1 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 vehicles^{[1, 2, 3, 4, 5]}. However,they are often subjected to various types of thermal shock while in service,due to their inherent brittleness^{[6, 7, 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, 7, 8, 25]}, thermal environments^{[6, 7, 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, 7, 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,E_z*,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,respectively. The subscript z* = z'/h 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) = T_I. The current temperature T (z,t) is only the function of z and is independent of x and y. Thus,transverse normal stresses σx and σy 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 ε_x= ε_y and ε_z 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,ε_x = ε_y should be linearly distributed across the thickness and can be expressed as follows^{[6, 9]}:

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 where E,ν,and ε_th 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]: where α is the coefficient of thermal expansion from the reference temperature T₀ (in the present work,T₀ = 20 ℃).

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:

If the elongation of the plane z = z' is constrained,that is,

the resultant couple According to (2),(6) becomes

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,

ε_x = ε_y = β_z.

The thermal stress field can be expressed as

The coefficient β can be calculated from the following equation: If the bending is constrained,that is,the only deformation that the plate can undergo in the plane is a uniform transverse expansion,

ε_x = ε_y = ε.

The thermal stress field can be expressed as

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: If both the elongation and bending are constrained,that is,the deformation that the plate can undergo in the plane is fully constrained,

ε_x = ε_y =0

The thermal stress field can be directly calculated from the following equation:

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) S_z* : 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.

(iv) C_z* : 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]

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 = 489GPa^[4],Tm = 3 245 ℃^[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]: where σ_th⁰ is the fracture strength at room temperature,and cp(T ) is the temperature-dependent specific heat at constant pressure. For the ZrB₂ ceramics,σ_th⁰ = 457MPa^[5].

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:

where T_c 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]: 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 ℃ 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.

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 = 10mm,T_I = 1 000 ℃,T_∞ = 20℃,

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,

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 ℃ for Fig. 4(a)).

The time-dependent thermal stress field of C_z* is nearly equal to that of S_z* because the mechanical boundary condition of S_z* is symmetrical about the x- and y-axes. Thus,C_z* has the almost same TSR as S_z* . The critical failure temperature difference and critical failure time of C_z* 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.