Chemical composition of 20MnMoNi55.
Failure mechanism of 20MnMoNi55 steel in the lower self of ductile to brittle transition (DBT) region is considered as brittle fracture but it has been observed from the experimental analysis of stress-strain diagram that clear plastic deformation is shown by the material before failure. Therefore, strain correction is implemented in the cleavage fracture model proposed by different researchers in the lower self of the DBT region with the help of finite element analysis. To avoid a huge number of experiments being performed, Monte Carlo simulation is used to generate a huge number of random data at different temperatures in the lower self of the DBT region for calibration of the cleavage parameters with the help of the master curve methodology. Fracture toughness calculated after strain correction through different models are validated with experimental results for the different probability of failures.
- fracture toughness
- plastic strain
- reactor pressure vessel
- master curve
- finite element analysis
Regular maintenance of the reactor pressure vessel (RPV) is an important criterion that has to be considered where safety is the prime requirement for any country. In that respect embrittlement of the RPV material has to be quantified concerning reference temperature
It has been observed in our previous work that references temperature (
Therefore, in the last few years, the main aim of the researchers was focused to study the constraint effects of reference temperature
Beremin model is entirely focused on brittle fracture where no strained effect is considered but it has been observed in the experimental stres-strain diagram for the material 20MnMoNi55 steel a huge plastic deformation is observed in the material before failure even at −110°C and the plastic region diminishes as it moves towards −150°C. Therefore, strain correction is required in the model for proper calibration of the Beremin parameters for the material. Beremin himself in his work felt the requirement for strain correction and he simultaneously developed a model considering the effect of strain in calculating Weibull Stress . Recently, Ruggieri in his work [24, 25] utilized the strain effect by different models for a similar type of RPV material A515 Gr 65 pressure vessel steel. But he uses the toughness scaling model to calibrate the Beremin model parameters.
In December 2017, Claudio Ruggieri, Robert H. Dodds Jr.  through their work focuses on the importance of plastic strain effects into the probabilistic framework in brittle fracture.
In November 2019, Claudio Ruggieri  through his work proposed a probabilistic, micromechanics-based model which incorporates plastic strain effects on cleavage fracture and its dependence on the microcrack distribution. The model utilized a plastic-strain based form of the Weibull stress to capture the differences in brittle fracture toughness for a reactor pressure vessel (RPV) steel due to constraint loss.
In the present work fracture toughness of 20MnMoNi55 steel is determined with the help of three-point bending (TPB) at −100°C, −110°C, −120°C, −130°C, −140°C, at reference temperature
In this work, the entire focus is being made on the brittle failure nature of German based reactor pressure vessel material (20MnMoNi 55 steel) at the lower self of DBT region, which is a very important study as far the safety of reactor pressure vessel is concerned. In recent years study on this material dealing with specific topics is not performed. Moreover, the application of Monte Carlo simulation to reduce the burden of performing a huge number of fracture experiments is overcome by this procedure. The author proves the success in the application of the statistical model by matching it with the experimental results in his previous work . In the present work, the author utilizes the statistical model along with FEA to study the effect of strain on brittle fracture through four different strain corrected brittle fracture models. In the end, the fracture toughness predicted from these models is compared with the established ASTM E1921 and master curve results which is a very challenging and interesting part of the work.
The material studied is German steel, used in the reactor pressure vessel of Indian PHWR and designated as 20MnMoNi55. The material used in this investigation has received from Bhabha Atomic Research Centre, Mumbai, India. The steel was received in the form of a rectangular block. The specimens were made from this block to determine the fracture toughness of the selected steel using J-integral analysis and the master curve methodology, to understand the fracture behavior of the steel. The RPV material properties during operation are defined by their initial values, material type, chemical composition, and operating stressors, mainly operating temperature and neutron influence. The chemical composition of 20MnMoNi55 is shown in Table 1.
|Name of element||C||Si||Mn||P||S||Al||Ni||Mo||Cr||Nb|
|Percentage composition (in weight)||0.20||0.24||1.38||0.011||0.005||0.068||0.52||0.30||0.06||0.032|
3.1 Calculation of reference temperature (
T0) and master curve analysis
Brittle fracture probability according to Wallin [1, 2, 3], is defined as
Scale parameter K0 which dependent on the test temperature and specimen thickness, and Kmin is equal to 20 MPa√m .
For single-temperature evaluation, the estimation of the scale parameter K0, is performed according to Eq. (4).
3.2 Modified Beremin model
According to the Beremin model , the probability of failure is given as,
n is the number of volumes
The classical model described above is applicable where plastic strain is negligible or zero for perfectly brittle materials but for ferritic steels where an appreciable amount of plastic strain is observed in the crack tip area this formula cannot capture the failure mechanism perfectly. To impose plastic strain effect on the failure mechanism a correction formulation has been introduced by Beremin .
is the strain in the direction of the maximum principal stress .
Throughout the paper, the Weibull stress is calculated according to Eq. (8).
3.3 Local approach to cleavage fracture incorporating plastic strain effects
This methodology is derived from the work done by Wallin and Laukkanen  which is based on the strain effect near the crack tip field. Here the Weibull stress is modified by taking into account a particular volume δV in the fracture process zone is subjected to a principal stress σ1 and associated with a plastic strain (ϵp). In this mode micro-crack formed by the cracking of brittle particles only participate in the fracture process. It is assumed by Ruggieri and Dodds  that a fraction represented by ψC of the total number of brittle particles present in FPZ is responsible for nucleating the micro cracks which propagate unstably. This fraction ψC is a function of plastic strain but does not depend on microcracks. Based on the weakest link concept limiting distribution for the cleavage fracture stress can be expressed as.
V0 represents a reference volume conventionally taken as a unit volume.
Now ψC is calculated as follows.
L represents the particle size; LN a reference particle size; σprs is the particle reference fracture stress; αp denotes the Weibull modulus shape parameter of particle distribution; and σpf represents the characteristics of fracture stress.
where σ1 represents maximum principal stress, ϵp denotes the Maximum plastic strain of those particles whose σ1 is calculated in the fracture process zone (where σ1 = 2 σy) and E represents the Youngs Modulus of the particle at different temperatures. Now as assumed by Rugeirri et al.  the size of a fracture particle takes the size of a Griffith-like micro-crack of the same size the probability distribution of the fracture stress with increase loading for a cracked solid is given by the following equation.
As ψc is independent of microcrack size so L/LN is considered to be 1. Therefore, σW takes the form.
3.4 Exponential dependence of eligible micro-cracks on ϵp
Bordet et al.  include plastic strain effects on cleavage fracture in terms of the probability of nucleating a carbide micro crack. The original model considered only freshly nucleated carbides to act as Griffith-like micro-cracks and have the eligibility to propagate unstably take part in the fracture process. But in our work, we considered a simplified model as considered by Bordet et al.  and adopt a Poisson distribution by introducing a parameter λ to define ψc given by the following equation.
λ is assumed as the average rate of fracture particles which becomes Griffith-like micro crack with small strain increment. The author has taken the strain increment inconsistency with the quasi-static process. Therefore, the probability of fracture and Weibull stress takes the following form.
3.5 Influence of plastic strain on microcrack density
Based upon the work of Brindley and Gurland [32, 33, 34] the direct effect of plastic strain on micro-cracking of ferritic steel at different temperatures alter the probability distribution in the FPZ as follows:
and the Weibull Stress becomes.
4. Test procedure
4.1 Fatigue pre-cracking
The fracture toughness tests in this investigation were planned on three-point bending (TPB) specimens in L-T orientation. Standard 1T TPB specimens were machined following the guidelines of ASTM E 399-90. The designed dimensions of the specimens were; thickness (B) 25 mm and width (W) = 25 mm which is constant for all the specimen tested and machined notch length (aN) = 10 mm to produce different a/W ratio of 0.5. Fatigue pre-cracking of the TPB specimens was carried out at room temperature at constant ΔK mode as described in ASTM standard E 647 on servo hydraulic INSTRON UTM (Universal Testing Machine) with 8800 controllers having 100 KN grip capacity using a commercial da/dN fatigue crack propagating software supplied by INSTRON Ltd. U.K. The crack lengths were measured by compliance technique using a COD gauge of 10 mm gauge length mounted on the load line of the specimen.
4.2 Fracture test
The estimation of J-integral values of the fabricated specimens was carried out using an INSTRON UTM (Universal Testing Machine) with an 8800 controller with 100 KN grip capacity as described earlier. Tests were done at different temperatures ranging from −100°C to – 140°C. The specimen used is a three-point Bending specimen. The nomenclature along with a picture of the specimen is shown in Figure 1.
The Instron FAST TRACK JIC Fracture Toughness Program was used to determine the value of the J integral. This program performs Fracture Toughness on metallic materials following the American Society for Testing and Materials (ASTM) Standard test method E813. The method is applied specifically to specimens that have notches or flaws that are sharpened with fatigue cracks. The loading rate was slow, and cracking caused by environmental factors was considered negligible.
4.3 A result of the tensile test and J1C at different temperatures in the lower self of the DBT region
From the experimental stress-strain results performed at different temperatures for 20MnMoNi55 steel, a clear plastic zone is observed before failure as shown in Figure 2 . The same plastic strain effect is reflected in the TPB specimen also at the lower self of the DBT region. This provoked us to perform the required strain correction in computing Weibull Stress through different strain correction models as discussed above.
The results of KJC values of TPB specimens at different temperatures are shown in Figure 3.
4.4 Finite element analysis
Finite element analysis of all the fracture tests is performed using ABAQUS 6.13. The material constitutive properties are defined by Young’s modulus E, Poisson’s ratio
|Temperature (°C)||Yield strength (MPa)||Ultimate strength (MPa)|
4.5 Validation of the FE model and material properties
Figures 7–9 gives a comparison between experimental load versus load line displacement (LLD) of TPB specimen with FE simulated results from Abaqus 6.13 at the −100°C,−110°C and −130°C temperatures The FEA results show a close match with experimental results which validate the used FE model and material parameters. Now for each analysis, the Weibull stress at the failure point can be computed from the FE simulated results.
4.6 Calculation of Weibull stress
Now the maximum principal stress and corresponding strain in the direction of principal stress is known for each element in the fracture process zone, we calculate the Weibull Stress for each model using Eqs. (8), (13), (16), and (18).
The success of the Beremin model for predicting brittle fracture mainly depends on the accuracy of the values of the Beremin material parameters m and σu. The Beremin model describes the failure mechanism as an outcome of the distribution of the weakest sites in the statistical material. Hence any material parameters to represent the failure behavior should be determined from a large sample containing variation in candidatures as much as possible. With this in mind, the values of m and σu have been determined from the experimental fracture toughness tests at −100°, −110°, −130° which is described as a direct calibration strategy. The process is described vividly for −110°C by K. Bhattacharyya et al. calibration of beremin parameters for 20MnMoNi55 Steel and prediction of reference temperature (
4.7 Calibration of Cm,n
Cm,n an important parameter used for the determination of KJC for different models has been calibrated for this material at different temperatures. The process of determination of Cm,n for our material is different from that as framed by Beremin. The entire process is described step by step for −110°C in the previous work done by the author . A similar procedure is used for the determination of Cm,n for other temperatures. Once Cm,n is calibrated the value of fracture toughness for 5%, 63.2%, and 95% can be determined by Eqs. (6), (9), (12), (15), and (17). These fracture toughness values were then plotted with the experimentally determined master curve methodology as shown in Figure 12.
5. Results and discussions
The calibration of Weibull modulus “m” using different models as described above is shown in Figure 10 and the Weibull scale parameter is shown in Figure 11. It is observed that the value of Weibull modulus for the four different models almost coincides at −100°C and −110°C and as the temperature decreases to −120°C to −140°C the variation of in the value of Weibull modulus predicted from the different model is pronounced and it increases with decrease in temperature. As the material moves from the lower self of DBT region to purely cleavage fracture the effect of ductile stretch due to plasticity affect vanishes. As all the four models are functions of plastic strain therefore as it approaches purely brittle failure the strain component almost vanishes therefore prediction capability of the models to some extent becomes biased.
Weibull modulus “m” and Weibull modulus scale parameter “σu” is calculated from different models as explained, now once Cm,n is calculated at different temperatures, fracture toughness can be predicted for different probabilities of failure.
It is observed that the prediction capability Beremin strain correction model is much better in comparison to the other three models when validated with the experimental results as shown in Figure 12.
Though Claudio Ruggieri and his co-workers in their work  showed that fracture toughness predicted from local criteria matches well with the experimental results for A515 Gr 65 pressure vessel steel but the results obtained in this study appeal to be contradictory with their work for the material 20MnMoNi55 steel.
Our study is focussed on the lower self of DBT region starting from −100°C to −140°C where a very small amount of ductile stretch is observed before failure but their work is focussed at −20°C where a huge amount of ductile stretch is observed before cleavage failure for our material. This could be one reason for the deviation of the results with them.
The main aim of this work is to establish the strain affect in the brittle failure-dominated portion of the DBT region, which is observed in the form of ductile stretch in experimental results.
With the help of finite element analysis, we have a better outlook in fracture process zone and we can bridge the gap between macroscopic observations (like J-integral and fracture toughness) with micro-cracks developed in the fracture process zone.
The effect of strain is established in the brittle failure dominated portion of DBT region through different strain correction model.
Utilization of statistical model (Monte Carlo simulation) proves to be very useful to reduce the huge cost of performing a large number of experiments at the cryogenic conditions.
The values of the Weibull modulus “m” and Weibull modulus scale parameter “σu” are calibrated through different brittle fracture models for different temperatures in the brittle failure dominated portion of DBT region for the concerned material.
Cm,n another important parameter is also calculated for different temperatures for the concerned material.
With the help of Weibull modulus “m” and Weibull modulus scale parameter “σu” and Cm,n the fracture toughness is predicted for different probabilities of failure.
The probabilities of failure are then compared with experimentally obtained results.
It is observed that the prediction capability Beremin strain correction model is much better in comparison to the other three models when validated with the experimental results.
Whenever fracture mechanics is used from specimen level to component level there is a constrain loss which affects the results. This causes a great lacuna in the application of fracture mechanics to the real engineering problems. This study to some extent put a step forward in overcoming the lacuna by using extensive finite element analysis and different brittle fracture models on specimen level and tried to predict the results in comparison with experimental counterpart. With the hope that in future application of fracture mechanics will not be limited to specimen level. This study will propel more research work in this field and the development of new models.