12-0098

Appl. Math. Mech. -Engl. Ed., 33(11), 1351–1360 (2012)DOI 10.1007/s10483-012-1627-xc©Shanghai University and Springer-Verlag

Berlin Heidelberg 2012

Applied Mathematicsand Mechanics(English Edition)

Properties and appropriate conditions of stress reduction factorand thermal shock resistance parameters for ceramics∗

Wei-guo LI (��)1, Tian-bao CHENG (��)1,Ru-bing ZHANG (��)2, Dai-ning FANG (��)2

(1. Department of Engineering Mechanics, College of Resources and Environmental

Science, Chongqing University, Chongqing 400030, P. R. China;

2. State Key Laboratory for Turbulence and Complex Systems, College of

Engineering, Peking University, Beijing 100871, P. R. China)

Abstract Through introducing the analytical solution of the transient heat conduction

problem of the plate with convection into the thermal stress field model of the elastic

plate, the stress reduction factor is presented explicitly in its dimensionless form. A

new stress reduction factor is introduced for the purpose of comparison. The proper-

ties and appropriate conditions of the stress reduction factor, the first and second ther-

mal shock resistance (TSR) parameters for the high and low Biot numbers, respectively,

and the approximation formulas for the intermediate Biot number-interval are discussed.

To investigate the TSR of ceramics more accurately, it is recommended to combine the

heat transfer theory with the theory of thermoelasticity or fracture mechanics or use

a numerical method. The critical rupture temperature difference and the critical rup-

ture dimensionless time can be used to characterize the TSR of ceramics intuitively and

legibly.

Key words stress reduction factor, thermal shock resistance (TSR) parameter, ceram-

ics, Biot number, Fourier number

Chinese Library Classification O343.6

2010 Mathematics Subject Classification 80A17, 00A71, 00A73, 80M10

1 Introduction

Ceramics are renowned for their high melting points, physical and chemical stability, andexcellent resistance to extreme environments including erosion-corrosion. However, due to theirinherent brittleness, the thermal shock resistance (TSR) of ceramics is poor. The thermalshock has been an important factor in catastrophic failures in thermal engineering for a longtime[1–10]. Therefore, the improvement of the TSR of ceramics has been the subject of extensiveresearch in the ceramic industry, and a highly accurate evaluation of the TSR of ceramics isthe foundation of the research.

∗ Received Feb. 29, 2012 / Revised Mar. 28, 2012Project supported by the National Natural Science Foundation of China (Nos. 90916009 and11172336)Corresponding author Dai-ning FANG, Professor, Ph.D., E-mail: [email protected]

1352 Wei-guo LI, Tian-bao CHENG, Ru-bing ZHANG, and Dai-ning FANG

The TSR is the comprehensive effect of mechanical and thermal properties, and it is influ-enced by heat transfer modes, boundary conditions, structural shapes, size parameters, tem-peratures, initial imperfections, external environments, etc.

Since the 1950s, the TSR of ceramics has been studied extensively by both theoretical andexperimental methods[1–6]. Attempts were made to define the factors affecting the TSR ofceramics through simplifying the models of the thermal stress field and the transient temper-ature distribution of the structures mathematically[1–2]. Besides, the stress reduction factor(or the dimensionless stress) was also introduced to simplify the analysis[1–2,5,8]. These stud-ies have divided the TSR of ceramics into two broad theories, i.e., the thermal shock fracture(TSF) theory[1–2] and the thermal shock damage (TSD) theory[3–4]. Numerous TSR param-eters were proposed in this period[1–8]. The TSR of ceramics was mainly evaluated by ex-periments and largely centered on the influence of the crack length and density[11–15] and thevarious additives[16–19]. In addition, the numerical simulation for complex problems was alsoconducted[20–21]. The excellent research has deepened the understanding of the thermal shockbehavior of ceramics. However, the properties and appropriate conditions of the stress reductionfactor and the TSR parameters are urgently needed to be clarified to well and truly characterizethe TSR of ceramics for different application environments.

The purpose of the present work is to clarify the properties and appropriate conditions ofthe stress reduction factor and the TSR parameters. Besides, the thermal shock behavior ofthe ceramic plate is investigated from both the theoretical model and the numerical simulation.The universal calculation and characterization methods for the TSR of ceramics are proposed.The study is useful for the calculation of the thermal stress of the plate subjected to the thermalshock and the characterization of the TSR of ceramics.

2 Temperature distribution and thermal stress of ceramic plate subjectedto thermal shock

We consider a thin rectangular plate of 2h with the xy-plane as its middle plane and z asthe thickness coordinate, as shown in Fig. 1(a) (Fig. 1(b) is its simplified model used in thenumerical simulation), which is initially at a uniform temperature TI. Its upper surface at z=hand lower surface at z=−h are suddenly exposed to the fluid of the temperature T∞ with theconvection coefficient ts at t= 0. Besides, the plate is free, continuous, homogenous, isotropic,and perfectly elastic and submits to the small deformation hypothesis. Because the thicknessis small relative to the lateral dimensions (length and width) of the plate, it is reasonable toassume that the conduction occurs exclusively in the z-direction[22]. The transient temperatureresponse of the plate can be obtained by directly solving the governing differential equation anddefinite conditions of this problem, and the resulting temperature field is as follows[22]:

Fig. 1 Schematics of free plate with its both surfaces being exposed to fluid of temperature T∞ withconvection coefficient ts and its simplified model used in numerical simulation

T (z, t) =( ∞∑

n=1

Cn exp(−λ2nFo) cos

(λnz

h

))(TI − T∞) + T∞, (1)

Properties and appropriate conditions of stress reduction factor and TSR parameters 1353

where Fo is the dimensionless Fourier number and equivalent to the dimensionless time t∗,i.e., Fo≡ ath−2 ≡ t∗, the coefficient Cn= 4(2λn+ sin (2λn))−1sinλn and the discrete valuesof λn (eigenvalues) are the positive roots of the transcendental equation λntanλn=Bi (whereBi ≡ htsk

−1 is known as the Biot number), and a and k are the thermal diffusivity and thethermal conductivity, respectively.

The thermal stress field of the plate with the initial uniform temperature TI and the currenttemperature T that is only the function of z and independent of x and y can be obtained andexpressed as[23]

σx = σy = − Eα(T − TI)1 − ν

+1

2h(1 − ν)

∫ h

−h

Eα(T − TI)dz

+3z

2h3(1 − ν)

∫ h

−h

Eα(T − TI)zdz, (2)

where σx and σy are the normal stresses in the x- and y-directions, respectively, E and νare Young’s modulus and Poisson’s ratio, respectively, and α is the coefficient of the thermalexpansion. In this paper, the plate is cooled symmetrically. Thus, the thermal stress of theplate is symmetrical with respect to the middle plane. Deducting the bending term in Eq. (2),the thermal stress field model can be obtained as[23]

σx = σy = −Eα(T − TI)1 − ν

+1

2h(1 − ν)

∫ h

−h

Eα(T − TI)dz. (3)

Hereto, the TSR of the ceramic plate with convection can be achieved. For the given time,the temperature distribution can be obtained from Eq. (1), and then the thermal stress field ofthe plate can be calculated from Eq. (3). Repeating the preceding steps, the thermal stress fieldat different time can be obtained. This can be conveniently implemented via the programs.

In fact, a more general approach to analyze the thermal stress is to use the stress reductionfactor[2,5,8], especially in the middle and late periods of the 20th century. The thermal stressproblem can be analyzed by consideration of the Biot number and the Fourier number, and thenthe TSR parameters can be defined. For the plate which is initially at the uniform temperatureTI, when its upper and lower surfaces are suddenly exposed to the fluid of the temperature T∞with the convection coefficient ts at t=0, the stress reduction factor can be defined as[2,5,8]

ϕ =1

(1 − ν)−1Eα(TI − T∞)σ(z, t), (4)

where σ(z, t) is the thermal stress field of the plate at the time t. The denominator is the possiblemaximum stress when the surface of the plate is cooled down from the initial temperature TI tothe medium temperature T∞ with the infinite ts. Then, the temperature of the surface initiallyat TI is instantly changed to T∞. However, the temperature of other regions is still maintainedat TI.

For the most cases, the maximum stress appears at the surface and is the tensile stress forcooling. Substituting Eqs. (1) and (3) into Eq. (4), the stress reduction factor of the surface canbe expressed as

ϕ = − 1(1 − ν)−1Eα(T∞ − TI)

(−(1 − ν)−1Eα(T (1) − TI) + (2(1 − ν))−1

∫ 1

−1

Eα(T − TI)dz∗)

=1

(T∞ − TI)

((T (1) − TI) − 1

2

∫ 1

−1

(T − TI)dz∗)


=1 −( ∞∑

n=1

Cn exp(−λ2nFo) cos λn

)

− 12

∫ 1

−1

(1 −

( ∞∑n=1

Cn exp(−λ2nFo) cos(λnz

∗)))

dz∗, (5)

where z∗ ≡ zh−1 is the dimensionless spatial coordinate. Equation (5) is the stress reductionfactor of the surface. It is the function of Bi and Fo, i.e., ϕ=ϕ(Bi, Fo), and is independent ofthe material properties, the medium temperature, the thermal shock initial temperature, andthe plate thickness. It is obtained when the material properties are temperature-independent.We can see that it is the percentage of the possible maximum stress represented by the actualthermal stress of the surface. It is between 0 and 1. The stress reduction factor for different Biand Fo is calculated and shown in Fig. 2. The similar graph can also be found in Refs. [2, 5, 8,24]. In addition, the similar stress reduction factor can be obtained for the middle plane.

Fig. 2 Stress reduction factor ϕ of plate surface with convection versus dimensionless time t∗ fordifferent Biot numbers Bi calculated from Eq. (5) (maximum value of stress reduction factorincreases with increase of Biot number, and minimum dimensionless time is 10−4)

We can see that the stress reduction factor first increases sharply and then reaches a maxi-mum value, and afterwards decreases gently as the dimensionless time increases. According tothe TSF theory[1–2], cracks will yield and result in the instant fracture in the material once thethermal stress caused by the temperature gradient is greater than the fracture strength of thematerial. Thus, the ceramic plate will fracture at some time in the incremental portion of thestress reduction factor. To accurately obtain the critical rupture dimensionless time, numericaliterations are usually needed. There is not a uniform critical rupture value of the stress reduc-tion factor for different Biot numbers and materials. Combining the heat transfer theory withthe theory of thermoelasticity or fracture mechanics is the recommended approach to obtainthe critical rupture temperature difference and the critical rupture dimensionless time.

To better understand the above stress reduction factor, a new stress reduction factor isintroduced. For the plate with convection, the new stress reduction factor can be defined as

ψ =1

(1 − ν)−1Eα(TI − T (h, t))σ(z, t). (6)

It is the proportion of the actual thermal stress field of the plate and the possible maximumstress yielded at the surface under the assumption that the plate is cooled down from theinitial temperature TI to the current temperature of the surface T (h, t) with the infinite ts.Substituting Eqs. (1) and (3) into Eq. (6), the new stress reduction factor of the surface can be


expressed as

ψ = − 1(1 − ν)−1Eα(T (1) − TI)

·(−(1 − ν)−1Eα(T (1) − TI) + (2(1 − ν))−1

∫ 1

−1

Eα(T − TI)dz∗)

=1 − 12(T (1)− TI)

∫ 1

−1

(T − TI)dz∗

=1 − 1

2(1 −

( ∞∑n=1

Cn exp(−λ2nFo) cos λn

))

·∫ 1

−1

(1 −

( ∞∑n=1

Cn exp(−λ2nFo) cos(λnz

∗)))

dz∗. (7)

The new stress reduction factor for different Bi and Fo is also calculated and shown in Fig. 3.Likewise, the similar new stress reduction factor can be obtained for the middle plane. Wecan see that different from the existing stress reduction factor, the new stress reduction factordecreases as the dimensionless time increases.

Fig. 3 New stress reduction factor ψ of plate surface with convection versus dimensionless time t∗

for different Biot numbers Bi calculated from Eq. (7) (new stress reduction factor decreaseswith increase of Biot number)

3 TSR parameters

It is important and useful to define a series of parameters to characterize the TSR of ce-ramics, especially in the earlier stage of the selection of materials[2,5,8]. The problem is thatthe thermal stress is closely related to the details in heat transfer[2,5,8]. In fact, because of thedevelopment of computer programs and finite element methods, it is convenient to investigatethe TSR of ceramics through numerical methods[20–21]. In this section, the definitions, origins,and limitations of the TSR parameters are discussed for correctly characterizing the TSR ofceramics.

The first TSR parameter R is defined for the following case: When the plate is heated up orcooled down rapidly, the surface temperature changes, and the inside temperature is held to be


constant[2,5,8]. According to Eq. (2) or Eq. (3), the maximum stress occurs on the surface, i.e.,

σmax = σsurf = − 11 − ν

Eα(Tsurf − TI). (8)

Let σmax be equal to the fracture strength of materials σth.The maximum allowable temperaturedifference, i.e., the parameter R, can be calculated as

R = ΔTc = abs (Tsurf − TI) =σth(1 − ν)

Eα. (9)

We can see that the condition of Eq. (9) is an ideal one, such as the ceramic plate which isunder the first type thermal boundary condition. Obviously, the parameter R only involves themechanical properties on the effects of the TSR of materials. It is the lower bound value of thecritical rupture temperature difference of the ceramic plate subjected to the thermal shock.

Because of the difficulty to obtain the analytical solution accurately in the middle and lateperiods of the 20th century, a number of authors proposed the approximation formulas for therelationship of the maximum stress and the Biot number[2]. For the relatively low values of Biwhich mainly refer to the gas convection and radiation cooling, the formula ϕ−1

max ≈ a+ bBi−1

was suggested and cited in Ref. [2]. In particular, ϕmax ≈ cBi, and the maximum stress can beexpressed as

σmax = − 11 − ν

Eα(T∞ − TI)chtsk. (10)

Let σmax be equal to the fracture strength of materials σth. The maximum allowable tempera-ture difference can be calculated as follows:

ΔTc = abs(T∞ − TI) =σthk(1 − ν)Eαchts

. (11)

The second TSR parameter R′ and the critical rupture temperature difference ΔTc can be,respectively, defined and expressed as[2,5,8]

R′ =σthk(1 − ν)

Eα, ΔTc =

R′

chts. (12)

We can see that the involved factor in the parameter R′, besides the mechanical properties, isthe thermal conductivity of materials. For the relatively low convective heat transfer coefficientunder the conditions of convection and radiation, Manson[25] found that ϕ−1

max≈ 3.25Bi−1, i.e.,c ≈ 0.31. This case was widely cited later[2,5,8,26].

For the high and low Biot numbers, the TSR is approximately proportional to R and R′,respectively, and no single TSR parameter can adequately characterize a material for variousconditions[2]. In addition, the shape factor should be included for other shapes of structures[2,5].For the intermediate Biot number-interval, the following approximation relation has been widelyused by a number of authors[6,14,25,27–30]:

ΔTc = Rf(Bi) =σth(1 − ν)

Eα

(1.5 +

B

Bi− 0.5 exp

(− C

Bi

)), (13)

where f(Bi) is some function of the Biot number, and B and C are the shape factors dependenton the specimen geometry. B= 3.25, and C = 16 for the plate[25]. Besides, the followingapproximation formula has also been used[18–19,28–29]:

ΔTc = Rf(Bi) =σth(1 − ν)

Eα

(1 +

A

Bi

), (14)


where A is the shape factor dependent on the specimen geometry. A= 3.75 for the plate[28].Equations (13) and (14) are usually be used to fit the quenching-temperature difference re-sulting in strength degradation[6,14,18–19,25,27–28,30]. It is well-known that the convection heattransfer coefficient depends on the conditions in the boundary layer, which are influenced bythe surface geometry, the nature of the fluid motion, and an assortment of fluid thermodynamicand transport properties[22]. It is difficult to accurately measure the convection heat transfercoefficient for ceramics quenched into the water bath[31].

4 Thermal shock behavior of ceramic plate

Take the hafnium diboride (HfB2) ceramics as an example. The thermal shock behavior ofthe ceramic plate with convection is studied from both the theoretical model and the numericalsimulation. The involved material properties in the numerical simulation are the same as thoseused in the theoretical calculation. They are the thermal conductivity k, the specific heat cp,and the density ρ for the transient heat transfer analysis, and the coefficient of the thermalexpansion α, Young’s modulus E, and Poisson’s ratio ν for the thermal stress analysis. Theseproperties and fracture strength of the HfB2 ceramics are shown in Table 1[32–34].

Table 1 Material properties of HfB2 ceramics[32–34]

k/(W·m−1·◦C−1) cp/(J·kg−1·◦C−1) ρ/(g·cm−3) α/(10−6◦C−1) E/GPa ν σth/MPa

104.058 240.828 10.5 6.48 440.733 0.12 448

The theoretical method has been discussed in Section 2. The numerical simulation is accom-plished in the large general-purpose finite element analysis software SIMULIA Abaqus v6.9.1using the sequentially coupled thermal-stress analysis. In order to eliminate the spurious os-cillations, the 8-node forced convection-diffusion (DCC3D8) elements are used in the transientheat transfer analysis. The DCC3D8 elements are the first-order elements, in which the heatcapacity terms are lumped, and they use the trapezoidal rule for the time integration. More-over, because of the stress concentration near the cooled surface in this problem, the 20-nodequadratic brick reduced integration (C3D20R) elements are used in the thermal stress analysis.Thus, the first-order temperature field calculated in the heat transfer analysis is consistent withthe first-order thermal strain field provided by the second-order stress-displacement elements.

In the numerical simulation, the meshes used in the heat transfer analysis are obtainedthrough refining those used in the thermal stress analysis one time. Thus, each corner nodeand midside node of the elements used in the thermal stress analysis can obtain the temperaturesdirectly from the corresponding corner nodes of the elements used in the heat transfer analysis.Due to the stress concentration near the cooled surface, the meshes in this region are refined.To improve the efficiency of calculation significantly, the unequal aspect ratio elements areadopted, and the aspect ratios are less than 100.

The temperature and stresses are symmetrical about not only the x- and y-axes but alsothe middle plane in this problem. Thus, the one-eighth plate can be analyzed only, as shown inFig. 1(b). The upper surface is exposed to the fluid of the temperature T∞ with the convectioncoefficient ts at t= 0, and the middle plane is insulated. The middle plane and two mutuallyperpendicular sides (x= 0 and y=0) are applied to the symmetric constrains. In this section,the temperature of the fluid T∞=20 ◦C, and the initial temperature of the plate before beingsubjected to the thermal shock TI=500 ◦C. Here, the ceramic plate fractures once the normalstress of the upper surface (the stress σx of the node B for the numerical simulation) is greaterthan the fracture strength of the material.

The impacts of the heat transfer condition (the product of the convective heat transfercoefficient and the half thickness of the plate) on the critical rupture temperature differenceand the critical rupture dimensionless time of the HfB2 ceramic plate are calculated and shown


in Fig. 4. The results show that both the critical rupture temperature difference and the criticalrupture dimensionless time first decrease sharply and then become to be gentle as the heattransfer condition increases, i.e., the TSR of ceramics decreases as the heat transfer conditionincreases. The critical rupture temperature difference and the critical rupture dimensionlesstime tend to the parameter R of the materials and zero, respectively, as the heat transfercondition increases. We can see that it is intuitive and perspicuous if both the critical rupturetemperature difference and the critical rupture dimensionless time are used to characterize theTSR of ceramics.

Fig. 4 Critical rupture temperature difference and critical rupture dimensionless time t∗c of HfB2

ceramic plate with convection versus heat transfer condition η3 (T∞= 20 ◦C and TI=500 ◦C)

5 Conclusions

The properties and appropriate conditions of the stress reduction factor, the first and secondTSR parameters for the high and low Biot numbers, respectively, and the approximation for-mulas for the intermediate Biot number-interval are discussed. It is recommended to combinethe heat transfer theory with the theory of thermoelasticity or fracture mechanics or use thefinite element method for more accurately investigating the TSR of ceramics. Taking the HfB2

ceramics as an example, the thermal shock behavior of the ceramic plate with convection isstudied from both the theoretical model and the numerical simulation. The TSR of ceramicsdecreases as the heat transfer condition increases. The critical rupture temperature differenceand the critical rupture dimensionless time can be used to characterize the TSR of ceramicsintuitively and legibly.

References

[1] Cheng, C. M. Resistance to thermal shock. Journal of the American Rocket Society, 21(6), 147–153(1951)

[2] Kingery, W. D. Factors affecting thermal stress resistance of ceramic materials. Journal of theAmerican Ceramic Society, 38(1), 3–15 (1955)

[3] Hasselman, D. P. H. Elastic energy at fracture and surface energy as design criteria for thermalshock. Journal of the American Ceramic Society, 46(11), 535–540 (1963)

[4] Hasselman, D. P. H. Unified theory of thermal shock fracture initiation and crack propagation inbrittle ceramics. Journal of the American Ceramic Society, 52(11), 600–604 (1969)

[5] Kingery, W. D., Bowen, H. K., and Uhlmann, D. R. Introduction to Ceramics, 2nd ed., JohnWiley and Sons, New York (1976)


[6] Lewis, D. Comparison of critical ΔTc values in thermal shock with the R parameter. Journal ofthe American Ceramic Society, 63(11-12), 713–714 (1980)

[7] Wang, H. and Singh, R. N. Thermal shock behaviour of ceramics and ceramic composites. Inter-national Materials Reviews, 39(6), 228–244 (1994)

[8] Green, D. J. An Introduction to the Mechanical Properties of Ceramics, Cambridge UniversityPress, Cambridge (1998)

[9] Fahrenholtz, W. G., Hilmas, G. E., Talmy, I. G., and Zaykoski, J. A. Refractory diborides ofzirconium and hafnium. Journal of the American Ceramic Society, 90(5), 1347–1364 (2007)

[10] Song, F., Meng, S. H., Xu, X. H., and Shao, Y. F. Enhanced thermal shock resistance of ceramicsthrough biomimetically inspired nanofins. Physical Review Letters, 104(12), 125502 (2010)

[11] Ghosh, M. K. and Kanoria, M. Generalized thermoelastic functionally graded spherically isotropicsolid containing a spherical cavity under thermal shock. Applied Mathematics and Mechanics(English Edition), 29(10), 1263–1278 (2008) DOI 10.1007/s10483-008-1002-2

[12] Meng, S. H., Liu, G. Q., and Sun, S. L. Prediction of crack depth during quenching test for anultra high temperature ceramic. Materials and Design, 31(1), 556–559 (2010)

[13] Shao, Y. F., Xu, X. H., Meng, S. H., Bai, G. H., Jiang, C. P., and Song, F. Crack patternsin ceramic plates after quenching. Journal of the American Ceramic Society, 93(10), 3006–3008(2010)

[14] Liang, J., Wang, Y., Fang, G. D., and Han, J. C. Research on thermal shock resistance of ZrB2-SiC-AlN ceramics using an indentation-quench method. Journal of Alloys and Compounds, 493(1-2),695–698 (2010)

[15] Han, J. C. and Wang, B. L. Thermal shock resistance of ceramics with temperature-dependentmaterial properties at elevated temperature. Acta Materialia, 59(4), 1373–1382 (2011)

[16] Levine, S. R., Opila, E. J., Halbig, M. C., Kiser, J. D., Singh, M., and Salem, J. A. Evaluation ofultra-high temperature ceramics for aeropropulsion use. Journal of the European Ceramic Society,22(14-15), 2757–2767 (2002)

[17] Zhang, X. H., Xu, L., Du, S. Y., Han, W. B., Han, J. C., and Liu, C. Y. Thermal shock behavior ofSiC-whisker-reinforced diboride ultrahigh-temperature ceramics. Scripta Materialia, 59(1), 55–58(2008)

[18] Liang, J., Wang, C., Wang, Y., Jing, L., and Luan, X. The influence of surface heat transferconditions on thermal shock behavior of ZrB2-SiC-AlN ceramic composites. Scripta Materialia,61(6), 656–659 (2009)

[19] Zhang, X. H., Wang, Z., Hu, P., Han, W. B., and Hong, C. Q. Mechanical properties and thermalshock resistance of ZrB2-SiC ceramic toughened with graphite flake and SiC whiskers. ScriptaMaterialia, 61(8), 809–812 (2009)

[20] Ozdemir, I., Brekelmans, W. A. M., and Geers, M. G. D. Modeling thermal shock damage inrefractory materials via direct numerical simulation (DNS). Journal of the European CeramicSociety, 30(7), 1585–1597 (2010)

[21] Li, W. G., Cheng, T. B., Li, D. Y., and Fang, D. N. Numerical simulation for thermal shockresistance of ultra-high temperature ceramics considering the effects of initial stress field. Advancesin Materials Science and Engineering, 2011, 757543 (2011)

[22] Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S. Fundamentals of Heat andMass Transfer, 6th ed., John Wiley and Sons, New York, 166–188 (2007)

[23] Timoshenko, S. and Goodier, J. N. Theory of Elasticity, 2nd ed., McGraw-Hill Book Company,New York, 399–404 (1951)

[24] Song, F., Liu, Q. N., Meng, S. H., and Jiang, C. P. A universal Biot number determining thesusceptibility of ceramics to quenching. Europhysics Letters, 87(5), 54001 (2009)

[25] Manson, S. S. Behavior of Materials Under Conditions of Thermal Stress, NACA Technical Note2933, Washington, D. C. (1953)

[26] Li, W. G. and Fang, D. N. Effects of thermal environments on the thermal shock resistance ofultra-high temperature ceramics. Modern Physics Letters B, 22(14), 1375–1380 (2008)

[27] Manson, S. S. Thermal stresses: I. Machine Design, 30, 114–120 (1958)


[28] Becher, P. F., Lewis, D., III, Garman, K. R., and Gonzalez, A. C. Thermal-shock resistance ofceramics-size and geometry-effects in quench tests. American Ceramic Society Bulletin, 59(5),542–545 (1980)

[29] Noda, N., Matsunaga, Y., Tsuji, T., and Nyuko, H. Thermal shock problems of elastic bodieswith a crack. Journal of Thermal Stresses, 12(3), 369–383 (1989)

[30] Collin, M. and Rowcliffe, D. Analysis and prediction of thermal shock in brittle materials. ActaMaterialia, 48(8), 1655–1665 (2000)

[31] Kim, Y., Lee, W. J., and Case, E. D. The measurement of the surface heat transfer coefficientfor ceramics quenched into a water bath. Materials Science and Engineering A, 145(1), L7–L11(1991)

[32] Opeka, M. M., Talmy, I. G., Wuchina, E. J., Zaykoski, J. A., and Causey, S. J. Mechanical,thermal, and oxidation properties of refractory hafnium and zirconium compounds. Journal of theEuropean Ceramic Society, 19(13-14), 2405–2414 (1999)

[33] Wuchina, E., Opeka, M., Causey, S., Buesking, K., Spain, J., Cull, A., Routbort, J., andGuitierrez-Mora, F. Designing for ultrahigh-temperature applications: the mechanical and ther-mal properties of HfB2, HfCx, HfNx, and αHf(N). Journal of Materials Science, 39(19), 5939–5949(2004)

[34] Loehman, R., Corral, E., Dumm, H. P., Kotula, P., and Tandon, R. Ultra High TemperatureCeramics for Hypersonic Vehicle Applications, SAND 2006–2925, Albuquerque, NM, USA (2006)

12-0098

Documents