage ratio variable are verified by the experimental stress–strain curves of cast iron under uniaxial stress states. By comparing the predicted values to 104 experimental data points of cast iron specimens, the failure envelopes of the proposed criterion can represent the failure behavior of cast iron under biaxial loading conditions. To facilitate the practical application in engineering, the further simplified expression of the four-parameter damage ratio strength criterion for cast iron under biaxial stress states is suggested and those empirical parameters are recommended. Compared with some main strength criteria, the simplified criterion is in better agreement with the experimental results under every stress state.

Experimental analysis and constitutive modeling of anisotropic creep damage in a wrought age-hardenable Al alloy

Many components of aerospace and automotive engineering manufactured from forged age-hardening aluminum alloys operate at high temperature in the range 0.3Tm–0.5Tm, where Tm is the melting temperature. Experimental data for uni-axial creep on specimens sampled from a forged AA2014 alloy component show that inelastic strain rates depend significantly on the orientation of loading with respect to the material flow direction during the manufacturing stage. Based on the recently developed constitutive model of anisotropic creep, hardening/recovery, softening and overageing processes, this paper addresses experimental analysis and modeling tertiary creep regime up to the final stage of creep rupture. The emphasis is placed on the damage evolution equation in order to account for different damage mechanisms depending on the loading mode and loading direction relative to grain axis: cavitation on elongated grain boundaries as well as decohesions and microcracks around and inside coarse grain particles. Experimental data of creep strain rates for longitudinal and transverse loading directions at 130, 150 and 170 °C have been used to calibrate the model. The validation of the model was carried out by comparing experimental and modeled creep curves in the case of a 30°angle between the loading and the material flow direction during forging. The results show that the assumed damage mechanisms and the developed damage evolution equation are able to reflect both the anisotropic accelerated creep and the time to creep rupture for different stress levels and loading directions.

Bending stress relaxation of microscale single-crystal copper at room temperature: An in situ SEM study

The bending stress relaxation of microscale single-crystal copper was studied by in situ SEM bending experiments to elucidate the time-dependent plasticity involving the effect of the strain gradient and the sample size. Three specimens with varied heights of square cross sections were fabricated using a focused ion beam. Interestingly, an obvious bending plateau was observed in the load-displacement curves. The calculated flow stress on the cross section indicated an inverse relationship between the specimen size and strength. In addition, the proportion of decay stress is approximately 2.7% for the beam with a height of h = 1.82 μm and 13.9% for that of h = 0.94 μm in the bending plateau. Moreover, during the dwell period, a sudden stress drop occurred, which is ascribed to the strain burst or stress jerky during the deformation of single-crystal metals. Microbending is usually related to a significant increase in geometrically necessary dislocations (GNDs), which is supposed to improve the resistance for further deformation. However, the bending plateau implied perfect plastic flow and an evolving balance in the microstructures. On the other hand, thermal activation theory was employed to interpret the time-dependent plasticity. The apparent activation volume of the dwell periods in the bending plateau suggested that cross slip dominates the inelastic strain rate evolution. Finally, the inelastic strain rate for stress relaxation was derived from the curve of decay stress, and it was inversely proportional to the height of the microbeam, which is in complete contrast with the trend observed in the previous compression tests of micropillars. In current microscale bending specimens, the thinner beam containing fewer initial dislocation sources exhibits a larger creep ductility with the assistance of GNDs.

Inelastic Behavior of Polyoxymethylene for Wide Strain Rate and Temperature Ranges: Constitutive Modeling and Identification.

The aim of this paper is to present experimental data and the constitutive model for the inelastic behavior of polyoxymethylene in wide strain rate and temperature ranges. To capture the non-linearity of the stress responses for both loading and unloading regimes, the composite model of inelastic deformation is utilized and further developed. The equivalent inelastic strain rate is described by the Prandtl–Eyring law, while the temperature dependence is characterized by the modified Arrhenius-type law. Generalized equivalent stress and the flow rule are formulated to capture pressure sensitivity, transverse strain and volumetric strain responses. The results obtained by the constitutive law are compared with experimental data for stress vs. axial strain from standard tension tests as well as with axial and transverse strains measured by digital image correlation. The developed composite model is able to capture the non-linearity of stress–strain curves for complex loading paths within the small strain regime. For higher strains, apart from geometrically non-linear theory, evolution laws for the volume fraction of the constituents should be modified and calibrated. For the small strain regime, the inelastic dilatation is negligible. For higher axial strain values, a decrease in Poisson’s ratio under tension and increase in it under compression are observed. The Drucker–Prager-type equivalent stress and the developed flow rule provide a better description of both the transverse and volumetric strains than that of the classical von Mises–Odqvist flow rules.

Modeling for cyclic plasticity and rate dependence of metal materials with a new non-unified viscoplasticity model

The classic unified viscoplasticity model has achieved great success in areas such as cyclic plasticity, time-dependent behavior, and creep–fatigue interaction. However, the unified viscoplasticity model may be inadequate to describe inelastic strain rate with static recovery term. Our previous study [Chen, Kitamura, Feng (Int J Fatigue 112:70–77, 2018)] of 316L stainless steel has shown that adding a static recovery term predicts an accelerating creep strain rate in the creep–relaxation interaction tests, which is inconsistent with the experimental test results. In this study, a modified non-unified viscoplasticity (NUV) model is proposed based on the mechanism of viscous glide. The parameter determination process and the role of different parts of the model are illustrated using experimental data drawn from Al 5083; then, the proposed model is verified by simulating the test of 316L stainless steel. The modeling results show that the proposed NUV model accurately predicts the strain rate evolution during the creep–relaxation interaction tests and other behaviors, i.e., cyclic plasticity and loading rate sensitivity. As with fewer parameters and more concise equations, the proposed NUV model is a good choice for engineering applications.

Inelastic strain in the hypocentral region of the 2000 Western Tottori earthquake (M 7.3) inferred from aftershock seismic moment tensors

Inelastic deformation due to seismic activity is an important signal that reflects fault evolution. In particular, aftershock sequences indicate the evolution of damage in a medium that has experienced a large earthquake. Herein, we discuss the inelastic strain rate surrounding the fault that produced the M 7.3 Western Tottori earthquake in 2000 using long-term aftershock analysis. To obtain high-resolution focal mechanisms 18 years after the earthquake occurrence, we conducted dense seismic observations in the focal area. The inelastic strain rate estimated from the aftershock seismic moment tensor data showed spatial variations within a range of 10−7–10−11 per year, 18 years after the main shock. By comparing the inelastic strain rates from immediately after the earthquake and 18 years later, we detected the increase in the spatial variations in the inelastic strain rate; the variations are as small as 102 (= 10−5/10−7) for the early stage but as large as 104 (= 10−7/10−11) for the later period. In addition, the decay of the rate during these two periods varied spatially from spatial bin to bin. Certain bins in the northern segment of the earthquake fault, southern edge of the fault, and surrounding the location of the preceding swarm activity to the M 7.3 event showed slower decay rates than the inverse of the lapse time since the occurrence of the M 7.3 earthquake. We modeled this decay rate change as the relaxation response of a power-law fluid to an elastic strain input from the large earthquake. Most parts of the fault can be explained by this model. However, the areas with low decay rates suggest the presence of a dragging mechanism, such as aseismic slip, at or around these locations.

A novel rate based methodology for creep fatigue life estimation of superalloys

In this present work, we present an accumulated inelastic strain rate based methodology to predict creep fatigue life of two different superalloys i.e. Haynes 282 and IN 718. The evolution of differential strain rate during creep and fatigue respectively motivated the present work. As creep is the rate controlling damage process in creep fatigue interaction, present mean strain rate based approach considers creep strain rate as a kinetic variable controlling the process. The other variable considered for this approach is accumulative cyclic strain during Creep-fatigue interaction. Our prediction method is purely based upon a relative accumulation rate of creep and fatigue strain. This method correlates the creep fatigue life with rate of interaction in strain-controlled regions successfully from the data of creep fatigue tests on those superalloys at two temperatures, 650 °C and 760 °C. We found suitable microstructure sensitive constants in a lifing model.

Detection of plastic strain using GNSS data of pre- and post-seismic deformation of the 2011 Tohoku-oki earthquake

In general, there are three mechanisms causing crustal deformation: elastic, viscous, and plastic deformation. The separation of observed crustal deformation to each component has been a challenging problem. In this study, we succeed in separating plastic deformation as well as viscous deformation in the northern Niigata–Kobe Tectonic Zone (NKTZ), central Japan, using GNSS data before and after the 2011 Tohoku-oki earthquake, under the assumptions that elastic deformation is principally caused by the plate coupling along the Japan trench and that plastic deformation ceased after the Tohoku-oki earthquake due to the stress drop caused by the earthquake. The cessation of plastic deformation can be understood with the concept of stress shadow used in the field of seismic activity. The separated strain rates are about 30 nanostrain/year both for the plastic deformation in the preseismic period and for the viscous deformation in both the pre- and post-seismic periods, which means that the inelastic strain rate in the northern NKTZ is about 60 and 30 nanostrain/year in the pre- and post-seismic periods, respectively. This result requires the revision of the strain-rate paradox in Japan. The strain rate was exceptionally faster before the Tohoku-oki earthquake due to the effect of plastic strain, and the discrepancy between the geodetic and geologic strain rates is much smaller in usual time, when the plastic strain is off. In order to estimate the onset timing of plastic deformation, the information on stress history is essentially important.

Inelastic strain rate and stress fields in and around an aseismic zone of Kyushu Island, Japan, inferred from seismic and GNSS data

SUMMARYUnderstanding earthquake processes and crustal deformation requires knowledge of the stress concentration process in the crust. With the enhancement of observation networks, it has become possible to consider in detail the relationships between localized deformation and seismic activity in island arcs and the process of stress concentration. According to previous studies, inelastic deformation in localized weak zones in the crust is considered to play an important role in the stress concentration process. Kyushu, located in southwest Japan, has a 20–30 km band-like active seismic activity and an enclosed aseismic zone. In particular, a part of the seismic active region called the Beppu-Simahara Graben, which is dominated by north–south extensional deformation, is characterized by high seismic activity and a remarkable aseismic zone. We identified the relationship between inelastic deformation and stress concentration processes in this area by using analyses of geodetic and seismic data. The results inverted from both the strain rate field obtained by the geodetic observations and the deviatoric stress field estimated from focal mechanism data reveal a large inelastic deformation zone ($\sim {10^{ - 7}} \,\mathrm{ yr}^{-1}$) beneath the area of active seismicity. From comparison with previous works, the inelastic deformation zone in the lower crust may correspond to an area with high temperature and/or fluid. This may suggest that inelastic deformation is in progress in the area where the strength of lower crustal rocks has reduced due to the presence of geothermics and/or fluids. Furthermore, we confirmed that this inelastic deformation causes stress concentrations of up to $10\,\,{\rm{kPa}}\,\,{\rm{yr}}^{-1}$ in the upper crust. These results show that stress concentration occurs locally in the upper crust, above the inelastic deformation zone in the weakened lower crust, owing to the presence of geothermal and/or fluid; this stress concentration induces seismic activity and crustal deformation.

Stress-regime-dependence of inelastic anisotropy in forged age-hardening aluminium alloys at elevated temperature: Constitutive modeling, identification and validation

Many structural materials exhibit stress-regime-dependent anisotropy of inelastic responses. Examples are age-hardening aluminium alloys forgings. Experimental creep curves indicate that inelastic strain rates depend significantly on the loading direction within the power law regime, while in the power law breakdown range the anisotropy is weak and can be neglected. The aim of this paper is to analyze anisotropic behavior of forged AA2014 alloy and to develop a constitutive model to describe inelastic response under multi-axial stress state. Microstructural observations suggest that the anisotropy is primarily caused by elongated grains and coarse particles on grain boundaries. To account for inhomogeneous inelastic deformation and stress redistribution in different microstructural zones a composite model is developed. Constitutive equations for inelastic-hard and inelastic-soft constituents and anisotropic rules for mechanical interactions between them are elaborated. Hardening/recovery and overageing processes in grain interiors are characterized by internal state variables and kinetic equations. The model is calibrated against families of creep curves for two loading directions in a wide stress and temperature ranges. For the validation, creep tests under the loading with the angle of 30∘ relative to the longitudinal grain axis as well as tensile tests are simulated by the model and results are compared with experimental data.

Nonlinear Rheology at Shallow Depths with Reference to the 2016 Kumamoto Earthquakes

Abstract Strong S waves produce dynamic stresses, which bring the shallow subsurface into nonlinear inelastic failure. We examine implications of nonlinear viscous flow, which may be appropriate for shallow muddy soil, and contrast them with those of Coulomb friction within a shallow reverberating uppermost layer with low‐seismic velocities. Waves refract into essentially vertical paths at the shallow layers and produce tractions on horizontal planes. The Coulomb ratio of shear traction to lithostatic stress for S waves equals the resolved horizontal acceleration normalized to the acceleration of gravity. The ratio of dynamic vertical normal traction to lithostatic stresses is the vertical normalized acceleration from P waves. The predicted viscous inelastic strain rate in muddy soil begins at low normalized accelerations and then increases mildly and nonlinearly with increasing normalized acceleration. Failure is unaffected when P waves decrease the vertical normal traction. Seismic waves recorded at KiK‐net station KMMH16 for the 2016 Kumamoto mainshock and strong foreshock show these effects. Inelastic deformation commences at a normalized horizontal acceleration of ∼0.25 and reduces S‐ and P‐wave velocities within the uppermost ∼15 m reverberating layer. Normalized horizontal accelerations and the Coulomb stress ratio reach ∼1.25. Strong S waves arrived even when strong P waves produced vertical tension on horizontal planes. In contrast, inelastic Coulomb failure commences at a normalized horizontal acceleration equal to the effective coefficient of friction; rapid inelastic strain precludes even higher accelerations. Furthermore, horizontal planes should fail from the stresses of strong S waves during the tensional cycle of strong P waves.

Viscoplastic Interpretation of Localized Compaction Creep in Porous Rock

AbstractRecent laboratory evidence shows that compaction creep in porous rocks may develop through stages of acceleration, especially if the material is susceptible to strain localization. This paper provides a mechanical interpretation of compaction creep based on viscoplasticity and nonlinear dynamics. For this purpose, a constitutive operator describing the evolution of compaction creep is defined to evaluate the spontaneous accumulation of pore collapse within an active compaction band. This strategy enables the determination of eigenvalues associated with the stability of the response which is able to differentiate decelerating from accelerating strain. This mathematical formalism was linked to a constitutive law able to simulate compaction localization. Material point simulations were then used to identify the region of the stress space where unstable compaction creep is expected, showing that accelerating strains correspond to pulses of inelastic strain rate. Such pulses were also found in full‐field numerical analyses of delayed compaction, revealing that they correspond to stages of inception and propagation of new bands across the volume of the simulated sample. These results illustrate the intimate relation between the spatial patterns of compaction and their temporal dynamics, showing that while homogeneous compaction develops with decaying rates of accumulation, localized compaction occurs through stages of accelerating deformation caused by the loss of strength taking place during the formation of a band. In addition, they provide a predictive modeling framework to simulate and explain the spatio‐temporal dynamics of compaction in porous sedimentary formations.

A unified constitutive model in simulating creep strains in addition to fatigue responses of Haynes 230

A unified constitutive model (UCM) specifies that its flow rule for inelasticity computes both the plastic and creep strains as a single state variable. A Chaboche framework based UCM with the modeling features of strain range-dependence, strain rate-dependence, static recovery and mean stress evolution was developed and experimentally validated against a broad set of fatigue and fatigue-creep responses of Haynes 230 (HA 230) under isothermal and anisothermal temperature conditions. This article demonstrates that this advanced Chaboche-based UCM can simulate the secondary minimum creep strain rates reasonably, but is unable to predict the tertiary creep strain responses. To simulate the tertiary creep strain responses a continuum damage model is needed to be coupled to the UCM. This study also evaluated three different unified flow rules, Norton's power law, exponential Norton and sine-hyperbolic Norton for calculating the inelastic strain rates. It is found that the choice of flow rule is important in simulating the stress amplitude saturation rate of fatigue responses, but has minimal effect in simulating the tertiary creep strains. However, the damage coupled UCM independent to the unified flow rules listed above can adequately simulate fatigue, fatigue-creep including the stress relaxation during strain dwell, and creep strain up to the tertiary range for HA 230. The drawbacks of the damage coupled UCM are the hysteresis loop softening at very high temperatures and asymptotic simulation at low creep temperatures, which are identified as challenges to be overcome towards developing a universal UCM for robust design and analysis of high temperature components.

Effects of Thermal Cycling on Creep of SnAgCu Solder Joints

The long-term life of properly optimized high-end microelectronics assemblies is commonly limited by fatigue of the solder joints. The most credible assessments of this rely on modeling-based extrapolation and generalization of accelerated test results. As the validity of extrapolations to long-term service can rarely if ever be verified experimentally, there is a strong need for mechanistically justified models. The models all rely on predictions of stresses, strains, and perhaps functions of those versus time and temperature in cycling. Finite-element-based predictions of these require the assumption of relatively complex constitutive relations capable of predicting inelastic strain rates under continuously varying stresses and temperatures. These are usually extracted from a series of creep or stress relaxation experiments and generalized under the implicit assumption that the inelastic deformation is dominated by dislocation creep. However, the resulting constitutive relations were recently shown to be strongly misleading. It was suggested that this is a result of the inelastic deformation at the stresses and temperatures of primary concern actually being dominated by dislocation core diffusion. This paper lends further support for this picture, showing additional contributions from diffusion on cycling induced dislocation structures to make for ongoing, usually nonmonotonic, variations in deformation rates from cycle to cycle. These effects are certain to have major consequences for modeling. In fact, it has been argued that the combination of wrong constitutive relations with a wrong damage function has led to a partial cancellation of errors across a limited parameter range, but that this cannot be counted on for more detailed predictions of accelerated test results and does not allow for interpretation of such results in terms of even qualitative trends in service.