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S0955-2219(17)30467-3 http://dx.doi.org/doi:10.1016/j.jeurceramsoc.2017.06.044 JECS 11348

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Journal of the European Ceramic Society

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Please cite this article as: Deng Yong, Li Weiguo, Shao Jiaxing, Zhang Xianhe, Kou Haibo, Geng Peiji, Zhang Xuyao, Li Ying, Ma Jianzuo.A novel theoretical model to predict the temperature-dependent fracture strength of ceramic materials.Journal of The European Ceramic Society http://dx.doi.org/10.1016/j.jeurceramsoc.2017.06.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A novel theoretical model to predict the temperature-dependent fracture strength of ceramic materials

Yong Deng, Weiguo Li*, Jiaxing Shao, Xianhe Zhang, Haibo Kou, Peiji Geng, Xuyao Zhang, Ying Li and Jianzuo Ma

State Key Laboratory of Coal Mine Disaster Dynamics and Control, College of Aerospace Engineering, Chongqing University, Chongqing 400030, China

*Corresponding

author’s address: College of Aerospace Engineering, Chongqing University, Chongqing 400030,

China. Phone number: +86 13452029896 (mobile phone) Fax: +86 023 65102421. E-mail address: [email protected]

Abstract A novel temperature-dependent fracture strength model for ceramic materials is developed, based on a critical fracture energy density associated with material fracture comprising strain energy, the corresponding equivalent potential energy, and kinetic energy of atoms per unit volume. It relates the fracture strength at high temperatures to that at the reference temperature, the temperature-dependent Young’s modulus, the temperature, and the melting point. The model is verified by comparison with experimental data of ceramic materials. The model predictions and the experimental data are in excellent agreement with each other. As the Young’s modulus can easily be obtained by experiments and the melting point can easily be obtained by materials

1

handbook, the model can easily predict the fracture strength of ceramic materials at arbitrary temperatures.

Keywords: Ceramic; Fracture strength; Temperature-dependent; Model

1. Introduction Ultra-high temperature ceramics are potential candidates for use in thermal protection systems and propulsion systems in aerospace applications, due to their excellent properties, including high melting temperature, low density, and good chemical and physical stability under high temperatures [1-4]. However, owing to the intrinsic brittleness of ceramics, low fracture toughness directly connected with the limited mechanical reliability is still a major obstacle for their wide use, especially for structural applications in extremely harsh environments [5]. The recent research indicates that the addition of a second phase such as fiber and particulate has been considered as the most promising approach for the improvement the mechanical behavior of ceramic composites [6,7]. As is known, for the sake of safety and reliability of high temperature structural components, fracture strength is a critical factor in extreme environments. Over the past decades, a large amount of experimental work has reported the temperature-dependent fracture strength of ceramic materials [3,6-12]. However, the strength test at ultra-high temperature is very difficult to conduct, and the current experimental temperature is hard to meet the demands of actual high temperature structural applications. Therefore, theoretical prediction for the temperature-dependent fracture strength of ceramic materials is of great importance. To our best knowledge, the theoretical 2

framework of strength has been well established in normal temperature ranges, there are many room temperature property models in the literature [13,14]. But ultra-high temperature strength predictions are lacking and one could count those models on fingers. Considering the significant differences between the mechanical properties and fracture mechanism of ceramic materials at high temperature and those at normal temperature, fracture strength model for ceramic materials which applied to high temperature needs to be established urgently. In view of a constant maximum storage of energy including both the strain energy and corresponding equivalent heat energy associated with the onset of material failure, Li et al. have proposed a model for ultra-high temperature ceramics to quantitatively analyze the effect of temperature on the fracture strength, which has the following form [15]:

1 0 E (T ) th (T ) th 1 T E0 0 C p (T )dT m

1

2 T 0 C p (T )dT

(1)

where th (T ) and th are the fracture strength at temperature T and at absolute zero, respectively; 0

E (T ) and E0 are the Young’s modulus at temperature T and at absolute zero, respectively; C p (T ) is the specific heat capacity at constant pressure p and temperature T ; Tm is the melting point. Li et al.’s model provided a good approach to characterize the quantitative effect of temperature on the fracture strength for ultra-high temperature ceramics. However, in some cases, considering the large difficulty of obtaining the specific heat capacity of ceramics, especially for ceramic matrix composites, it is not very convenient to use the model to predict the temperature-dependent fracture strength in engineering applications. In addition, for the ZrB2-SiC composites, a temperature-dependent fracture strength model was developed by Wang et al. [16], 3

which could well predict the fracture strength of ZrB2-SiC composites at high temperatures. The effects of microstructures, residual stress and temperature on the strength were included in their model. In this paper, a novel way is used to characterize the quantitative effect of temperature on the fracture strength of ceramic materials. A critical fracture energy density characterizing the total energy per unit volume associated with material failure is introduced, which comprises the strain energy, the corresponding equivalent kinetic energy, and potential energy of atoms. Thus, we propose a novel temperature-dependent fracture strength model for ceramic materials, which has no fitting parameters. A simple quantitative relationship between fracture strength, temperature, melting point, and Young’s modulus is obtained by this model. The specific heat capacity is not included in the novel temperature-dependent fracture strength model, compared with Li et al.’s model [15], our model is more convenient to predict the fracture strength of ceramic materials when the specific heat capacity is difficult to obtain. Particularly, the novel model also can be used to predict the temperature-dependent fracture strength of ceramic matrix composites, so our model has a much wider range of application compared with Li et al.’s and Wang et al.’s model. The model predictions are presented for some ultra-high temperature ceramics and ceramic matrix composites, which agree well with the experimental data. This research could provide a potential technical platform and theoretical basis for the design, application, and reliability assessment of ceramic materials in aerospace applications and structural applications. 2. Theoretical model From thermodynamics theory, the system’s internal energy includes the kinetic energy of atomic motion and the potential energy between atoms in the system. The potential energy 4

comprises those of the nuclear and chemical particle bonds, the strain energy owing to solids deformation, and the physical force fields in the system such as due to internal induced electric dipole moment. The system’s kinetic energy is provided by the sum of the motions of all particles within the system in regard to the center-of-mass frame. According to the viewpoint of energy in Li et al.’s and Zhang et al.’s work [15,17], a novel way is found to characterize the quantitative effect of temperature on the fracture strength of ceramic materials. It is assumed that there is a kind of equivalence between the kinetic energy of atomic motion and the potential energy between atoms during the fracture of material. In an isothermal process, the variation of the potential energy is equal to the strain energy. The ceramic materials will fracture when their strain energy reaches a critical value at a certain temperature. We assume there is a maximum storage energy density, WTOTAL, which is associated with the onset of material failure and independent on temperature. The maximum storage energy density characterizes the total energy including the strain energy, the corresponding equivalent kinetic energy and potential energy of atoms per unit volume. Based on the above assumptions and analysis, the maximum storage energy density, WTOTAL, can be expressed as:

WTOTAL =W th (T ) + Ek (T ) EP (T )

(2)

where W th (T ) is the temperature-dependent critical strain energy density associated with materials fracture; For ceramic materials, assumed as linear elastic solid, the critical strain energy density, W th (T ) , can be calculated by using the following formula:

W th (T ) =

th2 (T ) 2 E (T )

(3)

where th (T ) and E (T ) are the fracture strength and the Young’s modulus at temperature T , respectively. Ek (T ) is the temperature-dependent kinetic energy density of atomic motion, which 5

has the form:

Ek (T )

3 kNT 2

(4)

where k is the Boltzmann constant; N is the number of atoms in per unit volume; T is the current temperature (in Kelvin). , assumed constant, is the ratio coefficient between the strain energy and the kinetic energy; EP (T ) is the temperature-dependent potential energy density of atoms; is also a assumed constant which means the ratio coefficient between the strain energy and potential energy. The potential energy and kinetic energy due to the vibrating of atoms transform periodically in the material, and their average value are equal [17]. So EP (T ) can also be expressed as:

EP (T )

3 kNT 2

(5)

Substituting T T0 , T0 is an arbitrary reference temperature, into Eq. (2), we can obtain that:

WTOTAL

2 (T0 ) 3 th

2 E (T0 )

( + )kNT0 2

(6)

where E (T0 ) and th (T0 ) are the Young’s modulus and fracture strength at the arbitrary reference temperature, respectively. When the ceramic material is totally melted, the material in liquid state can’t sustain any external load. Thus th (Tm ) 0 , Tm is the melting point of the ceramic materials, there is no contribution from applied work:

W th (Tm ) 0

(7)

Substituting T Tm into Eq. (2) we can obtain that:

3 WT OTAL = ( + )kNTm 2

(8)

Combining Eqs. (6) and (8) yields

+ th2 (T0 ) 3kNE (T0 )(Tm T0 ) 6

(9)

Substituting Eq. (9) into (2) yields

WTOTAL

2 (T ) th

2 E (T )

2 (T0 )T th

2 E (T0 )(Tm T0 )

(10)

The combination of Eqs. (8) (9) and (10), the novel temperature-dependent fracture strength model without any fitting parameters for ceramic materials is proposed 12

E (T ) Tm T th (T ) th (T0 ) E ( T ) Tm T0 0

(11)

From the above formula, a simple quantitative relationship is revealed between fracture strength, temperature, melting point, and Young’s modulus. Particularly, the influences of microstructure and some micro mechanisms, whose influences on the fracture strength are complicated and weak dependent on temperature, have been tactfully considered in the model by the fracture strength at a reference temperature. It reduces the large difficulty of analyzing the complex influence of microstructure and micro mechanism on the fracture strength during modeling. However, the model cannot well predict the fracture strength of materials at high temperatures, when the evolutions of microstructure occur, whose influences on the fracture strength are significant and evolve with temperature. And it can be concluded that there is change or evolution of the microstructure in materials when the model predictions deviate from the experimental results. Considering the temperature-dependent Young’s modulus can easily be obtained by experiment and the melting point can easily be obtained by materials handbook, so the novel temperature-dependent fracture strength model can easily predict the fracture strength of ceramic materials at arbitrary temperatures. 3. Results and discussion Using the proposed temperature-dependent fracture strength model (Eq. (11)), the fracture 7

strengths at different temperatures of some ceramic materials are predicted and compared with experimental data. During calculations, all parameters including Young’s modulus and melting point are from the literature. For ceramic matrix composites, the Young’s modulus used in the model is determined by the Young’s modulus of composites, and the melting point used in the model is determined by the melting point of matrix. 3.1HfB2 The parameters used in the Eq. (11) are shown as follows [18]:

th (T0 ) =448MPa, E (T0 ) =441GPa, Tm =3380oC, T0 can be chosen arbitrary, 24oC was chosen as the reference temperature for convenience. The Young’s modulus at high temperatures of HfB2 is shown in Fig.1. The theoretical results obtained by Li et al.’s model (Eq. (1)) are also added to the Fig. 2 for comparison. Both the maximum data and minimum data of Young’s modulus are used to predict the fracture strength of HfB2. It can be seen from the Fig. 2, the theoretical values predicted by our model and Li et al.’s model are both in good agreement with the experimental results. 3.2 TiC The parameters used in the Eq. (11) are shown as follows [19]:

th (T0 ) =472MPa, E (T0 ) =444GPa, Tm =3016oC, 25oC was chosen as the reference temperature, the temperature dependent Young’s modulus of TiC is shown in Table 1. In the Fig. 3, the fracture strength of TiC is accurately predicted by both Li et al.’s model and our model. From the above, the difference between the two model predictions is not apparent. Particularly, it should be noted that the specific heat capacity is included in Li et al.’s model, which is not necessary in our model. So it is much more convenient for our model to predict the 8

temperature-dependent fracture strength of ceramic materials whose specific heat capacity is hard to obtain. Moreover, Li et al.’s model could not give a prediction on the fracture strength of ceramic matrix composites due to the great difficulty of obtaining the specific heat capacity. 3.3 ZrB2-SiC composite The parameters used in the Eq. (11) are shown as follows [20]:

th (T0 ) =451MPa, E (T0 ) =194GPa, Tm =3245oC, 20oC was chosen as the reference temperature, the temperature-dependent Young’s modulus of ZrB2-SiC composite is shown in Table 1. Fig. 4 indicates that the novel temperature-dependent fracture strength model can predict the fracture strength of ZrB2-SiC composites at elevated temperatures accurately. Moreover, for ZrB2-SiC composites, as the evolution of the microstructure as a function of temperature is included in Wang et al.’s model [16], their model can better predict the high temperature strength of ZrB2-SiC composites when microstructures evolutions occur. 3.4 B4C-ZrB2 ceramic The parameters used in the Eq. (11) are shown as follows [21, 22]:

th (T0 ) =576.3MPa, E (T0 ) =162.6GPa, Tm =2470oC, 1000oC was chosen as the reference temperature, the Young’s modulus of B4C-ZrB2 ceramic at high temperatures is shown in Table 1. As can be seen from the Fig. 5, the predicted results are in good agreement with the experimental results except it at 1400oC. While the temperature is 1400oC, the predicted value is larger than the experimental data. Cheng et al. reported that the severer oxidation could lead to the sudden drop of strength at 1400oC [21]. The evolution of oxidation mechanisms with temperature is not considered in our model so far. Thus, there is difference between the predicted data and the experimental data. 9

3.5 SiBN fiber reinforced BN composite The parameters used in the Eq. (11) are shown as follows [23,24]:

th (T0 ) =120MPa, E (T0 ) =46.6GPa, Tm =3000oC, 24oC was chosen as the reference temperature, the Young’s modulus of SiBNf/BN composite at high temperatures is shown in Table 1. Fig.6 shows that the excellent agreement is obtained between the model predictions and the experimental results. 3.6 SCS-9a SiC fiber reinforced ZrB2 plus 20 vol% SiC (ZSS) The parameters used in the Eq. (11) are shown as follows [25,26]:

th (T0 ) =130MPa, E (T0 ) =34.5GPa, Tm =3049.85oC, T0 was set as 25oC, the Young’s modulus of ZSS composites at high temperatures is shown in Table 1. Fig. 7 shows a comparison between the theoretical results and the experimental data of ZSS composites from room temperature to1300oC, the good agreement is obtained. 3.7 SiON fiber reinforced BN composites The parameters used in the Eq. (11) are shown as follows [24, 27]:

th (T0 ) =236.8MPa, E (T0 ) =35.96GPa, Tm =3000oC, 600oC was chosen as the reference temperature, the temperature-dependent Young’s modulus of SiONf/BN composites is shown in Table 1. It can be seen from the Fig. 8, our model can well describe the trend of fracture strength of SiONf/BN composite as function of temperature under relative low temperature. The model predictions are larger than the experimental data at 1200oC and 1300oC. Zou et al. indicated that the degradation of fiber caused by high temperature may accelerate the reduction of composite strength [27]. This factor is not included directly in our model. The influence of fiber on the fracture strength is mainly determined by the Young’s modulus of composite in the model, and the 10

high temperature did not result in the sharp reduction of Young’s modulus, which made the difference between the theoretical predictions and the experimental data. 3.8 2D CVI SiC/SiC composite The Young’s modulus of 2D CVI SiC/SiC composite at room temperature (23oC) and 1200oC are 207GPa and 212 GPa, respectively [28]; 23oC was chosen as the reference temperature,

th (T0 ) =324MPa [28], Tm =2700oC[29]. Fig. 9 indicates that the model can well predict the fracture strength of 2DSiC/SiC composite at high temperature. 3.9 Silicon Melt Infiltrated Ceramic Composites (HiPerComp) The Young’s modulus of HiPerComp at room temperature (25oC) and 1200oC are 285GPa and 243GPa, respectively [28]; 25oC was chosen as the reference temperature, th T0 =321MPa [28], Tm =2700oC[29]. Fig.10 shows that the excellent agreement is obtained between the predicted data and the experimental data. 3.10 2.5D Si3N4 fiber reinforced BN composite The parameters used in the Eq. (11) are shown as follows [24, 30]:

th (T0 ) =132.6MPa,

E (T0 ) =28.9GPa,

Tm =3000oC,

T0 was

set

as

25oC,

the

temperature-dependent Young’s modulus of 2.5DSi3N4f/BN composite is shown in Table 1. It can be seen from the Fig.11, the predicted results are in good agreement with experimental data. 3.11 B4C The parameters used in the Eq. (11) are shown as follows [22, 31]:

th (T0 ) =645MPa, Tm =2470oC, T0 was set as 25oC, E (T0 ) =430GPa, the temperature-dependent Young’s

modulus

of

B4C

is

determined

by

the

Ref.

[32],

and

E (T )=430 6.918 103 T 3.71106 T 2 . In Fig. 12, the predicted data by our model is 11

very close to the experimental data at high temperature. From the above results, it can be concluded that the novel temperature-dependent fracture strength model without any fitting parameters can accurately predict the fracture strength of ceramic materials at arbitrary temperatures. Further, it also shows that our model has obvious advantages compared with Li et al.’s model and Wang’s model from the aspects of applicability and convenience, considering the great difficulty in obtaining the specific heat capacity of some ceramic materials. 4. Conclusions In summary, a novel approach for modeling the quantitative influence of temperature on the fracture strength of ceramic materials is proposed. We introduce a critical fracture energy density as the maximum storage energy per unit volume associated with materials fracture, which includes the strain energy, potential energy and kinetic energy of atoms. It is assumed that there is a kind of equivalence relationship between strain energy, potential energy, and kinetic energy. On the basis of the critical fracture energy density and the equivalence relationship, a novel temperature-dependent fracture strength model for ceramic materials is developed. The model establishes the quantitative relationship between fracture strength, temperature, melting point, and Young’s modulus. It is verified by comparison with the experimental data of fracture strength for ceramic materials at different temperatures. Excellent agreement is obtained between the model predictions and the experimental data. As the Young’s modulus and the melting point in the model can easily be obtained by experiments and materials handbook, the temperature-dependent fracture strength of ceramic materials can easily be predicted by the model, avoiding the large difficulty of conducting strength tests at extremely high temperatures. 12

Acknowledgments This work was supported by the National Natural Science Foundation of China under GrantNos.11472066 and 11672050, the Program for New Century Excellent Talents in University under Grant No.ncet-13-0634, China Postdoctoral Science Foundation funded project (2016M592636).

13

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Figure captions Fig.1 Temperature dependent Young’s modulus of HfB2 [18] Fig. 2 Predicted and experimental values of temperature dependent fracture strength of HfB2 [18] Fig. 3 Predicted and experimental values of temperature dependent fracture strength of TiC [19] Fig. 4 Predicted and experimental values of temperature dependent fracture strength of ZrB2-SiC composite [20] Fig. 5 Predicted and experimental values of temperature dependent fracture strength of B4C-ZrB2 ceramic [21] Fig. 6 Predicted and experimental values of temperature dependent fracture strength of SiBNf/BN composite [23] Fig. 7 Predicted and experimental values of temperature dependent fracture strength of ZSS composite [25] Fig. 8 Predicted and experimental values of temperature dependent fracture strength of SiONf/BN composite [27] Fig. 9 Predicted and experimental values of temperature dependent fracture strength of 2D CVI SiC/SiC composite [28] Fig. 10 Predicted and experimental values of temperature dependent fracture strength of HiPerComp [28] Fig. 11 Predicted and experimental values of temperature dependent fracture strength of 2.5DSi3N4f/BN composite [30] Fig. 12 Predicted and experimental values of temperature dependent fracture strength of B4C [31]

18

Figure 1

Figure 2 19

Figure 3

Figure 4 20

Figure 7

Figure 8 21

Figure 9

Figure 10 22

Figure 11

Figure 12 23

Table captions Table 1 Young's modulus (in GPa) of ceramic materials at different temperatures Temper ature/oC

TiC [19]

ZrB2-S iC

rB2 [20]

Room

444

B4C-Z

SiBNf/ BN

[21]

ZSS [26]

[23]

194

46.6

SiONf/ BN

Si3N4/B N [30]

[27] 34.5

28.9

temperature 200

44.5

400

40.8

600

38.2

35.9

33.6

33.7

25.7

31

28

26.3

24

24.5

16.2

700 411 800 900 377 1000

137

163

1100 333 1127

33

1150

101

1200

126

1300 278 1327

31

1400

86

1600

30

24