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Thermal shock residual strength of functionally graded ceramics
Materials Science and Engineering A 435–436 (2006) 71–77
Thermal shock residual strength of functionally graded ceramics Z.-H. Jin ∗ , W.J. Luo Department of Mechanical Engineering, University of Maine, 5711 Boardman Hall, Orono, ME 04469, USA Received 8 June 2006; accepted 6 July 2006
Abstract When subjected to severe thermal shocks functionally graded ceramics (FGCs) suffer strength degradation due to microcrack growth, coalescence and macrocrack propagation induced by the shocks. This paper presents a thermo-fracture mechanics model to study the residual strength behavior of thermally shocked FGCs. It is assumed that thermal shock damage mainly results from the extension of a pre-existing edge crack initially located at the thermally shocked surface of an FGC specimen. Thermal shock threshold, or the critical thermal shock that causes precipitous strength drop, is determined. The paper describes applications of the model to calculate the thermal shock residual strength of FGCs as a function of thermal shock severity. Two FGC systems, i.e., Al2 O3 /Si3 N4 and TiC/SiC FGCs are considered in the numerical studies. The numerical results show that material gradation profile has a pronounced effect on the thermal shock threshold and the thermal shock residual strength of FGCs. © 2006 Elsevier B.V. All rights reserved. Keywords: Functionally graded materials; Ceramics; Fracture; Thermal shock; Residual strength
1. Introduction Ceramics represent one of the most promising materials in the future turbine engines, internal combustion engines, metal cutting, and other high temperature engineering applications because of their excellent properties at high temperatures and superior corrosion and wear resistance. One major limitation for the application of ceramic materials is their inherent brittleness which can result in catastrophic failure under severe thermal shocks. To overcome this disadvantage, considerable efforts have been made to toughen ceramics with some success. Alternatively, one may specifically design a ceramic or ceramic–ceramic composite to reduce the thermal stresses when subjected to thermal shocks thereby enhancing the thermal shock fracture resistance. This is one of the objectives to be fulfilled by the concept of functionally graded ceramics (FGCs) [1–3]. An FGC is a ceramic or ceramic–ceramic composite with graded microstructure and material properties. FGCs distinguish themselves with conventional, macroscopically homogeneous ceramic–ceramic composites by the gradual change in the microstructure and properties over the thickness of the material. Knowledge of the thermal shock resistance behavior of FGCs is critical for the material design of FGCs in high temperature
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applications. While there has been some progress in the thermal fracture of FGCs, the existing studies have been mainly concerned with the computation of thermal stress intensity factors (TSIFs) and the material property gradation effects on TSIFs. For example, under steady-state thermal loading conditions, Noda and Jin [4] studied a crack parallel to the boundaries of an infinite functionally graded material (FGM) strip. Tanigawa et al. [5] considered a penny-shaped crack problem in a nonhomogeneous material. Erdogan and Wu [6] investigated cracks perpendicular to the boundaries of an infinite FGM strip. Bleeck et al. [7] studied the effect of a graded layer on the TSIF in a joint. Nemat-Alla and Noda [8] analyzed an edge crack problem in a semi-infinite FGM plate with a bidirectional gradation in the coefficient of thermal expansion. Under transient thermal loading conditions, Jin and Noda [9] studied a crack parallel to the boundary of a nonhomogeneous half-plane. Jin and Batra [10] investigated an edge crack in an FGM strip. Choi et al. [11] investigated collinear cracks in a layered half-plane with a graded interfacial zone. Wang et al. [12] studied thermal crack problems using a laminated material model. Jin [13] considered surface cracking in an FGC coating. Other studies include experimental and computational investigations of thermal fracture of FGM coatings (for example, Balke et al. [14] and Kokini et al. [15]), evaluation of maximum thermal shock at material failure [16] and crack growth in FGM plates by a finite element method [17]. Tanigawa [18], Noda [19] and Jin [20] have written extensive reviews on thermal fracture in FGMs.
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Fig. 1. Thermal shock residual strength behavior of ceramics.
For crack growth under thermal shock conditions, the crack propagation may be arrested depending on the severity of thermal shock, thermal stress field characteristics and material properties. If one measure the strength of a thermally shocked specimen made of a ceramic, the material generally exhibits the behavior as shown in Fig. 1 [21]. The strength remains unchanged when the thermal shock T is less than a critical value, Tc , called the thermal shock threshold. At T = Tc , the strength σ R suffers a precipitous drop and then decreases gradually with increasing severity of thermal shock. Both material properties and specimen size influence the residual strength behavior of thermally shocked materials. It is clear that the thermal shock residual strength characterizes the thermal shock fracture resistance of ceramics. Moreover, the residual strength approach has some advantages over the TSIF approach because it characterizes the overall thermal shock resistance. While there have been a few experimental studies on the thermal shock residual strength behavior of FGCs [22], predictive models for FGCs have not been developed. In this paper, the thermal shock residual strength behavior of FGCs is studied. Section 2 introduces the thermofracture mechanics of FGCs including temperature fields, thermal stresses and TSIF formulations. Section 3 describes the modeling approach to evaluate the thermal shock threshold and thermal shock damage in FGCs. In Section 4, the residual strength of thermally shocked FGCs is determined using a fracture mechanics approach. Section 5 presents and discusses some numerical results of thermal shock threshold and residual strength for FGC specimens made of Al2 O3 /Si3 N4 and TiC/SiC FGC systems. Finally, Section 6 provides some concluding remarks.
microcracks will initiate and grow to form macrocracks, which results in strength degradation. Surface cracks are the dominant defects affecting the thermal shock behavior of FGCs because tensile stresses develop at the surfaces of an FGC when quenched. In their study of thermal shock damage of monolithic ceramics, Bahr and Weiss [23] showed that strength degradation of monolithic ceramics is initially dominated by one single crack and multiple surface crack propagation play a role under further severe thermal shock conditions. In this study, we only consider the strength degradation due to propagation of a single crack, which has proven reasonable in evaluating thermal shock residual strength of monolithic ceramics [24,25]. Effects of multiple crack propagation on strength degradation will be considered in the future study. Specifically, we consider a long FGC strip with an edge crack shown in Fig. 2, where b is the thickness of the strip and a0 is the crack length. The thermal properties of the FGC are arbitrarily graded in the thickness direction (x-direction). The strip is initially at a constant temperature T0 , and its surfaces x = 0 and x = b are suddenly cooled to temperatures Ta and Tb , respectively. This represents an idealized thermal shock loading case, i.e., the heat transfer coefficients on the surfaces of the FGC strip are infinitely large which corresponds to the most severe thermal stress induced in the strip. In other words, the thermal shock residual strength predicted by the current model would be lower than that using a finite heat transfer coefficient. 2.1. Temperature field For the thermal shock problem shown in Fig. 2, thermal stresses in the FGC specimen result from the transient tempera-
2. Thermo-fracture mechanics Microcracks inherently exist in an FGC. When the FGC is subjected to severe thermal shocks, some of the pre-existing
Fig. 2. An edge cracked FGC strip subjected to a thermal shock.
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ture gradients induced by the thermal shock. Jin [26] obtained a closed-form, short time asymptotic solution (τ → 0) of temperature field in an FGC strip with arbitrary spatial distributions of thermal properties using the Laplace transform and its asymptotic properties. The asymptotic temperature field, T(x, τ), has the following form T (x, τ) − T0 T0 − T a x ρ(0)c(0)k(0) 1/4 1 κ(0) √ =− erfc dx ρ(x)c(x)k(x) κ(x) 2b τ 0 ρ(b)c(b)k(b) 1/4 T 0 − Tb T0 − Ta ρ(x)c(x)k(x) b 1 κ(0) √ × erfc dx , κ(x) 2b τ x
where θ(x, τ) = T(x, τ) − T0 , E Young’s modulus, ν Poisson’s ratio, α = α(x) the coefficient of thermal expansion and Aij (i, j = 1, 2) and A0 constants, which can be found in Ref. [20]. Here we assume that the FGC strip undergoes plane strain deformations in the x–y plane and the strip is free from constraints at the far ends. 2.3. Thermal stress intensity factor (TSIF) The thermal fracture problem of the edge cracked FGC strip may be solved by various analytical and numerical methods. Here we adopt a singular integral equation approach and the integral equation has the form 1 1 ψ(s, τ) + K(r, s) √ ds 1−s −1 s − r
−
(1)
where τ = tκ(0)/b2 denotes the nondimensional time, t time, ρ(x) the mass density, c(x) the specific heat, k(x) the thermal conductivity, κ(x) = k/ρc the thermal diffusivity, ρ(0), c(0), k(0) and κ(0) the values of ρ(x), c(x), k(x) and κ(x) at x = 0, respectively, ρ(b), c(b), k(b) and κ(b) the values at x = b, respectively, and erfc() the complementary error function. The asymptotic solution in Eq. (1) holds for an FGC strip with continuous and piecewise differentiable thermal properties. The significance of the solution lies in the fact that thermal stress and thermal stress intensity factor (TSIF) in the FGC strip induced by the thermal shock reach their peak values in a very short time. Thus, Eq. (1) may be used to evaluate the peak values of thermal stress and TSIF which govern the thermal stress failure of the material. 2.2. Thermal stress
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2π(1 − ν2 ) T σyy (r, τ), |r| ≤ 1 (3) E where K(r, s) is a known kernel, r = 2x/a − 1, a the current crack length, ψ a continuous function related to the dislocation density along the crack surface: =−
ψ(r, τ) =
√ 1 − rϕ(r, τ),
ϕ=
∂ [ν(x, 0+ ) − ν(x, 0− )] ∂x (4)
T the with ν(x, y) being the displacement in the y-direction, σyy thermal stress induced in the FGC strip free of cracks given in Eq. (2). Once the solution of the above integral equation is obtained, the TSIF at the edge crack tip can be computed from
(1 − ν) (1 − ν)KI √ √ = Eα0 T πb Eα0 T πb
1 a 2π(x − a)σyy (x, 0, τ) = − limx→a+ ψ(r, τ)|r=1 , 2 b (5)
K∗ =
The transient temperature field (1) induces thermal stresses in the edge cracked FGC strip shown in Fig. 2. This work focuses where KI denotes the TSIF, K* the nondimensional TSIF, σ yy the on the effect of thermal property gradients on the thermal shock normal stress, T = T0 − Ta , and α0 = α(x)|x=0 . In Eq. (5), ψ(r, strength behavior of FGCs. We thus consider a special kind τ)|r=1 depends on the nondimensional crack length a/b. of FGCs with constant Young’s modulus and Poisson’s ratio. While this assumption imposes limitations on the application of 3. Thermal shock threshold and thermal shock damage the present model, there exist some FGC systems for which the Young’s modulus remains approximately constant. Examples 3.1. Thermal shock threshold include TiC/SiC, MoSi2 /Al2 O3 , and Al2 O3 /Si3 N4 FGC systems. The Young’s modulus of each of these FGCs may not change Eq. (5) indicates that the TSIF is proportional to the thermal significantly because the constituents have similar Young’s modshock T. Crack extension will not occur if the thermal shock uli. T has not reached a critical value Tc , the so-called therThe temperature filed, Eq. (1), induces thermal stresses in mal shock threshold, at which the peak value of TSIF reaches both longitudinal (perpendicular to the crack direction) and the fracture toughness of the material. At T = Tc , the peak transverse (perpendicular to the x–y plane) directions. However, value of TSIF reaches the fracture toughness and crack extension only the longitudinal stress determines crack growth. The longitudinal stress can be obtained as follows
b b Eαθ(x, τ) Eαθ(x, τ) Eαθ(x, τ) E T σyy (x, τ) = − (A22 − xA21 ) + dx − (A12 − xA11 ) x dx , (2) 1−ν (1 − ν2 )A0 1−ν 1−ν 0 0
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is initiated. When T > Tc , the crack will grow thereby resulting in severe damage in the FGC. The thermal shock threshold Tc can be determined by equating the peak TSIF to the material fracture toughness, i.e., Max{τ>0} {KI (τ, a0 , Tc } = KIc (a0 )
(6)
where KIc (a0 ) denotes the fracture toughness of the FGC at x = a0 , and a0 denotes the initial pre-existing crack length. Substitution of Eq. (5) into Eq. (6) yields Tc : √ (1 − ν)KIc (a0 )/(Eα0 πb) √ Tc = (7) Max{τ>0} {−(1/2) a0 /b ψ(1, τ)} Fracture toughness of an FGC generally depends on spatial position. Determination of fracture toughness of FGCs remains a challenging task. Jin and Batra [27] proposed two schemes to evaluate the fracture toughness of ceramic–metal FGMs, i.e., the simple rule of mixtures scheme and the crack bridging method. While the rule of mixtures significantly overestimates the fracture toughness of ceramic–metal FGMs, it may be employed to approximately determine the fracture toughness of an FGC when the constituents of the FGC have similar fracture toughness. The rule of mixture formula presented by Jin and Batra [27] for a thermally nonhomogeneous but elastically homogeneous two-phase FGC with the properties graded in x-direction has the following form 2 1/2
2
1 2 KIc (x) = {V1 (x)(KIc ) + V2 (x)(KIc ) }
(8)
where Vi (x) (i = 1, 2) denote the volume fractions of phases 1 1 and K 2 the fracture toughness of and 2, respectively, and KIc Ic the two phases, respectively. Here the crack surface bridging by unbroken grains is not considered. Substituting Eq. (8) into Eq. (7) leads to Tc =
1 )2 (1 − ν){V1 (a0 )(KIc
1/2 2 )2 } + V2 (a0 )(KIc
a = af − a0
(12)
4. Residual strength Once the damage length a is determined for a thermal shock T, the residual strength of thermally shocked FGC may be theoretically evaluated by subjecting the thermally shocked specimen with a crack length af to bending and considering subsequent crack extension. The singular integral equation method can still be used and the equation has the form 1 1 ψ(s) ds + K(r, s) √ 1−s −1 s − r 2π(1 − ν2 ) af =− (13) 1 − (1 + r) σb , |r| ≤ 1 E b where σ b is the bending strength corresponding to the crack length af . The residual strength is thus determined as the maximum strength during the subsequent crack growth. To determine σ b , the stress intensity factor is first evaluated as follows √ 1 KI = σb π af − ψ(r)|r=1 (14) 2 σ b thus can be evaluated by equating KI to the fracture toughness with the result 2
σb =
2 1/2
1 ) + V (a )(K 2 ) } {V1 (af )(KIc 2 f Ic √ π af {−(1/2)ψ(r)|r=1 }
(15)
5. Numerical results and discussion
√
/(Eα0 πb) √ Max{τ>0} {−(1/2) a0 /bψ(1, τ)} (9)
3.2. Thermal shock damage When T > Tc , the crack will grow in the FGC. The crack extension may be arrested depending on a number of factors such as thermal shock severity, thermal stress field and TSIF characteristics, material properties and specimen geometry. If the crack growth is arrested, the final crack length af can be determined by the following equation Max{τ>0} {KI (τ, af , T } = KIc (af )
The damage length caused by the thermal shock is thus determined as
(10)
Substituting Eqs. (5) and (8) into the above equation yields the relationship between the final crack length af and the thermal shock T: √ 1/2 1 )2 + V (a )(K 2 )2 } (1 − ν){V1 (af )(KIc /(Eα0 πb) 2 f Ic √ T = Max{τ>0} {−(1/2) af /bψ(1, τ)} (11)
In the numerical calculations, we first consider the alumina/silicon nitride (Al2 O3 /Si3 N4 ) FGC system. Al2 O3 coated Si3 N4 cutting tools for machining steels have been developed [28] to take advantages of the high temperature deformation resistance of Si3 N4 and to minimize chemical reactions of Si3 N4 with steels by the Al2 O3 coating layer. The conventional Al2 O3 /Si3 N4 tools may be prone to cracking due to the mismatch in thermal expansion at the interface between the coating and the substrate. This proneness to cracking may be alleviated by replacing the Al2 O3 -coating/Si3 N4 -substrate system with an Al2 O3 /Si3 N4 FGC system. The second example concerns a titanium carbide/silicon carbide (TiC/SiC) FGC system. TiC has been used as a cutting tool material for its excellent wear resistance and SiC has also been considered as reinforcements for cutting tools. The TiC/SiC FGC thus is a candidate material for cutting tools applications. The FGC systems are treated as two-phase composites with graded volume fractions of their constituents. Reiter and Dvorak [29] argued that the micromechanics models for conventional composite materials may be employed for FGCs with reasonable accuracy. Thus the thermal conductivity, coefficient of thermal expansion (CTE), specific heat and mass density of the FGC are
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Table 1 Material properties of Al2 O3 , Si3 N4 , TiC and SiC
Al2 O3 Si3 N4 TiC SiC
Young’s modulus (GPa)
Poisson’s ratio
CTE (×10−6 K−1 )
Thermal conductivity (W/m K)
Mass density (g/cm3 )
Specific heat (J/g K)
Fracture toughness (MPa m1/2 )
320 320 450 450
0.25 0.25 0.2 0.2
8.0 3.0 7.0 4.0
20 35 20 60
3.8 3.2 4.9 3.2
0.9 0.7 0.7 1.0
4 5 5 5
calculated from the conventional micromechanics models [30]. The thermal conductivity, k(x), is given by V (x)(ki − k0 ) k(x) = k0 1 + , (16) k0 + (ki − k0 )[1 − V (x)]/3 where subscript ‘0’ represents the properties of the ceramic phase that is exposed to the thermal shock, subscript ‘i’ stands for properties of the other phase, and V(x) denotes the volume fraction of phase i. For the Al2 O3 /Si3 N4 FGC, the thermally shocked surface is pure Al2 O3 . The thermally shocked surface consists of pure TiC for the TiC/SiC system. The Voigt rule of mixtures gives the mass density, ρ(x), and the coefficient of thermal expansion, α(x), because the two constituents have identical elastic bulk moduli. The specific heat, c(x), of the FGC is assumed to follow the Voigt rule of mixtures, i.e., c(x) = V (x)ci + [1 − V (x)]c0
(17)
Fig. 3. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.01).
Table 1 lists the properties of the constituent materials, i.e., Al2 O3 , Si3 N4 , TiC and SiC. The Al2 O3 considered is not fully dense and has a modulus matching that of Si3 N4 . This study assumes that the volume fraction of phase i follows a simple power function: x p V (x) = , (18) b
higher overall fraction of Si3 N4 ). The FGC should transition smoothly and rapidly from pure Al2 O3 at the thermally shocked surface to a Si3 N4 -rich structure to achieve optimal thermal shock resistance. Fig. 4 highlights the residual strength results for T > Tc . Figs. 5 and 6 show the residual strength of an Al2 O3 /Si3 N4 FGC specimen with a smaller initial crack length of a0 = 0.005b. The specimen size is still assumed as b = 5 mm. Similar residual strength behavior to that observed for the FGC specimen with a0 = 0.01b prevails. While both the thermal shock threshold Tc and the strength for T < Tc become higher when
where p is the power exponent. In the numerical calculations, we only consider the loading case of Tb = T0 , which means that only the cracked surface x = 0 of the FGC system is subjected to a temperature drop. Fig. 3 shows the residual strength of the Al2 O3 /Si3 N4 FGC versus the thermal shock T for various values of the exponent p. The specimen size is assumed as b = 5 mm. The pre-existing surface crack has a length of a0 = 0.01b. Several interesting observations can be made. First, the model qualitatively predicts the thermal shock residual strength behavior of FGCs observed in experiment [22], i.e., for a specific value of exponent p, the residual strength remains constant when the thermal shock has not reached the thermal shock threshold Tc . At T = Tc , the strength suffers a precipitous drop and then decreases gradually with increasing severity of thermal shock. Second, Tc increases with a decrease in the exponent p. For example, Tc is about 110 ◦ C for p = 1.0 and increases to about 156 ◦ C when p = 0.2. Finally, the residual strength also increases with decreasing p. At T = 200 ◦ C, for example, the residual strength is about 54 MPa for p = 1.0 and is enhanced to about 79 MPa for p = 0.2. These results imply that the bulk of the Al2 O3 /Si3 N4 FGC should be Si3 N4 (a smaller value of p corresponds to a
Fig. 4. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.01) (highlighted results around T = Tc ).
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Fig. 5. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.005).
compared with the case of a0 = 0.01b as expected, the strength suffers more severe degradation. For example, the strength for p = 0.2 decreases from about 442 to about 77 MPa when T crosses the threshold Tc , an 83% drop. This value for the case of a0 = 0.01b is about 72% (the strength decreases from 318 to 90 MPa). Fig. 7 shows the residual strength of the TiC/SiC FGC versus the thermal shock T for various values of the exponent p. The specimen size is still assumed as b = 5 mm and the pre-existing surface crack has a length of a0 = 0.005b. The thermally shocked surface consists of pure TiC. It is seen from the figure that the material gradation profile (described by p) does not influence the strength for T < Tc . This is because the fracture toughness of the TiC equals that of the SiC as assumed. It also appears that the thermal shock threshold and the residual strength can be enhanced with decreasing value of p. Again, these results imply that the bulk of the TiC/SiC FGC should be SiC (a smaller value of p corresponds to a higher overall fraction of SiC). The FGC should be made such that it transitions smoothly and rapidly
Fig. 7. Thermal shock residual strength of TiC/SiC (b = 5 mm, a0 /b = 0.005).
Fig. 8. Thermal shock residual strength of TiC/SiC (b = 5 mm, a0 /b = 0.005) (highlighted results around T = Tc ).
from pure TiC at the thermally shocked surface to a SiC-rich structure to achieve optimal thermal shock resistance. Fig. 8 highlights the residual strength results for T > Tc . 6. Concluding remarks
Fig. 6. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.005) (highlighted results around T = Tc ).
A thermo-fracture mechanics model is developed to predict the residual strength of thermally shocked FGCs. The model is capable of predicting qualitatively the thermal shock residual strength behavior of FGCs observed in experiments, i.e., the residual strength remains constant when the thermal shock T has not reached the thermal shock threshold Tc . At T = Tc , the strength suffers a precipitous drop and then decreases gradually with increasing severity of thermal shock. The material gradation profile has a pronounced effect on the thermal shock resistance behavior of FGCs. The application of the model to the Al2 O3 /Si3 N4 FGC implies that the bulk of the Al2 O3 /Si3 N4 FGC should be Si3 N4 . The FGC should transition smoothly and rapidly from pure Al2 O3 at the thermally shocked surface to a Si3 N4 -rich structure to achieve optimal thermal
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shock resistance. Similar remarks apply to the TiC/SiC FGC system for which the TiC mainly functions as a cutting surface material and a graduate, rapid transition to the SiC provides enhanced thermal shock resistance. Acknowledgment This work is supported by the University of Maine. References [1] T. Hirai, in: R.J. Brook (Ed.), Materials Science and Technology, vol. 17B: Processing of Ceramics, Part 2, VCH Verlagsgesellschaft mbH, Weinheim, Germany, 1996, pp. 292–341. [2] M. Koizumi, Compos. Part B: Eng. 28 (1997) 1–4. [3] S. Suresh, A. Mortensen, Fundamentals of Functionally Graded Materials, The Institute of Materials, IOM Communications Ltd., London, 1998. [4] N. Noda, Z.-H. Jin, Int. J. Solids Struct. 30 (1993) 1039–1056. [5] Y. Tanigawa, T. Muraki, R. Kawamura, JSME Int. J., Ser. A 39 (1996) 540–547. [6] F. Erdogan, B.H. Wu, J. Therm. Stress. 19 (1997) 237–265. [7] O. Bleeck, D. Munz, W. Schaller, Y.Y. Yang, Eng. Fract. Mech. 60 (1998) 615–623. [8] M. Nemat-Alla, N. Noda, Acta Mech. 144 (2000) 211–229. [9] Z.-H. Jin, N. Noda, Int. J. Solids Struct. 31 (1994) 203–218. [10] Z.-H. Jin, R.C. Batra, J. Therm. Stress. 19 (1996) 317–339. [11] H.J. Choi, T.E. Jin, K.Y. lee, Int. J. Fract. 94 (1998) 123–135. [12] B.L. Wang, J.C. Han, S.Y. Du, ASME J. Appl. Mech. 67 (2000) 87–95.
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[13] Z.-H. Jin, Surf. Coat. Technol. 179 (2004) 210–214. [14] H. Balke, H.-A. Bahr, A.S. Semenov, G. Kirchho, H.-J. Weiss, in: K. Trumble, K. Bowman, I. Reimanis, S. Sampath (Eds.), Ceramic Transactions, vol. 114: Functionally Graded Ceramics 2000, American Ceramic Society, Westerville, OH, 2001, pp. 201–212. [15] K. Kokini, J. DeJonge, S. Rangaraj, B. Beardsley, in: K. Trumble, K. Bowman, I. Reimanis, S. Sampath (Eds.), Ceramic Transactions, vol. 114: Functionally Graded Ceramics 2000, American Ceramic Society, Westerville, OH, 2001, pp. 213–221. [16] B.L. Wang, Y.-W. Mai, X.H. Zhang, J. Am. Ceram. Soc. 88 (2005) 683– 690. [17] N. Noda, J. Therm. Stress. 20 (1997) 373–387. [18] Y. Tanigawa, ASME Appl. Mech. Rev. 48 (1995) 287–300. [19] N. Noda, J. Therm. Stress. 22 (1999) 477–512. [20] Z.-H. Jin, in: D.Y. Gao, R.W. Ogden (Eds.), Advances in Mechanics and Mathematics: vol. II (AMMA Series, vol. 4), Kluwer Academic Publishers, Dordrecht, 2003, pp. 1–108. [21] D.P.H. Hasselman, J. Am. Ceram. Soc. 52 (1969) 600–604. [22] J. Zhao, X. Ai, J. Deng, J. Wang, J. Eur. Ceram. Soc. 24 (2004) 847– 854. [23] H.-A. Bahr, H.J. Weiss, Theor. Appl. Fract. Mech. 8 (1986) 33–39. [24] Z.-H. Jin, Y.-W. Mai, J. Am. Ceram. Soc. 78 (1995) 1873–1881. [25] B. Cotterell, S.W. Ong, C.D. Qin, J. Am. Ceram. Soc. 78 (1995) 2056–2064. [26] Z.-H. Jin, Int. Commun. Heat Mass Transfer 29 (2002) 887–895. [27] Z.-H. Jin, R.C. Batra, J. Mech. Phys. Solids 44 (1996) 1221–1235. [28] R. Komanduri, in: S. Jahanmir (Ed.), Friction and Wear of Ceramics, Marcel Dekker Inc., New York, 1994, p. 289. [29] T. Reiter, G.J. Dvorak, J. Mech. Phys. Solids 46 (1998) 1655–1673. [30] R.M. Christensen, Mechanics of Composite Materials, John Wiley & Sons, New York, 1979.
Thermal shock residual strength of functionally graded ceramics Z.-H. Jin ∗ , W.J. Luo Department of Mechanical Engineering, University of Maine, 5711 Boardman Hall, Orono, ME 04469, USA Received 8 June 2006; accepted 6 July 2006
Abstract When subjected to severe thermal shocks functionally graded ceramics (FGCs) suffer strength degradation due to microcrack growth, coalescence and macrocrack propagation induced by the shocks. This paper presents a thermo-fracture mechanics model to study the residual strength behavior of thermally shocked FGCs. It is assumed that thermal shock damage mainly results from the extension of a pre-existing edge crack initially located at the thermally shocked surface of an FGC specimen. Thermal shock threshold, or the critical thermal shock that causes precipitous strength drop, is determined. The paper describes applications of the model to calculate the thermal shock residual strength of FGCs as a function of thermal shock severity. Two FGC systems, i.e., Al2 O3 /Si3 N4 and TiC/SiC FGCs are considered in the numerical studies. The numerical results show that material gradation profile has a pronounced effect on the thermal shock threshold and the thermal shock residual strength of FGCs. © 2006 Elsevier B.V. All rights reserved. Keywords: Functionally graded materials; Ceramics; Fracture; Thermal shock; Residual strength
1. Introduction Ceramics represent one of the most promising materials in the future turbine engines, internal combustion engines, metal cutting, and other high temperature engineering applications because of their excellent properties at high temperatures and superior corrosion and wear resistance. One major limitation for the application of ceramic materials is their inherent brittleness which can result in catastrophic failure under severe thermal shocks. To overcome this disadvantage, considerable efforts have been made to toughen ceramics with some success. Alternatively, one may specifically design a ceramic or ceramic–ceramic composite to reduce the thermal stresses when subjected to thermal shocks thereby enhancing the thermal shock fracture resistance. This is one of the objectives to be fulfilled by the concept of functionally graded ceramics (FGCs) [1–3]. An FGC is a ceramic or ceramic–ceramic composite with graded microstructure and material properties. FGCs distinguish themselves with conventional, macroscopically homogeneous ceramic–ceramic composites by the gradual change in the microstructure and properties over the thickness of the material. Knowledge of the thermal shock resistance behavior of FGCs is critical for the material design of FGCs in high temperature
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applications. While there has been some progress in the thermal fracture of FGCs, the existing studies have been mainly concerned with the computation of thermal stress intensity factors (TSIFs) and the material property gradation effects on TSIFs. For example, under steady-state thermal loading conditions, Noda and Jin [4] studied a crack parallel to the boundaries of an infinite functionally graded material (FGM) strip. Tanigawa et al. [5] considered a penny-shaped crack problem in a nonhomogeneous material. Erdogan and Wu [6] investigated cracks perpendicular to the boundaries of an infinite FGM strip. Bleeck et al. [7] studied the effect of a graded layer on the TSIF in a joint. Nemat-Alla and Noda [8] analyzed an edge crack problem in a semi-infinite FGM plate with a bidirectional gradation in the coefficient of thermal expansion. Under transient thermal loading conditions, Jin and Noda [9] studied a crack parallel to the boundary of a nonhomogeneous half-plane. Jin and Batra [10] investigated an edge crack in an FGM strip. Choi et al. [11] investigated collinear cracks in a layered half-plane with a graded interfacial zone. Wang et al. [12] studied thermal crack problems using a laminated material model. Jin [13] considered surface cracking in an FGC coating. Other studies include experimental and computational investigations of thermal fracture of FGM coatings (for example, Balke et al. [14] and Kokini et al. [15]), evaluation of maximum thermal shock at material failure [16] and crack growth in FGM plates by a finite element method [17]. Tanigawa [18], Noda [19] and Jin [20] have written extensive reviews on thermal fracture in FGMs.
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Fig. 1. Thermal shock residual strength behavior of ceramics.
For crack growth under thermal shock conditions, the crack propagation may be arrested depending on the severity of thermal shock, thermal stress field characteristics and material properties. If one measure the strength of a thermally shocked specimen made of a ceramic, the material generally exhibits the behavior as shown in Fig. 1 [21]. The strength remains unchanged when the thermal shock T is less than a critical value, Tc , called the thermal shock threshold. At T = Tc , the strength σ R suffers a precipitous drop and then decreases gradually with increasing severity of thermal shock. Both material properties and specimen size influence the residual strength behavior of thermally shocked materials. It is clear that the thermal shock residual strength characterizes the thermal shock fracture resistance of ceramics. Moreover, the residual strength approach has some advantages over the TSIF approach because it characterizes the overall thermal shock resistance. While there have been a few experimental studies on the thermal shock residual strength behavior of FGCs [22], predictive models for FGCs have not been developed. In this paper, the thermal shock residual strength behavior of FGCs is studied. Section 2 introduces the thermofracture mechanics of FGCs including temperature fields, thermal stresses and TSIF formulations. Section 3 describes the modeling approach to evaluate the thermal shock threshold and thermal shock damage in FGCs. In Section 4, the residual strength of thermally shocked FGCs is determined using a fracture mechanics approach. Section 5 presents and discusses some numerical results of thermal shock threshold and residual strength for FGC specimens made of Al2 O3 /Si3 N4 and TiC/SiC FGC systems. Finally, Section 6 provides some concluding remarks.
microcracks will initiate and grow to form macrocracks, which results in strength degradation. Surface cracks are the dominant defects affecting the thermal shock behavior of FGCs because tensile stresses develop at the surfaces of an FGC when quenched. In their study of thermal shock damage of monolithic ceramics, Bahr and Weiss [23] showed that strength degradation of monolithic ceramics is initially dominated by one single crack and multiple surface crack propagation play a role under further severe thermal shock conditions. In this study, we only consider the strength degradation due to propagation of a single crack, which has proven reasonable in evaluating thermal shock residual strength of monolithic ceramics [24,25]. Effects of multiple crack propagation on strength degradation will be considered in the future study. Specifically, we consider a long FGC strip with an edge crack shown in Fig. 2, where b is the thickness of the strip and a0 is the crack length. The thermal properties of the FGC are arbitrarily graded in the thickness direction (x-direction). The strip is initially at a constant temperature T0 , and its surfaces x = 0 and x = b are suddenly cooled to temperatures Ta and Tb , respectively. This represents an idealized thermal shock loading case, i.e., the heat transfer coefficients on the surfaces of the FGC strip are infinitely large which corresponds to the most severe thermal stress induced in the strip. In other words, the thermal shock residual strength predicted by the current model would be lower than that using a finite heat transfer coefficient. 2.1. Temperature field For the thermal shock problem shown in Fig. 2, thermal stresses in the FGC specimen result from the transient tempera-
2. Thermo-fracture mechanics Microcracks inherently exist in an FGC. When the FGC is subjected to severe thermal shocks, some of the pre-existing
Fig. 2. An edge cracked FGC strip subjected to a thermal shock.
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ture gradients induced by the thermal shock. Jin [26] obtained a closed-form, short time asymptotic solution (τ → 0) of temperature field in an FGC strip with arbitrary spatial distributions of thermal properties using the Laplace transform and its asymptotic properties. The asymptotic temperature field, T(x, τ), has the following form T (x, τ) − T0 T0 − T a x ρ(0)c(0)k(0) 1/4 1 κ(0) √ =− erfc dx ρ(x)c(x)k(x) κ(x) 2b τ 0 ρ(b)c(b)k(b) 1/4 T 0 − Tb T0 − Ta ρ(x)c(x)k(x) b 1 κ(0) √ × erfc dx , κ(x) 2b τ x
where θ(x, τ) = T(x, τ) − T0 , E Young’s modulus, ν Poisson’s ratio, α = α(x) the coefficient of thermal expansion and Aij (i, j = 1, 2) and A0 constants, which can be found in Ref. [20]. Here we assume that the FGC strip undergoes plane strain deformations in the x–y plane and the strip is free from constraints at the far ends. 2.3. Thermal stress intensity factor (TSIF) The thermal fracture problem of the edge cracked FGC strip may be solved by various analytical and numerical methods. Here we adopt a singular integral equation approach and the integral equation has the form 1 1 ψ(s, τ) + K(r, s) √ ds 1−s −1 s − r
−
(1)
where τ = tκ(0)/b2 denotes the nondimensional time, t time, ρ(x) the mass density, c(x) the specific heat, k(x) the thermal conductivity, κ(x) = k/ρc the thermal diffusivity, ρ(0), c(0), k(0) and κ(0) the values of ρ(x), c(x), k(x) and κ(x) at x = 0, respectively, ρ(b), c(b), k(b) and κ(b) the values at x = b, respectively, and erfc() the complementary error function. The asymptotic solution in Eq. (1) holds for an FGC strip with continuous and piecewise differentiable thermal properties. The significance of the solution lies in the fact that thermal stress and thermal stress intensity factor (TSIF) in the FGC strip induced by the thermal shock reach their peak values in a very short time. Thus, Eq. (1) may be used to evaluate the peak values of thermal stress and TSIF which govern the thermal stress failure of the material. 2.2. Thermal stress
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2π(1 − ν2 ) T σyy (r, τ), |r| ≤ 1 (3) E where K(r, s) is a known kernel, r = 2x/a − 1, a the current crack length, ψ a continuous function related to the dislocation density along the crack surface: =−
ψ(r, τ) =
√ 1 − rϕ(r, τ),
ϕ=
∂ [ν(x, 0+ ) − ν(x, 0− )] ∂x (4)
T the with ν(x, y) being the displacement in the y-direction, σyy thermal stress induced in the FGC strip free of cracks given in Eq. (2). Once the solution of the above integral equation is obtained, the TSIF at the edge crack tip can be computed from
(1 − ν) (1 − ν)KI √ √ = Eα0 T πb Eα0 T πb
1 a 2π(x − a)σyy (x, 0, τ) = − limx→a+ ψ(r, τ)|r=1 , 2 b (5)
K∗ =
The transient temperature field (1) induces thermal stresses in the edge cracked FGC strip shown in Fig. 2. This work focuses where KI denotes the TSIF, K* the nondimensional TSIF, σ yy the on the effect of thermal property gradients on the thermal shock normal stress, T = T0 − Ta , and α0 = α(x)|x=0 . In Eq. (5), ψ(r, strength behavior of FGCs. We thus consider a special kind τ)|r=1 depends on the nondimensional crack length a/b. of FGCs with constant Young’s modulus and Poisson’s ratio. While this assumption imposes limitations on the application of 3. Thermal shock threshold and thermal shock damage the present model, there exist some FGC systems for which the Young’s modulus remains approximately constant. Examples 3.1. Thermal shock threshold include TiC/SiC, MoSi2 /Al2 O3 , and Al2 O3 /Si3 N4 FGC systems. The Young’s modulus of each of these FGCs may not change Eq. (5) indicates that the TSIF is proportional to the thermal significantly because the constituents have similar Young’s modshock T. Crack extension will not occur if the thermal shock uli. T has not reached a critical value Tc , the so-called therThe temperature filed, Eq. (1), induces thermal stresses in mal shock threshold, at which the peak value of TSIF reaches both longitudinal (perpendicular to the crack direction) and the fracture toughness of the material. At T = Tc , the peak transverse (perpendicular to the x–y plane) directions. However, value of TSIF reaches the fracture toughness and crack extension only the longitudinal stress determines crack growth. The longitudinal stress can be obtained as follows
b b Eαθ(x, τ) Eαθ(x, τ) Eαθ(x, τ) E T σyy (x, τ) = − (A22 − xA21 ) + dx − (A12 − xA11 ) x dx , (2) 1−ν (1 − ν2 )A0 1−ν 1−ν 0 0
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is initiated. When T > Tc , the crack will grow thereby resulting in severe damage in the FGC. The thermal shock threshold Tc can be determined by equating the peak TSIF to the material fracture toughness, i.e., Max{τ>0} {KI (τ, a0 , Tc } = KIc (a0 )
(6)
where KIc (a0 ) denotes the fracture toughness of the FGC at x = a0 , and a0 denotes the initial pre-existing crack length. Substitution of Eq. (5) into Eq. (6) yields Tc : √ (1 − ν)KIc (a0 )/(Eα0 πb) √ Tc = (7) Max{τ>0} {−(1/2) a0 /b ψ(1, τ)} Fracture toughness of an FGC generally depends on spatial position. Determination of fracture toughness of FGCs remains a challenging task. Jin and Batra [27] proposed two schemes to evaluate the fracture toughness of ceramic–metal FGMs, i.e., the simple rule of mixtures scheme and the crack bridging method. While the rule of mixtures significantly overestimates the fracture toughness of ceramic–metal FGMs, it may be employed to approximately determine the fracture toughness of an FGC when the constituents of the FGC have similar fracture toughness. The rule of mixture formula presented by Jin and Batra [27] for a thermally nonhomogeneous but elastically homogeneous two-phase FGC with the properties graded in x-direction has the following form 2 1/2
2
1 2 KIc (x) = {V1 (x)(KIc ) + V2 (x)(KIc ) }
(8)
where Vi (x) (i = 1, 2) denote the volume fractions of phases 1 1 and K 2 the fracture toughness of and 2, respectively, and KIc Ic the two phases, respectively. Here the crack surface bridging by unbroken grains is not considered. Substituting Eq. (8) into Eq. (7) leads to Tc =
1 )2 (1 − ν){V1 (a0 )(KIc
1/2 2 )2 } + V2 (a0 )(KIc
a = af − a0
(12)
4. Residual strength Once the damage length a is determined for a thermal shock T, the residual strength of thermally shocked FGC may be theoretically evaluated by subjecting the thermally shocked specimen with a crack length af to bending and considering subsequent crack extension. The singular integral equation method can still be used and the equation has the form 1 1 ψ(s) ds + K(r, s) √ 1−s −1 s − r 2π(1 − ν2 ) af =− (13) 1 − (1 + r) σb , |r| ≤ 1 E b where σ b is the bending strength corresponding to the crack length af . The residual strength is thus determined as the maximum strength during the subsequent crack growth. To determine σ b , the stress intensity factor is first evaluated as follows √ 1 KI = σb π af − ψ(r)|r=1 (14) 2 σ b thus can be evaluated by equating KI to the fracture toughness with the result 2
σb =
2 1/2
1 ) + V (a )(K 2 ) } {V1 (af )(KIc 2 f Ic √ π af {−(1/2)ψ(r)|r=1 }
(15)
5. Numerical results and discussion
√
/(Eα0 πb) √ Max{τ>0} {−(1/2) a0 /bψ(1, τ)} (9)
3.2. Thermal shock damage When T > Tc , the crack will grow in the FGC. The crack extension may be arrested depending on a number of factors such as thermal shock severity, thermal stress field and TSIF characteristics, material properties and specimen geometry. If the crack growth is arrested, the final crack length af can be determined by the following equation Max{τ>0} {KI (τ, af , T } = KIc (af )
The damage length caused by the thermal shock is thus determined as
(10)
Substituting Eqs. (5) and (8) into the above equation yields the relationship between the final crack length af and the thermal shock T: √ 1/2 1 )2 + V (a )(K 2 )2 } (1 − ν){V1 (af )(KIc /(Eα0 πb) 2 f Ic √ T = Max{τ>0} {−(1/2) af /bψ(1, τ)} (11)
In the numerical calculations, we first consider the alumina/silicon nitride (Al2 O3 /Si3 N4 ) FGC system. Al2 O3 coated Si3 N4 cutting tools for machining steels have been developed [28] to take advantages of the high temperature deformation resistance of Si3 N4 and to minimize chemical reactions of Si3 N4 with steels by the Al2 O3 coating layer. The conventional Al2 O3 /Si3 N4 tools may be prone to cracking due to the mismatch in thermal expansion at the interface between the coating and the substrate. This proneness to cracking may be alleviated by replacing the Al2 O3 -coating/Si3 N4 -substrate system with an Al2 O3 /Si3 N4 FGC system. The second example concerns a titanium carbide/silicon carbide (TiC/SiC) FGC system. TiC has been used as a cutting tool material for its excellent wear resistance and SiC has also been considered as reinforcements for cutting tools. The TiC/SiC FGC thus is a candidate material for cutting tools applications. The FGC systems are treated as two-phase composites with graded volume fractions of their constituents. Reiter and Dvorak [29] argued that the micromechanics models for conventional composite materials may be employed for FGCs with reasonable accuracy. Thus the thermal conductivity, coefficient of thermal expansion (CTE), specific heat and mass density of the FGC are
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Table 1 Material properties of Al2 O3 , Si3 N4 , TiC and SiC
Al2 O3 Si3 N4 TiC SiC
Young’s modulus (GPa)
Poisson’s ratio
CTE (×10−6 K−1 )
Thermal conductivity (W/m K)
Mass density (g/cm3 )
Specific heat (J/g K)
Fracture toughness (MPa m1/2 )
320 320 450 450
0.25 0.25 0.2 0.2
8.0 3.0 7.0 4.0
20 35 20 60
3.8 3.2 4.9 3.2
0.9 0.7 0.7 1.0
4 5 5 5
calculated from the conventional micromechanics models [30]. The thermal conductivity, k(x), is given by V (x)(ki − k0 ) k(x) = k0 1 + , (16) k0 + (ki − k0 )[1 − V (x)]/3 where subscript ‘0’ represents the properties of the ceramic phase that is exposed to the thermal shock, subscript ‘i’ stands for properties of the other phase, and V(x) denotes the volume fraction of phase i. For the Al2 O3 /Si3 N4 FGC, the thermally shocked surface is pure Al2 O3 . The thermally shocked surface consists of pure TiC for the TiC/SiC system. The Voigt rule of mixtures gives the mass density, ρ(x), and the coefficient of thermal expansion, α(x), because the two constituents have identical elastic bulk moduli. The specific heat, c(x), of the FGC is assumed to follow the Voigt rule of mixtures, i.e., c(x) = V (x)ci + [1 − V (x)]c0
(17)
Fig. 3. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.01).
Table 1 lists the properties of the constituent materials, i.e., Al2 O3 , Si3 N4 , TiC and SiC. The Al2 O3 considered is not fully dense and has a modulus matching that of Si3 N4 . This study assumes that the volume fraction of phase i follows a simple power function: x p V (x) = , (18) b
higher overall fraction of Si3 N4 ). The FGC should transition smoothly and rapidly from pure Al2 O3 at the thermally shocked surface to a Si3 N4 -rich structure to achieve optimal thermal shock resistance. Fig. 4 highlights the residual strength results for T > Tc . Figs. 5 and 6 show the residual strength of an Al2 O3 /Si3 N4 FGC specimen with a smaller initial crack length of a0 = 0.005b. The specimen size is still assumed as b = 5 mm. Similar residual strength behavior to that observed for the FGC specimen with a0 = 0.01b prevails. While both the thermal shock threshold Tc and the strength for T < Tc become higher when
where p is the power exponent. In the numerical calculations, we only consider the loading case of Tb = T0 , which means that only the cracked surface x = 0 of the FGC system is subjected to a temperature drop. Fig. 3 shows the residual strength of the Al2 O3 /Si3 N4 FGC versus the thermal shock T for various values of the exponent p. The specimen size is assumed as b = 5 mm. The pre-existing surface crack has a length of a0 = 0.01b. Several interesting observations can be made. First, the model qualitatively predicts the thermal shock residual strength behavior of FGCs observed in experiment [22], i.e., for a specific value of exponent p, the residual strength remains constant when the thermal shock has not reached the thermal shock threshold Tc . At T = Tc , the strength suffers a precipitous drop and then decreases gradually with increasing severity of thermal shock. Second, Tc increases with a decrease in the exponent p. For example, Tc is about 110 ◦ C for p = 1.0 and increases to about 156 ◦ C when p = 0.2. Finally, the residual strength also increases with decreasing p. At T = 200 ◦ C, for example, the residual strength is about 54 MPa for p = 1.0 and is enhanced to about 79 MPa for p = 0.2. These results imply that the bulk of the Al2 O3 /Si3 N4 FGC should be Si3 N4 (a smaller value of p corresponds to a
Fig. 4. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.01) (highlighted results around T = Tc ).
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Fig. 5. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.005).
compared with the case of a0 = 0.01b as expected, the strength suffers more severe degradation. For example, the strength for p = 0.2 decreases from about 442 to about 77 MPa when T crosses the threshold Tc , an 83% drop. This value for the case of a0 = 0.01b is about 72% (the strength decreases from 318 to 90 MPa). Fig. 7 shows the residual strength of the TiC/SiC FGC versus the thermal shock T for various values of the exponent p. The specimen size is still assumed as b = 5 mm and the pre-existing surface crack has a length of a0 = 0.005b. The thermally shocked surface consists of pure TiC. It is seen from the figure that the material gradation profile (described by p) does not influence the strength for T < Tc . This is because the fracture toughness of the TiC equals that of the SiC as assumed. It also appears that the thermal shock threshold and the residual strength can be enhanced with decreasing value of p. Again, these results imply that the bulk of the TiC/SiC FGC should be SiC (a smaller value of p corresponds to a higher overall fraction of SiC). The FGC should be made such that it transitions smoothly and rapidly
Fig. 7. Thermal shock residual strength of TiC/SiC (b = 5 mm, a0 /b = 0.005).
Fig. 8. Thermal shock residual strength of TiC/SiC (b = 5 mm, a0 /b = 0.005) (highlighted results around T = Tc ).
from pure TiC at the thermally shocked surface to a SiC-rich structure to achieve optimal thermal shock resistance. Fig. 8 highlights the residual strength results for T > Tc . 6. Concluding remarks
Fig. 6. Thermal shock residual strength of Al2 O3 /Si3 N4 (b = 5 mm, a0 /b = 0.005) (highlighted results around T = Tc ).
A thermo-fracture mechanics model is developed to predict the residual strength of thermally shocked FGCs. The model is capable of predicting qualitatively the thermal shock residual strength behavior of FGCs observed in experiments, i.e., the residual strength remains constant when the thermal shock T has not reached the thermal shock threshold Tc . At T = Tc , the strength suffers a precipitous drop and then decreases gradually with increasing severity of thermal shock. The material gradation profile has a pronounced effect on the thermal shock resistance behavior of FGCs. The application of the model to the Al2 O3 /Si3 N4 FGC implies that the bulk of the Al2 O3 /Si3 N4 FGC should be Si3 N4 . The FGC should transition smoothly and rapidly from pure Al2 O3 at the thermally shocked surface to a Si3 N4 -rich structure to achieve optimal thermal
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shock resistance. Similar remarks apply to the TiC/SiC FGC system for which the TiC mainly functions as a cutting surface material and a graduate, rapid transition to the SiC provides enhanced thermal shock resistance. Acknowledgment This work is supported by the University of Maine. References [1] T. Hirai, in: R.J. Brook (Ed.), Materials Science and Technology, vol. 17B: Processing of Ceramics, Part 2, VCH Verlagsgesellschaft mbH, Weinheim, Germany, 1996, pp. 292–341. [2] M. Koizumi, Compos. Part B: Eng. 28 (1997) 1–4. [3] S. Suresh, A. Mortensen, Fundamentals of Functionally Graded Materials, The Institute of Materials, IOM Communications Ltd., London, 1998. [4] N. Noda, Z.-H. Jin, Int. J. Solids Struct. 30 (1993) 1039–1056. [5] Y. Tanigawa, T. Muraki, R. Kawamura, JSME Int. J., Ser. A 39 (1996) 540–547. [6] F. Erdogan, B.H. Wu, J. Therm. Stress. 19 (1997) 237–265. [7] O. Bleeck, D. Munz, W. Schaller, Y.Y. Yang, Eng. Fract. Mech. 60 (1998) 615–623. [8] M. Nemat-Alla, N. Noda, Acta Mech. 144 (2000) 211–229. [9] Z.-H. Jin, N. Noda, Int. J. Solids Struct. 31 (1994) 203–218. [10] Z.-H. Jin, R.C. Batra, J. Therm. Stress. 19 (1996) 317–339. [11] H.J. Choi, T.E. Jin, K.Y. lee, Int. J. Fract. 94 (1998) 123–135. [12] B.L. Wang, J.C. Han, S.Y. Du, ASME J. Appl. Mech. 67 (2000) 87–95.
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