Appl. Math. Mech. -Engl. Ed. DOI 10.1007/s10483-015-1909-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2015

Applied Mathematics and Mechanics (English Edition)

Effects of mechanical boundary conditions on thermal shock resistance of ultra-high temperature ceramics∗ Tianbao CHENG1,2 , Weiguo LI1,2,† , Yushan SHI1,2 , Wei LU1,2 , Daining FANG3 1. Chongqing Key Laboratory of Heterogeneous Material Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400030, China; 2. State Key Laboratory of Coal Mine Disaster Dynamics and Control, College of Resources and Environmental Science, Chongqing University, Chongqing 400030, China; 3. State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China

Abstract The effects of mechanical boundary conditions, often encountered in thermalstructural engineering, on the thermal shock resistance (TSR) of ultra-high temperature ceramics (UHTCs) are studied by investigating the TSR of a UHTC plate with various types of constraints under the first, second, and third type of thermal boundary conditions. The TSR of UHTCs is strongly dependent on the heat transfer modes and severity of the thermal environments. Constraining the displacement of the lower surface in the thickness direction can significantly decrease the TSR of the UHTC plate, which is subject to the thermal shock at the upper surface. In contrast, the TSR of the UHTC plate with simply supported edges or clamped edges around the lower surface is much better. Key words thermal shock resistance (TSR), ultra-high temperature ceramic (UHTC), mechanical boundary condition, temperature-dependent material property, thermal environment Chinese Library Classification O343.6 2010 Mathematics Subject Classification

1

80A17, 00A71, 00A73, 80M10

Introduction

Both reentry and hypersonic vehicles, regardless of their specific designs, require control surfaces with sharp nose cones and sharp leading edges to be maneuverable at hypersonic velocities[1] . Such design features pose a severe challenge to the thermal protection materials and structures of vehicles, especially for the reliability of materials under thermal-mechanical coupling conditions. Available thermal protection materials will not survive such extreme environments[1]. Ultra-high temperature ceramics (UHTCs) are potential candidates for the thermal protection materials of nose cones and wing leading edges of reentry and hypersonic ∗ Received Mar. 26, 2014 / Revised Jul. 28, 2014 Project supported by the National Natural Science Foundation of China (Nos. 11472066 and 11172336), the Chongqing Natural Science Foundation (No. cstc2013jcyjA50018), the Program for New Century Excellent Talents in University (No. ncet-13-0634), and the Fundamental Research Funds for the Central Universities (Nos. CDJZR13240021 and CDJZR14328801) † Corresponding author, E-mail: [email protected]

2

Tianbao CHENG, Weiguo LI, Yushan SHI, Wei LU, and Daining FANG

vehicles[1–5] . However, they are often subjected to various types of thermal shock while in service, due to their inherent brittleness[6–8] . Since the 1950s, the thermal shock resistance (TSR) of ceramics has been studied extensively by both theoretical and experimental methods. These studies divided the TSR of ceramics into two broad theories: the thermal shock fracture theory[9−10] and the thermal shock damage theory[11–12] . Numerous TSR parameters were proposed and compiled in Refs. [10,13–14]. Currently, the TSR of ceramics is mainly evaluated by experimental methods, such as quenchingstrength test[15−18] , arc-discharge technique[19] , moving electron beam heating[20] , lamp radiation heating[21] , hydrogen-oxygen torch[3] , and electric resistance methods[22–23] . Efforts were made to improve the TSR of ceramics from the manufacturing processes[16,18,24]. The theories mainly focus on the influence of the temperature dependence of the material properties[6–8,25] , thermal environments[6–8] , length and density of cracks[26], and non-Fourier effect[27] on the TSR of ceramics. Critical failure temperature difference, critical failure (dimensionless) time, and critical heat transfer condition were introduced to characterize the TSR of ceramics[6–8] . These studies have deepened the understanding of the thermal shock behavior of ceramics. As a part of the thermal protection system (TPS), UHTCs must connect with other parts in some ways. At this point, the TSR of UHTCs is not only the TSR of the material its own, but also significantly affected by the constraint conditions. However, the effects of mechanical boundary conditions that are often encountered in the thermal-structural engineering on the TSR of materials and how to optimize the constraint conditions in terms of TSR are still open questions that need to be addressed. In this work, taking the zirconium diboride (ZrB2 ) ceramics as an example, the effects of mechanical boundary conditions that are often encountered in the thermal-structural engineering on the TSR of UHTCs are studied by investigating the TSR of F, E−1 , S−1 , C−1 , B, FL, and EB under the first, second, and third type thermal boundary condition. F, Ez∗ , Sz∗ , Cz∗ , B, FL, and EB denote the unconstrained (free) plate, plate with the elongation of the plane z = z  being constrained, plate with simply supported edges around the plane z = z  , plate with clamped edges around the plane z = z  , plate with the bending being constrained, plate with fixed lower surface, and plate with both the elongation and bending being constrained, respec tively. The subscript z ∗ = zh is the dimensionless coordinate which indicates the location of the constraint, that is, E−1 , S−1 , and C−1 denote plate with the elongation of the lower surface being constrained, plate with simply supported edges around the lower surface, and plate with clamped edges around the lower surface, respectively. The upper surface of the plate is subject to thermal shock. The involved factors are the first TSR parameter, critical failure temperature difference, critical failure time, surface heat flux, and convective heat transfer coefficient. The study is useful for the design and evaluation of UHTCs in the thermal-structural engineering.

2

Analysis

Consider a thin rectangular plate of thickness 2h, shown in Fig. 1. Cartesian coordinates are embedded at the center of the plate which is initially at a uniform temperature T (z, 0) = TI . The current temperature T (z, t) is only the function of z and is independent of x and y. Thus, transverse normal stresses σ x and σ y are present and equal. Longitudinal normal stress σ z and shear stresses τ yz , τ xz , and τ xy vanish. The nonvanishing strain components are εx = εy and εz which is only the function of z and is independent of x and y. To meet the strain compatibility equations of the theory of elasticity, εx = εy should be linearly distributed across the thickness and can be expressed as follows[6,9] : εx = εy = ε + βz,

(1)

Effects of mechanical boundary conditions on thermal shock resistance

Fig. 1

3

Schematic diagram of thin rectangular plate which is subject to thermal shock at upper surface

where ε and β are two constants for the given temperature distribution. According to the stress-strain relations of the thermoelasticity, the thermal stress field of the plate can then be expressed as σx = σy =

E (ε + βz − εth ), 1−ν

(2)

where E, ν, and εth are Young’s modulus, Poisson’s ratio, and the thermal strain due to free thermal expansion, respectively. They are functions of temperature. Moreover, εth also depends on the initial temperature and reference temperature, and can be generated according to the following formula[28] : εth = α(T )(T − T0 ) − α(TI )(TI − T0 ),

(3)

where α is the coefficient of thermal expansion from the reference temperature T0 (in the present work, T0 = 20 ◦ C). If the plate is not constrained, the resultants and resultant couples of the transverse normal stresses of the four lateral sides of the plate are zero. Then, ε and β can be calculated from the following equations: ⎧  h  h  h E Ez E ⎪ ⎪ ⎪ ε dz + β dz = εth dz, ⎪ ⎨ −h 1 − ν 1 − ν 1 − ν −h −h (4)  h  h  h ⎪ ⎪ Ez Ez 2 Ez ⎪ ⎪ dz + β dz = εth dz. ⎩ε −h 1 − ν −h 1 − ν −h 1 − ν If the elongation of the plane z = z  is constrained, that is, εx = εy = ε + βz  = 0,

(5)

the resultant couple 

h −h

σx abs(z − z  )dz = 0.

(6)

According to (2), (6) becomes  ε

h

−h

E (z − z  ) dz + β 1−ν



h

−h

E z (z − z  ) dz = 1−ν



h

−h

E εth (z − z  ) dz. 1−ν

(7)

The coefficients ε and β can be calculated from (5) and (7). Then, the thermal stress field can be calculated from (2). In particular, for z  = 0, εx = εy = βz.

4

Tianbao CHENG, Weiguo LI, Yushan SHI, Wei LU, and Daining FANG

The thermal stress field can be expressed as σx = σy =

E (βz − εth ). 1−ν

(8)

The coefficient β can be calculated from the following equation: β = h



1

Ez 2 dz −h 1−ν

h

−h

Ez εth dz. 1−ν

(9)

If the bending is constrained, that is, the only deformation that the plate can undergo in the plane is a uniform transverse expansion, εx = εy = ε. The thermal stress field can be expressed as σx = σy =

E (ε − εth ). 1−ν

(10)

The coefficient ε can be calculated from the following equation because the resultants of the transverse normal stresses of the four lateral sides of the plate are zero:  h E 1 εth dz. (11) ε = h E dz −h 1 − ν −h 1−ν

If both the elongation and bending are constrained, that is, the deformation that the plate can undergo in the plane is fully constrained, εx = εy = 0. The thermal stress field can be directly calculated from the following equation: σx = σy = −

E εth . 1−ν

(12)

The transient temperature distribution of the plate under different thermal environments can be conveniently calculated using finite volume method[8] when the temperature dependence of the material properties is included. Then, the thermal stress field of the plate with different mechanical boundary conditions can be calculated by combining the models of thermal stress field presented above. The above thermal stress problem can also be analyzed by means of direct numerical simulation. The temperature and stresses are symmetric about the x- and y-axes so that only one-quarter of the plate can be analyzed. The upper surface (z = h) is subject to thermal shock. The four lateral sides and the lower surface (z = –h) are insulated. The mechanical boundary conditions are described as follows. (i) F: Lateral sides x = 0 and y = 0 are applied to symmetric constraints. The displacement of the common point of the two constrained lateral sides and the lower surface is constrained in the z-direction. (ii) Ez∗ : Besides the mechanical boundary conditions applied in the case of F, the displacements of the plane z = z  in the x- and y-directions are constrained. (iii) Sz∗ : Lateral sides x = 0 and y = 0 are applied to symmetric constraints. The displacements of the edges around the plane z = z  in the x- (for the edge being parallel to the y-axis) (or y-(for the edge being parallel to the x-axis)) and z-directions are constrained.

Effects of mechanical boundary conditions on thermal shock resistance

5

(iv) Cz∗ : Lateral sides x = 0 and y = 0 are applied to symmetric constraints. The displacements of the edges around the plane z = z  in the x-, y-, and z-directions are constrained. (v) B: Lateral sides x = 0 and y = 0 and the lower surface are applied to symmetric constraints. (vi) FL: Lateral sides x = 0 and y = 0 are applied to symmetric constraints. The displacements of the lower surface in the x-, y-, and z-directions are constrained. (vii) EB: Besides the mechanical boundary conditions applied in the case of F, the normal displacements of the other two lateral sides are constrained. In this problem, solid models are used, and thus the rotational degrees of freedom are not meaningful. Other information about the numerical simulation can be found in Ref. [6].

3

Results and discussion

Taking the ZrB2 ceramics as an example, the effects of mechanical boundary conditions on the TSR of the UHTC plate under the first, second, and third type thermal boundary conditions are studied. The UHTC plate fails when the normal stress of the upper surface is greater than or equal to the fracture strength of material at the current temperature. (For simplicity, we have assumed that UHTCs are tension and compression isotropic, which gives a more conservative prediction for the TSR of UHTCs.) The temperature-dependent material properties of the ZrB2 ceramics are shown in Table 1[1,4–5] , where the temperature-dependent Young’s modulus can be expressed as[29]  T m E(T ) = E0 − B0 T exp − T

 T m + B1 (T − B2 Tm + abs(T − B2 Tm )) exp − , T

(13)

where E0 is Young’s modulus at room temperature, Tm is the melting point, and B0 , B1 , and B2 are the material constants. For the ZrB2 ceramics, E0 = 489 GPa[4] , Tm = 3 245 ◦ C[4] , B0 = 2.54, B1 = 1.9, and B2 = 0.363[29]. In addition, the temperature-dependent fracturestrength model used in the calculation is presented as follows[30] : 0 σth (T ) = σth

 T  1  12 1 E(T ) 1 −  Tm cp (T )dT , E0 cp (T )dT 0

(14)

0

0 where σth is the fracture strength at room temperature, and cp (T ) is the temperature-dependent 0 specific heat at constant pressure. For the ZrB2 ceramics, σth = 457 MPa[5].

Table 1 Material property m−1

◦ C−1 )

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

Temperature-dependent material properties of ZrB2 ceramics Value and expression 60.225 – 0.011 28T + 1.244 56 × 10−6 T 2 210.75 ln(T ) – 746.28 6.119 6.52 + 6.811 4 × 10−4 T See (13) 0.15

The TSR of the UHTC plate under the first type thermal boundary condition can be characterized by the first TSR parameter[10,13–14] as follows: R = ΔTc = abs(Tc − TI ),

(15)

6

Tianbao CHENG, Weiguo LI, Yushan SHI, Wei LU, and Daining FANG

where Tc is the critical failure temperature of the surface that is subject to thermal shock and can be calculated from the following equation by numerical iterative computation[7] :

σth (Tc ) =

E(Tc ) (α(TI )(TI − T0 ) − α(Tc )(Tc − T0 )). 1−ν

(16)

Note that the first TSR parameter is independent of the mechanic boundary conditions. The results in Fig. 2 show that the first TSR parameter has a danger temperature range about the thermal shock initial temperature. This is the result of the comprehensive action of mechanical properties varying with temperature. Thus, the thermal down-shock resistance is better than (and is worse than) the thermal up-shock resistance for low temperature (and for high temperature) (before (and after) 800 ◦ C in Fig. 2). Note that this conclusion is drawn based on the assumption that UHTCs are tension and compression isotropic. For ceramics that compressive strength is greater than tensile strength, the thermal up-shock resistance is usually better than the thermal down-shock resistance.

Fig. 2

First TSR parameter of ZrB2 ceramic plate versus thermal shock initial temperature

The critical failure temperature difference and critical failure time of the ZrB2 ceramic plate are calculated with different mechanical boundary conditions under the second type (see Fig. 3) and third type (see Fig. 4) thermal boundary conditions h = 10 mm,

TI = 1 000 ◦C,

T∞ = 20 ◦ C,

where T∞ is the fluid temperature. In this work, the results of F, B, and EB are calculated from the theoretical model, and those of E−1 , S−1 , and FL are calculated from the numerical simulation. It can be seen that the TSR of the plate initially has the following order: F > E−1 > S−1 > B > FL > EB,

Effects of mechanical boundary conditions on thermal shock resistance

Fig. 3

7

(a) Critical failure temperature difference and (b) critical failure time versus surface heat flux

and then the critical failure temperature difference and critical failure time tend to approach the first TSR parameter and zero, respectively, as the surface heat flux (or convective heat transfer coefficient) increases. One can also see that the TSR of ceramics is strongly dependent on the heat transfer modes and severity of the thermal environments. In many engineering applications, constraints are applied to the lower surface. Constraining the displacement of the lower surface in the z-direction can decrease the TSR of ceramics significantly (B and FL in Figs. 3 and 4). However, the TSRs of S−1 and C−1 (discussed later) are much better than those of B and FL. That is, simply supported constraint and clamped constraint can reduce the incidences of failures and extend the operating horizons of UHTCs in the thermal-structural engineering compared with that constraining the displacement of the lower surface in the thickness direction. UHTCs are the potential candidates for the thermal protection materials of nose cones and wing leading edges on both reentry and hypersonic vehicles. These vehicles, regardless of their specific designs, require control surfaces with sharp nose cones and sharp leading edges if they are to be maneuverable at hypersonic velocities[1] . (Such design features could produce more agile vehicles that would open up to a greater range of hypersonic flight paths and reentry trajectories[1].) Thus, UHTCs without warping may be desirable when we are designing the TPS but this can also decrease the TSR of UHTCs significantly. Figures 3(a) and 4(a) also show that the critical failure temperature difference of EB remains approximately constant as the surface heat flux (or convective heat transfer coefficient) increases. This constant is the corresponding R parameter shown in Fig. 2 (102.08˚C for Fig. 3(a) and 96.64 ◦ C for Fig. 4(a)).

8

Tianbao CHENG, Weiguo LI, Yushan SHI, Wei LU, and Daining FANG

Fig. 4

(a) Critical failure temperature difference and (b) critical failure time versus convective heat transfer coefficient

The time-dependent thermal stress field of Cz∗ is nearly equal to that of Sz∗ because the mechanical boundary condition of Sz∗ is symmetrical about the x- and y-axes. Thus, Cz∗ has the almost same TSR as Sz∗ . The critical failure temperature difference and critical failure time of Cz∗ are not shown in Figs. 3 and 4 to make the figures clearly visible. The TSR of E−1 is very close to those of S−1 and C−1 because their time-dependent thermal stresses of the upper surface are very similar although those of other places are apparently different.

4

Conclusions

The effects of mechanical boundary conditions that are often encountered in the thermalstructural engineering on the TSR of UHTCs are studied by investigating the TSR of F, E−1 , S−1 , C−1 , B, FL, and EB under the first, second, and third type thermal boundary condition. The TSR of the UHTC plate initially has the following order: F > E−1 > S−1 ≈ C−1 > B > FL > EB, and then the critical failure temperature difference and critical failure time tend to approach the first TSR parameter and zero, respectively, as the surface heat flux (or convective heat transfer coefficient) increases. The TSR of ceramics is strongly dependent on the heat transfer modes and severity of the thermal environments. Constraining the displacement of the lower surface in the thickness direction can significantly decrease the TSR of the UHTC plate which is subject to thermal shock at the upper surface. In contrast, the TSR of the UHTC plate with simply supported edges or clamped edges around the lower surface is much better.

Effects of mechanical boundary conditions on thermal shock resistance

9

References [1] Loehman, R., Corral, E., Dumm, H. P., Kotula, P., and Tandon, R. Ultra High Temperature Ceramics for Hypersonic Vehicle Applications, Sandia Corporation, Albuquerque, 1–46 (2006) [2] 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 the European Ceramic Society, 19(13-14), 2405–2414 (1999) [3] Levine, S. R., Opila, E. J., Halbig, M. C., Kiser, J. D., Singh, M., and Salem, J. A. Evaluation of ultra-high temperature ceramics for aeropropulsion use. Journal of the European Ceramic Society, 22(14-15), 2757–2767 (2002) [4] Fahrenholtz, W. G., Hilmas, G. E., Talmy, I. G., and Zaykoski, J. A. Refractory diborides of zirconium and hafnium. Journal of the American Ceramic Society, 90(5), 1347–1364 (2007) [5] Guo, S. Q., Kagawa, Y., and Nishimura, T. Mechanical behavior of two-step hot-pressed ZrB2 based composites with ZrSi2 . Journal of the European Ceramic Society, 29(4), 787–794 (2009) [6] Cheng, T. B., Li, W. G., and Fang, D. N. Thermal shock resistance of ultra-high-temperature ceramics under aerodynamic thermal environments. AIAA Journal, 51(4), 840–848 (2013) [7] Cheng, T. B., Li, W. G., Zhang, C. Z., and Fang, D. N. Unified thermal shock resistance of ultrahigh temperature ceramics under different thermal environments. Journal of Thermal Stresses, 37(1), 14–33 (2014) [8] Cheng, T. B., Li, W. G., Lu, W., Shi, Y. S., and Fang, D. N. Thermal shock resistance of ultrahigh-temperature ceramic thermal protection system. Journal of Spacecraft and Rockets, 51(3), 986–990 (2014) [9] Cheng, C. M. Resistance to thermal shock. Journal of the American Rocket Society, 21(6), 147–153 (1951) [10] Kingery, W. D. Factors affecting thermal stress resistance of ceramic materials. Journal of the American Ceramic Society, 38(1), 3–15 (1955) [11] Hasselman, D. P. H. Elastic energy at fracture and surface energy as design criteria for thermal shock. Journal of the American Ceramic Society, 46(11), 535–540 (1963) [12] Hasselman, D. P. H. Unified theory of thermal shock fracture initiation and crack propagation in brittle ceramics. Journal of the American Ceramic Society, 52(11), 600–604 (1969) [13] Wang, H. and Singh, R. N. Thermal shock behaviour of ceramics and ceramic composites. International Materials Reviews, 39(6), 228–244 (1994) [14] Li, W. G., Cheng, T. B., Zhang, R. B., and Fang, D. N. Properties and appropriate conditions of stress reduction factor and thermal shock resistance parameters for ceramics. Applied Mathematics and Mechanics (English Edition), 33(11), 1351–1360 (2012) DOI 10.1007/s10483-012-1627-x [15] Collin, M. and Rowcliffe, D. Analysis and prediction of thermal shock in brittle materials. Acta Materialia, 48(8), 1655–1665 (2000) [16] Zhang, X. H., Xu, L., Du, S. Y., Han, W. B., Han, J. C., and Liu, C. Y. Thermal shock behavior of SiC-whisker-reinforced diboride ultrahigh-temperature ceramics. Scripta Materialia, 59(1), 55–58 (2008) [17] Liang, J., Wang, C., Wang, Y., Jing, L., and Luan, X. The influence of surface heat transfer conditions on thermal shock behavior of ZrB2 -SiC-AlN ceramic composites. Scripta Materialia, 61(6), 656–659 (2009) [18] Song, F., Meng, S. H., Xu, X. H., and Shao, Y. F. Enhanced thermal shock resistance of ceramics through biomimetically inspired nanofins. Physical Review Letters, 104(12), 125502 (2010) [19] Sato, S., Sato, K., Imamura, Y., and Kon, J. I. Determination of the thermal shock resistance of graphite by arc discharge heating. Carbon, 13(4), 309–316 (1975) [20] Schubert, C., Bahr, H. A., and Weiss, H. J. Crack propagation and thermal shock damage in graphite disks heated by moving electron beam. Carbon, 24(1), 21–28 (1986) [21] Schneider, G. A. and Petzow, G. Thermal shock testing of ceramics — a new testing method. Journal of the American Ceramic Society, 74(1), 98–102 (1991)

10

Tianbao CHENG, Weiguo LI, Yushan SHI, Wei LU, and Daining FANG

[22] Meng, S. H., Qi, F., Chen, H. B., Wang, Z., and Bai, G. H. The repeated thermal shock behaviors of a ZrB2 -SiC composite heated by electric resistance method. International Journal of Refractory Metals and Hard Materials, 29(1), 44–48 (2011) [23] Yang, L., Zhou, Y. C., and Lu, C. Damage evolution and rupture time prediction in thermal barrier coatings subjected to cyclic heating and cooling: an acoustic emission method. Acta Materialia, 59(17), 6519–6529 (2011) [24] Zhang, R. B., Chen, G. Q., and Han, W. B. Synthesis, mechanical and physical properties of bulk Zr2 Al4 C5 ceramic. Materials Chemistry and Physics, 119(1-2), 261–265 (2010) [25] Han, J. C. and Wang, B. L. Thermal shock resistance of ceramics with temperature-dependent material properties at elevated temperature. Acta Materialia, 59(4), 1373–1382 (2011) [26] Han, J. C. and Wang, B. L. Thermal shock resistance enhancement of functionally graded materials by multiple cracking. Acta Materialia, 54(4), 963–973 (2006) [27] Chang, D. M. and Wang, B. L. Transient thermal fracture and crack growth behavior in brittle media based on non-Fourier heat conduction. Engineering Fracture Mechanics, 94, 29–36 (2012) [28] Dassault Systemes Simulia Corporation. Heat transfer and thermal-stress analysis. Abaqus Analysis User’s Manual, Dassault Systemes Simulia Corporation, Providence (2012) [29] Li, W. G., Wang, R. Z., Li, D. Y., and Fang, D. N. A model of temperature-dependent Young’s modulus for ultrahigh temperature ceramics. Physics Research International, 2011, 791545 (2011) [30] Li, W. G., Yang, F., and Fang, D. N. The temperature-dependent fracture strength model for ultra-high temperature ceramics. Acta Mechanica Sinica, 26(2), 235–239 (2010)