A Thermodamage Strength Theoretical Model of Ceramic Materials Taking into Account the Effect of Residual Stress

A thermodamage strength theoretical model taking into account the effect of residual stress was established and applied to each temperature phase based on the study of effects of various physical mechanisms on the fracture strength of ultrahigh-temperature ceramics. The effects of SiC particle size, crack size, and SiC particle volume fraction on strength corresponding to different temperatures were studied in detail. This study showed that when flaw size is not large, the bigger SiC particle size results in the greater effect of tensile residual stress in the matrix grains on strength reduction, and this prediction coincides with experimental results; and the residual stress and the combined effort of particle size and crack size play important roles in controlling material strength.


Introduction
As the most promising materials at the ultrahigh-temperature, the ultrahigh-temperature ceramics (UHTCs) are a family of ceramic composites that have melting points higher than 3000 • C, can be used in high temperature and oxidizing environment, and have good chemical and physical stability [1][2][3][4].Improving fracture strength of UHTCs has always been a key goal.The study of this aspect of UHTCs is essential for design andimproving reliability in applications.Significant improvements have been made [2,[5][6][7][8][9].Gasch et al. [2] revealed that adding SiC particles can significantly improve the oxidation resistance of UHTCs, especially when the added volume fraction is 20%.Endo et al. [5] studied the mechanical behavior of the TiC ceramics and showed that they can be toughened and strengthened with the addition of SiC particles.Song et al. [6] studied the thermomechanical properties of carbon fiber-reinforced TiC matrix composites.Hu et al. [7] studied the fracture behavior of the ZrB 2 -15 vol.% SiC UHTCs at 1800 • C showed that the strength of composites at ultrahigh-temperature significantly depends on grain size and SiC content.Zhang et al. [8] studied the effects of sintering method and initial added ZrB 2 and SiC particle size on the material properties.Li et al. [9] obtained a temperature-dependent strength model according to a quantitative equivalent relationship between heat energy and strain energy, and the calculated results agree well with experimental results.
As regards particle-reinforced material, the fracture strength is associated with the residual stress.Many experiments and theories have showed that the cracking occurs around second-phase particles due to thermal expansion mismatch between the matrix and adding particle [10,11].But it is very difficult to measure residual stress.And current studies of UHTCs focus mainly on experimental researches, but experiments in the laboratory do not yet meet the demands of actual aerospace applications, and strengths attained with current experimental techniques barely satisfy the strong desire to understand the behavior of materials in high-temperature environment.Moreover, fewer theories consider effects of residual stress and temperature on the strength of UHTCs meanwhile.Based on the above situation and difficulties, the theoretical research of this part of UHTCs is necessary.
In this current paper, a thermodamage strength theoretical model taking into account the effect of residual stress was applied to each stage of temperature based on the study of effects of various physical mechanisms on the fracture strength of UHTCs.The effects of SiC particle size, crack size, and SiC particle volume fraction on strength corresponding Advances in Materials Science and Engineering to different temperatures were studied in detail.This study will provide a theoretical basis and guidance to the design and preparation of UHTCs.

Thermodamage Strength Theoretical Model
In this current paper, we studied the fracture strength of TiC containing numbers of spherical SiC particles with annular flaws (as shown in Figure 1).
The following is the stress intensity factor due to the applied stress σ a suffered by material [12]: where, R is the SiC particle radius and s is the crack size.
The stress intensity factor due to the residual stress P [11,13] is as follows: Then, the total stress intensity factor considering the combined effects of the applied stress and residual stress is obtained as follows: The Griffice fracture criterion states that material damage will occur at the crack edge with the largest stress intensity factor.Thus, the stress intensity factor is calculated using (3), and let , σ a = σ f , the fracture strength is obtained as follows: where where A is a constant; α m , α p are the thermal expansion coefficients of the matrix and particle, respectively; E m , E p are the Young's modulus of the matrix and particle, respectively; ν m , ν p are the Poisson's ratio of the matrix and particle, respectively; T s is the sintering temperature; T r is the room temperature; γ is the fracture surface energy of the matrix.Now, we use the Cornwall's research [14] to extend to be (4) applicable for a multiple-particle material: where where p is the SiC particle volume fraction.
As temperature increases, the residual stress will be changed, therefore, we modify (6) as follows: where T is the current temperature.
Finally, based on a temperature-dependent strength model according to the quantitative equivalent relationship between heat energy and strain energy proposed by Li et al. [9], we introduce the effect of temperature on the fracture strength.The model is expressed as follows: where where σ th (T) is the temperature-dependent fracture strength given at initial damage state [9], σ f 0 is the damage-dependent fracture strength with respect to some reference temperature, and σ 0 th is the fracture strength at the same initial damage state and reference temperature.In the second expression, C p (T) is the specific heat capacity for constant p pressure and temperature T while ΔH M is the latent heat of melting.

Results and Discussion
Table 1 shows the relative parameters [6,8,[15][16][17][18][19][20] obtained from experiments or extrapolated from known values at other temperatures.Using the thermodamage strength theoretical model expression above, the strength behaviors of material at different temperatures were calculated and analyzed.
Figure 2(a) shows that, when the SiC particle size is not large, no matter what value the crack size is, the strength of material decreases as the temperature increases.The sensitivity of material to the crack size decreases as temperature increases.Figure 2(b) indicates that, when the SiC particle size is 38.5 μm and the value of crack size s is very little, the strength increases firstly and then decreases with the increase of temperature, yet when crack size s reaches a certain value, the strength decreases with the increase of temperature.We can analyze that when the SiC particle size is very large compared to the crack size, the residual stress will decrease with a certain level of increase of temperature, so the stress concentration near the crack tip gets weaker, thus the strength increases.However, when the temperature reaches a certain value, temperature plays a much more important role in controlling the strength of the composite.
As can be seen from Figures 2(a) and 2(b), corresponding to a certain SiC particle size, when the crack size s is less than a certain value and the temperature is in a certain range, the strength decreases firstly and then increases with the increase of the value of crack size s; while the temperature reaches a certain value, the strength decreases as the value of crack size s increases.We can know that when the value of crack size s is in a certain range, the stress intensity factor of crack tip increases firstly and then decreases with the increase of the value of crack size s.Therefore, the stress concentration near the crack tip will get stronger with a little increase of crack size s, so the strength decreases; while with the continued increase of crack size s, the stress concentration near the crack tip will get weaker instead, so the strength will increase; but when the value of crack size s is large enough, as the damage size is also too large, the damage size plays a much more important role in controlling the strength of the composite, thus the strength decreases with the increase of crack size s.
In summary, the strength of material is controlled by many factors, and the effects of factors on the strength are different under different conditions.
As can be seen from Figure 3(a), when the R + s equals to a certain value, the strength increases as the SiC particle size decreases in a certain level.That is because the stress concentration near the crack tip due to the residual stress will get weaker (as shown in Figure 3  studied the effect of SiC particle size on the sintered ZrB 2 -SiC ceramics by experiment and showed that the bigger SiC particle size resulted in greater effect of tensile residual stress in the matrix grains on strength reduction.From this we can also predict that the effect of SiC particle size on the strength is greater than that of crack size on the strength.However, when the SiC particle size reduces to a certain value, the strength will decrease with the decrease of SiC particle size.This is because if the particle size is too small or even approaches zero, the spherical SiC particle with an annular flaw will approach a penny crack.Figures 4(a) and 4(b) show that if the crack size is smaller or not so larger than the SiC particle size, strength will decrease with the increase of particle size, as the stress concentration near the crack tip will get stronger.However, Figure 4(c) indicates that strength increases as the SiC particle size increases.We know through research that when the crack size is relatively larger than the SiC particle size, with decrease of SiC particle size, the stress concentration near crack tip will get stronger, so the crack is more likely to expand.Figures 4(a), 4(b) and 4(c) also reveal that if the magnitude of increase of particle size is the same as that of decrease, the effect of decrease on strength is more significant than that of increase on strength, while the effect is more significant with the bigger magnitude of change.We can see from the above conclusions that the role that SiC particle size controls the strength will be affected by the nearby cracks.
In summary of all of the above conclusions, as we cannot avoid the flaws nearby the SiC particles and the residual stress, when we process UHTCs, we should reduce the SiC Advances in Materials Science and Engineering particle size, but the value of which should not be too little, as this can retain the high strength property while the flaws expand.As the certain value of SiC particle size depends on the preparation technology, according to the present study [7,21], the better starting SiC particle size should be in the range 1∼2 μm.
As can be seen from Figures 5(a) and 5(b), the strength decreases as the SiC particle volume fraction increases.The conclusion can be analyzed that the strength property of SiC particle is lower than that of TiC material, and as the SiC particle volume fraction increases, the tensile residual stress field and crack volume fraction will increase too.We can also know from Figures 5(a) and 5(b) that when the SiC particle size is set, the bigger s/R ratio is, the more sensitive the strength to the SiC particle volume fraction.

Conclusions
In this current paper, a thermo-damage strength theoretical model accounting for the effect of residual stress was applied to each temperature phase based on the study of effects of various physical mechanisms on the fracture strength of UHTCs.The effects of SiC particle size, crack size, and SiC particle volume fraction corresponding to different temperatures on the strength were studied in detail.The study showed that the effects of SiC particle size R and crack size s on strength are seriously dependent on the residual stress and s/R ratios; and the particle size, crack size, and their ratios will affect the roles in which the SiC particle volume fraction controls the strength; and when we process UHTCs, we should reduce the appearance of flaws nearby the SiC particles; meanwhile we should reduce the SiC particle size, but the value of which should not be too little, as this can retain the high strength property while the flaws expand; in summary, we should take into account the important roles that the residual stress and the combined effort of particle size and crack size control the strength.

Figure 1 :
Figure 1: Floor plan of the model containing numbers of spherical particles with annular flaws.

TTFigure 3 :
Figure 3: (a) SiC particle size dependence of strength of TiC-20 vol.% SiC at different temperatures (R+s = 17 μm).(b) SiC particle size dependence of the stress intensity factor due to the residual stress at different temperatures (R + s = 17 μm).

Table 1 :
Temperature-dependent material properties of TiC.