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Thermal shock resistance analysis of a semi-infinite ceramic foam
International Journal of Engineering Science 62 (2013) 22–30
Contents lists available at SciVerse ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Thermal shock resistance analysis of a semi-infinite ceramic foam Y.X. Zhang a, B.L. Wang a,b,⇑ a b
Graduate School at Shenzhen, Harbin Institute of Technology, Harbin 150001, PR China School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 28 April 2012 Received in revised form 18 July 2012 Accepted 19 July 2012 Available online 23 September 2012 Keywords: Ceramic foams Thermal shock resistance Stress intensity factor Crack propagation
a b s t r a c t This paper considers the fracture mechanics of a ceramic foam under sudden thermal load. The ceramic foam is of semi-infinite and contains an edge crack perpendicular to its surface. The temperature field and transient thermal stress field in un-cracked medium are calculated first. Then, the stresses are used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. Numerical results for the stress and stress intensity factor are calculated as the functions of the thermal shock time, the crack length and the relative density of the foam. Crack propagation behavior and the thermal shock resistance of the ceramic foam are discussed in details. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Ceramic foams have been regarded as promising candidates as the catalytic substrates, heat engines and thermal insulation due to their low bulk density, catalytic activity, high permeability, erosion resistance as well as high temperature stability (Green & Colombo, 2003; She, Deng, Daniel-Doni, & Ohji, 2002). In these applications, ceramic foams usually experience severe thermal shock. Therefore, it is necessary to investigate the thermal shock behavior of these advanced materials. The factors affecting the thermal shock resistance and fracture behavior of medium include the dimensions, strength, elastic modulus, pre-existing crack length, etc. (Feng & Su, 2007; Hasselman, 1963, 1966, 1969; Kingery, 1955; Song, Meng, Xu, & Shao, 2010). In recent years, thermal shock resistance of ceramic foams has been investigated by many researchers. A number of interesting observations were made during the research works aiming at investigating the thermal shock resistance. The thermal shock resistance analysis of ceramic media was mainly focused on the measuring and calculating of Young’s modulus and retained strength after thermal shock (Dimitrijevic, Posarac, Majstorovic, Volkov-Husovic, & Matovic, 2009; Ding, Zeng, & Jiang, 2006; Kan, Wei, Meng, & Xu, 2009; Orenstein & Green, 1992; Rendtorff, Garrido, & Aglietti, 2011; Vedula, Green, & Hellman, 1998; You et al., 2005), the length of crack growth (Panda, Kannan, Dubois, Olagnon, & Fantozzi, 2002; Yuan, 2008), and thermal stress resistance parameter (R0f ) (Vedula, Green, & Hellman, 1999). It is essential to establish theoretical models for the evaluation of thermal shock resistance behavior of these advanced materials. The theoretical model in this paper can be applied to the thermal shock prediction of thermal protection system (Kubota & Uchida, 1999). Generally, the thermal shock resistance of a material is obtained based on two distinct failure criteria: (i) A local tensile stress criterion: fracture occurs when the maximum tensile stress attains the failure strength of the ceramic foam. ⇑ Corresponding author at: Graduate School at Shenzhen, Harbin Institute of Technology, Harbin 150001, PR China. E-mail addresses: [email protected], [email protected] (B.L. Wang). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.07.002
Y.X. Zhang, B.L. Wang / International Journal of Engineering Science 62 (2013) 22–30
23
(ii) A fracture toughness criterion: the largest pre-existing crack propagates when the maximum stress intensity factor attains the fracture toughness of the ceramic foams. In most practical circumstances, it has been demonstrated that this criterion is important if the cracks in the materials are not small. Considered in this paper is the cracking of a semi-infinite ceramic foam subjected to a sudden cooling on its surface. During rapid cooling, the temperature of the surface of the ceramic foam is much lower than the inside. Thus, the surface of the foam tends to shrink. This shrinkage may be constrained inside the foam, resulting in a state of tension in the surface. The stresses of un-cracked ceramic foam are used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. The mode I stress intensity factor is calculated from the transient thermal stress field. Crack propagation behavior is discussed. Based on stress-controlled and fracture-controlled criterion, the maximum jump in surface temperature to withstand fracture of the ceramic foam is calculated. Numerical results demonstrate that the ceramic foams with lower relative density can withstand higher thermal shock. It is also noted that for given length of cell size and relative density, the thermal shock resistance of ceramic foams becomes stronger if the length of the pre-existing crack in the foam decreases. 2. Temperature distribution The problem under consideration is a semi-infinite ceramic foam which is exposed to a sudden temperature drop on its surface as shown in Fig. 1. The foam is initially at a uniform temperature zero, and its boundary is exposed to an environment whose temperature is T0H(t), where H(t) is the Heaviside function. Since the problem is a cold shock event, T0 is negative. In the problem, the heat conduction is one dimensional and a straight crack does not obstruct the heat flow. Suppose the temperature at any time is T(z, t). The temperature field is governed by
kz
@ 2 Tðz; tÞ @Tðz; tÞ ¼ qc v @z2 @t
ð1Þ
which has a well-known solution (Wang, Noda, Han, & Du, 2002):
TðzÞ ¼ T 0 erfc
! z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tkz =ðqcv Þ
ð2Þ
where kz is the thermal conductivity of the ceramic foam in the z-direction, q is the density of the foam, cv is the specific heat of the foam. The thermal diffusivity a is a = kz/(qcv). Since a is generally not affected by the porosity provided the pores contain gases only, it can be dealt with constant (Lu & Fleck, 1998). 3. Thermal stress in the un-cracked foam Once T(z) is known, the stresses can be found from the constitutive relations:
8 9 > < rxx > =
2
C 11 6 rzz ¼ 4 C 13 > > : 0 sxz ;
C 13 C 33 0
9 9 8 38 > < k11 > < exx > = = > 7 0 5 ezz k33 T > > > > : cxz ; : 0 ; C 44 0
ð3Þ
where Cij are elastic constants and kij are the stress-temperature coefficients (i, j = 1, 3). Suppose the foam is free of surface traction on its top surface. It may be shown that rzz = 0 and exx = 0. It follows from Eq. (3) that the transient thermally induced stress rxx(z, t) associated with the temperature distribution T(z, t) is
rxx ðz; tÞ ¼
C 13 k33 k11 T C 33
ð4Þ
Substitute Eq. (2) into Eq. (4), the stress is rewritten as
x
c
z Fig. 1. A semi-infinite ceramic foam with an edge crack.
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rxx ðz; tÞ ¼
! C 13 k33 z k11 T 0 erfc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 33 2 tkz =ðqcv Þ
ð5Þ
Consider the isotropic ceramic foam for simple whose material property parameters become
Ef 1 v 2f Ef v f C 13 ¼ v f C 11 ¼ 1 v 2f Ef af k11 ¼ k33 ¼ 1 vf C 11 ¼ C 33 ¼
ð6Þ
where Ef, af and vf stand, respectively, for the elastic modular, the coefficient of thermal expansion and the Poisson’s ratio of the ceramic foams. For ceramic foams, the coefficients of thermal expansion and Poisson’s ratio are not affected by porosity (Lu & Fleck, 1998; Vedula et al., 1999) and are the same as the corresponding values of their dense materials. Assuming the bending of the pores wall is the dominant deformation mechanism under both tensile and compressive loading, Zhang and Ashby (1989) derived the equation: Ef = 0.5Es(q/qs)2, where Es is the elastic modular of dense materials, q is the density of the ceramic foam and qs is the density of the dense materials. Therefore, it follows from Eqs. (5) and (6) that
rxx ðz; tÞ ¼ 0:5Es
q qs
2
as T 0 erfc
z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tkz =ðqcv Þ
! ð7Þ
Eq. (7) is a closed form expression of the transient thermal stress in semi-infinite ceramic foams. 4. Numerical results and discussion 4.1. Thermal stress field The calculated distributions of stress for the un-cracked ceramic foam are shown in Figs. 2–4. Time dependence is formally replaced by the penetration depth of cooling, d(t) = 2tkz/(qcv). Under cold shock, the surface of ceramic foam experiences a transient tensile stress when z is small, no matter the values of relative density of the ceramic foam. This fact is in accord with the thermal shock experiment of Yuan (2008), for the crack is found on the surface of medium. The maximum tensile stress is attained at the surface of the ceramic foam which can be seen in Fig. 2. It is also noted that the stress the ceramic foam experienced tends to zero when z is sufficiently large, no matter the values of time and relative density of the ceramic foam. That is not difficult to understand since the temperature does not change when the location z is far away from the surface of the ceramic foam. Fig. 3 demonstrates that the stress increases with time at selected positions and reaches a constant when the time becomes infinity. That is true as the temperature of the foam tends to uniform when the time goes infinity. Fig. 4 shows that the stress increases with the increasing value of relative density. This is in good agreement with the experimental observation (Table IV in the reference) of Orenstein and Green (1992). 4.2. Thermal stress intensity factors In this section, the thermal stress intensity factor corresponding to fracture mechanics criterion is analyzed. For the crack problem, the opposite value of the stress rxx(z, t), obtained from Eq. (7) will be used as the crack surface traction to form the
Fig. 2. Variation of the thermal stress rxx(z, t) with coordinate z at selected relative densities (r0 = EsasT0, d(t) = 1.0).
Y.X. Zhang, B.L. Wang / International Journal of Engineering Science 62 (2013) 22–30
25
Fig. 3. Variation of the thermal stress rxx(z, t) with time at selected positions (r0 = EsasT0, q/qs = 0.25).
Fig. 4. Variation of the thermal stress rxx(z, t) with relative densities at selected positions (r0 = EsasT0, d(t) = 2.0 mm).
crack problem. Such approach has been adopted by Erdogan and Rizk (1992) and Rizk (2006). Therefore, the problem is equivalent to a crack subjected to a crack face compressive stress of magnitude rxx(z, t). The transient thermal stress intensity factors can be obtained by numerical integration such that (Evans & Charles, 1977)
K¼2
rffiffiffiffi Z c
p
c
0
zi rxx ðz; tÞ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ F dz c c 2 z2
ð8Þ
where F is a function given by
F
z 2 z4 z 6 z 8 z ¼ 1 0:2945 0:3912 þ 0:7685 0:9942 þ 0:5094 c c c c c c
z
ð9Þ
Numerical results for the thermal stress intensity factor K are shown in Figs. 5 and 6, where K is normalized according to K/ r0, in which
r0 ¼ Es as T 0
ð10Þ
The values of K/r0 have the same sign with K. As time passes, the normalized stress intensity factor increases and reaches a certain value for a surface crack subjected to a uniform compressive stress EfasT0 on its surface as shown in Fig. 5, where a representative value 0.15 of the relative density is selected. If the crack length is very small, the values of stress on the crack surface can be considered as the same. The stress intensity factor for infinite time t is obtained as
K ¼ 1:12Ef as T 0
pffiffiffiffiffiffi pc
ð11Þ
In fact, as the cooling time approaches infinity, the crack surface will undergo a uniform tensile stress r0 = EfasT0. It is well known that the normalized stress intensity factor is 1.12, Fig. 5 and Eq. (11) do conform this result. In addition, it can also be noted that the stress intensity factor increases with the increasing relative density for any given time and crack length (Fig. 6). Therefore, in practical application, selection of proper relative density of ceramic foams is very important for thermal shock application.
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Fig. 5. Variation of thermal stress intensity factors with time at selected crack lengths (r0 = EsasT0, q/qs = 0.15).
Fig. 6. Variation of thermal stress intensity factors with relative densities at different thermal shock time (r0 = EsasT0, c = 1.5 mm).
4.3. Thermal crack growth analyses Figs. 7–9 plot the variations of thermal stress intensity factors with crack lengths for selected values of thermal shock time and relative densities of the ceramic foams. It can be seen that for any given value of crack length c and relative density, the stress intensity factor vanishes if time approaches to zero. For any given time and relative density, the value of the thermal stress intensity factor increases to a maximum value and then decreases with increasing crack length. This suggests that if the ceramic foam contains a pre-existing crack, it tends to grow at the beginning of the thermal shock of ceramic foams. Therefore, a pre-existing crack of length c0 will start propagating when the thermal stress intensity factor K(c0) reaches the
Fig. 7. Variation of thermal stress intensity factors with crack lengths with different time (r0 = EsasT0, q/qs = 0.15).
Y.X. Zhang, B.L. Wang / International Journal of Engineering Science 62 (2013) 22–30
27
Fig. 8. Variation of thermal stress intensity factors with crack lengths with different time (r0 = EsasT0, q/qs = 0.1).
Fig. 9. Variation of thermal stress intensity factors with crack lengths with different time (r0 = EsasT0, q/qs = 0.05).
fracture toughness KIC. The initial crack will grow instantaneously at least to a new length c1, where the thermal stress intensity factor K(c1) again reaches the fracture toughness KIC of the ceramic foam. After c = c1, since the stress intensity factor increases with increasing time, the crack will continue to grow in a stable way. This behavior is plotted in Fig. 10 for crack growth trajectory d(t) versus crack length c for an initial crack length c0 = 0.3 mm. In the figure, for the relative density 0.1 and the thermal shock temperature T0, the fracture toughness is assumed to be equals to the maximum thermal stress intenpffiffiffiffiffiffiffiffiffi sity factor and is K IC ¼ 0:0049r0 mm. The related crack growth trajectory is the curve (a) in Fig. 10. According to the relationship between the fracture toughness of ceramic foams and their dense materials (Morgan, Wood, & Bradt, 1981)
Fig. 10. Crack growth trajectory d(t) vs. c (c0 = 0.3 mm, r0 = EsasT0).
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K IC ¼ 0:65rfs ðpLÞ1=2 ðq=qs Þ1:5
ð12Þ
where L is the cell size of ceramic foam, rfs the tensile strength of the dense materials, the fracture toughness will be pffiffiffiffiffiffiffiffiffi K IC ¼ 0:009r0 mm for the relative density 0.15 and the thermal shock temperature T0. Accordingly, the crack grow trajectory is the curve (b) in Fig. 10. In addition, when the relative density is 0.15 and the thermal shock temperature 1.125T0, the pffiffiffiffiffiffiffiffiffi fracture toughness is K IC ¼ 0:009r0 mm, and the crack grow trajectory is the curve (c) in Fig. 10. The purpose of Fig. 10 is to establish the effects of relative density and thermal shock environment on the crack propagation behavior of the ceramic foams. It can be seen that, for each curve, there is a time at which the crack begins to grow suddenly from the initial length c0; to a new length c1 or c2, then stably propagate as time increases. Obviously, the values of c1 or c2 depends on the fracture toughness, initial crack length, relative density and the thermal shock temperature T0. It can be seen that when relative density of the ceramic foams is fixed, the time corresponding to the initial crack growth unstably for a severe cold shock is smaller than that for a less severe cold shock. In addition and the crack length is longer for a severe cold shock after crack propagation. Again, the result is in agreement with the experiment observation of Yuan (2008) (Fig. 9 of the reference) and the experiment of Panda et al. (2002) (Fig. 2 in the reference). It is also noted that for a same cold shock environment, a ceramic foam of larger relative density will result in an earlier unstably crack growth. This demonstrates the fact that the ceramic foams with a low relative density can sustain severer thermal shock, which is in accordance with the experiment observation of Vedula et al. (1999) for a relative density less than 0.15. The phenomenon can also be explained from the results in Fig. 9 which is plotted for a relative density of 0.05 and a normalized fracture toughness of 0.00173r0. Clearly, no crack growth can happen at any time in this situation.
4.4. Thermal shock resistance Thermal shock resistance is a major issue in the design of engineering ceramic foams for the thermal application. A critical problem in designing a medium against thermal shock is the identification of appropriate material selection criterion in order to select the most shock-resistant material for a given application. Basically, material performance indices are summarized for both stress-controlled and toughness-controlled failure (Lu & Fleck, 1998). A stress-based failure criterion for thermal shock is that the maximum thermal stress at the surface of the ceramic foam attains the stress value rcr, which is (Morgan et al., 1981)
rcr ¼ 0:2rfs
q qs
3=2 ð13Þ
where rfs is the strength of the dense material of the ceramic foam. For a given material ceramic foam, the values of rfs and qs are specified. The maximum temperature jump sustained by the ceramic foam follows
DT c ¼ DT cstress ¼
rcr Ef as
ð14Þ
A similar strategy can be employed to rank the foam on the basis of failure form a dominant crack by thermal shock. The toughness-based fracture criterion is taken to be that the maximum thermal stress intensity factor Kmax attains the fracture toughness KIC, which is expressed in Eq. (12). Obviously, the maximum temperature jump sustainable depends on the length of the crack for a given ceramic foam. It follows from Kmax = KIC that, for the ceramic foam under consideration,
DT c ¼ DT cfracture ¼
K IC pffiffiffiffiffiffi 1:12Ef as pc
ð15Þ
It is instructive to plot the sustainable temperature jump as a function of crack length. The results of DTc with crack length based on fracture toughness are depicted in Fig. 11. It is clear that the thermal shock resistance predicted by the fracture toughness-based criterion exhibit a strong dependence on preexisting crack length. The stress intensity factor is zero for un-cracked medium. Accordingly, the admissible temperature jump based on fracture toughness will be infinity at crack length c=0. Therefore, when the crack length tends to zero, the fracture toughness-based criterion will be inappropriate. Thus, the failure of the ceramic foam will be governed by the stress-based criterion for small crack lengths. A transient crack length of c1 exists for which DTc is equal for toughness-based failure and stress-based failure. This length can be obtained for Eqs. (12)–(15). The result is
c1 ¼ 9L
ð16Þ
as mentioned above, L is the cell size of the ceramic foam. Therefore, the thermal shock strength of ceramic foams with crack smaller than c1 will be controlled by the stress-based criterion; while a ceramic foam with crack length larger than c1 will be controlled by the fracture-based criterion. If the crack length of ceramic foam is c1, its thermal shock resistance can be established according to the stress-based or the fracture-based criterion.
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29
Fig. 11. The relationship of thermal shock resistance based on stress and fracture toughness criterion with crack length of ceramic foam.
5. Conclusions The thermal shock resistance of ceramic foams has been investigated theoretically for a semi-infinite ceramic foam containing an edge crack. The surface of the foam undergoes a sudden cooling. The closed-form solutions to the temperature and stress fields were obtained. The results of the stress demonstrated that the maximum stress appears at the surface of the foam. This may cause the most severe thermal shock damage on the surface of the foam, agreeing well with the thermal shock experiments (Yuan, 2008). The propagation manner of pre-existing cracks in the ceramic foam was discussed. After the stress intensity factor of a pre-existing crack reaches the fracture toughness of the foam, the crack jumps suddenly to a new crack length and than grows in a stable way with increasing thermal shock time. In addition, the studies of the effect of the relative density of the ceramic foam have demonstrated that the ceramic foam with lower relative density can sustain severe thermal shock. In addition, the thermal shock resistance of ceramic foams was evaluated according to the stress based and the fracture toughness based criteria. The maximum temperature that the ceramic foam can sustain without catastrophic failure was analyzed according to the maximum local tensile stress criterion and fracture toughness criterion. A transient crack length c1 was identified for which the admissible temperature jump is equal for the stress-controlled failure and the fracture-controlled failure. Thermal shock resistance of ceramic foams with a crack smaller than c1 will be controlled by the stress-based criterion, while a crack larger than c1 will be controlled by the fracture-based criterion. Finally, it should be mentioned that the configuration considered in this paper is relative simple yet important for real structures. Firstly, the thermal shock resistance of a real material is mainly determined by the crack initiation behavior. Therefore, an infinitesimal crack (or a semi-infinite foam medium) considered by the paper is an appropriate approach. Secondly, the thermal shock resistance behavior of the foam is expressed in terms of the relative density of the foam and is presented in non-dimensional form. This avoids to identify the range of real microstructures of materials, which are diverse. Acknowledgments This research was supported by the National Science Foundation of China (projects #10972067, #11172081). BLW is an ARC Future Fellowship at professorial level in the University of New South Wales supported by the Australian Research Council. Reference Dimitrijevic, M., Posarac, M., Majstorovic, J., Volkov-Husovic, T., & Matovic, B. (2009). Behavior of silicon carbide/cordierite composite material after cyclic thermal shock. Ceramics International, 35, 1077–1081. Ding, S. Q., Zeng, Y. P., & Jiang, D. l. (2006). Thermal shock resistance of in situ reaction bonded porous silicon carbide ceramics. Materials Science and Engineering A, 425, 326–329. Erdogan, F., & Rizk, A. A. (1992). Fracture of coated plates and shells under thermal-shock. International Journal of Fracture, 53, 159–185. Evans, A. G., & Charles, E. A. (1977). Structural integrity in severe thermal environments. Journal of the American Ceramic Society, 60, 22–28. Feng, W. J., & Su, R. K. L. (2007). Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate. European Journal of Mechanics A/ Solids, 26, 363–379. Green, D. J., & Colombo, P. (2003). Cellular ceramics: intriguing structures, novel properties and innovative applications. Materials Research Society Bulletin, 28, 296–300. Hasselman, D. P. H. (1963). Elastic energy at fracture and surface energy as design criteria for thermal shock. Journal of the American Ceramic Society, 46, 535–540. Hasselman, D. P. H. (1966). Theory of thermal shock resistance of semitransparent ceramics under radiation heating. Journal of the American Ceramic Society, 49, 103–104. Hasselman, D. P. H. (1969). Unified theory of thermal shock fracture initiation and crack propagation in brittle ceramics. Journal of the American Ceramic Society, 52, 600–604.
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Kan, J., Wei, L. M., Meng, S. H., & Xu, C. H. (2009). Real-time observation of damage nucleation in 3D braided carbon/carbon composites. Advanced Composites Letters, 18, 131–137. Kingery, W. D. (1955). Factors affecting thermal stress resistance of ceramic materials. Journal of the American Ceramic Society, 38, 3–15. Kubota, H., & Uchida, S. (1999). Thermal protection system with use of porous media for a hypersonic reentry vehicle. Journal of Porous Media, 2, 71–86. Lu, T. J., & Fleck, N. A. (1998). The thermal shock resistance of solids. Acta Materialia, 46, 4755–4768. Morgan, J. S., Wood, J. L., & Bradt, R. C. (1981). Cell size effects on the strength of foamed glass. Materials Science and Engineering, 47, 37–42. Orenstein, R. M., & Green, D. J. (1992). Thermal shock behavior of open-cell ceramic foams. Journal of the American Ceramic Society, 75, 1899–1905. Panda, P. K., Kannan, T. S., Dubois, J., Olagnon, C., & Fantozzi, G. (2002). Thermal shock and thermal fatigue study of alumina. Journal of the European Ceramic Society, 22, 2187–2196. Rendtorff, N. M., Garrido, L. B., & Aglietti, E. F. (2011). Thermal shock resistance and fatigue of zircon-mullite composite materials. Ceramics International, 37, 1427–1434. Rizk, A. E. F. A. (2006). An elastic strip with periodic surface cracks under thermal shock. International Journal of Engineering Science, 44, 807–818. She, J. H., Deng, Z. Y., Daniel-Doni, J., & Ohji, T. (2002). Oxidation bonding of porous silicon carbide ceramics. Journal of Materials Science, 37, 3615–3622. Song, F., Meng, S. H., Xu, X. H., & Shao, Y. F. (2010). Enhanced thermal shock resistance of ceramics through biomimetically inspired nanofins. Physic Review Letters, 104, 245502. Vedula, V. R., Green, D. J., & Hellman, J. R. (1998). Thermal fatigue resistance of open cell ceramic foams. Journal of European Ceramic Society, 18, 2073–2080. Vedula, V. R., Green, D. J., & Hellman, J. R. (1999). Thermal shock resistance of ceramic foams. Journal of American Ceramic Society, 82, 649–656. Wang, B. L., Noda, N., Han, J. C., & Du, S. Y. (2002). Surface thermal shock fracture of a semi-infinite piezoelectric medium (poling axis parallel to the crack plane). Mechanics of Materials, 34, 135–144. You, X. Q., Si, T. Z., Liu, N., Ren, P. P., Xu, Y. D., & Feng, J. P. (2005). Effect of grain size on thermal shock resistance of Al2O3–TiC ceramics. Ceramic International, 31, 33–38. Yuan, C. (2008). The effect of porosity in thermal shock. Journal of Materials Science, 43, 4099–4106. Zhang, J., & Ashby, M. F. (1989). Theoretical studies on isotropic foams. Cambridge University Publication.
Contents lists available at SciVerse ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Thermal shock resistance analysis of a semi-infinite ceramic foam Y.X. Zhang a, B.L. Wang a,b,⇑ a b
Graduate School at Shenzhen, Harbin Institute of Technology, Harbin 150001, PR China School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 28 April 2012 Received in revised form 18 July 2012 Accepted 19 July 2012 Available online 23 September 2012 Keywords: Ceramic foams Thermal shock resistance Stress intensity factor Crack propagation
a b s t r a c t This paper considers the fracture mechanics of a ceramic foam under sudden thermal load. The ceramic foam is of semi-infinite and contains an edge crack perpendicular to its surface. The temperature field and transient thermal stress field in un-cracked medium are calculated first. Then, the stresses are used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. Numerical results for the stress and stress intensity factor are calculated as the functions of the thermal shock time, the crack length and the relative density of the foam. Crack propagation behavior and the thermal shock resistance of the ceramic foam are discussed in details. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Ceramic foams have been regarded as promising candidates as the catalytic substrates, heat engines and thermal insulation due to their low bulk density, catalytic activity, high permeability, erosion resistance as well as high temperature stability (Green & Colombo, 2003; She, Deng, Daniel-Doni, & Ohji, 2002). In these applications, ceramic foams usually experience severe thermal shock. Therefore, it is necessary to investigate the thermal shock behavior of these advanced materials. The factors affecting the thermal shock resistance and fracture behavior of medium include the dimensions, strength, elastic modulus, pre-existing crack length, etc. (Feng & Su, 2007; Hasselman, 1963, 1966, 1969; Kingery, 1955; Song, Meng, Xu, & Shao, 2010). In recent years, thermal shock resistance of ceramic foams has been investigated by many researchers. A number of interesting observations were made during the research works aiming at investigating the thermal shock resistance. The thermal shock resistance analysis of ceramic media was mainly focused on the measuring and calculating of Young’s modulus and retained strength after thermal shock (Dimitrijevic, Posarac, Majstorovic, Volkov-Husovic, & Matovic, 2009; Ding, Zeng, & Jiang, 2006; Kan, Wei, Meng, & Xu, 2009; Orenstein & Green, 1992; Rendtorff, Garrido, & Aglietti, 2011; Vedula, Green, & Hellman, 1998; You et al., 2005), the length of crack growth (Panda, Kannan, Dubois, Olagnon, & Fantozzi, 2002; Yuan, 2008), and thermal stress resistance parameter (R0f ) (Vedula, Green, & Hellman, 1999). It is essential to establish theoretical models for the evaluation of thermal shock resistance behavior of these advanced materials. The theoretical model in this paper can be applied to the thermal shock prediction of thermal protection system (Kubota & Uchida, 1999). Generally, the thermal shock resistance of a material is obtained based on two distinct failure criteria: (i) A local tensile stress criterion: fracture occurs when the maximum tensile stress attains the failure strength of the ceramic foam. ⇑ Corresponding author at: Graduate School at Shenzhen, Harbin Institute of Technology, Harbin 150001, PR China. E-mail addresses: [email protected], [email protected] (B.L. Wang). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.07.002
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(ii) A fracture toughness criterion: the largest pre-existing crack propagates when the maximum stress intensity factor attains the fracture toughness of the ceramic foams. In most practical circumstances, it has been demonstrated that this criterion is important if the cracks in the materials are not small. Considered in this paper is the cracking of a semi-infinite ceramic foam subjected to a sudden cooling on its surface. During rapid cooling, the temperature of the surface of the ceramic foam is much lower than the inside. Thus, the surface of the foam tends to shrink. This shrinkage may be constrained inside the foam, resulting in a state of tension in the surface. The stresses of un-cracked ceramic foam are used as the crack surface traction with opposite sign to formulate the mixed boundary value problem. The mode I stress intensity factor is calculated from the transient thermal stress field. Crack propagation behavior is discussed. Based on stress-controlled and fracture-controlled criterion, the maximum jump in surface temperature to withstand fracture of the ceramic foam is calculated. Numerical results demonstrate that the ceramic foams with lower relative density can withstand higher thermal shock. It is also noted that for given length of cell size and relative density, the thermal shock resistance of ceramic foams becomes stronger if the length of the pre-existing crack in the foam decreases. 2. Temperature distribution The problem under consideration is a semi-infinite ceramic foam which is exposed to a sudden temperature drop on its surface as shown in Fig. 1. The foam is initially at a uniform temperature zero, and its boundary is exposed to an environment whose temperature is T0H(t), where H(t) is the Heaviside function. Since the problem is a cold shock event, T0 is negative. In the problem, the heat conduction is one dimensional and a straight crack does not obstruct the heat flow. Suppose the temperature at any time is T(z, t). The temperature field is governed by
kz
@ 2 Tðz; tÞ @Tðz; tÞ ¼ qc v @z2 @t
ð1Þ
which has a well-known solution (Wang, Noda, Han, & Du, 2002):
TðzÞ ¼ T 0 erfc
! z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tkz =ðqcv Þ
ð2Þ
where kz is the thermal conductivity of the ceramic foam in the z-direction, q is the density of the foam, cv is the specific heat of the foam. The thermal diffusivity a is a = kz/(qcv). Since a is generally not affected by the porosity provided the pores contain gases only, it can be dealt with constant (Lu & Fleck, 1998). 3. Thermal stress in the un-cracked foam Once T(z) is known, the stresses can be found from the constitutive relations:
8 9 > < rxx > =
2
C 11 6 rzz ¼ 4 C 13 > > : 0 sxz ;
C 13 C 33 0
9 9 8 38 > < k11 > < exx > = = > 7 0 5 ezz k33 T > > > > : cxz ; : 0 ; C 44 0
ð3Þ
where Cij are elastic constants and kij are the stress-temperature coefficients (i, j = 1, 3). Suppose the foam is free of surface traction on its top surface. It may be shown that rzz = 0 and exx = 0. It follows from Eq. (3) that the transient thermally induced stress rxx(z, t) associated with the temperature distribution T(z, t) is
rxx ðz; tÞ ¼
C 13 k33 k11 T C 33
ð4Þ
Substitute Eq. (2) into Eq. (4), the stress is rewritten as
x
c
z Fig. 1. A semi-infinite ceramic foam with an edge crack.
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rxx ðz; tÞ ¼
! C 13 k33 z k11 T 0 erfc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 33 2 tkz =ðqcv Þ
ð5Þ
Consider the isotropic ceramic foam for simple whose material property parameters become
Ef 1 v 2f Ef v f C 13 ¼ v f C 11 ¼ 1 v 2f Ef af k11 ¼ k33 ¼ 1 vf C 11 ¼ C 33 ¼
ð6Þ
where Ef, af and vf stand, respectively, for the elastic modular, the coefficient of thermal expansion and the Poisson’s ratio of the ceramic foams. For ceramic foams, the coefficients of thermal expansion and Poisson’s ratio are not affected by porosity (Lu & Fleck, 1998; Vedula et al., 1999) and are the same as the corresponding values of their dense materials. Assuming the bending of the pores wall is the dominant deformation mechanism under both tensile and compressive loading, Zhang and Ashby (1989) derived the equation: Ef = 0.5Es(q/qs)2, where Es is the elastic modular of dense materials, q is the density of the ceramic foam and qs is the density of the dense materials. Therefore, it follows from Eqs. (5) and (6) that
rxx ðz; tÞ ¼ 0:5Es
q qs
2
as T 0 erfc
z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tkz =ðqcv Þ
! ð7Þ
Eq. (7) is a closed form expression of the transient thermal stress in semi-infinite ceramic foams. 4. Numerical results and discussion 4.1. Thermal stress field The calculated distributions of stress for the un-cracked ceramic foam are shown in Figs. 2–4. Time dependence is formally replaced by the penetration depth of cooling, d(t) = 2tkz/(qcv). Under cold shock, the surface of ceramic foam experiences a transient tensile stress when z is small, no matter the values of relative density of the ceramic foam. This fact is in accord with the thermal shock experiment of Yuan (2008), for the crack is found on the surface of medium. The maximum tensile stress is attained at the surface of the ceramic foam which can be seen in Fig. 2. It is also noted that the stress the ceramic foam experienced tends to zero when z is sufficiently large, no matter the values of time and relative density of the ceramic foam. That is not difficult to understand since the temperature does not change when the location z is far away from the surface of the ceramic foam. Fig. 3 demonstrates that the stress increases with time at selected positions and reaches a constant when the time becomes infinity. That is true as the temperature of the foam tends to uniform when the time goes infinity. Fig. 4 shows that the stress increases with the increasing value of relative density. This is in good agreement with the experimental observation (Table IV in the reference) of Orenstein and Green (1992). 4.2. Thermal stress intensity factors In this section, the thermal stress intensity factor corresponding to fracture mechanics criterion is analyzed. For the crack problem, the opposite value of the stress rxx(z, t), obtained from Eq. (7) will be used as the crack surface traction to form the
Fig. 2. Variation of the thermal stress rxx(z, t) with coordinate z at selected relative densities (r0 = EsasT0, d(t) = 1.0).
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Fig. 3. Variation of the thermal stress rxx(z, t) with time at selected positions (r0 = EsasT0, q/qs = 0.25).
Fig. 4. Variation of the thermal stress rxx(z, t) with relative densities at selected positions (r0 = EsasT0, d(t) = 2.0 mm).
crack problem. Such approach has been adopted by Erdogan and Rizk (1992) and Rizk (2006). Therefore, the problem is equivalent to a crack subjected to a crack face compressive stress of magnitude rxx(z, t). The transient thermal stress intensity factors can be obtained by numerical integration such that (Evans & Charles, 1977)
K¼2
rffiffiffiffi Z c
p
c
0
zi rxx ðz; tÞ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ F dz c c 2 z2
ð8Þ
where F is a function given by
F
z 2 z4 z 6 z 8 z ¼ 1 0:2945 0:3912 þ 0:7685 0:9942 þ 0:5094 c c c c c c
z
ð9Þ
Numerical results for the thermal stress intensity factor K are shown in Figs. 5 and 6, where K is normalized according to K/ r0, in which
r0 ¼ Es as T 0
ð10Þ
The values of K/r0 have the same sign with K. As time passes, the normalized stress intensity factor increases and reaches a certain value for a surface crack subjected to a uniform compressive stress EfasT0 on its surface as shown in Fig. 5, where a representative value 0.15 of the relative density is selected. If the crack length is very small, the values of stress on the crack surface can be considered as the same. The stress intensity factor for infinite time t is obtained as
K ¼ 1:12Ef as T 0
pffiffiffiffiffiffi pc
ð11Þ
In fact, as the cooling time approaches infinity, the crack surface will undergo a uniform tensile stress r0 = EfasT0. It is well known that the normalized stress intensity factor is 1.12, Fig. 5 and Eq. (11) do conform this result. In addition, it can also be noted that the stress intensity factor increases with the increasing relative density for any given time and crack length (Fig. 6). Therefore, in practical application, selection of proper relative density of ceramic foams is very important for thermal shock application.
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Fig. 5. Variation of thermal stress intensity factors with time at selected crack lengths (r0 = EsasT0, q/qs = 0.15).
Fig. 6. Variation of thermal stress intensity factors with relative densities at different thermal shock time (r0 = EsasT0, c = 1.5 mm).
4.3. Thermal crack growth analyses Figs. 7–9 plot the variations of thermal stress intensity factors with crack lengths for selected values of thermal shock time and relative densities of the ceramic foams. It can be seen that for any given value of crack length c and relative density, the stress intensity factor vanishes if time approaches to zero. For any given time and relative density, the value of the thermal stress intensity factor increases to a maximum value and then decreases with increasing crack length. This suggests that if the ceramic foam contains a pre-existing crack, it tends to grow at the beginning of the thermal shock of ceramic foams. Therefore, a pre-existing crack of length c0 will start propagating when the thermal stress intensity factor K(c0) reaches the
Fig. 7. Variation of thermal stress intensity factors with crack lengths with different time (r0 = EsasT0, q/qs = 0.15).
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Fig. 8. Variation of thermal stress intensity factors with crack lengths with different time (r0 = EsasT0, q/qs = 0.1).
Fig. 9. Variation of thermal stress intensity factors with crack lengths with different time (r0 = EsasT0, q/qs = 0.05).
fracture toughness KIC. The initial crack will grow instantaneously at least to a new length c1, where the thermal stress intensity factor K(c1) again reaches the fracture toughness KIC of the ceramic foam. After c = c1, since the stress intensity factor increases with increasing time, the crack will continue to grow in a stable way. This behavior is plotted in Fig. 10 for crack growth trajectory d(t) versus crack length c for an initial crack length c0 = 0.3 mm. In the figure, for the relative density 0.1 and the thermal shock temperature T0, the fracture toughness is assumed to be equals to the maximum thermal stress intenpffiffiffiffiffiffiffiffiffi sity factor and is K IC ¼ 0:0049r0 mm. The related crack growth trajectory is the curve (a) in Fig. 10. According to the relationship between the fracture toughness of ceramic foams and their dense materials (Morgan, Wood, & Bradt, 1981)
Fig. 10. Crack growth trajectory d(t) vs. c (c0 = 0.3 mm, r0 = EsasT0).
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K IC ¼ 0:65rfs ðpLÞ1=2 ðq=qs Þ1:5
ð12Þ
where L is the cell size of ceramic foam, rfs the tensile strength of the dense materials, the fracture toughness will be pffiffiffiffiffiffiffiffiffi K IC ¼ 0:009r0 mm for the relative density 0.15 and the thermal shock temperature T0. Accordingly, the crack grow trajectory is the curve (b) in Fig. 10. In addition, when the relative density is 0.15 and the thermal shock temperature 1.125T0, the pffiffiffiffiffiffiffiffiffi fracture toughness is K IC ¼ 0:009r0 mm, and the crack grow trajectory is the curve (c) in Fig. 10. The purpose of Fig. 10 is to establish the effects of relative density and thermal shock environment on the crack propagation behavior of the ceramic foams. It can be seen that, for each curve, there is a time at which the crack begins to grow suddenly from the initial length c0; to a new length c1 or c2, then stably propagate as time increases. Obviously, the values of c1 or c2 depends on the fracture toughness, initial crack length, relative density and the thermal shock temperature T0. It can be seen that when relative density of the ceramic foams is fixed, the time corresponding to the initial crack growth unstably for a severe cold shock is smaller than that for a less severe cold shock. In addition and the crack length is longer for a severe cold shock after crack propagation. Again, the result is in agreement with the experiment observation of Yuan (2008) (Fig. 9 of the reference) and the experiment of Panda et al. (2002) (Fig. 2 in the reference). It is also noted that for a same cold shock environment, a ceramic foam of larger relative density will result in an earlier unstably crack growth. This demonstrates the fact that the ceramic foams with a low relative density can sustain severer thermal shock, which is in accordance with the experiment observation of Vedula et al. (1999) for a relative density less than 0.15. The phenomenon can also be explained from the results in Fig. 9 which is plotted for a relative density of 0.05 and a normalized fracture toughness of 0.00173r0. Clearly, no crack growth can happen at any time in this situation.
4.4. Thermal shock resistance Thermal shock resistance is a major issue in the design of engineering ceramic foams for the thermal application. A critical problem in designing a medium against thermal shock is the identification of appropriate material selection criterion in order to select the most shock-resistant material for a given application. Basically, material performance indices are summarized for both stress-controlled and toughness-controlled failure (Lu & Fleck, 1998). A stress-based failure criterion for thermal shock is that the maximum thermal stress at the surface of the ceramic foam attains the stress value rcr, which is (Morgan et al., 1981)
rcr ¼ 0:2rfs
q qs
3=2 ð13Þ
where rfs is the strength of the dense material of the ceramic foam. For a given material ceramic foam, the values of rfs and qs are specified. The maximum temperature jump sustained by the ceramic foam follows
DT c ¼ DT cstress ¼
rcr Ef as
ð14Þ
A similar strategy can be employed to rank the foam on the basis of failure form a dominant crack by thermal shock. The toughness-based fracture criterion is taken to be that the maximum thermal stress intensity factor Kmax attains the fracture toughness KIC, which is expressed in Eq. (12). Obviously, the maximum temperature jump sustainable depends on the length of the crack for a given ceramic foam. It follows from Kmax = KIC that, for the ceramic foam under consideration,
DT c ¼ DT cfracture ¼
K IC pffiffiffiffiffiffi 1:12Ef as pc
ð15Þ
It is instructive to plot the sustainable temperature jump as a function of crack length. The results of DTc with crack length based on fracture toughness are depicted in Fig. 11. It is clear that the thermal shock resistance predicted by the fracture toughness-based criterion exhibit a strong dependence on preexisting crack length. The stress intensity factor is zero for un-cracked medium. Accordingly, the admissible temperature jump based on fracture toughness will be infinity at crack length c=0. Therefore, when the crack length tends to zero, the fracture toughness-based criterion will be inappropriate. Thus, the failure of the ceramic foam will be governed by the stress-based criterion for small crack lengths. A transient crack length of c1 exists for which DTc is equal for toughness-based failure and stress-based failure. This length can be obtained for Eqs. (12)–(15). The result is
c1 ¼ 9L
ð16Þ
as mentioned above, L is the cell size of the ceramic foam. Therefore, the thermal shock strength of ceramic foams with crack smaller than c1 will be controlled by the stress-based criterion; while a ceramic foam with crack length larger than c1 will be controlled by the fracture-based criterion. If the crack length of ceramic foam is c1, its thermal shock resistance can be established according to the stress-based or the fracture-based criterion.
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Fig. 11. The relationship of thermal shock resistance based on stress and fracture toughness criterion with crack length of ceramic foam.
5. Conclusions The thermal shock resistance of ceramic foams has been investigated theoretically for a semi-infinite ceramic foam containing an edge crack. The surface of the foam undergoes a sudden cooling. The closed-form solutions to the temperature and stress fields were obtained. The results of the stress demonstrated that the maximum stress appears at the surface of the foam. This may cause the most severe thermal shock damage on the surface of the foam, agreeing well with the thermal shock experiments (Yuan, 2008). The propagation manner of pre-existing cracks in the ceramic foam was discussed. After the stress intensity factor of a pre-existing crack reaches the fracture toughness of the foam, the crack jumps suddenly to a new crack length and than grows in a stable way with increasing thermal shock time. In addition, the studies of the effect of the relative density of the ceramic foam have demonstrated that the ceramic foam with lower relative density can sustain severe thermal shock. In addition, the thermal shock resistance of ceramic foams was evaluated according to the stress based and the fracture toughness based criteria. The maximum temperature that the ceramic foam can sustain without catastrophic failure was analyzed according to the maximum local tensile stress criterion and fracture toughness criterion. A transient crack length c1 was identified for which the admissible temperature jump is equal for the stress-controlled failure and the fracture-controlled failure. Thermal shock resistance of ceramic foams with a crack smaller than c1 will be controlled by the stress-based criterion, while a crack larger than c1 will be controlled by the fracture-based criterion. Finally, it should be mentioned that the configuration considered in this paper is relative simple yet important for real structures. Firstly, the thermal shock resistance of a real material is mainly determined by the crack initiation behavior. Therefore, an infinitesimal crack (or a semi-infinite foam medium) considered by the paper is an appropriate approach. Secondly, the thermal shock resistance behavior of the foam is expressed in terms of the relative density of the foam and is presented in non-dimensional form. This avoids to identify the range of real microstructures of materials, which are diverse. Acknowledgments This research was supported by the National Science Foundation of China (projects #10972067, #11172081). BLW is an ARC Future Fellowship at professorial level in the University of New South Wales supported by the Australian Research Council. Reference Dimitrijevic, M., Posarac, M., Majstorovic, J., Volkov-Husovic, T., & Matovic, B. (2009). Behavior of silicon carbide/cordierite composite material after cyclic thermal shock. Ceramics International, 35, 1077–1081. Ding, S. Q., Zeng, Y. P., & Jiang, D. l. (2006). Thermal shock resistance of in situ reaction bonded porous silicon carbide ceramics. Materials Science and Engineering A, 425, 326–329. Erdogan, F., & Rizk, A. A. (1992). Fracture of coated plates and shells under thermal-shock. International Journal of Fracture, 53, 159–185. Evans, A. G., & Charles, E. A. (1977). Structural integrity in severe thermal environments. Journal of the American Ceramic Society, 60, 22–28. Feng, W. J., & Su, R. K. L. (2007). Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate. European Journal of Mechanics A/ Solids, 26, 363–379. Green, D. J., & Colombo, P. (2003). Cellular ceramics: intriguing structures, novel properties and innovative applications. Materials Research Society Bulletin, 28, 296–300. Hasselman, D. P. H. (1963). Elastic energy at fracture and surface energy as design criteria for thermal shock. Journal of the American Ceramic Society, 46, 535–540. Hasselman, D. P. H. (1966). Theory of thermal shock resistance of semitransparent ceramics under radiation heating. Journal of the American Ceramic Society, 49, 103–104. Hasselman, D. P. H. (1969). Unified theory of thermal shock fracture initiation and crack propagation in brittle ceramics. Journal of the American Ceramic Society, 52, 600–604.
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