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: W T O T A L = W K C ( T ) + α [ E k ( T ) + E P ( T ) ] (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 C ( T ) = [ 1 − ν 2 ( T ) ] K I c 2 ( T ) E ( T ) (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 k ( T ) = 3 2 k B N T (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 P ( T ) = 3 2 k B N T (4)

Combining Eqs. 1–4, the maximum storage energy density for a particular material is obtained as W T O T A L = [ 1 − ν 2 ( T ) ] K I c 2 ( T ) E ( T ) + 3 α k B N T (5)

Substituting an arbitrary reference temperature T ₀ into Eq. 5, W_TOTAL can also be expressed as W T O T A L = [ 1 − ν 2 ( T 0 ) ] K I c 2 ( T 0 ) E ( T 0 ) + 3 α k B N T 0 (6) where K _Ic (T ₀) denotes the fracture toughness at T ₀ and v (T ₀) and E (T ₀), respectively, denote the Poisson’s ratio and Young’s modulus at T ₀.

At melting point T _m , the material loses its ability to carry any external stress, so W K C ( T m ) = 0 (7)

Substituting T=T _m and Eq. 7 into Eq. 5, we can obtain W T O T A L = 3 α k B N T m (8)

Combining Eqs. 6, 8, we can deduce that α = [ 1 − ν 2 ( T 0 ) ] K I c 2 ( T 0 ) 3 k B N E ( T 0 ) ( T m − T 0 ) (9)

Substituting Eq. 9 into Eq. 5 yields W T O T A L = [ 1 − ν 2 ( T ) ] K I c 2 ( T ) E ( T ) + [ 1 − ν 2 ( T 0 ) ] K I c 2 ( T 0 ) T E ( T 0 ) ( T m − T 0 ) (10)

Finally, combining Eqs. 8–10, the temperature-dependent fracture toughness model is obtained: K I c ( T ) = K I c ( T 0 ) [ E ( T ) E ( T 0 ) × 1 − ν 2 ( T 0 ) 1 − ν 2 ( T ) × T m − T T m − T 0 ] 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 (T ₀) in 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 T ₀ 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).

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 (T ₀)=162.5 M P a ⋅ m 1 / 2 , E ( T ) = e − 19.916 × ( 2.27 × 10 − 9 T 2 + 9.44 × 10 − 6 T + 12.333 ) (Vratnica et al., 2013), T _m =1400°C (Zhang et al., 2017a), and T ₀=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 (T ₀)=103 MPa.m^1/2, E ( T ) = e − 28.217 × ( 3.131 × 10 − 9 T 2 + 11.43 × 10 − 6 T + 12.207 ) (Li et al., 2018), T _m =1400 °C (Zhang et al., 2017a), and T ₀=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 T ₀=25°C. The experimental value at room temperature was set as K _Ic (T ₀) 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 (T ₀)=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 T ₀=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).

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 (T ₀). 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.

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 ρ ₀, K _Ic is independent on ρ. When ρ is larger than ρ ₀, K _Ic increases linearly with ρ ^1/2, namely (Carolan et al., 2011), K I c ( ρ ) = { K I c 0   ( ρ ≤ ρ 0 ) K I c 0 + m ( ρ − 2 d ) 1 / 2   ( ρ > ρ 0 ) (12) where K I c 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 I c ( T , ρ ) = { K I c 0 ( T 0 , ρ 0 ) [ E ( T ) E ( T 0 ) × 1 − ν 2 ( T 0 ) 1 − ν 2 ( T ) × T m − T T m − T 0 ] 1 / 2   ( ρ ≤ ρ 0 ) [ K I c 0 ( T 0 , ρ 0 ) + m ( ρ − 2 d ) 1 / 2 ] × [ E ( T ) E ( T 0 ) × 1 − ν 2 ( T 0 ) 1 − ν 2 ( T ) × T m − T T m − T 0 ] 1 / 2   ( ρ > ρ 0 ) (13) where K I c 0 ( T 0 , ρ 0 ) is the material fracture toughness at reference temperature T ₀ below the critical notch root radius ρ ₀. 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.

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 (T ₀) of aluminum alloy at room temperature (Srinivas and Kamat, 2000) and from Eq. 12, we can obtain the values of m and K I c 0 ( T 0 ) . Also, Eq. 13 has the following expression for aluminum alloy: K I c ( T , ρ ) = { 13 [ E ( T ) E ( T 0 ) × T m − T T m − T 0 ] 1 / 2   ( ρ < 80 μ m ) [ 0.47 ( ρ − 4 ) 1 / 2 + 13 ] × [ E ( T ) E ( T 0 ) × T m − T T m − T 0 ] 1 / 2   ( 80 μ m ≤ ρ ≤ 500 μ m ) (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), T ₀=25°C, and d=2 μ 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.