Application of T 33-Stress to Predict the Lower Bound Fracture Toughness for Increasing the Test Specimen Thickness in the Transition Temperature Region

This work was motivated by the fact that although fracture toughness of a material in the ductile-to-brittle transition temperature region J c exhibits the test specimen thickness (TST) effect on J c , frequently described as J c ∝ (TST)−1/2, experiences a contradiction that is deduced from this empirical formulation; that is, J c = 0 for large TST. On the other hand, our previous works have showed that the TST effect on J c could be explained as a difference in the out-of-plane constraint and correlated with the out-of-plane


Introduction
The cleavage fracture toughness   of a material in the ductile-to-brittle transition (DBT) temperature region, which is important in the assessment of aging steel structures and reactor pressure vessels, has been known to exhibit test specimen size effects, even when tested using a standardized specimen [1][2][3][4][5][6][7][8][9].For example,   obtained using a shallow cracked specimen exhibits a higher value than that obtained using a deep cracked specimen.Another known size effect is the test specimen thickness (TST) effect on   , hereafter abbreviated as the TST effect on   , which is described as   ∝  (−1/2) ( ≡ TST) [2,10].The two most physically logical explanations in general are the statistical weakest link (SWL) size effect and the loss of the crack-tip constraint [2].Both explanations lead to an increasing toughness with decreasing TST.The difference in   obtained with a different planar specimen configuration, including the crack depth [4], has been explained as the differences in the crack-tip constraint or the hydrostatic stress triaxiality, which  fails to describe [3,[5][6][7][8][9].
However, the TST effect has been explained in terms of the SWL size effect being dominant [6,[11][12][13], even though   does not decrease indefinitely with thickness [6], which contradicts the prediction from the SWL size effect [2].
Based on the above, the authors believed that the contribution of the crack-tip constraint to the TST effect on   could be demonstrated if the TST effect (especially the bounded nature of   with increasing TST) was demonstrated using a series of nonstandard test specimens whose planar configurations are identical but whose thickness-to-width ratios, /, are changed to realize different thickness specimens and if the test results were reproduced using finite element analysis (FEA).This use of nonstandard test specimens was prompted by the inability to predict the bounded nature of   using the SWL formulation.This prediction was thought to be enabled by these specimens because the out-of-plane cracktip constraint will increase and saturate with increasing /, but the in-plane crack-tip constraint will not change.The fracture toughness tests for a series of nonstandard compacttension (CT) and three-point-bend (3PB, also named as  SE(B) specimen) specimens for 0.55% carbon steel S55C [14][15][16] and 0.40% carbon chromium molybdenum steel SCM440 [17] validated the noticeable contribution of the outof-plane crack-tip constraint to the TST effects on   , and the constraint parameter  33 -stress was demonstrated to be effective for correlating this out-of-plane crack-tip constraint with the TST effects on   [14][15][16][17].These results indicated a possibility of correlating the fracture toughness of a test specimen and the crack-like flaws in the structure more accurately by considering  33 .
This work is an extension of our previous studies regarding the point that the contribution of the out-of-plane cracktip constraint to the TST effect on   was demonstrated for the decommissioned Shoreham reactor vessel steel, ASTM A533 Grade B Class 1 (A533B) [1], which is experimentally formulated as   [N/mm] = 2.3 ⋅ | 33 | 0.6 (80 ≤ | 33 | ≤ 320 MPa) to describe the   decreasing tendency for increasing TST.Because the Shoreham data included a lower bound of   for increasing TST, a new finding was made that  33 successfully predicted the lower bound   with increasing TST.This lower bound   prediction with  33 resolves the contradiction that the empirical   ∝ (TST) −1/2 predicts   = 0 for large TST.

TST Effect on 𝐽 𝑐 Described by the 𝑇 33 -Stress
2.1.-Stress.In an isotropic linear elastic body containing a crack subjected to symmetric (mode I) loading, the leading two terms in a series expansion of the stress field very near to the crack front are [18] where  and  are the in-plane polar coordinates of the plane normal to the crack front, as shown in Figure 1, and  I is the local mode I stress intensity factor (SIF) at location A.Here  1 is the direction formed by the intersection of the plane normal to the crack front and the crack plane.The terms  11 and  33 are the amplitudes of the second-order terms in the three-dimensional series expansion of the crack front stress field in the  1 and  3 directions, respectively.
2.2.TST Effect on   Described by  33 -Stress.In our previous works [14,15,17], the following relationships were obtained for 0.55% carbon steel S55C [14,15] and 0.40% carbon chromium molybdenum steel SCM440 [17] with both CT and 3PB specimens: The object of these works was to demonstrate that the outof-plane crack-tip constraint has a noticeable contribution to the TST effect on   and that the TST effect can be correlated with a mechanical parameter  33 (expressing the out-of-plane crack-tip constraint).
Because the bounded nature of   with increasing TST could not be realized with the tested specimens of thicknessto-width ratios / = 0.25, 0.4, and 0.5, the tested results with large / were searched in the published documents, and the decommissioned Shoreham reactor vessel steel data [1] were found to fulfill our requirement.In the following, Shoreham's   data were compiled to validate the relationship   ∝ | 33 |  (: material constant) and, in particular, to correlate the bounded nature of   for increasing TST with  33 .

Compilation of the Decommissioned Shoreham Reactor Vessel Steel Fracture Toughness Test Data from the Standpoint of the Out-of-Plane Constraint.
To determine whether the relationship   = | 33 |  is valid for other materials and especially whether the lower bound   min can be predicted by  33 , the decommissioned Shoreham reactor vessel steel [1] A533B was selected in this work because a large amount of fracture toughness test data for A533B with various thickness 3PB specimens at a common temperature −91 ∘ C (located in the DBT temperature region) was published.A more detailed description for the fracture toughness tests can be found in [1].
Here, the fracture toughness test data for 3PB specimens with width  = 25.4 and 50.8 mm whose thicknesses  = 8, 15.9, 31.8, and 63.5 mm (thickness-to-width ratio / = 0.157∼2.5)were recompiled from the published results [1] on the standpoint of the out-of-plane crack-tip constraint.Although the eight replicate fracture toughness test results reported in [1] for these 3PB specimens were considered to be valid overall, some of the individual   datum still appeared to deviate greatly from the remainder in each / set.Considering the fact that the   scatter from eight replicate tests always exceeded the guideline value as given in ASTM E1921 [19], we thought it was necessary to recompile these test results because the impact of the apparent deviated   datum for each / set was considered non-negligible in studying the TST effect on the cleavage fracture toughness.Therefore, the cases with maximum and minimum   values were excluded, with the test results of the remaining cases summarized in Tables 1 and 2.
The   in the tables was obtained as the SIF  corresponding to the fracture load   from the following equation in ASTM E1921 [19]: Here,  = 4 is the support span, and  is a function of /, which is given in the ASTM E1921 [19].
in the table is the fracture toughness in terms of the SIF.  was calculated from   as   =  2  ⋅ (1 − ] 2 )/, where the value of Young's modulus of  = 207.9GPa and the value of Poisson's ratio of ] = 0.29 were used, as specified in [20]. 33 , which reflects the fracture load and the actual crack length, was calculated from the  33 solutions of elastic FEA, as summarized in the Appendix. and Σ are the average and standard deviation of each value, respectively.2Σ/ is a reference value that was used to represent the magnitude of the data scatter.
It is seen from Tables 1 and 2 that, except for the case of  = 50.8mm with a very thin thickness  = 8mm (2Σ/ = 63.8%), the reference value 2Σ/ of   was in the range from 33.1% to 45.6% for the selected specimens, which satisfied the guideline for 2Σ/ given in ASTM E1921 [19] for   .Here the guideline for 2 Σ/ is 56(1-20/)% with the range from 40.7% to 47.4% for the data in Tables 1 and 2.
As a result, it could be concluded that the scatter in the   data of the selected specimens summarized in the tables was acceptable in an engineering sense.One interesting fact was that the change in   , that is, the SIF for the fracture load   , exhibited a relatively small dependence on /, although a significant change in the fracture toughness   was observed.The average   for each / was in the range from 67.2 to 80.7 MPa m 1/2 for  = 25.4 mm and 75.8 to 95.7 MPa m 1/2 for  = 50.8mm.This result was similar to the experience with S55C [14-16] and SCM440 [17], which validated one of the assumptions used to predict the lower bound of   for large TST proposed in Section 3.1.
The relationship between   and  33 for A533B is shown in Figure 3; note that  33 reflects the fracture load and the actual crack length for each /, as summarized in Table 1 and 2. The solid marks represent the average for each /.The difference in  was distinguished by the color of the marks.As shown in Figure 3, all the data in Tables 1 and 2 are fitted to the power law expression for A533B tested using 3PB specimens at −91 ∘ C.   seemed to be bounded for 2| 33 | < 100 MPa.The bounded value of   in Figure 3 for the case of  = 25.4 mm was obtained from Table 1 as an average   for the specimens of / = 1.25 and 2.5.For the case of  = 50.8mm, the bounded   was obtained from Table 2 as an average for / = 1.25.
On the other hand, if the method to predict the lower bound   min for increasing TST proposed in Section 3.1 is applied, for the case of  = 25.4 mm as an example, first | 33∞ | is calculated with | 33∞ | = 0.2 for the case of / = 0.5 and   = 79.7 MPa m 1/2 (the averaged SIF for B/W = 0.315∼1.25 was used from Table 1, considering the fact that   exhibited a very small dependence on TST) as | 33∞ | = | 33∞ | ⋅   /() 1/2 = 0.2 × 79.7/( 0.0127) 1/2 = 79.8MPa.Then, the lower bound   min is predicted from (3) as   min = 2.3 × |79.8| 0.6 = 31.8N/mm, and it was close to experimental average 36.0N/mm.In case of  = 50.8mm, by the same method,   min = 27.2N/mm was obtained and was also very close to the experimental average 27.9 N/mm.In summary, the TST effect on   of A533B could be described by  33 , as   = 2.  the experimental average value, which indicated that  33 can successfully predict the bounded nature of   .

Discussion
In this work, the TST effect and the bounded nature of   observed for the decommissioned Shoreham reactor vessel steel, A533B, at −91 ∘ C, which is in the DBT range [1], were compiled by  33 -stress in the general form of (5).In (5), the similar power law relationship between   and  33 was also valid for the combination of S55C [14,15] and SCM440 [17] tested using both CT and 3PB specimens.In addition,  33 , which seemed to be useful for predicting the bounded nature of   for S55C [16], has also been proven to be valid for A533B.In these empirical equations, the TST effect and the bounded nature of   were described with a single out-of-plane elastic parameter  33 taken at the specimen midplane.Although the depicted relationship between the fracture toughness   of a material and  33 must be validated for other materials and other types of test specimen configurations, using  33 as a relevant out-of-plane constraint parameter is definitely worthy of further investigation.
It could be argued that the relationship   ∝  (−1/2) ∝ | 33 | 0.6 (Figure 3) is similar to the formulation deduced from the SWL model, but no more than what is predicted by the SWL model (  ∝  (−1/2) ) [2], because | 33 | first approaches to 0 for large TST (Note: with increase in TST for 3PB specimen, negative  33 first increases, crosses 0 and converges to V 11 ).As Anderson et al. indicated, as a contradiction of the SWL model, the "fracture toughness does not decrease indefinitely with thickness [6]." On the point that  33 exhibits a saturating tendency for large TST,  33 has also been proven to be able to predict the bounded behavior of   (Figure 3).The advantage of using  33 is that  33 has the characteristic to not only describe the TST effect on   but to also predict the bounded nature of   .This advantage of  33 successfully avoids the contradiction deduced from the SWL model; that is,   → 0 for  → ∞.
ASTM E1921 [19] presents a method to adjust   for CT's TST change by considering the empirical relationship   ∝  (−1/2) , under the assumption that 1-inch (1T) thickness CT toughness data exists.The presented method in this paper for a 3PB specimen can be generally applied to any type of test specimens, if a curve similar to Figure 2 is obtained.The fact that 1T CT test data are not necessary for our method can help practitioners in their works.
When the proposed general formulation of ( 3) is practically used for determining the lower bound of   for a specific material tested with a fracture toughness test specimen, the material constants  and  should be first determined by conducting measurements on at least two different-sized   specimens.Nevertheless, if measurements on only one size of specimen are conducted, (3) can also be simply but not accurately applied for predicting the lower bound fracture toughness just by assuming  = 1/2 in the relationship   ∝ | 33 |  for that one size of specimen considered, because the material constant  = 1/2 has been verified for the materials S55C and SCM440 tested with both CT and 3PB specimens [14,15,17]; in addition, this work validated that the approximated  = 0.6 which is close to  = 1/2 was applicable for the material A533B tested using 3PB specimens.
The normalized  33 -stress solutions used in this work were taken at the specimen midplane.It is true that these values are distributed in the specimen thickness direction [21].There are many possibilities to treat this 3D effect, but, considering the fact that the fracture tends to initiate at the specimen midplane, the values at the specimen midplane were chosen to represent the characteristic intensity of these values.

Conclusions
This paper demonstrated for the decommissioned Shoreham reactor vessel steel A533B [1] that the out-of-plane crack-tip constraint has a noticeable contribution to the TST effect on   and that the magnitude of this out-of-plane cracktip constraint can be described by the elastic  33 -stress.The experimental expression of the TST effect on   using  33stress, which was proposed for 0.55% carbon steel S55C and 0.40% carbon chromium molybdenum steel SCM440 with both CT and 3PB specimens in our previous work [14,15,17], was shown to be a correct description for A533B.In concrete, the experimental relationship for A533B was compiled as   [N/mm] = 2.3 ⋅ | 33 | 0.6 (80 ≤ | 33 | ≤ 320 MPa) to describe the   decreasing tendency for increasing TST.Because the Shoreham data included a lower bound   for increasing TST, a new discovery was that  33 successfully predicted the lower bound of   with increasing TST.This lower bound of   prediction with  33 resolved the contradiction that the empirical   ∝ (TST) −1/2 predicts   = 0 for large TST.

Appendix
The normalized  33 solutions used to calculate  33 in Tables 1 and 2 were obtained from the elastic FEA.In the present FEA, all the 3PB specimen dimensions were specified in accordance with those recorded in [1], and the material properties were set to be consistent with those specified in [20] for A533B.
The typical FEA model of the 3PB specimen used in the present elastic analysis is shown in Figure 4, with the details for the generated mesh being summarized in Table 3.The details of the elastic FEA procedure can be found in our recent work [17].The normalized  33 -stress,  33 , at the specimen midplane is summarized in Table 4, which is in a good agreement with the interpolated solutions from our previous results [22].A v e r a g ev a l u e ]: Poisson's ratio   : S t r e s sc o m p o n e n t s( ,  = 1, 2, 3).

Figure 1 :
Figure 1: Three-dimensional coordinate system for the region along the crack front.

6 Advances
for the case of a/W = 0.52|T 33∞ | = 2| 33∞ | • K c /√a J c avg J c avg J c avg (N/mm) J c min

Figure 4 :
Figure 4: Typical finite element model of the 3PB specimen.
e r i a lc o n s t a n t( s e e( 3)) : Y o u n g ' sm o d u l u s : -integral   and   avg : Fracture toughness and its average   min : Lower bound fracture toughness  I : Local mode I stress intensity factor (SIF)   : F r a c t u r et o u g h n e s s(  = [ ⋅  c /(1 − ] 2 )]

2
Advances in Materials Science and Engineering

Decommissioned Shoreham Reactor Vessel Steel Fracture Toughness Test Data from the Standpoint of Out-of-Plane Constraint
[17]Prediction of a LowerBound of   for Increasing TST with  33 .From our recent elastic FEA results for the nonstandard 3PB specimen with various / values, as shown in Figure2(a), the midplane  33 normalized in the form of  33 =  33 () 1/2 / I exhibited a strong dependence on /[17]. 33 was negative for / < 1.5, whereas it was positive and approached ] 11 ( 11 =  11 () 1/2 / I ) for increasing TST.
11 ,  33 : Normalized forms of the -stresses  33∞ : Bo undedval ueo f 33 : M a t e r i a lc o n s t a n t( s e e( 3)) :