Modeling the effect of temperature and notch root radius on fracture toughness

In this work, a physics-based analytical model without any fitting parameters for fracture toughness depending on temperature was proposed. The quantitative relationship between fracture toughness, melting point, Young’s modulus, Poisson’s ratio, and temperature was established. Furthermore, the effect of notch root radius on fracture toughness was investigated, and then we developed a simple theoretical model to characterize the combined effect of temperature and notch root radius on the fracture toughness. Satisfactory agreement between the available experimental measurements and our model predictions was achieved. This study not only advances our in-depth understanding of the quantitative relationship between the temperature and fracture toughness but also provides a powerful and convenient tool to predict the temperature-dependent fracture toughness at different notch root radius.

1 Introduction

Fracture toughness (K_Ic), as a vital factor of materials, plays a crucial role in structure integrity assessment and damage tolerance design since it can determine whether a crack reaches an unstable state in a particular material (Mu et al., 2015; Jia et al., 2019). Currently, the demand of metallic materials in high-temperature applications becomes more and more intense, such as in nuclear power reactors and components in aerospace (Tong et al., 2014). The working temperature of metals is getting increasingly higher in service, and several studies have indicated that temperature has a significant influence on K_Ic (Wen et al., 2008; Byun et al., 2013; Chandra Rao et al., 2008). Meanwhile, several experimental studies proved that the notch root radius of a specimen also affected the value of fracture toughness (Fujita et al., 2007; Mourad et al., 2013; Vratnica et al., 2013). Therefore, investigating the influence of temperature and notch root radius on K_Ic is important and necessary for the design and application of materials working at high temperatures.

In recent years, a large number of valuable experimental studies have reported the K_Ic of metals at different temperatures (Wen et al., 2008; Byun et al., 2013; Chandra Rao et al., 2008), (Srinivas and Kamat, 2000; Harimon et al., 2017; Li et al., 2018; Jia et al., 2019; Pan et al., 2019; Li et al., 2020; Zhang et al., 2020). Wen et al. (2008) demonstrated that temperature has a complex effect on the fracture toughness of nickel-based, single-crystal superalloys and found that the fracture mode of the specimens transitioned from brittle to ductile as the temperature increased. Byun et al. (2013) reported the fracture toughness of HT9 steel from room temperature to 504°C. They found that it decreased with the temperature increasing for non-irradiated specimens. Srinivas and Kamat (2000) found that the fracture toughness of a dispersion-strengthened aluminum alloy was significantly affected by temperature and notch root radius, and it increased linearly with the square root of notch root radius beyond a critical value and decreased with increasing test temperature. Jia et al. (2019) investigated the impact of temperature on fracture toughness of Ti60 alloy and its fracture mechanism, and the result showed that the fracture toughness increased from room temperature to 400°C but decreased at 600°C due to the area of the crack tip around the plastic zone. The abovementioned experimental research provides a solid basis for understanding and revealing the effects of the relevant factors on the high-temperature fracture toughness of materials. However, a high temperature test for K_Ic lacks the standard unified method as yet, and the experimental values of K_Ic from different literatures are quite different. Moreover, measuring K_Ic is difficult and complex at high temperatures, which also causes considerable consumption of time and money (Li et al., 2018; Pan et al., 2019). Currently, it is difficult to provide a systematic knowledge of temperature’s influence on K_Ic only by experiments. Thus, establishing a theoretical model for uncovering the quantitative influence of temperature on K_Ic is urgently needed.

The existing temperature-dependent fracture toughness models mainly focus on empirical or semi-empirical models (Jia et al., 2019; Amar and Pineau, 1985; Margolin et al., 2003a; Margolin et al., 2003b; Shao et al., 2016), which lack a detailed physical mechanism. They all rely on large number of experimental data at different temperatures to fit the fitting parameters in models. Thus, the use of these models including fitting parameters that predict temperature-dependent K_Ic is inconvenient. In addition, Wallin (1984) put forward the Master Curve method (Wallin, 2010) to calculate temperature-dependent K_Ic of ferritic steel. Recently, Li et al. (2018) established a temperature-dependent fracture toughness model for superalloys, which developed the relationship between fracture toughness, Young’s modulus, and strain-hardening exponent at different temperatures. However, it depends on extensive experiments to determine the temperature-dependent strain-hardening exponent when K_Ic is predicted at high temperatures, which seriously weakens the predictability of their model.

In view of the current situation, the objective of the present work is to reveal and quantify the influence of temperature and notch root radius on K_Ic. To this end, we first proposed a concept called the maximum storage energy density associated with material failure and established a temperature-dependent fracture toughness model without any fitting parameters. A comparison was made between the model predictions and the available experimental measurements, which validated our model well. On the basis of the model, we further considered the effect of notch root radius on K_Ic and established a temperature- and notch root radius-dependent model, which offers a good approach to predict K_Ic at different temperatures and notch root radius. At the end, discussions and conclusions were obtained.

2 Temperature-dependent fracture toughness model

2.1 Derivation of the model

In this work, we consider the fracture process as a process of bond breaking in the fractured parts to create a new material surface. Volokh’s research (Volokh, 2007) demonstrated that there exists a maximum strain energy density related to material fracture. But, temperature’s influence was not included in his work. Based on thermodynamics theory, the internal energy of a particular system is composed of kinetic energy due to atomic motion in the system and potential energy between atoms (Zhang et al., 2017a; Deng et al., 2017). To investigate the quantitative influence of temperature on K_Ic, we assumed that there is a temperature-independent maximum storage energy density associated with material fracture (Li et al., 2010), which includes the critical strain energy density arising from externally applied stress and internal energy density. Moreover, it was assumed that the critical strain energy density and the system’s internal energy density have an equivalent relationship, based on their different contribution to material fracture. From the abovementioned discussion, the maximum storage energy density W_TOTAL has the following form:

${\mathrm{W}}_{\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{A}\mathrm{L}}=W_{K_{\mathrm{C}}}\left(T\right)+\alpha \left[E_{\mathrm{k}}\left(T\right)+E_{\mathrm{P}}\left(T\right)\right](1)$

where T denotes the ambient temperature, W_Kc(T) is the critical strain energy density at temperature T, and E_p(T) and E_k(T), respectively, denote the potential energy density and kinetic energy density at temperature T. α is a hypothetical constant, which represents the ratio coefficient between the critical elastic energy density and internal energy density. Based on the classical fracture mechanics theory, the critical strain energy density for the plain strain W_Kc(T) under linearly elastic conditions can be expressed as

$W_{K_{\mathrm{C}}}\left(T\right)=\frac{\left[1-{\nu }^2\left(T\right)\right]K_{Ic}^2\left(T\right)}{E\left(T\right)}(2)$

where W_Kc(T) is the temperature-dependent fracture toughness and v(T) and E(T) are the temperature-dependent Poisson’s ratio and Young’s modulus, respectively.

According to the thermodynamics theory, the kinetic energy density can be described as

$E_{\mathrm{k}}\left(T\right)=\frac{3}{2}{k}_{B}NT(3)$

where N denotes the number of atoms per unit volume and k_B denotes the Boltzmann constant.

Moreover, the kinetic energy and potential energy in the material transforms periodically, and their mean value is equivalent (Zhang et al., 2017a). Thus, E_p(T) can be described as

$E_{\mathrm{P}}\left(T\right)=\frac{3}{2}{k}_{B}NT(4)$

Combining Eqs. 1–4, the maximum storage energy density for a particular material is obtained as

${\mathrm{W}}_{\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{A}\mathrm{L}}=\frac{\left[1-{\nu }^2\left(T\right)\right]K_{Ic}^2\left(T\right)}{E\left(T\right)}+3\alpha k_{B}NT(5)$

Substituting an arbitrary reference temperature T₀ into Eq. 5, W_TOTAL can also be expressed as

${\mathrm{W}}_{\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{A}\mathrm{L}}=\frac{\left[1-{\nu }^2\left(T_0\right)\right]K_{Ic}^2\left(T_0\right)}{E\left(T_0\right)}+3\alpha k_{B}N{T}_0(6)$

where K_Ic(T₀) denotes the fracture toughness at T₀ and v (T₀) and E (T₀), respectively, denote the Poisson’s ratio and Young’s modulus at T₀.

At melting point T_m, the material loses its ability to carry any external stress, so

$W_{K_{\mathrm{C}}}\left(T_m\right)=0(7)$

Substituting T=T_m and Eq. 7 into Eq. 5, we can obtain

${\mathrm{W}}_{\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{A}\mathrm{L}}=3\alpha k_{B}N{T}_m(8)$

Combining Eqs. 6, 8, we can deduce that

$\alpha =\frac{\left[1-{\nu }^2\left(T_0\right)\right]K_{Ic}^2\left(T_0\right)}{3k_{B}NE\left(T_0\right)\left(T_m-T_0\right)}(9)$

Substituting Eq. 9 into Eq. 5 yields

${\mathrm{W}}_{\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{A}\mathrm{L}}=\frac{\left[1-{\nu }^2\left(T\right)\right]K_{Ic}^2\left(T\right)}{E\left(T\right)}+\frac{\left[1-{\nu }^2\left(T_0\right)\right]K_{Ic}^2\left(T_0\right)T}{E\left(T_0\right)\left(T_m-T_0\right)}(10)$

Finally, combining Eqs. 8–10, the temperature-dependent fracture toughness model is obtained:

$K_{Ic}\left(T\right)=K_{Ic}\left(T_0\right){\left[\frac{E\left(T\right)}{E\left(T_0\right)}\times \frac{1-{\nu }^2\left(T_0\right)}{1-{\nu }^2\left(T\right)}\times \frac{T_m-T}{T_m-T_0}\right]}^{1/2}(11)$

The physics-based model without any fitting parameters reveals the inherent relationship between fracture toughness, melting point, Poisson’s ratio, Young’s modulus, and temperature. Considering that these material parameters in Eq. 11 can be obtained easily from the existing literatures or material handbooks, this model provides a convenient approach to predict K_Ic(T) at different temperatures. In particular, influence factors such as test approach and size and shape of specimen and microstructure, whose effect on K_Ic is complex and less sensitive to temperature, are considered by the K_Ic(T₀) in the model.

2.2 Validation of the model

To validate our temperature-dependent fracture toughness model, the available experimental data of K_Ic(T) for GH4720Li alloy, GH4169 alloy, HT9 steel, sintered steel, and aluminum alloy were compared with those of our model predictions (Figures 1–4). All material parameters used in the calculations were obtained from the existing literatures. The reference temperature T₀ in the model can be selected arbitrarily. For convenience, it was set at room temperature in this study. In addition, Poisson’s ratio was assumed to be a constant at different temperatures due to its weak temperature dependence (Li et al., 2016). For more details about the materials and experiment method, interested readers could refer to Refs. (Shan and Leng, 1999; Srinivas and Kamat, 2000; High Temperature Materials Session of China Metal Institute, 2012; Byun et al., 2013; Baek et al., 2014; Li et al., 2018).

FIGURE 1

FIGURE 1. Temperature-dependent fracture toughness of (A) GH4720Li (Li et al., 2018) and (B) GH4169 (High Temperature Materials Session of China Metal Institute, 2012).

FIGURE 2

FIGURE 2. Temperature-dependent fracture toughness of HT9 Steel (Byun et al., 2013; Baek et al., 2014).

FIGURE 3

FIGURE 3. Temperature-dependent fracture toughness of (A) sintered steel (B) SiC (Xu et al., 1992). (C) Al₂O₃ (Sglavo et al., 1999) and (D) YSZ coating (Qu et al., 2018).

FIGURE 4

FIGURE 4. Temperature- and notch root radius-dependent fracture toughness of aluminum alloy (Srinivas and Kamat, 2000).

First, we used our model to predict the temperature-dependent K_Ic(T) of GH4720Li alloy. The material parameters in the calculation are given as follows: K_Ic(T₀)=162.5$\mathrm{M}\mathrm{P}\mathrm{a}\cdot {\mathrm{m}}^{1/2}$,$E\left(T\right)=e^{-19.916\times \left(2.27\times {10}^{-9}{T}^2+9.44\times {10}^{-6}T+12.333\right)}$ (Vratnica et al., 2013), T_m=1400°C (Zhang et al., 2017a), and T₀=24°C. The prediction made by our model achieved good agreement with the experimental data (Li et al., 2018) (Figure 1A). For comparison, Li’s model prediction (Li et al., 2018) was also depicted in Figure 1A, and his model also can predict well with K_Ic(T) of GH4720Li. However, it relies on a large number of experiments to determine the fitting parameters in Li’s model. Thus, it is much more practical and convenient to predict K_Ic(T) using our model than Li’s model.

Figure 1B showed the comparison between the experimental data and theoretical prediction of K_Ic(T) for the GH4169 alloy. The needed material parameters are given as follows: K_Ic(T₀)=103 MPa.m^1/2,$E\left(T\right)=e^{-28.217\times \left(3.131\times {10}^{-9}{T}^2+11.43\times {10}^{-6}T+12.207\right)}$ (Li et al., 2018), T_m=1400 °C (Zhang et al., 2017a), and T₀=24°C. Our fracture toughness model predicted well the trend of fracture toughness of GH4169 materials with temperature (Fig. 1(b)). At relatively low temperatures and 650°C, the model predictions agreed well with the experimental data (High Temperature Materials Session of China Metal Institute, 2012). When the temperature is 550 and 600°C, the predicted data are a little lower than those of the experimental data. It is likely due to the microstructure evolution at high temperatures (An et al., 2019), which is not included in our current model. Similarly, although Li’s model prediction has a better agreement with the experimental data above 300°C, the fitting parameters in his model seriously weaken the predictability.

Figure 2 displayed the comparison between the theoretical predictions and experimental measurement of four HT9 steel. The needed material parameters for HT9 steel are given as follows: E(T)=216.2–0.0692T (Shan and Leng, 1999), T_m=1480°C (Klug et al., 1996), and T₀=25°C. The experimental value at room temperature was set as K_Ic(T₀) for each HT9 steel. The predictions by our model achieved good consistency with the experimental results (Byun et al., 2013; Baek et al., 2014) (Figure 2). However, in Figure 2D, the model prediction is larger than the experimental data at 600°C, which is likely due to the interaction between gliding dislocations and light elements (Baek et al., 2014).

Then, we predicted the K_Ic(T) of sintered steel from room temperature to 300°C. The needed material parameters are given as follows: K_Ic(T₀)=28.8 MPa.m^1/2, E(T)=128.5–0.11T (Shan and Leng, 1999), T_m=1538°C (Shan and Leng, 1999), and T₀=23°C. An excellent agreement was obtained between the experimental results (Shan and Leng, 1999) and our model predictions (Figure 3A).

In addition, the K_Ic(T) at high temperatures of some ceramic materials (SiC, Al₂O_3, and YSZ coating) was also predicted by our model. The relative material parameters are listed in Table 1. As can be seen from Figures 3B–D, our model predictions achieved satisfactory agreement with the corresponding experiment measurement (Xu et al., 1992; Sglavo et al., 1999; Qu et al., 2018).

TABLE 1

TABLE 1. Temperature-dependent Young’s modulus and melting point (Sglavo et al., 1999; Deng et al., 2016; Qu et al., 2018).

Based on the abovementioned results (Figures 1–3), it can be concluded that our temperature-dependent fracture toughness model (Eq. 11) can well-predict the K_Ic(T) at different temperatures. It is worth noting that the factors which have an effect on K_Ic and does not evolve with temperature are included in the K_Ic(T₀). However, the factors with temperature evolution, which have influence on K_Ic(T), such as plastic deformation, phase transition, gliding dislocations, and microstructure evolution are not included in the current work, which will be considered in our next work.

3 Temperature- and notch root radius-dependent fracture toughness model

3.1 Derivation of the model

It is well-known that K_Ic is not only temperature-dependent but also has a strong dependence on notch root radius of test specimen (ρ). In general, when ρ is smaller than critical data ρ₀, K_Ic is independent on ρ. When ρ is larger than ρ₀, K_Ic increases linearly with ρ^1/2, namely (Carolan et al., 2011),

$K_{Ic}\left(\rho \right)=\{\begin{array}{l}{K}_{Ic}^0\left(\rho \le {\rho }_0\right)\\ K_{Ic}^0+m(\rho -2d{)}^{1/2}(\rho >{\rho }_0)\end{array}(12)$

where $K_{Ic}^0$ is the material fracture toughness below the critical notch root radius, m is called the notch sensitivity, and d is the characteristic length, which equals to the microstructural grain size for some material. Thus, combining Eqs. 11, 12, the theoretical model for revealing the effect of notch root radius on fracture toughness and their evolution with temperature can be expressed as

$K_{Ic}\left(T,\rho \right)=\{\begin{array}{l}{K}_{Ic}^0\left(T_0,{\rho }_0\right){\left[\frac{E\left(T\right)}{E\left(T_0\right)}\times \frac{1-{\nu }^2\left(T_0\right)}{1-{\nu }^2\left(T\right)}\times \frac{T_m-T}{T_m-T_0}\right]}^{1/2}\left(\rho \le {\rho }_0\right)\\ \left[K_{Ic}^0\left(T_0,{\rho }_0\right)+m{\left(\rho -2d\right)}^{1/2}\right]\times {\left[\frac{E\left(T\right)}{E\left(T_0\right)}\times \frac{1-{\nu }^2\left(T_0\right)}{1-{\nu }^2\left(T\right)}\times \frac{T_m-T}{T_m-T_0}\right]}^{1/2}\left(\rho >{\rho }_0\right)\end{array}(13)$

where $K_{Ic}^0\left(T_0,{\rho }_0\right)$ is the material fracture toughness at reference temperature T₀ below the critical notch root radius ρ₀. This temperature- and notch root radius-dependent fracture toughness model offers a powerful means for predicting the notch root radius-dependent K_Ic at different temperatures.

3.2 Validation of the model

Srinivas and Kamat (2000) have reported the value of K_Ic(T,ρ) of aluminum alloy at different notch root radius (ρ) from room temperature to 350 °C. To validate our temperature- and notch root radius-dependent fracture toughness model, the K_Ic(T,ρ) of aluminum alloy at different notch root radius (from 0 to 500 μm) and different temperatures was predicted by Eq. 13. Based on the experimental data of K_Ic(T₀) of aluminum alloy at room temperature (Srinivas and Kamat, 2000) and from Eq. 12, we can obtain the values of m and $K_{Ic}^0\left(T_0\right)$. Also, Eq. 13 has the following expression for aluminum alloy:

$K_{Ic}\left(T,\rho \right)=\{\begin{array}{l}13{\left[\frac{E\left(T\right)}{E\left(T_0\right)}\times \frac{T_m-T}{T_m-T_0}\right]}^{1/2}\left(\rho <80\mathrm{\mu }\mathrm{m}\right)\\ \left[0.47{\left(\rho -4\right)}^{1/2}+13\right]\times {\left[\frac{E\left(T\right)}{E\left(T_0\right)}\times \frac{T_m-T}{T_m-T_0}\right]}^{1/2}\left(80\mathrm{\mu }\mathrm{m}\le \rho \le 500\mathrm{\mu }\mathrm{m}\right)\end{array}(14)$

Figure 4 presented the comparison between the experimental (Srinivas and Kamat, 2000) and theoretical predictions of aluminum alloy. In the calculation, Young’s modulus of aluminum alloy was obtained by the existing yield strength in the literature (Srinivas and Kamat, 2000), and the temperature-dependent yield strength model is established by Zhang et al. (2017a) due to lack of its values at elevated temperatures. Also T_m=660°C (Zedalis et al., 1989), T₀=25°C, and d=2 $\mu$m. Figure 4 shows that our temperature- and notch root radius-dependent fracture toughness model (Eq. 13) can well-predict the K_Ic of aluminum alloy at different notch root radius from room temperature to 350°C. The evolution of notch root radius’s influence on K_Ic was neglected, which led to a small difference between the predicted and experimental data above room temperature when ρ equals 500 μm.

4 Discussion

As discussed previously, high-temperature plastic deformation significantly affects the fracture toughness of materials when the temperature is higher than the brittle–ductile transition temperature. In this case, it is necessary to consider its effect when modeling. Thus, the temperature-dependent fracture toughness includes two parts above brittle–ductile transition temperature:

$K_{Ic}\left(T\right)=K_1\left(T\right)+K_2\left(T\right)(15)$

where K₁(T) and K₂(T) are the initial fracture toughness and its incremental value due to plastic deformation, respectively. K₁(T) is expressed by Eq. 11. The next critical step is to give the formula of K₂(T). Based on the yielding fracture mechanics (Egan, 1973; Larsson, 2011) and our previous work (Wang et al., 2019), the fracture toughness plasticity component (K₂(T)) is inversely proportional to the yield strength (${\sigma }_y\left(T\right)$):

$K_2\left(T\right)\propto 1/{\sigma }_y\left(T\right)(16)$

As a result, K₂(T) can be expressed as (Zhang et al., 2017b):

$K_2\left(T\right)=K_2\left(T_1\right){\left[\frac{E\left(T\right)}{E\left(T_1\right)}\times \frac{1-{\nu }^2\left(T_1\right)}{1-{\nu }^2\left(T\right)}\times \frac{T_m-T}{T_m-T_1}\right]}^{-1/2}(17)$

where T₁ is the brittle–ductile transition temperature and K₂ (T₁) is the fracture toughness plasticity component at T₁.

Then, the temperature-dependent fracture toughness model, considering the effect of plastic deformation at temperature higher than T₁, is expressed as

$K_{Ic}\left(T\right)=K_{Ic}\left(T_0\right){\left[\frac{E\left(T\right)}{E\left(T_0\right)}\times \frac{1-{\nu }^2\left(T_0\right)}{1-{\nu }^2\left(T\right)}\times \frac{T_m-T}{T_m-T_0}\right]}^{1/2}+K_2\left(T_1\right){\left[\frac{E\left(T\right)}{E\left(T_1\right)}\times \frac{1-{\nu }^2\left(T_1\right)}{1-{\nu }^2\left(T\right)}\times \frac{T_m-T}{T_m-T_1}\right]}^{-1/2}(18)$

Moreover, the proposed fracture toughness model indicates that Young’s modulus has a significant effect on the fracture toughness of materials. The influencing factor of fracture toughness was analyzed by the theoretical model, and the effect of Young’s modulus on the fracture toughness of GH4720Li and GH4169 in the temperature range of 25–650°C was analyzed, as shown in Figure 5. The sensitivity of fracture toughness to Young’s modulus decreases with increasing temperature, and Figure 5 also indicates that improving Young’s modulus is helpful in increasing the fracture toughness of alloys.

FIGURE 5

FIGURE 5. Influence of Young’s modulus on fracture toughness. (A) GH4270Li and (B) GH4169.

5 Conclusion

In summary, based on the equivalent relationship between the critical strain energy density associated with material fracture and the system’s internal energy density, we established a physics-based model without any fitting parameters for predicting the temperature-dependent fracture toughness. The quantitative relationship between fracture toughness, melting point, Young’s modulus, Poisson’s ratio, and temperature was uncovered. The comparisons were made between the available experimental results and the model predictions, which showed good agreement. Considering that the material parameters in the model can be obtained easily, the fracture toughness at high temperatures can be predicted conveniently, avoiding difficult and laborious high-temperature experiments. Moreover, by further considering the influence of notch root radius on fracture toughness, a theoretical model of fracture toughness dependent on temperature and notch root radius was developed. It was validated by comparison with experimental values at different notch root radius. This work provides a practical and convenient technical means for predicting the fracture toughness at different temperature and notch root radius.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding authors.

Author contributions

YD: conceptualization, methodology, and writing—original draft. CZ: modeling, supervision, and validation. JS: formal analysis and visualization. WL: supervision and writing—review and editing.

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 12102354, 12002288, and 12002223]; the Shanghai Sailing Program [grant number 21YF1450900]; the Natural Science Basic Research Plan in Shaanxi Province of China [grant number 2021JQ-073]; and the Fundamental Research Funds for the Central Universities [grant number G2021KY05111].

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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