C∗ estimation for cracks in mismatched welds and finite element validation

C∗ estimation for cracks in mismatched welds and finite element validation

© 2004 Kluwer Academic Publishers Printed in the Netherlands.

validation

FU-ZHEN XUAN, SHAN-TUNG TU and ZHENGDONG WANG

School of Mechanical Engineering, East China University of Science and Technology, 130 Meilong Road, PO Box 402, Shanghai 20037, China (e-mail: fzxuan@ecust.edu.cn)

Received 28 July 2003; accepted in revised form 10 February 2004

Abstract A C∗integral estimation method is proposed for a crack located in the weld with a mismatch in mechan-ical properties from the surrounding base material The method involves the definition of an equivalent stress-creep strain rate (ESCSR) relationship based on the mechanical properties of both the weld and base materials and the geometrical dimension of welding seam The value of creep fracture mechanics parameter C∗, for the mismatched weldment, is then estimated using the proposed ESCSR in conjunction with the reference stress (RS) method where the reference stress is defined based on the plastic limit load and the GE/EPRI estimation scheme Referring

to the equivalent stress-plastic strain (ESPS) curve in R6 and SINTAP procedures, an approximate solution for the ESCSR relationship has been obtained Detailed formulae for the compact tension (CT) specimens have been derived on the basis of limit load solutions Nonlinear finite element analysis of 48 cases with various degrees

of mismatch in creep behaviour and different dimension of welding seam has been performed for CT specimens Overall good agreement between the ESCSR method and the FE results provides confidence in the use of the proposed method in practice.

Key words: Creep crack, C∗integral estimation, mismatched welds, reference stress method.

1 Introduction

Mismatch frequently occurs in structure welds operating at high temperature In order to carry out a more precise assessment of the integrity of a cracked welded structure, knowledge of the creep crack initiation and creep crack growth is needed In engineering practice, a global parameter C* in the extensive creep regimes is often computed and then used in an empir-ical correlation for crack initiation or crack growth assessment (Dogan and Petrovski, 2001) Previous results indicate that the CCG behaviour in a weldment is dependent not only on the crack growth resistance along the crack path, but also on the deformation response of the complete welded component (Andersson et al., 2000; Segle et al., 2000) With respect to the creep fracture mechanics parameter calculation of mismatched welded structures, however, the traditional way is simply to consider that the welded structure is made of a homogeneous material with the lowest mechanical properties among those of the base material, weld mater-ial and heat affected zone (HAZ) This method, generally, yields safe results However, it may

be over-conservative for some cases and non-conservative for some others (Tu, 2002; Tu and Yoon, 1999) To calculate C∗for a crack in such a complicated component more accurately, the finite element method (FEM) is, currently, the best method because it can simulate various weld geometries and mismatching variables (Segle et al., 2000; Tu and Yoon, 1999) However,

it is also useful to have simplified estimation methods for engineering calculations

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For homogeneous materials, the engineering estimation schemes of C∗include the GE/EPRI method (Kumar et al., 1981; Kumar, 1982; Kumar and German, 1988), the RS approach originally proposed by Ainsworth (1984) and the enhanced reference stress method recently proposed by Kim et al (2001) These methods, however, cannot be directly used in C∗ estima-tion of inhomogeneous materials To approximately estimate the C∗parameter in mismatched welds, recently, some attempts have been made using the weighted average strain rate by Budden et al (2000) and the corrected reference stress by Assire et al (2001) However, no engineering applicable solutions are provided in these results, which are limited to specific case

This paper presents an engineering estimation method for C∗for a crack in a mismatched weld The weld model considered is an idealized bi-material ‘sandwich’ structure without HAZ and residual stress The two materials, weld and base material, have the same elastic modulus, E and Poisson’s ratio The term ‘mismatch’ then refers to the two materials having different creep coefficient and creep exponent in a Norton law or having different stress-creep strain rate relationships First, the concept of an ‘equivalent stress-creep strain rate relation-ship’ for the weld model is introduced This is then used to develop estimates of C∗for the CT specimen Finite element results for this type of specimen are then presented and compared with the estimates using the proposed equivalent stress-creep strain rate relationship

2 Summary of engineering C∗estimation methods for homogeneous materials

For two-dimensional cracked structures, the steady-state creep fracture mechanics parameter

C∗is derived from the Rice contour integral applied to a viscous problem

C∗=

where the notation Ŵ is used to denote the integral path enclosing the crack tip and traversing anticlockwise; Ti are the outward traction vectors on ds; ˙ui are the displacement rate vector components; x and y are coordinates in a rectangular coordinate system, and ds is the incre-ment on the contour path Ŵ,w˙ is the strain energy rate density and given byw˙ =˙ε

c ij

0 σijd˙εijc, where σij and ˙εcij denote the stress and creep strain rate tensors, respectively

Generally, the theoretical expressions of C∗for complexity structures are very difficult to obtain according to the above definition Even when the analytical solutions of C∗are built for certain structures, they are usually showing expressions of high complexity and unsuitable for engineering application In practice, two engineering methods, i.e., GE/EPRI scheme and RS method, are often employed to estimate the value of C∗and further predict the residual life of defective structures

2.1 GE/EPRIMETHOD

From the analogy between plasticity and creep, the J estimation scheme in the GE/EPRI handbooks (Kumar et al., 1981; Kumar, 1982; Kumar and German, 1988) can be used in principle to estimate the C* integral, by replacing the strain with the strain rate For a material

of which creep deformation properties follow the power law (Norton’s law) relation

where ˙εc denotes the creep strain rate; B and n are material constants, namely, creep coeffi-cient and creep exponent, respectively

The estimation formulation for C∗is then derived from GE/EPRI method in the following form

C∗= Blh1(n, a) F σT

0.2

FL

n +1

where the function l represents the ligament thickness of the cracked structures; the non-dimensional function h1(n, a)is plastic influence function calibrated from detailed FE results

as a function of the component geometry, the crack length, the loading mode and the creep exponent n F is the generalized load; and FLis a reference generalized load for normalization The reference load FLcan be chosen arbitrary, and it is typically chosen as the plastic limit load of the cracked component corresponding to the proof stress σ0.2T

It should be noted that, although the proof stress σT

0.2appears in Equation (3), FLdepends linearly on σ0.2T and thus it is eliminated in Equation (3) Thus the choice of σ0.2T does not affect C* as long as a consistent value of σT

0.2is used to evaluate FL

As the non-dimensional function h1(n, a) are calibrated from FE results for Ramberg-Osgood materials, to apply this method, the creep deformation data of the material of interest should be idealized in the form of the power law creep For non R-O materials, the piecewise estimation method proposed by Lei et al (1994) may be suitable for C∗estimation

2.2 REFERENCE STRESS METHOD

Following Ainsworth (1984), the steady state C∗ integral for creep can be estimated on the basis of the RS approach

where ˙εcref is the creep strain rate at σ = σref, determined from creep-deformation data for

a material of interest, and K is the elastic stress intensity factor The reference stress σrefis defined by

σref= F σ0.2T /FLσT

where F represents the magnitude of the applied load and FLis the corresponding magnitude

at plastic collapse for the proof stress σT

0.2 corresponding to 0.2% inelastic strain and crack size a Again, as FL depends linearly on the proof stress σ0.2T , σref does not depends on σ0.2T , and thus the choice of σ0.2T does not affect C∗results

There are advantages of the RS method over the GE/EPRI method Firstly, the RS method

is not restricted to idealized power law creep materials, and can be applied to materials exhib-iting general creep behaviour In addition, it is quite simple to use, and is easy to generalize to more complex cases, provided that the plastic limit load solutions are available for the cracked components of interest

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Figure 1. The ‘sandwich’ model of a cracked mismatched bi-material weld and fictitious equivalent material.

3 C∗estimation for crack in mismatched welds

From the two recommended estimation methods for C∗integral, it is worth noting that both RS and GE/EPRI estimation methods are related to the creep resistance described by the stress-creep strain rate relationship of the materials For mismatched weld problems, however, this resistance is neither represented by the stress-creep strain rate relationship of the weld nor that

of the base material, but results from contributions from the full weldment For elastic-plastic fracture analysis of the cracked mismatched weld at ambient temperature, an ‘equivalent stress-plastic strain relationship’ is proposed by Lei and Ainsworth (1997) to describe the characteristic of the plastic deformation of the mismatched weld, and such that J estimation for the cracked mismatched structure reduces to J calculation for the homogeneous structure made of the ‘equivalent’ material Recently, such equivalent material approach has been in-corporated in the R6 (2000) and SINTAP procedures (Kim et al., 2000) In consideration of the analogy between plasticity and creep as well as the definitions of J and C∗, it is expected

to find an equivalent stress-creep strain rate relationship and then proceed to establish a C∗ estimation formulation for the cracked mismatched structures

3.1 EQUIVALENT STRESS-CREEP STRAIN RATE RELATIONSHIP

The actual mismatched weld, with a crack parallel to the interface of the weld and base mater-ial and located in the middle of the weld, is usually simplified as an idealized bi-matermater-ial

‘sandwich’ structure without HAZ and residual stress, as shown in Figure 1a In R6 and SINTAP procedures, this problem is treated as an equivalent homogeneous material with the same configuration as shown in Figure 1b which follows the constitutive equation

σeq(εp)= (FLmis/FLb− 1) σw(ε

p)+ (M − FLmis/FLb) σb(εp)

where FLmis represents the limit load for the mismatched weld; FLb is the limit load of the cracked homogeneous structure for base metal; σb(εp)and σw(εp)are respectively the stress-plastic strain relationship of base metal and of the weld metal; M denotes the mismatch

ratio defined at a number of plastic strain values, M = σw(εp)/σb(εp) For εp = 0.2%,

Mcorresponds to the 0.2% proof stress ratio

From the phenomenological point of view, no difference exists between plasticity and creep in metal and both of them are often classified as the inelastic strains In terms of different causes of deformation, total strain of the metal at high temperature is split into three components: elastic strain, creep strain and plastic strain and can be expressed by

ε(σ, t)= εe(σ )+εp(σ )+ εc(σ )ine, (7) where εe(σ ), εp(σ )and εc(σ )is the elastic strain, plastic strain and creep strain, respectively

At a given value of time t and taking the creep strain as plastic strain, the equivalent rela-tionship described by Equation (6) should hold for all the inelastic strains Under the condition

of long term creep state, the total strain of Equation (7) is dominated by creep component and the plastic strain component is often negligible In this instance, the constitutive equation for the fictitious equivalent material is

σeq(εine)= (FLmis/FLb) σw(εine)+ (M − FLmis) σb(εine)

For both base and weld materials obeying the elastic-power creep law, the inelastic strain

in both materials is expressed as

εine= ˙εct= Bbσnb

εine= ˙εct= Bwσnw

where Bb, nb and Bw, nw are, respectively the material constants for base metal and weld metal; t denotes the service time

In terms of Equations (9) and (10), the mismatch ratio M can be calculated by the following formulation for high temperature application

M= (εine/t)nb−nwnb nw (Bw)−nw1 (Bb)nb1 (11) Equation (11) indicates that the mismatch ratio M is a time dependent function at high temperature For base metal and weld metal with the same value of creep exponent, nb = nw,

Mthen reduces to a function of material constants which is independent of service time t Substitute Equations (9) and (10) into Equation (8), the equivalent stress-creep strain rate relationship for the mismatched weld is then obtained by canceling out the variable t

σeq= ˙εc

Bb

1

nb M − FLmis/FLb

˙εc

Bw

1

nw FLmis/FLb− 1

3.2 C∗ INTEGRAL ESTIMATION

Now that the actual creep resistance in the weldment is reflected by the ESCSR of a fictitious material (Figure 1b), it is reasonable to assume that C∗ can be estimated from previously developed methods for homogeneous materials but using the equivalent stress-creep strain rate relationship developed above

For the cases of ESCSR of mismatched welds exhibiting power law creep behaviour

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The C∗ integral for mismatched welds can be estimated using the GE/EPRI-scheme based approach Taking the reference load FL in Equation (3) as the limit load of the mismatched welded structures and inserting Equation (5) into Equation (3), the C∗integral for mismatched welds can be obtained from

If nb = nw, the equivalent stress-creep strain rate relationship will have the same creep exponent (nb = nw = neq) as that of weld or base metal In this instance, creep coefficient

Beqwill reduce to

where

fb= M− FMLmis/FLb

− 1 , fw =

FLmis/FLb− 1

Substitute Equation (15) into Equation (14) and replace neqwith nb, the GE/EPRI-scheme based C∗estimation formulation is then rearranged as

C∗= Bblh1(nb, a)(σref)nb +1fb+ fw(Bb/Bw)1/nb−n b

(17) For the cases of ESCSR relationship exhibiting general creep behaviour, i.e., not always confined to the power law creep materials, C∗integral for mismatched welds can be estimated from the RS based approach, that is

Under the condition of base metal and weld metal with the same creep exponent, nb = nw, Equation (18) is thus simplified as

C∗= Bb(σref)nb −1K2fb+ fw(Bb/Bw)1/nb−n b

(19)

It should be noted that, the calculations of elastic stress intensity factor K and reference stress σref in Equations (14) and (18) are identical with that for the homogeneous material

at ambient temperature because the difference of elastic properties of different zone in weld

is neglected Although K and σref depend on the load magnitude, load type and structural geometry, the ratio of K/σrefinvolved in C∗calculation is independent of load magnitude

4 Application to mismatched welded CT specimen

Kim and Schwalbe (2001) have developed a series of limit load solutions for mismatched welded CT specimen (Figure 2) based on the FE results Inserting those limit load solutions into Equation (12) proposed in this work, the corresponding ESCSR for the mismatched CT specimen is then established

For the under-matching weld, M < 1, the equivalent stress-creep strain rate relationship is

σeq= ˙εc

Bb

1

nb M − min(x1, x2)

˙εc

Bw

1

nw min(x1, x2)− 1

Figure 2. Schematic figure for mismatched welded CT specimen.

where

C1e−C2 (ψ −2)+ C3 for 2
ψ

x2=









M[1.094 − 1.017(ψ/10) + 3.129(ψ/10)2

− 1.952(ψ/10)3

] for 2
ψ
7

ψ = (W − a)/ h, C1= 9(M − 1)/10, C2= 1/(20(1 − M)), C3= (M + 9)/10, W, a and h are the geometrical parameters of CT specimen, as shown in Figure 2

For the over-matching weld, M > 1, the equivalent stress-creep strain rate relationship of

CT specimen is

σeq= ˙εc

Bw

1 nw

0
ψ
ψ1 (21)

σeq= ˙εc

Bb

1

nb M − x3

M− 1 +

˙εc

Bw

1

nw x3− 1

where ψ1= 1.9e−(M−1)/3, A= M50+ 49, B = 49(M50− 1)− C, C = 0.3(M − 1)√M− 1 Coalescing Equation (17) and Equations (20), (21) or (22), the GE/EPRI-scheme based estimation of C∗ for welded CT specimen is then obtained Furthermore, substitute Equa-tions (20), (21) or (22) into EquaEqua-tions (18) and (19), we can get the approximate values of C∗ integral based on the RS method

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Table 1. Mechanical property utilized in C∗calculation.

Materials Stress-strain relationship Elastic modulus B (MPa−nh−1) n

E(MPa)

Table 2. Cases calculation for mismatched CT specimens and their equivalent stress-creep strain rate relationship (a/W = 0.5).

Case 2h/mm ψ M Base Weld FLmis/FLb Equivalent stress-creep strain rate relationship

material material No.1 3 8 1.29 Mt1 Mt2 1.09 σeq = 463.594(˙εc) 9.031

No.2 3 8 0.77 Mt1 Mt3 0.92 σeq = 392.393(˙εc) 9.031

No.3 4 6 1.29 Mt1 Mt2 1.12 σeq = 477.137(˙εc) 9.031

No.4 4 6 0.77 Mt1 Mt3 0.89 σeq = 379.659(˙εc) 9.031

No.5 2 12 1.29 Mt1 Mt2 1.06 σ eq = 450.635(˙εc) 9.031

No.6 2 12 0.77 Mt1 Mt3 0.95 σeq = 405.974(˙εc) 9.031

No.7 3 8 ≈0.55 Mt1 Mt4 0.71 σ eq = 150.336(˙εc)1/9.03+ 153.465(˙εc)1/9.36 No.8 3 8 ≈1.81 Mt4 Mt1 1.15 σeq = 76.993(˙εc)1/9.03+ 192.621(˙εc)1/9.36

Four materials, denoted Mt1 to Mt4 in Table 1, with different creep coefficients and creep exponents have been chosen to examine the predictions of the above C∗estimation approach The data of Mt1 and Mt4 are respectively those of 1Cr0.5Mo and 1.25Cr0.5Mo from the work of Yoon and Kim (1999) The other two materials are idealized to produce a range of stress-creep strain response

Eight different mismatching cases have been considered The mismatching conditions and the ESCSR relationship calculated from Equation (20) or Equations (21) and (22) are given in Table 2 The results indicate that ESCSR relationships of mismatched weld depend not only

on the mechanical properties of the weldment constituents, but also on the geometry of the weldment and its constituents C∗ integral values are estimated using the GE/EPRI-scheme based approach Equation (17) and the RS based method Equations (18) and (19) Typical results are compared later with results from finite element analyses which are described next The detailed comparisons between the ESCSR for both over-matching (M > 1) and under-matching (M < 1) weldment and the stress- creep strain rate curves of weld metal and base metal are depicted in Figure 3 It can be seen from the Figure 3 that, the ESCSR is mainly dominated by the creep behaviuor of base metal no matter whether it is under-matching weld

or over-matching weld and affected by the value of weld width h When the value of welding seam width h decreases, the ESCSR curve of mismatched weld from Equation (12) closes to the stress-creep strain rate curve of homogeneous specimen for base metal While with the

value of h increasing, the ESCSR curve gradually closes to the stress-creep strain rate curve

of homogeneous specimen made of weld metal, as seen in Figure 3a and 3b It is expected according to the above results that as long as the welding seam is wide enough, the weld metal will control the creep resistance of mismatched weld and the influence of base metal can be negligible in this instance

5 Finite element validation

To investigate the reliability of engineering estimation equations for C∗, described in the Section 3, extensive elastic-creep FE analysis are performed for the welded CT specimens The general purpose FE program ABAQUS was used to calculate the C∗ integral A small geometry change continuum FE model was employed In order to avoid problems associated with material incompressibility, the 8-node reduced integration elements (ABAUQS element type CPE8R) were used (ABAQUS User’s Manual, 1996) The number of elements used was

918 and 2941 for the nodes Figure 4 depicts typical FE meshes for CT specimens, employed

in the present work

The overall specimen width is 30 mm and height is 28.8 mm with a thickness of 13 mm,

W = 24 mm and crack length of a = 12 mm to make a/W = 0.5, and the width of welding seam 2h = 2, 3, 4 mm, respectively The static loads applied to the nodes at the upper half

of the pin hole, as shown in Figure 4, are taken as F = 3000, 3920, 5000, 6000, 7000,

7500 N, respectively The node at the specimen end with the arrow heads is fixed as a boundary condition The material properties used in the finite element analysis are shown in Table 2

In the elastic-creep FE analysis, a mechanical load was firstly applied to the FE model using an elastic calculation at time t = 0 The mechanical load was then held constant and subsequent time-dependent creep calculations were performed For power-law creep, ABAQUS provides an in-built routine to calculate the C(t) integral which is again denoted

as C∗under the steady state condition with time exceeding the redistribution time, tred, (Kim

et al., 2002) Total 48 cases were analyzed and C∗ integral values were calculated on five contours surrounding the crack tip and average values are shown in Figure 5 The C∗values calculated at any contour differ by less than 5% from the average values

6 Discussion

The C∗ values estimated by the equivalent stress-creep strain rate method proposed in this paper are depicted in Figure 5 and show good agreement with those calculated by the FEM for situations of both over-matching, Figure 5a, and under-matching, Figure 5b The cases considered cover variation in weld width h In Figure 5, all values of C∗ of weldements are

in between of the results of homogeneous structures made of base metal and weld metal and also affected by geometry of welding seam For the weldments with smaller h, the value of

C∗is closer to of homogeneous specimens for base metal than to of homogeneous specimens for weld metal With the width h of welding seam increasing, the C∗of weldment is gradually nearing of homogeneous specimens for weld metal

Summarizing results in Figure 5, it can be concluded that predictions based on the GE/EPRI method are best for the idealized power law creep This is not surprising as the h1function in the GE/EPRI method were calibrated from detailed FE results using power law plasticity, and thus the GE/EPRI method should be accurate in principle However, for cases in Figure 5c

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Figure 3. Comparison between the equivalent stress-creep strain rate curve and the stress-creep strain rate curves

of homogeneous specimens for base metal and weld metal (for CT specimen with a/W = 0.5).