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Procedía Engineering 130 (2015) 1677 - 1685

Procedía Engineering

www.elsevier.com/locate/proeedia

14th International Conference on Pressure Vessel Technology

Unified Correlation of In-Plane and Out-of-Plane Creep Constraints

with Creep Crack Growth Rate

H.S. Maa, G.Z. Wang3 *, F.Z. Xuana, S.-T. Tua

aKey Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237,

Abstract

In this paper, the equivalent creep strain distributions ahead of crack tips in different specimens were calculated by extensive finite element analyses, and the creep crack growth (CCG) rates of these specimens were simulated over a wide range of C*. The capability and applicability of the constraint parameter Ac for characterizing both in-plane and out-of-plane creep crack-tip constraints and establishing a unified correlation with CCG rate of a steel were investigated. Base on the parameter Ac, the unified correlation formulas of in-plane and out-of-plane constraints with CCG rate of a steel have been obtained. © 2015 The Authors.PublishedbyElsevierLtd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICPVT-14

Keywords:In-plane constraint; out-of-plane constraint; creep crack growth rate; equivalent creep strain; unified correlation; specimens

1. Introduction

Under creep conditions, a lot of experimental and theoretical evidences have shown that crack-tip constraint can affect creep crack growth (CCG) rate [1-8]. To accurately predict the creep life and achieve structural integrity assessments for high temperature components, it is important to quantify the creep crack-tip constraint levels. However, the studies for the creep constraint effects and the two-parameter or three-parameter characterization of creep crack-tip fields are very limited.

* Corresponding author. Tel.: +86-21-64252681; fax: +86-21-64253153. E-mail address: gzwang@ecust.edu.cn

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

doi:10.1016/j.proeng.2015.12.337

In our previous work [9], by analogy to the unified constraint parameter Ap based on the equivalent plastic strain for elastic-plastic fracture condition, a unified creep constraint parameter based on crack-tip equivalent creep strain was defined as follows [9]:

Where Aceeq is the area surrounded by the equivalent creep strain (£c) isoline ahead of a crack tip and Are/is the reference area surrounded by the £c in a standard specimen, t is creep time, and the tred is stress redistribution time. The Ac represents a ratio of the Aceeq of a specimen to the reference area Aref at the same £c isoline, the same creep time t/tred=l and C* level. An increase in the parameterAc implies a loss ofconstraint.

In this work, the equivalent creep strain sc distributions ahead of crack tips for specimens with different geometries and loading configurations (different in-plane and out-of-plane constraints) were calculated by FEM, and the CCG rates of these specimens are simulated over a wide range of C* by using stress dependent creep ductility and strain rate model in a ductility exhaustion based damage model. Based on the results, the capability and applicability of the parameter Ac for characterizing both in-plane and out-of-plane creep crack-tip constraints and establishing a unified correlation with CCG rate for specimens with different geometries and loading configurations were investigated.

2. Finite element model and numerical procedures

The material used in this work is a Cr-Mo-V steel (Chinese 25Cr2NiMolV steel). The true stress-strain curve of the steel at 566°Cwas shown in Fig. 1 [10].The Young's modulus E and yield stress ayof the steel at 566°C is 160GPa and 383MPa, respectively. For the Cr-Mo-V steel, the two-regime Norton (2RN) creep model in Eq. (2) has been developed in our previous work [10] by using the relation between minimum creep rate and stress, and the different Norton model parameters of (A1, n1) in low stress regime and (A2, n2) in high stress regime are listed in Table 1 [10]. This 2RN behavior of the Cr-Mo-V steel is similar to that ofthe 316H steel [11, 12].

Ac = Aceeq / Aref at t/tred=l

2.1. Material

2RN model

0.0 0.2 0.4 0.6 0.8 True strain

Fig. 1. The true stress-strain curve of the Cr-Mo-V steel at 566°C [10].

Tablel.The 2RN creep model parameters[10].

Stress A(MPa"°h"1) n

a<250MPa Ay=7.26xl0-26 n1=8.75

a>250MPa A2=3.53xlQ-36 n2=13.08

2.2. Specimens geometry and Finite element model

Three sets of specimens with different loading configurations and geometries were used in FEM analyses. To investigate the in-plane constraint effect, four values of crack depths denoted as a!W=- 0.2, 0.35, 0.5, and 0.7 were set for the two-dimensional (2D) plane strain SEN(B), SEN(T), and M(T) specimens. To investigate the out-of-plane constraint effect, four values of specimen thickness denoted as BIW= 0.1, 0.25, 0.5, and 1 were set for the three-dimensional (3D) specimens.

Due to symmetry in geometry, only half of the 2D plane strain M(T), SEN(B) and SEN(T) specimens and a quarter of the 3D M(T), SEN(B) and SEN(T) specimens were modeled. The typical finite element meshes for the 2D and 3D SEN(T) specimens (W = 20 mm, B =10 mm and a/W = 0.5) are illustrated in Figs. 2(a) and (b), respectively. The typical model in Fig. 2(a) contains 7751 four-node linear plane strain elements (CPE4H) and 7785 nodes, while it contains 63788 eight-node linear 3-D elements (C3D8H) and 72905 nodes in Fig. 2(b).The analyses have been carried out using ABAQUS code [13]. The local mesh distribution around the crack tip is shown in Fig. 2(c).

Fig. 2.Typical meshes in finite element models of 2D SEN(T) specimen (a), 3D SEN(T) specimen (b), typical local meshes around the crack region (c) and typical local meshes along the CCG region (d)

2.3. Creep damage model and creep crack growth simulation

The creep crack growth rate of the specimens with different loading configurations and geometries (different inplane and out-of-plane constraints) was simulated. The FEM models are the same as Figs. 2(a) and (b). The element

size in the crack growth zone is usually taken to be similar to grain size of the steels examined [14, 15]. For the Cr-Mo-V steel used in this study, the average grain size is estimated to be around lOO^m [10, 16], thus the element size of lOO^m is uniformly set in the crack growth zone, as shown in Fig. 2(d).

The ductility exhaustion based creep damage model has been widely used in the simulations of CCG rate [10, 1416]. In this damage model, the rate of damage m is defined by the ratio of equivalent creep strain rate sc and multiaxial creep ductility £^[10, 14-16].

A® = — (3)

And the total damage at any time is calculated from the time integral of creep damage accumulation rate in Eq. (4) [10, 14-16]:

ft ft s

I audi = \ —^dt (4)

Jo Jo £ f v 7

where m is the creep damage rate, £cis the equivalent creep strain rate, and sf is the multiaxial creep ductility. The value of damage parameterwis between 0 and 1.

The Cocks and Ashby model [17] in Eq. (5) was widely used which is based on the cavity growth theory by power-lawcreep [10, 14-16]:

£f —— = sin ' 2 f n - 0.5 ^ / sinh (n - 0.5 ^ CTm

— 2

£f _ 3 v n + 0.5 j v n + 0.5 j Oe

where the sf and £f are multiaxial and uniaxial creep ductility, respectively, and the n is the creep exponent (for power law creep). The uniaxial creep ductility £f in Eq. (5) is usually assumed to be a constant for a given temperature. However, a lot of experimental results and analyses have shown that the creep fracture mechanism depends on stress levels (strain rates), which leads to the stress-regime dependence of the creep ductility of materials [18-23]. The creep ductility can be expressed by Eq. (6) in a wide range of stress [10, 16].

sfl a <225MPa

■clam + c2 225 < a <275MPa ef, 275MPa <a

where £f1 and £f2 are lower shelf and upper shelf creep ductility, respectively, and c1, c2 and m are constants (ci=1.14xl0"14, c =5.37xl0"3 and m=5.80). The lower shelf and upper shelf creep ductility were estimated to be £f1 = 0.5 and£y2 = 1-6, respectively. Eq. (6) was implemented in the ABAQUS code by a combination of user subroutine USDFLD and CREEP, and the CCG rates in a wide range of C* were simulated by using FEM for the specimens with different loading configurations and geometries .The true stress-strain curve of the steel in Fig. 1 and the 2RN creep model parameters in Table 1 were used.

3. Results and discussion

3.1. Creep crack growth rates in specimens with different in-plane and out-of-plane constraints

To establish unified correlation of in-plane and out-of-plane creep constraints with CCG rates, the CCG rates in specimens with different in-plane and out-of-plane constraints need to be obtained for a wide range of C*. Figs. 3-5

(a) show the simulated da/dt-C* curves for the 3D SEN(B), SEN(T) and M(T) specimens with different out-of-plane constraints (different B). Figs. 3-5 (b) show the simulated da/dt-C* curves for the SEN(B), SEN(T) and M(T) specimens with different in-plane constraints (different a/W) in 2D plane strain (PE) . For a given specimen, a comparison between Figs. 3-5 show that the effect of out-of-plane constraint induced by specimen thickness B on CCG rate is more obvious than that of in-plane constraint induced by crack depth a/W. Thus, in CCG tests and life assessments, it should pay more attention on the effect of out-of-plane dimensions of specimens or components on CCG rate than that of in-plane dimensions, especially for the high constraint specimen or component geometries (such as SEN(B) and C(T) specimens).

3.2. Unified correlation of in-plane and out-of-plane creep constraints with creep crack growth rate

It has been identified that the constraint parameters T, Q, A2 and stress triaxiality factor h based on crack-tip stress fields are not a good unified representative of constraints, and a monotonic trend curve does not exist between these parameters and fracture toughness for various specimens with different in-plane and out-of-plane constraints under elastic-plastic fracture conditions [24-28, 29].It is necessary to identify whether or not the parameter Ac can be used to establish the unified correlation of creep constraints with creep crack growth rates for the different specimen geometries with a wide of range in-plane and out-of-plane constraints.

□ B=2mm

o Z?=5mm

A 5= 10mm

V 5=20mm

1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.1 C*, MPa.m/h

(b) 10

SEN(B)-PE

Turning point 2

Turning point 3

Turning point 1

□ a/W=02

o a/W= 0.35

A a/W= 0.5

V a/W=0.1

1E-7 1 E-6 1E-5 1E-41E-3 0.01 0.1 C*, MPa.m/h

Fig. 3. Simulated CCG rate curves for the SEN(B) specimens (a) different out-of-plane constraints; (b) different in-plane constraints.

I 1E-3 a

^r 1E-4 -§ 1 E-5 1 E-6 1 E-7

a/W= 0.5

Turning point 2

Turning point 3

Turning point 1

□ 2?=2mm

o Z?=5mm

• SEN(T)5-Experiment

A B= 10mm

V ß=20mm

1E-7 1E-6 1 E-5 1E-4 1 E-3 0.01 0.1 C* MPa.m/h

□ a/W=0.2

V a/W= 0.35

A a/W= 0.5

O a/W=0J

1 E-7 1 E-6 1 E-5 1E-4 1E-3 0.01 0.1 C* MPa.m/h

Fig. 4. Simulated CCG rate curves for the SEN(T) specimens (a) different out-of-plane constraints; (b) different in-plane constraints.

' 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.1 C*, MPa.m/h

0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7

M(T)-PE

Turning point 2

Turning point 3

Turning point 1

□ ci/tV=0.2

0 aW= 0.35

A a/W= 0.5

V a/W= 0.7

1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0. C* MPa.m/h

Fig. 5. Simulated CCG rate curves for the M(T) specimens (a) different out-of-plane constraints; (b) different in-plane constraints.

Based on the unified constraint parameter Ac, the relations between CCG rates and overall constraint levels in specimens or structures with different in-plane and out-of-plane constraints may be obtained. The CCG rates are usually expressed in the form [30],

a = D o C *q (7)

where a is CCG rate and in mm/h, C* is crack-tip fracture parameter and in MPa.m/h. The Do and q are material constants which are often measured experimentally. When the constraint effect is considered, the CCG rate can be expressed as a function of C* and constraint parameter. The correlation lines calculated between CCG rates da/dt and Ac for two typical C* values at low C* region (3xl0~7 MPa.m/h and 3xl0~6 MPa.m/h) and transition C* region (3x10 s MPa.m/h and 3xl0~4 MPa.m/h) are shown in Figs. 6 (a) and (b), respectively. The data of the C(T) specimens in our previous work [9] are also included in Fig. 6.

(a) 0.01 1E-3

1 1E-4 1E-5

□ C(T) C*=3xl0'7MPa.m/h o M(T) C*=3xlO'7MPa.m/h a SEN(T) C*=3x 1 o 'MPa.m/h v SEN(B) C*=3x 1 o'MPa.m/h 0 C(T) C*=3xl0"6MPa.m/h < M(T) C'=3x 1 O^MPa.in/h > SEN(T) C*=3xlO"6MPa.m/h 0 SEN(B)C*=3xl0'*MPa.m/li

..........i........i........

Fig. 6. The relations between CCG rates daldt and constraint parameterAcfor two typical C* values at low C* region (3xl07MPa.m/h and 3*106 MPa.m/h) (a) and transitionC* region (3 x10 s MPa.m/h and 3x10^ MPa.m/h) (b) for all specimens with various in-plane and out-of-plane constraint levels (the standard specimen is the C(T) specimen with W = 20mm and a/W =0.5 in plane strain)

The constraint dependent CCG rate a can be described in the following form [31],

a = a» f (Ac) (8)

where a0 is the CCG rate from the standard specimen with high constraint, andf Ac) is a function of Ac. If the CCG rate in the high constraint C(T) specimen with B=10mm, W=2B=20mm and a/W=0.5 is taken to be the normalized CCG rate a0 and the CCG rate ratio a/a0is calculated for the four C* levels in Fig. 6, the relations between a/a0 and the parameter Ac can be established for the two C* regions, as shown in Fig. 7. It can be seen from Fig. 7 that there exists a linear relation on log-log scale between the a /a0 and Ac regardless of the specimens with different geometries, loading configurations, crack sizes and load levels C*.

Power law relation can be used to fit the a/a0- Ac curves in Fig. 7. The fitting a/a0- Ac relations for the low C* region and transition C* region are shown in Eqs. (9) and (10).

f (Ac) = a / a0 = 1.312Ac ~1 623 , a0 = 0.793C *0611 (9)

f{Ac) = a/ a0 = 1.221 Ac~1008 , a0 = 0.157C *0484 (10)

The Eqs. (9) and (10) represent the relations between the CCG rates and the overall constraint levels composed of in-plane and out-of-plane constraints induced by different specimen geometries, loading configurations and crack sizes for a given material. These formulas may be used in constraint-dependent CCG life assessments of high-temperature structures with any in-plane and out-of-plane constraint levels.

(a) 100 10

□ C(T) C*=3xlO'1MPa.m/h o M(T) C*-3xl0 7MPa.m/h A SEN(T) C*=3xlO'7MPa.m/h v SEN(B) C*=3xlO"7MPa.m/h O C(T) C"*=3xlO"6MPa.m/h < M(T)C*=3xlO'6MPa.m/h t> SEN(T) C*=3xlO"6MPa.m/h 0 SEN(B) C*=3xl0^MPa.m/h

H—SgS^ïDy . _ JO

□ C(T) C*=3xlO'5MPa.m/h o M(T) C*=3x 10"5MPa.m/h a SEN(T) C*=3xl0'5MPa.m/h v SEN(B)C=3xlO"!MPo.m/h o C(T)C*=3xlO"4MPa.m/h < M(T) C*=3xlO'4MPa.m/h > SEN(T)C*=3xl0JMPa.m/h o SEN(B)C*=3xl0JMPa.m/h

Fig. 7.The relations between à /à0 andAcfor low C* region (a) and transition C* region (b) (the standard specimen is the C(T) specimen with W = 20mm, B = 10mm and a/W = 0.5)

4. Conclusion

In this work, the equivalent creep strain sc distributions ahead of crack tips for specimens with different geometries and loading configurations (different in-plane and out-of-plane constraints) were calculated by FEM, and the CCG rates of these specimens are simulated over a wide range of C* by using stress dependent creep ductility and strain rate model in a ductility exhaustion based damage model. The capability and applicability of the parameterAc for characterizing both in-plane and out-of-plane creep crack-tip constraints and establishing a unified correlation with CCG rate for specimens with different geometries and loading configurations were investigated. The main results obtained are as follows:

(1) In low(below turning point l)and transition C* (between turning points 1 and 2) regions, the out-of-plane (induced by specimen thickness B) and in-plane (induced by crack depth alW) constraints have effects on CCG rates, and with increasing the out-of-plane and in-plane constraints, the CCG rate increases. The effect of out-of-plane constraint on CCG rate is more obvious than that of in-plane constraint.

(2) The out-of-plane and in-plane constraint effects on the CCG rates are related to specimen geometries. The CCG rate of high constraint specimen geometry (such as SEN(B)) is more sensitive to the out-of-plane and in-plane constraints than that of low constraint specimen geometry (such as M(T)).

(3) The monotonic correlation lines between the CCG rate da/dt and the constraint parameter Ac on log-log scale can be formed for all specimens with different geometries, crack sizes and loading configurations. This further suggests that the parameter Ac is a unified characterization parameter for a wide of specimen geometries and in-plane and out-of-plane creep constraints.

Acknowledgments

This work was financially supported by the Projects of the National Natural Science Foundation of China (51375165, 51325504)

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