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A two-stage model for predicting crack growth due to repeated thermal shock
Engineering Fracture Mechanics 70 (2003) 721–730 www.elsevier.com/locate/engfracmech
A two-stage model for predicting crack growth due to repeated thermal shock B. Kerezsi, J.W.H. Price, R. Ibrahim
*
Department of Mechanical Engineering, Monash University, P.O. Box 197, Caulfield East, Vic. 3145, Australia Received 30 November 2001; received in revised form 4 March 2002; accepted 16 April 2002
Abstract The growth of cracks in equipment that is exposed to repeated thermal down shocks presents a complex problem of analysis. The transient, highly non-linear nature of the stress profiles that are developed during the shock in addition to localized plasticity and environmental interactions makes difficult any accurate analytical predictions. The use of current analysis techniques based on linear stress approximations can result in overly conservative results that may lead to unnecessary and costly component replacements. This paper outlines results from an experimental investigation into crack growth in notched, flat plate specimens exposed to repeated one-dimensional thermal shocks. Analysis of the results shows that a simple two-stage growth model may be applicable for describing the crack growth. The model is comprised of a high strain fatigue region where crack growth is in the plastic range and a region where growth is described by linear elastic fracture mechanics. Allowances for the effects of mean loads and environment on the crack growth are also included in the model. The model is currently limited to the consideration of carbon steel components, operating at temperatures below the creep range. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermal shock; Crack growth; High strain fatigue; Linear elastic fracture mechanics; Corrosion assisted fatigue
1. Introduction The initiation and growth of cracks due to repeated thermal shock (RTS) is caused by the restraint of thermal expansions and contractions of a material as it is exposed to rapid changes in temperature. This restraint (either external or internal in nature) results in strains and associated stresses in the material. If the changes in temperature are severe (especially in the presence of stress concentration factors such as abrupt changes in geometry), and the resulting strains large enough, local plastic deformation of the component will occur. Repeated application of this loading leads to rapid crack initiation as per low-cycle mechanical fatigue. Further application of the loading leads to small crack and then large crack growth. In some cases, this crack growth can finally lead to component failure.
*
Corresponding author. Tel.: +61-3-9903-2484; fax: +61-3-9903-2766. E-mail address: [email protected] (R. Ibrahim).
0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 8 9 - 9
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B. Kerezsi et al. / Engineering Fracture Mechanics 70 (2003) 721–730
Crack growth due to RTS is a recognized damage mechanism in fossil fuel power plants where thermal shocks in boiler equipment are an inevitable side effect of normal and cyclic operation. Examples of thermal shock cracking include waterside-initiated cracking in carbon steel economiser headers, damage in valves or spray stations, depressurization vessels and safety valves. The highly non-linear transient stresses characteristic of thermal shock make prediction of crack growth using codes such as the ASME Boiler and Pressure Vessel Code and BS 7910 extremely difficult. Unfortunately, thermal shocks in boiler equipment are an inevitable side effect of normal operation, especially when water is injected or there is sudden depressurization. Start-up and shutdown procedures are likely to be especially damaging. Cyclic operation of traditionally base-loaded units only increases the severity of the problem. The boiler tube damage mechanism of waterside-initiated cracking in carbon steel economiser headers is related to this practice [1]. A typical shutdown due to such a failure can (depending on the market price of electricity at the time) cost a utility over A$1,000,000 for replacement power. Another cost associated with RTS is the replacement of equipment with cracks. This equipment may be designated unfit for continued operation by ‘‘fitness for service’’ codes such as the British Standard BS 7910 [2] and Section XI of the ASME Boiler and Pressure Vessel Code [3] and require immediate repair or replacement. The fitness for service codes mentioned above use models based on isothermal fatigue tests and simplified stress profiles. The conservatism of these codes when analyzing RTS cracks is of concern. A particular reference to the conservatism of the ASME code can be found in work by Czuck et al. [4] where growth at the tip of a crack exposed to RTS loading was found to be an order of magnitude less than that predicted by using the code. As mentioned previously, mechanical loading and environmental conditions are very important considerations when analyzing RTS. Existing codes deal with these two issues only in a simplified manner. This paper outlines results from an experimental investigation into crack growth in notched, flat plate specimens exposed to repeated one-dimensional thermal shocks. Analysis of the results shows that a twostage growth model may be applicable for describing the crack growth. The model is based on high strain fatigue (HSF) and linear elastic fracture mechanics (LEFM) considerations. The crack growth occurs firstly by rapid crack growth acceleration in the plastic zone near the starting notch, followed by a steady deceleration in the surrounding elastic region. Allowances for the effects of mean loads and environment on the crack growth are included in the model.
2. Experimental procedure Crack growth results used in this investigation have been obtained using a thermal fatigue test rig, purpose-built for the investigation of crack growth due to RTS. The test rig consists of a convection furnace, static loading structure and quenching system. The design allows for the monitored growth of cracks for a wide variety of component geometries. A thorough analysis of the development of the test rig and specimen design, including a review of previous trends in the experimental investigations of thermal shock cracking can be found in [5]. The crack growth data used in this analysis was obtained through the testing of, large-scale flat plate specimens. The notched specimen design is shown in Fig. 1. Note that the inclusion of the ‘‘attached masses’’ in the design ensures that one-dimensional cooling occurs when a thermal shock is applied to the front face. During all tests, the maximum specimen temperature was limited to 370 °C to remove any creep effects. Primary mechanical loads of 90 MPa were applied to half of the specimens, while the remainder had no mechanical loading. Variation of the dissolved oxygen (DO) level in the cooling water between 8 and 2 ppm in the tests simulated altered environmental conditions. Water was recycled during individual tests and replaced for each new specimen. In all tests the pH of the water was held steady at around 8.0.
B. Kerezsi et al. / Engineering Fracture Mechanics 70 (2003) 721–730
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Fig. 1. Specimen design.
Table 1 Results from high temperature tensile tests of AS1548-7-430R carbon steel Test temperature (°C)
Upper yield stress, ry (MPa)
Ultimate tensile stress, ru (MPa)
24 50 100 200
324 295 289 288
456 446 460 –
The specimen material used for these investigations was boiler grade carbon steel, AS1548-7-430R [6]. Results from elevated temperature tensile tests of the material are summarised below in Table 1. For all testing, the Young’s Modulus can be satisfactorily estimated at 200 GPa. To simulate a thermal shock, the specimens were heated to a set temperature in a convection furnace and then sprayed with room temperature water (25 °C) for a period of 7 s. The full cycle time being around 15 min. Thermocouples were used to record the temperature profiles of the specimens during testing with elastic stress profiles calculated using the elastic theory [7]. A weight function technique as outlined in Section XI of the ASME Boiler and Pressure Vessel Code [3] was then used to calculate the resulting stress intensity factor profiles for cracks in the range 0:05 < a=W < 0:3 (where a is the crack length and W is the depth of the specimen 65 mm in this case). Typical temperature profiles as recorded by the thermocouples during a thermal shock from 370 °C and the corresponding elastic stress profiles are shown in Fig. 2. The corresponding maximum stress intensity factors as a function of crack length (including notch depth of 3.5 mm) for cracks outside of the influence of the notch are given in Fig. 3. 3. Experimental results Visual inspections of the crack development using a stereo microscope (10–100 mag.) were completed at intervals of 500 cycles. Crack lengths were measured on either side of the specimen, the average of the
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Fig. 2. Temperature and elastic stress profiles recorded during a quench lasting 7 s from a set temperature of 370 °C.
Fig. 3. Maximum stress intensity factor profiles during 7 s shock from 370 °C, with and without 90 MPa primary load (note that crack length a includes the notch depth of 3.5 mm).
two values taken as the through depth. The use of edge values for crack depth was justified through observations of the post fracture surface at the conclusion of testing. It should be noted however that this method prevented any accurate crack growth rate measurements for crack depths of less than 1 mm (not including the notch depth) from being obtained. The crack growth data from the experiments is presented as Fig. 4. The data is presented as recorded during the testing and has not been adjusted. Crack lengths include the notch depth of 3.5 mm. The general trend for all data is a period of crack growth acceleration followed by a period of deceleration. The crack growth data as presented in Fig. 4 is not very useful for developing an inclusive analysis of crack growth trends during RTS. The large number of variables included in the testing makes it difficult to complete any useful generalisation. Fortunately, as shown by previous analyses [8–10], LEFM methods may be successfully applied to RTS crack growth analysis as long as the plastic zone surrounding the crack tip is smaller than the LEFM plastic zone allowance. In the case when this plastic zone is too large however, other methods must be used. As shown in Fig. 2, thermal shock stresses larger than yield will usually be confined to a region quite close to the shocked surface. For this reason, crack growth analysis has been divided into two regions. The first region is when the crack is small, and the tip is still near the shocked surface where plastic stresses/
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Fig. 4. Raw crack length data, T ¼ maximum cycle temperature (°C), P ¼ primary mechanical load (MPa), DO ¼ dissolved oxygen level (ppm).
strains are dominant. The second region is when the crack is longer, the bulk of the material is behaving elastically and LEFM may be applied.
4. Crack growth analysis 4.1. Short crack growth Fig. 5 shows the crack growth rate versus crack length results for a number of testing variable combinations. This data clearly shows the general trend of the crack growth as a period of acceleration followed by a period of deceleration. It is clear however that some cracks accelerate over a greater depth than others do. This depth of acceleration has a rough correlation with the size of the plastic zone generated during the thermal shock. The more severe the surface stress during a thermal shock, the greater the depth at which plastic strains develop and the deeper the region of crack acceleration. When plastic strains dominate the region surrounding the crack tip, cyclic loading is sometimes referred to as HSF. As reported by Skelton [11], for small cracks (a=W < 0:1), the crack growth rate in a HSF region can be linearly related to crack length as shown in Eq. (1):
Fig. 5. Crack growth rate versus crack length (includes notch depth) for a number of cracks grown by RTS. T ¼ maximum cycle temperature (°C), P ¼ primary mechanical load (MPa), DO ¼ dissolved oxygen level (ppm).
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da ¼ Ba dN
ð1Þ
where da=dN is the crack growth per load cycle and B is a constant related to the plastic strain range. Fitting a linear curve over the region of crack acceleration yields the straight line shown in Fig. 5. As expected, the linear curve correlates well with the data during the period of crack growth acceleration. This reinforces the assumption that plastic stresses dominate in this region. Note that when fitting Eq. (1) to the first region of Fig. 4, an additional constant L has been incorporated. This is because the crack length a includes the notch depth. The new form of Eq. (1) becomes: da ¼ Bða LÞ for ða LÞ=W < 0:1 dN
ð2Þ
Substituting the values for B and L obtained from a best-fit analysis, the following relationship is obtained to describe the crack growth in the HSF region of a notched specimen exposed to RTS: da ¼ 4:88 104 ða 4:05 103 Þ m=cycle dN
for a > 4:05 mm
ð3Þ
where a is the crack length in metres. Eq. (3) is represented in Fig. 5 by the full dark line. We can also safely assume that as long as the crack total crack length a remains less than 6.5 mm, the limit of a=W ¼ 0:1 will not be exceeded and the HSF relation can still be applied. It is worth noting that the development of Eq. (3) is dominated by the results from the testing conducted at 370 °C (where most crack acceleration was observed). This is due to the lack of reliable measurements for cracks of less than 1 mm. The number of variable in Eq. (2) can be reduced by setting L to the original notch depth (in this case 3.5 mm). The effect of doing so is to produce a conservative prediction curve for the HSF region as shown by the broken line in Fig. 5. The appropriate equation for this curve is given below: da ¼ 4:88 104 ða 3:5 103 Þ m=cycle dN
for a > 3:5 mm
ð4Þ
In previous experimental work analysing crack growth in the HSF region, Marsh [12] developed an empirical relationship that suggested the constant B in Eq. (1) was dependent on the material properties and the cyclic plastic strain range. The results shown here however suggest that cracks grown in a range of thermal shock severities (and therefore plastic strain ranges) may be approximated by one curve and hence a single value of B (4:88 104 ). Particular observations can be made about the effect of DO and primary load on the crack accelerations observed during the 370 °C thermal shocks (corresponding to the worst case in our testing). First of all, the level of DO in the cooling water seems to have little effect on crack growth rates in the HSF region. Secondly, while the primary loads do not affect the relation of the growth to Eq. (3), they do extend the plastic zone, increasing the size of the HSF region and hence the depth to which the cracks accelerate. 4.2. Long crack growth Once the crack has grown out of the HSF region and growth begins to decelerate, crack length alone is not sufficient as a parameter for describing the growth relationship. Providing the limitations on notch influence and plastic zone size are satisfied, and elastic stresses are dominant, the cyclic change in elastic stress intensity factor (DK) may become a valid parameter for describing the stress field at the crack tip. As the previous work in this area has suggested, the stress intensity factor can be related to crack growth via Paris law type equations. For the work reported here, the size of the notch influence is quite small. As defined pffiffiffiffiffiffiffiin experimental work by Smith and Miller [13], the size of notch influence may be approximated by 0.13 Dq, where D is the
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Fig. 6. Change in stress intensity factor versus crack growth rate for RTS.
notch depth and q is notch radius. Using the values for D and q used in the experimental work, the maximum notch influence is around 0.13 mm. This crack length is well below any region in which LEFM will be applied. Additionally, the plastic zone developed at the tip of a crack during RTS can p be considered adequately small (less than 0.5 mm) for application of LEFM if DK is kept below 35 MPa m. This value is based on the plastic zone size developed at a crack tip during cycling being much smaller than the specimen thickness and crack length, a general requirement for LEFM validity. Plotting of the cyclic change in stress intensity factor versus the crack growth rate is shown in Fig. 6. Data is categorised by both R-ratio (minimum stress intensity factor divided by maximum stress intensity factor in the loading cycle) and the DO level of the cooling water. Also included on the figure are arrows that indicate the direction of growth of the cracks, reinforcing the fact that crack deceleration is occurring. On first analysis, the data from Fig. 6 seems fairly scattered. However after grouping, several trends can be observed: (1) When primary loads are low (R < 0:3), crack growth rate can be roughly related to stress intensity factor by means of a single power law. Changes in the DO level have minimal effect on the overall growth rate. (2) When primary loads are high (R > 0:3), crack growth rates are accelerated and cannot be modelled by a single power law fit. An increase in DO results in an increase in crack growth rate. It is assumed that this accelerated growth is evidence of an environmental interaction. (3) The increased levels of crack growth observed at high R-ratios and DO levels displays a region where the crack growth is independent of the applied DK. This suggests an environmental assistance to the crack growth that may be dominated by a corrosion mechanism. Observations of the crack surface after the completion of testing confirmed a high level of corrosion to be present in this region [14]. (4) An eventual arrest trend for all cracks seems to be occurring as growth is in a reducing DK field (as shown in Fig. 3). The large arrow on the Fig. 6 indicates this trend. Unlike observations in the HSF region, both primary loads and the DO level in the cooling water seem to influence the crack growth rate in the LEFM region. This complicates the matter of predicting crack growth in this region. Some success has been obtained by using an environmentally assisted crack growth model developed by Gabetta et al. [15] to describe the growth in this region. This equation, reproduced below, allows for the environmental assistance by adding a time dependent corrosion term to the wellknown Paris law equation:
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da da m ¼ CðDKÞ þ s dN dt C
ð5Þ
where ðda=dN Þ ¼ total crack growth per load cycle due to fatigue and corrosion, s ¼ rise time of loading cycle (7 s in this work), ðda=dtÞC ¼ time rate of crack growth due to corrosion processes (found by curve fitting). The last term of Eq. (5) represents the contribution of corrosion to the crack growth process and is present only when the stress intensity factor range is above a critical value termed DKc . A formula for determining DKc is provided in Gabetta et al. [15]: DKc2 ¼
1 aASCR
ry Es
1 R da 1 þ R dt C
ð6Þ
where aASCR is a constant that relates the ‘‘active surface creation rate’’ (ASCR) to the rate of change of crack mouth opening distance during a load cycle and ry and E are the material yield and elastic modulus respectively. The experimental results produced to date in this work are not sufficient to distinguish all of the factors of Eq. (6), however sufficient data has been produced to allow the following interpretation for our conditions: DKc2 ¼ 1:45 103
1R 1þR
ð7Þ
Curve fitting the data for DK < DKc to determine C and m, the following solution for crack growth due to repeated 7 s long thermal shocks is obtained: da 5:89 ¼ 5:28 1016 ðDKÞ dN
for DK < DKc
da 5:89 ¼ 5:28 1016 ðDKÞ þ 7 107 dN
for DK > DKc
ð8Þ ð9Þ
Figs. 7 and 8 show Eqs. (8) and (9) plotted against smoothed test data for cooling water DO of 8 and 2 ppm. Considering the number of simplifications made in this analysis (including the values for da=dtSCF and a), the Gabetta et al. model provides a fairly good representation of the observed environmental assisted crack
Fig. 7. Smoothed experimental crack growth data plotted against a Gabetta et al. [15] model prediction, allowing for the effects of environment and primary load. Experimental data for DO ¼ 8 ppm plotted.
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Fig. 8. Smoothed experimental crack growth data plotted against a Gabetta et al. [15] model prediction, allowing for the effects of environment and primary load. Experimental data for DO ¼ 2 ppm plotted.
growth. p Note however that the continued validity of the Gabetta et al. model at values of DK above 35 MPa m is yet to be verified.
5. Discussion––a simple combined growth model A combined law for explaining RTS crack growth through both the HSF and LEFM regions can be proposed from this work. The model is divided into two sections, the first when crack growth is inside the plastic zone close to the specimen surface and the second is the elastic region that is encountered at further depths. For crack growth in the plastic zone, a linear acceleration of crack growth rate with crack depth is suggested (Eq. (2)). For crack growth outside the plastic zone, the crack growth rate (typically of a decelerating nature) is predicted using a LEFM growth law such as that proposed by Gabetta et al. [15] (i.e. Eq. (6)). The results from such a two-stage analysis are summarised in Fig. 9, which shows the crack growth rate as a function of crack depth for two cases of thermal shock loading. The figure shows a good prediction of the acceleration and deceleration observed in the specimens both with and without primary loading.
Fig. 9. Crack growth as a function of crack length for two specimens along with crack growth predictions using both HSF and LEFM methods. T ¼ maximum cycle temperature (°C), P ¼ primary mechanical load (MPa), DO ¼ dissolved oxygen level (ppm).
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Fig. 9 also shows that the crack depth at which the crossover between the HSF and LEFM growth laws occurs is a critical parameter. Because crack growth will accelerate until this depth is reached, a crack of ‘‘crossover depth’’ can be assumed to rapidly appear during RTS loading. The smaller this depth is, the less chance that appreciable crack growth can occur in the ensuing LEFM region. The crossover depth seems to be related to the plastic stress depth developed by both the thermal shock and steady state primary stresses. A theoretical proposal for determining this depth is not currently possible. However, a possible suggestion, based on an approach that has approximately fitted our experimental data is given below as Eq. (10): 2 1 Kmax ac ¼ ð10Þ bp ry where ac is the crossover depth, Kmax is the maximum stress intensity factor during a cycle, ry is the material yield stress and b is a constant determined by curve fitting (the test data of this work supports a value of approximately 1.2).
6. Conclusions In this paper, crack growth in a heated flat plate specimen exposed to repeated one-dimensional thermal shocks (RTS) has been analysed. Experimental results reveal that crack growth occurs in a stage of acceleration followed by a stage of deceleration. The relative magnitude of the stages is dependent on the presence of primary mechanical loads and the testing environment. A two-stage growth model has been suggested to represent the observed crack growth based on HSF and LEFM techniques. Allowances for environmental and primary steady state stress interactions in the LEFM region have been made through an advanced corrosion fatigue type assessment.
References [1] Dooley B. Don’t let those boiler tubes fail again: Part 2. Power Engineering 1997;(August):31–40. [2] BSI. BS 7910: 1999––Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures. London: BSI; 1999. [3] ASME. ASME Boiler and Pressure Vessel Code, Section XI, Rules for Inservice Inspection of Nuclear Power Plant Components. New York: American Society of Mechanical Engineers; 1998. [4] Czuck G, Mattheck C, Munz D, Stamm H. Crack growth under cyclic thermal shock loading. Nucl Eng Des 1985;84:189–99. [5] Kerezsi BB, Kotousov A, Price JWH. Experimental apparatus for thermal shock fatigue investigations. Int J Pressure Vessels Piping 2000;77:425–34. [6] AS. AS 1548–1995: Australian Standard: steel plates for pressure equipment. Homebush: Standards Australia; 1995. [7] Timoshenko SP, Goodier JN. Theory of elasticity. third ed. London: McGraw Hill; 1982. [8] Mowbray DF, Woodford DA, Brandt DE. Thermal fatigue characterization of cast cobalt and nickel-base superalloys. In: Fatigue at Elevated Temperatures, ASTM STP 520. Philadelphia: American Society for Testing and Materials; 1973. p. 416–26. [9] Gemma AE, Langer BS, Leverant GR. Thermomechanical fatigue crack propagation in an anisotropic (directionally solidified) nickel-base superalloy. In: Thermal Fatigue of Materials and Components, ASTM STP 612. Philadelphia: American Society for Testing and Materials; 1976. p. 199–213. [10] Skelton RP. Crack initiation and growth in simple metal components during thermal cycling. In: Skelton RP, editor. Fatigue at high temperature, 1983. p. 1–62. [11] Skelton RP. Growth of short cracks during high strain fatigue and thermal cycling. In: Low Cycle Fatigue and Life Prediction, ASTM STP 770. Philadelphia: American Society for Testing and Materials; 1982. p. 337–81. [12] Marsh DJ. A thermal shock fatigue study of type 304 and 316 stainless steels. Fatig Eng Mater Struct 1981;4(2):179–95. [13] Smith RA, Miller KJ. Fatigue cracks at notches. Int J Mech Sci 1977;19:11–22. [14] Kerezsi BB, Price JWH, Kotousov A. Features of fatigue crack growth due to repeated thermal shock. Fatig Fract Eng Mater Struct 2002;25:215–22. [15] Gabetta G, Rinaldi C, Pozzi D. A model for environmentally assisted crack growth rate. In: Environmentally Assisted Cracking: Science and Engineering, ASTM STP 1049. Philadelphia: American Society for Testing and Materials; 1990. p. 266–82.
A two-stage model for predicting crack growth due to repeated thermal shock B. Kerezsi, J.W.H. Price, R. Ibrahim
*
Department of Mechanical Engineering, Monash University, P.O. Box 197, Caulfield East, Vic. 3145, Australia Received 30 November 2001; received in revised form 4 March 2002; accepted 16 April 2002
Abstract The growth of cracks in equipment that is exposed to repeated thermal down shocks presents a complex problem of analysis. The transient, highly non-linear nature of the stress profiles that are developed during the shock in addition to localized plasticity and environmental interactions makes difficult any accurate analytical predictions. The use of current analysis techniques based on linear stress approximations can result in overly conservative results that may lead to unnecessary and costly component replacements. This paper outlines results from an experimental investigation into crack growth in notched, flat plate specimens exposed to repeated one-dimensional thermal shocks. Analysis of the results shows that a simple two-stage growth model may be applicable for describing the crack growth. The model is comprised of a high strain fatigue region where crack growth is in the plastic range and a region where growth is described by linear elastic fracture mechanics. Allowances for the effects of mean loads and environment on the crack growth are also included in the model. The model is currently limited to the consideration of carbon steel components, operating at temperatures below the creep range. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermal shock; Crack growth; High strain fatigue; Linear elastic fracture mechanics; Corrosion assisted fatigue
1. Introduction The initiation and growth of cracks due to repeated thermal shock (RTS) is caused by the restraint of thermal expansions and contractions of a material as it is exposed to rapid changes in temperature. This restraint (either external or internal in nature) results in strains and associated stresses in the material. If the changes in temperature are severe (especially in the presence of stress concentration factors such as abrupt changes in geometry), and the resulting strains large enough, local plastic deformation of the component will occur. Repeated application of this loading leads to rapid crack initiation as per low-cycle mechanical fatigue. Further application of the loading leads to small crack and then large crack growth. In some cases, this crack growth can finally lead to component failure.
*
Corresponding author. Tel.: +61-3-9903-2484; fax: +61-3-9903-2766. E-mail address: [email protected] (R. Ibrahim).
0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 8 9 - 9
722
B. Kerezsi et al. / Engineering Fracture Mechanics 70 (2003) 721–730
Crack growth due to RTS is a recognized damage mechanism in fossil fuel power plants where thermal shocks in boiler equipment are an inevitable side effect of normal and cyclic operation. Examples of thermal shock cracking include waterside-initiated cracking in carbon steel economiser headers, damage in valves or spray stations, depressurization vessels and safety valves. The highly non-linear transient stresses characteristic of thermal shock make prediction of crack growth using codes such as the ASME Boiler and Pressure Vessel Code and BS 7910 extremely difficult. Unfortunately, thermal shocks in boiler equipment are an inevitable side effect of normal operation, especially when water is injected or there is sudden depressurization. Start-up and shutdown procedures are likely to be especially damaging. Cyclic operation of traditionally base-loaded units only increases the severity of the problem. The boiler tube damage mechanism of waterside-initiated cracking in carbon steel economiser headers is related to this practice [1]. A typical shutdown due to such a failure can (depending on the market price of electricity at the time) cost a utility over A$1,000,000 for replacement power. Another cost associated with RTS is the replacement of equipment with cracks. This equipment may be designated unfit for continued operation by ‘‘fitness for service’’ codes such as the British Standard BS 7910 [2] and Section XI of the ASME Boiler and Pressure Vessel Code [3] and require immediate repair or replacement. The fitness for service codes mentioned above use models based on isothermal fatigue tests and simplified stress profiles. The conservatism of these codes when analyzing RTS cracks is of concern. A particular reference to the conservatism of the ASME code can be found in work by Czuck et al. [4] where growth at the tip of a crack exposed to RTS loading was found to be an order of magnitude less than that predicted by using the code. As mentioned previously, mechanical loading and environmental conditions are very important considerations when analyzing RTS. Existing codes deal with these two issues only in a simplified manner. This paper outlines results from an experimental investigation into crack growth in notched, flat plate specimens exposed to repeated one-dimensional thermal shocks. Analysis of the results shows that a twostage growth model may be applicable for describing the crack growth. The model is based on high strain fatigue (HSF) and linear elastic fracture mechanics (LEFM) considerations. The crack growth occurs firstly by rapid crack growth acceleration in the plastic zone near the starting notch, followed by a steady deceleration in the surrounding elastic region. Allowances for the effects of mean loads and environment on the crack growth are included in the model.
2. Experimental procedure Crack growth results used in this investigation have been obtained using a thermal fatigue test rig, purpose-built for the investigation of crack growth due to RTS. The test rig consists of a convection furnace, static loading structure and quenching system. The design allows for the monitored growth of cracks for a wide variety of component geometries. A thorough analysis of the development of the test rig and specimen design, including a review of previous trends in the experimental investigations of thermal shock cracking can be found in [5]. The crack growth data used in this analysis was obtained through the testing of, large-scale flat plate specimens. The notched specimen design is shown in Fig. 1. Note that the inclusion of the ‘‘attached masses’’ in the design ensures that one-dimensional cooling occurs when a thermal shock is applied to the front face. During all tests, the maximum specimen temperature was limited to 370 °C to remove any creep effects. Primary mechanical loads of 90 MPa were applied to half of the specimens, while the remainder had no mechanical loading. Variation of the dissolved oxygen (DO) level in the cooling water between 8 and 2 ppm in the tests simulated altered environmental conditions. Water was recycled during individual tests and replaced for each new specimen. In all tests the pH of the water was held steady at around 8.0.
B. Kerezsi et al. / Engineering Fracture Mechanics 70 (2003) 721–730
723
Fig. 1. Specimen design.
Table 1 Results from high temperature tensile tests of AS1548-7-430R carbon steel Test temperature (°C)
Upper yield stress, ry (MPa)
Ultimate tensile stress, ru (MPa)
24 50 100 200
324 295 289 288
456 446 460 –
The specimen material used for these investigations was boiler grade carbon steel, AS1548-7-430R [6]. Results from elevated temperature tensile tests of the material are summarised below in Table 1. For all testing, the Young’s Modulus can be satisfactorily estimated at 200 GPa. To simulate a thermal shock, the specimens were heated to a set temperature in a convection furnace and then sprayed with room temperature water (25 °C) for a period of 7 s. The full cycle time being around 15 min. Thermocouples were used to record the temperature profiles of the specimens during testing with elastic stress profiles calculated using the elastic theory [7]. A weight function technique as outlined in Section XI of the ASME Boiler and Pressure Vessel Code [3] was then used to calculate the resulting stress intensity factor profiles for cracks in the range 0:05 < a=W < 0:3 (where a is the crack length and W is the depth of the specimen 65 mm in this case). Typical temperature profiles as recorded by the thermocouples during a thermal shock from 370 °C and the corresponding elastic stress profiles are shown in Fig. 2. The corresponding maximum stress intensity factors as a function of crack length (including notch depth of 3.5 mm) for cracks outside of the influence of the notch are given in Fig. 3. 3. Experimental results Visual inspections of the crack development using a stereo microscope (10–100 mag.) were completed at intervals of 500 cycles. Crack lengths were measured on either side of the specimen, the average of the
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Fig. 2. Temperature and elastic stress profiles recorded during a quench lasting 7 s from a set temperature of 370 °C.
Fig. 3. Maximum stress intensity factor profiles during 7 s shock from 370 °C, with and without 90 MPa primary load (note that crack length a includes the notch depth of 3.5 mm).
two values taken as the through depth. The use of edge values for crack depth was justified through observations of the post fracture surface at the conclusion of testing. It should be noted however that this method prevented any accurate crack growth rate measurements for crack depths of less than 1 mm (not including the notch depth) from being obtained. The crack growth data from the experiments is presented as Fig. 4. The data is presented as recorded during the testing and has not been adjusted. Crack lengths include the notch depth of 3.5 mm. The general trend for all data is a period of crack growth acceleration followed by a period of deceleration. The crack growth data as presented in Fig. 4 is not very useful for developing an inclusive analysis of crack growth trends during RTS. The large number of variables included in the testing makes it difficult to complete any useful generalisation. Fortunately, as shown by previous analyses [8–10], LEFM methods may be successfully applied to RTS crack growth analysis as long as the plastic zone surrounding the crack tip is smaller than the LEFM plastic zone allowance. In the case when this plastic zone is too large however, other methods must be used. As shown in Fig. 2, thermal shock stresses larger than yield will usually be confined to a region quite close to the shocked surface. For this reason, crack growth analysis has been divided into two regions. The first region is when the crack is small, and the tip is still near the shocked surface where plastic stresses/
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Fig. 4. Raw crack length data, T ¼ maximum cycle temperature (°C), P ¼ primary mechanical load (MPa), DO ¼ dissolved oxygen level (ppm).
strains are dominant. The second region is when the crack is longer, the bulk of the material is behaving elastically and LEFM may be applied.
4. Crack growth analysis 4.1. Short crack growth Fig. 5 shows the crack growth rate versus crack length results for a number of testing variable combinations. This data clearly shows the general trend of the crack growth as a period of acceleration followed by a period of deceleration. It is clear however that some cracks accelerate over a greater depth than others do. This depth of acceleration has a rough correlation with the size of the plastic zone generated during the thermal shock. The more severe the surface stress during a thermal shock, the greater the depth at which plastic strains develop and the deeper the region of crack acceleration. When plastic strains dominate the region surrounding the crack tip, cyclic loading is sometimes referred to as HSF. As reported by Skelton [11], for small cracks (a=W < 0:1), the crack growth rate in a HSF region can be linearly related to crack length as shown in Eq. (1):
Fig. 5. Crack growth rate versus crack length (includes notch depth) for a number of cracks grown by RTS. T ¼ maximum cycle temperature (°C), P ¼ primary mechanical load (MPa), DO ¼ dissolved oxygen level (ppm).
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da ¼ Ba dN
ð1Þ
where da=dN is the crack growth per load cycle and B is a constant related to the plastic strain range. Fitting a linear curve over the region of crack acceleration yields the straight line shown in Fig. 5. As expected, the linear curve correlates well with the data during the period of crack growth acceleration. This reinforces the assumption that plastic stresses dominate in this region. Note that when fitting Eq. (1) to the first region of Fig. 4, an additional constant L has been incorporated. This is because the crack length a includes the notch depth. The new form of Eq. (1) becomes: da ¼ Bða LÞ for ða LÞ=W < 0:1 dN
ð2Þ
Substituting the values for B and L obtained from a best-fit analysis, the following relationship is obtained to describe the crack growth in the HSF region of a notched specimen exposed to RTS: da ¼ 4:88 104 ða 4:05 103 Þ m=cycle dN
for a > 4:05 mm
ð3Þ
where a is the crack length in metres. Eq. (3) is represented in Fig. 5 by the full dark line. We can also safely assume that as long as the crack total crack length a remains less than 6.5 mm, the limit of a=W ¼ 0:1 will not be exceeded and the HSF relation can still be applied. It is worth noting that the development of Eq. (3) is dominated by the results from the testing conducted at 370 °C (where most crack acceleration was observed). This is due to the lack of reliable measurements for cracks of less than 1 mm. The number of variable in Eq. (2) can be reduced by setting L to the original notch depth (in this case 3.5 mm). The effect of doing so is to produce a conservative prediction curve for the HSF region as shown by the broken line in Fig. 5. The appropriate equation for this curve is given below: da ¼ 4:88 104 ða 3:5 103 Þ m=cycle dN
for a > 3:5 mm
ð4Þ
In previous experimental work analysing crack growth in the HSF region, Marsh [12] developed an empirical relationship that suggested the constant B in Eq. (1) was dependent on the material properties and the cyclic plastic strain range. The results shown here however suggest that cracks grown in a range of thermal shock severities (and therefore plastic strain ranges) may be approximated by one curve and hence a single value of B (4:88 104 ). Particular observations can be made about the effect of DO and primary load on the crack accelerations observed during the 370 °C thermal shocks (corresponding to the worst case in our testing). First of all, the level of DO in the cooling water seems to have little effect on crack growth rates in the HSF region. Secondly, while the primary loads do not affect the relation of the growth to Eq. (3), they do extend the plastic zone, increasing the size of the HSF region and hence the depth to which the cracks accelerate. 4.2. Long crack growth Once the crack has grown out of the HSF region and growth begins to decelerate, crack length alone is not sufficient as a parameter for describing the growth relationship. Providing the limitations on notch influence and plastic zone size are satisfied, and elastic stresses are dominant, the cyclic change in elastic stress intensity factor (DK) may become a valid parameter for describing the stress field at the crack tip. As the previous work in this area has suggested, the stress intensity factor can be related to crack growth via Paris law type equations. For the work reported here, the size of the notch influence is quite small. As defined pffiffiffiffiffiffiffiin experimental work by Smith and Miller [13], the size of notch influence may be approximated by 0.13 Dq, where D is the
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Fig. 6. Change in stress intensity factor versus crack growth rate for RTS.
notch depth and q is notch radius. Using the values for D and q used in the experimental work, the maximum notch influence is around 0.13 mm. This crack length is well below any region in which LEFM will be applied. Additionally, the plastic zone developed at the tip of a crack during RTS can p be considered adequately small (less than 0.5 mm) for application of LEFM if DK is kept below 35 MPa m. This value is based on the plastic zone size developed at a crack tip during cycling being much smaller than the specimen thickness and crack length, a general requirement for LEFM validity. Plotting of the cyclic change in stress intensity factor versus the crack growth rate is shown in Fig. 6. Data is categorised by both R-ratio (minimum stress intensity factor divided by maximum stress intensity factor in the loading cycle) and the DO level of the cooling water. Also included on the figure are arrows that indicate the direction of growth of the cracks, reinforcing the fact that crack deceleration is occurring. On first analysis, the data from Fig. 6 seems fairly scattered. However after grouping, several trends can be observed: (1) When primary loads are low (R < 0:3), crack growth rate can be roughly related to stress intensity factor by means of a single power law. Changes in the DO level have minimal effect on the overall growth rate. (2) When primary loads are high (R > 0:3), crack growth rates are accelerated and cannot be modelled by a single power law fit. An increase in DO results in an increase in crack growth rate. It is assumed that this accelerated growth is evidence of an environmental interaction. (3) The increased levels of crack growth observed at high R-ratios and DO levels displays a region where the crack growth is independent of the applied DK. This suggests an environmental assistance to the crack growth that may be dominated by a corrosion mechanism. Observations of the crack surface after the completion of testing confirmed a high level of corrosion to be present in this region [14]. (4) An eventual arrest trend for all cracks seems to be occurring as growth is in a reducing DK field (as shown in Fig. 3). The large arrow on the Fig. 6 indicates this trend. Unlike observations in the HSF region, both primary loads and the DO level in the cooling water seem to influence the crack growth rate in the LEFM region. This complicates the matter of predicting crack growth in this region. Some success has been obtained by using an environmentally assisted crack growth model developed by Gabetta et al. [15] to describe the growth in this region. This equation, reproduced below, allows for the environmental assistance by adding a time dependent corrosion term to the wellknown Paris law equation:
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da da m ¼ CðDKÞ þ s dN dt C
ð5Þ
where ðda=dN Þ ¼ total crack growth per load cycle due to fatigue and corrosion, s ¼ rise time of loading cycle (7 s in this work), ðda=dtÞC ¼ time rate of crack growth due to corrosion processes (found by curve fitting). The last term of Eq. (5) represents the contribution of corrosion to the crack growth process and is present only when the stress intensity factor range is above a critical value termed DKc . A formula for determining DKc is provided in Gabetta et al. [15]: DKc2 ¼
1 aASCR
ry Es
1 R da 1 þ R dt C
ð6Þ
where aASCR is a constant that relates the ‘‘active surface creation rate’’ (ASCR) to the rate of change of crack mouth opening distance during a load cycle and ry and E are the material yield and elastic modulus respectively. The experimental results produced to date in this work are not sufficient to distinguish all of the factors of Eq. (6), however sufficient data has been produced to allow the following interpretation for our conditions: DKc2 ¼ 1:45 103
1R 1þR
ð7Þ
Curve fitting the data for DK < DKc to determine C and m, the following solution for crack growth due to repeated 7 s long thermal shocks is obtained: da 5:89 ¼ 5:28 1016 ðDKÞ dN
for DK < DKc
da 5:89 ¼ 5:28 1016 ðDKÞ þ 7 107 dN
for DK > DKc
ð8Þ ð9Þ
Figs. 7 and 8 show Eqs. (8) and (9) plotted against smoothed test data for cooling water DO of 8 and 2 ppm. Considering the number of simplifications made in this analysis (including the values for da=dtSCF and a), the Gabetta et al. model provides a fairly good representation of the observed environmental assisted crack
Fig. 7. Smoothed experimental crack growth data plotted against a Gabetta et al. [15] model prediction, allowing for the effects of environment and primary load. Experimental data for DO ¼ 8 ppm plotted.
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Fig. 8. Smoothed experimental crack growth data plotted against a Gabetta et al. [15] model prediction, allowing for the effects of environment and primary load. Experimental data for DO ¼ 2 ppm plotted.
growth. p Note however that the continued validity of the Gabetta et al. model at values of DK above 35 MPa m is yet to be verified.
5. Discussion––a simple combined growth model A combined law for explaining RTS crack growth through both the HSF and LEFM regions can be proposed from this work. The model is divided into two sections, the first when crack growth is inside the plastic zone close to the specimen surface and the second is the elastic region that is encountered at further depths. For crack growth in the plastic zone, a linear acceleration of crack growth rate with crack depth is suggested (Eq. (2)). For crack growth outside the plastic zone, the crack growth rate (typically of a decelerating nature) is predicted using a LEFM growth law such as that proposed by Gabetta et al. [15] (i.e. Eq. (6)). The results from such a two-stage analysis are summarised in Fig. 9, which shows the crack growth rate as a function of crack depth for two cases of thermal shock loading. The figure shows a good prediction of the acceleration and deceleration observed in the specimens both with and without primary loading.
Fig. 9. Crack growth as a function of crack length for two specimens along with crack growth predictions using both HSF and LEFM methods. T ¼ maximum cycle temperature (°C), P ¼ primary mechanical load (MPa), DO ¼ dissolved oxygen level (ppm).
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Fig. 9 also shows that the crack depth at which the crossover between the HSF and LEFM growth laws occurs is a critical parameter. Because crack growth will accelerate until this depth is reached, a crack of ‘‘crossover depth’’ can be assumed to rapidly appear during RTS loading. The smaller this depth is, the less chance that appreciable crack growth can occur in the ensuing LEFM region. The crossover depth seems to be related to the plastic stress depth developed by both the thermal shock and steady state primary stresses. A theoretical proposal for determining this depth is not currently possible. However, a possible suggestion, based on an approach that has approximately fitted our experimental data is given below as Eq. (10): 2 1 Kmax ac ¼ ð10Þ bp ry where ac is the crossover depth, Kmax is the maximum stress intensity factor during a cycle, ry is the material yield stress and b is a constant determined by curve fitting (the test data of this work supports a value of approximately 1.2).
6. Conclusions In this paper, crack growth in a heated flat plate specimen exposed to repeated one-dimensional thermal shocks (RTS) has been analysed. Experimental results reveal that crack growth occurs in a stage of acceleration followed by a stage of deceleration. The relative magnitude of the stages is dependent on the presence of primary mechanical loads and the testing environment. A two-stage growth model has been suggested to represent the observed crack growth based on HSF and LEFM techniques. Allowances for environmental and primary steady state stress interactions in the LEFM region have been made through an advanced corrosion fatigue type assessment.
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