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Three-dimensional finite element analyses of in-plane and out-of-plane creep crack-tip constraints for different specimen geometries
Engineering Fracture Mechanics 133 Supplement 1 (2015) 264–280
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Three-dimensional finite element analyses of in-plane and out-of-plane creep crack-tip constraints for different specimen geometries S. Liu, G.Z. Wang ⇑, F.Z. Xuan, S.T. Tu Key Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 15 January 2015 Received in revised form 14 May 2015 Accepted 6 October 2015 Available online 13 October 2015 Keywords: In-plane creep constraint Out-of-plane creep constraint Finite element analysis Specimen Crack
a b s t r a c t Based on extensive three-dimensional finite element analyses, the creep crack-tip constraints for three specimen geometries (C(T), SEN(T) and M(T)) were quantified by using three constraint parameters (namely R*, h and Tz) for 316H steel at 550 °C and steady-state power law creep. The three parameters were comparatively analyzed with respect to their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them. The results show that the specimen geometries with higher constraint strengthen the out-of-plane creep constraint effect. The parameters R* and h can quantify in-plane creep constraint, but they are inadequate to characterize higher out-of-plane creep constraint. The Tz can adequately incorporate out-of-plane creep constraint effect. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction It is well known that the constraint contains in-plane and out-of-plane constraints. The in-plane constraint is directly affected by the length of the un-cracked ligament of a specimen, while the out-of-plane constraint is affected by the specimen dimension parallel to crack front, that is, the specimen thickness. Because the constraint can dramatically alter fracture behavior of materials and structures, it is indispensable to quantify constraint and consider it in structural integrity assessments. The quantification of constraint has been widely studied within the elastic–plastic fracture mechanics frame for a long time, and different constraint parameters and fracture theories have been put forward in the last few decades, such as the two-parameter concepts K–T [1], J–Q [2,3] and J–A2 [4]. By introducing the out-of-plane stress constraint factor Tz, Guo and his colleagues derived out the 3-D asymptotic fields near tensile crack border in power law plastic (the Tz term has been proven to enter the singular term of the crack tip fields) [5–7]. The stress triaxiality parameter h also is usually used to characterize the constraint [8]. Most of these constraint parameters (T, Q, A2, Tz and h) were only used to quantify the inplane or out-of-plane constraint separately. However, in the actual engineering structures, both in-plane and out-of-plane constraints exist simultaneously. The variety of available constraint parameters provokes new questions regarding their applicability, comparability and ease of use [9]. In the work of Hebel et al. [9], several two-parameter concepts have been analyzed experimentally and numerically with respect to their capabilities of characterizing in-plane and out-of-plane
⇑ Corresponding author. Tel./fax: +86 21 64252681. E-mail address: [email protected] (G.Z. Wang). http://dx.doi.org/10.1016/j.engfracmech.2015.10.009 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
S. Liu et al. / Engineering Fracture Mechanics 133 Supplement 1 (2015) 264–280
Nomenclature a A A2 Ap B C* C*avg C(t) E h H J K KJc L n Q Q* r R R* t tred T Tz W z e_ 0
ep
h
r0 r22 r22,CT Dr
re rm ri rii rxx, ryy rzz t
crack depth coefficient in the power-law creep strain rate expression constraint parameter unified characterization parameter of in-plane and out-of-plane constraint specimen thickness C* integral analogous to the J integral average C* integral C(t) integral Young’s modulus stress triaxiality factor loading point height J-integral stress intensity factor fracture toughness characteristic length power-law creep stress exponent constraint parameter under elastic–plastic or creep condition a parameter for correlating creep crack growth rate distance from a crack tip creep constraint parameter load-independent creep constraint parameter creep time creep redistribution time T-stress constraint parameter under elastic condition out-of-plane constraint parameter specimen width distance from specimen center along specimen thickness creep strain rate at stress r0 equivalent plastic strain polar coordinate at the crack tip normalizing stress, usually taken as yield stress opening stress opening stress of C(T) specimen under plane strain opening stress difference von Mises effective stress hydrostatic stress principal stresses (i = 1, 2 and 3) stresses along x, y and z axis for the Cartesian coordinate (i = 1, 2 and 3) in-plane stresses out-of-plane stress Poisson’s ratio
Abbreviations 2-D two-dimensional 3-D three-dimensional CCG creep crack growth C(T) compact tension FEM finite element method HRR Hutchinson–Rice–Rosengren LSC large-scale creep M(T) middle cracked tension NSW Nikbin–Smith–Webster SEN(T) single-edge notched tension SSC small-scale creep
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crack-tip constraint effects for brittle fracture of the 22NiMoCr3-7 steel. The second term T-stress of the linear-elastic crack tip stress field, a higher term A2 of the power-law hardening crack-tip stress field, a hydrostatic correction term Q for a reference stress field, and the stress triaxiality parameter h were compared. It was found that with respect to their capabilities of quantifying combined in-plane and out-of-plane constraint effects, the investigated concepts differ significantly. A monotonic trend curve (toughness loci) cannot be formed between these constraint parameters and brittle fracture toughness KJc. This is caused by the underlying theory of these parameters [9]. The parameters T, A2, Q and h may be mainly capable of describing in-plane constraint effect. The parameters h and Q may partially incorporate the out-of-plane constraint effect due to the more reasonable toughness loci obtained by them than by T-stress and A2 [9]. In the work of Tkach and Burdekin [10,11], an extensive series of three-dimensional elastic–plastic finite element analyses has been carried out to investigate the influence of specimen geometry on the near-tip stress fields and constraint levels based on the constraint parameters Q and h. It has been shown that the in-plane constraint effect is controlled not only by the ratio of the crack length over the specimen width (a/W), but also by the absolute ligament length of the specimens. In recent work of Wang et al. [12], based on three-dimensional (3-D) elastic–plastic finite element analyses, the in-plane and out-of-plane constraint parameters (namely Q, A2, h and Tz) for single-edge notched tension (SEN(T)) specimens were investigated. Their results indicated that Q is essentially equivalent to h and correlated to both the in-plane and out-of-plane constraints, whereas A2 is more suitable to characterize the in-plane constraint, and Tz is more appropriate to characterize the out-of-plane constraint [12]. In recent work of authors [13–16], a constraint parameter Ap based on the areas surrounded by equivalent plastic strain (ep) isolines ahead of crack tips has been proposed to characterize both in-plane and out-of-plane constraints for ductile fracture [13–15] and brittle fracture [16]. The parameter Ap was comparatively analyzed with the two constraint parameters T and Q, and the results also showed that T and Q are mainly in-plane constraint parameters and Q can partially incorporate the out-of-plane constraint effect [16]. Under creep conditions, a lot of experimental and theoretical evidences have shown that the constraint can affect creep crack growth (CCG) rate [17–22]. For a given C* value (creep fracture mechanics parameter), the model predications showed that the CCG rates in plane strain are significantly greater than those in plane stress [19]. The experimental results of Tabuchi et al. [20] and Tan et al. [21] have shown that there is an effect of specimen thickness on the creep crack growth rate, and the specimens with larger thickness exhibit higher creep crack growth rate. It also has been found that at the same C* value the creep crack growth rates measured in the middle tension (M(T)) specimens are lower than those obtained from deep crack compact tension (C(T)) specimens for the austenitic stainless steels [17] and ferritic steels [22]. Yokobori et al. proposed a parameter Q* for correlating creep crack growth rate [23–27], and their work shown that the creep crack growth rate for a thick specimen is higher than that of a thin specimen [24]. The creep ductility and constraint effects can be estimated by using the parameter Q* [25], which were defined as ‘‘structural brittleness” [26]. To accurately predict the creep life and achieve structural integrity assessments for high temperature structures, it is important to quantify the creep crack-tip constraint levels in specimens or structures. However, the studies for the creep crack-tip constraint effects and the two-parameter or three-parameter characterization of creep crack-tip fields are very limited. The creep crack-tip stress and strain rate fields are often described by the C*–Q two-parameter under plane strain or plane stress conditions, and the parameter Q was used to quantify the constraint [28–30], and its effect on CCG has been examined [29]. Combined the C*–Q two-parameter concept with the Nikbin–Smith–Webster (NSW) model, Nikbin [19] investigated the effect of constraint on the CCG rate. Based on the C*–Q two-parameter concept and finite element analysis, Bettinson et al. [28] examined the effects of specimen type and load level on the Q from short to long term creep conditions for elastic–creep materials. The two-parameter C(t)–Tz and the three-parameter C(t)–Tz–Q descriptions for crack-tip fields were proposed by Xiang et al. [31] and Xiang and Guo [32] for small and extensive creeping, respectively. The in-plane constraint was characterized by the parameter Q, and the out-of-plane constraint was characterized by the parameter Tz. In the previous work of authors [33], it has been suggested that the HRR stress field may not be suitable to be a reference field for defining the constraint parameter under creep condition due to larger crack-tip blunting and damage. Based on the reference field of the standard C(T) specimen in plane strain with high constraint and deep crack (a/W = 0.5), a constraint parameter R was proposed to characterize the creep crack-tip constraint, and the constraint effects induced by the crack depths [33,34], specimen thicknesses [35] and loading configurations [36] have been investigated in detail. For the convenience of application, a load-independent creep constraint parameter R* has been defined by modifying the parameter R [37]. Based on the parameter R*, the characterization and correlation of two-dimensional and three-dimensional creep constraint between axially cracked pipelines and test specimens were studied [38,39], and the constraint parameter R* solutions for axially cracked pipes with different geometries and semi-elliptical surface crack sizes for the 316H steel have been obtained [40]. However, the existing creep constraint parameters have not been comparatively analyzed and systematically investigated with respect to their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them. In this work, based on extensive 3-D FEM analyses, the in-plane and out-of-plane creep crack-tip constraints for the three specimen geometries (C(T), SEN(T) and M(T)) were quantified by using three constraint parameters (namely R*, h and Tz) for 316H steel at 550 °C and steady-state power law creep. These constraint parameters were comparatively analyzed to understand their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them.
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2. Creep crack-tip constraint parameters In the previous work of authors [33], it was shown that the HRR field is not suitable to be as a reference field for defining constraint parameter under creep condition. The stress singularity near the crack tips does not exist due to the crack-tip blunting caused by larger creep deformation and damage. Thus, a constraint parameter R has been defined based on the reference field of C(T) specimen (a/W = 0.5) with high constraint in plane strain state as follows [33]:
R¼
r22 ðr22 ÞCT r0
at h ¼ 0; r ¼ distance from a crack tip
ð1Þ
where r22 is opening stress, r0 is the normalizing stress, h is polar coordinate at the crack tip (it usually taken to be h = 0 for defining constraint parameter due to the maximum opening stress there), and r is distance from a crack tip. The stress r22 are obtained at h = 0 and certain distance r from the crack tip. The R represents opening stress difference Dr = r22 (r22)CT between the under evaluated specimen or component (r22) and the C(T) specimen with a/W = 0.5 in plane strain ((r22)CT) at the same C* value. For resolving the problem of R parameter in application, in a recent study of authors [37], a load-independent creep constraint parameter R* at steady-state creep has been defined as follows:
R ¼ R
1 nþ1 C r0 e_ 0 L
for h ¼ 0 and r ¼ distance from crack tip
ð2Þ
where r0 is the normalizing stress (it is usually taken as the yield stress), e_ 0 is corresponding strain rate at the stress r0, n is power-law creep stress exponent, C* is the contour integral at steady-state creep, L is a characteristic length (it is usually set to be 1 m [41]) and (r, h) are the polar coordinates centered at the crack tip. The influences of the crack-tip creep blunting on the crack-tip stress field and r value have been analyzed in detail in the previous work of authors [38] for a wide range specimen and pipe geometries, crack sizes and material properties in a wide range of C*. The results (as shown in Figs. 4–7 in Ref. [38]) showed that the r = 0.2 mm is located outside the creep blunting region at t/tred = 1 (the beginning time of steady-state creep), and the load independence of R* can hold. Therefore, it is reasonable to define the load independent constraint parameter R* at r = 0.2 mm and t/tred = 1. It also has been found that the parameter R* under steady state creep (t/tred = 1) (Eq. (2)) can be easily calculated at different C* values [37,38]. Then, the creep constraint parameter R* can be expressed and calculated in Eq. (3) [38]:
R ¼ r22
1 1 nþ1 C 1 nþ1 C r22;CT 2 AL AL
at r ¼ 0:2 mm; h ¼ 0; t=t red ¼ 1
ð3Þ
where the C*1 is the C* value in the under evaluated cracked specimen or component, and the C*2 is the C* value in the standard C(T) specimens for obtaining reference stress field. The parameter R* is similar to the parameter Q within the elastic–plastic fracture mechanics frame. The Q is suitable for characterizing the in-plane constraint effect, but may not be adequate to characterize the out-of-plane constraint effect [9,12,16,42–44]. The stress triaxiality factor h is usually used to quantify the in-plane and out-of-plane constraint effect in both elastic– plastic [8,9,12,13] and creep fracture mechanics [35]. The triaxiality factor h is defined as the ratio of hydrostatic stress ðrm Þ to the von Mises effective stress ðre Þ:
h¼
1 ðr1 þ r2 þ r3 Þ rm 3 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi re 1 ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr3 r1 Þ2 2
ð4Þ
where ri (i = 1, 2 and 3) are the principal stresses. The parameter h incorporates both in-plane and out-of-plane principal stresses, thus it is expected to incorporate both in-plane and out-of-plane constraint effects. However, under creep condition, the capability of the triaxiality factor h for characterizing both in-plane and out-of-plane constraints has not been analyzed for different specimen geometries. The constraint parameter Tz was introduced to characterize the 3-D crack border fields by Guo [5–7] and defined as follows:
Tz ¼
r33 r11 þ r22
ð5Þ
where rii (i = 1, 2 and 3) are the stresses along x, y and z axis for the Cartesian coordinate, with z axis along the direction tangential to the crack front line. The parameter Tz is related to both out-of-plane stress ðrzz Þ and in-plane stresses (rxx and ryy ). The value of Tz is located between 0 (plane stress) and 0.5 (plane strain) for elastic–plastic analyses. The Tz term has been proven to enter the singular term of the crack tip fields [5–7]. Furthermore, the three-parameter J–Tz–Q solution for plastic solids by Guo [45] and the C(t)–Tz–Q solution for creeping solids by Xiang et al. [31] were proposed to consider both the in-plane and out-of-plane constraints. For the convenience of application, Xiang and Guo [32] formulized the 3-D theoretical solutions into a set of empirical explicit formulae in the whole range of out-of-plane stress constraint from Tz = 0 at plane stress state to Tz = 0.5 at plane strain state. In some recent work [44,46], it also has been shown that Tz can
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adequately characterize the out-of-plane constraint effect. However, in creep fracture mechanics, the parameter Tz and its correlation with other constraint parameters (such as R* and h) have not been systematically analyzed for different specimen geometries. 3. Finite element models and calculations of creep crack-tip constraint parameters 3.1. Material The material used in this work was the 316H stainless steel which is the same as that used in the literature [40,47]. An elastic–plastic-power law creep material model was used, and the creep strain rate e_ is given by:
e_ ¼ Arn ¼ e_ 0 ðr=r0 Þn
ð6Þ
The creep and elastic–plastic material parameters of the 316H stainless steel at 550 °C in the literature [47] were used. The values of n and A are 11.3 and 3 1034 MPan h1, respectively. The Young’s Modulus E of the steel at 550 °C is 140 GPa, and the Poisson’s ratio v was taken to be 0.3. The normalizing stress r0 was taken as 170 MPa which is the yield stress of the steel at 550 °C, and the corresponding e_ 0 is 4.80E9 h1. The true stress–strain curve of the 316H stainless steel at 550 °C is shown in Fig.1 [47]. 3.2. Finite element models of different specimen geometries Three fracture specimens with different geometries and crack sizes have been modeled using 3-D finite element analyses, including C(T) specimens with H/W = 0.71, SEN(T) specimens with H/W = 2 and M(T) specimens with H/W = 2. The 2-D geometries of the specimens are shown in Fig. 2. To investigate in-plane and out-of-plane creep constraint effects and their interaction for the three specimens, the crack depth a/W (for changing in-plane constraint) and thickness-to-width ratio B/W (for changing out-of-plane constraint) vary systematically for all specimens with a/W = 0.2, 0.3, 0.5 and 0.7 and B/W = 1/4, 1/2, 1, 2 and 4. Here, a is the crack depth, B is the specimen thickness, W is the specimen width which is fixed to be 25 mm and H is the distance between the clamps. Due to symmetry in geometry, only a quarter of the C(T), SEN(T) and M(T) specimens was modeled. The symmetry boundary condition is applied on the un-cracked ligament and middle plane (z/B = 0) of the specimens. The load is applied to the load hole as a distributed load for the specimens. The typical 3-D finite element models constructed for the C(T), SEN(T) and M(T) specimens with a/W = 0.5 and B/W = 1/2 are illustrated in Fig. 3(a–c), respectively, and the local mesh distribution around the crack tip is shown in Fig. 3(d). There is a single crack-tip node in the finite element meshes, and the crack tip is initially sharp. The analyses were carried out using ABAQUS code [48], and the eight-node linear 3-D elements (C3D8) was used for all models. The mesh sensitivity studies have been done for seven smaller element sizes of 30 lm, 40 lm, 50 lm, 60 lm, 70 lm, 80 lm and 100 lm around the crack tips. The results show that when the smallest element size is less than 50 lm, the FEM results of the C(t), C* and stresses do not essentially change with the element sizes. Hence the smallest element size of 40 lm around the crack tips is used in this study. The finite element analyses were performed using large-displacement with the NLGEOM option ON. The contour integral C(t) and C* were evaluated by using the in-built ABAQUS routines. Ten contours were set around the crack tips. The C* is usually path-independent, and C(t) is not path-independent, but with increasing time the C(t) gradually becomes pathindependent. The C(t) was defined as the value from the tenth contour near the crack tip for the different specimens with the same size of meshes and contours (The selection of other contours can give similar result). The constraint parameter R*, Tz and h ahead of the crack tips were calculated at steady-state creep.
Fig. 1. True stress–strain curve of 316H stainless steel at 550 °C [47].
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Fig. 2. Geometries and dimensions of the specimens: (a) C(T), (b) SEN(T), and (c) M(T).
Fig. 3. Typical finite element meshes for specimens with a/W = 0.5 and B/W = 1/2, (a) C(T), (b) SEN(T), (c) M(T) and (d) local meshes around the crack tip.
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3.3. Calculations of creep crack-tip constraint parameters The three-dimensional FEM calculations were conducted to obtain the values of the three creep crack-tip constraint parameters (R*, h and Tz) along crack fronts for the C(T), SEN(T) and M(T) specimens with different a/W and B/W described in Section 3.2. The parameter R* was calculated by using Eq. (3) at r = 0.2 mm, h = 0 and t/tred = 1 (steady-state creep). To facilitate the analysis and interpretation of results, the parameters h and Tz were also calculated and evaluated at r = 0.2 mm, h = 0 and t/tred = 1 by using the Eqs. (4) and (5), respectively. 4. Results and discussion 4.1. Distributions of parameters R*, h and Tz ahead of crack tips The parameters R*, h and Tz were calculated and analyzed at steady-state creep. Fig. 4 shows the distributions of R*, h and Tz ahead of crack tips at the middle plane of the SEN(T) specimen with a/W = 0.5 and B/W = 1 at h = 0, t/tred = 1 and average C*avg 1E8 MPa m h1 along specimen thickness. Fig. 4(a–c) shows that the R*, h and Tz decrease with increasing the distance r from the crack tip. These results are similar to those of Q, h and Tz in elastic–plastic analyses [12]. Fig. 4 also shows that r = 0.2 mm is located outside the blunted crack-tip region (the region with lower R*, h and Tz values near the crack tip due to blunting and damage), thus the R*, h and Tz values at r = 0.2 mm were calculated to characterize creep crack-tip constraint.. It should be noted that similar distributions of these constraint parameters ahead of the crack tips were observed for the C(T), SEN(T) and M(T) specimens with different a/W, B/W, and load level C*. 4.2. Distributions of parameter R* along specimen thickness Figs. 5–7 show the distributions of constraint parameter R* at average C*avg 1E8 MPa m h1 along specimen thickness for C(T), SEN(T) and M(T) specimens with different crack depths (a/W = 0.2, 0.3, 0.5 and 0.7) and thickness-to-width ratios (B/ W = 1/4, 1/2, 1, 2 and 4). The z/B = 0 is the middle plane, and the z/B = 0.5 is the free surface of the specimens. It can be seen from Figs. 5–7 that the center region (around z/B = 0) of all specimens has higher constraint, and the surface region (near z/B = 0.5) has lower constraint. For all specimens with different a/W, the size of the center region with uniform distribution of R* increases with increasing B/W, i.e. the R* distributes more uniformly along crack fronts as the B/W increases. With increasing specimen thickness (B/W), the R* of the surface regions in all specimens increases, but that of the center regions depends on specimen geometries and crack depth a/W. For the C(T) specimens in Fig. 5, the R* at the center region
Fig. 4. Distributions of R* (a), h (b) and Tz (c) ahead of crack tips at the middle plane of SEN(T) specimen with a/W = 0.5 and B/W = 1 at h = 0, t/tred = 1 and average C*avg 1E8 MPa m h1 along specimen thickness.
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Fig. 5. The distributions of R* along specimen thickness at C*avg 1E8 MPa m h1 for C(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d).
Fig. 6. The distributions of R* along specimen thickness at C*avg 1E8 MPa m h1 for SEN(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d).
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Fig. 7. The distributions of R* along specimen thickness at C*avg 1E8 MPa m h1 for M(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d).
increases with increasing the B/W from 1/4 to 1, and then it essentially does not change with B/W. Fig. 5 also shows that the distributions of the parameter R* along specimen thickness are similar for the C(T) specimens with different a/W ratios. This implies that the in-plane constraint effect induced by crack depth is small for C(T) geometry specimens with high constraint. Fig. 6 shows that the center regions for the SEN(T) specimens with B/W from 1/4 to 1 have slightly higher R* than those with B/W = 2 and 4. Fig. 6 also indicates that the distributions of the R* along SEN(T) specimen thickness are related to crack depth a/W, and the R* increases with increasing a/W. This implies that the in-plane dimension induced by crack depth has an effect on the constraint R* of SEN(T) specimens. These results of R* for the SEN(T) specimens are similar to the results of Q for the SEN(T) specimens in elastic–plastic analyses in the recent literature [12]. Fig. 7 shows that the distributions of R* in M(T) specimens with a/W = 0.2 and 0.3 are almost the same, and the R* at the center regions essentially does not change with the B/W. The distributions of R* in M(T) specimens with a/W = 0.5 and 0.7 are also almost the same, and the R* at the center regions with B/W = 1/4 and 1/2 is slightly higher than that of B/W = 1, 2 and 4. In general, the crack depth a/W has small effect on the R* distributions in M(T) specimens. These results imply that the constraint R* of the M(T) specimens is insensitive to both in-plane (a/W) and out-of-plane (B/W) dimensions. A comparison of Figs. 5–7 shows that for a given a/W, the variation amount of R* with the B/W ratio increases with increasing specimen constraint levels from the M(T) to C(T), i.e., the B/W ratio of high constraint specimens has more obvious effect on the R* than that of low constraint specimens. This implies that the specimen geometries with higher constraint strengthen the out-of-plane constraint effect, and the effect of specimen thickness on creep crack growth (CCG) rate of high constraint C(T) specimens may be more obvious than that of low constraint M(T) specimen. This inference has been confirmed by the recent experimental and simulation results of CCG rate by authors [49,50] which show that the effect of the specimen thickness B on the CCG rate of C(T) specimens is more obvious than that of M(T) specimens. It should be noted that the R* in the center region for all specimens does not consistently increase for higher B/W as B/W increases, which implies that R* may be inadequate to characterize the out-of-plane creep constraint for higher B/W. This result is similar to that of the constraint parameter Q within elastic–plastic fracture mechanics in the literature [12,42–44]. Fig. 8 shows the values of R* at the middle plane (z/B = 0) of all specimens with various a/W at the loading level C*avg 1E8 MPa m h1 for two typical B/W = 0.25 and 2. It can be seen that the C(T) specimen has the highest constraint, and that of the M(T) specimen has the lowest constraint. The constraint R* of C(T) and M(T) specimens is insensitive to crack depth a/W, and that of SEN(T) specimen increases with the increase of a/W. The similar results were observed for the specimens with other B/W. In general, the effects of crack depths a/W (in-plane dimension) on the R* is smaller than those of the specimen thickness ratios B/W (out-of-plane dimension) for all specimens. This implies that for a given specimen geometries, the effect of crack depths on creep crack growth (CCG) rate of a material may be smaller than those of specimen thickness. This inference has been confirmed by the recent CCG simulation results of authors [50] which show that the effects of the specimen thickness B on the CCG rate are more obvious than those of crack depth a/W for the C(T) and M(T) specimens.
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Fig. 8. The values of R* at the middle plane (z/B = 0) of all specimens with various a/W at C*avg 1E8 MPa m h1 for two typical B/W ratios, (a) B/W = 0.25 and (b) B/W = 2.
Fig. 9. The distributions of h along the thickness of C(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
4.3. Distributions of triaxiality factor h along specimen thickness Figs. 9–11 show the distributions of triaxiality factor h along specimen thickness at C*avg 1E8 MPa m h1 for C(T), SEN (T) and M(T) specimens with crack depth a/W = 0.2, 0.3, 0.5 and 0.7 and thickness-to-width ratio B/W = 1/4, 1/2, 1, 2 and 4. It can be seen that the distributions of triaxiality factor h for each specimen with different a/W and B/W are similar to the corresponding distributions of R* in Figs. 5–7. The only difference is that the h is more sensitive to the B/W ratio than the R*, which implies that the parameter h may incorporate more out-of-plane constraint effect than the R*. For the low constraint M(T) geometry, the distributions of h are not sensitive to both B/W and a/W, and this is almost the same as the case of the R* of M(T) specimens in Fig. 7. For the middle constraint SEN(T) geometry, the effect of out-of-plane dimension B/W on h is smaller than that of C(T) geometry, and is larger than that of M(T) geometry. The distributions of h along specimen thickness do not consistently increase for higher B/W with increasing the B/W (Figs. 9–11), which implies that the parameter h cannot incorporate all out-of-plane constraint effect. This observation is consistent with the recent result of 3-D elastic–plastic analysis for the SEN(T) specimens [12]. The results of the h in Figs. 9–11 indicate that the higher constraint geometries
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Fig. 10. The distributions of h along the thickness of SEN(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
Fig. 11. The distributions of h along the thickness of M(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
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strengthen the out-of-plane constraint effect, and the h is also inadequate to incorporate the out-of-plane constraint effect for higher B/W. Similar to the R* results in Figs. 5–7, Figs. 9–11 also show the h is insensitive to the crack depth a/W for the C(T) and M(T) specimens, but the h of the SEN(T) specimens increases with increasing a/W. 4.4. Distributions of parameter Tz along specimen thickness Figs. 12–14 show the distributions of Tz along the thickness at C*avg 1E8 MPa m h1 for C(T), SEN(T) and M(T) specimens with crack depths a/W = 0.2, 0.3, 0.5 and 0.7 and thickness-to-width ratios B/W = 1/4, 1/2, 1, 2 and 4. The Tz distributes more uniformly along crack fronts as the B/W increases for all specimens. For a given specimen geometry, the Tz consistently increases as B/W (out-of-plane constraint) increases regardless of a/W. This result indicates that compared with the parameters R* and h, the Tz is adequate to incorporate the out-of-plane constraint effect, thus it is a good out-of-plane constraint parameter (it is insensitive to the in-plane constraint effect induced by crack depths). These results of the parameter Tz are similar to the elastic–plastic analysis results which show that compared with parameters Q and h, the Tz is more appropriate to characterize the out-of-plane constraint [12,44]. Figs. 12–14 also show that the increase rate of Tz decreases when the B/W value is larger than 1, which implies that the increase rate of out-of-plane constraint decreases after the specimen thickness B increases to a larger value than the width W. In general, the Tz values of the C(T) specimens are higher than those of SEN(T) and M(T) specimens. This also indicates that the higher constraint geometries strengthens the out-of-plane constraint effect. It should be noted that the creep constraint parameters above are calculated and analyzed by using C* at steady-state creep. The crack-tip stress fields characterized by the C(t) and the constraint parameter R* at transition creep have been investigated in the previous work [37] of authors. It has been shown that during the early stage of creep (0 < t/tred < 0.1), the R* increases with creep time, and then it becomes independent on the t/tred. Because the period of 0 < t/tred < 0.1 is very short and R* is low in this period, the parameter R* under steady-state creep can be used to evaluate the constraint level with little conservatism during the whole creep time [37]. 5. Interactions between in-plane and out-of-plane creep constraint parameters To further analyze the capabilities of different parameters for characterizing in-plane and out-of-plane creep constraints, the interactions between constraint parameters R*, h and Tz are analyzed for all thickness layers from middle plane (z/B = 0) to free surface (z/B = 0.5) of C(T), SEN(T) and M(T) specimens with a/W = 0.2 and 0.5 and B/W = 1/4, 1/2, 1, 2 and 4 at C*avg 1E8 MPa m h1.
Fig. 12. The distributions of Tz along the thickness of C(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
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Fig. 13. The distributions of Tz along the thickness of SEN(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
Fig. 14. The distribution of Tz along the thickness of M(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
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Fig. 15. Interaction between constraint parameter R* and h for all thickness layers of C(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 16. Interaction between constraint parameter R* and h for all thickness layers of SEN(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 17. Interaction between constraint parameter R* and h for all thickness layers of M(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Figs. 15–17 show the interaction between h and R* of C(T), SEN(T) and M(T) specimens for two typical crack depths a/W = 0.2 and 0.5 (note that the results for other crack depths are similar). The R* and h values are corresponding to the different locations along crack front (different z/B) for different B/W in Figs. 5–7 (for R*) and Figs. 9–11 (for h). It can be seen that there essentially exists a linear correlation between h and R* regardless of a/W and B/W for all specimens with R* < 0. For the C(T) specimens with B/W > 1, the R* at the center region of specimens is close or slightly larger than zero (the constraint level is around that of the standard C(T) specimen in plane strain with R* = 0), the linear correlation between h and R* loses due to the increase of h with small or no change in R*. These results imply that for all specimens with R* < 0 (the constraint level is lower than that of the standard high constraint C(T) specimen in plane strain), the parameter R* is equivalent to the triaxiality factor h. The interaction between the h and R* is similar to that between h and Q in elastic–plastic fracture mechanics [12,51]. The factor h can quantify both in-plane and out-of-plane constraints [12,42,43,51], thus it can be inferred the
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Fig. 18. Interaction between constraint parameter R* and Tz for all thickness layers of C(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 19. Interaction between constraint parameter R* and Tz for all thickness layers of SEN(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 20. Interaction between constraint parameter R* and Tz for all thickness layers of M(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
parameter R* may also characterize in-plane and out-of-plane constraints of lower constraint geometries. This confirms the capability of R* for characterizing both in-plane and out-of-plane constraints for lower constraint geometries which may usually occur in high temperature components (such as pressure pipes [38–40]). The h and Q parameters can quantify both in-plane and out-of-plane constraints in elastic–plastic fracture mechanics, but they are not equally sensitive to in-plane and out-of-plane constraint [13–16] and only partially incorporate the out-of-plane constraint effect. Thus, it can be inferred that the parameter R* may also not be equally sensitive to in-plane and out-of-plane constraint. This has been verified by a recent study of authors, and a unified parameter of in-plane and out-of-plane creep constraint which is equally sensitive to in-plane and out-of-plane constraint has been proposed [52]. Figs. 18–20 show the interaction between Tz and R* of C(T), SEN(T) and M(T) specimens for two typical crack depths a/W = 0.2 and 0.5 (note that the results for other crack depths are similar). The R* and Tz values are corresponding to the
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different locations along crack front (different z/B) for different B/W in Figs. 5–7 (for R*) and Figs. 12–14 (for Tz). It can be seen that there also essentially exists a linear correlation between Tz and R* for all specimens with Tz < 0.42, but the linear correlation is influenced by B/W and regardless of a/W. The effects of B/W of SEN(T) and M(T) specimens on the linear correlation are more obvious than those of C(T) specimens, and with decreasing B/W the correlation lines have a trend moving down. For the all specimens with B/W > 1 and Tz > 0.42, the linear correlation between Tz and R* loses due to the increase of Tz with small or no change in R*. Because Tz is an out-of-plane constraint parameter, the approximate linear correlation of Tz with R* for Tz < 0.42 implies that the parameter R* also can characterize out-of-plane constraint. This further confirms the capability of R* for characterizing both in-plane and lower out-of-plane constraint. But the R* is insensitive to very high outof-plane constraint with Tz > 0.42 and B/W > 1, which implies that the R* is inadequate to characterize higher out-of-plane creep constraint. The dependence of the correlation lines on B/W in Figs. 18–20 may be due to that the Tz mainly incorporates out-of-plane constraint and the R* incorporates both in-plane and out-of-plane constraint. 6. Conclusion Extensive three-dimensional finite element analyses have been conducted for C(T), SEN(T) and M(T) specimens with various crack depth (a/W) and thickness-to-width ratio (B/W), and the creep crack-tip constraints for the three specimen geometries were quantified by using three constraint parameters (namely R*, h and Tz) for 316H steel at 550 °C and steady-state power law creep. Based on the results, these constraint parameters have comparatively investigated with respect to their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them. The main results obtained are as follows: (1) The parameters R* and h both correctly show that the specimen order for constraint from high to low is C(T), SENT(T) and M(T). For a given crack depth a/W, the sizes of the center regions with uniform distributions of parameters R*, h and Tz for all specimens increase with increasing B/W. The B/W ratio has more obvious effects on the parameters R* and h of higher constraint specimens, which implies that the specimen geometries with higher constraint strengthen the out-of-plane constraint effect. (2) The constraints R* and h of C(T) and M(T) specimens are insensitive to crack depth a/W, but those of SEN(T) specimens increase with the increase of a/W. In general, the effects of crack depths a/W (in-plane dimensions) on the R* and h (constraint levels) are smaller than those of specimen thickness ratios B/W (out-of-plane dimensions) for all specimens. (3) The parameters R* and h can quantify in-plane creep constraint, but they are inadequate to characterize higher out-ofplane creep constraint. Compared with the R* and h, the Tz is adequate to incorporate the out-of-plane creep constraint effect. (4) For all specimens with R* < 0 (the constraint level is lower than that of the standard high constraint C(T) specimen in plane strain), the parameter R* is equivalent to the triaxiality factor h. There also exists approximate linear correlation of Tz with R* for Tz < 0.42.
Acknowledgment This work was financially supported by the Projects of the National Natural Science Foundation of China (51375165, 51575184 and 51325504). References [1] Larsson SG, Carlsson AJ. Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 1973;21:263–77. [2] O’Dowd NP, Shih CF. Family of crack tip fields characterized by a triaxiality parameter-I. Structure of fields. J Mech Phys Solids 1991;39:989–1015. [3] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter-II. Fracture applications. J Mech Phys Solids 1992;40:939–63. [4] Chao YJ, Yang S, Sutton MA. On the fracture of solids characterized by one or two parameters: theory and practice. J Mech Phys Solids 1994;42:629–47. [5] Guo W. Elastoplastic three dimensional crack border field—I. Singular structure of the field. Engng Fract Mech 1993;46:93–104. [6] Guo W. Elastoplastic three dimensional crack border field—II. Asymptotic solution for the field. Engng Fract Mech 1993;46:105–13. [7] Guo W. Elasto-plastic three-dimensional crack border field—III. Fracture parameters. Engng Fract Mech 1995;51:51–71. [8] Clausmeyer H, Kussmaul K, Roos E. Influence of stress state on the failure behavior of cracked components made of steel. Appl Mech Rev 1991;44:77–92. [9] Hebel J, Hohe J, Friedmann V, Siegele D. Experimental and numerical analysis of in-plane and out-of-plane crack tip constraint characterization by secondary fracture parameters. Int J Fract 2007;146:173–88. [10] Tkach Y, Burdekin FM. A three-dimensional analysis of fracture mechanics test pieces of different geometries – Part 1 Stress-state ahead of the crack tip. Int J Pres Ves Pip 2012;93–94:42–50. [11] Tkach Y, Burdekin FM. A three-dimensional analysis of fracture mechanics test pieces of different geometries Part 2 – Constraint and material variations. Int J Pres Ves Pip 2012;93–94:51–6. [12] Wang E, Zhou W, Shen G. Three-dimensional finite element analysis of crack-tip fields of clamped single-edge tension specimens – Part II: Crack-tip constraints. Engng Fract Mech 2014;116:144–57. [13] Yang J, Wang GZ, Xuan FZ, Tu ST. Unified characterisation of in-plane and out-of-plane constraint based on crack-tip equivalent plastic strain. Fatigue Fract Engng Mater Struct 2013;36:504–14.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Three-dimensional finite element analyses of in-plane and out-of-plane creep crack-tip constraints for different specimen geometries S. Liu, G.Z. Wang ⇑, F.Z. Xuan, S.T. Tu Key Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 15 January 2015 Received in revised form 14 May 2015 Accepted 6 October 2015 Available online 13 October 2015 Keywords: In-plane creep constraint Out-of-plane creep constraint Finite element analysis Specimen Crack
a b s t r a c t Based on extensive three-dimensional finite element analyses, the creep crack-tip constraints for three specimen geometries (C(T), SEN(T) and M(T)) were quantified by using three constraint parameters (namely R*, h and Tz) for 316H steel at 550 °C and steady-state power law creep. The three parameters were comparatively analyzed with respect to their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them. The results show that the specimen geometries with higher constraint strengthen the out-of-plane creep constraint effect. The parameters R* and h can quantify in-plane creep constraint, but they are inadequate to characterize higher out-of-plane creep constraint. The Tz can adequately incorporate out-of-plane creep constraint effect. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction It is well known that the constraint contains in-plane and out-of-plane constraints. The in-plane constraint is directly affected by the length of the un-cracked ligament of a specimen, while the out-of-plane constraint is affected by the specimen dimension parallel to crack front, that is, the specimen thickness. Because the constraint can dramatically alter fracture behavior of materials and structures, it is indispensable to quantify constraint and consider it in structural integrity assessments. The quantification of constraint has been widely studied within the elastic–plastic fracture mechanics frame for a long time, and different constraint parameters and fracture theories have been put forward in the last few decades, such as the two-parameter concepts K–T [1], J–Q [2,3] and J–A2 [4]. By introducing the out-of-plane stress constraint factor Tz, Guo and his colleagues derived out the 3-D asymptotic fields near tensile crack border in power law plastic (the Tz term has been proven to enter the singular term of the crack tip fields) [5–7]. The stress triaxiality parameter h also is usually used to characterize the constraint [8]. Most of these constraint parameters (T, Q, A2, Tz and h) were only used to quantify the inplane or out-of-plane constraint separately. However, in the actual engineering structures, both in-plane and out-of-plane constraints exist simultaneously. The variety of available constraint parameters provokes new questions regarding their applicability, comparability and ease of use [9]. In the work of Hebel et al. [9], several two-parameter concepts have been analyzed experimentally and numerically with respect to their capabilities of characterizing in-plane and out-of-plane
⇑ Corresponding author. Tel./fax: +86 21 64252681. E-mail address: [email protected] (G.Z. Wang). http://dx.doi.org/10.1016/j.engfracmech.2015.10.009 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
S. Liu et al. / Engineering Fracture Mechanics 133 Supplement 1 (2015) 264–280
Nomenclature a A A2 Ap B C* C*avg C(t) E h H J K KJc L n Q Q* r R R* t tred T Tz W z e_ 0
ep
h
r0 r22 r22,CT Dr
re rm ri rii rxx, ryy rzz t
crack depth coefficient in the power-law creep strain rate expression constraint parameter unified characterization parameter of in-plane and out-of-plane constraint specimen thickness C* integral analogous to the J integral average C* integral C(t) integral Young’s modulus stress triaxiality factor loading point height J-integral stress intensity factor fracture toughness characteristic length power-law creep stress exponent constraint parameter under elastic–plastic or creep condition a parameter for correlating creep crack growth rate distance from a crack tip creep constraint parameter load-independent creep constraint parameter creep time creep redistribution time T-stress constraint parameter under elastic condition out-of-plane constraint parameter specimen width distance from specimen center along specimen thickness creep strain rate at stress r0 equivalent plastic strain polar coordinate at the crack tip normalizing stress, usually taken as yield stress opening stress opening stress of C(T) specimen under plane strain opening stress difference von Mises effective stress hydrostatic stress principal stresses (i = 1, 2 and 3) stresses along x, y and z axis for the Cartesian coordinate (i = 1, 2 and 3) in-plane stresses out-of-plane stress Poisson’s ratio
Abbreviations 2-D two-dimensional 3-D three-dimensional CCG creep crack growth C(T) compact tension FEM finite element method HRR Hutchinson–Rice–Rosengren LSC large-scale creep M(T) middle cracked tension NSW Nikbin–Smith–Webster SEN(T) single-edge notched tension SSC small-scale creep
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crack-tip constraint effects for brittle fracture of the 22NiMoCr3-7 steel. The second term T-stress of the linear-elastic crack tip stress field, a higher term A2 of the power-law hardening crack-tip stress field, a hydrostatic correction term Q for a reference stress field, and the stress triaxiality parameter h were compared. It was found that with respect to their capabilities of quantifying combined in-plane and out-of-plane constraint effects, the investigated concepts differ significantly. A monotonic trend curve (toughness loci) cannot be formed between these constraint parameters and brittle fracture toughness KJc. This is caused by the underlying theory of these parameters [9]. The parameters T, A2, Q and h may be mainly capable of describing in-plane constraint effect. The parameters h and Q may partially incorporate the out-of-plane constraint effect due to the more reasonable toughness loci obtained by them than by T-stress and A2 [9]. In the work of Tkach and Burdekin [10,11], an extensive series of three-dimensional elastic–plastic finite element analyses has been carried out to investigate the influence of specimen geometry on the near-tip stress fields and constraint levels based on the constraint parameters Q and h. It has been shown that the in-plane constraint effect is controlled not only by the ratio of the crack length over the specimen width (a/W), but also by the absolute ligament length of the specimens. In recent work of Wang et al. [12], based on three-dimensional (3-D) elastic–plastic finite element analyses, the in-plane and out-of-plane constraint parameters (namely Q, A2, h and Tz) for single-edge notched tension (SEN(T)) specimens were investigated. Their results indicated that Q is essentially equivalent to h and correlated to both the in-plane and out-of-plane constraints, whereas A2 is more suitable to characterize the in-plane constraint, and Tz is more appropriate to characterize the out-of-plane constraint [12]. In recent work of authors [13–16], a constraint parameter Ap based on the areas surrounded by equivalent plastic strain (ep) isolines ahead of crack tips has been proposed to characterize both in-plane and out-of-plane constraints for ductile fracture [13–15] and brittle fracture [16]. The parameter Ap was comparatively analyzed with the two constraint parameters T and Q, and the results also showed that T and Q are mainly in-plane constraint parameters and Q can partially incorporate the out-of-plane constraint effect [16]. Under creep conditions, a lot of experimental and theoretical evidences have shown that the constraint can affect creep crack growth (CCG) rate [17–22]. For a given C* value (creep fracture mechanics parameter), the model predications showed that the CCG rates in plane strain are significantly greater than those in plane stress [19]. The experimental results of Tabuchi et al. [20] and Tan et al. [21] have shown that there is an effect of specimen thickness on the creep crack growth rate, and the specimens with larger thickness exhibit higher creep crack growth rate. It also has been found that at the same C* value the creep crack growth rates measured in the middle tension (M(T)) specimens are lower than those obtained from deep crack compact tension (C(T)) specimens for the austenitic stainless steels [17] and ferritic steels [22]. Yokobori et al. proposed a parameter Q* for correlating creep crack growth rate [23–27], and their work shown that the creep crack growth rate for a thick specimen is higher than that of a thin specimen [24]. The creep ductility and constraint effects can be estimated by using the parameter Q* [25], which were defined as ‘‘structural brittleness” [26]. To accurately predict the creep life and achieve structural integrity assessments for high temperature structures, it is important to quantify the creep crack-tip constraint levels in specimens or structures. However, the studies for the creep crack-tip constraint effects and the two-parameter or three-parameter characterization of creep crack-tip fields are very limited. The creep crack-tip stress and strain rate fields are often described by the C*–Q two-parameter under plane strain or plane stress conditions, and the parameter Q was used to quantify the constraint [28–30], and its effect on CCG has been examined [29]. Combined the C*–Q two-parameter concept with the Nikbin–Smith–Webster (NSW) model, Nikbin [19] investigated the effect of constraint on the CCG rate. Based on the C*–Q two-parameter concept and finite element analysis, Bettinson et al. [28] examined the effects of specimen type and load level on the Q from short to long term creep conditions for elastic–creep materials. The two-parameter C(t)–Tz and the three-parameter C(t)–Tz–Q descriptions for crack-tip fields were proposed by Xiang et al. [31] and Xiang and Guo [32] for small and extensive creeping, respectively. The in-plane constraint was characterized by the parameter Q, and the out-of-plane constraint was characterized by the parameter Tz. In the previous work of authors [33], it has been suggested that the HRR stress field may not be suitable to be a reference field for defining the constraint parameter under creep condition due to larger crack-tip blunting and damage. Based on the reference field of the standard C(T) specimen in plane strain with high constraint and deep crack (a/W = 0.5), a constraint parameter R was proposed to characterize the creep crack-tip constraint, and the constraint effects induced by the crack depths [33,34], specimen thicknesses [35] and loading configurations [36] have been investigated in detail. For the convenience of application, a load-independent creep constraint parameter R* has been defined by modifying the parameter R [37]. Based on the parameter R*, the characterization and correlation of two-dimensional and three-dimensional creep constraint between axially cracked pipelines and test specimens were studied [38,39], and the constraint parameter R* solutions for axially cracked pipes with different geometries and semi-elliptical surface crack sizes for the 316H steel have been obtained [40]. However, the existing creep constraint parameters have not been comparatively analyzed and systematically investigated with respect to their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them. In this work, based on extensive 3-D FEM analyses, the in-plane and out-of-plane creep crack-tip constraints for the three specimen geometries (C(T), SEN(T) and M(T)) were quantified by using three constraint parameters (namely R*, h and Tz) for 316H steel at 550 °C and steady-state power law creep. These constraint parameters were comparatively analyzed to understand their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them.
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2. Creep crack-tip constraint parameters In the previous work of authors [33], it was shown that the HRR field is not suitable to be as a reference field for defining constraint parameter under creep condition. The stress singularity near the crack tips does not exist due to the crack-tip blunting caused by larger creep deformation and damage. Thus, a constraint parameter R has been defined based on the reference field of C(T) specimen (a/W = 0.5) with high constraint in plane strain state as follows [33]:
R¼
r22 ðr22 ÞCT r0
at h ¼ 0; r ¼ distance from a crack tip
ð1Þ
where r22 is opening stress, r0 is the normalizing stress, h is polar coordinate at the crack tip (it usually taken to be h = 0 for defining constraint parameter due to the maximum opening stress there), and r is distance from a crack tip. The stress r22 are obtained at h = 0 and certain distance r from the crack tip. The R represents opening stress difference Dr = r22 (r22)CT between the under evaluated specimen or component (r22) and the C(T) specimen with a/W = 0.5 in plane strain ((r22)CT) at the same C* value. For resolving the problem of R parameter in application, in a recent study of authors [37], a load-independent creep constraint parameter R* at steady-state creep has been defined as follows:
R ¼ R
1 nþ1 C r0 e_ 0 L
for h ¼ 0 and r ¼ distance from crack tip
ð2Þ
where r0 is the normalizing stress (it is usually taken as the yield stress), e_ 0 is corresponding strain rate at the stress r0, n is power-law creep stress exponent, C* is the contour integral at steady-state creep, L is a characteristic length (it is usually set to be 1 m [41]) and (r, h) are the polar coordinates centered at the crack tip. The influences of the crack-tip creep blunting on the crack-tip stress field and r value have been analyzed in detail in the previous work of authors [38] for a wide range specimen and pipe geometries, crack sizes and material properties in a wide range of C*. The results (as shown in Figs. 4–7 in Ref. [38]) showed that the r = 0.2 mm is located outside the creep blunting region at t/tred = 1 (the beginning time of steady-state creep), and the load independence of R* can hold. Therefore, it is reasonable to define the load independent constraint parameter R* at r = 0.2 mm and t/tred = 1. It also has been found that the parameter R* under steady state creep (t/tred = 1) (Eq. (2)) can be easily calculated at different C* values [37,38]. Then, the creep constraint parameter R* can be expressed and calculated in Eq. (3) [38]:
R ¼ r22
1 1 nþ1 C 1 nþ1 C r22;CT 2 AL AL
at r ¼ 0:2 mm; h ¼ 0; t=t red ¼ 1
ð3Þ
where the C*1 is the C* value in the under evaluated cracked specimen or component, and the C*2 is the C* value in the standard C(T) specimens for obtaining reference stress field. The parameter R* is similar to the parameter Q within the elastic–plastic fracture mechanics frame. The Q is suitable for characterizing the in-plane constraint effect, but may not be adequate to characterize the out-of-plane constraint effect [9,12,16,42–44]. The stress triaxiality factor h is usually used to quantify the in-plane and out-of-plane constraint effect in both elastic– plastic [8,9,12,13] and creep fracture mechanics [35]. The triaxiality factor h is defined as the ratio of hydrostatic stress ðrm Þ to the von Mises effective stress ðre Þ:
h¼
1 ðr1 þ r2 þ r3 Þ rm 3 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi re 1 ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr3 r1 Þ2 2
ð4Þ
where ri (i = 1, 2 and 3) are the principal stresses. The parameter h incorporates both in-plane and out-of-plane principal stresses, thus it is expected to incorporate both in-plane and out-of-plane constraint effects. However, under creep condition, the capability of the triaxiality factor h for characterizing both in-plane and out-of-plane constraints has not been analyzed for different specimen geometries. The constraint parameter Tz was introduced to characterize the 3-D crack border fields by Guo [5–7] and defined as follows:
Tz ¼
r33 r11 þ r22
ð5Þ
where rii (i = 1, 2 and 3) are the stresses along x, y and z axis for the Cartesian coordinate, with z axis along the direction tangential to the crack front line. The parameter Tz is related to both out-of-plane stress ðrzz Þ and in-plane stresses (rxx and ryy ). The value of Tz is located between 0 (plane stress) and 0.5 (plane strain) for elastic–plastic analyses. The Tz term has been proven to enter the singular term of the crack tip fields [5–7]. Furthermore, the three-parameter J–Tz–Q solution for plastic solids by Guo [45] and the C(t)–Tz–Q solution for creeping solids by Xiang et al. [31] were proposed to consider both the in-plane and out-of-plane constraints. For the convenience of application, Xiang and Guo [32] formulized the 3-D theoretical solutions into a set of empirical explicit formulae in the whole range of out-of-plane stress constraint from Tz = 0 at plane stress state to Tz = 0.5 at plane strain state. In some recent work [44,46], it also has been shown that Tz can
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adequately characterize the out-of-plane constraint effect. However, in creep fracture mechanics, the parameter Tz and its correlation with other constraint parameters (such as R* and h) have not been systematically analyzed for different specimen geometries. 3. Finite element models and calculations of creep crack-tip constraint parameters 3.1. Material The material used in this work was the 316H stainless steel which is the same as that used in the literature [40,47]. An elastic–plastic-power law creep material model was used, and the creep strain rate e_ is given by:
e_ ¼ Arn ¼ e_ 0 ðr=r0 Þn
ð6Þ
The creep and elastic–plastic material parameters of the 316H stainless steel at 550 °C in the literature [47] were used. The values of n and A are 11.3 and 3 1034 MPan h1, respectively. The Young’s Modulus E of the steel at 550 °C is 140 GPa, and the Poisson’s ratio v was taken to be 0.3. The normalizing stress r0 was taken as 170 MPa which is the yield stress of the steel at 550 °C, and the corresponding e_ 0 is 4.80E9 h1. The true stress–strain curve of the 316H stainless steel at 550 °C is shown in Fig.1 [47]. 3.2. Finite element models of different specimen geometries Three fracture specimens with different geometries and crack sizes have been modeled using 3-D finite element analyses, including C(T) specimens with H/W = 0.71, SEN(T) specimens with H/W = 2 and M(T) specimens with H/W = 2. The 2-D geometries of the specimens are shown in Fig. 2. To investigate in-plane and out-of-plane creep constraint effects and their interaction for the three specimens, the crack depth a/W (for changing in-plane constraint) and thickness-to-width ratio B/W (for changing out-of-plane constraint) vary systematically for all specimens with a/W = 0.2, 0.3, 0.5 and 0.7 and B/W = 1/4, 1/2, 1, 2 and 4. Here, a is the crack depth, B is the specimen thickness, W is the specimen width which is fixed to be 25 mm and H is the distance between the clamps. Due to symmetry in geometry, only a quarter of the C(T), SEN(T) and M(T) specimens was modeled. The symmetry boundary condition is applied on the un-cracked ligament and middle plane (z/B = 0) of the specimens. The load is applied to the load hole as a distributed load for the specimens. The typical 3-D finite element models constructed for the C(T), SEN(T) and M(T) specimens with a/W = 0.5 and B/W = 1/2 are illustrated in Fig. 3(a–c), respectively, and the local mesh distribution around the crack tip is shown in Fig. 3(d). There is a single crack-tip node in the finite element meshes, and the crack tip is initially sharp. The analyses were carried out using ABAQUS code [48], and the eight-node linear 3-D elements (C3D8) was used for all models. The mesh sensitivity studies have been done for seven smaller element sizes of 30 lm, 40 lm, 50 lm, 60 lm, 70 lm, 80 lm and 100 lm around the crack tips. The results show that when the smallest element size is less than 50 lm, the FEM results of the C(t), C* and stresses do not essentially change with the element sizes. Hence the smallest element size of 40 lm around the crack tips is used in this study. The finite element analyses were performed using large-displacement with the NLGEOM option ON. The contour integral C(t) and C* were evaluated by using the in-built ABAQUS routines. Ten contours were set around the crack tips. The C* is usually path-independent, and C(t) is not path-independent, but with increasing time the C(t) gradually becomes pathindependent. The C(t) was defined as the value from the tenth contour near the crack tip for the different specimens with the same size of meshes and contours (The selection of other contours can give similar result). The constraint parameter R*, Tz and h ahead of the crack tips were calculated at steady-state creep.
Fig. 1. True stress–strain curve of 316H stainless steel at 550 °C [47].
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Fig. 2. Geometries and dimensions of the specimens: (a) C(T), (b) SEN(T), and (c) M(T).
Fig. 3. Typical finite element meshes for specimens with a/W = 0.5 and B/W = 1/2, (a) C(T), (b) SEN(T), (c) M(T) and (d) local meshes around the crack tip.
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3.3. Calculations of creep crack-tip constraint parameters The three-dimensional FEM calculations were conducted to obtain the values of the three creep crack-tip constraint parameters (R*, h and Tz) along crack fronts for the C(T), SEN(T) and M(T) specimens with different a/W and B/W described in Section 3.2. The parameter R* was calculated by using Eq. (3) at r = 0.2 mm, h = 0 and t/tred = 1 (steady-state creep). To facilitate the analysis and interpretation of results, the parameters h and Tz were also calculated and evaluated at r = 0.2 mm, h = 0 and t/tred = 1 by using the Eqs. (4) and (5), respectively. 4. Results and discussion 4.1. Distributions of parameters R*, h and Tz ahead of crack tips The parameters R*, h and Tz were calculated and analyzed at steady-state creep. Fig. 4 shows the distributions of R*, h and Tz ahead of crack tips at the middle plane of the SEN(T) specimen with a/W = 0.5 and B/W = 1 at h = 0, t/tred = 1 and average C*avg 1E8 MPa m h1 along specimen thickness. Fig. 4(a–c) shows that the R*, h and Tz decrease with increasing the distance r from the crack tip. These results are similar to those of Q, h and Tz in elastic–plastic analyses [12]. Fig. 4 also shows that r = 0.2 mm is located outside the blunted crack-tip region (the region with lower R*, h and Tz values near the crack tip due to blunting and damage), thus the R*, h and Tz values at r = 0.2 mm were calculated to characterize creep crack-tip constraint.. It should be noted that similar distributions of these constraint parameters ahead of the crack tips were observed for the C(T), SEN(T) and M(T) specimens with different a/W, B/W, and load level C*. 4.2. Distributions of parameter R* along specimen thickness Figs. 5–7 show the distributions of constraint parameter R* at average C*avg 1E8 MPa m h1 along specimen thickness for C(T), SEN(T) and M(T) specimens with different crack depths (a/W = 0.2, 0.3, 0.5 and 0.7) and thickness-to-width ratios (B/ W = 1/4, 1/2, 1, 2 and 4). The z/B = 0 is the middle plane, and the z/B = 0.5 is the free surface of the specimens. It can be seen from Figs. 5–7 that the center region (around z/B = 0) of all specimens has higher constraint, and the surface region (near z/B = 0.5) has lower constraint. For all specimens with different a/W, the size of the center region with uniform distribution of R* increases with increasing B/W, i.e. the R* distributes more uniformly along crack fronts as the B/W increases. With increasing specimen thickness (B/W), the R* of the surface regions in all specimens increases, but that of the center regions depends on specimen geometries and crack depth a/W. For the C(T) specimens in Fig. 5, the R* at the center region
Fig. 4. Distributions of R* (a), h (b) and Tz (c) ahead of crack tips at the middle plane of SEN(T) specimen with a/W = 0.5 and B/W = 1 at h = 0, t/tred = 1 and average C*avg 1E8 MPa m h1 along specimen thickness.
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Fig. 5. The distributions of R* along specimen thickness at C*avg 1E8 MPa m h1 for C(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d).
Fig. 6. The distributions of R* along specimen thickness at C*avg 1E8 MPa m h1 for SEN(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d).
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Fig. 7. The distributions of R* along specimen thickness at C*avg 1E8 MPa m h1 for M(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d).
increases with increasing the B/W from 1/4 to 1, and then it essentially does not change with B/W. Fig. 5 also shows that the distributions of the parameter R* along specimen thickness are similar for the C(T) specimens with different a/W ratios. This implies that the in-plane constraint effect induced by crack depth is small for C(T) geometry specimens with high constraint. Fig. 6 shows that the center regions for the SEN(T) specimens with B/W from 1/4 to 1 have slightly higher R* than those with B/W = 2 and 4. Fig. 6 also indicates that the distributions of the R* along SEN(T) specimen thickness are related to crack depth a/W, and the R* increases with increasing a/W. This implies that the in-plane dimension induced by crack depth has an effect on the constraint R* of SEN(T) specimens. These results of R* for the SEN(T) specimens are similar to the results of Q for the SEN(T) specimens in elastic–plastic analyses in the recent literature [12]. Fig. 7 shows that the distributions of R* in M(T) specimens with a/W = 0.2 and 0.3 are almost the same, and the R* at the center regions essentially does not change with the B/W. The distributions of R* in M(T) specimens with a/W = 0.5 and 0.7 are also almost the same, and the R* at the center regions with B/W = 1/4 and 1/2 is slightly higher than that of B/W = 1, 2 and 4. In general, the crack depth a/W has small effect on the R* distributions in M(T) specimens. These results imply that the constraint R* of the M(T) specimens is insensitive to both in-plane (a/W) and out-of-plane (B/W) dimensions. A comparison of Figs. 5–7 shows that for a given a/W, the variation amount of R* with the B/W ratio increases with increasing specimen constraint levels from the M(T) to C(T), i.e., the B/W ratio of high constraint specimens has more obvious effect on the R* than that of low constraint specimens. This implies that the specimen geometries with higher constraint strengthen the out-of-plane constraint effect, and the effect of specimen thickness on creep crack growth (CCG) rate of high constraint C(T) specimens may be more obvious than that of low constraint M(T) specimen. This inference has been confirmed by the recent experimental and simulation results of CCG rate by authors [49,50] which show that the effect of the specimen thickness B on the CCG rate of C(T) specimens is more obvious than that of M(T) specimens. It should be noted that the R* in the center region for all specimens does not consistently increase for higher B/W as B/W increases, which implies that R* may be inadequate to characterize the out-of-plane creep constraint for higher B/W. This result is similar to that of the constraint parameter Q within elastic–plastic fracture mechanics in the literature [12,42–44]. Fig. 8 shows the values of R* at the middle plane (z/B = 0) of all specimens with various a/W at the loading level C*avg 1E8 MPa m h1 for two typical B/W = 0.25 and 2. It can be seen that the C(T) specimen has the highest constraint, and that of the M(T) specimen has the lowest constraint. The constraint R* of C(T) and M(T) specimens is insensitive to crack depth a/W, and that of SEN(T) specimen increases with the increase of a/W. The similar results were observed for the specimens with other B/W. In general, the effects of crack depths a/W (in-plane dimension) on the R* is smaller than those of the specimen thickness ratios B/W (out-of-plane dimension) for all specimens. This implies that for a given specimen geometries, the effect of crack depths on creep crack growth (CCG) rate of a material may be smaller than those of specimen thickness. This inference has been confirmed by the recent CCG simulation results of authors [50] which show that the effects of the specimen thickness B on the CCG rate are more obvious than those of crack depth a/W for the C(T) and M(T) specimens.
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Fig. 8. The values of R* at the middle plane (z/B = 0) of all specimens with various a/W at C*avg 1E8 MPa m h1 for two typical B/W ratios, (a) B/W = 0.25 and (b) B/W = 2.
Fig. 9. The distributions of h along the thickness of C(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
4.3. Distributions of triaxiality factor h along specimen thickness Figs. 9–11 show the distributions of triaxiality factor h along specimen thickness at C*avg 1E8 MPa m h1 for C(T), SEN (T) and M(T) specimens with crack depth a/W = 0.2, 0.3, 0.5 and 0.7 and thickness-to-width ratio B/W = 1/4, 1/2, 1, 2 and 4. It can be seen that the distributions of triaxiality factor h for each specimen with different a/W and B/W are similar to the corresponding distributions of R* in Figs. 5–7. The only difference is that the h is more sensitive to the B/W ratio than the R*, which implies that the parameter h may incorporate more out-of-plane constraint effect than the R*. For the low constraint M(T) geometry, the distributions of h are not sensitive to both B/W and a/W, and this is almost the same as the case of the R* of M(T) specimens in Fig. 7. For the middle constraint SEN(T) geometry, the effect of out-of-plane dimension B/W on h is smaller than that of C(T) geometry, and is larger than that of M(T) geometry. The distributions of h along specimen thickness do not consistently increase for higher B/W with increasing the B/W (Figs. 9–11), which implies that the parameter h cannot incorporate all out-of-plane constraint effect. This observation is consistent with the recent result of 3-D elastic–plastic analysis for the SEN(T) specimens [12]. The results of the h in Figs. 9–11 indicate that the higher constraint geometries
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Fig. 10. The distributions of h along the thickness of SEN(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
Fig. 11. The distributions of h along the thickness of M(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
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strengthen the out-of-plane constraint effect, and the h is also inadequate to incorporate the out-of-plane constraint effect for higher B/W. Similar to the R* results in Figs. 5–7, Figs. 9–11 also show the h is insensitive to the crack depth a/W for the C(T) and M(T) specimens, but the h of the SEN(T) specimens increases with increasing a/W. 4.4. Distributions of parameter Tz along specimen thickness Figs. 12–14 show the distributions of Tz along the thickness at C*avg 1E8 MPa m h1 for C(T), SEN(T) and M(T) specimens with crack depths a/W = 0.2, 0.3, 0.5 and 0.7 and thickness-to-width ratios B/W = 1/4, 1/2, 1, 2 and 4. The Tz distributes more uniformly along crack fronts as the B/W increases for all specimens. For a given specimen geometry, the Tz consistently increases as B/W (out-of-plane constraint) increases regardless of a/W. This result indicates that compared with the parameters R* and h, the Tz is adequate to incorporate the out-of-plane constraint effect, thus it is a good out-of-plane constraint parameter (it is insensitive to the in-plane constraint effect induced by crack depths). These results of the parameter Tz are similar to the elastic–plastic analysis results which show that compared with parameters Q and h, the Tz is more appropriate to characterize the out-of-plane constraint [12,44]. Figs. 12–14 also show that the increase rate of Tz decreases when the B/W value is larger than 1, which implies that the increase rate of out-of-plane constraint decreases after the specimen thickness B increases to a larger value than the width W. In general, the Tz values of the C(T) specimens are higher than those of SEN(T) and M(T) specimens. This also indicates that the higher constraint geometries strengthens the out-of-plane constraint effect. It should be noted that the creep constraint parameters above are calculated and analyzed by using C* at steady-state creep. The crack-tip stress fields characterized by the C(t) and the constraint parameter R* at transition creep have been investigated in the previous work [37] of authors. It has been shown that during the early stage of creep (0 < t/tred < 0.1), the R* increases with creep time, and then it becomes independent on the t/tred. Because the period of 0 < t/tred < 0.1 is very short and R* is low in this period, the parameter R* under steady-state creep can be used to evaluate the constraint level with little conservatism during the whole creep time [37]. 5. Interactions between in-plane and out-of-plane creep constraint parameters To further analyze the capabilities of different parameters for characterizing in-plane and out-of-plane creep constraints, the interactions between constraint parameters R*, h and Tz are analyzed for all thickness layers from middle plane (z/B = 0) to free surface (z/B = 0.5) of C(T), SEN(T) and M(T) specimens with a/W = 0.2 and 0.5 and B/W = 1/4, 1/2, 1, 2 and 4 at C*avg 1E8 MPa m h1.
Fig. 12. The distributions of Tz along the thickness of C(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
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Fig. 13. The distributions of Tz along the thickness of SEN(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
Fig. 14. The distribution of Tz along the thickness of M(T) specimens with a/W = 0.2 (a), a/W = 0.3 (b), a/W = 0.5 (c) and a/W = 0.7 (d) at C*avg 1E8 MPa m h1.
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Fig. 15. Interaction between constraint parameter R* and h for all thickness layers of C(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 16. Interaction between constraint parameter R* and h for all thickness layers of SEN(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 17. Interaction between constraint parameter R* and h for all thickness layers of M(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Figs. 15–17 show the interaction between h and R* of C(T), SEN(T) and M(T) specimens for two typical crack depths a/W = 0.2 and 0.5 (note that the results for other crack depths are similar). The R* and h values are corresponding to the different locations along crack front (different z/B) for different B/W in Figs. 5–7 (for R*) and Figs. 9–11 (for h). It can be seen that there essentially exists a linear correlation between h and R* regardless of a/W and B/W for all specimens with R* < 0. For the C(T) specimens with B/W > 1, the R* at the center region of specimens is close or slightly larger than zero (the constraint level is around that of the standard C(T) specimen in plane strain with R* = 0), the linear correlation between h and R* loses due to the increase of h with small or no change in R*. These results imply that for all specimens with R* < 0 (the constraint level is lower than that of the standard high constraint C(T) specimen in plane strain), the parameter R* is equivalent to the triaxiality factor h. The interaction between the h and R* is similar to that between h and Q in elastic–plastic fracture mechanics [12,51]. The factor h can quantify both in-plane and out-of-plane constraints [12,42,43,51], thus it can be inferred the
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Fig. 18. Interaction between constraint parameter R* and Tz for all thickness layers of C(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 19. Interaction between constraint parameter R* and Tz for all thickness layers of SEN(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
Fig. 20. Interaction between constraint parameter R* and Tz for all thickness layers of M(T) specimens with a/W = 0.2 (a) and a/W = 0.5 (b) at C*avg 1E8 MPa m h1.
parameter R* may also characterize in-plane and out-of-plane constraints of lower constraint geometries. This confirms the capability of R* for characterizing both in-plane and out-of-plane constraints for lower constraint geometries which may usually occur in high temperature components (such as pressure pipes [38–40]). The h and Q parameters can quantify both in-plane and out-of-plane constraints in elastic–plastic fracture mechanics, but they are not equally sensitive to in-plane and out-of-plane constraint [13–16] and only partially incorporate the out-of-plane constraint effect. Thus, it can be inferred that the parameter R* may also not be equally sensitive to in-plane and out-of-plane constraint. This has been verified by a recent study of authors, and a unified parameter of in-plane and out-of-plane creep constraint which is equally sensitive to in-plane and out-of-plane constraint has been proposed [52]. Figs. 18–20 show the interaction between Tz and R* of C(T), SEN(T) and M(T) specimens for two typical crack depths a/W = 0.2 and 0.5 (note that the results for other crack depths are similar). The R* and Tz values are corresponding to the
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different locations along crack front (different z/B) for different B/W in Figs. 5–7 (for R*) and Figs. 12–14 (for Tz). It can be seen that there also essentially exists a linear correlation between Tz and R* for all specimens with Tz < 0.42, but the linear correlation is influenced by B/W and regardless of a/W. The effects of B/W of SEN(T) and M(T) specimens on the linear correlation are more obvious than those of C(T) specimens, and with decreasing B/W the correlation lines have a trend moving down. For the all specimens with B/W > 1 and Tz > 0.42, the linear correlation between Tz and R* loses due to the increase of Tz with small or no change in R*. Because Tz is an out-of-plane constraint parameter, the approximate linear correlation of Tz with R* for Tz < 0.42 implies that the parameter R* also can characterize out-of-plane constraint. This further confirms the capability of R* for characterizing both in-plane and lower out-of-plane constraint. But the R* is insensitive to very high outof-plane constraint with Tz > 0.42 and B/W > 1, which implies that the R* is inadequate to characterize higher out-of-plane creep constraint. The dependence of the correlation lines on B/W in Figs. 18–20 may be due to that the Tz mainly incorporates out-of-plane constraint and the R* incorporates both in-plane and out-of-plane constraint. 6. Conclusion Extensive three-dimensional finite element analyses have been conducted for C(T), SEN(T) and M(T) specimens with various crack depth (a/W) and thickness-to-width ratio (B/W), and the creep crack-tip constraints for the three specimen geometries were quantified by using three constraint parameters (namely R*, h and Tz) for 316H steel at 550 °C and steady-state power law creep. Based on the results, these constraint parameters have comparatively investigated with respect to their capabilities for characterizing in-plane and out-of-plane creep constraints as well as the interaction between them. The main results obtained are as follows: (1) The parameters R* and h both correctly show that the specimen order for constraint from high to low is C(T), SENT(T) and M(T). For a given crack depth a/W, the sizes of the center regions with uniform distributions of parameters R*, h and Tz for all specimens increase with increasing B/W. The B/W ratio has more obvious effects on the parameters R* and h of higher constraint specimens, which implies that the specimen geometries with higher constraint strengthen the out-of-plane constraint effect. (2) The constraints R* and h of C(T) and M(T) specimens are insensitive to crack depth a/W, but those of SEN(T) specimens increase with the increase of a/W. In general, the effects of crack depths a/W (in-plane dimensions) on the R* and h (constraint levels) are smaller than those of specimen thickness ratios B/W (out-of-plane dimensions) for all specimens. (3) The parameters R* and h can quantify in-plane creep constraint, but they are inadequate to characterize higher out-ofplane creep constraint. Compared with the R* and h, the Tz is adequate to incorporate the out-of-plane creep constraint effect. (4) For all specimens with R* < 0 (the constraint level is lower than that of the standard high constraint C(T) specimen in plane strain), the parameter R* is equivalent to the triaxiality factor h. There also exists approximate linear correlation of Tz with R* for Tz < 0.42.
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