Constitutive relationships for anisotropic high-temperature alloys

A constitutive theory is presented for representing the anisotropic viscoplastic behavior of high-temperature alloys that possess directional properties resulting from controlled grain growth or solidification. The theory is an extension of a viscoplastic model that has been applied in structural analyses involving isotropic metals. Anisotropy is introduced through the definition of a vector field that identifies a preferential (solidification) direction at each material point. Following the development of a full multiaxial theory, application is made to homogeneously stressed elements in pure shear and to a uniaxially stressed rectangular block in plane stress with the stress direction oriented at an arbitrary angle with the material direction. It is shown that an additional material parameter introduced to characterize the degree of anisotropy can be determined on the basis of simple creep tests.     

Creep failure time of 20% cold-worked type 316 stainless steel

The Robinson failure criterion is examined for its accuracy to predict the creep failure time of 20% cold-worked type 316 stainless steel under uniaxial and multiaxial stresses. Observed changes of slope in the log-log plots of maximum tensile stress versus isothermal rupture life are neglected, and large errors in prediction of creep failure time are found. A failure criterion that best explains the experimental behavior of 20% cold-worked type 316 stainless steel in uniaxial and multiaxial creep conditions is developed by incorporating the effective stress responsible for crack initiation and the maximum tensile stress responsible for crack propagation into the creep failure time equation.     

Creep in phosphorus alloyed copper during power-law breakdown

During the first phase of storage, creep will take place in the copper canisters in the KBS-3 package for nuclear waste. The temperatures are below 100 °C, and the creep is well inside the power-law breakdown regime. Creep models for this situation have been developed. The analysed material is pure copper with about 50 ppm phosphorus. Constitutive equations for creep and other plastic deformation have been set up based on a generalised Norton expression and Kocks-Mecking’s model for the back stress. A model for the minimum creep rate based on fundamental principles for climb and glide has been derived. This model gives the correct order of magnitude for the creep rate in the temperature range from 400 to 20 °C without the use of fitted parameters. The creep exponent varies from 5 to 105 in this interval. The constitutive equations have also been formulated for multiaxial stress states.     

Description of creep-plasticity interaction with non-unified constitutive equations: application to an austenitic stainless steel

We present constitutive equations able to account for time independent plasticity together with creep and creep-plasticity interaction. A classical decomposition of the inelastic strain into a time independent plastic strain and a time dependent viscoplastic part is assumed. The coupling between both deformation modes (i.e. creep and plasticity) is obtained through an interaction between the plastic and viscoplastic state variables. In a first part, the capabilities of the model are described, and qualitative identifications are given in order to characterize the behaviour of the model. The practical applicability of the model is then tested, mainly using test results from the literature, but also specific data including creep, relaxation and tensile tests with various loading rates, as reported in the paper. The model is found able to discriminate between the increase of hardening produced by plasticity or creep. The effect of the loading rate on the subsequent amount of relaxation is correctly described and a good general agreement is observed between experiment and model predictions, even for complex loading paths (monotonic with temporary unloading periods, multiaxial loading paths in the stress space).     

Intergranular damage in AISI 316L(N) austenitic stainless steel at 600 °C: Pre-strain and multiaxial effects

This study is devoted to the effect of a multiaxial stress state and of pre-straining on the creep properties of an austenitic stainless steel. Creep tests on both smooth and notched specimens have been carried out on type 316L(N) steel at 600 °C. In comparison to the annealed state, pre-straining caused a substantial increase in creep lifetime but also a dramatic drop in intergranular damage resistance. The effect of a pre-strain on creep ductility was so strong that compact tension specimens in pre-strained state tested under relaxation conditions cracked, whereas specimens in annealed state were not prone to cracking. A model taking into account both pre-strain and multiaxial effects was developed and identified on the basis of local intergranular micro-cracks measurements on notched specimens. It satisfactorily predicts the results of relaxation crack propagation tests. This model may also provide a useful estimation of the relaxation cracking risk of 316L(N) as a function of pre-strain level and stress triaxiality ratio.     

Influence of prior dislocation structure on anisotropy of thermal creep of cold-worked Zr-2.5Nb tubes

Normally, creep anisotropy of hcp metals is thought to be controlled by the crystallographic texture. Here, we show that the creep anisotropy of cold-worked Zr-2.5Nb tubes is also very dependent on the anisotropic dislocation structures introduced by cold-work. The contribution of each slip system to the creep deformation of an individual grain orientation depends upon the activity of that slip system during prior cold-work. This conclusion is reached by comparing the self-consistant visco-plastic polycrystalline models with thermal creep tests performed on internally pressurized thin-wall capsules with different textures under a transverse stress of 300 MPa at 350 °C, where dislocation creep is the dominant operating mechanism. The non-uniform dislocation distributions prior to creep were derived by simulating the cold-work process of Zr-2.5Nb tubes from an Elasto-Plastic Self-Consistent (EPSC) model.     

The various creep models for irradiation behavior of nuclear graphite

Graphite is a widely used material in nuclear reactors, especially in high temperature gascooled reactors (HTRs), in which it plays three main roles: moderator, reflector and structure material. Irradiation-induced creep has a significant impact on the behavior of nuclear graphite as graphite is used in high temperature and neutron irradiation environments. Thus the creep coefficient becomes a key factor in stress analysis and lifetime prediction of nuclear graphite. Numerous creep models have been established, including the visco-elastic model, UK model, and Kennedy model. A Fortran code based on user subroutines of MSC.MARC was developed in INET in order to perform three-dimensional finite element analysis of irradiation behavior of the graphite components for HTRs in 2008, and the creep model used is for the visco-elastic model only. Recently the code has been updated and can be applied to two other models—the UK model and the Kennedy model. In the present study, all three models were used for calculations in the temperature range of 280–450 °C and the results are contrasted. The associated constitutive law for the simulation of irradiated graphite covering properties, dimensional changes, and creep is also briefly reviewed in this paper. It is shown that the trends of stresses and life prediction of the three models are the same, but in most cases the Kennedy model gives the most conservative results while the UK model gives the least conservative results. Additionally, the influence of the creep strain ratio is limited, while the absence of primary creep strain leads to a great increase of failure probability.     

Mathematical model for creep and thermal shrinkage of concrete at high temperature

Based on the existing limited test data, it is possible to set up an approximate constitutive model for creep and shrinkage at temperatures above 100°C, up to about 400°C. The model presented here describes the effect of various constant temperatures on the creep rate and the rate of aging, similar effects of the specific water content, the creep increase caused by simultaneous changes in moisture content, the thermal volume changes as well as the volume changes caused by changes in moisture content (drying shrinkage or thermal shrinkage), and the effect of pore pressure produced by heating. Generalizations to time-variable stresses and multiaxial stresses are also given. The model should allow more realistic analysis of nuclear reactor vessels and containments for accident situations, of concrete structures subjected to fire, of vessels for coal gassification or liquefaction, etc.     

Mechanical modeling of creep, swelling and damage under irradiation for polycrystalline metals

A constitutive equation of creep, swelling and damage under irradiation for polycrystalline metals applicable to structural analyses in multiaxial state of stress is developed. After reviewing microscopic mechanisms of irradiation creep and swelling, the relevant theories proposed so far from the view point of metallurgical physics and their applicability are discussed first. Then a constitutive model is developed by assuming that creep under irradiation can be decomposed into irradiation-affected thermal creep and irradiation-induced creep. By taking account of the Stress-Induced Preferential Absorption (SIPA) mechanism, the irradiation-induced creep is represented by an isotropic tensor function of order one and zero with respect to stress, which is, at the same time, the function of neutron flux and neutron fluence. The volumetric part of the irradiation-induced creep is identified with swelling. The irradiation-affected thermal creep is described by modifying Kachanov-Rabotnov theory for stress-controlled creep and creep damage by incorporating the effect of irradiation. Finally irradiation creep and swelling of 20% cold-worked type 316 stainless steel at elevated temperature are predicted by the proposed constitutive equations, and the numerical results are compared with the corresponding experimental results.     

Life prediction techniques for combined creep and fatigue

This paper contains a review of the techniques that are currently available for predicting the life of metallic materials when they fail at high temperatures under the combined action of creep and fatigue. The work concentrates on those theories that are relevant to engineering design without specifically employing fracture mechanics concepts. As a consequence this generally limits the prerequisite data to that of low cycle fatigue under high rate strain cycling conditions and monotonic creep at a given temperature. Some techniques require additional or alternative data on tensile ductility, high cycle fatigue, cyclic plasticity and creep.Each technique is appraised from the results of laboratory experiments. It is shown that life may be predicted quite reliably in one instance but not in another. Some attempt is made to provide explanations and recommendations where these anomalies occur. The range of application is also clarified. Many approaches are limited to strain controlled cycling under uniaxial stress conditions. Only a limited number may further be applied to predict life for cycles in which ratcheting occurs. At present it appears that very few techniques are available to predict the creep-fatigue life of materials under high temperature multiaxial stress states. under high temperature multiaxial stress states.     

Multiaxial creep–fatigue rules

Within the UK, a comprehensive procedure, called R5, is used to assess the high temperature response of structures. One part of R5 deals with creep–fatigue initiation, and in this paper we describe developments in this part of R5 to cover multiaxial stress states.To assess creep–fatigue, damage is written as the linear sum of fatigue and creep components. Fatigue is assessed using Miner's law with the total endurance split into initiation and growth cycles. Initiation is assessed by entering the curve of initiation cycles vs. strain range using a Tresca equivalent strain range. Growth is assessed by entering the curve of growth cycles vs. strain range using a Rankine equivalent strain range. The number of allowable cycles is obtained by summing the initiation and growth cycles. In this way the problem of defining an equivalent strain range applicable over a range of endurance is avoided.Creep damage is calculated using ductility exhaustion methods. In this paper we address two aspects: first, the effect of multiaxial stress on creep ductility; secondly, the nature of stress relaxation and, hence, accumulated creep strain in multiaxial stress fields.     

Modeling dislocation structure development and creep-swelling coupling in neutron irradiated stainless steel

Anisotropic nucleation and growth of multi-classes of dislocation loops under the combined actions of fast-neutrons and an external applied stress are considered in modeling dislocation structure development in metals and alloys. The stochastic nature of the nucleation kinetics is formulated via the Fokker-Planck equation. The strain derived from the climb of the anisotropic dislocation structure is separable into volumetric and deviatoric components, corresponding respectively to swelling and creep. The creep contribution resulting from the development of the stress-induced dislocation anisotropy is found to be very significant and exhibits a strong correlation with swelling. For stainless steel, our model explains very well the complex deformation behavior observed in a wide variety of in-reactor experiments.     

Microstructure dependence of irradiation creep and growth of zirconium alloys

Zirconium alloys exhibit both irradiation creep and irradiation growth. The mechanisms governing these processes determine the sensitivity of their rates to variables such as temperature, stress, neutron flux, and microstructure. In this paper I compare the observed relationships between creep and growth of zirconium alloys and dislocation density, grain structure, and crystallographic texture with the predictions of theoretical models. The approximately linear dependence of growth on dislocation density and its dependence on texture and grain shape are consistent with a model in which there is a net flux of interstitials to edge dislocations, and vacancies arrive at grain boundaries by pipe diffusion down screw dislocations. The insensitivity of irradiation creep to dislocation density and its dependence on texture are consistent with a climb-plus-glide model in which dislocations climb out of sub-boundaries and glide across the subgrains.     

Influence of variations in creep curve on creep behavior of a high-temperature structure

It is one of the key issues for a high-temperature structural design guideline to evaluate the influence of variations in creep curve on the creep behavior of a high-temperature structure. In the present paper, a comparative evaluation was made to clarify such influence.The evaluation results showed that, in almost all cases of a creep behavior pattern in which creep strain accumulated during a stress cycle caused significant relaxation of the corresponding deformation-controlled stress in the following cycles, the variations in creep behavior with the creep curve were qualitatively similar to those in fundamental creep properties. On the other hand, in many cases of another creep behavior pattern in which creep strain accumulated during a cycle doesn't cause the significant stress-relaxation in the following cycles, the variations in creep behavior for earlier cycles are different from those in fundamental creep properties. Even in these cases, however, those get qualitatively similar after several cycles when their stress-time histories are stabilized.Additional consideration was given to the influence of the relationship between creep rupture life and minimum creep rate, i.e., the Monkman-Grant's relationship, on the creep damage evaluation.The consideration suggested that the Monkman-Grant's relationship be taken into account in evaluating the creep damage behavior, especially the creep damage variations. However, it was clarified that the application of the creep damage evaluation rule of ASME B.&P.V. Code Case N-47 to the “standard case”which was predicted from the average creep property would predict the creep damage on the safe side.     

Coupled thermal structural analysis of LWR vessel creep failure experiments

Considering the hypothetical core melt down scenario for a light water reactor (LWR) the failure mode of the reactor pressure vessel (RPV) has to be investigated to determine the loadings on the containment. The failure of reactor vessel retention (FOREVER)-experiments, currently underway, are simulating the thermal and pressure loadings on the lower head for a melt pool with internal heat sources. Due to the multi-axial creep deformation of the vessel with a non-uniform temperature field these experiments are an excellent source of data for validation of numerical creep models. Therefore, a finite element (FE) model has been developed based on a commercial multi-purpose code. Using the computational fluid dynamics (CFD) module the temperature field within the vessel wall is evaluated. The transient structural mechanical calculations are performed using a new numerical approach, which avoids the use of a single creep law employing constants derived from the data for a limited stress and temperature range. Instead of this a three-dimensional array is developed where the creep strain rate is evaluated according to the values of the actual total strain, temperature and equivalent stress. Care has to be exercised performing post-test calculations particularly in the comparisons of the measured data and the numerical results. Considering the experiment FOREVER-C2, for example, the recorded creep process appears to be tertiary, if a constant temperature field is assumed. But, small temperature increase during the creep deformation stage could also explain the observed creep behavior. Such considerations provide insight and better predictive capability for the vessel creep behavior during prototypic severe accident scenarios.     

Transferability of time-dependent data and constitutive equations to cases of multiaxial load

The transferability of materials data obtained in uniaxial tests to multiaxial load conditions depends on the nature of the multiaxial load and the type of semi-finished product. Whereas in the stationary creep range there is good agreement in the deformation behaviour of rod and tube material, tubes with slight wall thicknesses display considerable strengthening in the primary creep range. Lifetime prediction on the basis of the deviatoric stress alone is not sufficient. The time course of the largest elongation component must rather be considered in determining the specimen life.     

Interconnection between irradiation creep and interstitial loop formation in fcc metals

The creep strain resulting from stress-assisted, preferential loop nucleation was — following an idea proposed by Lewthwaite —calculated for both the uniaxial and the biaxial stress state. Taking the texture into account the calculated creep is about one half as large as the observed creep. This small discrepancy can be reduced further by assuming that more than just three interstitials constitute a nucleus and/or by also taking into account the stress-assisted loop growth. A contribution to the creep strain may further arise from stress-assisted, anisotropic ‘swelling’. It is concluded that these processes might account for all or a larger part of irradiation-induced creep.