1 - University of Manchester



General Time-Dependent C(t) and J(t) Estimation Equations for

Elastic-Plastic-Creep Fracture Mechanics Analysis

Han-Sang Lee, Jin-Ho Je, Yun-Jae Kim*

Department of Mechanical Engineering, Korea University

5Ka Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea

Robert A Ainsworth

The University of Manchester, Manchester M13 9PL, UK

Peter J Budden

Assessment Technology Group, EDF Energy, Barnwood

Gloucester GL4 3RS, UK

* Corresponding author

☎ +82-2-3290-3372

(Fax) +82-2-926-9290

(E-Mail) kimy0308@korea.ac.kr

Manuscript for Submission to

Fatigue and Fracture of Engineering Materials and Structures

Submission Date: December 2015

Revision date: January 2016

(Revised parts are shown in red)

Abstract

This paper presents equations for estimating the crack tip characterizing parameters C(t) and J(t), for general elastic-plastic-creep conditions where the power-law creep and plasticity stress exponents differ, by modifying the plasticity correction term in published equations. The plasticity correction term in the newly proposed equations is given in terms of the initial elastic-plastic and steady state creep stress fields. The predicted C(t) and J(t) results are validated by comparison with systematic elastic-plastic-creep FE results. Good agreement with the FE results is found.

Keywords

Crack-tip stress fields; Elastic-plastic-creep; Plasticity correction term; Time-dependent C(t) and J(t) estimation equations

Nomenclature

a crack length (or half length)

A, B material constants for plasticity and creep, see Eq. (1) and Eq. (2)

b specimen thickness

C(t), C* C-integrals for transient and steady state creep conditions

D normalized opening stress at initial conditions (t=0), HRR fields

E Young’s modulus

F normalized opening stress under steady state creep conditions (t→∞), RR fields

Im, In integration constants that depend on m (or n)

J(t), J(0) J-integrals for transient creep and initial (t=0) conditions

K linear elastic stress intensity factor

Lr parameter related to plastic yielding, see Eq. (3)

m strain hardening exponent, Eq. (1)

n creep exponent, Eq. (2)

Q, QL applied general load, plastic limit load

r, θ polar coordinates at the crack tip

t, tred time, redistribution time, Eq. (8)

Ti traction vector

[pic] displacement rate components (i=1,2)

x, y Cartesian coordinates

W specimen width (or half width)

[pic] strain energy rate density

α coefficient in the Ramberg-Osgood model, Eq. (1)

Г contour of integration around crack tip

ε, εp, εe strain, plastic strain, elastic strain

[pic] creep strain rate

ν Poisson’s ratio

τ normalized time, =t/tred

σ stress

σo yield (0.2% proof) strength

σref reference stress

[pic] stress field normal to crack

[pic] dimensionless function of m (or n) and θ

φ plasticity correction factor, Eq. (9)

φ’ proposed plasticity correction factor, Eq. (15)

FE finite element

M(T) middle-cracked tension

MPC multi-point constraint

SE(B) single-edge-cracked bend

1 Introduction

To assess creep crack incubation and growth of existing crack-like flaws, time-dependent fracture mechanics analysis should be performed. It has been shown that creep crack growth rates can be quantified by the C(t)-integral which characterizes the amplitude of the singular stress and strain fields.1-3 Note that the singular terms in the stress and strain fields hold very near the crack tip and are functions of time, t, following initial loading.4,5 At the steady-state (t(() or under widespread creep conditions, the notation C* is used for the value of C(t). Thus estimates of C(t) and C* are needed to assess creep crack growth in conjunction with creep crack growth rate data determined in terms of C(t) or C* from specimen tests. To predict the incubation and early stages of creep crack growth, on the other hand, the time-dependent failure assessment diagram approach3,6,7 has been developed as a simplified method. This latter approach is an extension of the J-based failure assessment diagram method for low temperature applications8 and requires an estimate of the time-dependent J integral, J(t), for high temperature creep problems.

For elastic-power law creep problems, a relaxation curve for C(t)/C* was proposed by Ehlers and Riedel.9 A slightly different expression was then developed by Ainsworth and Budden,10 which was extended to include the effect of initial plasticity by Ainsworth and Joch.11 The relaxation curves developed by Ainsworth and co-workers include the creep exponent explicitly. However, based on detailed elastic-creep and elastic-plastic-creep finite element (FE) analyses of three typical cracked specimens, Kim et al.12 showed that the relaxation curves for C(t)/C* were not very insensitive to the creep exponent. Note that the equations11,12 which take account of initial plasticity are valid only for equal power law stress exponents, i.e., the plastic hardening and creep exponents are the same (see Eqs. (1) and (2) below for the definition of creep and plastic stress exponents). As materials often have unequal stress exponents, a more general equation applicable to general stress exponent cases needs to be developed.

For estimating the time-dependent J(t)-integral, Ainsworth and Budden10 assumed a simple linear summation of the initial value of J and the steady state increase related to C*, that is C(t)=J(0)+C*t. However, based on detailed elastic-plastic-creep FE analysis, Kim et al.12 showed that FE J(t) results were larger than the estimated J(t) values. Accordingly a simple bounding estimate of J(t) was proposed based on the FE results. As for C(t), the proposed equations for estimating J(t) are supposed to be valid only for equal creep and plasticity stress exponents and thus more general equation applicable to general stress exponent cases should be developed.

The present work presents estimation equations for C(t) and J(t) for general elastic-plastic-creep conditions where the creep and plastic stress exponents are different. The proposal is made simply by modifying the plasticity-correction term in the existing equations. The proposed equations are validated against elastic-plastic-creep FE analysis results for plane strain single-edge-cracked bend (SE(B)) and middle-cracked (M(T)) specimens. Section 2 describes the FE analyses. Sections 3 and 4 present the proposed C(t) and J(t) estimation equations and comparison with FE results, respectively. Conclusions are given in Section 5.

2 Finite element analysis

2.1 Geometry and Loading

Two typical geometries with very different crack-tip constraint levels are considered in this work: (1) plane strain single-edge-cracked bend (SE(B)) specimen and (2) plane strain middle cracked tension (M(T)) specimen, as depicted in Fig. 1. The specimen width (or half-width), W, was taken to be W=50mm with a relative crack length of a/W=0.5.

2.2 Material Properties

An isotropic elastic-plastic material was considered, characterized by the following power-law equation:

[pic] (1)

where σ denotes stress (MPa); ε, εe and εp denote total, elastic and plastic strain, respectively; A and m are material constants. For elastic properties, the following values were used: Young’s modulus E=200GPa and Poisson’s ratio ν=0.3. For plastic properties, the yield (0.2% proof) strength σo was assumed to be σo=300MPa with two values of the strain hardening exponent, m=5 and 10.

For creep analyses, the material was assumed to follow power-law creep:

[pic] (2)

where [pic]is the creep strain rate; B and n are constants. Two values of the creep exponent n were considered, n=5 and 10. For the creep constant, the following values were assumed, B=3.2x10-15 (MPa)–n h-1 for n=5 and B=3.2x10-25 (MPa)–n h-1 for n=10. However, the choices of B do not affect the results as these are presented in a normalized manner.

2.3 Elastic-Plastic-Creep Analysis

Elastic-plastic-creep FE analyses were performed using ABAQUS.13 Figure 2 depicts the FE mesh. Eight-noded plane strain elements with reduced integration (CPE8R within ABAQUS) were used to avoid problems associated with incompressibility. The crack tip was designed with collapsed elements, and a ring of wedge-shaped elements was used in the crack-tip region. To calculate accurate stress fields even under transient creep conditions, sufficiently fine meshes were used at the crack tip. The element size at the crack tip was 0.0002 times the uncracked ligament size, W-a. Validity of the crack-tip mesh design will be discussed later. The numbers of elements and nodes in the FE meshes were about 4,542 and 14,055, respectively. Small displacements are modelled in the FE analyses.

To apply pure bending or tension loading conditions, the MPC (multi-point constraint) option within ABAQUS was used. To quantify the applied loading magnitude, a parameter related to plastic yielding, Lr, is introduced:

[pic] (3)

where Q denotes the generalized load; QL is the plastic limit load based on the von Mises yield condition; and σref is the reference stress. The following expressions are used for plane strain SE(B) and M(T) specimens:2

[pic] (4)

where b is the specimen thickness. Three different values of Lr, Lr=0.5, 0.8 and 1.0, were considered in the present work to examine the effect of the initial loading magnitude on the crack-tip stress fields.

Elastic-plastic-creep FE analyses were performed as follows. In the FE model, constant loading was applied in the first step (at time t=0). The load was then held constant for t>0 and subsequent time-dependent creep calculations were performed. For time-dependent creep calculations, an implicit method was selected within ABAQUS for numerical efficiency. From ABAQUS, values of the C(t)-integral can be extracted as a function of time t. The C(t)-integral is defined by

[pic] (5)

where Ti are components of the traction vector, [pic] are the displacement rate vector components, and [pic] is strain energy rate density. According to the definition of the C(t)-integral, values of C(t) should be extracted close to the crack tip. Thus a very refined mesh around the crack tip is required for accurate FE results and values of C(t) were extracted from the second contour from the crack tip.12,14 At longer times, when the stress has re-distributed within the cracked body, a steady state condition is achieved, the ABAQUS C(t)-integral results show path-independence and the notation C* is used. For a power-law creeping material, the value of the C*-integral is constant for a given load, geometry and material creep properties. Figure 3 shows FE C(t)-integral values extracted from different contours of the SE(B) specimen. The contours are shown in Fig. 2. It shows that, in the steady state condition (t>tred), the C* value is path independent. At shorter times, C(t) values are path dependent and the value closer to the crack tip is larger.

For efficient FE calculations, the mesh density should be determined according to the time scale. For instance, when the steady state C* integrals are of interest, relatively coarser crack-tip mesh would be sufficient. However, when the transient C(t) values at very small times are needed, very fine mesh is needed.

2.4 Validity of FE Mesh

To ensure confidence in the mesh, the asymptotic behaviour of C(t) at very short times was evaluated. Elastic-creep FE analysis was performed. At short times (t→0), values of elastic-creep C(t) should satisfy:1,9

[pic] (6)

where K denotes the stress intensity factor due to initial loading and tred is an estimate of the time to redistribute to steady-state conditions. Figure 4 compares the FE results with Eq. (6) suggesting that the FE mesh used in the present work can provide accurate C(t) values even at times 4 order of magnitude less than the redistribution time (i.e. normalized times t/tred=10-4).

3. C(t) Estimation Equations and Comparison with FE Results

3.1 Existing C(t) Estimation Equations

The following relaxation curve for C(t) is proposed in terms of time t and steady-state C* by Ehlers and Riedel:9

[pic] (7)

where the normalized time τ is given in terms of the redistribution time tred

[pic] (8)

and J(0) denotes the initial FE value of J(t) at time t=0. Note that Eq. (7) can be applied only to elastic-creep cases, not to elastic-plastic ones.

Joch and Ainsworth11 proposed another approximation by matching the near-tip stress and strain fields described by C(t) with estimates of the time-dependent J(t):

[pic] (9)

where the constants A and B in Eq. (9) are defined according to Eqs. (1) and (2), respectively. The plasticity correction factor φ in Eq. (9) reflects the influence of initial plasticity on C(t). As 0≤ φ ≤1, C(0)=C*/(1-φ) is finite,11 except for the elastic-creep case (φ=1). Moreover, φ (0 for extensive plasticity so that C(t)/C*=1.11 For the elastic-creep case, Eq. (9) reduces to the result of Ainsworth and Budden10 with a singular asymptotic behaviour at short times given by Eq. (6). An important point to note is that Eq. (9) was derived based on the assumption of equal stress exponents for plasticity and creep (n=m). When the stress exponents are different (n≠m), Eq. (9) cannot be applied, which will be shown later.

Based on detailed elastic-plastic-creep FE analysis of three typical cracked specimens, Kim et al.12 showed that relaxation curves of C(t)/C* are not sensitive to the creep exponent n, and (n+1) in Eq. (9) can be replaced by a constant value of 4.5:

[pic] (10)

3.2 Validity of Existing C(t) Estimation Equations

The three approximations given in Section 3.1 are compared with elastic-creep FE results for the SE(B) specimen in Fig. 5, for two different n values, n=5 and 10. Among the three approximations, the predictions using Eq. (7) agree closest with the FE results and tend to overestimate the FE data, but Eq. (7) cannot be applied to elastic-plastic-creep problems. The predictions using Eq. (10) overestimate the FE results for short times, whereas those using Eq. (9) are underestimates. Although the predictions using Eq. (10) seem to be better than those using Eq. (9), Eq. (9) will be used subsequently for the following reasons. Firstly it is an analytical equation derived by matching near-tip stress and strain fields. The second point is that in practice steady state C* values tend to be overestimated. Then predicted C(t) values using Eq. (9) can be much closer to the FE results in Fig. 5b for n=10. For instance, when the C* value is overestimated by 10% for n=10, predicted C(t) values using Eq. (9) are close to the FE results in Fig. 5b.

For elastic-plastic creep cases with m=n, the φ values in Eq. (9) can be calculated using FE J(0) and C* results. The values of FE J(0) and C* are tabulated in Tables 1 and 2 and calculated values of φ are listed in Table 3. Note that the results in Table 1 are independent of the creep exponent n whereas those in Table 2 are independent of the plasticity exponent, m. Figure 6 compares C(t)/C* predicted using Eq. (9) with FE results. The figure shows that the relaxation curves for C(t)/C* are not very sensitive to the specimen geometry and loading conditions. Furthermore the predictions can capture the plasticity effect on the initial C(t) values reasonably well.

For unequal stress exponent cases, n≠m, the calculated φ values using the FE J(0) and C* results are also shown in Table 3. This shows that, for the case where n>m, the calculated φ values are finite but negative and, for the case where m>n, they are unity regardless of Lr. This confirms that Eq. (9), which was developed for cases with n=m, cannot simply be used for n≠m cases. The trends in φ shown in Table 3 can be rationalized by noting that, from the reference stress approximations to J(0) and C*,3,8 [pic] so that [pic], 0 as σref (∞ for n>m, n0, the normalized opening stress under creep conditions is similarly given by

[pic] (12)

where Im (or In) is an integration constant that depend on m (or n), and [pic] in Eqs. (11) and (12) denotes a dimensionless function of m (or n) and θ. Note that Eq. (12) is strictly valid for r→0 at fixed time, t>0. At long times under steady state-creep conditions, the normalized opening stress becomes a function of r and is denoted as F:17

[pic] (13)

Using Eqs. (9) and (13), Eq. (12) can be re-written as

[pic] (14)

By matching Eq. (11) and Eq. (14) at time t=0, we can obtain that[1]

[pic] (15)

With φ’ a functions of r. Equation (15) has the same form as Eq. (9) but the plasticity-correction factor is different. In the proposed equation, Eq. (15), the plasticity-correction factor φ’ is given in terms of the stress states at the initial and steady-state conditions, whereas in Eq. (9), it is given in terms of fracture mechanics parameters, J(0) and C*. In the cases of n=m, the equations are the same. It will be shown that Eq. (15) can be applied even for the case of n≠m.

For evaluation of φ’ in Eq. (15), its dependence on the position r should be emphasised. The φ value in Eq. (9) does not depend on the location, as J(0) and C* values are path independent. However, the normalized opening stresses at initial and steady state conditions do depend on r, see Eqs. (11) and (13). For n=m, dependence on the position r disappears and the φ’ value does not depend on r. However, for n≠m, it does depend on r. In practice, the value of r can be chosen to be a suitable fracture process zone size (for instance, related to the grain size for creep crack growth).

The φ’ values in Eq. (15) for n≠m cases are given in Tables 4 and 5 at five different distance from the crack tip; r/a ranging from 0.0025 to 0.02. (Noting that the crack length a in the problem is a=25mm, r/a=0.0025 and 0.02 correspond to r=62.5μm and 500μm, respectively). These are calculated using Eqs. (11) and (13) using the FE J(0) and C* values. Values of Im (or In) and [pic] are found from Ref. [15, 16]. Values of [pic] for the SE(B) and M(T) specimens are similar. For the (m=5, n=10) case, the φ’ values decrease with increasing r/a. For the (m=10, n=5) case, on the other hand, the trend is opposite; the φ’ values increase with increasing r/a.

3.4 Comparison with FE Results

Predicted C(t)/C* relaxation curves using Eq. (15) for the case of m=5 and n=10 are compared with FE results in Fig. 7. Two predictions are shown; one using r/a=0.005 and the other r/a=0.015. Corresponding results for the case of m=10 and n=5 are shown in Fig. 8. Overall, the FE C(t)/C* results agree well with predicted relaxation curves. For the case of m=5 and n=10, predictions using r/a=0.015 agree slightly better than those using r/a=0.005. For the case of m=10 and n=5, the trend is the opposite; predictions using r/a=0.005 agree better than those using r/a=0.015. Overall the effect of the choice of r is not particularly significant and as noted a fracture process zone size can be used to approximately define r.

4. J(t) Estimation Equations and Comparison with FE Results

4.1 J(t) Estimation Equations

In deriving Eq. (9), Ainsworth and Budden10 assumed the following simple approximation for J(t):

[pic] (16)

Note that Eq. (16) is asymptotically correct both for t=0 and for the long time, t((. Later, based on detailed elastic-plastic-creep FE analysis of three typical cracked specimens, Kim et al.12 showed that FE J(t) results are consistently higher than predictions using Eq. (16). Typical variations of FE J(t) with normalized time for the m=n cases are shown in Fig. 9. This shows that the differences between the FE results and Eq. (16) increase with decreasing Lr and with increasing n. Although not shown in Fig. 9, it was also found that the geometry and loading condition affect the differences. Kim et al.12 proposed the following simple bounding estimate of J(t):

[pic] (17)

where φ is given by Eq. (9). As for Eq. (9), Eq. (17) is valid only for the m=n cases.

In Section 3.3, a new parameter φ’ was derived, which is valid for unequal stress exponents. Thus Eq. (17) can be generalized simply by replacing φ with φ’:

[pic] (18)

where φ’ is given by Eq. (15).

4.2 Comparison with FE Results

Predicted J(t) values using Eq. (18) for the equal stress exponent cases (m=n) are compared with FE results in Fig. 10. Results are presented using the ratio of estimated value, J(t)est, from Eq. (18) and the FE results J(t)FE. For m=n=5, the predictions are slightly higher than the FE results for transient creep conditions. For m=n=10, the differences are greater but are less than 25%. At steady state creep conditions, the predictions are closer to the FE results. In all cases, the predicted J(t) values are larger than the FE results, suggesting that Eq. (18) gives conservative J(t) estimates.

Corresponding results for the unequal stress exponent cases (n≠m) are shown in Fig. 11. Two predictions are also shown; one using r/a=0.005 and the other r/a=0.015. For the case of m=5 and n=10, the overall trends are similar to those for the m=n=10 case. For the case of m=10 and n=5, they are similar to those for the m=n=5 case. The choice of r/a seems to be not particularly significant.

5. Conclusions

In the present work, estimation equations for C(t) and J(t) are proposed for general power law elastic-plastic-creep conditions where the creep and plastic stress exponents are different. The proposed expressions are derived simply by modifying the plasticity-correction term in existing equations which are valid only when the creep and plastic stress exponents are equal. In these previous equations, the plasticity-correction term is expressed in terms of the initial J and steady state C* values. In the newly proposed equations, on the other hand, it is given in terms of the initial elastic-plastic and steady state creep stress fields at some distance from the crack tip; these can be calculated from analytical Hutchinson-Rice-Rosengren and Riedel-Rice field expressions. In the case when creep and plastic stress exponents are the same, the proposed equations recover the earlier relationships.

To validate the proposed equations, the predicted C(t) and J(t) results are compared with systematic elastic-plastic-creep FE results for plane strain single-edge-cracked bend and middle-cracked specimens. In the FE analysis, the creep and plastic stress exponents are varied systematically. It is found that the proposed equations provide good agreement with the FE results, even when the creep and plastic stress exponents are different.

In this work, the proposed equations for C(t) and J(t) are validated against two-dimensional plane strain FE results. Although such validation is believed to be sufficient, it would still be valuable to validate against three-dimensional FE results for more realistic cases such as through-wall or surface cracks in plates or in pipes. Further validation against three-dimensional FE results are in progress and the results will be reported separately. Another discussing point is that visco-plastic constitutive equations for typical materials can be very complicate and thus should be characterized by non power-law creep. The proposed equation can be easily extended to non-power law materials using the reference stress approach embedded in R5.3

Acknowledgements

This research was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning. (NRF- 2013M2A7A1076396, NRF-2013M2B2B1075733)

References

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2] Webster, G. A. and Ainsworth, R. A. (1994) High Temperature Component Life Assessment, Chapman & Hall, UK.

3] R5, Issue 3: Assessment procedure for the high temperature response of structures (2003) with updates to Revision 002, 2014, EDF Energy, Gloucester, UK.

4] Gallo, P., Berto, F. and Glinka, G. (2015) Generalized approach to estimation of strains and stresses at blunt V-notched under non-localized creep. Fatigue Fract. Eng. Mater. Struct., (electronic version available)

5] Tang, K. K. and Li. S. H. (2015) Interactive creep–fatigue crack growth of 2024-T3 Al sheets: selective transitional functions. Fatigue Fract. Eng. Mater. Struct., 38, 597–609.

6] Ainsworth, R. A. (1993) The use of a failure assessment diagram for initiation and propagation of defects at high temperatures. Fatigue Fract. Eng. Mater. Struct., 16, 1091-1108.

7] Ainsworth, R. A., Hooton, D. G. and Green, D. (1999) Failure assessment diagrams for high temperature defect assessment. Eng. Fract. Mech., 62, 95-109.

8] R6, Revision 4: Assessment of the integrity of structure containing defects (2001) with updates to Amendment 11, 2015, EDF Energy, Gloucester, UK.

9] Ehlers, R. and Riedel, H. (1981) A finite element analysis of creep deformation in a specimen containing a macroscopic crack. Proceedings 5th International Conference on Fracture, Cannes, France, Pergamon, 2, 691-698.

10] Ainsworth, R. A. and Budden, P. J. (1990) Crack tip fields under non-steady creep conditions-I. Estimates of the amplitudes of the fields. Fatigue Fract. Eng. Mater. Struct., 13, 263-276.

11] Joch, J. and Ainsworth, R. A. (1992) The Effect of geometry on the development of creep singular fields for defects under step-load controlled loading. Fatigue Fract. Eng. Mater. Struct., 15, 229-240.

12] Kim, Y. J., Dean, D. W. and Budden, P. J. (2001) Finite element analysis to assess the effect of initial plasticity on transient creep for defects under mechanical loading. Int. J. Press. Vessels Piping, 78, 1021-1029.

13] ABAQUS version 6. 13. (2013) User’s manual, Inc. and Dassault Systems.

14] Song, T. K., Kim, Y, J., Nikbin, K. and Ainsworth, R. A. (2010) Estimation of the transient creep parameter C(t) under combined mechanical and thermal stresses. Eng. Fract. Mech., 77, 685-704.

15] Huchinson, J. W. (1968) Singular Behavior at End of a Tensile Crack Tip in a Hardening Material. J. Mech. Phys. Solids, 16, 13-31.

16] Rice, J. R. and Rosengren, G.F. (1968) Plane Strain Deformation near a Crack Tip in a Power-Law Hardening Material. J. Mech. Phys. Solids, 16, 1-12.

17] Riedel, H. and Rice, J. R. (1980) Tensile Cracks in Creeping Solids, ASTM STP 700, Philadelphia, 112-130.

Table 1 Values of FE J(0) (Unit: MPa∙mm).

| | |SE(B) | |M(T) |

| | |Lr=0.5 |Lr=|Lr=1.0 |

| | | |0.8| |

| | |Lr=0.5 |Lr=|Lr=1.0 |

| | | |0.8| |

| |Lr=0.5 |Lr=0.8 |Lr=1.0 | |Lr=0.5 |Lr=0.8 |Lr=1.0 | |m=n=5 | |0.96 |0.77 |0.59 | |0.94 |0.73 |0.53 | |m=n=10 | |1.00 |0.93 |0.65 | |1.00 |0.91 |0.60 | |m=5, n=10 | |(-2.92E9) |(-1.53E11) |(-8.49E11) | |(-3.40E9) |(-1.73E11) |(-9.21E11) | |m=10, n=5 | |(1.00) |(1.00) |(1.00) | |(1.00) |(1.00) |(1.00) | |

Table 4 Values of φ’ in Eq. (15) for the case of m=5 and n=10.

r/a |Lr=0.5 | |Lr=0.8 | |Lr=1.0 | | |SE(B) |M(T) | |SE(B) |M(T) | |SE(B) |M(T) | |0.0025 |1.00 |1.00 | |0.99 |0.99 | |0.98 |0.98 | |0.0050 |1.00 |1.00 | |0.99 |0.99 | |0.97 |0.96 | |0.0100 |1.00 |1.00 | |0.98 |0.98 | |0.95 |0.93 | |0.0150 |1.00 |1.00 | |0.98 |0.97 | |0.93 |0.90 | |0.0200 |1.00 |1.00 | |0.97 |0.96 | |0.91 |0.88 | |

Table 5 Values of φ’ in Eq. (15) for the case of m=10 and n=5.

r/a |Lr=0.5 | |Lr=0.8 | |Lr=1.0 | | |SE(B) |M(T) | |SE(B) |M(T) | |SE(B) |M(T) | |0.0025 |0.89 |0.89 | |0.06 |0.01 | |0.00 |0.00 | |0.0050 |0.92 |0.92 | |0.31 |0.28 | |0.00 |0.00 | |0.0100 |0.94 |0.94 | |0.50 |0.48 | |0.00 |0.00 | |0.0150 |0.95 |0.95 | |0.58 |0.56 | |0.00 |0.00 | |0.0200 |0.96 |0.96 | |0.63 |0.62 | |0.13 |0.11 | |

[pic][pic]

(a) (b)

Fig. 1 Schematics of specimens considered in this work: (a) plane strain SE(B) and (b) plane strain M(T). Note that the height of the specimen was taken as equal to 2W.

[pic]

Fig. 2 FE mesh (include a close-up of the detailed crack tip mesh).

[pic]

Fig. 3 Comparison of C(t)/C* from elastic-creep FE calculations of SE(B) specimen for different contour.

[pic][pic]

(a) (b)

Fig. 4 Variations of C(t)/C* from elastic-creep FE calculations with normalized time: (a) SE(B) specimen and (b) M(T) specimen.

[pic][pic]

(a) (b)

Fig. 5 Comparison of FE C(t)/C* from elastic-creep FE calculations of the SE(B) specimen with three estimation equations: (a) n=5 and (b) n=10. Note that φ=1 in Eqs. (9) and (10).

[pic][pic]

(a) (b)

[pic][pic]

(c) (d)

[pic][pic]

(e) (f)

Fig. 6 Comparison of predicted C(t)/C* with elastic-plastic-creep FE results for the m=n cases: (a)-(b) Lr=0.5; (c)-(d) Lr=0.8; and (e)-(f) Lr=1.0.

[pic][pic]

(a) (b)

[pic][pic]

(c) (d)

[pic][pic]

(e) (f)

Fig. 7 Comparison of predicted C(t)/C* with elastic-plastic-creep FE results for the case of m=5 and n=10: (a)-(b) Lr=0.5; (c)-(d) Lr=0.8; and (e)-(f) Lr=1.0.

[pic][pic]

(a) (b)

[pic][pic]

(c) (d)

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(e) (f)

Fig. 8 Comparison of predicted C(t)/C* with elastic-plastic-creep FE results for the case of m=10 and n=5: (a)-(b) Lr=0.5; (c)-(d) Lr=0.8; and (e)-(f) Lr=1.0.

[pic][pic]

(a) (b)

Fig. 9 Variations of FE J(t) with normalized time: (a) SE(B) and (b) M(T).

[pic][pic]

(a) (b)

Fig. 10 Comparison of predicted J(t) with elastic-plastic-creep FE results for the case of m=n: (a) m=n=5 and (b) m=n=10.

[pic][pic]

(a) (b)

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(c) (d)

Fig. 11 Comparison of predicted J(t) with elastic-plastic-creep FE results for the case of m≠n: (a)-(b) m=5 and n=10; and (c)-(d) m=10 and n=5.

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[1] Note that this is not exactly correct because Eq. (12) or Eq. (14) are valid only for small r at t→0. However, this is a plausible assumption also made in previous works.10,11

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