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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/249956819 Experimental Evaluation of the J or C Article in Journal of ASTM International · January 2006 DOI: 10.1520/JAI13221 CITATIONS READS 49 4 authors, including: C.M Davies Noel O'Dowd Imperial College London University of Limerick 103 PUBLICATIONS 535 CITATIONS 155 PUBLICATIONS 3,272 CITATIONS SEE PROFILE SEE PROFILE Kamran Nikbin Imperial College London 264 PUBLICATIONS 1,821 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Improving the Accuracy of Elevated Temperature Design and Life Assessment Procedures View project All content following this page was uploaded by Noel O'Dowd on 15 June 2015 The user has requested enhancement of the downloaded file All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately Journal of ASTM International, April 2006, Vol 3, No Paper ID JAI13221 Available online at www.astm.org C M Davies,1 M Kourmpetis,1 N P O’Dowd,1 and K M Nikbin1 Experimental Evaluation of the J or C* Parameter for a Range of Cracked Geometries ABSTRACT: In the current ASTM Standard Test Method for Measurement of Creep Crack Growth Rates in Metals ͑E 1457͒ the experimental C* parameter is related to the load and creep load line displacement rate through the geometry related factor In this work factors for a range of geometries are presented The geometries examined are the compact tension specimen, C͑T͒, single edge notch specimen in tension, SEN͑T͒, and bending, SEN͑B͒, double edge notch specimen in tension, DEN͑T͒, middle crack specimen in tension, M͑T͒ and the C-shaped specimen in tension CS͑T͒ Calculations have been performed for a linear elastic-power law hardening material but the resulting factors are applicable to either power law plastic or power law creeping materials Values for LLD and CMOD, based on the load line displacement and crack mouth opening displacement, respectively, have been determined A wide range of crack depths, 0.1ഛa/Wഛ0.7, where a is crack length and W is specimen width, and hardening exponents, ഛN ഛ10, under plane stress and plane strain conditions have been examined using the finite element method The influence of specimen length, crack length, material properties and out of plane stress state on the factor has also been considered It has been found that for shallow cracks the value of depends quite strongly on the exponent, N in the material power law, regardless of whether is defined based on the load line displacement or crack mouth opening displacement The LLD factor has also been found to be strongly sensitive to plane stress/strain conditions imposed, a/W and specimen length, whereas CMOD depends more weakly on a/W and is almost independent of specimen length for the cases examined There is, however, no clear trend in these variations over the range of specimen geometries and a/W examined These results are found to be consistent with those in the literature Recommendations are made regarding the most appropriate values for , depending on the specimen type and geometry while taking into account the variability due to the material properties, out of plane stress state and variations between the numerical analyses KEYWORDS: eta factor, creep crack growth, finite element, specimen geometry, C*, J, load line displacement, crack mouth opening displacement Nomenclature ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ HCMOD ϭ HLLD ϭ J ϭ a b n x A Ac B , Bn C Ce C* E EЈ Crack length or crack half length Uncracked ligament size ͑=W − a͒ Power-law creep stress exponent Ratio of crack length to specimen width, a / W Appropriate area on a load displacement record Area of the cracked surface Specimen thickness and ͑net͒ thickness between side grooves Coefficient in the power-law creep strain rate relation Elastic compliance function Steady-state creep fracture mechanics parameter Elastic ͑Young’s͒ modulus Effective elastic modulus, E / ͑1 − 2͒ for plane strain and E for plane stress Geometric function to calculate C* from crack mouth opening displacement rate Geometric function to calculate C* from load line displacement rate Elastic-plastic fracture mechanics parameter Manuscript received April 28, 2005; accepted for publication January 30, 2006; published online March 2006 Presented at ASTM Symposium on Fatigue and Fracture Mechanics: 35th, on 18–20 May 2005 in Reno, NV; R E Link and K M Nikbin, Guest Editors Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK Copyright © 2006 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959 2 JOURNAL OF ASTM INTERNATIONAL Je , J p K L N P Ri Ro W X Y͑a / W͒ ␣ 0 ˙ c CMOD LLD 0 ⌬ e, ⌬ p ⌬˙ c LLD ˙ LLD ⌬ ,⌬ CMOD ˙ CMOD ⌬ ,⌬ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ Elastic and plastic components of J Linear elastic stress intensity factor Gage length over which load line displacement, ⌬LLD, is measured Power-law plasticity hardening exponent Applied load Inner radius of CS͑T͒ specimen Outer radius of CS͑T͒ specimen Specimen width or half width Load hole offset in CS͑T͒ specimen Linear elastic shape factor Constant in Ramberg-Osgood law Strain Normalizing strain in the Ramberg-Osgood law Creep strain rate Geometric factor to calculate C* from crack mouth opening displacement rate Geometric factor to calculate C* from load line displacement rate Poisson’s ratio Normalizing stress in the Ramberg-Osgood law Stress Elastic and plastic components of displacement Creep displacement rate Load line displacement, load line displacement rate Crack mouth opening displacement, crack mouth opening displacement rate Introduction Creep crack growth ͑CCG͒ is often characterized by the C* parameter ͓1͔, which is analogous to J for rate independent nonlinear materials ͓2͔ In a CCG test the value of C* is estimated experimentally from load-load line displacement rate ͑LLD͒ records C* can also be estimated from experimental load-crack mouth opening displacement rate ͑CMOD͒ records, which, under certain conditions, has been found to provide a more robust measurement than LLD ͓3͔ The ASTM Standard Test Method for measurement of Creep Crack Growth Rates in Metals ͑E 1457͒ ͓4͔ proposes the Compact Tension, C͑T͒, specimen geometry While the use of the C͑T͒ specimen for testing and analysis is expected to provide a safe, upper bound to the CCG rate of the material, a wider range of test geometries is needed in order to apply the results from experimental laboratory test specimens to the range of conditions experienced by engineering components under service conditions This trend towards CCG testing using nonstandard specimens has drawn attention to the need for accurate determination of C* in order to analyze the data ͓1͔ In order to determine the value of C* from the displacement rate, a dimensionless parameter, , which depends on specimen geometry but is independent of load magnitude ͑see, e.g., ͓5͔͒, is required In this study the geometric factors, used to derive experimental C* values, have been evaluated for a number of geometries: the compact tension, C͑T͒, C-shaped tension, CS͑T͒, double edge notched tension, DEN͑T͒, single edge notch bend, SEN͑B͒, single edge notched tension, SEN͑T͒, and middle cracked tension, M͑T͒ ͑sometimes referred to as center crack tension, CC͑T͒͒ These geometries are shown schematically in Fig For the SEN͑B͒ specimen, loading is under three point bending Values of the factor have been obtained from the literature and additional values have been generated using finite element analysis A brief review of the analysis methods for the derivation of factors is first presented The application to a nonlinear power law plastic material is provided and the generalization to a power law creeping material is then considered The effect of material properties, specimen geometry and out-of-plane stress state ͑plane stress or plane strain͒ on the value of is examined On the basis of the analyses presented, recommendations for mean and upper/lower bound values of factors are made for each geometry examined This will allow a user to determine values of C* for a wide range of cracked geometries DAVIES ET AL ON THE J OR C* PARAMETER FIG 1—Specimen geometry definition Material Behavior In this work, the elastic-plastic behavior is assumed to be represented by a Ramberg-Osgood type material law The uniaxial stress-strain response is represented as = ͩ ͪ 0 +␣ E 0 E N ͑1͒ where E is the Young’s modulus, 0 denotes a normalizing stress, often taken to be the yield strength, N is the hardening exponent and ␣ is the yield offset For a power law creeping material, the creep strain rate, ˙ c, may be described by ˙ c = Cn ͑2͒ where the creep stress exponent, n, and the constant C, are temperature dependent material properties Estimation of J and C* from Load-Displacement History The nonlinear elastic crack tip parameter, J, may be evaluated from a line integral along a contour surrounding a crack tip ͓2͔ Alternatively the value of J may be obtained from the potential energy release rate of a cracked body, JOURNAL OF ASTM INTERNATIONAL FIG 2—Definition of the (a) plastic displacement ⌬ p and plastic area A p, under the load-displacement curve, and (b) plastic secant area Asec, under the load-plastic displacement curve J=− ץU ץAc ͑3͒ where U is the potential energy stored in the system, and Ac is the crack area For geometries with a single crack tip, e.g., SEN͑T͒, C͑T͒, CS͑T͒, the crack area, Ac = B ϫ a, whereas for geometries with two crack tips, e.g., M͑T͒, DEN͑T͒, Ac = B ϫ 2a, where B is the crack front length ͑specimen thickness for a throughthickness crack͒ and a is the crack length or half crack length ͑see Fig 1͒ The potential energy, U, may be related to the load, P, and the displacement along the load line, ⌬LLD For a nonlinear material, in order to obtain the value of J directly from a load-displacement history, it is convenient to split J into elastic and plastic parts ͓6͔, J = Je + J p ͑4͒ The elastic component of J, Je, is related to the linear elastic stress intensity factor, K, and the effective elastic modulus of the material, EЈ, ͑E in plane stress and E / ͑1 − 2͒ in plane strain͒ by the well known relation, Je = K2 EЈ ͑5͒ The plastic component of J, J p may be estimated using the equation, Jp = A B͑W − a͒ ͑6͒ where the factor relates J p to an appropriate area, A, on the load displacement record generated during a fracture toughness test The dimensionless factor, , for a given geometry, is generally considered to be a function of crack dimension, a / W only, where W is the specimen width or half width However, may also depend on material hardening exponent, N, normalized length, L / W, and specimen thickness B The displacement, ⌬, used to determine the area, A, in Eq may be separated into linear elastic, ⌬e, and plastic, ⌬ p, components, ⌬ = ⌬e + ⌬ p ͑7͒ The linear elastic component of the displacement of a cracked geometry is directly related to the applied load, P, through the linear elastic compliance function, Ce, which is a function of crack length, a, ⌬e = Ce͑a͒P ͑8͒ The plastic component of dispacement, ⌬ p, may then be determined as shown graphically in Fig 2͑a͒ In order to obtain values of that are consistent with those used historically, i.e., = for a tension specimen and = for a bend specimen ͓5͔, in Eq is related to an appropriately chosen area of the load-displacement curve for each specimen Two areas are used, the plastic area, A p, and plastic secant area, Asec, shown in Figs 2͑a͒ and 2͑b͒, respectively These areas may be expressed as DAVIES ET AL ON THE J OR C* PARAMETER TABLE 1—Definitions of H, HЈ and A for each specimen Specimen Type C͑T͒ SEN͑T͒ DEN͑T͒ M͑T͒ SEN͑B͒ CS͑T͒ HLLD N / ͑N + 1͒ N / ͑N + 1͒ ր ͑N − 1͒͑N + 1͒ ր ͑N − 1͒ / ͑N + 1͒ N / ͑N + 1͒ HCMOD N / ͑N + 1͒ N / ͑N + 1͒ ր ͑N − 1͒ / ͑N + 1͒ ր ͑N − 1͒ / ͑N + 1͒ Associated Area, A Ap Ap Asec Asec Ap HЈLLD 1 1 HЈCMOD 1 1 2L / W N / ͑N + 1͒ N / ͑N + 1͒ Ap 1 ͑ 2L ր W ͒N / ͑N + 1͒ Ap = ͵ ⌬p Pd⌬ p ͑9͒ Asec = ͵ ⌬p Pd⌬ p − P⌬ p ͑10͒ and may be obtained from the load-displacement record produced during a test For a power-law hardening material these areas can be calculated from Ap = Asec = N P⌬ p N+1 ͑11͒ 1N−1 P⌬ p 2N+1 ͑12͒ Equations 11 and 12 are only appropriate for materials where the plastic component of displacement, ⌬ p, can be represented by a power law as in the Ramberg-Osgood material law, Eq Replacing the area A in Eq with Eq 11 or 12, the solution for J p, Eq 6, can be rewritten as Jp = P⌬ p H B͑W − a͒ ͑13͒ The parameter, H, is included in Eq 13, to simplify the definition of for a range of geometries, and in general, H will depend on N and specimen geometry During testing, the displacement of the specimen is often measured along the load line, ⌬LLD, which is generally taken at the point of load application Alternatively the specimens crack mouth opening displacement, ⌬CMOD, may be measured These quantities are defined for the geometries of interest in Fig As will be discussed later, the value of H and will depend on whether J is calculated using load line or crack mouth opening displacement While ⌬CMOD is defined unambiguously, if J is to be evaluated using ⌬LLD, the value of H and may depend on the point at which ⌬LLD is measured Note that the length 2L in Fig for the M͑T͒, DEN͑T͒, and SEN͑T͒ specimen is strictly speaking not the specimen length but the gage length over which ⌬LLD is measured For SEN͑B͒ under three point loading, 2L is the spacing between the support points The function H is defined through Eqs and 11–13 For the SEN͑B͒ specimen an additional normalizing factor is included in H to account for the dependency of ⌬CMOD on L / W The values of HLLD and HCMOD for each specimen examined are provided in Table along with the appropriate area to use in Eq The superscripts LLD and CMOD are used with H to indicate that they are associated with the load line and crack mouth opening displacement, respectively The values of H in Table are appropriate for a power law hardening material More generally the plastic component of J, J p, in Eq 13 may be determined from Jp = A H Ј B͑W − a͒ ͑14͒ where A represents the associated area used in the calculation of and HЈ is an additional normalizing parameter The value of HЈ for each geometry examined is also included in Table 6 JOURNAL OF ASTM INTERNATIONAL The steady-state creep crack tip characterizing parameter, C*, defined under widespread ͑steady-state͒ creep conditions, is analogous to J p for a nonlinear elastic material Hence for a power law creeping material ͑Eq 2͒ C* may be determined from the load displacement rate record during a creep crack growth test using the relations, C* = ˙ P⌬ c H B͑W − a͒ ͑15͒ In Eq 15 the value of H and for a power law creeping material, with creep exponent n, is the same as that for a power law plastic material with exponent, N = n Review of Factors in the Literature A review has been performed of factors available in the literature Summaries are provided in Tables 2–4 for studies based on numerical and analytical methods, respectively Some variations in the values of from each source have been found, which may be due to differences in the numerical code used ͑all analyses reviewed have used the finite element method͒, the mesh design ͑e.g., mesh size and/or element type͒, the application of boundary conditions ͑e.g., point loads/distributed load, the method of modeling the loading pin, displacement or load controlled analysis, etc.͒ and in the precise method used to calculate Also, the displacement measuring positions and the material properties used vary among the analyses Numerical Analysis In this work numerical methods have been employed to estimate factors for different crack geometries, using the commercial finite element ͑FE͒ package, ABAQUS ͓7͔ The specimen geometries analyzed are a standard compact tension, C͑T͒, middle tension, M͑T͒, C-shaped tension, CS͑T͒, double edge notched tension, DEN͑T͒ and the single edge notch bend specimen under three point bend loading, SEN͑B͒ The Ramgerg-Osgood material model ͑Eq 1͒ has been employed throughout The J-integral value is obtained using a domain integral within the FE software ͓7͔ and the total displacements at the relevant positions on the specimen, ⌬, are extracted In an elastic-plastic analyses, the elastic components, Je and ⌬e, are then subtracted from the total values to obtain J p and ⌬ p, respectively, for use in Eq 13 Linear-elastic analyses were performed for each case in order to obtain the elastic compliance value Ce͑a͒ and the linear elastic stress intensity factor, K, for each crack length considered, enabling Je and ⌬e to be determined in the elastic-plastic analyses The values obtained are found to be in good agreement with those available in the literature ͓4,8,9͔ The value of has then been calculated according to the relation = B͑W − a͒J p HP⌬ p ͑16͒ Two-dimensional, plane stress, and plane strain analyses have been performed Each specimen has been modeled with a focused mesh and multiple nodes at the crack tip to capture the crack tip singularity Small geometry change conditions are used Symmetry conditions are fully employed For all specimens except the CS͑T͒, four noded plane strain and plane stress continuum elements ͑type CPE4H and CPS4͒ are used For the CS͑T͒ specimen eight-noded elements have been used A typical mesh consists of over 2000 elements and nodes and the J integral and K values have been averaged over 41 contours Three values of N = 3, 5, and 10 are examined in order to determine the sensitivity of to N ͑or n͒ The value of obtained from Eq 16 initially shows some dependence on load but reaches a constant value once the plastic strains are sufficiently high that power law material behavior prevails throughout the specimen This constant value is taken as the value for the specimen It is assumed that N is the only material property that effects as it is evaluated where plasticity dominates and J p ӷ Je Results The results of the FE analysis are presented in this section The factor values determined in this work are compared with the values found in the literature under various conditions For the SEN͑T͒, M͑T͒, DEN͑T͒, DAVIES ET AL ON THE J OR C* PARAMETER TABLE 2—Summary of numerical studies in literature Author Kim et al ͓3͔ Specimen Type DEN͑T͒ N Material Properties LLD / CMOD E / 0 = 667, 1000, 500 a/W 0.25 Dimensions LLD 0.5 B / W = 0.5, CMOD 0.75 1, 5, 10 P 10 P ϱ Kim et al ͓10͔ Panontin et al ͓16͔ Nevalainen and Dodds ͓19͔ O’Dowd et al ͓20͔ O’Dowd et al ͓21͔ M͑T͒ SEN͑B͒-3PB SEN͑T͒ C͑T͒ 3-D 0.9 0.1 E / 0 = 500 LLD ␣=1 CMOD 10 10 20 E / 0 = 800 E / 0 = 500 E / 0 = 250 SEN͑T͒ 10 20 E / 0 = 500 = 0.3 ␣=1 Bimaterial SEN͑B͒:LLD CMOD C͑T͒:LLD LLD 0.25 CC͑T͒ E / 0 = 500 10 20 ␣=1 CMOD P 0.25, 0.125 0.5 0.4 0.45 0.5 0.6 0.7 SEN͑B͒: 0.1,0.5 C͑T͒:0.6 3-D P L/W=2 B / W = 1, 0.5, 0.25 L / W = 2.5 3-D P P 0.1 P LLD Kirk and Dodds ͓22,23͔ SEN͑B͒-3PB Joyce et al ͓24͔ SEN͑T͒ DEN͑T͒ SEN͑B͒-3PB 10 50 yield offset 0 = 0 / E ␣=1 LLD CMOD High strength structural steel y = 747 L/W=2 0.5 0.05 0.15 0.25 0.5 0.7 L/Wϭ2 SEN͑T͒: 0.12–0.8 LLD UTS= 877 MPa E / 0 = 388 = 0.3 E / 0 = 500 SEN͑B͒-3PB 0.25 P = 0.3 Lei et al ͓25͔ P B / W = 0.5, 0.3 C͑T͒ SEN͑B͒-3PB C͑T͒ P / Pa or 3-D P P DEN͑T͒ 0.72,0.84,0.9 ␣=1 LLD = 0.3 CMOD 10 0.1– 0.7 L/W=2,3 P Ro = 18.29 3-D Ri = 9.14 P 20 Kim et al ͓26͔ Wang and Gordon ͓27,28͔ Griffiths ͓29͔ CS͑T͒ Steel 304L SEN͑B͒-3PB LLD 0.54 21-6-9 0.1–0.5 Bimaterial LLD 0.25 SEN͑T͒ L / W = 2.5, P 5, 7.5 a Pϵ Plane stress, Pϵ Plane strain and SEN͑B͒ a summary highlighting the effects of normalized length, L / W on the factor are also presented 8 JOURNAL OF ASTM INTERNATIONAL TABLE 3—Summary of and J estimation studies in literature Specimen Type C͑T͒ LLD / CMOD LLD a/W Dimensions P / Pa or 3-D Comments Limit Load Analysis C͑T͒ LLD 0.45– 0.62 Provides a linear fit to the analysis in Ref ͓14͔ CS͑T͒ LLD X / W = 0.5 M͑T͒ C͑T͒ SEN͑B͒-3PB LLD CMOD M͑T͒, SE͑B͒: 0.1–0.7 C͑T͒:0.45– 0.7 Kim and Budden ͓30͔ SEN͑T͒ LLD CMOD 0.1– 0.8 M͑T͒: P P C͑T͒ & SE͑B͒: P P P Sumpter ͓31͔ SEN͑B͒-3PB CMOD 0.1 – 0.5 Wu et al ͓32͔ 3PB CC͑T͒ SEN͑T͒ DEN͑T͒ LLD 0.125 – 0.875 L/W=2 Load-Separation Analysis Reviews accuracyof J estimation from various solutions Limit load solutions, comparisons with FE analysis of Ref ͓20͔ Fit made to limited range of data Slip Line Field & EPRI Solutions Author Merkle and Corten ͓14͔ Clarke and Landes ͓13͔ Junnan ͓18͔ Kim and Schwalbe ͓28͔ a Pϵ Plane stress, Pϵ Plane strain Single Edge Notched Tension Specimen, SEN(T) The effect of normalized length, L / W, on LLD and CMOD for the SEN͑T͒ specimen are shown in Fig for the two extremes of crack length examined ͑a / W = 0.1 and 0.7͒ Note that this specimen is modeled with uniform remote stress and thus bending is allowed in the specimen, which can be significant for deep cracks Figure 3͑a͒ provides the results for a shallow crack, a / W = 0.1 and Fig 3͑b͒ the results for a deeply cracked specimen, a / W = 0.7 The results shown are for hardening exponent N = 10 under plane stress and plane strain conditions The trends given for N = are similar and thus not shown here It may be seen that LLD is strongly sensitive to L / W for the shallow crack, a / W = 0.1, but for the deep crack, a / W = 0.7, is insensitive to L / W This sensitivity for a shallow crack is related to the significant amount of remote plasticity which increases with increasing specimen length It may be seen in Fig that the value of CMOD is insensitive to specimen length for both shallow and deep cracks Similar trends are seen under plane stress and plane strain conditions ͑the plane strain and plane strain results for CMOD are almost coincident in Fig 3͑b͒ and is insensitive to the plane stress/strain assumption Figure illustrates the influence of the hardening exponent, N, on LLD and CMOD for the SEN͑T͒ specimen over a range of crack lengths For deep cracks, a / W ജ 0.6, N has a weak effect on LLD ͑Fig 4͑a͒͒ and CMOD is almost independent of N over the range of a / W ͑Fig 4͑b͒͒ The same trend is seen under plane stress conditions, since, as shown in Fig 3, there is little variation in the values of LLD and CMOD from a plane stress or plane strain analysis Figure shows the full set of results for LLD and CMOD from the current analysis ͑indicated by circles͒ and data taken from Ref ͓10͔ for the SEN͑T͒ specimen The values shown cover the range of conditions examined ͑i.e., plane stress/plane strain, 0.1ഛ a / W ഛ 0.7; ഛ N ഛ 10; ഛ L / W ഛ 10͒ It may be seen in Fig 5͑a͒ that there is some irregularity in the values of LLDat a / W = 0.6 This is consistent with the trend shown by existing solutions ͑see, e.g., ͓11͔͒ In Fig 5͑a͒ and 5͑b͒ piecewise linear fits are made to the combined data sets The line labeled “mean fit” is an approximate fit to the mean value of the dataset at each value of a / W, and the upper bound and lower bound line have been estimated to cover between one and two standard deviations of the dataset within the region of the linear fit These simple fits and associated bounds quantify the variation in under DAVIES ET AL ON THE J OR C* PARAMETER FIG 3—Specimen length effects on LLD and CMOD under plane stress and plane strain conditions for (a) shallow and (b) deeply cracked SEN(T) specimens, N = 10 various conditions; no attempt has been made to describe the individual dependence on material or geometric parameters ͑apart from a / W͒ The equations for the mean fit and the measure of uncertainty are given in Appendix A As seen in Fig 5͑a͒ the value of LLD is strongly sensitive to crack length for shallow cracks ͑a / W Ͻ 0.5͒ and approaches 2.5 for a / W Ͼ 0.6 This high value of LLD for deep cracks is an indication of the level of bending in the specimen ͑for a tension specimen LLD is generally taken to be 1͒ The variability in LLD ͑indicated by the upper/lower bound lines in Fig 5͑a͒͒ can be over 80 % of the mean value for shallow cracks ͑a / W = 0.1͒ This variability is mostly indicative of the strong sensitivity to specimen length, L / W, as indicated in Fig 3͑a͒ It is seen in Fig 5͑b͒ that CMOD is almost independent of crack length over the range considered The variability is approximately % of the mean value for the cases examined The recommended solutions for LLD and CMOD are provided in Appendix A The bilinear fit provided for LLD in Appendix A has been restricted to the range ഛ L / W ഛ 3, since the majority of the data presented are for L / W = As the value of CMOD has a weak dependence on problem parameters for the SEN͑T͒ specimen, it is recommended that, where practicable, J or C* be estimated using ⌬CMOD rather than ⌬LLD Middle Cracked Tension Specimen, M(T) Following the findings for the SEN͑T͒ specimens the effect of normalized length, L / W, has only been examined under plane strain conditions and N = 10 for the M͑T͒ specimen Figure shows that the dependence on L / W is similar to that seen for the SEN͑T͒ specimen, with LLD for the shallow crack, a / W = 0.1, being strongly sensitive to specimen length, while CMOD is unaffected by the specimen length Note that for a / W = 0.7 ͑Fig 6͑b͒͒ LLD and CMOD are independent of specimen length and their values almost coincide This indicates that for deep cracks the majority of the deformation is occurring in the vicinity of the crack and ⌬LLD and ⌬CMOD are almost the same FIG 4—Example of the influence of N on (a) LLD and (b) CMOD, under plane strain conditions, over a range of crack lengths for a SEN(T) specimen 10 JOURNAL OF ASTM INTERNATIONAL FIG 5—Variation in (a) LLD and (b) CMOD with crack length for SEN(T) specimen data from various sources In Fig the sensitivity to N and to plane stress or plane strain conditions is examined It is seen, that as for the SEN͑T͒ specimen the influence of out of plane constraint is insignificant compared to the influence of N For deep cracks, a / W ജ 0.5, LLD and CMOD are relatively insensitive to N The full set of results, including those taken from the literature, is shown in Fig Good agreement is found between the values of LLD and CMOD obtained here with those in the literature As for the SEN͑T͒ specimen LLD shows a strong dependence on crack length, a / W, and for shallow cracks has a high variability, indicated by the upper and lower bounds in Fig 8͑a͒ ͑this variability is mainly due to dependence on length, L / W, and N͒ It may be noted that for deep cracks, a / W Ͼ 0.4, LLD is close to unity A bilinear fit is made for LLD, while a single linear fit adequately describes the dependence of CMOD on a / W The same specimen length restrictions are imposed as for the SEN͑T͒ specimen In Fig 8͑b͒ it is seen that some of the data for a / W = 0.1 and 0.3 lie above the upper bound line These data correspond to results for N = ͑see also Fig 7͑b͒͒ and thus the additional restriction, that N Ͼ 3, is imposed on this specimen calibration It may be seen in Fig that variability in CMOD is somewhat less than that for LLD particularly for shallow cracks It is therefore recommended that, where practicable, J or C* be estimated using ⌬CMOD rather than ⌬LLD Double Edge Notched Tension Specimen, DEN(T) Figures and 10 illustrate the sensitivity to specimen length, out of plane constraint and strain hardening for the DEN͑T͒ specimen The trends are similar to those for the M͑T͒ specimen, though the dependence on strain hardening, N, is somewhat greater for deep cracks and out-of-plane constraint effects on the value of are also greater A bilinear fit is made to both LLD and CMOD in Fig 11 The uncertainty bands are considered constant over the range of crack lengths considered The variability in CMOD is on the order of 20 % of FIG 6—Specimen length effects on LLD and CMOD under plane strain conditions for (a) shallow and (b) deeply cracked M(T) specimens DAVIES ET AL ON THE J OR C* PARAMETER 11 FIG 7—Example of the influence of N on (a) LLD and (b) CMOD, under plane stress and plane strain conditions, over a range of crack lengths for a M(T) specimen the mean fit over the range of crack lengths For LLD the variability is between approximately 75 % for shallow cracks to approximately 25 % for deep cracks Since, both LLD and CMOD have a relatively large variability and strong dependence on a / W, care should be used when determining J or C* from this geometry For shallow cracks it is recommended that J or C* be evaluated using ⌬CMOD Single Edge Notched Bend Specimen, SEN(B) The specimen has been analyzed under three point bending Results under plane strain conditions are shown in Fig 12 for three specimen lengths, L / W There is some influence of L / W on LLD, though this effect is weaker than that seen for the tension specimens ͑DEN͑T͒, SEN͑T͒, and M͑T͒͒ Through the use of the additional normalizing factor included in HCMOD, length effects on CMOD have been eliminated for the cases examined, indicated by the constant value of CMOD in Fig 12͑b͒ The out of plane stress state and N effects on LLD and CMOD are presented for L / W = 4, in Fig 13 The main source of variation in LLD is due to the hardening exponent, N, particularly for shallow cracks, a / W Ͻ 0.5, for LLD and a / W Ͻ 0.3 for CMOD For deeper cracks ͑a / W Ͼ 0.5͒, LLD is essentially independent of out-of-plane stress state and strain hardening and is close to 2, as expected for a deeply cracked bend geometry For a / W Ͻ 0.3, CMOD for the SEN͑B͒ specimen is independent of out of plane stress state and strain hardening and has a near linear variation with a / W In Fig 14͑a͒ the available data for LLD are shown The data fall into two regions: for a / W Ͻ 0.4 there is significant variability, due primarily to dependence on strain hardening, and LLD depends strongly on a / W; for a / W Ͼ 0.3 the variability is lower and LLD is more weakly dependent on a / W On average, the values of LLD deviate from the mean by about 35 % for 0.1ഛ a / W ഛ 0.3, but variability is less than % for a / W Ͼ 0.3 The value of CMOD decreases linearly with crack length over the whole range of a / W considered The relatively small variations at a given a / W are mainly due to the dependence on N Over FIG 8—Variation in (a) LLD and (b) CMOD with crack length for M(T) specimen data from various sources 12 JOURNAL OF ASTM INTERNATIONAL FIG 9—Specimen length effects on LLD and CMOD under plane strain conditions for (a) shallow and (b) deeply cracked DEN(T) specimens the range 0.1ഛ a / W ഛ 0.7 the value of CMOD deviates from the mean value by about 10 % for the cases examined The value of CMOD for N = and a / W = 0.1 falls outside the upper bound line ͑see Fig 14͑b͒͒ Therefore, the equation fitting the data is restricted to N Ͼ The results indicate that when a / W Ͻ 0.3 the value of J or C* may be more accurate if ⌬CMOD is used, while for a / W Ͼ 0.3, it is expected that a more accurate value will be obtained using ⌬LLD Compact Tension Specimen, C(T) The C͑T͒ specimen is the only specimen examined where the crack mouth opening displacement is measured along the load line, as opposed to the front face as, for example, in the SEN͑T͒ specimen ͑see Fig 1͒ Thus, the displacement measured at the crack mouth is often considered to be equivalent to that measured at the point of load application This assumption has been confirmed by the analyses carried out here: displacement measurements from the crack mouth ͑along the load line͒ and those at the loading point have been compared, and these are almost identical Note that due to differences between the specimen and finite element model geometry for the cases examined here, the position of the node used to determine ⌬CMOD is not precisely coincident with the point indicated in Fig Note further that for the majority of the results presented here the pin hole indicated in Fig was not modeled explicitly The ASTM Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials ͑E 399͒ ͓9͔ and ASTM E 1457 ͓4͔ is solution for the C͑T͒ specimen is widely used and is given by = + 0.522 W−a W ͑17͒ Equation 17 was first proposed in Ref ͓12͔ and was taken from a linear fit in Ref ͓13͔, over the region 0.45ഛ a / W ഛ 0.62, to the fully plastic limit load analysis of Ref ͓14͔ This formula is derived based on the FIG 10—Example of the influence of N on (a) LLD and (b) CMOD, under plane stress and plane strain conditions, over a range of crack lengths for a DEN(T) specimen DAVIES ET AL ON THE J OR C* PARAMETER 13 FIG 11—Variation in (a) LLD and (b) CMOD with crack length for DEN(T) specimen data from various sources assumption that the displacement is measured at the loading point, though in Refs ͓12,15͔ J values found experimentally using ⌬CMOD were in excellent agreement with values for J obtained from multi specimen techniques Equation 17 has since been widely used to determine using displacements measured at both the load point and at the crack mouth Figure 15 shows the influence of out-of-plane stress state and N on over a range of crack lengths It is seen that there is no general trend but the largest differences are due to the variation in N ͑though these are small͒ Figure 16 presents the data from the current analysis and from the literature No distinction is made between CMOD and LLD, though the data presented form the current analysis were obtained using ⌬LLDwhile those from Refs ͓10,16͔ used ⌬CMOD The results for obtained from the current analysis are in good agreement with those of Ref ͓16͔, except at a / W = 0.4 The reason for this discrepancy has not been identified but the trend indicated by the current data ͑open circles͒ seems more reasonable Equation 17, which was originally fitted within the range 0.45ഛ a / W ഛ 0.62 and is indicated in Fig 16 by the dash line, provides an upper bound to the data at a / W = 0.45 and a lower bound at a / W = 0.7 Thus an adequate fit is provided by the ASTM formula, which may be used to evaluate using either ⌬LLD or ⌬CMOD for 0.45ഛ a / W ഛ 0.7 However, following the approach for the other geometries examined here a mean line fit has been made to the dataset, in the range of 0.45ഛ a / W ഛ 0.7, which is independent of crack length ͑ = 2.2͒ The upper and lower bounds are less than % of the mean fit value in the range of crack lengths considered C-shaped Specimen, CS(T) The final specimen examined is the CS͑T͒ specimen, which has a somewhat similar geometry to the C͑T͒ specimen The results presented here are only considered valid for a CS͑T͒ specimen of outer radius Ro = 50 mm, with Ri / Ro = 0.5 and X / W = 0.5 FIG 12—Specimen length effects on (a) LLD and (b) CMOD under plane strain conditions for SEN(B) specimens of a / W = 0.5, N = 14 JOURNAL OF ASTM INTERNATIONAL FIG 13—Example of the influence of N on (a) LLD and (b) CMOD, under plane stress and plane strain conditions, over a range of crack lengths for a SEN(B) specimen The variation in LLD from the plane stress and plane strain analyses is relatively small compared with the variation due to N, as seen in Fig 17͑a͒ In Fig 17͑b͒ it is demonstrated that for a / W = 0.1, N has the greatest influence on CMOD; however, for a / W ജ 0.4 the influence of out-of-plane stress state dominates This result is similar to that observed for the DEN͑T͒ specimen A very limited amount of data are available in the literature for the CS͑T͒ specimen No appropriate numerical data for comparison have been found hence only the values determined from the current analysis are shown in Fig 18 As before, mean, upper, and lower bound lines are provided, though the data are relatively sparse In Ref ͓17͔ a 2-D plane strain and a 3-D analysis were performed for a specimen of somewhat different dimensions ͑see Table 2͒ to that considered in this work, for a single crack length of a / W = 0.54 Values of LLD of approximately 2.24 and 2.44 were obtained from 2-D and 3-D analyses, respectively These results are consistent with the values shown in Fig 18͑a͒ An analytical estimate for LLD, based on the load separation technique, is provided in Ref ͓18͔, values from which are consistent with the upper bound for a / W = 0.4 and the mean fit at a / W = 0.55 However, the solution is only relevant for deep cracks and there is a large discrepancy between this solution and the data for shallower cracks No solutions were found in the literature for CMOD Our analyses indicate that for the CS͑T͒ specimen, unlike the C͑T͒ specimen, there is a significant difference between CMOD and LLD for the cases examined as seen in Fig 18 Figure 18͑b͒ shows that CMOD decreases with a / W, and has a variability of approximately % of the mean value This variability is mainly due to the dependence on N for shallow cracks, and dependence on out of plane stress state for deeper cracks ͑see Fig 17͑b͒͒ The limited number of crack depths examined here makes it difficult to provide definitive recommendations for the CS͑T͒ specimen It may be seen that there is a large variability in LLD for 0.2ഛ a / W Ͻ 0.4 ͑the variability ranges from to 79 to 30 % of the mean value͒ A more definite trend is shown for CMOD and the variability is significantly less than that for LLD in the region 0.2 ഛ a / W Ͻ 0.4 The variability in CMOD and LLD are comparable for a / W Ͼ 0.4 FIG 14—Variation in (a) LLD and (b) CMOD with crack length for SEN(B) specimen data from various sources DAVIES ET AL ON THE J OR C* PARAMETER 15 FIG 15—Example of the influence of N on LLD under plane stress and plane strain conditions, over a range of crack lengths for a C(T) specimen Due to the limited data available for this specimen and the relatively large variability for shallow cracks, it is recommended that J or C* values for the CS͑T͒ specimen are evaluated using the crack mouth opening displacement, ⌬CMOD Discussion Analyses have been reported which allow the fracture mechanics parameters J and C* to be evaluated from load-displacement and load-displacement rate histories, respectively The geometry parameters and H used to evaluate J or C* have been tabulated and values plotted for a range of crack geometries using either the load line displacement or crack mouth opening displacement The influence of material response ͑power law exponent͒, out-of-plane stress state ͑plane stress/plane strain͒, and specimen length on the parameters LLD and CMOD have been examined and results compared with existing solutions in the literature, where available For most specimens LLD is sensitive to crack length especially for shallow cracks A lower sensitivity to crack length is generally seen for CMOD, though this is not always the case Most of the results presented here are from a 2-D analysis There may also be an influence from specimen thickness, B / W, and specimen side-grooving which has not been fully explored It is expected that the plane stress and plane strain solutions for will bound the values for different thickness specimens Some 3-D calculations and discussion on thickness and side-grooving effects can be found in Refs ͓3,10,17,19͔ In an analysis, the appropriate bound of should be used to provide a conservative prediction When material properties are being determined the lower bound will provide a conservative measure of the material toughness If a structural assessment is being carried out using the factor then a conservative assessment is obtained using the upper bound value Alternatively, the mean value of can be used and the uncertainty, evaluated as a percentage of the mean, should be quantified in each case Since C* is linearly FIG 16—Variation in with crack length for C(T) specimen data from various sources 16 JOURNAL OF ASTM INTERNATIONAL FIG 17—Example of the influence of N on (a) LLD and (b) CMOD, under plane stress and plane strain conditions, over a range of crack lengths for a CS(T) specimen related to ͑see Eq 15͒ then, for example, a 20 % uncertainty in would lead to a 20 % uncertainty in C*, in addition to other experimental uncertainties In some experiments both ⌬LLD and ⌬CMOD may be available The values of J or C* obtained from the two measurements should agree within the quoted variability for the specimen Note that, agreement between C* values from ⌬CMOD and ⌬LLD does not imply that the “exact” value of J or C* has been obtained, since the degree of uncertainty in both LLD and CMOD are sometimes comparable If an exact value is required then the value for the particular geometry ͑accounting for the effects of a / W, B / W, and L / W͒ and particular material should be obtained from FE analysis or otherwise The analysis carried out here indicates that for most specimens the variability in the factors is sufficiently low that a separate specimen calibration is not required Note, finally, that within the FE analysis certain assumptions have been made, e.g., in most cases the rigid pin has not been modeled and for the SEN͑T͒, M͑T͒, and DEN͑T͒ specimens a remote uniform stress has been applied to the specimen at the specimen boundaries In an experiment a minimum gage length is required for a pin loaded structure to ensure a relatively uniform stress distribution to be achieved in the region of the crack plane For most of the specimens it has been found that typically the variability of CMOD is much less than that for LLD Thus it is recommend that, where practicable, C* or J values should be obtained using the crack mouth opening displacements and the geometric functions presented here Conclusions Values for LLD and CMOD, based on the load line displacement and crack mouth opening displacement, respectively, have been presented for a range of geometries The geometries examined are the compact tension specimen, C͑T͒, single edge notch specimen in tension, SEN͑T͒, and bending, SEN͑B͒, double edge notch specimen in tension, DEN͑T͒, middle crack specimen in tension, M͑T͒ and the C-shaped FIG 18—Variation in (a) LLD and (b) CMOD, with crack length for CS(T) specimen data of X / W = 0.5, Ri / Ro = 0.5 DAVIES ET AL ON THE J OR C* PARAMETER 17 TABLE 4—Details of the numerical models for each specimen specimen tension CS͑T͒ Calculations have been performed for a Ramberg-Osgood ͑power law hardening͒ material and the resulting factors are applicable to either power law plastic or power law creeping materials A wide range of crack depths, 0.1ഛ a / W ഛ 0.7, and hardening exponents, ഛ N ഛ 10, under plane stress and plane strain conditions have been examined using the finite element method Specimen length has been found to have a strong influence on LLD for shallow cracked SEN͑T͒, M͑T͒, and DEN͑T͒, but little or no influence on CMOD The influence of stress state on the factor varies with specimen type and crack length These results are generally consistent with those in the literature Mean linear fits, that are dependent on crack length, are provided and the variations in due to N, length, out-of-plane stress state, and other effects, are accommodated by providing an upper and lower bound to the data Generally, more confidence is expected in J or C* values derived from crack mouth opening displacement measurements Acknowledgments The mesh for the CS͑T͒ specimen was kindly provided by VTT Industrial Systems, Finland Helpful discussions with Prof C E Turner are gratefully acknowledged APPENDIX A—Equations for the Mean Fit to Including a Measure of Uncertainty Equations of the mean fits to the data, describing the dependency of LLD and CMOD on the normalized crack length, x = a / W, are given below, together with an approximate absolute measure of the uncertainty in the data, thus providing an upper and lower bound estimate The restrictions on specimen geometry and hardening exponent, N, are also specified 18 JOURNAL OF ASTM INTERNATIONAL SEN(T) 0.1 ഛ x ഛ 0.5 LLD = ͑5.0x − 0.06͒ ± 0.38 ഛ L/W ഛ 3ഛNഛϱ ͑A-1͒ 0.5 Ͻ x ഛ 0.7 LLD = 2.44 ± 0.38 ഛ L/W ഛ 3ഛNഛϱ ͑A-2͒ 0.1 ഛ x ഛ 0.7 CMOD = 1.0 ± 0.05 ഛ L/W ഛ 3ഛNഛϱ ͑A-3͒ M(T) 0.1 ഛ x ഛ 0.35 0.35 Ͻ x ഛ 0.7 0.1 ഛ x ഛ 0.7 ഛ L/W ഛ 3ഛNഛϱ LLD = ͑2.32x + 0.18͒ ± 0.22 LLD = 0.99 ± 0.10 ഛ L/W ഛ 3ഛNഛϱ ഛ L/W ഛ 3ϽNഛϱ CMOD = ͑1.26 − 0.36x͒ ± 0.15 ͑A-4͒ ͑A-5͒ ͑A-6͒ DEN(T) 0.1 ഛ x ഛ 0.3 LLD = ͑1.43 + 0.15͒ ± 0.22 ഛ L/W ഛ 3ഛNഛϱ ͑A-7͒ 0.3 Ͻ x ഛ 0.7 LLD = ͑0.53x + 0.42͒ ± 0.22 ഛ L/W ഛ 3ഛNഛϱ ͑A-8͒ 0.1 ഛ x Ͻ 0.5 CMOD = ͑1.26 − 0.80x͒ ± 0.20 0.5 ഛ x ഛ 0.7 CMOD = 0.86 ± 0.20 ഛ L/W ഛ 3ϽNഛϱ ഛ L/W ഛ 3ഛNഛϱ ͑A-9͒ ͑A-10͒ SEN(B)–3PB 0.1 ഛ x ഛ 0.3 LLD = ͑4.06x + 0.60͒ ± 0.35 3ഛNഛϱ ͑A-11͒ 0.3 Ͻ x ഛ 0.7 LLD = ͑0.56x + 1.65͒ ± 0.07 3ഛNഛϱ ͑A-12͒ 0.1 ഛ x ഛ 0.7 CMOD = ͑0.92 − 0.46x͒ ± 0.06 3ϽNഛϱ ͑A-13͒ C(T) 0.45 ഛ x ഛ 0.7 LLD = CMOD = 2.2 ± 0.1 3ഛNഛϱ ͑A-14͒ DAVIES ET AL ON THE J OR C* PARAMETER 19 CS(T) 0.2 ഛ x Ͻ 0.4 0.4 ഛ x ഛ 0.55 0.2 ഛ x ഛ 0.55 LLD = ͑7x − 0.64͒ ± 0.06 Ri/Ro = 0.5 X/W = 0.5 ഛ N ഛ 10 ͑A-15͒ = ͑0.4x + 2͒ ± 0.15 Ri/Ro = 0.5 X/W = 0.5 ഛ N ഛ 10 ͑A-16͒ LLD CMOD = ͑4.6 − 1.6x͒ ± 0.2 Ri/Ro = 0.5 X/W = 0.5 ഛ N ഛ 10 ͑A-17͒ References ͓1͔ ͓2͔ ͓3͔ ͓4͔ ͓5͔ ͓6͔ ͓7͔ ͓8͔ ͓9͔ ͓10͔ ͓11͔ ͓12͔ ͓13͔ ͓14͔ ͓15͔ ͓16͔ ͓17͔ Webster, G A and Ainsworth, R A., High Temperature Component Life Assessment, 1st ed., Chapman and Hall, London, 1994 Rice, J R., “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks,” ASME Trans J Appl Mech., Vol 35, 1968, pp 379–386 Kim, Y.-J., Son, B.-G., and Kim, Y.-J., “Elastic-Plastic Finite Element Analysis for Double-Edge Cracked Tension ͑DE͑T͒͒ Plates,” Eng Fract Mech Vol 71, 2004, pp 945–966 ASTM Standard E 1457, “Standard Test Method for Measurement of Creep Crack Growth Rates in Metals,” , Annual Book of ASTM Standards, Vol 3, No 1, 2001, pp 936–950 Turner, C E., The Ubiquitous Factor, in Fracture Mechanics: Twelfth Conference, ASTM STP 700, ASTM International, West Conshohocken, PA, 1980, pp 314–337 Ainsworth, R A., “The Assessment of Defects in Structures of Strain Hardening Material”, Eng Fract Mech., Vol 19, 1984, pp 633–642 ABAQUS, Users Manual, in Version 6.2 2001 Tada, H., Paris, P C., and Irwin, G R., The Stress Analysis of Cracks Handbook, Paris Productions & ͑Del Research Corp.͒, Saint Louis, 1985 ASTM Standard E 399-90, “Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials”, Annual Book of ASTM Standards, Vol 3, No 1, 2001, pp 434–464 Kim, Y.-J., Kim, J.-S., Cho, S.-M., and Kim, Y.-J., “3-D Constraint Effects on J Testing and Crack Tip Constraint in M͑T͒, SE͑B͒, SE͑T͒ and C͑T͒ Specimens: Numerical Study,” Vol 71, No 9, 2004, pp 1203–1219 Kanninen, M F and Popelar, C H., Advanced Fracture Mechanics, The Oxford Engineering Science Series, Clarendon Press, Oxford, 1985 Ernst, H A., Paris, P C., and Landes, J D., Estimations on J-Integral and Tearing Modulus T from a Single Specimen Test Record, in Fracture Mechanics: Thirteenth Conference, ASTM STP 743, R Roberts, Ed., ASTM International, West Conshohocken, PA, 1981, pp 476–502 Clarke, G A and Landes, J D., “Evaluation of the J Integral for the Compact Specimen, J Test Eval., Vol 7, No 5, 1979, pp 264–269 Merkle, J G and Corten, H T., “A J Integral Analysis for the Compact Specimen, Considering Axial Force as Well as Bending Effects,” ASME J Pressure Vessel Technol., Vol 96, 1974, pp 286–292 Landes, J D., Walker, H., and Clarke, G A., “Evaluation of Estimation Procedures Used in J-Integral Testing, in Elastic-Plastic Fracture, ASTM STP 668,” J D Landes, J A Begley, and G A Clarke, Eds., ASTM International, West Conshohocken, PA, 1979, pp 286–287 Panontin, T L., “Crack Tip Opening Displacement Estimation Formulae for C͑T͒ Specimens,” Eng Fract Mech., Vol 67, No 3, 2000, p 293 Kim, Y., Chao, Y J., Pechersky, M J., and Morgan, M J., “C-Specimen Fracture Toughness Testing: Effect of Side Grooves and Factor,” J Pressure Vessel Technol., Vol 126, No 3, 2004, pp 20 JOURNAL OF ASTM INTERNATIONAL 293–299 ͓18͔ Junnan, C., “The Elastic-plastic Parameters of the C-shaped Specimen,” Eng Fract Mech., Vol 37, No 1, 1990, pp 119–125 ͓19͔ Nevalainen, M and Dodds, R H., Jr., “Numerical Investigation of 3-D Constraint Effects on Brittle Fracture in SE͑B͒ and C͑T͒ Specimens, International Journal of Fracture, Vol 74, No 2, 1995, pp 131–161 ͓20͔ O’Dowd, N P., Budden, P J., and Griffiths, E R J., “Finite Element Analysis of a Bi-Material SENT Specimen under Elastic-Plastic Loading, in Proceedings IUTAM Symposium Nonlinear Analysis of Fracture, Kluwer Academic Publisher, Dordrecht, The Netherlands, 1997, pp 33–42 ͓21͔ O’Dowd, N P., Kolednik, O., and Naumenko, V P., “Elastic-Plastic Analysis of Biaxially Loaded Center-Cracked Plates,” Int J Solids Struct., Vol 36, 1999, pp 5639–5661 ͓22͔ Kirk, T M and Dodds, R H J., “J and CTOD Estimation Equations for Shallow Cracks in Single Edge Notch Bend Specimens”, Department of Civil Engineering, University of Illinois at Urbana, Champaign Urbana, Illinois, No 570, January 1992 ͓23͔ Kirk, T M and Dodds, R H J., “Effect of Weld Strength Mismatch on Elastic-Plastic Fracture Parameters”, Department of Civil Engineering, University of Illinois at Urbana, Champaign Urbana, Illinois, No 570, August 1992 ͓24͔ Joyce, J A., Hackett, E M., and Roe, C., “Effects of Crack Depth and Mode of Loadings on the J-R Curve Behavior of High Strength Steel,” in Constraint Effects in Fracture, ASTM STP 1171, E M Hackett, K.-H Schwalbe, and R H Dodds, Eds., ASTM International, West Conshohocken, PA, 1993, pp 239–263 ͓25͔ Lei, Y., Webster, G A., and O’Dowd, N P., Investigation of Residual Stresses in Welded Joints Part II: Finite Element Modelling, Department of Mechanical Engineering, Imperial College London, GR/L42926, December 1998 ͓26͔ Kim, Y., “C-Specimen Fracture Toughness Testing: Effect of Side Grooves and Factor, J Pressure Vessel Technol., Vol 126, No 3, 2004, pp 293–299 ͓27͔ Wang, Y.-Y and Gordon, J R., “The Limits of Applicability of J and CTOD Estimation Procedures for Shallow-Cracked SENB Specimens,” in International Conference on Shallow Crack Fracture Mechanics, Toughness Test and Applications, M G Dawes, Ed., 1992, Cambridge, UK ͓28͔ Kim, Y.-J and Schwalbe, K.-H., “On the Sensitivity of J Estimation to Materials’ Stress-Strain Curves in Fracture Toughness Testing Using the Finite Element Method,” J Test Eval., Vol 29, No 1, 2001, pp 18–31 ͓29͔ Griffiths, E R J., “Finite Element Modelling of Cracks in Homogeneous and BiMaterial SENT Specimens,” 3M Project Report, Department of Mechanical Engineering, Imperial College London, 1994 ͓30͔ Kim, Y.-J and Budden, P J., “Plastic Eta Factor Solutions of Homogeneous and Bi-Material SE͑T͒ Specimens for Toughness and Creep Crack Growth Testing,” Fatigue and Fracture of Engineering Materials and Structures, Vol 24, No 11, 2001, pp 751–751 ͓31͔ Sumpter, J D G., “Jc Determination for Shallow Notch Welded Bend Specimens,” Fatigue and Fracture of Engineering Materials and Structures, Vol 10, No 6, 1987, pp 479–193 ͓32͔ Wu, S.-X., Mai, Y.-W., and Cottrell, B., “Plastic -Factor ͑p͒ of Fracture Specimens with Deep and Shallow Cracks,” International Journal of Fracture, Vol 45, 1990, pp 1–18 View publication stats ... the J or C* Parameter for a Range of Cracked Geometries ABSTRACT: In the current ASTM Standard Test Method for Measurement of Creep Crack Growth Rates in Metals ͑E 1457͒ the experimental C* parameter. .. Geometric function to calculate C* from crack mouth opening displacement rate Geometric function to calculate C* from load line displacement rate Elastic-plastic fracture mechanics parameter Manuscript... is related to an appropriately chosen area of the load-displacement curve for each specimen Two areas are used, the plastic area, A p, and plastic secant area, Asec, shown in Figs 2? ?a? ? and 2͑b͒,
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