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1 Materials 2013, 6, ; doi: /ma Artile OPEN ACCESS materials ISSN Theoretial Researh on Thermal Shok Resistane of Ultra-High Temperature Ceramis Fousing on the Adjustment of Stress Redution Fator Dengjian Li 1, Weiguo Li 1, *, Dingyu Li 1, Yushan Shi 1 and Daining Fang State Key Laboratory of Coal Mine Disaster Dynamis and Control, College of Resoures and Environmental Siene, Chongqing University, Chongqing , China; s: wldj@qu.edu.n (De.L.); lidingyu@qu.edu.n (Di.L.); ysshi@qu.edu.n (Y.S.) State Key Laboratory for Turbulene and Complex Systems, College of Engineering, Peking University, Beijing , China; fangdn@pku.edu.n * Author to whom orrespondene should be addressed; wgli@qu.edu.n; Tel.: ; Fax: Reeived: 27 November 2012; in revised form: 30 January 2013 / Aepted: 31 January 2013 / Published: 18 February 2013 Abstrat: The thermal shok resistane of eramis depends on not only the mehanial and thermal properties of materials, but also the external onstraint and thermal ondition. So, in order to study the atual situation in its servie proess, a temperature-dependent thermal shok resistane model for ultra-high temperature eramis onsidering the effets of the thermal environment and external onstraint was established based on the existing theory. The present work mainly foused on the adjustment of the stress redution fator aording to different thermal shok situations. The influenes of external onstraint on both ritial rupture temperature differene and the seond thermal shok resistane parameter in either ase of rapid heating or ooling onditions had been studied based on this model. The results show the neessity of adjustment of the stress redution fator in different thermal shok situations and the limitations of the appliable range of the seond thermal shok resistane parameter. Furthermore, the model was validated by the finite element method. Keywords: ultra-high temperature eramis; stress redution fator; the seond thermal shok resistane parameter; onstraint

2 Materials 2013, Introdution Ultra-high temperature eramis (UHTCs) are a family of erami-based omposites mainly onsisting of transition metal ompounds, suh as ZrB 2, TaC, HfN and HfB 2, whih have melting points higher than 3,000 C and an be potentially used at temperatures above 2,000 C in an oxidizing environment. As the most promising andidates for high temperature appliations of thermal protetion systems (TPS), UHTCs are attrating more and more attention urrently [1 4]. The thermal shok resistane (TSR) is one of the most important parameters in UHTCs haraterizations, sine it determines their performanes in many appliations. Due to their inherent brittleness and poor TSR performane, atastrophi failure may our under severe thermal shok, whih is one of the most important reasons for erami frature [5]. Therefore, improving the TSR of eramis has been one of the most important foal points in the eramis field. Signifiant progress has been made in the understanding of the thermal shok behavior of erami materials, with great efforts of theories and experiments sine the 1950s [5 9]. Theoretial researh mainly foused on the fators that affet the TSR of eramis by simplifying the models of the thermal stress field and the transient temperature field. Thus, the stress redution fator was introdued in order to simplify the analysis proess [5 7]. At present, the researh of TSR mostly foused on the effets of surfae defets, temperature, indentation rak length [10,11], partile reinfored [12], whisker reinfored [13] or initial stress field [14] on TSR performane to explain the mehanisms of thermal shok failure. However, few experiments have onsidered the influenes of external onstraint onditions, beause they are diffiult to indut. It is known to all that the UHTCs are always a part of the TPS; it must be onstrained by other parts. So, the TSR performane of the UHTCs is sensitive to the onstraint. Thus, the TSR of the material annot be simply onsidered on its own, but needs to take the external onstraint onditions and the thermal environment into full aount. However, in the urrent experiment it is diffiult to simulate the thermal environment and external onstraint onditions suffered by the UHTCs, whih were used as thermal protetion materials. Due to the restritions of urrent experiments, in the present investigation, a TSR model onsidering the effets of the thermal environment and external onstraint had been established. The adjustment of stress redution fator was onsidered and the influenes of external onstraint on both ritial rupture temperature differene and the seond TSR parameter in either ase of rapid heating or ooling onditions had been studied. The present work was limited to establishing the TSR theoretial model and its validation by finite element simulation, whih the experimental validation deferred to for future work. 2. Derivation of the Theoretial Model The geometri model is shown in Figure 1. Assumptions that have been adopted are given below. 1. The model is a two well-bonded plate, whih doesn t onsider the interfae damage. 2. The upper is the UHTC plate, and the lower is the matrix base. These two plates are assumed to have the same plane geometry size for the onveniene of theoretial model derivation.

3 Materials 2013, There is no heat exhange between the UHTC plate and the matrix base, and the temperature of the matrix base is onstant, being equal to the predefined room temperature field of 25 C. 4. The plate is ontinuous, homogenous, isotropi, elasti and submits to small deformation hypothesis. Figure 1. The geometri model. So, the stress field of the erami layer is only the funtion of thikness diretion when it suffers thermal shok. One the thermal stress is greater than the frature strength of the eramis, raks will be generated and result in instant frature [5,7]. Besides, the maximum stress appears at the surfae for most ases, so it is very reasonable to regard the UHTC plate rupture one the thermal stress of the upper surfae aused by thermal shok is greater than the frature strength of the material orresponding to the urrent temperature Heating Thermal Shok Conditions The initial stress field is set up by slow heating from the predefined room temperature of 25 C to the thermal shok initial temperature T uniformly without an internal temperature gradient [14]. Then, the model is subjeted to a heating thermal shok. If the temperature of the erami plate hanges without external restrition, the elongation is L 1. Considering the onstraint of the base plate, the expansion of the erami plate will be restrited, pressure stress σ will our in the erami and the elongation of the plate aused by the pressure stress will be L σ. So, the total elongation of the erami plate L should be the sum of both: σ L= L1 + Lσ = α( T Ti) ( 1 ν) L E (1) where both the length and width of the erami plate are equal to L. α is the thermal expansion oeffiient, and T i is the predefined room temperature of 25 C. Known from the third Newtonian law, a tensile stress will be produed in the base material, the magnitude of whih is also σ. Then, the elongation of the matrix under the tensile will be L. By the assumption that the two plates are well-bonded and not bending, L and L are equal, so σ an be derived as follows:

4 Materials 2013, σ = E EE B α ( T Ti) ( 1 ν ) + E ( 1 ν ) B B (2) When onsidering the effets of temperature on the UHTC s material properties, the formula will be: B ( ) α ( ) ( i) ( 1 ν ) + ( ) ( 1 ν ) EE T T T T σ = E E T B where E B and v B are Young s modulus and the Poisson ratio of the base plate, respetively. E (T) and α(t) are Young s modulus and the thermal expansion oeffiient of the erami material at temperature T, respetively. The relationship between Young s modulus and temperature is assumed to satisfy the following relation [15]: Tm T ( m m ) E= E BTe + B T BT + T BT e (4) where E 0 is Young s modulus at 0 C, T m is the melting point and B 0, B 1, B 2 are material onstants. The temperature dependent strength σ f (T) of the UHTCs is shown in Equation (5) [16]: σ f ( ) 0 ( σ th ) B Tm T 2 1 T T = E ( T ) 1 C T ( ) m 0 p T dt E 0 C ( ) 0 p T dt 0 where σ th is the frature strength at the referene temperature, E 0 is Young s modulus at the referene temperature of the material and E(T) is the temperature-dependent Young s modulus. C p (T) is the speifi heat apaity for onstant pressure, and T m is the melting point of material. Combining the Equation (3) and the seond TSR parameter, R, onsidering only the effet of the thermal environment from referene [17], it is easy to know that the seond TSR parameter, R, onsidering the effets of the thermal environment and the external onstraint, and an be solved out as Equation (6): ( ) α( ) ( i ) ( 1 ) ( ) ( 1 ) 1 2 ( )( 1 ν ) ( ) ( ) EE B T T T T kt+ T R = σ f ( T + T) EB ν + E T ν E T + T α T + T B where σ f (T + T ) and k(t + T ) are the temperature-dependent frature strength and thermal ondutivity at temperature T + T, respetively. Generally, the stress redution fator is used to analyze the thermal stress so as to study the TSR of the materials [5,6,18]. To obtain the ritial rupture temperature differene, the stress redution fator needs to be evaluated using the Biot number and dimensionless time. For an infinite erami plate after a sudden temperature hange T, the stress redution fator of the surfae an be expressed as follows [5,6,18]: σ () t φ = 1 (1 ν) αe T where ϕ is the stress redution fator and σ(t) is the atual thermal stress field of the plate surfae at time t. The expression of the numerator, σ(t), ontains the Biot number, β, and dimensionless time, F 0, (3) (5) (6) (7)

5 Materials 2013, whih are used in the evaluation of stress redution fator [19]. The denominator represents the possible maximum stress when the temperature of the surfae is instantly hanged to the external temperature, while the other regions remain unhanged. The Biot number, β, is defined as follows [18]: hts β = (8) k where h is the thikness of erami plate, t s is the surfae heat transfer oeffiient and k is thermal ondutivity. The dimensionless time is defined as [18,20]: kt F = (9) 0 2 CPρh where C p is the speifi heat apaity for onstant pressure, ρ is density and t is time. Normally, when the temperature hange was relatively slow, Manson found that the value of the stress redution fator is 0.31β [20], and later, this onlusion was widely ited [5 6,18]. So, the ritial rupture temperature differene is as follows [18,20]: R ' T =. (10) 0.31hts However, it an be seen from its definition formula [Equation (7)] that the stress redution fator represents the ratio of the atual thermal stress and the possible maximum stress of the surfae. So, parameter ϕ ranges from 0 to 1. When the value of the Biot number is larger, the temperature hanges infinitely fast, ϕ is equal to 1, and it dereases as the temperature hange slows down. So, depending on different thermal shok proesses, the value of the stress redution fator needs to be adjusted. As it is subjet to the ombined effet of the Biot number and dimensionless time, we an assume that: φ = Cβ (11) where C is a onstant, whih reflets the impat of the Biot number and dimensionless time, orresponding to different thermal shok proesses. Based on Equation (10), the ritial rupture temperature differene T orresponding to R an be alulated using the following equation: R ' T = (12) Chts The ombination of Equations (6) and (12) yields: ( ) α( ) ( i ) ( 1 ) ( ) ( 1 ) ( )( 1 ν ) ( ) ( ) 1 EE B T T T T kt+ T T = σ f ( T + T) Chts EB ν + E T ν E T + T α T + T B (13)

6 Materials 2013, Cooling Thermal Shok Conditions For a ooling proess, we set the temperature of an UHTC plate heating up from the predefined room temperature of 25 C to the thermal shok initial temperature T uniformly [14]. Then, the model is subjeted to a ooling thermal shok The seond TSR parameter, R, an be solved by Equation (14): ' R σ f T T ( ) α( ) ( ) ( 1 ) ( )( 1 ) ( )( 1 ν ) ( ) ( ) EE B T T T T i kt T = ( ) + E ν + E T ν E T T α T T B B (14) The ritial rupture temperature differene, T, an be solved as follows: ( ) α( ) ( ) ( 1 ) ( )( 1 ) ( )( 1 ν ) ( ) ( ) 1 EE T T T T kt T T = σ ( T T ) + Cht E E T E T T T T B i f s B ν + νb α (15) 2.3. Finite Element Model Due to the lak of experimental data, the finite element method was used to validate the theoretial model. The numerial simulation was aomplished by using the software SIMULIA Abaqus Aording to the symmetry of the model, one-fourth of the plate is used in the numerial simulation. The omputational mesh is shown in Figure 2 (length and width are equal to 150 mm; the thikness of matrix base is 50 mm, and the thikness of UHTC plate is 7 mm). The C3D20RT element is used for the UHTC plate and the C3D8T element for the matrix base. The left and lower surfaes are restrited by applying the symmetri onstraint, and the displaement of matrix base in the thikness diretion is zero. Figure 2. The top view, right view and partial enlarged view of the omputational mesh. 3. Results and Disussion The relative parameters were obtained from experiments [3,21] or extrapolated from known values at other temperatures, as shown in Table 1. The Poisson ratio of the matrix base was equal to the UHTC plate, and the surfae heat transfer oeffiient was fixed in the alulation. The thermal shok

7 Materials 2013, behavior of HfB 2 was alulated and analyzed by using the TSR parameter expression, onsidering the effets of the thermal environment and the external onstraint above. Table 1. Temperature-dependent material properties of HfB 2 [3,21]. Material parameter Values and expressions E(T) (GPa) See Equation (4) E 0 (GPa) B 0, B 1, B , 1.9, σ th (MPa) 448 ν 0.12 T m ( C) 3400 C p (T) [J/(kg C)] (T ) (T ) 2 k [W/(m C)] lnt Α ( C 1 ) (2lnT 5) 10 6 If the value of the Biot number and dimensionless time were determined, we ould easily get the value of the stress redution fator [19]. However, when taking into aount the effets of temperature on the material properties, both the Biot number and dimensionless time are funtions of temperature, whih hange ontinuously in the entire thermal shok proess: hts k( T ) t β = ; F = kt ( ) C( T) h (16) 0 2 P ρ So, we an t obtain the aurate value of onstant C during eah speifi proess. However, at the same time, we found that the range of C is extremely small by alulation. For instane, in view of the ooling thermal shok proess of whih the initial thermal shok temperature was 1,300 C, we alulated the value of C in the ondition of T (max), T (min) and T (avg), respetively. As shown in Table 2, its value hanges in the range of ±1%. Aording to Equation (12), the value of the ritial rupture temperature differene T is diretly proportional to 1/C, so only extremely small hanges our in the alulated values of T. So, the temperature dependene of the Biot number and dimensionless time an be negligible in the alulation of the follow-up questions, beause the range of the ritial rupture temperature differene hanges in small sope. In the atual proess of thermal shok, with the inrease of dimensionless time, F 0, stress redution fator φ inreases rapidly and then dereases slowly, orresponding to the determined Biot number, β. Besides, frature ours at a time when φ approahes its maximum [18 20]. Beause the range of F 0 is extremely small in this proess, a urve, whih shows the relationship between φ max and β, is fitted aording to the relationship of the stress redution fator, the Biot number and dimensionless time (Figure 3). Also, as shown in Table 3, the orresponding value of oeffiient C in the ase of different plate thiknesses is onluded and summarized, based on Figure 3.

8 Materials 2013, Table 2. The related material parameters of HfB 2 in the ondition of T (max), T (min) and T (avg), respetively (the ooling rate is assumed to be 200 C s 1, and the initial temperature is 1,300 C). Material parameters Values Values Values T ( C) h (m) t s [W /(m 2 C)] ρ (kg/m 3 ) k [W/(m C)] C p (T) [J/(kg C)] β F C Figure 3. Relationship between the Biot number, β, and the maximum value of stress redution fator, φ max Adjustment of Stress Redution Fator Table 3. Coeffiient C [18,19]. Plate thikness: h(m) C In either ase of heating or ooling onditions, there were two obvious unreasonable aspets of the ritial rupture temperature differene represented by the urve of the unmodified situation in Figures 4 and As the plate thikness inreased, the ritial rupture temperature differene gradually dereased and slowly approahed zero.

9 Materials 2013, There was a big differene between the theoretial value and the numerial simulation value of the ritial rupture temperature differene in the unmodified situation. Figure 4. Relationship between ritial rupture temperature differene, T, and plate thikness, h, inluding modified, unmodified and numerial simulation situations under different initial thermal shok temperature of temperature elevated onditions (FEM represents finite element method). Figure 5. Relationship between ritial rupture temperature differene, T, and plate thikness, h, inluding modified, unmodified and numerial simulation situations under different initial thermal shok temperature of ooling onditions (FEM represents finite element method). However, the adjustment of the stress redution fator led to the optimized situation: 1. The theoretial and simulation results shared the same trend in different thermal shok onditions, and the value range of the modified situation was more similar to the simulation value.

10 Materials 2013, As the plate thikness inreased, the theoretial value of the ritial rupture temperature differene gradually dereased and slowly approahed a onstant (nonzero). Besides, the differene between the theoretial and simulation results also gradually dereased, and the entire ontrol results tended toward onvergene. Generally, for problems under the onditions of onvetion and radiation, the ritial rupture temperature differene is negatively orrelated with the thikness [5,20,22 26]. Thus, it is quite reasonable that the values of the ritial rupture temperature differene derease with the inrease of plate thikness. Normally, when the temperature hange was relatively slow, Manson found that the value of the stress redution fator is 0.31β [20]. However, it is not appliable to all ases. It would bring large deviations and errors when not used in aordane with the thermal environment. So, it is absolutely neessary to adjust the stress redution fator based on the different thermal shok situations. Nevertheless, further experimental validation is also needed in the near future Limitations of the Appliable Range of the Seond TSR Parameter In the ase of ooling onditions, Figure 5 shows that the ritial rupture temperature differene dereased gradually and slowly approahed a onstant as the plate thikness inreased, while the seond TSR parameter hanged in a totally opposite way in Figure 6: as the plate thikness inreased, the value of R inreased slightly and, finally, slowly approahed a onstant. Figure 6. Relationship between the seond thermal shok resistane (TSR) parameter, R, and plate thikness, h, under different initial thermal shok temperature of ooling onditions. In the ase of heating onditions, Figures 7 and 8 show the same onlusion: the seond TSR parameter and ritial rupture temperature differene hanged in totally different ways. Besides, the theoretial results agreed well with the numerial simulation ones in Figures 4, 5 and 7.

11 Materials 2013, Figure 7. Relationship between ritial rupture temperature differene, T, and plate thikness, h, inluding modified and numerial simulation situations under different initial thermal shok temperature of temperature elevated onditions. Figure 8. Relationship between the seond TSR parameter, R, and plate thikness, h, under different initial thermal shok temperature of temperature elevated onditions. Determined from the definition of the seond TSR parameter, R reflets the diffiulty of the material damage of TSR. The greater the R, the more diffiult it is to initiate raking and the better the TSR is. Moreover, R an desribe the atual thermal shok proess better, and thus, it is superior to the first TSR parameter [18]. However, in this model, the seond TSR parameter ouldn t reflet the diffiulty of the material damage of TSR fatually and even showed the opposite hange state in either ase of heating or ooling onditions. Thus, there were limitations to the appliable range of the seond TSR parameter, and it was unreasonable when R was used blindly to reflet all the situations of the diffiulty of the

12 Materials 2013, material damage of TSR. Certainly, it is quite neessary to verify these onlusions by experimental researh in the near future A Danger Region of Thermal Shok Initial Temperature As it an be seen from Figures 4 and 7, the 100 C heating ondition has a similar level of the ritial rupture temperature differene as those for the 1,600 C heating ondition, while the 800 C heating ondition has a muh lower ritial rupture temperature differene. This is beause a danger thermal shok initial temperature region exists [14,17] when the ritial rupture temperature differene is used to alulate the TSR of the UHTCs. Also, the phenomenon is aused by the temperature dependene of UHTC s material properties. The data that was alulated after the adjustment of the stress redution fator reflet the existene of the danger region well and, thus, indiate that the thermal shok initial temperature of the erami plate should be as far away as possible from the danger region in the proess of atual servie. 4. Conlusions In this paper, a temperature-dependent TSR model for UHTCs, onsidering the effets of the thermal environment and external onstraints, was established based on the existing theory. The adjustment of the stress redution fator aording to different thermal shok situations was onsidered and mainly studied in this model, and the influenes of external onstraint on both ritial rupture temperature differene and the seond TSR parameter in either ase of rapid heating or ooling onditions had been studied in detail. The results show the neessity of the adjustment of the stress redution fator: it leads to the optimization of the theoretial values in different thermal shok situations ompared with the unmodified ones; it also reflets the existene of the danger region well and, thus, indiates that the thermal shok initial temperature of the erami plate should be as far away as possible from the danger region in the proess of atual servie. There are limitations on the appliable range of the seond TSR parameter, and it is unreasonable when R is used blindly to reflet all the situations of the diffiulty of the material damage of TSR. Aknowledgments The authors are grateful for support from the National Natural Siene Foundation of China under Grant Nos and Referenes 1. Wang, C.R.; Yang, J.M.; Hoffman, W.P. Thermal stability of refratory arbide/boride omposites. Mater. Chem. Phys. 2002, 74, Gash, M.; Ellerby, D.; Irby, E.; Bekman, S.; Gusman, M.; Johnson, S. Proessing properties and ar jet oxidation of hafnium diboride-silion arbide ultra high temperature eramis. J. Mater. Si. 2004, 39,

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