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Detection of cracks in ceramic matrix composites based on surface temperature
Composite Structures 48 (2000) 71±77
Detection of cracks in ceramic matrix composites based on surface temperature Larry W. Byrd a, Victor M. Birman b,* b
a Wright-Patterson Air Force Base, Dayton, OH, USA Engineering Education Center, University of Missouri-Rolla, 8001 Natural Bridge Road, St. Louis, MO 63121, USA
Abstract Ceramic matrix composites are processed at high temperatures and experience signi®cant residual thermal stresses upon cooling to room temperature. These stresses often result in matrix cracking prior to the application of external loads. Matrix cracks may also appear as a result of thermomechanical loading. It is important to detect these matrix cracks using a nondestructive technique. The method proposed in this paper is based on measurements of the surface temperature of a ceramic matrix material subjected to cyclic stresses. The elevated surface temperature is due to friction between the ®bers and the matrix that occurs in the presence of bridging matrix cracks. The solution presents a relationship between surface temperature and matrix crack spacing that can be used to identify the extent of the damage. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Ceramic matrix composites; Matrix cracks; Nondestructive testing
Nomenclature E DE Ec ÿ E f h k k1 k2 L Nu hL=k r Ra s t T T Ts ÿ Ta V *
modulus of elasticity (GPa) change in modulus of elasticity with crack spacing (GPa) load frequency (Hz) unit surface heat transfer coecient (W/m2 K) air thermal conductivity (W/mK) ÿ6.735 GPa constant in Eq. (14) 6.2754 GPaámm constant in Eq. (14) characteristic length used in Nu number area/perimeter for ¯at surfaces (m) Nusselt number given by Eq. (18) to calculate heat transfer coecient ®ber radius (m) Rayleigh number given by Eq. (19) to calculate Nu for free convection crack spacing (m) specimen thickness (m) temperature (K) surface temperature rise (K) volume fraction
Corresponding author. Tel.: +1-314-516-5431; fax: +1-314-5165434. E-mail address: [email protected] (V.M. Birman).
W w_ x0 Greek a a0 b b0 de e r s m Subscripts a A B C c f m o
0263-8223/00/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 7 5 - 6
instantaneous work done by friction on one ®ber (J) rate of energy dissipation (W) slip length as shown in Fig. 1 (m) coecient of thermal expansion (Kÿ1 ) thermal diusivity of the air (m2 /s) volumetric thermal expansion coecient for the air (Kÿ1 ) 5.67(10ÿ8 )Stefan±Boltzmann constant dynamic strain strain normal stress (Pa) shear stress (Pa) kinematic viscosity of the air (m2 /s) ambient temperature position associated with left-hand crack x 0, Fig. 1 position associated with the end of the left hand slip region, Fig. 1 position associated with the start of the right-hand slip region Fig. 1. composite ®ber matrix operating temperature
72
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
p s
processing temperature surface temperature
Superscripts T
residual thermal stress
1. Introduction The fact that ceramic matrix composites (CMCs) are processed at a high temperature implies signi®cant thermally-induced residual stresses. These stresses can result in damage even before the material is subject to an external load. For example, Bischo et al. [1] and Nishiyama et al. [2] observed post-processing cracking in CMCs. Additional cracks are formed when the material is subject to an external load. The cracks usually form a regular pattern with the spacing that can be assumed constant. Long cracks perpendicular to the ®bers, similar to those observed by Marshall and Evans [3] and other investigators, are called ``bridging cracks'', because the ®bers bridge the cracks without breaking. It is important to be able to determine the extent of damage that can be associated with the density of matrix cracks. This is because matrix cracks degrade the strength and stiness of the material in the ®ber direction. In addition, in CMCs, matrix cracks serve as pathways for oxygen to the ®bers. At high temperatures, this results in oxidation of the ®ber±matrix interface and a dramatic embrittlement of the material [4±7]. In the presence of matrix cracks, the ®bers slide relative to the matrix in the regions adjacent to the planes of the cracks. This sliding that occurs when the material experiences dynamic bending or tension is accompanied with an increase of temperature due to friction between the ®bers and the matrix. This phenomenon was observed by Holmes and his associates [8,9] in their experiments on carbon±®ber SiC matrix composites. Cho et al. [10] developed a solution that related the interfacial shear stress to the rise of temperature of the specimen. It was suggested that the interfacial shear stress along the ®ber±matrix interface can be monitored as a
Fig. 1. Distribution of stresses in the ®bers and in the matrix during cycling (not to scale).
function of temperature. In the present paper, the analytical foundation is developed for prediction of the spacing of post-processing matrix cracks (and the interfacial shear stress) as a function of the surface temperature of a vibrating unidirectional CMC. Simple modi®cations explained in the text enable us to extend the analysis to the case where matrix cracks emerged in the intact material during lifetime. This technique can be applied to a nondestructive testing of CMC components. 2. Analysis The purpose of the present solution is to determine a relationship between the post-processing or load-induced matrix crack spacing in a unidirectional CMC and its surface temperature during a nondestructive dynamic test. The amplitudes of cyclic stresses are assumed below the matrix cracking limit of the material so that cycling does not change the matrix crack spacing. An elevated temperature due to frictional heating is triggered by relative movement of bridged ®bers with respect to the matrix. The solution assumes the mode of cracking employed in the theories of Aveston±Cooper± Kelly [11] and Budiansky±Hutchnison±Evans [12], i.e., long, regularly spaced cracks. Shown in Fig. 1, the problem is formulated in terms of the crack spacing (s), the length of the sliding distance (x0 ), and a distribution of stresses in the ®bers and the matrix. The solution involves the following steps. First, the modulus of elasticity of the material is determined as a function of the interfacial shear stresses, matrix crack spacing and the residual thermal stresses in the ®ber using a modi®ed approach of Pryce and Smith [13]. Residual thermal stresses are evaluated accounting for the eect of temperature on the properties of the constituent materials. Then the experimental ®ndings of Karandnikar and Chou [14] are used to justify a simple relationship between the modulus of elasticity and the matrix spacing. Combining the two solutions referred to above, the modulus of elasticity can be eliminated and a single equation relating the matrix crack spacing to the interfacial shear stress obtained. Subsequently, a balance between the rate of heat ¯ow and the rate of dissipation of the frictional energy is employed, as suggested by Cho et al. [10], to obtain a relationship between the surface temperature and the interfacial shear stress. This procedure enables us to evaluate both the shear stress and the matrix crack spacing as functions of the surface temperature. A distribution of stresses in a specimen subjected to an external stress rc is shown in Fig. 1. Note that this distribution corresponds to a partial slip, i.e., 2x0
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
the matrix can be evaluated based on their values at the points A and B: rc rfA ; Vf rc E f rfB rTf ; Ec 1 rmA 0; rc E m rTm ; rmB Ec where the subscripts ``f'' and ``m'' refer to the ®bers and matrix, respectively, rc is the stress applied to the composite, Vf is the volume fraction of the ®bers, Ef , Em and Ec are the moduli of elasticity of the ®bers, matrix and undamaged composite, respectively, and rTf and rTm are the residual thermal post-processing stresses outside the slippage region. Note that during cycling the composite stress rc varies continuously. Therefore, the stresses given by Eq. (1) represent instantaneous values, although dynamic (viscous) eects are not included in the present analysis. It can be immediately observed that the equilibrium of forces in the cross sections outside the slippage region is satis®ed. The stresses within the slippage region are also balanced at each cross section. Within the slippage region, the stresses in the ®ber and the matrix are linear functions of the distance from the plane of the crack, i.e., x. The stress in the ®ber is [12] rc 2sx ; rf ÿ Vf r
2
where r is the ®ber radius and s is the interfacial shear stress which is assumed constant, as in the theories of Aveston±Cooper±Kelly and Budiansky±Hutchinson± Evans. Note that the ®nite element solution of Sorensen et al. [15] showed that the maximum variation of the interfacial shear stress is 15% which supports the theoretical solutions based on the assumption of constant shear stress. The stress in the matrix within the slippage region is Em x T rm ; 3 rm rc Ec x0 where x0 is the half-length of the slippage region that can be determined according to Pryce and Smith [13] as r V m Em T ÿ rf 4 rc x0 V f Ec 2s Vm being the matrix volume fraction. The residual thermal stresses in the ®ber and in the matrix can be found from the force equilibrium that should be preserved at an arbitrary cross section, i.e., Ef Vf e ÿ af DT Em Vm e ÿ am DT 0;
5
where af and am are the coecients of thermal expansion of the ®bers and the matrix, respectively, and DT is a
73
dierence between the processing and operating temperatures. The strain, e, can be immediately evaluated from Eq. (5). However, considering the fact that the processing temperature of CMCs is usually above 1200°C, it may be necessary to account for an eect of the temperature on the properties of the materials of the ®bers and the matrix. In this case, the strain will be found from Z To Ef T Vf af T Em T Vm am T e dT ; 6 Ef T Vf Em T Vm Tp where the integration is carried out from the processing (Tp ) to the operating (To ) temperature. An example of analytical expressions for the moduli of elasticity and the coecients of thermal expansion of CMCs as functions of temperature can be found in the report of NASA TM 106789. The residual stresses in the region that is not aected by the slip are found as rTf Ef e ÿ af DT Z To Em T Vm am T ÿ af T dT ; Ef To Ef T Vf Em T Vm Tp rTm Em e ÿ am DT Z To Ef T Vf af T ÿ am T dT : Em To Ef T Vf Em T Vm Tp
7
It is now necessary to evaluate an instantaneous modulus of elasticity of the material. This procedure follows the approach adapted by Pryce and Smith [13], although the problem of post-processing cracking considered in the present solution is dierent which dictates the modi®cations outlined below. The instantaneous mean strain in the ®ber is found by averaging the strains over the spacing length, i.e., ef
2x0 s ÿ 2x0 eAB eBC ; s s
8
where the mean strain within the slippage region eAB and the strain outside the slippage region eBC can be expressed in terms of the ®ber stresses. The substitution of these strains into Eq. (8) yields x0 rc 2Vf Ef Vm Em rTf ef s Ef V f E f Ec s ÿ 2x0 rc rTf : 9 s Ec E f Note that the strain given by Eq. (9) includes the post-processing residual and the additional cycling components. The increase of the strain due to cyclic loading is represented by the latter component. The mean residual strain is found from an analog of Eq. (8) where the mean strain in the region AB is given by rTf / 2Ef , while the strain within the region BC is rTf =Ef (see Fig. 2). Accordingly,
74
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
s
Ec ÿ k1 ÿ k2 =s k1 k2 =s
r Em V m Em V m ÿ rTf : 2rc Ec V f 2s Ef Vf 15
Fig. 2. Thermal residual stresses in a unidirectional material with postprocessing cracks (not to scale).
eTf
s ÿ x0 rTf : s Ef
10
Note that this expression diers from the mean residual strain for the initially uncracked material used in [13] The additional strain associated with cyclic loading can now be found as a dierence of the strains given by Eqs. (9) and (10), i.e. 3 2 Vm Em T r ÿ r E V c m m f Vf Ec r 5 rc : e 41 11 V f Ef Ec 2ss as
The instantaneous modulus of elasticity is determined
E
drc de
yielding ÿ1 r Em V m Em V m ÿ rTf 2rc Ec : E 1 Ec V f 2ss Ef Vf
12
2 q h Ts ÿ Ta e0 b0 Ts4 ÿ Ta4 : t 13
If the material remained intact after processing and the cracks developed later during lifetime, the thermal residual stress in the right side must be multiplied by 2 [13]. Note that Eq. (13) presents the elastic modulus of a cracked material that is not aected by cycling. Accordingly, it is possible to compare this result to experimental ®ndings of Karandikar and Chou [14] who showed that the change of the modulus of a unidirectional Nicalon ®ber CAS matrix composite is a linear function of the matrix crack density (1/s) DE Ec ÿ E k1 k2 1=s;
Now it is necessary to ®nd a relationship between the interfacial shear stress and the surface temperature of the specimen. This problem can be addressed by considering the equilibrium between the rate of the steady state heat loss from a unit volume of the specimen and the rate of work performed by the interfacial friction within this volume. The former quantity was presented in the paper of Cho et al. [10] for the general case where the heat loss occurs through conduction in the ®ber direction and free convection and radiation from the surface. In the case of a uniform crack distribution and small-amplitude vibrations excited in the course of a nondestructive test prior to any additional fatigue loading, the temperature may be assumed independent of the axial coordinate oriented along the ®bers. Therefore, no conduction takes place in the ®ber direction. At the sea level the radiation is typically small compared to free convection (although the relative contribution of radiation increases with altitude). Therefore, radiation from the surface may be neglected, although this assumption is not necessary for the solution. Retaining the radiation contribution, the rate of the heat loss from the element with the surface area As (including both surfaces of the specimen) and the volume V As t/2 where t is the thickness becomes
14
where k1 and k2 are constants. For Nicalon ®ber CAS matrix, k1 ÿ6.7350 GPa and k2 6.2754 GPamm. Note that the present approach to the solution remains valid as long as an arbitrary analytical relationship DE f(s) is available. A combination of Eqs. (13) and (14) yields the relationship between the interfacial shear stress and the matrix crack spacing
16
In Eq. (16), Ts and Ta are the surface and ambient air temperatures, respectively, h is the heat transfer coecient, e0 is the emissivity and b0 5.67 ´ 10ÿ8 W/m2 K4 is the Stefan±Boltzman constant. As indicated above, the second term in the square brackets in Eq. (16) may be negligible. Note that the heat transfer coecient refers to the mean properties of the ®lm adjacent to the surface of the specimen, i.e., its evaluation requires the knowledge of T Ts Ta =2. The heat transfer coecients from the surfaces of a representative element can be determined as hi
Nu k ; L
17
where the subscript identi®es the surface, Nu is the Nusselt number, L is the ratio of the surface area of the element to its perimeter and k is the thermal conductivity of the ®lm. The Nusselt number can be found as a function of the Rayleigh number (see, for example, Incropera, F.P. and DeWitt, D.P., ``Fundamentals of Heat and Mass Transfer'', 4th edition, Wiley, New York, 1996). In particular, if the surfaces losing heat are horizontal, the following formulae apply:
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
Upper surface : Nu 0:27 Ra1=4 Lower surface : Nu 0:54 Ra
1=4
0:15 Ra1=3
for 105 < Ra < 1010 ; 4
for 10 < Ra < 10
7
for 107 < Ra < 1011 : 18
The Rayleigh number can be calculated by gb Ts ÿ Ta L3 ; 19 mao where g is the gravity acceleration, b is the air volumetric thermal expansion coecient, m is the kinematic viscosity and ao is the thermal diusivity of the ®lm at its average temperature. The instantaneous work produced by the interfacial friction on the slippage length of one ®ber is obtained as [10] Z x0 2p r uf ÿ um dx; 20 W 2
Ra
0
where uf and um are dynamic components of the axial displacements of the ®ber and the matrix due to cyclic loading that are functions of the x-coordinate. Note that the upper limit of integration given by Eq. (20) is affected by the magnitude of the interfacial shear stress. The dierence between dynamic components of the axial ®ber and matrix displacements can be found as 1 21 uf ÿ um def x ÿ dem x x0 ÿ x; 2 where def and dem are the dynamic strains at the cross section x. This equation re¯ects the fact that the ®ber and the matrix experience identical axial displacements at x x0 , and the change of the length of the element (x0 ÿ x) within the slippage region can be found as the mean strain within this element multiplied by its length. The dynamic strain in the ®ber is determined as a dierence between the total and residual strains. The former can be found from Eq. (2) but it is more convenient to use the following expression that immediately follows from Fig. 1 rc rc rc Ef x T ÿ ÿ Eÿ1 : ÿ rf 22 ef x Vf Vf Ec x0 f The residual strain is of course (see Fig. 2), (rTf =Ef x=x0 ). Obviously, if cracks developed in the intact material during lifetime, the residual strain is constant and independent of x. Now the dynamic components of the strains in the ®bers and in the matrix can be found as 1 1 E f x rc ÿ ÿ def x Vf V f Ec x 0 E f 23 x rc dem x : x 0 Ec Substituting dynamic strains given by Eq. (23) into Eq. (21) and subsequently, integrating Eq. (20) one obtains
W
2p rs rc 2 x: 3Vf Ef 0
75
24
Note that the composite stress in Eq. (24) represents the maximum stress during the cycle, while the minimum stress is assumed equal to zero. In the case of a stress ratio dierent from zero, the composite stress should be replaced with the stress range. The rate of the frictional energy dissipation per unit volume can be found as recommended by Cho et al. [10], i.e., w_ 2fW = p r2 s=Vf ;
25
where f is the frequency of loading and the factor 2 in the numerator accounts for the fact that equal amounts of energy are generated during the loading and unloading phases of each cycle. The solution can be obtained by prescribing the surface temperature. Then a relationship between the matrix crack spacing and the interfacial shear stress can be obtained from the requirement that the rate of the heat loss given by Eq. (16) must be equal to the rate of the frictional energy dissipation according to Eq. (25). This relationship should be considered together with Eq. (15) to specify both the interfacial shear stress as well as the matrix crack spacing.
3. Results A parametric study of the model was undertaken with Em 100 GPa and Ef 200 GPa. The heat transfer coecient was approximated as a constant h 6 W/m2 K. The coecients of thermal expansion were assumed constant with values of af 3(10ÿ6 ) and am 4.5(10ÿ6 ) Kÿ1 . The dierence between the processing and use temperature was assumed to be 600°K. The ®ber radius was 10 lm. The surface temperature rise, DT Ts ÿ Ta , was then found as a function of the crack spacing s, the mean composite stress rc , the frequency of loading f, and the ®ber volume fraction Vf as shown in Figs. 3±6. The interfacial shear was also calculated as a function of the mean composite stress as shown in Fig. 7. It should be noted that the form of the empirical relation between the decrease in YoungÕs modulus and the crack density places some restraints on the values of s. For DE to remain positive k1 + k2 /s must be positive as well. This restricts s to values less than 0.932 mm which is in agreement with the recommendation of Karandikar and Chou [14]. There is also a minimum crack spacing which can be calculated from Eq. (4) once the shear stress has been determined. The slippage half length x0 cannot be greater than s/2 or the analysis is invalid. From Fig. 7, s 0.7662rc ÿ 7.37. For rc larger than 100 MPa this can be approximated as s 0.7662rc with less than a 10% error. Substituting this into Eq. (4) the
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L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
Fig. 6. Surface temperature rise as a function of ®ber volume fraction. s 0:4 mm; rc 200 MPa; f 10 Hz. Fig. 3. Surface temperature rise as a function of crack spacing. Vf 0:35; f 10Hz; rc 200 MPa.
Fig. 7. Shear stress as a function of mean composite stress. Vf 0:35; s 0:4 mm; f 10 Hz.
Fig. 4. Surface temperature rise as a function of mean composite stress. Vf 0:35; s 0:4 mm; f 10 Hz.
Fig. 5. Surface temperature rise as a function of loading frequency. Vf 0:35; s 0:4 mm; rc 200 MPa.
minimum crack spacing as a function of the load and residual stress is approximately: r Vm Em rTf ÿ : 26 s> rc 0:7662 Vf Ef While this equation is very speci®c to the particular parameter values that were studied, the relationship between the variables in Eq. (26) should be qualitatively
correct. For this study for rc 200 MPa, the minimum crack spacing is on the order of 0.019 mm. This is considerably smaller than minimum value used in Fig. 3 so the range of the crack spacing considered there should be valid. Fig. 3 shows that DT increases as the crack spacing decreases. This is expected because there is increasing damage and more sliding between the matrix and ®bers as the crack density increases. As seen in Fig. 4, DT increases with the mean composite stress. This is expected because more work is being done as the composite stress increases. Fig. 5 illustrates that the surface temperature rise increases linearly with the frequency. This is expected because the heat transfer coecient has been approximated as a constant and the frequency appears as a linear factor in the energy balance. Fig. 6 shows that DT decreases as the ®ber fraction increases. This is because the mean composite stress has been held constant. As the ®ber fraction increases there is less sliding because the composite modulus increases. As shown in Fig. 7 the interfacial shear stress increases linearly for rc > 50 MPa. It does not go to zero at rc 0 because of the residual thermal stress.
4. Discussion and conclusions The solution presented in this paper outlines the theoretical background for a nondestructive detection of
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
the presence and density of matrix cracks in unidirectional ceramic matrix composite materials. The detailed analysis is presented for the case of post-processing cracks. However, this analysis can easily be extended for cracks developed in the intact material during lifetime. The density of the matrix cracks and the interfacial shear stress can be evaluated using this solution, based on the measurement of the surface temperature of the component subjected to forced periodic vibrations. The solution is obtained in a closed-form, i.e., it is accurate as long as the basic assumptions incorporated into the analysis are valid. In particular, these assumptions include the form of matrix cracking, i.e., long bridging cracks, and the presumed analytical relationship between the change of the modulus of elasticity and the matrix crack spacing. However, these assumptions are justi®ed by available experimental evidence. Note that vibrations of the component are assumed to take place in such manner that the stresses are uniform throughout the thickness. Another factor that will be considered in the future research is an eect of viscosity of the matrix (and possibly the ®bers) on the frictional heating.
[2]
[3] [4]
[5] [6] [7] [8] [9] [10]
Acknowledgements This paper was written with the support of the Air Force Oce of Scienti®c Research and Wright Laboratory, (Contract number F49620-93-C-0063). The program manager is Dr. Brian sanders.
[11] [12] [13] [14]
References [1] Bischo E, Ruhle M, Sbaizero O, Evans AG. Microstructural studies of the interface zone of a SiC-®ber-reinforced lithium
[15]
77
aluminum silicate glass±ceramic. J Am Ceramic Soc 1989; 72:741±5. Nishiyama K, Umekawa S, Yuasa M, Fukumoto I. Eect of ®ber coating on interfacial phenomena in tungsten-®ber/boron carbide composites. In: Proceedings of the Fourth Japan-US Conference on Composite Material, 1988, Technomic, Lancaster, PA, pp. 776±788. Marshall DB, Evans AG. Failure mechanisms in ceramic±®ber/ ceramic±matrix composites. J Am Ceramic Soc 1985;68:225±31. Wetherhold RC, Zawada LP. Fractography of glasses and ceramics. In: Frechete VD, Varner JP, editors. Ceramic Transactions, vol. 17. American Ceramic Society, Westerville, Ohio, 1991:391. Zawada LP, Wetherhold RC. The eect of thermal fatigue on a SiC ®bre-reinforced nitrogen glass matrix composites. J Mater Sci 1991;26:648±54. Kahraman R. A microdebonding study of the high-temperature oxidation embrittlement of a cross-ply glass±ceramic/SiC composite. Compos Sci Technol 1996;56:1453±9. Westwood ME, Webster JD, Day RJ, Hayes FH, Taylor R. Review. oxidation protection for carbon ®bre composites. J Mater Sci 1996;31:1389±97. Holmes JW, Shuler SF. Temperature rise during fatigue of ®brereinforced ceramics. J Mater Sci Lett 1990;9:1290±1. Holmes JW, Cho C. Experimental observations of frictional heating in ®ber-reinforced ceramics. J Am Ceramic Soc 1992;75:929±38. Cho C, Holmes JW, Barber JR. Estimation of interfacial shear in ceramic composites from frictional heating measurements. J Am Ceramic Soc 1991;74:2802±8. Aveston J, Kelly A. Theory of multiple fracture of ®brous composites. J Mater Sci 1973;8:352±62. Budiansky B, Hutchinson JW, Evans AG. Matrix fracture in ®ber-reinforced ceramics. J Mech Phys Solids 1986;34:167±89. Pryce AW, Smith PA. Matrix cracking in unidirectional ceramic matrix composites under quasi-static and cyclic loading. Acta Mettalurgica et Materialia 1993;41:1269±81. Karandikar PG, Chou T-W. Microcracking and elastic moduli reductions in unidirectional Nicalon-CAS composite under cyclic fatigue loading. Ceramic Eng Sci Proc 1992;13:882±8. Sorensen BF, Talreja R, Sorensen OT. Micromechanical analysis of damage mechanics in ceramic±matrix composites during mechanical and thermal cycling . Composites 1993;24:129±40.
Detection of cracks in ceramic matrix composites based on surface temperature Larry W. Byrd a, Victor M. Birman b,* b
a Wright-Patterson Air Force Base, Dayton, OH, USA Engineering Education Center, University of Missouri-Rolla, 8001 Natural Bridge Road, St. Louis, MO 63121, USA
Abstract Ceramic matrix composites are processed at high temperatures and experience signi®cant residual thermal stresses upon cooling to room temperature. These stresses often result in matrix cracking prior to the application of external loads. Matrix cracks may also appear as a result of thermomechanical loading. It is important to detect these matrix cracks using a nondestructive technique. The method proposed in this paper is based on measurements of the surface temperature of a ceramic matrix material subjected to cyclic stresses. The elevated surface temperature is due to friction between the ®bers and the matrix that occurs in the presence of bridging matrix cracks. The solution presents a relationship between surface temperature and matrix crack spacing that can be used to identify the extent of the damage. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Ceramic matrix composites; Matrix cracks; Nondestructive testing
Nomenclature E DE Ec ÿ E f h k k1 k2 L Nu hL=k r Ra s t T T Ts ÿ Ta V *
modulus of elasticity (GPa) change in modulus of elasticity with crack spacing (GPa) load frequency (Hz) unit surface heat transfer coecient (W/m2 K) air thermal conductivity (W/mK) ÿ6.735 GPa constant in Eq. (14) 6.2754 GPaámm constant in Eq. (14) characteristic length used in Nu number area/perimeter for ¯at surfaces (m) Nusselt number given by Eq. (18) to calculate heat transfer coecient ®ber radius (m) Rayleigh number given by Eq. (19) to calculate Nu for free convection crack spacing (m) specimen thickness (m) temperature (K) surface temperature rise (K) volume fraction
Corresponding author. Tel.: +1-314-516-5431; fax: +1-314-5165434. E-mail address: [email protected] (V.M. Birman).
W w_ x0 Greek a a0 b b0 de e r s m Subscripts a A B C c f m o
0263-8223/00/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 7 5 - 6
instantaneous work done by friction on one ®ber (J) rate of energy dissipation (W) slip length as shown in Fig. 1 (m) coecient of thermal expansion (Kÿ1 ) thermal diusivity of the air (m2 /s) volumetric thermal expansion coecient for the air (Kÿ1 ) 5.67(10ÿ8 )Stefan±Boltzmann constant dynamic strain strain normal stress (Pa) shear stress (Pa) kinematic viscosity of the air (m2 /s) ambient temperature position associated with left-hand crack x 0, Fig. 1 position associated with the end of the left hand slip region, Fig. 1 position associated with the start of the right-hand slip region Fig. 1. composite ®ber matrix operating temperature
72
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
p s
processing temperature surface temperature
Superscripts T
residual thermal stress
1. Introduction The fact that ceramic matrix composites (CMCs) are processed at a high temperature implies signi®cant thermally-induced residual stresses. These stresses can result in damage even before the material is subject to an external load. For example, Bischo et al. [1] and Nishiyama et al. [2] observed post-processing cracking in CMCs. Additional cracks are formed when the material is subject to an external load. The cracks usually form a regular pattern with the spacing that can be assumed constant. Long cracks perpendicular to the ®bers, similar to those observed by Marshall and Evans [3] and other investigators, are called ``bridging cracks'', because the ®bers bridge the cracks without breaking. It is important to be able to determine the extent of damage that can be associated with the density of matrix cracks. This is because matrix cracks degrade the strength and stiness of the material in the ®ber direction. In addition, in CMCs, matrix cracks serve as pathways for oxygen to the ®bers. At high temperatures, this results in oxidation of the ®ber±matrix interface and a dramatic embrittlement of the material [4±7]. In the presence of matrix cracks, the ®bers slide relative to the matrix in the regions adjacent to the planes of the cracks. This sliding that occurs when the material experiences dynamic bending or tension is accompanied with an increase of temperature due to friction between the ®bers and the matrix. This phenomenon was observed by Holmes and his associates [8,9] in their experiments on carbon±®ber SiC matrix composites. Cho et al. [10] developed a solution that related the interfacial shear stress to the rise of temperature of the specimen. It was suggested that the interfacial shear stress along the ®ber±matrix interface can be monitored as a
Fig. 1. Distribution of stresses in the ®bers and in the matrix during cycling (not to scale).
function of temperature. In the present paper, the analytical foundation is developed for prediction of the spacing of post-processing matrix cracks (and the interfacial shear stress) as a function of the surface temperature of a vibrating unidirectional CMC. Simple modi®cations explained in the text enable us to extend the analysis to the case where matrix cracks emerged in the intact material during lifetime. This technique can be applied to a nondestructive testing of CMC components. 2. Analysis The purpose of the present solution is to determine a relationship between the post-processing or load-induced matrix crack spacing in a unidirectional CMC and its surface temperature during a nondestructive dynamic test. The amplitudes of cyclic stresses are assumed below the matrix cracking limit of the material so that cycling does not change the matrix crack spacing. An elevated temperature due to frictional heating is triggered by relative movement of bridged ®bers with respect to the matrix. The solution assumes the mode of cracking employed in the theories of Aveston±Cooper± Kelly [11] and Budiansky±Hutchnison±Evans [12], i.e., long, regularly spaced cracks. Shown in Fig. 1, the problem is formulated in terms of the crack spacing (s), the length of the sliding distance (x0 ), and a distribution of stresses in the ®bers and the matrix. The solution involves the following steps. First, the modulus of elasticity of the material is determined as a function of the interfacial shear stresses, matrix crack spacing and the residual thermal stresses in the ®ber using a modi®ed approach of Pryce and Smith [13]. Residual thermal stresses are evaluated accounting for the eect of temperature on the properties of the constituent materials. Then the experimental ®ndings of Karandnikar and Chou [14] are used to justify a simple relationship between the modulus of elasticity and the matrix spacing. Combining the two solutions referred to above, the modulus of elasticity can be eliminated and a single equation relating the matrix crack spacing to the interfacial shear stress obtained. Subsequently, a balance between the rate of heat ¯ow and the rate of dissipation of the frictional energy is employed, as suggested by Cho et al. [10], to obtain a relationship between the surface temperature and the interfacial shear stress. This procedure enables us to evaluate both the shear stress and the matrix crack spacing as functions of the surface temperature. A distribution of stresses in a specimen subjected to an external stress rc is shown in Fig. 1. Note that this distribution corresponds to a partial slip, i.e., 2x0
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
the matrix can be evaluated based on their values at the points A and B: rc rfA ; Vf rc E f rfB rTf ; Ec 1 rmA 0; rc E m rTm ; rmB Ec where the subscripts ``f'' and ``m'' refer to the ®bers and matrix, respectively, rc is the stress applied to the composite, Vf is the volume fraction of the ®bers, Ef , Em and Ec are the moduli of elasticity of the ®bers, matrix and undamaged composite, respectively, and rTf and rTm are the residual thermal post-processing stresses outside the slippage region. Note that during cycling the composite stress rc varies continuously. Therefore, the stresses given by Eq. (1) represent instantaneous values, although dynamic (viscous) eects are not included in the present analysis. It can be immediately observed that the equilibrium of forces in the cross sections outside the slippage region is satis®ed. The stresses within the slippage region are also balanced at each cross section. Within the slippage region, the stresses in the ®ber and the matrix are linear functions of the distance from the plane of the crack, i.e., x. The stress in the ®ber is [12] rc 2sx ; rf ÿ Vf r
2
where r is the ®ber radius and s is the interfacial shear stress which is assumed constant, as in the theories of Aveston±Cooper±Kelly and Budiansky±Hutchinson± Evans. Note that the ®nite element solution of Sorensen et al. [15] showed that the maximum variation of the interfacial shear stress is 15% which supports the theoretical solutions based on the assumption of constant shear stress. The stress in the matrix within the slippage region is Em x T rm ; 3 rm rc Ec x0 where x0 is the half-length of the slippage region that can be determined according to Pryce and Smith [13] as r V m Em T ÿ rf 4 rc x0 V f Ec 2s Vm being the matrix volume fraction. The residual thermal stresses in the ®ber and in the matrix can be found from the force equilibrium that should be preserved at an arbitrary cross section, i.e., Ef Vf e ÿ af DT Em Vm e ÿ am DT 0;
5
where af and am are the coecients of thermal expansion of the ®bers and the matrix, respectively, and DT is a
73
dierence between the processing and operating temperatures. The strain, e, can be immediately evaluated from Eq. (5). However, considering the fact that the processing temperature of CMCs is usually above 1200°C, it may be necessary to account for an eect of the temperature on the properties of the materials of the ®bers and the matrix. In this case, the strain will be found from Z To Ef T Vf af T Em T Vm am T e dT ; 6 Ef T Vf Em T Vm Tp where the integration is carried out from the processing (Tp ) to the operating (To ) temperature. An example of analytical expressions for the moduli of elasticity and the coecients of thermal expansion of CMCs as functions of temperature can be found in the report of NASA TM 106789. The residual stresses in the region that is not aected by the slip are found as rTf Ef e ÿ af DT Z To Em T Vm am T ÿ af T dT ; Ef To Ef T Vf Em T Vm Tp rTm Em e ÿ am DT Z To Ef T Vf af T ÿ am T dT : Em To Ef T Vf Em T Vm Tp
7
It is now necessary to evaluate an instantaneous modulus of elasticity of the material. This procedure follows the approach adapted by Pryce and Smith [13], although the problem of post-processing cracking considered in the present solution is dierent which dictates the modi®cations outlined below. The instantaneous mean strain in the ®ber is found by averaging the strains over the spacing length, i.e., ef
2x0 s ÿ 2x0 eAB eBC ; s s
8
where the mean strain within the slippage region eAB and the strain outside the slippage region eBC can be expressed in terms of the ®ber stresses. The substitution of these strains into Eq. (8) yields x0 rc 2Vf Ef Vm Em rTf ef s Ef V f E f Ec s ÿ 2x0 rc rTf : 9 s Ec E f Note that the strain given by Eq. (9) includes the post-processing residual and the additional cycling components. The increase of the strain due to cyclic loading is represented by the latter component. The mean residual strain is found from an analog of Eq. (8) where the mean strain in the region AB is given by rTf / 2Ef , while the strain within the region BC is rTf =Ef (see Fig. 2). Accordingly,
74
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
s
Ec ÿ k1 ÿ k2 =s k1 k2 =s
r Em V m Em V m ÿ rTf : 2rc Ec V f 2s Ef Vf 15
Fig. 2. Thermal residual stresses in a unidirectional material with postprocessing cracks (not to scale).
eTf
s ÿ x0 rTf : s Ef
10
Note that this expression diers from the mean residual strain for the initially uncracked material used in [13] The additional strain associated with cyclic loading can now be found as a dierence of the strains given by Eqs. (9) and (10), i.e. 3 2 Vm Em T r ÿ r E V c m m f Vf Ec r 5 rc : e 41 11 V f Ef Ec 2ss as
The instantaneous modulus of elasticity is determined
E
drc de
yielding ÿ1 r Em V m Em V m ÿ rTf 2rc Ec : E 1 Ec V f 2ss Ef Vf
12
2 q h Ts ÿ Ta e0 b0 Ts4 ÿ Ta4 : t 13
If the material remained intact after processing and the cracks developed later during lifetime, the thermal residual stress in the right side must be multiplied by 2 [13]. Note that Eq. (13) presents the elastic modulus of a cracked material that is not aected by cycling. Accordingly, it is possible to compare this result to experimental ®ndings of Karandikar and Chou [14] who showed that the change of the modulus of a unidirectional Nicalon ®ber CAS matrix composite is a linear function of the matrix crack density (1/s) DE Ec ÿ E k1 k2 1=s;
Now it is necessary to ®nd a relationship between the interfacial shear stress and the surface temperature of the specimen. This problem can be addressed by considering the equilibrium between the rate of the steady state heat loss from a unit volume of the specimen and the rate of work performed by the interfacial friction within this volume. The former quantity was presented in the paper of Cho et al. [10] for the general case where the heat loss occurs through conduction in the ®ber direction and free convection and radiation from the surface. In the case of a uniform crack distribution and small-amplitude vibrations excited in the course of a nondestructive test prior to any additional fatigue loading, the temperature may be assumed independent of the axial coordinate oriented along the ®bers. Therefore, no conduction takes place in the ®ber direction. At the sea level the radiation is typically small compared to free convection (although the relative contribution of radiation increases with altitude). Therefore, radiation from the surface may be neglected, although this assumption is not necessary for the solution. Retaining the radiation contribution, the rate of the heat loss from the element with the surface area As (including both surfaces of the specimen) and the volume V As t/2 where t is the thickness becomes
14
where k1 and k2 are constants. For Nicalon ®ber CAS matrix, k1 ÿ6.7350 GPa and k2 6.2754 GPamm. Note that the present approach to the solution remains valid as long as an arbitrary analytical relationship DE f(s) is available. A combination of Eqs. (13) and (14) yields the relationship between the interfacial shear stress and the matrix crack spacing
16
In Eq. (16), Ts and Ta are the surface and ambient air temperatures, respectively, h is the heat transfer coecient, e0 is the emissivity and b0 5.67 ´ 10ÿ8 W/m2 K4 is the Stefan±Boltzman constant. As indicated above, the second term in the square brackets in Eq. (16) may be negligible. Note that the heat transfer coecient refers to the mean properties of the ®lm adjacent to the surface of the specimen, i.e., its evaluation requires the knowledge of T Ts Ta =2. The heat transfer coecients from the surfaces of a representative element can be determined as hi
Nu k ; L
17
where the subscript identi®es the surface, Nu is the Nusselt number, L is the ratio of the surface area of the element to its perimeter and k is the thermal conductivity of the ®lm. The Nusselt number can be found as a function of the Rayleigh number (see, for example, Incropera, F.P. and DeWitt, D.P., ``Fundamentals of Heat and Mass Transfer'', 4th edition, Wiley, New York, 1996). In particular, if the surfaces losing heat are horizontal, the following formulae apply:
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
Upper surface : Nu 0:27 Ra1=4 Lower surface : Nu 0:54 Ra
1=4
0:15 Ra1=3
for 105 < Ra < 1010 ; 4
for 10 < Ra < 10
7
for 107 < Ra < 1011 : 18
The Rayleigh number can be calculated by gb Ts ÿ Ta L3 ; 19 mao where g is the gravity acceleration, b is the air volumetric thermal expansion coecient, m is the kinematic viscosity and ao is the thermal diusivity of the ®lm at its average temperature. The instantaneous work produced by the interfacial friction on the slippage length of one ®ber is obtained as [10] Z x0 2p r uf ÿ um dx; 20 W 2
Ra
0
where uf and um are dynamic components of the axial displacements of the ®ber and the matrix due to cyclic loading that are functions of the x-coordinate. Note that the upper limit of integration given by Eq. (20) is affected by the magnitude of the interfacial shear stress. The dierence between dynamic components of the axial ®ber and matrix displacements can be found as 1 21 uf ÿ um def x ÿ dem x x0 ÿ x; 2 where def and dem are the dynamic strains at the cross section x. This equation re¯ects the fact that the ®ber and the matrix experience identical axial displacements at x x0 , and the change of the length of the element (x0 ÿ x) within the slippage region can be found as the mean strain within this element multiplied by its length. The dynamic strain in the ®ber is determined as a dierence between the total and residual strains. The former can be found from Eq. (2) but it is more convenient to use the following expression that immediately follows from Fig. 1 rc rc rc Ef x T ÿ ÿ Eÿ1 : ÿ rf 22 ef x Vf Vf Ec x0 f The residual strain is of course (see Fig. 2), (rTf =Ef x=x0 ). Obviously, if cracks developed in the intact material during lifetime, the residual strain is constant and independent of x. Now the dynamic components of the strains in the ®bers and in the matrix can be found as 1 1 E f x rc ÿ ÿ def x Vf V f Ec x 0 E f 23 x rc dem x : x 0 Ec Substituting dynamic strains given by Eq. (23) into Eq. (21) and subsequently, integrating Eq. (20) one obtains
W
2p rs rc 2 x: 3Vf Ef 0
75
24
Note that the composite stress in Eq. (24) represents the maximum stress during the cycle, while the minimum stress is assumed equal to zero. In the case of a stress ratio dierent from zero, the composite stress should be replaced with the stress range. The rate of the frictional energy dissipation per unit volume can be found as recommended by Cho et al. [10], i.e., w_ 2fW = p r2 s=Vf ;
25
where f is the frequency of loading and the factor 2 in the numerator accounts for the fact that equal amounts of energy are generated during the loading and unloading phases of each cycle. The solution can be obtained by prescribing the surface temperature. Then a relationship between the matrix crack spacing and the interfacial shear stress can be obtained from the requirement that the rate of the heat loss given by Eq. (16) must be equal to the rate of the frictional energy dissipation according to Eq. (25). This relationship should be considered together with Eq. (15) to specify both the interfacial shear stress as well as the matrix crack spacing.
3. Results A parametric study of the model was undertaken with Em 100 GPa and Ef 200 GPa. The heat transfer coecient was approximated as a constant h 6 W/m2 K. The coecients of thermal expansion were assumed constant with values of af 3(10ÿ6 ) and am 4.5(10ÿ6 ) Kÿ1 . The dierence between the processing and use temperature was assumed to be 600°K. The ®ber radius was 10 lm. The surface temperature rise, DT Ts ÿ Ta , was then found as a function of the crack spacing s, the mean composite stress rc , the frequency of loading f, and the ®ber volume fraction Vf as shown in Figs. 3±6. The interfacial shear was also calculated as a function of the mean composite stress as shown in Fig. 7. It should be noted that the form of the empirical relation between the decrease in YoungÕs modulus and the crack density places some restraints on the values of s. For DE to remain positive k1 + k2 /s must be positive as well. This restricts s to values less than 0.932 mm which is in agreement with the recommendation of Karandikar and Chou [14]. There is also a minimum crack spacing which can be calculated from Eq. (4) once the shear stress has been determined. The slippage half length x0 cannot be greater than s/2 or the analysis is invalid. From Fig. 7, s 0.7662rc ÿ 7.37. For rc larger than 100 MPa this can be approximated as s 0.7662rc with less than a 10% error. Substituting this into Eq. (4) the
76
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
Fig. 6. Surface temperature rise as a function of ®ber volume fraction. s 0:4 mm; rc 200 MPa; f 10 Hz. Fig. 3. Surface temperature rise as a function of crack spacing. Vf 0:35; f 10Hz; rc 200 MPa.
Fig. 7. Shear stress as a function of mean composite stress. Vf 0:35; s 0:4 mm; f 10 Hz.
Fig. 4. Surface temperature rise as a function of mean composite stress. Vf 0:35; s 0:4 mm; f 10 Hz.
Fig. 5. Surface temperature rise as a function of loading frequency. Vf 0:35; s 0:4 mm; rc 200 MPa.
minimum crack spacing as a function of the load and residual stress is approximately: r Vm Em rTf ÿ : 26 s> rc 0:7662 Vf Ef While this equation is very speci®c to the particular parameter values that were studied, the relationship between the variables in Eq. (26) should be qualitatively
correct. For this study for rc 200 MPa, the minimum crack spacing is on the order of 0.019 mm. This is considerably smaller than minimum value used in Fig. 3 so the range of the crack spacing considered there should be valid. Fig. 3 shows that DT increases as the crack spacing decreases. This is expected because there is increasing damage and more sliding between the matrix and ®bers as the crack density increases. As seen in Fig. 4, DT increases with the mean composite stress. This is expected because more work is being done as the composite stress increases. Fig. 5 illustrates that the surface temperature rise increases linearly with the frequency. This is expected because the heat transfer coecient has been approximated as a constant and the frequency appears as a linear factor in the energy balance. Fig. 6 shows that DT decreases as the ®ber fraction increases. This is because the mean composite stress has been held constant. As the ®ber fraction increases there is less sliding because the composite modulus increases. As shown in Fig. 7 the interfacial shear stress increases linearly for rc > 50 MPa. It does not go to zero at rc 0 because of the residual thermal stress.
4. Discussion and conclusions The solution presented in this paper outlines the theoretical background for a nondestructive detection of
L.W. Byrd, V.M. Birman / Composite Structures 48 (2000) 71±77
the presence and density of matrix cracks in unidirectional ceramic matrix composite materials. The detailed analysis is presented for the case of post-processing cracks. However, this analysis can easily be extended for cracks developed in the intact material during lifetime. The density of the matrix cracks and the interfacial shear stress can be evaluated using this solution, based on the measurement of the surface temperature of the component subjected to forced periodic vibrations. The solution is obtained in a closed-form, i.e., it is accurate as long as the basic assumptions incorporated into the analysis are valid. In particular, these assumptions include the form of matrix cracking, i.e., long bridging cracks, and the presumed analytical relationship between the change of the modulus of elasticity and the matrix crack spacing. However, these assumptions are justi®ed by available experimental evidence. Note that vibrations of the component are assumed to take place in such manner that the stresses are uniform throughout the thickness. Another factor that will be considered in the future research is an eect of viscosity of the matrix (and possibly the ®bers) on the frictional heating.
[2]
[3] [4]
[5] [6] [7] [8] [9] [10]
Acknowledgements This paper was written with the support of the Air Force Oce of Scienti®c Research and Wright Laboratory, (Contract number F49620-93-C-0063). The program manager is Dr. Brian sanders.
[11] [12] [13] [14]
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aluminum silicate glass±ceramic. J Am Ceramic Soc 1989; 72:741±5. Nishiyama K, Umekawa S, Yuasa M, Fukumoto I. Eect of ®ber coating on interfacial phenomena in tungsten-®ber/boron carbide composites. In: Proceedings of the Fourth Japan-US Conference on Composite Material, 1988, Technomic, Lancaster, PA, pp. 776±788. Marshall DB, Evans AG. Failure mechanisms in ceramic±®ber/ ceramic±matrix composites. J Am Ceramic Soc 1985;68:225±31. Wetherhold RC, Zawada LP. Fractography of glasses and ceramics. In: Frechete VD, Varner JP, editors. Ceramic Transactions, vol. 17. American Ceramic Society, Westerville, Ohio, 1991:391. Zawada LP, Wetherhold RC. The eect of thermal fatigue on a SiC ®bre-reinforced nitrogen glass matrix composites. J Mater Sci 1991;26:648±54. Kahraman R. A microdebonding study of the high-temperature oxidation embrittlement of a cross-ply glass±ceramic/SiC composite. Compos Sci Technol 1996;56:1453±9. Westwood ME, Webster JD, Day RJ, Hayes FH, Taylor R. Review. oxidation protection for carbon ®bre composites. J Mater Sci 1996;31:1389±97. Holmes JW, Shuler SF. Temperature rise during fatigue of ®brereinforced ceramics. J Mater Sci Lett 1990;9:1290±1. Holmes JW, Cho C. Experimental observations of frictional heating in ®ber-reinforced ceramics. J Am Ceramic Soc 1992;75:929±38. Cho C, Holmes JW, Barber JR. Estimation of interfacial shear in ceramic composites from frictional heating measurements. J Am Ceramic Soc 1991;74:2802±8. Aveston J, Kelly A. Theory of multiple fracture of ®brous composites. J Mater Sci 1973;8:352±62. Budiansky B, Hutchinson JW, Evans AG. Matrix fracture in ®ber-reinforced ceramics. J Mech Phys Solids 1986;34:167±89. Pryce AW, Smith PA. Matrix cracking in unidirectional ceramic matrix composites under quasi-static and cyclic loading. Acta Mettalurgica et Materialia 1993;41:1269±81. Karandikar PG, Chou T-W. Microcracking and elastic moduli reductions in unidirectional Nicalon-CAS composite under cyclic fatigue loading. Ceramic Eng Sci Proc 1992;13:882±8. Sorensen BF, Talreja R, Sorensen OT. Micromechanical analysis of damage mechanics in ceramic±matrix composites during mechanical and thermal cycling . Composites 1993;24:129±40.