Multiscale-constraint-based Model to predict uniaxial/multiaxial creep damage and crack growth in 316-H steels

1K. Nikbin , 2S. Liu

1Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK

2ECUST University, Shanghai 200237, China

Abstract

A new failure ductility/multiscale constraint strain-based model to predict creep damage, rupture and crack growth under uniaxial and multiaxial conditions is developed for 316H Type stainless steels by linking globally uniform failure strains with a multiaxial constraint factor. The model identifies a geometric constraint and a time-dependent local constraint at the sub-grain level. Uniaxial and notched 316H steel as-received and pre-compressed data at various load levels and temperatures with substantial scatter were used to derive the appropriate constitutive equations by using the proposed empirical/mechanistic approach. Constrained hydrostatic development of creep damage at the sub-grain level is assumed to directly relate to the uniform lowerbound creep steady state region of damage development measured at the global level. Uniaxial and notched bar rupture at long terms is predicted based on the initial short-term creep or a representative tensile strength and a multiaxial constraint factor. The model is consistent with the well-known NSW remaining multiaxial ductility creep crack growth model which predicts crack growth bounds over the plane strain/stress states. This model, therefore, unifies the creep process response over the whole range of uniaxial, notched and crack growth processes which is extremely consequential to simple long term failure predictions of components at elevated temperatures.

Keywords: Uniaxial, Multiaxial, Creep, Damage, Cracks, Constraint, Ductility, Strain

* Corresponding author.

Email address: k.nikbin@imperial.ac.uk (K. Nikbin)

Nomenclature

a', b, A’, A”, α,

material constants

a

crack extension

initial crack growth rate

steady state crack growth rate

A

creep stress coefficient in power law creep

C*

steady state creep fracture mechanics parameter

D’

Material constant related to the failure strain

eQ/RT

the temperature activation terms

h

constraint parameter (as the ratio between mean and equivalent stress)

ho

value of h under plane stress state

normalised h from plane stress ho

In

dimensionless function of n

n

creep stress exponent in power law creep

rc

creep process zone

ti

crack initiation time

tr

time to rupture in uniaxial creep test

1

principal stress

m

mean stress

e

von Mises stress (MPa)

t

failure stress at time t

to

upper bound rupture stress at short term

f

uniaxial failure strain

fMG

MG uniaxial failure strain

max

upper shelf ductility

min

lower shelf ductility

multiaxial failure strain

MG multiaxial failure strain

multiaxial strain factor,

RA

rupture failure strain reduction in area

creep strain rate

creep strain rate at which ductility is equal to (max+min)/2

AR

Asreceived material

AGR

Advanced Gascooled Reactor

CCG

creep crack growth

CDM

continuum damage model

HAZ

heat affected zone

MG

MonkmanGrant strain

MSF

Multiaxial Strain Factor

NSW

Nikbin, Smith and Webster model

PC

Precompressed material

Introduction

Higher temperatures and pressures are required for more efficient energy usage in modern industries in future decades. Creep failures is likely to further dominate in components operating at elevated temperatures [1]. Quantification and understanding of creep damage have been a longstanding aim in creep life assessment and prediction of failures in components [2][2]. Though many creep stress/strainbased models [6] have been proposed in uniaxial rupture predictions for the multiaxial creep behaviour, the problem has not been fully dealt with. The two different approaches for multiaxial creep damage are the multiaxial ductilitybased models [9] and the continuum damage models (CDM) [12]. Due to complex constitutive relationships and many variables in creep damage predictions, the CDM models are difficult to implement in an industrial code for life assessment. Furthermore, the CDM models do not have an explicit explanation for multiaxiality and therefore are not able to predict long term failure response from accelerated short term tests. In addition, large creep data scatter also makes it impractical to predict long term failure behaviours in components using models that need many input variables.

In this paper a simplified ductility exhaustion model is developed which links two levels of constraint, based on the multiaxial response of the geometry and a time dependent local constraint due to subgrain microstructure microstructural anomalies and local materials property variations which allow the development void growth under local multiaxial stress states. The appropriate creep strain is at first identified for creep life at low stresses where the secondary creep usually dominates, and diffusional processes operate. The minimum strain rate prevails in long term tests while the primary and tertiary creep are limited. The uniform failure strain in this steady state creep region is called the MonkmanGrant (MG) failure strain [18]. The MG failure strain in isotropic materials is an inherent measure of the strain and damage accumulating uniformly with time at the subgrain micro scale [9]. From this understanding a simple and practical method is proposed to predict creep damage, rupture and crack related behaviours by using a failure ductilitybased approach linked to local time dependent constraint. The model is presented and verified using tensile strength and failure strain as the important variables derived from uniaxial and multiaxial creep tests. By using a parametric approach using a large dataset for 316H steel with substantial scatter it is shown that long term creep damage under uniaxial, notched bars and crack imitation and growth in fracture mechanics geometries can be predicted.

Micro Damage Growth Models

A general understanding of stress level and stress state on creep ductility is related to void growth and coalescence. Numerous methods to describe the effect of cavity growth mechanism under multiaxial stresses have been proposed [9] previously. In effect the models suggest that voids grow faster under hydrostatic environment leading to lower failure ductility.

Various void growth models have been proposed to derive the multiaxial failure strain based on uniaxial failure strain, such as Cocks and Ashby [10], Rice and Tracy [9] and Spindler [11] models. The ratio of multiaxial strain to uniaxial strain is denoted as with respect to triaxial factor h in those models which have been widely used to describe creep damage and rupture [9]. These relationships are presented in Fig. 1 based the following equations below

(1)

(2)

(3)

where n is the creep exponent in the Norton’s creep law, is multiaxial failure strain and is the uniaxial failure strain and multiaxial strain factor is given as . The uniaxial failure strain is usually taken as the elongation or area reduction RA. In fact, the MG strain is the proper measure of failure strain in these models since the constrain measure of strain at the global level correlated solely with development of damage at the subgrain level. From these equations, it is seen that an inverse relationship exists between the multiaxial strain factor (MSF) and triaxiality factor h. This means that an increasing h at the microscale produces the localized creep damage which leads to material failure at lower strains.

For most engineering materials, the creep exponent n in Norton’s creep law is usually at a range of 5 to 15. Then Eq. (1) can be given in an approximation relationship [4] as

(4)

This lower bound equation removes the sensitivity to n. Considering the above equations, the stress state at the local microscale level within the global geometry can be characterized by the local constraint term h and the MG failure strain assuming uniform damage development under uniaxial conditions and local damage development in multiaxial conditions. Using this assumption, it is argued that a geometric multiaxial stress state which is time independent during steady state creep side by side with a continually varying time dependent multiaxial stress state at the microstructural level. Both these effect failure times under creep. Based on this argument at the microstructural level the multiaxial failure strain factor (MSF) as a function of constraint factor h in the above models can be expressed in a general term as:

(5)

where and are the multiaxial and uniaxial MG failure strains in void growth models, respectively. The MG failure strain fMG at different temperatures are appropriately described by

(6)

where tr is the rupture time, A and n are material constants in Norton’s creep law, eQ/RT is the temperature activation term. The MG failure strain fMG is shown to be a direct measure of local strains in creep diffusion process and can be regarded as the measure of local creep damage at microstructural level.

Fig. 1 shows the relationship for the normalised multiaxial ductility versus h in Eqs. (1)(3). The values of is shown in the extremes to be in a range of 1/30 to 1 [21] under plane strain and stress conditions. However, for most engineering components in practical circumstances, the noramalised multiaxial ductility has been shown to be in the range 0.1 ≤ ≤ 1 [20]. Under plane stress condition where plasticity prevails, the normalised multiaxial ductility equals 1 and then the constraint factor h is 1/3 according to Eqs. (1)(3). Fig. 2 shows the relationship of versus the nomalised constraint factor ho/h normalised by ho =1/3 which shows the normalized ductility seems to be relatively insensitive to.

A simple relationship for as function of can be given Fig. 2 as:

(7)

where a and b are material constants. The mean fit to relationships between and in Fig. 2 shows that a is near unity and b <<1. Then, a simplified equation can be expressed as

(8)

suggesting parity between the two normalised variables consisting of a measurable variable of uniform strain related to a constraint factor describing the local subgrain stress state. This is consistent with the inverse relation of constraint and remaining multiaxial ductility derived analytically by several models shown in Fig. 1. Clearly the local constraint cannot be simply calculated as it involves a random process estimation across the microstructure. However, based on the model in Eq. 4 and the parity relation in Eq. 8 there is a direct interchange between measured strain and constraint and are interchangeable. An upper/lowerbound constraint level based on the upper/lower shelf ductility can therefore be identified from short to long term creep data. These bounds could possibly be theoretically and analytically explained but for the present remit an experimental approach is adopted using a large database.

Fig. 1. Range of versus h for several models and creep properties.

Fig. 2. The normalised ductility versus (=ho/h) taken from models in Fig. 1.

Analysis of Creep Data

Substantial numbers of experimental tests of uniaxial bars at short and longterm data for 316H have been collected from various databases [3] over a wide range of applied stresses and temperatures. The level of scatter which is normal in such tests is an indication of the unknowns and uncertainties that need to be considered for predictive modelling purposes. The effects of stress state and stress level on the creep ductility have been investigated in detail for different models [24]. Based on the stressdependent creep ductility and creep strain rate, both creep initiation and crack growth behavior have been numerically predicted for various fracture specimens [25] and notched bars [32].

Fig. 3 shows the uniaxial 316H data over a temperature range of 550700 oC. It is presented in terms of failure strain normalized by extrapolated uppershelf failure strain at time zero and rupture stress normalized by the yield stress at the appropriate temperature. Generally, given the level of scatter in the data, the data still suggests a lower/upper shelf with a transition region. The shortterm rupture mechanism, which occurs at or above the yield stress, is controlled by plastic hole growth whilst at the lower stress level diffusioncontrolled cavity growth [33] dominates which is reflected by a decreasing creep failure strain at the transition region. At very low stress levels, the constrained cavity growth model [34] predicts that the creep failure strain is independent of stress level, leading to a lower shelf.

The data at 550 oC, which is the normal operating temperatures for the AGR plant, is highly scattered mainly short term and in the region of the yield stress levels. This is partly since the creep activation starts to drop substantially from this temperature level and therefore higher loads are necessary to induce creep cracking. There are data for this temperature at very low stresses which are not in the public domain [3]. Therefore the extrapolative approach assumed to predict the lowershelf strains at 550 oC using the upper/lower/transition regions data from higher temperature tests is speculative to a certain degree. However, it has been shown to be consistent with long term data not available in the public domain [25].

Fig. 3 shows the normalised failure strain data for four temperatures highlighting substantial scatter. Each temperature set has been fitted to a mean Fermi’s function [35] shown in Eq. (9). The variation in creep failure ductility with applied loads can therefore be assumed to be a function of the creep strain rate, using:

(9)

where and are the upper and lower shelf ductility. denotes the creep strain rate at which the ductility is equal to the average of the upper ductility and the lower ductility: . α is a constant set at 0.9 in this case, derived by nonlinear data fitting. This is shown in a normalised form in Fig. 3 where it can be seen for the mean fit that in general there is a factor of 10 between the upper and lower shelves regardless of temperature. Clearly the scatter in the data has further implications with respect to the accuracies of the fitting procedure. However, for the present purpose the data is assumed to support the trends described by Eq. (9).

X10

Fig. 3. Fitted functions for the normalised strain and stress level versus failure strain for 316H at various temperatures with data taken from [25].

(a) (b)

(c)

Fig. 4. Profiles of ruptured uniaxial bars for 316H at 550℃ (a) short term tests <2000 h and (b) long term tests 20,000h and (c) long term tests at 700℃ [3].

Microstructural Constraint Criteria Model

To highlight the physical differences between the lower/upper shelves failures, example microstructure profiles of rupture bars for 316H at 550℃ and 700℃ are shown in Fig. 4. Reduced failure strains and final necking occur as test time increases which means a reduction in failure strain. The creep ductility reaches to a lower shelf at low load level and longer time with intergranular failure mode in control. In a similar manner, the creep crack growth morphology of C(T) specimens for 316H at short term and longterm tests are shown in Fig. 5. Large crack tip deformation occurs for short term creep crack growth tests and ductile intergranular fracture dominates whilst brittle intergranular facture behaviour and limited crack tip deformation dominate in long term crack growth tests. This behaviour implicates constraint at the subgrain and global level as the dominant criteria for creep damage and cracking. Thus the paper focuses on using the relation between ductility and micro constraint as the driving force in the failure modes in these steels to predict longterm creep failure.

(a) (b)

Fig. 5. Creep crack growth of C(T) specimens in relatively (a) short term, 2000 h and (b) long term test >20,000 h for 316H steel at 550℃ [36].

The model identifies, therefore, the effect of time and material dependent metallurgical constraint at the subgrain level controlling creep damage. This local constraint will arise from any inclusions, voids, grain boundary triple junction or any other anomalies that are present at the microstructural level. Thus, it is argued that even in a uniaxial test where voids develop, and creep failure strain occurs this local constraint play a unique part in the level of failure strain with respect to the applied stress especially at long term test times.

The model, at its basis, considers two critical material properties. These are the very shortterm creep strength equivalence to the material tensile strength and at the long term the appropriate uniaxial uniform failure strain (meaning strain measured prior any necking) that can best be described by the MonkmanGrant failure strain relationship described above [20]. From the available creep constitutive relations, a combined geometric and microstructural constraint/remaining ductility approach is arrived at which unifies the creep uniaxial, multiaxial and crack growth failure processes for very long test times [20].

Using large uniaxial database [3], notched tests [37] and precompressed tests data [38] for type 316H steel over a wide range of temperatures and test times the model is developed and validated based on the statistical bounds of the 316H material constitutive properties. There will clearly be a wide measure of scatter and a level of uncertainty in the data due to its timescale, number of tests, test temperatures, different batches of materials and test performed in different laboratories. However, as suggested previously, the paper will consider the bounds derived from the test data rather than assess smaller data sets.

Fig. 6 and Fig. 7 show the relationship, respectively, between the uniaxial and notched bar and precompressed [3] rupture stress tnormalised by the extrapolated upper bound shortterm rupture stress to. For uniaxial data in Fig. 6, it can be seen the relationship shows a temperature dependence. The upper bound stress to for short time tends an upper limiting stress which is found to be close to the yield stress or tensile stress at various temperatures. For long times above 100,000 hours, the temperature dependence can be dealt with using Eq. (6) by a temperature activation energy term. For notch bar and precompressed data in Fig. 7, the rupture time of precompressed specimen is substantially shorter than that of notch bar and uniaxial specimen due to limited plasticity and increased notch tip constraint for the limited data shown. Furthermore, it can be observed that notched data, show longer rupture times as compared to the more uniaxial precompressed data typical of stainlesssteel behaviour observed suggesting a notch strengthening effect when plotted against netsection stress.

Fig. 6. Normalised stress rupture plot for 316H uniaxial data at different temperatures.

Fig. 7. Normalised stress rupture plot for 316H notch bar data and precompressed test data [37] at 550℃.

Analysis of MG Failure Strains in 316H Database

Fig. 8 shows the best measure of MG failure ductility versus rupture time for 316H uniaxial data at temperatures 550℃750℃. Note that for all data at various temperatures extreme levels of scatter dominate the database. This scatter is partly due to different batches of steels and other test uncertainties which even under controlled circumstances cannot be fully eradicated. The MG ductility at 550℃ is also shown to be slightly lower compared with that at other temperatures. It is also clear that an upper bound MG ductility exists for ductility data at 650℃750℃.

If all uniaxial tests data at 550℃750℃ is taken as one dataset, the upper bound MG ductility 0.3 can be adopted to normalise the failure strains as shown in Fig. 9 in which the normalised MG strains are cross plotted against the rupture stresses normalised by the extrapolated failure stress at or near time zero. Another approach for fitting the data is to use Eq. 9 and identify the mean transition region with an upper lowershelf as shown in Fig. 10. In both cases a linear transition region with different slopes gives the best possible outcome. Both the figures include 316H uniaxial, as well as MG strains derived from notched and precompressed data in the manner of Eq. (5). The MG derived for notched bars are effectively an approximation of the accumulated average creep strains during the steady state creep period. This is still consistent with a pseudo MG strain as the tertiary strains are not considered. In fact, it has been shown that notch bars show very little tertiary as the geometric constraint increases does not allow it. In the case of notched bars, it should be noted that both the geometric constrain and the local time dependent microstructural constraint are present. However, it is argued that the time dependent driver for fusion growth is based on the latter and the stress multiplication factor in the notch region depends on the former.

A simplified linear fit in Fig. 9 of the data gives the best trends. It can also be assumed based on the lower shelf ductility in Fig. 3 a lowest value of 1% could be a practical limiting failure strain. It should be noted that the derivation of the MG strain for the notched bar are at best an estimate of the average strains measured at the notch root. To derive the constitutive relationship, an important emphasis should be on the longerterm tests which already exhibit substantial scatter. Based on the data in Fig. 6 and Fig. 7, the best relationship between t/to and the normalised MG failure strain that can be conveniently derived and given as:

(10)

where is a material constant related to creep strain and temperature. Over the range 0.1≤ ≤1, the stress to failure can therefore be defined as t = f() where is given by Eq. (5) and derived from the MG strains.

Considering this analysis and arguments presented above, the globally measured MG failure strains are directly affected by the constraint at the microstructural level. This approach demonstrates the scalability of this relationship and gives an engineering approach to be developed based on physical void growth model. As a result, by substituting into Eq. (5) a relationship for rupture stress t prediction at time t can then be derived as a simple normalized form in the practical range of 0.1 1 and shown as:

(11)

where β as a measure of a constraint factor could vary between 0.1 to 1, depending on the material creep ductility, metallurgical factors, temperature and variations of models shown in Fig. 9 and Fig. 10 Effectively the higher the β the more the sensitivity of the reduction in failure strain with local constraint.

Clearly the characterisation of uniaxial experiment test data of MG failure strain with time which is plotted in Fig. 7 will show substantial scatter and temperature dependence. This may be alleviated in the future when standardised batch specific tests with ductile and brittle material properties were conducted under controlled testing conditions. In this way, the difference of creep failure strain with time to say the lower shelf ductility may be used to derive Fig 10 for the present model.

Fig. 8. MG ductility versus time to rupture for 316H steel at various temperatures.

Fig. 9. Normalised rupture stress versus normalised MG failure strains for 316H at different temperatures.

Fig. 10. Normalised rupture stress versus normalised MG failure strains for 316H at different temperatures fitted using Eq. 9.

Fig. 11 shows the plot of MG failure times versus the normalised MG failure strains for 316H steel uniaxial, notched and precompressed specimens at different temperatures. Due to a wide range of scatter, it is difficult to illustrate individual trends. In fact, there is difficulty in establishing any trend even for a specific material set. What is clear, however, is the stress failure strain sensitivity over very long test times. Therefore, to perform an analysis of MG failure strains, an assessment was made based on the mean/upper/lower bounds in Fig. 11 using an exponential fit for times to reach /10 of 1000,10000 and 100000 to investigate the sensitivity to predictions of the parameters employed in the model.

The notch bar tests of 316H at 550℃ given in Fig. 11 are conducted in accordance to testing standard [40]. This code covers several notch profiles including blunt, semicircular, parallelsided notches and vnotches. Not only guidance is given on how to interpret the test data and select a notch profile to represent the stress state in the considered region but also advice is also provided on how to determine the influence of state of stress on creep deformation and rupture behaviour of material [40]. In calculating the appropriate strain, a degree of inaccuracy of the normalised local strains measured at the notch throat may be generated. However, at present only available data in the literature can be utilised to develop the model. Therefore, the available estimated MG ductility tests data for 316H notched bars at 550℃ is also included in Fig. 11 [37]. The normalised MG strain is plotted against the normalised rupture stress t/to in Fig. 9. Based on similar approach taken for the uniaxial tests, the versus t/to for 316H notched and precompressed specimens with different plastic precompression level ranging between 48 at 550℃ are also plotted in Fig. 9. Once again, the inclusion of the additional data which in themselves had substantial scatter only confirmed the general trend of a reduction in failures strain with time whilst shown no clearcut differences.

It is, therefore, difficult to characterise for 316H steels individual MG strains versus time due to the large scatter caused by batch to batch and compositional variabilities for 316H steel. Hence, a general relationship between failure time tr and the normalised strain in Fig. 11 can be expressed as:

(12)

where A’ and A” are material constants. The mean, upper and lower bound values for the constants in Eq. (12) for 316H steel are listed in Table 1. Based on the relationships proposed in the model given in Eqs. (11) and (12), appropriate and conservative predictions may be possibly made for the rupture behaviour of 316H uniaxial, notched and precompressed tests.

Fig. 11. Normalised MG ductility for 316H versus time bounded by exponential mean, upper and lower bound fits.

Table 1 Material Constants in Eqs. (11) and (12) and Figs. 911

Material

Fitting

A’

A”

β

to (MPa)

550℃

600℃

650℃

700℃

750℃

316H

Figs 911

Upper

2.5E5

8.5

0.2 to 0.7

440

265

216

157

108

Mean

2E4

6

Lower

1700

3.5

Predictions of Creep Rupture Times in Uniaxial and Notched bars

As there is a substantial scatter of the data and hence variabilities in the constitutive relationships, the rupture time predictions shown in Fig. 12 to Fig. 15 use the upper/mean/lower bounds allowing for parameter sensitivity analysis to be conducted. The model proposed above based on MG failure strain and its implicit relationship to a local constraint can be used to predict creep damage and creep rupture behaviour for steels with different material conditions and stress states. Predictions using Eqs. (11) and (12) for 316H uniaxial tests with the variables in Table 1, for notched and precompressed tests are shown in Fig. 12 to Fig. 15. Although substantial divergence exists in the predictions, it can be seen that safe predictions can be made depending on the constitutive relationship that is used. The figures below therefore highlight the sensitivity of the predictions with respect to the input variables used.

(a)

(b)

(c)

Fig. 12. Predicted rupture times for 316H uniaxial tests using Eqs. (11) and (12) with the values in Table 1 (a) 550℃, (b) 600℃ and (c) 750℃.

Fig. 12 shows the rupture time predictions for 316H uniaxial tests at 550℃, 650℃ and 750℃. It can be seen the lower bound predictions are too conservative for both short term and longterm tests. By comparison, the mean prediction agrees well with short term tests data while it is conservative for long term tests data. The upper bound prediction shows a good trend although it is nonconservative. Note that the rupture time predictions in Fig. 12 based on the upper/mean/lower bound values in Fig. 11 with a fixed value of 0.45 in Eq. (11). If an appropriate value is determined, more accurate rupture time predictions may be obtained. The predictions sensitivity to parameter is investigated and shown below.

(a)

(b)

Fig. 13. Predicted rupture times for 316H at 550 oC using Eqs. (11) and (12) with the input values in Table 1 showing sensitivity to β in Eq. (11) (a) based on mean prediction and (b) based on upper bound prediction.

(a)

(b)

Fig. 14. Predicted rupture times for 316H notched bars at 550 oC using Eqs. (11) and (12) with the values in Table 1 (a) based on mean prediction and (b) based on upper bound prediction and a fixed value of 0.45 in Eq. (11).

Fig. 13 shows the rupture time predictions sensitivity to parameter β in Eq. (11) for 316H uniaxial data at 550oC based on the mean and upper bound in Fig. 11. For both mean and upper bound predictions, the variation of parameter β has a conspicuous effect on the prediction lines. The conservatism of prediction increases with the increasing β. If a more conservative result is required, it can be obtained with higher β value highlighting the sensitivity of the predicitons to the value of β. Compared with the mean predictions in Fig. 13, the upper bound predictions give a more reliable and practical rupture time for long term tests. Therefore, the rupture time predictions based on the upper bound values in Fig. 11 are more suitable for a wide range time scale test. Considering the huge scatter of database, it is difficult to estimate a better batch in all tests. Therefore, the creep damage and rupture behaviour can be approximately predicted with various β values which are derived from the relationship of description of the multiaxial strain factor to constraint level. shows the rupture time predictions based on the model for 316H notch bars [37] assuming a fixed value of 0.45 in Eq. (11). Comparing Fig. 13 and Fig. 14 for uniaxial and unnotched samples it should be noted that when the correlation is made using netsection stress then the a notch strengthening effect is shown suggesting that notch bar failure times are higher than uniaxial samples. The main reason for this phenomenon is the correlation using netsection stress which may not necessarily be representative of the stress state at the notch throat [40]. The rupture times can be predicted for notched bars by both mean and upper bound predictions with acceptable conservatism. An accelerated failure occurs due to the presence of notch which weakens the material. In fact, the geometric constraint and microstructural constraint are the main reasons leading to the accelerated failure. Considering the relatively short rupture times of notched bars, both the mean and upper bound predictions based on the model in this paper can be well used for rupture time assessments of notched bars.

As a further extension of the model’s capabilities, data for 316H precompressed tests [38] were also analysed in terms of time to rupture and shown in Fig. 15 assuming a fixed value of 0.45 in Eq. (11). It has been shown previously [38] that the precompression increases the yield stress and reduces the nonlinear deformation. The creep crack growth (CCG) rate of precompressed tests are similar to that of longterm tests due to lower creep ductility. Under a given applied load, the creep ductility and rupture time decreased with the increasing prestrain levels [39]. It can be seen the rupture times of precompressed tests can also be adequately predicted using this model. All relevant material properties in Eqs. (11) and (12) to predict creep rupture behaviours for 316H uniaxial, notched and precompressed tests are listed in Table 1.

(a)

(b)

Fig. 15. Predicted rupture times for 316H precompressed tests at 550 oC using Eqs. (11) and (12) with the values in Table 1 (a) based on mean prediction and (b) based on upper bound prediction and a fixed value of 0.45 in Eq. (11).

Prediction of Creep Crack Initiaiton and Growth

It is well known that viscoplasticity plays an important role in creep damage development at high stress levels whilst the local subgrain stress state at low stress level dominates the creep failure. As set out for the uniaxial and notched bar case, creep crack initiation and growth will have the same mechanisms depending on the length of the test time. In the same way creep cracking will exhibit an upper/lower shelf described by the plane strain/stress bounds. The models that implemented this idea is the wellknown NSW (Nikbin, Smith and Webster) model [21] based on Eqns. 3,4. In this section the direct link between the arguments set out for the global/local constraint can be shown to correspond with the upper/lower bound predictions in the NSW model thus unifying the whole range of creep failure mechanisms.

7.1 NSW Crack Growth Model

Under steady state creep conditions, the uniaxial creep ductility data are used in crack initiation and crack growth predictions based on the multiaxial ductility NSW model [21] assuming a creep process zone rc. For the uniaxial, notched and precompressed tests, the crack tip stress field is characterized with C* fracture parameter while the local constraint controlling the stress state is modelled with parameter h given in Eqs. (1) to (4). The NSW model used to predict cracking rate as a function of multiaxial ductility and C* is given as:

(13)

where A is a material creep constant, n is the power law stress exponent, In is a dimensionless stress state constant dependent on n , rc is the creep process zone which is relatively insensitive due to the small fractional power and is taken as an upperbound failure strain in short term tests under plane stress and a fraction of 1/30 of it under plain strain lower bound condition. In this case, either the reduction area or elongation failure strain in short term tests determined under plane stress condition is used in Eq. (13) [41]. When material properties are not available, an approximate solution to Eq. (13) for predicting crack growth rate for most engineering materials [41] can be given in the form:

= (14)

where was given as 3 and 90 for a lower/upper bound plane stress/strain prediction for a wide range of alloys [41]. Eq. (14) can be rewritten to reflect the multiaxial constraint factor due to ductility and the cracked geometry giving as:

= (15)

where is taken as 1 for plane stress predicting the lower bound cracking rate and a maximum of 1/30 for plane strain as the extreme creep brittle upper bound. However, for creep ductile 316H steel a factor of 1/10 should be enough in predicting the plane strain upper bound. If is used instead of uniaxial failure strain or reduction in area rA, then the lower bound prediction line is a more conservative prediction. The relationship in Eq. (15) can be changed by replacing ,for which at short test times ≈3 for 316H, giving:

= (16)

As an approximation it has been shown for most alloys D’ ≈ 3 [31] and it has been shown that for most steels which gives an approximate relationship for crack growth rate as inversely proportional to . Furthermore, by taking Eq. (8) as the approximate relationship between and the normalised constraint factor then Eq. (16) can also be expressed as:

= (17)

where the value for which should give the lower bound plane stress predictions in Eq. (17). From this equation, cracking rate is shown to be strongly proportional to the crack tip constraint. The derivation of h could be performed independently by numerical means, but this will be complicated since the degree of accuracy for its derivation is dependent on several factors such as geometry, thickness, ductility and material creep properties. This also highlights a different approach using a twoparameter method to attempt to predict the constraint effects [42].

7.2 NSW Crack Initiation Model

The creep crack initiation time ti is defined as the time to achieve a measurable small amount of crack extension. In this paper, the small crack extension is taken as a=0.2mm [47]. The estimates of initiation time can be obtained by assuming a constant crack growth rate over a. Then the upper and lower bound crack initiation times in plane stress may be obtained by substituting appropriate initial crack growth rate and the steady state crack growth rate . Then the initiation time can be expressed as:

(18)

If it is assumed that C* value remains constant during the small crack extension a, the initiation time ti can be calculated by NSW model in Eq. (16) shown as:

(19)

Eq. (19) above gives the bounds for the initiation times for the plane stress. Assuming for 316H steel, based on Eqs. (14)(19) using =0.1, crack initiation and growth rates over the plane stress/strain bound predictions for 316H steel are shown in Fig. 16 and Fig. 17, respectively.

For the crack initiation predictions Eq. (19) bounds the upper bound plane stress range as shown in Fig. 16. For the extreme lower bound under plane strain there would be an additional factor of 10 or 30, as discussed previously, depending on the level of conservatism required. Most initiation tests data are bounded by the upper and lower bound predictions. The initiation time for HAZ and precompressed tests, which have lower ductilites are for short term tests and fall close to long term tests data if extrapolated data are adopted. Fig. 17 for crack growth rates shows the plane stress predictions using Eq. (16). A factor of 10 would suffice for the upper bound predictions.

UB-Plane stress

LB-Plane strain

Fig. 16. Approximate NSW upper and lower bound predictions of 316H precompressed tests, HAZ, short term and longterm creep crack initiation at 550oC. The transition shift use Eq. (9) and the experimental data from [36].

UB-Plane stress

LB-Plane strain

Fig. 17. Approximate NSW upper and lower bound predictions of 316H precompressed tests, HAZ, short term and longterm creep crack growth at 550 oC. The transition shift uses Eq. (9) and the experimental data from [36].

It should be noted that for the HAZ weldment which is more creep brittle than the as received a faster cracking rate at the higher C* level is measured. This corresponds with the plane strain predictions and is also comparable with the long term as received cracking rate of 316H steel. This further validates the importance of the effect of creep ductility on crack growth rate.

Also, by using Eq.(8) which established a simple relationship between ductility and constraint and deriving the ratios of upper/lower bound uniaxial failure strains from Fig. 3 which show a factor of 1015 for 316H over the 550700 oC temperature range, an upper bound in cracking rate can be derived for each test temperature. For 550 oC a factor of 10 is taken. It could be representative of the bounds for multiaxial failure strain range between plane stress and plain strain. Once the upper shelf ductility is obtained from short term tests, the lower bound CCG rate at short term can be determined in terms of C* from Eqs. (16) or (17). Then the upper bound CCG rate at long term can be approximately estimated by a multiply factor 10. Accordingly, the transition shift from low to high C* is given based on the experimental data of 316H [36] as shown in Fig. 16. Assuming the same transition shift C* value from low to high, the crack initiation transition behavior is also determined with Eq. (19) and shown in Fig. 17 Thus, a simplified and robust engineering approach to predict the upper bound long term CCG rate and initiation time can be easily determined from short term tests without the need for detailed numerical modelling.

Discussions and Conclusions

A uniform creep strain/multiscale constraintbased model which takes into account a geometrical constraint and a time dependent subgrain microstructural constraint is proposed. An approximate parity inverse relation of MG strain against constraint is formed based on multiaxial void growth models indicating that microstructural constraint is the main determinant in the longterm creep damage development based on diffusion growth. As the model is an approximation the sensitivity of MG strain to constraint based on experimental data is found to vary be between 0.39 in the controlling equation.

Based on an assessment of an extensive dataset of uniaxial, notched and precompressed tests of 316H steel over a temperature range of 550700 oC, the proposed unified strain/constraintbased model is empirically verified considering the MG failure strain/stress dependence of creep damage. Given the substantial scatter in the available 316H data presented, especially at the operating temperature of 550 oC, the material characterisation is considered to identify bounds between plane/stress to plain strain longterm trends, in their creep response. These were chosen as they would exhibit different levels of ductility with time. Unfortunately, the only data available for notched and crack growth samples was at 550 oC, a temperature characterised by low creep energy driving force and few longterm data. In effect, the longterm failure mechanism, regardless of geometry, assumes that the uniform MG strains are inversely proportional to the timedependent microstructural constraint levels distributed at stress concentrations within the substructure.

For uniaxial, notched and precompressed tests, the longterm failure behaviour can be predicted by considering the normalised rupture stress and time dependent creep ductility. The MG failure strain is an inherent measure of the strain linked with the local creep damage. The simplified linear inverse relationship between the normalised MG strain and the local constraint that is developed in this model to derive the rupture stress t based on the local constraint level follows the NSW approach. Specifically, the rupture times of uniaxial tests can be conservatively predicted using the upper bound fit while the rupture times of notched bars and precompressed tests can be predicted by both mean and upper bound fits of experimental MG strains normalised by the upperbound plane/stress failure strain. The model is then further extended to show that crack initiation and growth behaviour of fracture mechanics specimens, as predicted by the wellknown NSW model, which gives lower and upper bound at plain stress and plain strain state, fail on the same principals as the uniaxial and notched bar tests. Once the upper shelf ductility is obtained from short term rupture tests, the upper bound cracking rate and initiation time at long term can be easily determined by an appropriate multiplying factor. The transition from upper/lower bound initiation and cracking can also be determined form the MG strain/stress relationship. This factor is approximately 10 and is independent of temperature.

The simplified relationship using MG strains and constraint at the subgrain level can be used to predict failure bounds by rupture and/or crack growth at the testpiece level by identifying two variables. These are the failure strain at the tensile yield or very shortterm creep test stress levels extrapolated to time equals zero and the MG failure strain/stress/time relationships over short test times that could be extrapolated linearly. The upper shelf for the failure strain could therefore be determined from very shortterm tests and the lower shelf could be set as time taken to 1% strain which could take as much as 100,000 hours. For detailed verification and validation of the model, which would be needed to implement in life assessment predictions, a testing programme of pedigree single batch samples testing, the tensile, and medium test creep of uniaxial and notched bars should be setup to reduce the scatter in the data. It would be then possible to determine the longterm MG failure strain properties from much shorterterm creep tests.

It is also important to note that the lower ductility precompressed specimens and HAZ ductility can be used in accelerated test program to predict the upper bound cracking rate which is seen in the asreceived at very long test times (>20,000 hours). These accelerated tests which can be completed at a much shorter timeperiods would be a means to derive conservative cracking rates in these steels at shorter test times. It is also clear that with controlled batch testing of pedigree material the upper bound could be further improved to reduce excessive conservativism.

Acknowledgements

This work was performed at the EDF Energy High Temperature Centre, Imperial College, London. Thanks is given to Albert Liu of ECUST university for the data analysis.

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