Norton Power Law Research Articles

Elasto-plastic phase field modelling of oxidation of zirconium alloys

A phase field model is proposed to study the oxidation behaviour of zirconium alloys at high temperatures. The oxidation diffusion process is described by the time-dependent Cahn–Hilliard equation, and the elasto-plastic deformation is predicted using an elastic-perfectly plastic model and the Norton power law creep equation. The corresponding finite element formulations are derived using the Galerkin approach and the Newton–Raphson iteration scheme. The phase field modelling results are validated using a one-dimensional oxidation case, and the results are in good agreement with published experimental results. Some numerical examples are carried out, and the influences of an initial rough surface and microvoids on the mechanical behaviour of the oxide layer are analysed and discussed in detail. The mechanical failure mechanisms of the protective oxide layer are discussed and compared with related experimental observations.

Microstructure evolution and fracture mechanism of a novel 9Cr tempered martensite ferritic steel during short-term creep

In this work, the microstructure evolution and fracture mechanism of a novel 9% chromium tempered martensite ferritic steel G115 were investigated over the temperature range of 625–675°C using uniaxial creep tests. The creep curves consist of a primary transient stage followed by an apparent secondary stage, and an accelerated tertiary creep regime. The relationship between the minimum creep rate and the applied stress followed Norton's power law. Based on the EBSD analysis, there were no obvious textural features formed after creep deformation, and with the increase in creep time, the number of subgrains slightly increased, and then sharply increased, indicating dynamic recrystallization (DRX) occurs after creep deformation. In addition, three types of precipitates can be observed after creep deformation: W-rich Laves phase, Nb-rich MX, and Cu-rich precipitates. The Nb-rich MX with a square shape and Cu-rich precipitates with an ellipsoidal shape remain very stable. However, the W-rich Laves phases distributed mainly on the grain boundaries have rod-like, chain-like, and bulky shape, which are coarsened significantly. Representative fractographs of the G115 steel after creep deformation exhibit significant necking with an elliptical shape. A dense array of deep and equiaxed dimples appear in the central region under the tested creep conditions. Ductile fracturing is the dominant fracture mechanism during short-term creep deformation.

CAF—a simplified approach to calculate springback in Al 7050 alloys

Aeronautical industries have looked for news fabrication processes to reduce the costs and the waste of the material during milling operations. One of these new processes is the creep age forming (CAF). The purpose of this work was based on springback analysis during the forming of single- and double-curved sheets of Al 7050 by the process of CAF. A simplified model based on a Norton power law was used in order to, alongside with a program of finite elements, allow the calculation of springback after the process. The experimental verification was carried out. Beside the springback results, other results were a decrease in the Young’s module of 11.5% in the creep aging temperature, in relation with ambient temperature during the CAF process and the variation of aluminum alloy’s yield stress during the process. The springback effect increased on the basis of aging time and then decreased due to intense aging of the alloy; an ideal time of 8 h is estimated for the Al 7050 alloy to carry out the CAF process. The research is limited to the use of the simplified model and its applicable results to Al 7050 alloys. Al 7050 is aligned with its vast use in the aeronautical industry, and the simplified model’s application may rapidly offer the necessary values of springback for the tooling project.

Study of the upper die clamping conditions in the small punch test

A parametric study of small punch tests on miniaturized discs under constant deflection rate and constant force has been performed to study the influence of various upper die conditions on the test results. A comparison between the experimental results and the simulations by means of the finite element method is presented, under different clamping conditions. Heat resistant steel P22 has been selected for this investigation. The elasto-plastic behaviour of the disc was described by multilinear isotropic hardening. Norton power-law and exponential creep constitutive relationships have been applied in the ANSYS FE model of the SPT arrangement under creep conditions. The investigation confirms relatively small influence of the upper die conditions for both types of small punch tests for the steel under investigation.

Thermal Stability of Residual Stresses in Ti-6Al-4V components

The need for light weight design while maintaining a high safety is essential for many components, especially in the aircraft industry. Therefore, it's important to consider every aspect to reduce weight, improve fatigue life and maintain safety of crucial components. Residual stresses are a major factor which can positively influence components and fulfil all three requirements. However, due to the inconstancy of the behaviour of residual stresses during the life time of a component, residual stresses are often neglected. If the behaviour of residual stresses could be described reliably over the entire life time of a component, residual stresses could be taken into account and components could be optimized even further. Mechanical and thermal loads are the main reason for relaxation of residual stresses. This work covers the thermal stability of residual stresses in Ti-6Al-4V components. Therefore, exposure tests at raised temperatures were performed on specimens with different surface conditions. Residual stresses were measured by x-ray diffraction before and after testing. Creep tests were also carried out to describe the creep behaviour and thereby the ability for residual stress relaxation. A correlation between the creep rate and amount of relaxed stress was found. The creep behaviour of the material was described by using a combination of the Norton Power law and the Arrhenius equation. The Zener-Wert-Avrami model was used to describe the residual stress relaxation. With these models a satisfying correlation between measured and calculated data was found. Hence, the relaxation of residual stresses due to thermal load was described reliably.

Creep deformation and rupture behaviour of thermal aged P92 steel

Creep deformation and rupture behaviour of thermal aged P92 steel have been investigated over a stress range of 110–130 MPa at 923 K. Laves phase was observed in the thermal aged steel. Laves phase predominantly coarsens towards the intra-lath region assisted by substructure within the lath region. The stress dependence of minimum creep rate followed Norton's power law of creep with comparable apparent stress exponent value in the thermal aged, and normalised and tempered steels. The creep rate and creep rupture lives of both the steels were relatively comparable under lower stress level. The steel obeyed Monkman–Grant and modified Monkman–Grant ductility relationships. Enhanced precipitation of Laves phase has been observed at the necked region in both the conditions of the steel, which experienced higher plastic flow. Thermal aging of P92 steel has not significantly influenced the creep strength under longer creep duration.

Creep deformation and rupture behavior of Alloy 617

A series of creep data was obtained from creep tests at different applied stresses at the temperatures of 850°C, 900°C, and 950°C for Alloy 617, which is a leading candidate material for high-temperature components in Gen-IV nuclear reactor systems. The creep deformation and rupture behavior were investigated in terms of Norton's power law, Monkman–Grant relation (MGR), modified Monkman–Grant relation (MMGR), creep damage tolerance factor (λ), Zener–Hollomon Parameter (Z), and fracture behavior. Alloy 617 did not exhibit textbook creep behavior and revealed somewhat differences from typical heat resistant steels. The MMGR provided improved correlation between creep rate and rupture life in Alloy 617. The Z parameter obeyed a good agreement for a function of Z=2.30×1033 (σ/E)5.87, and the same creep mechanism was operative within the ranges tested in the present study. The value of λ for Alloy 617 was found to be 2.40, and this was in agreement with materials exhibiting typical cavitation damage. The creep failure analysis revealed a dominant intergranular fracture mode, which proceeds via initiation, linking, and incorporation of the cavities.

An assessment of creep deformation and rupture behaviour of 9Cr–1.8W–0.5Mo–VNb (ASME grade 92) steel

Creep deformation and rupture behaviour of 9Cr–1.8W–0.5Mo–VNb steel have been investigated at 873K, 923K and 973K over a stress range of 80–220MPa. The absence of clear primary creep regime and prolonged secondary stage of creep deformation have been noticed under lower stress level at 973K. The variation of minimum creep rate with applied stress obeyed Norton's power law of creep. The apparent stress exponents of 15.2, 12.3 and 5.8, and apparent activation energy of 619kJ/mole have been estimated for creep deformation of the steel. The apparent stress exponents and activation energy have been rationalised on the basis of threshold stress. The threshold stress values of 137.5MPa, 83.3MPa and 29.7MPa were obtained at 873K, 923K and 973K respectively. The threshold stress compensated true stress exponent of 4 and true activation energy of 244kJ/mole, and threshold stress normalised by Orowan stress confirms that the lattice diffusion assisted localised climb of dislocation is the rate controlling of creep deformation in the steel. The steel obeyed Monkman and modified Monkman–Grant relationships. Damage tolerance factor of 6 in the steel demonstrates that the microstructural degradation such as coarsening of precipitates and subgrain structure is the dominant creep damaging mechanism in the steel.

Creep deformation and rupture behavior of CLAM steel at 823 K and 873 K

China Low Activation Martensitic (CLAM) steel is selected as the candidate structural material in Fusion Design Study (FDS) series fusion reactor conceptual designs. The creep property of CLAM steel has been studied in this paper. Creep tests have been carried out at 823K and 873K over a stress range of 150–230MPa. The creep curves showed three creep regimes, primary creep, steady-state creep and tertiary creep. The relationship between minimum creep rate (ε̇min) and the applied stress (σ) could be described by Norton power law, and the stress exponent n was decreased with the increase of the creep temperature. The creep mechanism was analyzed with the fractographes of the rupture specimens which were examined by scanning electron microscopy (SEM). The coarsening of precipitates observed with transmission electron microscope (TEM) indicated the microstructural degradation after creep test.

FE modeling of creep in linear and non-linear FGM cylinder under internal pressure

Purpose – The purpose of this paper is to investigate creep in an internally pressurized thick-walled, closed ends cylinder made of functionally graded composite, having linear and non-linear distribution of reinforcement, using finite element (FE) analysis. Design/methodology/approach – FE-based Abaqus software is used to investigate creep behavior of a functionally graded cylinder. The cylinder is made of composite containing linear and non-linearly varying distributions of reinforcement along the radius. The creep behavior has been described by Norton's power law. The creep stresses and strains have been estimated in linear and non-linear functionally graded materials (FGM) cylinders and compared with those estimated for a similar composite cylinder but having uniform distribution of reinforcement. Findings – The radial stress in the composite cylinder is observed to decreases over the entire radius upon imposing linear or non-linear reinforcement gradients. However, the tangential stress in the cylinder increases near the inner radius but decreases toward the outer radius, on imposing linear or non-linear reinforcement gradients. The creep strains in the FGM cylinders are significantly lower than those observed in a uniform composite cylinder. Originality/value – The creep strains in an internally pressurized functionally graded thick composite cylinder could be reduced significantly by employing non-linear distribution of reinforcement along the radial direction.

Biaxial creep deformation behavior of Fe–14Cr–15Ni–Ti modified austenitic stainless steel fuel cladding tube for sodium cooled fast reactor

Twenty percent cold worked Fe–14Cr–15Ni–Ti modified austenitic stainless steel is used as the cladding tube material for the fuel pins of the Prototype Fast Breeder Reactor in India. Biaxial creep properties of the tubes have been studied at 973K by carrying out creep tests by internally pressurizing the tubes. Hoop and axial components of creep strain were measured and found to be significantly different. For a given gas pressure, steady state hoop rate was higher than the axial rate. Steady state hoop and axial creep rates followed Norton's power law with the same stress exponent n=7. Steady state hoop rates determined from biaxial creep tests agreed with the steady state creep rates determined from uniaxial creep tests. For a thin walled closed tube under internal pressure, significant axial deformation along with hoop deformation is indicative of anisotropic deformation of the material.

Analysis of Creep Behavior of Alloy 617 for Use of VHTR System

Alloy 617 is the one of the leading candidate materials for intermediate heat exchangers (IHX) application of a Very High Temperature Reactor (VHTR) system for economic production of electricity and hydrogen. Creep rupture data for Alloy 617 were obtained from a series of creep tests with different applied stresses at 850oC, 900oC and 950oC. On the basis of a systematic analysis of the experimental data, creep behaviour for Alloy 617 was analyzed using various creep relationships and laws, such as Norton's power law, Monkman-Grant Relationships (MGR), Modified Monkman-Grant Relationships (MMGR), creep damage tolerance factor λ, and Zener-Hollomon Parameter (Z), and the creep constants used in each equation were reasonably determined. The creep curve of Alloy 617 was somewhat different from those of typical heat-resistance steels, and it did not exhibit a textbook creep curve. The secondary creep stage is unclear, and it appeared to exhibit a predominantly accelerated creep rate from the start of a tertiary creep stage in short time ranges. The creep deformation was dominantly developed from the formation and growth of cavity in creep damage process. The MMGR appeared to be more narrowed in data scattering than the MGR, and it followed well a straight line of m ≅ 1.0 as m=0.97. In the plot of the Z parameter vs. stress, it obeyed a straight line with a slope of n’=5.87 regardless of the three different temperatures. It would be inferred that the same creep mechanism was operative within the present stress and temperature ranges, and the creep damage tolerance factor of Alloy 617 was found to be 2.40.

Modeling and Experimental Study of Long Term Creep Damage in Austenitic Stainless Steels

Austenitic Stainless Steels (SSs) are presently being investigated as appropriate candidates for structural components for the future Generation IV nuclear reactors. Austenitic SSs of different grades will operate at high temperature and suffer low stress loading for decades. At the laboratory, austenitic SSs have been subjected to creep tests at various stresses and temperatures between 500°C to 700°C, up to nearly 50·103h. Interrupted creep tests show an acceleration of the reduction in cross-section only during the last 15% of creep lifetime which may be called macroscopic necking. The modeling of necking using a modified Norton power-law allows lifetime predictions in agreement with experimental data up to a few thousand hours only. And the experimental results show that, the extrapolation of the 'stress lifetime curves obtained at high stress leads to large overestimations of lifetimes at low stress. After FEGSEM observations, these overestimates are mainly due to additional intergranular cavitation along grain boundaries. The modeling of cavity growth by vacancy diffusion along grain boundaries coupled with continuous nucleation proposed by Riedel has been carried out. Lifetimes for long term creep are rather correctly predicted with respect to experimental lifetimes. The lifetime curves predicted by either the necking model or the creep cavity one cross each other, defining transition times of five to ten thousand hours for temperatures between 600°C and 700°C, in agreement with experimental curves.

Small Punch Creep Studies for Optimization of Nitrogen Content in 316LN SS for Enhanced Creep Resistance

Small punch creep (SPC) studies have been carried out to evaluate the creep properties of 316LN stainless steel (SS) at 923 K (650 °C) at various stress levels. The results have been compared with uniaxial creep rupture data obtained from conventional creep tests. The minimum deflection rate was found to obey Norton power law. SPC rupture life was correlated with uniaxial creep rupture life. The influence of nitrogen content on the creep rupture properties of 316LN SS was investigated in the range of 0.07 to 0.14 wt pct. SPC rupture life increased and the minimum deflection rate decreased with the increase in nitrogen content. The trends were found to be in agreement with the results obtained from uniaxial creep rupture tests. These studies have established that SPC is a fast and reliable technique to screen creep properties of different experimental heats of materials for optimizing the chemical composition for developing creep-resistant materials.

Application of small punch creep testing to a thermally sprayed CoNiCrAlY bond coat

High velocity oxy-fuel thermal spraying was used to prepare free-standing CoNiCrAlY (Co–31.7% Ni–20.8% Cr–8.1% Al–0.5% Y (wt%)) bond coat alloy samples approximately 0.5mm thick. Creep tests were conducted at 750°C on these samples using a small punch (SP) creep test method. The samples were characterised before and after creep testing using scanning electron microscopy with electron backscatter diffraction (EBSD). EBSD revealed a two phase fcc γ-Ni and bcc B2 β-NiAl microstructure with grain sizes ~1–2μm for both phases, which did not change significantly following testing. The constant temperature SP test data were characterised by a minimum creep strain rate, ε̇min, and a total time to failure, tf, at different applied stresses. The data are fitted to conventional power law equations with a stress exponent for creep close to 8 in the Norton power law and between 7 and 10 in the Monkman–Grant creep rupture law. Creep rupture was predominantly due to creep cavitation voids nucleating at both the γ–β interphase boundaries and the γ–γ grain boundaries leading to final failure by void linkage. However, rupture life was influenced by the quantity of oxide entrained in the coating during the spray deposition process.

Creep Deformation and Rupture Behaviour of P92 Steel at 923 K

Abstract Creep deformation and rupture behaviour of P92 steel has been examined at 923 K for stresses ranging from 75 to 150 MPa. The steel exhibited well defined primary, secondary characterized by minimum creep rate and prolonged tertiary creep stages. The stress dependence of minimum creep rate obeyed Norton's power law and exhibited distinct stress regimes characterised by separate values of stress exponents in low and high stress regimes. Similarly, the stress dependence of rupture life also obeyed power law and displayed two stress regimes with separate stress exponent values. The steel displayed decrease in creep ductility with increase in rupture life in the low stress regime and followed generalised Monkman-Grant relation interrelating minimum creep rate and rupture life. Modified Monkman-Grant relation has been found to be valid for the steel. Fractographic examination indicated dominance of transgranular fracture on the fracture surfaces of tested specimens.

Transient and residual stresses in large castings, taking time effects into account

Casting of large scale steel and iron parts leads to long solidification and cooling times. Solid mechanical calculations for these castings have to take the time scale of the process into account, in order to predict the transient and residual stress levels with a reasonable accuracy. This paper presents a study on the modelling of the thermo-mechanical conditions in the cast material using a unified approach to describe the constitutive behaviour. This means a classical splitting of the mechanical strain into an elastic and an inelastic contribution, where the inelastic strain is only formulated in the deviatoric space in terms of the J2 invariant. At high temperatures, creep is pronounced. Since the cooling time is long, the model includes a type of Norton's power law to integrate the significant contribution of creep to the inelastic strains. At these temperature levels, annealing effects are also dominant and hence no hardening is modelled. However, at intermediate and lower temperature levels, hardening is more pronounced and isotropic hardening is considered. Different hardening models have been studied and selected based on their ability to describe the behaviour at the different temperature levels. At the lower temperature levels, time effects decrease and the formulation reduces to a time independent formulation, like classical J2-flow theory. Several tensile and creep experiments have been made at different temperature levels to provide input data for selecting the appropriate contributions to the material model. The measurements have furthermore been used as input for extracting material data for the model. The numerical model is applied on different industrial examples to verify the agreement between measured and calculated deformations.

Creep of Sn-9Zn as measured by a bending test method

A bending creep setup was introduced and its feasibility to evaluate creep behavior of materials was experimentally verified on pure Sn. Thereby, the creep of eutectic Sn-9Zn was studied. It was identified that high and low stress regimes exist in the creep of Sn-9Zn, and in both stress regimes, the data obtained well fit the modified equation of Norton power law. Parameters n and Q in the modified Norton equation obtained are very closed to those reported by other researchers by means of conventional tensile creep tests. The mechanisms governing the creep deformation of Sn-9Zn in both stress regimes were also discussed.

Modelling and experimental study of the tertiary creep stage of Grade 91 steel

This article addresses experimental studies and analytical simulations of the tertiary creep stage of Grade 91 steel tested at various stresses and temperatures between 500°C (up to 160 × 103 h) and 600°C (up to 94 × 103 h). The strain rate increases after its minimum mainly because of the softening of the material which microstructure evolves strongly during creep deformation. An interrupted creep test shows that necking significantly affects the acceleration of the reduction in cross-section only during the last 10% of the creep lifetime. The Hoff model based on homogeneous reduction of cross-section correctly predicts lifetimes only for high applied stress. The Hart necking model using the Norton power-law allows fair predictions of lifetimes up to 60 × 103 h at 500°C. The necking model using a modified Norton power-law combined with a material softening term allows predictions of lifetimes for all creep tests, differing from the experimental results by less than 50%, which is consistent with the experimental scatter. The evolution of the cross-section predicted by this model is in agreement with measurements carried out during the interrupted creep test. Two prediction rules for the lifetime prediction are deduced from the necking model that takes into account the material softening. For a large number of tempered martensitic steels, these two criteria bound the experimental lifetimes up to 200 × 103 h at 500–700°C.

Creep Properties of High Cr Resisting Weld Metal Alloyed with Co

A 9-12% Cr ferritic steel weld metal containing 1% Co partially substituted for nickel was prepared by submerged arc welding processing(SAW). The microstructures and creep properties of the weld metal have been investigated. The microstructure exhibited a fully tempered martensitic structure free of δ-ferrite. The creep properties of the obtained weld metal are inferior to that of the P92 base metal at 600 and 650 . The values of A and n for weld metal in Norton power law constitution at 650 are 1.12×10-21 and 8.14, respectively.