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Procedía Engineering 130 (2015) 853 - 860

Procedía Engineering

www.elsevier.com/locate/procedia

14th International Conference on Pressure Vessel Technology

Unified Correlation of Constraint and Strength Mismatch with Fracture Toughness ofBimetallic Joint

J. Yanga'*

aSchool of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093;China

Abstract

A unified constraint characterization parameter Ap has been defined and a unified correlation line of in-plane and out-of-plane constraint with fracture toughness has been obtained based on the homogeneous material A508 in the previous study of authors. In order to study whether the unified correlation of constraint with fracture toughness can be established by using the constraint parameter Ap for the bimetallic joint, in this paper, a bimetallic joint constituted with A508 ferritic steel and Alloy52M was selected, the finite element numerical simulation method was used to model the equivalent plastic strain (sp) distributions ahead of crack tips for specimens with different in-plane and out-of-plane constraints, and the ./-resistance curves of specimens were obtained by the finite element method (FEM) simulations based on GTN (Gurson-Tvergaard-Needleman) damage model. Unified correlation of constraint with fracture toughness of bimetallic joint has been investigated. The results show that there exists a sole linear relation between the /ic//ref and A regardless of the in-plane and out-of-plane constraint. The /ic//ref- A line can be

regarded as a unified reference line to characterize the dependence of material's fracture toughness on constraint. So the Ap is a

suitable unified constraint parameter which can characterize both in-plane and out-of-constraint for bimetallic joint. For there

exists heterogeneous of fracture property and strength mismatch in bimetallic joint, the Ap can be also regarded as a suitable

unified constraint parameter which can characterize both constraint and strength mismatch.

©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; Fracture toughness; Strength mismatch; Bimetallic joint

* Corresponding author. Tel.: +86-021-55272320. E-mail address: yangjie@usst.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.2015.12.209

1. Introduction

Constraint is the resistance of a structure against plastic deformation, the loss of constraint increases the load bearing capacity of cracked components [1]. It is known that constraint can be divided into two conditions of inplane and out-of-plane according to the crack plane. The in-plane constraint relates to the specimen dimension in the direction of growing crack, while the out-of-plane constraint relates to the specimen dimension parallel to the crack front. However, constraint is not only a function of geometry, but also of strength mismatch. Strength mismatch is also exists in the structure [2]. As constraint can significantly alter the material's fracture toughness, it is important to develop a clear understanding of its effect on the fracture behavior of material.

Nomenclature

Â2 Ap

B E fo fc fF

fN, EN, Sn

q1, q2 ,q3

T TZ W Aa,.

3D FEM GTN SENB

crack length

parameter quantifying second and third term of stress relative to the first term in a cracked elastic-plastic body

a new unified parameter for quantifying both in-plane and out-of-plane constraints area surrounded by equivalent plastic strain isoline

area surrounded by equivalent plastic strain isoline at fracture measured in a standard test specimen thickness Young's Modulus

initial void volume fraction in GTN model critical void volume fraction in GTN model final failure parameter in GTN model the void nucleation parameters in GTN model J-integral

fracture toughness characterized by J-integral fracture toughness measured in a standard test stress intensity factor loading span

Strain hardening exponent

the constitutive parameters in the GTN model

a constraint parameter under elastic-plastic condition

T-stress constraint parameter under elastic condition

factor of the stress-state in 3D cracked body

specimen width

crack growth extension

Strain hardening coefficient

a total constraint parameter

equivalent plastic strain

Poisson's ratio

yield stress

tensile strength

a unified constraint parameter defined by plastic region area three dimension finite element method Gurson-Tvergaard-Needleman

single-edge notched bend_

Within the framework of geometry constraint (in-plane and out-of-plane constraints), some constraint parameters, such as K-T [3], J-Q [4, 5], J-A2 [6], Tz [7-9] etc. have been suggested to characterize the constraint effect. These parameters were successfully used to quantify the in-plane or out-of-plane constraint separately. In order to characterize both in-plane and out-of-plane constraints together, Mostafavi et al. [10-13] defined a unified constraint parameter q> which quantified constraint by the size of the plastic region at the onset of fracture. Unfortunately, the constraint parameter ^ has its limitation in characterizing constraint at higher J-integral for the ductile material with higher fracture toughness [14]. The authors in the previous study [14] have defined a unified constraint parameter Ap by modified the parameter ^ as follows:

A _ PEEQ

where Apeeq is the areas surrounded by the equivalent plastic strain (ep) isolines ahead of the crack tip and Are/is the reference area surrounded by the Sp in a standard test. This unified constraint parameter has been indentified to be suitable for specimens with different in-plane and out-of-plane constraints [14, 15]. And a unified correlation line (Jic/Jref- A line) of in-plane and out-of-plane constraints with fracture toughness was obtained [15]. However, the

results mentioned above were only obtained from a homogeneous material (A508 steel). It is not clear that whether the constraint parameter Ap can be used to quantify strength mismatch and the unified correlation of in-plane and out-of-plane constraint with fracture toughness can be established for the bimetallicjoint with strength mismatch.

In this paper, the bimetallic joint constituted with A508 ferritic steel and Alloy52M was selected. The areas surrounded by the Sp isolines ahead of crack tips in the specimens under different constraint conditions were calculated by finite element method (FEM). The material's fracture toughness under various constraint conditions were obtained by the FEM simulation based on GTN damage model. And then, the parameter Ap was analyzed as a unified measure of constraint for bimetallic joint and the unified correlation of in-plane and out-of-plane constraints with fracture toughness of the bimetallicjoint was established.

2. Finite element analysis

2.1. Materials of the bimetallic joint

Two common low-alloy steels A508 and Alloy52Mb for making nuclear pressure vessels were used to constitute the bimetallic joint. The measured average mechanical property data are listed in Table 1, and the true stress-strain curves are shown in Fig.l.

C3 1000

c*> 600

QJ 400

-□-A508 —o—Alloy 52Mb

0.0 0.2 0.4 0.6 0.8 True strain s

Fig. 1. True stress-strain curves of the two materials at room temperature.

2.2. Specimen geometry

The single-edge-notched bend (SENB) specimen was used in FEM analyses. The specimen with W=32mm, B=16mm and a/W=0.5 is considered as standard specimen. The interface is located in the center of the specimen and the crack is located in A508/52Mb interface. To investigate the in-plane constraint quantitatively, five crack depths denoted as a/W=0.2, 0.3, 0.5, 0.6 and 0.7 were chosen. To investigate the out-of-plane constraint quantitatively, four specimen thicknesses denoted as B=4, 8, 16 and 32mm were chosen. The loading configuration and geometries of the specimen are illustrated in Fig.2 (a).

Table 1. Mechanical property data ofthe two materials at room temperature.

Material Young's modulus E(MPa) Poisson's ratio v yield stress a0 (MPa) Strain hardening coefficient a Strain hardening exponent n

A508 202410 0.3 514 3.89 7.04

Alloy52Mb 178130 0.3 495 10.73 4.28

2.3. Finite element model

The commercial finite element code, ABAQUS, was used to calculate the J-integral and equivalent plastic strain (£p) distributions ahead of crack tips. The three-dimensional (3D) finite element models for the specimens with different in-plane and out-of-plane constraints were built and eight-node isoperimetric elements with reduced integration (C3D8R) were used for all the specimens. The typical finite element meshes for the standard specimen are illustrated in Fig.2 (b). Since the crack-tip region contains steep stress and strain gradients, the mesh refinement should be more at the crack tip. Thus a conventional mesh configuration having a focused ring of elements surrounding the crack front was used with a small initial root radius (2pm) at the crack tip (blunt tip) to enhance convergence of the nonlinear iterations, as shown in Fig.2 (c). The typical model in Fig.2 (b) contains 59472 elements and 181704 nodes. The loading is applied at the up center of the SENB specimens by prescribing a displacement of 6mm. The J-integral and the areas surrounded by the sp isolines ahead of crack tips were calculated.

Fig. 2. (a) loading configuration and geometries ofthe SENB specimen; (b)the typical meshes in the finite element model ofspecimen; (c) the typical local meshes around the crack tip region; (d) the typical local meshes along the crack growth region in the GTN finite element models.

2.4. GTN damage model

Ductile crack growth in metals is a result of nucleation, growth and coalescence of microvoids. In order to obtain the crack growth resistance curves and ductile fracture toughness of the bimetallic joint under different constraint conditions, the FEM simulation based on GTN damage model was chosen. The local approach based on the GTN damage model has been widely used to simulate ductile crack growth and obtain fracture resistance of materials, and its applicability for specimens with different crack tip triaxiality has been validated in the literature [2, 14-17]. The FEM model with GTN model was the same as Fig.2 (b), and a more fine mesh was arranged along the crack growth region, as typically shown in Fig.2 (d). The mesh sizes of O.lmmxO.lmm were used [14].

The GTN model contains nine parameters in general: the constitutive parameters qi7 q2 and qs7 the void nucleation parameters^, snand Sn the initial void volume fraction/?, the critical void volume fraction/c and the final failure parameter/^. To simplify the complex calibration process for these parameters, the values of qi, q2 and q3 are usually fixed to be qi=1.5, qi=\7 q3=qi2=2.25 and the values of £n=0.3 and Sn=0.1 have been used for low alloy steels in most investigations [17]. The parameters /o,/n, / and/ are commonly obtained by fitting the numerical result of J-resistance curve with experimental result [2, 17]. In the previous work of authors [14], these parameters /o,/n,/ and/F) of the A508 and 52Mb have been obtained, and their applicability for specimens with different constraints has also been validated by experiments. The GTN model parameters obtained for the A508 steel are/0=O.OOO2,/N=O.OO2,/c=O.O4 and/F=0.17, while the 52Mb are/0=O.OOOO8,/N=O.OO2,/c=O.O5 and/F=0.2.

The load versus load-line displacement curves can be obtained from the FEM simulations. With instantaneous crack lengths obtained at each loading point, a crack growth resistance curve {J- Aa curve) can be determined in reference to the resistance curve procedure where multiple points are determined from a single-specimen test, as specified in ASTM E1820 [18]. The crack extension length Aa, corresponding to each loading point i can be measured from the crack growth simulation step. As the ductile crack path is zigzag in the simulation, the crack growth extension, Aa,, is defined as a distance of the projection of the actual crack growth path down on the initial crack plane. The ductile crack growths of the SENB specimens under different constraint conditions were numerically simulated by using the GTN model based FEM, and the J- Aa curves were obtained. In order to include the blunting effect, the numerical resistance curves were corrected by adding half of the CTOD value to Aa (Aa corrected= Aa +CTOD/2) [2]. The ductile fracture toughness Jjc of the specimens were determined by the 0.2mm offset line in the J- Aa curves.

3. Numerical results and discussion

3.1. J-resistance curves and/racture toughness

Fig. 3 shows the J-resistance curves measured from the SENB specimens with different in-plane and out-of-plane constraints. It can be seen from Fig. 3 (a) that the J-resistance curves decrease with increasing the in-plane (crack depth), Fig. 3 (b) shows that with increasing specimen thickness from 4mm to 8mm J-resistance curves increase, and then they decrease with increasing specimen thickness from 8 to 16mm. The values of the ductile fracture toughness Jic of the different specimens with various in-plane and out-of-plane constraint were determined by the 0.2mm offset lines in the J-resistance curves, as shown in Fig. 3.

a 1800 M 1600 1400

3 1200 1000 oc 800 e 600 ¡5 400 200 0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

( rack extension A a, in in Crack extension A a, mm

Fig.3. (a) J-resistance curves for SENB specimens with different in-plane constraint; (b) with different out-of-plane constraint.

3.2. The areas surrounded by £p isolines under different magnitudes o/ J-integral

The specimen with a/W=0.5, B=16mm and W=32mm (W=2B) is considered as a standard specimen. The different equivalent plastic strain Sp isolines (£p=0.01, 0.1, 0.2 and 0.3) at J=50KJ/m2 and J=500KJ/m2 for standard specimen were calculated, as shown in Figs.4 (a) and (b), respectively. It can be seen that the distribution of equivalent plastic

strain ahead of crack tip is asymmetric due to the local heterogeneous mechanical properties, this is different from the sp isolines at J=50KJ/m2 and J=500KJ/m2 for homogeneous material (A508 steel), as shown in Figs.4 (c) and (d).

The areas surrounded by ep isolines around the crack tip is small at J=50KJ/m2, when the J-integral reached 500KJ/m2, the general yield has been produced in the specimen, and the ep=0.01 isoline ahead of the crack tip impinged into the plastic zone induced by loading point on the specimen surface.

Fig.4. (a) the ep contours at J=50KJ/m2 for bimetallic joint; (b) the ep contours at J=500KJ/m2 for bimetallic joint; (c) the ep contours at J=50KJ/m2 for homogeneous material (A508 steel); (d) the ep contours at J=500KJ/m2 for homogeneous material (A508 steel).

3.3. Correlations of fracture toughness with Apeeq for specimens with different constraints

In order to examine whether or not the Apeeq (area surrounded by equivalent plastic strain isoline) is a unified measure of constraint for bimetallic joint, the Apeeq values in the middle plane of the specimen for various ep isolines (ep=0.01, 0.1, 0.2 and 0.3) ahead of crack tips at J=JIC were calculated. The relations between fracture toughness JIC and Apeeq for the specimens with different in-plane and out-of-plane constraints were plotted. The Apeeq in Fig.5 (a) is for the ep=0.01 isoline, and that in Fig.5 (b) is for the ep=0.1, 0.2 and 0.3 isolines. It can be seen that there is a weak monotonic relation between JIC and the Apeeq areas surrounded by ep=0.01 due to the scatter in data, while a good monotonic correlation between JIC and the areas Apeeq surrounded by ep=0.1, 0.2 and 0.3 can be seen.

Fig.5. (a) fracture toughness versus the Apeeq surrounded by ep=0.01 isoline forthe specimens with different constraints; (b) ep= 0.1, 0.2 and 0.3

The reason for this is that the general yielding and larger plastic deformations have been produced prior to fracture. The areas surrounded by ep=0.01 isoline have been apparently influenced by the plastic deformations outside of the crack-tip region at fracture (Fig.4 (b)), it can't accurately characterize the crack-tip plastic deformation which is related to crack-tip constraint. However, the ep=0.1, 0.2 and 0.3 isolines still exist in the crack-tip region and are not affected by the plastic deformations outside of the crack-tip region (Fig.4 (a)). It is interesting

to note that all the date for the specimens with different in-plane and out-of-plane constraints are on a single Jic -Apeeq line for the £p=0.1, 0.2 and 0.3 isolines separately. This means that there is a unified relation between JIC and the Apeeq for the specimens with different in-plane and out-of-plane constraints. This suggests that the Apeeq may be a reliable unified measure ofconstraint, as it is equally sensitive to both in-plane and out-of-plane constraints.

3.4. Unified correlation of in-plane and out-of-plane constraint with fracture toughness for bimetallic joint

After being normalized by the reference toughness Jref and reference area Aref of the standard specimen, the normalized fracture toughness Jic/Jref against the square root of the constraint parameter Ap for the £p=0.1

isoline can be plotted, and as shown in Fig.6. It can be seen that there is a linear relation between the Jic/Jrefand regardless of the in-plane and out-of-plane constraints. So the^A is a suitable unified constraint parameter

which can characterize both the in-plane and out-of-plane constraints.

Fig.6. Normalized fracture toughness JIC/Jre/versus the JA f°r the specimens with different constraint levels.

It should be noted that there exists heterogeneous fracture properties and thus strength mismatch in bimetallic joint. Thus, the Ap is also a suitable unified constraint parameter which can characterize both geometry constraint (in-plane and out-of-constraint) and strength mismatch. Compared with the total constraint parameter fij in Ref. [19], the Ap can suit for more wider range. It not only suits for small and large scale yielding conditions, but also suits for undermatching, homogeneous and overmatching conditions, which have limitation for the parameter ^t [19].

The Jic/Jref - line in Fig.6 is a unified correlation line of in-plane and out-of-plane constraint with fracture

toughness of bimetallic joint. It can be used to determine constraint dependent fracture toughness or structurally relevant fracture toughness. If the reference fracture toughness Jref of standard specimen with high constraint and the unified constraint parameters A of the other specimens are obtained by FEM calculations or experiments, the

Jic/Jref -reference line (such as in Fig.6) can be built. And the constraint dependent fracture toughness or

structurally relevant fracture toughness can be determined from the reference line. It may be used to assess the safety of a cracked component with any constraint levels. The JIC/Jref - .^JA reference line and its use need to be

further investigated.

4. Conclusion

In this study, the J-resistance curves and fracture toughness of SENB specimens with different in-plane and out-of-plane constraints were obtained through the FEM based on GTN damage model, and the FEM was used to calculate the sp distributions ahead of crack tips for the specimens with different constraints. Furthermore, the Ap was analyzed as a unified measure of constraint to obtain the unified correlation of constraint with fracture toughness ofabimetallicjoint. The main results obtained are summarized as following.

(1) The parameter Ap has a good correlation with fracture toughness regardless of the levels of the in-plane and out-of-plane constraints. It is a suitable unified constraint parameter which can characterize both in-plane and out-of-plane constraints for bimetallic joint.

(2) There exists heterogeneous of fracture property and strength mismatch in bimetallic joint. Thus, the Ap can be regarded as a suitable unified constraint parameter which can characterize both geometry constraint and strength mismatch.

(3) There exists a sole linear relation between the JIC/Jre/ and regardless of the constraint levels for

bimetallic joint. The JIC/Jre/ - JA can regarded as a unified reference line to characterize the

dependence of material's fracture toughness on constraint, and can be used to obtain constraint dependent fracture toughness or structurally relevant fracture toughness. The JIC/Jre/ - ,JA reference line and its use

need to be further investigated.

Acknowledgements

This work was financially sponsored by Natural Science Foundation of Shanghai (15ZR1429000) and the Youth Foundation of Shanghai Province (ZZslgl5013).

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