Norton Power Law Research Articles

High temperature tensile and creep properties of the new cladding steel of 15-15Ti-Y

The 15-15Ti austenitic stainless steel is commonly used as a fast reactor cladding material due to its favorable comprehensive properties. To further enhance its high temperature strength, yttrium (Y) modified 15-15Ti (15-15Ti-Y) steel was developed. In this study, the tensile properties of 15-15Ti-Y were evaluated across the temperature range of 200–700 °C, and the constant load creep properties of 20 % cold worked (CW) 15-15Ti-Y were measured at temperatures of 550–700 °C under various applied stresses. The results indicated that Y addition led to enhanced intragranular precipitates and delayed intergranular precipitates, which resulted in a YS (YS) that was ∼ 100 MPa higher than that of Y-free steel at temperatures ranging from 200 to 600 °C, without compromising the plasticity and ductility. The creep fracture life of 15-15Ti-Y steel ranged from 21 to 5761 h, and the deformation and damage processes were found to adhere to the Norton power-law and Modified Monkman Grant relationship. The stress exponent revealed that the deformation was dislocation creep mechanism, while the creep damage tolerance parameter suggested that the creep fracture was caused by necking. These findings provide data and theoretical support for the application of Y-modified 15-15Ti steel as a fast reactor cladding material in high temperature environments.

Assessment of thermal and radiation induced creep in the dual cooled lead lithium blanket

The creep in recently designed blanket for the fusion nuclear science facility is described by decomposing it into thermal, irradiation and cavity swelling creep. Initial estimates of thermal creep using elastic high temperature rules showed that the recent heat transfer improvements at the first wall (FW) is feasible, and the FW can withstand heat flux of 0.35 MW/m2 at 3000 hrs without rupture. A viscoplastic model that is based on Norton's power law and relies on Multiphysics coupling of solid mechanics and heat transfer modules is used to capture the inelastic deformation in the blanket. Results showed that the design peak heat flux of 0.25 MW/m2 produced maximum thermal creep strain of 0.45 % and the relaxation of the thermal stress at the FW. Irradiation creep is prescribed to be proportional to the displacement damage dose and applied stress. The displacements from irradiation creep radially decrease from the FW to the back wall, but the maximum deformation is found at the back wall that is connected to the Helium manifold due to the high stresses at the region. Cavity swelling creep is dominant at steady state and combined with radiation creep to produce displacement of about 8 mm at the FW.

Exploring the creep and oxidation behaviors of four types of FeCrAl alloys through small punch test at 600 °C: Experiments and simulations

Four types of iron-chromium-aluminum (FeCrAl) alloys, composed of different elements, were used to investigate the creep performance and high-temperature oxidation resistance. The microstructures in the undeformed regions after the small punch creep test (SPCT) remained consistent with the as-received (AR) FeCrAl alloys, while the grain aspect ratio (GAR) decreased and grains elongated in the deformed regions. Amongst the four alloys, Q1 exhibited the worst oxidation resistance, whereas Q4 showed superior oxidation resistance due to the addition of Si. Precipitates were observed in the deformed regions after SPCT but not in undeformed regions, indicating that deformation had a positive effect on element segregation. Creep models including Norton power law, Monkman-Grant relationship, Larson-Miller Parameters, and Fracture-Time law were obtained. Calculation results from these creep models revealed that under low loads Q4 had lower minimum creep rate δm and longer creep fracture time tr compared to other alloys; however, Q3 demonstrated better performance in terms of δm and tr under high loads. This suggests a strong correlation between creep performance and loads. Additionally, finite element method (FEM) was employed to explore the effects of minor variation (±0.005 mm) in specimen thickness on SPT curves and SPCT curves. FEM results indicated that this slight variation alone did not account for discrepancies observed in load-deflection curves; deviations of ±6.1% for δm and ±6.3% for tr were found corresponding to specimens with 0.5 ± 0.005 mm thickness.

Deformation behavior, microstructure evolution, and rupture mechanism of the novel G115 steel welded joint during creep

To promote the wide application of the G115 steel in ultra-supercritical (USC) power plants, creep deformation, microstructure evolution and rupture mechanism of the G115 welded joint were systematically investigated. The fabricated G115 welded joint exhibits superior creep strength to the P92 or MARBN steels. The deformation behaviors are analyzed by the Norton's power law, proving that the deformation mechanism is closely associated with the stress levels. During short-term creep under higher stress, the weld metal (WM) and heat-affected zone (HAZ) exhibit outstanding microstructure stability, and the rupture occurs within the base metal (BM) due to the lath fracture. While, during long-term creep under lower stress, the brittle Type IV cracking occurs within the inter-critical heat-affected zone (ICHAZ) or fine-grained heat-affected zone (FGHAZ) due to the formed micro-voids along boundaries, obviously deteriorating the creep strength. The weld strength reduction factors are calculated under different stresses, and prove that synergistic action of back stress and dislocation motion can enhance the creep resistance of the G115 welded joint. Additionally, the allowable service stress of G115 weldments should be designed according to the creep strength of the welded joints under expected performance life.

Creep deformation behaviour of thermomechanically treated India specific reduced activation ferritic martensitic steel

In the present work, the influence of thermomechanical treatment (TMT) on microstructural evolution and creep deformation behaviour of India specific reduced activation ferritic martensitic (INRAFM) steel was investigated. Firstly, the TMT was carried out on INRAFM steel by reducing its thickness to 25% at warm rolling temperature (973 K), followed by tempering for 90 min at 1033 K. The resultant microstructural characteristics were compared with those attained in normalized and tempered (N+T) condition. TMT treated steel shows a refinement in lath structure with an increment in population of finer M23C6 precipitates along the hierarchal PAG/packet/block boundaries and MX carbonitrides in the intralath region. After TMT, creep tests were performed on the steel over the stress range of 200–300 MPa at temperature of 823 K. The rupture time of the steel significantly increased upon TMT, due to refinement in microstructural features that enhanced the creep resistance. Dependence of minimum creep rate on applied stress obeys Norton's power law and the stress exponent values indicated the deformation mechanism as dislocation creep. Modified Monkman-Grant relation shows a good relation between rupture time/strain to failure with minimum creep rate. Thereafter, the effect of TMT on the transient, secondary and tertiary creep behaviour was also examined and presented in this work. The TMT treated steel shows lower rate of exhaustion, higher threshold stress and prolonged tertiary region.

Creep lifetime of Al 6061-T6 pressurized rotating friction stir welded tube subjected to internal pressure and rotational velocity: Welding and creep lifetime optimization

Aluminum 6061-T6 alloys have numerous applications in the industry due to their unique properties such as higher corrosion resistance and strength-to-weight ratio. Friction stir welding is also a technique that can be used in aluminum joints because of its numerous advantages. In this paper, optimum friction stir welding parameters that lead to a higher failure load are obtained using tensile test data and analysis of variance technique. According to the analysis of variance results, it is found that the tool rotational speed has a major effect among the welding parameters on the failure load. Then, creep tests are carried out on non-welded and welded specimens to predict their creep behavior. The Norton power law equation is used as a creep constitutive equation to model the creep behavior of the specimens. The model is used to predict the creep lifetime of the welded and non-welded rotating pressurized vessels. The results show that an optimally welded vessel can only sustain up to 0.75 of pressure for the non-welded vessel with the same rotating speed. Since this comparison is not straightforward because of the number of affecting parameters. Hence, the reference stress method is used to produce a set of diagrams that provide design conditions and optimum creep lifetime for the friction stir welded Al 6061-T6 rotating pressurized vessels.

Numerical implementation and application of an internal state variable model to analyze the time-dependent behavior of mining excavations in rock salt

The geomechanical behavior of rock salt has been investigated extensively over the years. Experimental studies have identified distinctive features associated with nonlinear inelastic response that shows strong time and loading history dependencies. In this paper, a relatively simple constitutive model is presented and implemented into the numerical code FLAC, and then applied to analyze the time-dependent behavior of excavations in salt mines. The unified creep-plasticity model includes an internal state variable (ISV) with an evolution law that induces progressive strain hardening (SH) until a steady state is reached. Numerical analyses are performed with the proposed ISV–SH model to evaluate material parameters, based on creep test results on natural rock salt, and to simulate the response of a circular opening and of rectangular mining excavations created in sequence. The results are analysed and compared with those obtained with the well-known Norton power law equation, commonly used in salt mine engineering. The time-dependent calculation results illustrate key aspects of rock salt behavior and highlight the major influence of transient inelastic behavior and stress redistribution on the response of underground openings. The proposed modelling approach constitutes an advantageous alternative to analyses based on stationary creep laws, often used in rock salt mining operations.

Compressive creep behavior of spherical pressure hull scale model for full-ocean-depth manned submersible

The phenomenon of room temperature creep in titanium alloys has drawn great attention for decades. With the increasing application of titanium alloys to the manned submersible pressure hull, the traditional calculation and analysis based on strength and stability cannot meet the requirement of structure design and safety evaluation for the spherical pressure hull completely. In this paper, a series of compressive creep tests were performed to reveal the change rule and distribution characteristics for creep behavior of the spherical pressure hull model. Results show that there is no measurable creep strain for pressure hull under maximum operating pressure and steady-state creep appeared in high stress areas of the pressure hull under 1.25 times maximum operating pressure, with the characteristics of internal creep strain and strain rate being greater than the external counterparts. The stress threshold value of the pressure hull model for steady-state creep is 853.7 MPa (equivalent to 0.88Rp0.2), and a creep constitutive equation is proposed based on improved Norton power law model. The results of this paper can provide support for the safety assessment and optimization design of manned submersibles for long-term service.

Obtaining optimum creep lifetime of Al 7075-T6 rotating pressurized vessel based on the experimental data, using reference stress method (RSM)

Most of the mechanical components are exposed to environmental attacks and different loading conditions, e.g., internal pressure, high-temperature gradient, and angular velocity. Therefore, investigating their mechanical behavior under operating conditions is necessary to gain higher efficiency and less mechanical failure. Due to the vast industrial applications of Aluminum (Al) alloys, in this paper, the tensile and creep behavior of Al 7075-T6 were studied at elevated temperatures. The Norton-Power Law (NPL) equation was considered as a creep constitutive equation to estimate the creep behavior of Al 7075-T6 alloy. To this aim, experimental uniaxial creep tests were carried out to derive creep constitutive parameters. These parameters were used to predict the creep behavior of different structures, such as Al vessels. The reference stress method (RSM) and developed constitutive model were utilized to plot graphs that provide design conditions and optimum creep lifetime for Al vessel subjected to combined rotation and internal pressure. These graphs can be employed in industrial applications to facilitate the design process of Al vessels.

Irreversible deformation of a rotating disk having angular acceleration

This paper is devoted to the study of deformation of a disk rotating with changing angular velocity under consecutive accumulation of creep and plastic strains. The differential equations which help to calculate the fields of stresses, strains, displacements and velocities are obtained; the method of finite difference is used for solving the problem. The evolution of the viscoplastic flow region is investigated. The Norton power law is used as creep law; the von Mises’s generalized yield criterion is applied as viscoplastic flow condition. The obtained solution is compared to the classical elastoplasticity case. A hollow disk and a disk with a rigid inclusion are studied.

A practical power law creep modeling of alloy 690 SG tube materials

A new practical modeling of the Norton's power law creep is proposed and implemented to analyze the high temperature behaviors of Alloy 690 SG tube material. In the model, both the stress exponent n and the rate constant B are simply treated as the temperature dependent parameters. Based on the two-step optimization procedure, the temperature function of the rate constant B(T) was determined for the data set of each B value after fixing the stress exponent n value by using the prior optimized function at each temperature. This procedure could significantly reduce the numerical errors when using the power law creep equations. Based on the better description of the steady-state creep rates, the experimental rupture times could also be well predicted by using the Monkman-Grant relationship. Furthermore, the difference in tensile strengths at high temperatures could be very well estimated by assuming the imaginary creep stress related to the given strain rate after correcting the temperature effects on the elastic modulus.

Anisotropic steady-state creep behavior of Single-crystal β-Sn: A continuum constitutive model based on crystal viscoplasticity

Body-center tetragonal (BCT) β-Sn crystals exhibit highly anisotropic properties such as stiffness and thermal expansion, which significantly affect their thermal-mechanical behavior. The homologous temperature of β-Sn is relatively high under common applications due to its low melting point, which renders Sn and Sn-based alloys viscoplastic even at room temperature. The orientation-dependent creep behavior of β-Sn specimens have been previously measured by different research groups. A dislocation mechanics based crystal viscoplasticity model is applied in this study to describe this anisotropic steady-state creep behavior of β-Sn single crystals. The model constants are calibrated with single-crystal creep test results available in the literature. The resulting creep behavior of β-Sn single-crystal is also represented with a homogenized continuum-scale finite element approach based on the use of a combined Hill-Norton approach where the creep anisotropy is represented with Hill's anisotropic potential and the creep flow rule is represented with Norton's power-law model. Estimation of the six Hill's constants for β-Sn requires multiple creep tests under specific stress states, for single crystals along crystal principal directions. In this study, these physical creep tests are replaced with ‘virtual tests’ conducted with the developed dislocation-based crystal-viscoplasticity model. To assess the ability of the Hill-Norton finite element approach to represent dislocation creep, the finite element simulation results are compared with results of: (i) physical tests on single crystal specimens reported in the literature; and (ii) crystal-viscoplasticity modeling along many crystal orientations (beyond the fundamental calibration cases conducted along crystal principal directions). In future studies, this approach will be used for anisotropic finite element modeling of creep in polycrystalline specimens.

Creep deformation analysis of gas storage in salt caverns

The creep shrinkage of a salt cavern with constant gas pressures was studied. The gas pressures have constant values, but their positions (i.e. cavern radii) change with time, which is negligible for small deformations rather than for large deformations. Based on whether the displacement of the cavern wall is considered in boundary conditions, small and large deformation solutions for cavern creep shrinkage were developed. To validate the solutions, the analytical results were compared with the numerical results. The results of the large deformation solutions showed good agreement with the numerical results in the case of small and large deformations. However, the small deformation solutions only applied when their results were sufficiently small; otherwise, the cavern shrinkage was overestimated. In addition, the sensitivity of cavern shrinkage to factors, such as time, creep parameters, and the difference between the geostatic pressure and gas pressure (pressure difference), was discussed. In some cases, the results of the large deformation solutions showed that the cavern shrinkage grows over time at a declining rate instead of a constant rate. For caverns of rock salt with great creep parameters, such as D and n in the Norton power law, the cavern shrinkage was fast. The cavern shrinkage rate decreased as the pressure difference decreased. The minimum internal pressure obtained by the large deformation solutions was lower than that obtained by the small deformation solutions, which can help to reduce the cushion gas in salt caverns. The large deformation solutions can provide useful references for gas storage in salt caverns.

Revisiting creep test on polyethylene pipe—Data analysis and deformation mechanisms

AbstractCreep tests have long been used to determine long‐term hydrostatic strength (LTHS) for polyethylene (PE) pipe. In view that current standard requires some long‐term creep tests of over 9000 h to determine LTHS, this paper explores the possibility of using relatively short‐term creep tests for the same purpose. The study found that trend line for applied stress versus failure time changes at the failure time around 10 h, and the data trend in the low‐stress regime is consistent with that from the standard test. The trend line change is believed to be caused by the change of the involvement of amorphous and crystalline phases in PE in the deformation process, that is, sequential involvement of the amorphous phase first and then the crystalline phase in the low‐stress regime, and simultaneous involvement of both phases from the beginning in the high‐stress regime. Activation energies based on the Eyring's model and the Norton power law are determined to examine the proposed concept. Both models suggest that additional activation energy is needed for deformation in the high‐stress regime, which provides some support to the proposed concept and the possibility of using short‐term creep tests to determine LTHS.

Physical and geometrical nonlinear forced oscillations of beams

The paper presents a calculation method and the results of modeling the nonlinear forced planar oscillations of a beam. The calculation approach is based on the method of weighted residuals in the Galerkin form in combination with numerical methods of integration over time. A sequential analysis of elastic linear and geometrically nonlinear oscillations is performed and the case of irreversible deformation due to the occurrence of physically nonlinear creep strains is considered. To describe it, the Norton power law is used. Cases of hinge supported and a cantilever beam are considered. When solving the problem of a hinge supported beam, the sine system was used as the basis functions, and the Krylov functions were used for the cantilever beam’s problem. The results of numerical modeling are presented in the form of the dependence of the beam deflections on time and on the coordinate at a given point in time. The influence of geometric nonlinearity is demonstrated. The increase in deflection with time due to an increase in creep strains is analyzed.

Nonlinear damage creep model based on fractional theory for rock materials

In this study, based on the first application of Caputo fractional derivative with respect to the Mittag–Leffler and power functions, two improved time-varying viscoelastic constitutive models are presented. And the Norton power law and damage factor are applied to build a new damage function. Then the new nonlinear damage creep models 1 and 2 with time-varying viscoelasticity and time-dependent damage are proposed to describe the creep mechanical behavior, which are based on the two improved time-varying viscoelastic constitutive models. Then a series of creep experiments are conducted on the sandstone to obtain the creep experimental data. And based on the creep experimental data of sandstone and creep data of salt rock and mudstone obtained from other papers, the applicability of proposed creep models 1 and 2 is verified. The proposed creep model 2 with power function has a higher accuracy than model 1 with Mittag–Leffler function in describing creep mechanical behavior. And the predicted creep from model 2 is in good agreement with experimental data. Finally, it is shown that compared with creep model 1, model 2 needs fewer parameters to depict the creep mechanical behavior.

A nonlinear damage creep model for sandstone based on fractional theory

In this study, an improved viscoelastic model is proposed, which is based on the application of Caputo fractional derivatives of one function in regard to exponential function. This new mathematical way is effective in the construction of viscoelastic model with time dependence. And based on the type of Norton power law and damage factor, we propose a new damage model. By combining the improved viscoelastic model and proposed damage model, a nonlinear fractional damage creep model is presented to describe the creep mechanical behavior of sandstone. A series of creep experiments were conducted on the sandstone by the step loading way. The decaying, steady, and accelerating creep stage all appear in the experiments. Based on those experimental data, the fitting curves of the presented creep model are in well agreement with those of the experimental data. And the reasonability of fitting parameters also has been analyzed in detail.

A comparative study of different model families for the constitutive simulation of viscous clays

SummaryThe simulation of the viscous behavior of some clays is of high importance in many geotechnical problems. The literature offers a vast amount of constitutive models able to simulate the rate dependence observed on these materials. Although most of these models are calibrated to very similar experimental observations and share similar definitions of material parameters, some discrepancies of their response have been detected, which are related to their mathematical formulations. In this work, the causes of these discrepancies are carefully studied. To that end, four different model families are analyzed, namely, nonstationary flow surface (NSFS) models, viscoplasticity with overstress function (OVP), viscoplasticity with Norton's power law (NVP), and visco‐hypoplasticity (VHP). For the sake of a fair comparison, single constitutive models using the same set of material parameters, and following other requirements, are developed for each model family. Numerical implementations of the four resulting models are performed. Their response at different tests are carefully analyzed through simulation examples and direct examination of their constitutive equations. The set includes some basic tests at isotropic stress states and others as responses envelopes, undrained creep rupture, and an oedometer test with loading, unloading‐reloading, creep, and relaxation. The article is concluded with some remarks about the observed discrepancies of these model families.

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.

Creep constitutive equations for as-received 9Cr1Mo steel

9Cr1Mo has a wide application in petrochemical and power generation plants, prediction of its creep behavior is a major concern in these industries. In this paper, the creep behavior and creep lifetime of test samples machined from 15 years of service exposed boiler tubes in an oil refinery plant, are studied using uni-axial creep tests. Tests are carried out according to ASTM E139 at different stress and temperature levels (from 80 MPa to 190 MPa and at 550 °C, 600 °C and 650 °C). Based on the experimental data, the creep constitutive coefficients for three types of creep laws: Norton, Plrandtl and Garofalo are obtained together with Monkman-Grant and Larson Miller parameters. It has been shown that the constitutive equations in the form of Garofalo law predict the creep behavior of the material more accurately. The results show that the creep life relationships obtained based on both Garofalo and Norton power law give acceptable estimation. Also, the results are implemented in reference stress method to produce a set of loci for the optimal geometry of the tubes under combined loadings with maximum creep lifetime.