**2. Material**

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**.


#### **Table 1.**

*Chemical composition of 20MnMoNi55.*

## **3. Methods**

#### **3.1 Calculation of reference temperature (***T***0) and master curve analysis**

Brittle fracture probability according to Wallin [1–3], is defined as *Pf* for a specimen having fracture toughness *KJC* in the transition region is described by a three-parameter Weibull model as shown by.

$$P\_f = 1 - \exp\left[-\left(\frac{K\_{f\text{C}} - K\_{\text{min}}}{K\_0 - K\_{\text{min}}}\right)^4\right] \tag{1}$$

where,

$$K\_{fC} = \sqrt{\frac{J\_c E}{(1 - \nu^2)}}\tag{2}$$

Scale parameter K0 which dependent on the test temperature and specimen thickness, and Kmin is equal to 20 MPa√m [29].

For single-temperature evaluation, the estimation of the scale parameter K0, is performed according to Eq. (4).

$$K\_0 = \left[\sum\_{i=1}^{N} \frac{\left(K\_{fC(i)} - K\_{\min}\right)^4}{N}\right]^{1/4} + K\_{\min} \tag{3}$$

$$K\_{f\mathcal{C}(median)} = K\_{\min} + (K\_0 - K\_{\min})(\ln \mathcal{Z})^{\vee\_4} \tag{4}$$

Here,*T*<sup>0</sup> is the temperature at which the value of KJC(median) is 100 MPa√m and is known as the reference temperature. *T*<sup>0</sup> can be calculated from.

$$T\_0 = T\_{test} - \frac{1}{0.019} \ln \left[ \frac{K\_{fC(median)} - 30}{70} \right] \tag{5}$$

#### **3.2 Modified Beremin model**

According to the Beremin model [21], the probability of failure is given as,

$$P\_f = 1 - \exp\left(-\left(\frac{\sigma\_w}{\sigma\_u}\right)^m\right) \tag{6}$$

$$
\sigma\_w = \sqrt[m]{\left(\sum\_{j=1}^n \sigma\_1^j\right)^m \frac{V\_j}{V\_0}} \tag{7}
$$

n is the number of volumes *Vj*, or elements in a FEM calculation, and *σ<sup>1</sup> j* the maximal principal stress of the element *j* and *Vj*/*V*<sup>0</sup> is just a scaling based on the assumption that the probability scales with the volume. V0 is the reference volume which is normally taken as a cubic volume containing about 8 grains i.e., 50 � 50 � 50 μm.

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 [21].

*Investigation of Strain Effect on Cleavage Fracture for Reactor Pressure Vessel Material DOI: http://dx.doi.org/10.5772/intechopen.101245*

$$\sigma\_{w} = \sqrt[m]{\sum\_{j} \left(\sigma\_{1}^{j}\right)^{m} \frac{V\_{j}}{V\_{0}} \exp\left(-\frac{m\varepsilon\_{1}^{j}}{2}\right)} \tag{8}$$

*ε j* <sup>1</sup> is the strain in the direction of the maximum principal stress *<sup>σ</sup> <sup>j</sup>* 1. 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 [30] 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 <sup>δ</sup> V in the fracture process zone is subjected to a principal stress σ<sup>1</sup> 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 [30] that a fraction represented by ψ<sup>C</sup> of the total number of brittle particles present in FPZ is responsible for nucleating the micro cracks which propagate unstably. This fraction ψ<sup>C</sup> 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.

$$P\_f(\sigma\_1, \varepsilon\_p) = \mathbf{1} - \exp\left[ -\frac{1}{\nu\_0} \int\_{\Omega} \mu\_\epsilon \sigma\_1^m d\Omega \right]^{\frac{1}{\nu\_0}} \tag{9}$$

V0 represents a reference volume conventionally taken as a unit volume.

ῼ is the volume of the near-tip fracture process zone where *σ*<sup>1</sup> ≥*λσy*. λ = 2 [30, 31] is normally taken as twice of yield stress for the material.

Now ψ<sup>C</sup> is calculated as follows.

$$\boldsymbol{\nu}\_c = \mathbf{1} - \exp\left[-\left(\frac{L}{L\_N}\right)^3 \left(\frac{\sigma\_{pf}}{\sigma\_{prs}}\right)^{ap}\right] \tag{10}$$

L represents the particle size; LN a reference particle size; σprs is the particle reference fracture stress; α<sup>p</sup> denotes the Weibull modulus shape parameter of particle distribution; and σpf represents the characteristics of fracture stress.

$$
\sigma\_{pf} = \sqrt{\mathbf{1.3} \sigma\_1 \epsilon\_p E} \tag{11}
$$

where σ<sup>1</sup> represents maximum principal stress, ϵ<sup>p</sup> denotes the Maximum plastic strain of those particles whose σ<sup>1</sup> is calculated in the fracture process zone (where σ<sup>1</sup> = 2 σy) and E represents the Youngs Modulus of the particle at different temperatures. Now as assumed by Rugeirri et al. [25] 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.

$$P\_f(\sigma\_1, \varepsilon\_p) = 1 - \exp\left[ -\frac{1}{v\_0} \left[ \left\{ 1 - \exp\left[ -\left(\frac{\sigma\_{pf}}{\sigma\_{p\pi}}\right)^m \right] \right\} \left(\frac{\sigma\_1}{\sigma\_u}\right) d\Omega \right]^{\natural\_n} \tag{12}$$

As ψ<sup>c</sup> is independent of microcrack size so L/LN is considered to be 1. Therefore, σ<sup>W</sup> takes the form.

$$
\sigma\_W = \left[\frac{1}{V\_0} \int\_{\Omega} \left\{ 1 - \exp\left[ -\left(\frac{\sigma\_{pf}}{\sigma\_{pr}}\right)^{a\_p} \right] \right\} \sigma\_1^m d\Omega \right]^{\natural\_m} \tag{13}
$$

## **3.4 Exponential dependence of eligible micro-cracks on ϵ<sup>p</sup>**

Bordet et al. [31] 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. [31] and adopt a Poisson distribution by introducing a parameter λ to define ψc given by the following equation.

$$\boldsymbol{\mu}\_{\boldsymbol{c}} = \mathbf{1} - \exp\left(\lambda \boldsymbol{e}\_{p}\right) \tag{14}$$

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.

$$P\_f(\sigma\_1, \varepsilon\_p) = 1 - \exp\left[ -\frac{1}{v\_0} \left\{ \left( 1 - \exp\left( -\lambda \varepsilon\_p \right) \right) \left( \frac{\sigma\_1}{\sigma\_u} \right) d\Omega \right\}^{\prime \prime\_{\text{m}}} \tag{15}$$

$$\sigma\_W = \left[ \frac{1}{V\_0} \int\_{\Omega} \left\{ 1 - \exp\left( -\lambda \varepsilon\_p \right) \right\} \sigma\_1^m d\Omega \right]^{\prime \prime\_{\text{m}}} \tag{16}$$

#### **3.5 Influence of plastic strain on microcrack density**

Based upon the work of Brindley and Gurland [32–34] the direct effect of plastic strain on micro-cracking of ferritic steel at different temperatures alter the probability distribution in the FPZ as follows:

$$P\_f(\sigma\_1, \varepsilon\_p) = 1 - \exp\left[ -\frac{1}{v\_0} \left[ \epsilon\_p^\beta \left( \frac{\sigma\_1}{\sigma\_u} \right)^m d\Omega \right] \tag{17}$$

and the Weibull Stress becomes.

$$
\sigma\_W = \left[\frac{\mathbf{1}}{V\_0} \int\_{\Omega} \epsilon\_p^{\beta} \, , \sigma\_1^m d\Omega \right]^{\natural\_m} \tag{18}
$$

#### **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

*Investigation of Strain Effect on Cleavage Fracture for Reactor Pressure Vessel Material DOI: http://dx.doi.org/10.5772/intechopen.101245*

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** [35]. 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**.

**Figure 1.** *1T TPB specimen.*

**Figure 2.**

*Stress-strain diagram of 20MnMoNi55 steel at different temperatures [27].*

**Figure 3.** *KJC calculated from fracture toughness test at* �*110°C.*

#### **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 *ν*, and yield stress versus plastic strain obtained from tensile test data performed at different cryogenic temperatures [27]. **Figure 4** shows the stress versus plastic strain diagram at different temperatures and **Table 2** gives the yield stress and ultimate stress versus temperature for 20MnMoNi55 steel at different temperatures in the Brittle Dominated DBT region which is used as a material input parameter for elastoplastic finite element analysis. Isotropic elastic and isotropic hardening plastic material behavior are considered for the material used. 3-D finite element modeling is done for quarter TPB specimen at different temperatures to calculate the Weibull stress for the specimen and hence to calculate *T*<sup>0</sup> from the Beremin model. The FE model has meshed with 8-node isoparametric hexahedral elements with 8 Gauss points taken for all calculations as referred by

*Investigation of Strain Effect on Cleavage Fracture for Reactor Pressure Vessel Material DOI: http://dx.doi.org/10.5772/intechopen.101245*

#### **Figure 4.**

*Engineering stress versus plastic strain for 20MnMoNi55 steel at different temperatures in the brittle dominated DBT region.*


#### **Table 2.**

*Yield stress and ultimate stress versus temperature for 20MnMoNi55 steel at different temperatures in the brittle dominated DBT region.*

IAEA-TECDOC-1631 [36]. Reduced integration with full Newtonian non-linear analysis computation is carried out for all the specimens. In the region ahead of the crack tip the mesh was refined with an element volume of 0.05 0.05 0.05 mm<sup>3</sup> . To facilitate the calculation of Vj, the element size is kept constant near the crack tip [36, 37]. Since a large strain is expected in the crack tip field, a finite strain (large deformation theory) method is used. As the crack extension during the experiment is found to be very small, the crack growth is not simulated in this FE analysis. **Figure 5** shows the boundary conditions and the mesh of the specimen. **Figure 6** shows the region where maximum principal stress exceeded twice the yield stress at that temperature. This region is known as the fracture process zone (FPZ). For such elements, the strain in the direction of the maximum principal stress is also calculated.

#### **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

**Figure 5.** *Quarter TPB specimen model along with boundary conditions.*

#### **Figure 6.**

*Maximum principal stress (MPa) distribution in the fracture process zone.*

**Figure 7.** *Comparison of load vs. load line displacement (LLD)* �*100°C.*

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

*Investigation of Strain Effect on Cleavage Fracture for Reactor Pressure Vessel Material DOI: http://dx.doi.org/10.5772/intechopen.101245*

**Figure 8.** *Comparison of load vs. load line displacement (LLD)* �*110°C.*

**Figure 9.** *Comparison of load vs. load line displacement (LLD)* �*130°C.*

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 σ<sup>u</sup> 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 (*T*0) for different thicknesses and a/W ratios [28]. But testing such a huge number of specimens are very expensive so the author used to develop a random number of data with the help of master curve and Monte Carlo simulation [24]. This process is called an indirect calibration strategy. The entire process is described by the author in their previous work [23], step-wise description only the Step 7 of Article No.4 the Weibull Stress is calculated for different models using Eqs. (8), (13), (16), and (18). The validation of the results simulated from Monte Carlo simulation with the experimental work is also validated by the author in their previous work [24].

#### **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 [28]. 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.

#### **Figure 10.** *Variation of Weibull modulus "m" for different temperatures using different strain correction model is shown.*

#### **Figure 11.**

*Variation of Weibull modulus scale parameter "σu" for different temperatures using different strain correction model is shown.*

*Investigation of Strain Effect on Cleavage Fracture for Reactor Pressure Vessel Material DOI: http://dx.doi.org/10.5772/intechopen.101245*

#### **Figure 12.**

*Master curve determined from experimental results of* �*120°C compared with the fracture toughness determined from various models.*

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 [25] 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 failuredominated portion of the DBT region, which is observed in the form of ductile stretch in experimental results.

#### **6. Conclusion**


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.
