Determination of fiber-matrix interphase effective elastic moduli from ultrasonic phase velocity and attenuation data

Composrtes Engineering, Vol. 5. No. 6, pp. 113-733. 1995 Copyright 0 1995 Elscvier Science Ltd Printed in Great Britain. All rights reserved 0961-9526/95 S9.50+ .LW

Pergamon

0961-9526(95) 00038-o

DETERMINATION OF FIBER-MATRIX INTERPHASE EFFECTIVE ELASTIC MODULI FROM ULTRASONIC PHASE VELOCITY AND ATTENUATION DATA S. I. Rokhlin, Y. C. Chu and W. Huang Nondestructive Evaluation Program, Ohio State University, Columbus, OH 43210, U.S.A. (Received 1 November 1994;final version accepted 1 January 1995) Abstract-This paper discusses two potential ultrasonic methods for characterization of fiber-matrix interphases in composites: measurement of phase velocity and attenuation. The characterization procedures of these two methods, including different implementations using analytical or numerical models, are reviewed with emphasis on the applicability of these methods. Examples of ultrasonic characterization of interphases in ceramic and intermetallic matrix composites are given and the relation between the measured and actual interphase moduli is discussed. Application of these techniques for interfacial damage assessment is also demonstrated, including characterization of oxidation damage in ceramic matrix composites and fatigue damage in metal matrix composites. The experimental results show that degradation of the interphasial layer significantly affects ultrasonic wave attenuation and velocity. Thus both methods are very useful for assessment of interfacial damage. 1. INTRODUCTION

It is well established that the fiber-matrix interphase plays an important role in determining composite performance. The interphase not only allows load transfer between fibers and matrix but also provides matching of chemical and thermal compatibility between the constituents. In metal and intermetallic matrix composites, special interfacial reaction barrier coatings and compliant coatings are introduced to improve chemical and thermal compatibility. In ceramic matrix composites, the interphase is designed to provide frictional sliding contact between fiber and matrix to prevent fiber fracture due to matrix cracking. The interphase microstructure and its reaction with other composite constituents have received increasing attention. Despite great efforts in the development of special fiber coatings to tailor the interphase, interphase properties remain difficult to measure and interpret. The complexity of such interphases will become even greater if the properties are altered during manufacturing or in service by chemical reaction between the constituents, or if micromechanical defects occur. This paper reviews experimental and modeling efforts to determine effective elastic moduli of the interphases in composites. Here the interphase is considered to be a layer of bulk material between the fiber and matrix with distinct properties and thickness. Two ultrasonic methods for interphase characterization are discussed: phase velocity and attenuation. Other ultrasonic methods, such as measurement of local reflectivity from fiber-matrix interface (Matikas and Karpur, 1993; Blatt et al., 1993) and measurement of leaky guided waves along fibers (Nagy, 1994), are beyond the scope of this paper and are not discussed here. The basic idea of interphase characterization is illustrated schematically in Fig. 1. One method uses ultrasonic phase velocities to determine composite elastic moduli from which the interphasial moduli are deduced via micromechanical analysis. Another method uses the frequency dependence of the ultrasonic attenuation in the composite. By modeling the scattering of ultrasonic waves by the fiber-matrix interphases the attenuation data can be related to the interphase properties. Examples for ultrasonic characterization of fiber-matrix interphases in ceramic and intermetallic matrix composites reinforced with Sic SCS-6 fibers will be given. The fiber-matrix interphases in these composites are 3 pm thick carbon-rich coatings on the outer surface of the Sic shell. Emphasis will be placed on the connection between the measured and actual interphase moduli and applicability of the characterization method. 713

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Fig. 1. Flow chart for ultrasonic interphase characterization.

Further application of the technique is demonstrated by characterization of interphasial oxidation damage in a SiC/Si,Nq ceramic composite and fatigue damage in a Tibased metal matrix composite. It is shown that degradation of the interphasial layer significantly affects ultrasonic wave attenuation and velocity, and thus can be used for interphasial damage characterization. 2. INTERPHASE CHARACTERIZATION

USING PHASE VELOCITY DATA

2.1. Procedure overview The use of ultrasonic phase velocity data to characterize the fiber-matrix interphase was first suggested by Chu and Rokhlin (1992) and has been implemented using various micromechanical schemes (Gosz and Achenbach, 1993; Chu and Rokhlin, 1994c, 1995; Ma1 and Yang, 1995). As illustrated in Fig. 1, the idea is to measure phase velocities in composites using low frequency ultrasound and to find the composite moduli from the velocity data. These composite moduli are then used to determine the interphasial moduli via inversion of micromechanical models. It is worth noting that static micromechanical models are used in the analysis because the composite moduli determined by low frequency ultrasound are close to the static limit and the inversion can be greatly simplified in this case. Figure 2 shows typical phase velocity data for a Sic/%, N4 ceramic matrix composite in planes parallel and perpendicular to the fibers. The relation between the elastic constants C, of an anisotropic medium and the phase velocities of ultrasonic waves in the medium is given by the Christoffel equation (Auld, 1990). To determine the composite elastic constants from the ultrasonic phase velocity measurements, one inverts the Christoffel equation applying a nonlinear least-squares optimization technique (Rokhlin and Wang, 1989, 1992). It is shown (Chu and Rokhlin, 1994a; Chu et al., 1994) that the relative error in the elastic constants so determined is similar to that in the velocity measurements, which is less than one percent (Chu and Rokhlin, 1994b). The solid curves in Fig. 2 are theoretical velocities calculated using the reconstructed elastic constants. The good agreement between the calculations and experimental data indicates the accuracy of the reconstruction. In general, composites may be slightly anisotropic in the plane transverse to the fibers due to an uneven fiber distribution or matrix texture. Thus, in order to utilize

71s

Fiber-matrix interphase elastic moduli

(4

9

00000 -

DATA

EXPERIMENTAL THEORY

a

2 1 0 REFRACTION

ANGLE

(deg.)

lb) 9

00000 -

a

0

15

EXPERIMENTAL THEORY

30 REFRACT;ON

DATA

45

60 ANGLE

75

90

(deg.)

Fig. 2. The measured (points) and calculated (curves) phase velocities for SiURBSN matrix composites in planes (a) parallel and (b) normal to the fibers.

ceramic

micromechanical models for a transversely isotropic composite, one must find the composite average transverse moduli first. This can be done using averages of the upper and lower bounds which are determined from the composite elastic constants and compliances (Chu and Rokhlin, 1993). In addition to the composite moduli, moduli of composite constituents (fiber and matrix) must be measured prior to the determination of interphasial moduli. For composites reinforced with Sic SCS-6 fibers, multi-phase microstructure must be considered due to the presence of the fiber core (carbon), shell (Sic) and carbon-rich coating. Different models have been proposed for inversion and their results are very similar. One approach (Chu and Rokhlin, 1994~) replaces the core-shell combination with a so-called “equivalent” fiber. This simplifies the microstructure and allows the use of three-phase models for inversion. Another approach (Gosz and Achenbach, 1993) utilizes finite element analysis where the fiber core and shell are considered as different phases. In both approaches the interphasial stiffnesses (it, G: and Gj) are determined by equating the calculated composite moduli (G,C,G: and K’) with those determined experimentally and inverting for the unknown interphasial stiffnesses as illustrated in Fig. 3. More recently, Chu and Rokhlin (1995) have derived analytical expressions for the transverse shear moduli of composites with multi-phase fibers based on a multi-phase generalized self-consistent (MGSC) model (see Fig. 4) and a transfer matrix formulation. This development allows the interphasial moduli to be determined from inversion of the analytical multi-phase model without the use of finite element analysis or the equivalent fiber concept. cof5-6-J

S. I. Rokhlin et 1

716

EQUIVALENT

FIBER

INTERPHASE

’ TO

BE DETERMINED

/

FROMLMICROMECHANICAL MODEL: THREE PHASE MODEL FOR /cc, G; GENERALIZED SELF CONSISTENT MODEL FOR G’

u

EXPERIMENTAL DATA

Fig. 3. Procedure for determination of effective interphasial moduli by inversion of three.-phase micromechanical models.

-- my7 EFFECTIVE

MEDIUM

PhaseINTERPHASE

Fig. 4. MGSC model representation

of a multi-phase fiber composite embedded in an infinite effective medium.

The success of the described approach depends on the sensitivity of the composite moduli to variation of the interphasial moduli. As an example, Fig. 5 shows the composite axial shear modulus as a function of the interphasial axial shear modulus for different fiber/matrix stiffness ratios. The calculations were done using the multi-layered models. The solid line represents the fiber/matrix stiffness ratio (G,f/G,” = 3.5) for typical ceramic and metal matrix composites. The dashed lines are for composites with various fiber/matrix stiffness ratios adjusted by changing the fiber stiffness. The two vertical dashed lines shown in the figure represent typical interphasial moduli for ceramic and metal matrix composites with Sic fibers and carbon coating. It is clear from the figure that when the matrix is much stiffer than the fiber, the composite moduli are no longer sensitive to the interphasial moduli and one cannot find the interphasial moduli from the measured composite moduli. When the fiber is stiffer than the matrix, the composite moduli depend strongly on the interphasial moduli (for interphases with stiffness less than

711

Fiber-matrix interphase elastic moduli

G.‘/Gm=3.5 .. .

_.._______._______.............. -.-... G.‘/Gm=

1 .O

___________--_-_ Q

G,‘/Gm=O.

/

220

0.0

1

j

0.2 0.3 INTERPHASIAL

0.1

0.4

0.5 0.6 MODULUS

0.7 0.8 G.‘/G”

0.9

1.0

Fig. 5. Composite axial shear modulus versus interphasial axial shear modulus for a composite with 30% fiber fraction and different fiber/matrix stiffness ratios.

of the matrix stiffness). Therefore for composites with a compliant layer between fibers and matrix, the interphasial moduli can be determined by inversion of micromechanical models from the composite moduli. We will have additional discussion of sensitivity to interphasial moduli in Section 2.3 [see also Chu and Rokhlin (1994c, 1995)]. The inversion computation can be simplified substantially using an alternative spring formulation. In this case normal and transverse springs connecting the fiber and matrix replace the interphasial layer and the spring constants are directly related to the interphasial moduli by

20 percent

kt = G:/h,

k, = (K’ + G;)/h

= Cf,/h

(1)

where h is the thickness of the interphase. In this approximation the interphasial radial stiffness CC,‘,)is found directly in terms of the measured composite, matrix and fiber moduli: /cf(UrnKCKrn - &cm + um/cmGtm- /cCGtm) C,‘, z Xr a (1 - P)(KCKrn - K~G~~)+ /cCGtm+ UmKCKf - KfK” - um/PGtm

(2)

where a is the radius of the equivalent fiber, urnis the matrix volume fraction and h is the interphasial layer thickness. The superscripts m, f, I and c denote matrix, fiber, interphase and composite moduli. Similarly the interphasial axial shear modulus (Gf) can be obtained in the form

G&h

G,“G,f[(v” - 2)G,’ + u”G,“] + v”G,“* - PG,CGaf ’

a G,CG,“(vm - 2) + (2 - #‘)G,“G,

(3)

The derivation of the equation for the interphasial transverse shear modulus (G:) was performed using symbolic computation and after additional simplification reduced to 2h G:N

G:=aY

where the parameters N and D are given in Appendix A. The results of inversion using the spring formulation are found to be in good agreement with the more accurate layer formulation provided that the interphase thickness is much smaller than the fiber radius and the fiber moduli and much higher than the interphasial moduli. Table 1 shows typical interphasial moduli determined for ceramic and intermetalic matrix composites. Both composite systems are reinforced with the same fibers and the

718

S. 1. Rokhlin et al. Table 1. Interphasial elastic moduli (in GPa) obtained using a three phase model for (a) ceramic and (b) intermetallic matrix composites. Data in parentheses are interphasial moduli obtained using spring approximations Transverse bulk modulus K’

Axial shear modulus G:

Transverse shear modulus G,’

Radial modulus C:,

0.38 (0.34) 0.51 (0.41) 1.68 (1.53)

6.73 (6.89) 6.85 (7.23) 7.76 (8.19)

(b) SiC/Ti-24-11 intermetallic matrix composites 1 49.9 (46.7) 5.57 (5.72) 3.66 (3.40) 2 56.1 (52.2) 4.99 (5.12) 2.87 (2.64)

53.5 (65.3) 59.0 (73.9)

Sample

(a) SiC/Si,N, ceramic matrix composites 1 6.35 (6.0) 1.62 (1.65) 2 6.34 (6.23) 1.78 (1.82) 3 6.08 (6.12) 2.51 (2.57)

interphase is formed by carbon-rich coating on the fiber outer surfaces. As one can see, the results for ceramic matrix and intermetallic matrix composites are very different. Since the interphases in both composites are the same, the difference in the interphasial moduli raises serious questions on how to interpret the experimental data. A clear answer to this question requires establishing the connection between the measured and actual moduli of the interphasial layer. This issue is discussed in the next section. 2.2. Interpretation of the measured interphasial moduli To address the relation between the measured and actual moduli of the interphasial layer, knowledge of the actual interphasial moduli is necessary. Although there are no experimental data available for the fiber coating, the actual moduli of the interphasial layer can nevertheless be estimated based on its microstructure. It is known (Ning et al., 1990) that this carbon coating is composed of two different microstructural zones. One zone has randomly oriented basic structural units (BSU), which is similar to the structure of the carbon core. The other zone has a structure similar to pyrolytic carbon with a preferred basal plane orientation normal to the radial direction (along the circumferential direction). The moduli of the carbon interphasial layer may be calculated from the moduli of the carbon core and pyrolytic graphite using the rules of mixture (assuming the two different zones have the same thickness). In addition, the carbon coating contains very fine (2-4 nm diameter) Sic particles which introduce additional reinforcement. When the Sic particles are taken into account, the interphasial moduli are estimated to be CJ’,= 56 and Gt = 4.7 GPa for carbon layers with 25% Sic particles [details of the estimation are given in Chu and Rokhlin (1994c)]. The estimations are comparable to those measured for intermetallic matrix composites while being greater than those for ceramic matrix composites. This difference is due to imperfect contact between the interphasial carbon and the porous matrix. As a result of compaction of RBSN, the interfacial region is often highly porous compared to the matrix phase (Bhatt et al., 1992) thus affecting the state of mechanical contact between the interphase and the matrix. We use measurements on a ceramic composite without the fiber-matrix interphasial layers to illustrate this effect. The composite panel is made with the RBSN matrix and SCS-0 fibers having 24% fiber fraction and 18% matrix porosity. The SCS-0 fiber is identical to the SCS-6 fiber except without the outer carbon coating. Figure 6 shows the measured ultrasonic phase velocities for SCS-O/RBSN composites in the plane parallel to the fibers. The solid lines in the figure are theoretical calculations for the SCS-O/RBSN composite with a perfect interface between the fibers and matrix. It is clear that the theoretical velocities agree with experimental data near the fiber direction while the difference is greater in the direction transverse to the fibers. This difference is due to the fact that the contact between fiber and matrix in the SCS-O/RBSN composite is different from the perfect contact assumed in calculations. The theoretical and experimental velocity data for a SCSd/RBSN composite are also shown in the figure. The difference between the data for SCS-O/RBSN and SCSd/RBSN composites clearly indicates the effect of the interphasial layer.

Fiber-matrix interphase elastic moduli

719

MATRIX CARBON + SIC PARTICLES

Fig. 6. The measured (points) and calculated (dashed curves) ultrasonic phase velocities versus refraction angle for an SCS-O/RBSN composite in the plane parallel to the fibers. The solid lines are theoretical calculations for the SCS-O/RBSN composite with a perfect interface, which deviate from the experimental data due to excessive porosity at the interfacial region as shown in the upper part of the figure.

One may calculate the spring stiffnesses from ultrasonic measurements using spring boundary conditions to model the imperfect fiber-matrix contact in the SCS-O/RBSN composite. The resultant spring stiffnesses are k, = 17.1 and /q = 0.62 GPa/pm (for perfect contact both stiffnesses should be infinite). The finite stiffness values point to loss of joint integrity between fibers and matrix due to excessive interfacial porosity. Similar excessive interfacial porosity also occurs in SCSd/RBSN composites, thus reducing the measured effective interphasial moduli. This may be modeled as an imperfect interface between the carbon layer and the homogeneous matrix using springs as shown schematically in Fig. 7. It is shown (Chu and Rokhlin, 1994c) that the effective interphasial stiffnesses become Ck = 13 GPa and G,’ = 1.24 GPa for SCSd/RBSN composites with 30% porosity. This is comparable to those measured for the ceramic composites given in Table 1. From the above discussion, it is evident that the interphasial moduli obtained from SCS4/RSSN

POROUS MATRIX

MATRIX

11 OOOCO -----

10 ? \ sE

9

z

7

u S Y

6

Experimental Theoretical

C

8

5

2 0

k,=

17.1

GPalpm,

15

30 REFRACTION

k,=0.62

GPa/pm

45 60 ANGLE (deg.)

75

90

for SCS-OIRESN

Fig. 7. Schematic of fiber cross-section with imperfect contact between interphasial carbon and matrix in ceramic matrix composites.

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S. I. Rokhlin et al.

ultrasonic velocity data are “effective” moduli which depend on the state of contact between the fiber coating and the adjoining fiber and matrix. For better contact, such as the case of intermetallic matrix composites, the effective properties are close to the actual properties of the interphasial layer, while vanishing moduli correspond to complete fiber debond from the matrix. Thus deviation of the measured interphasial moduli from the desired (design) moduli may serve as a measure of the contact between the interphasial layer and the matrix, as in the case of ceramic matrix composites. Within the ceramic composite group, sample #3 has higher interphasial moduli than the other two. This is caused by variation in the interfacial porosity and by different compaction during processing (sample #3 is consolidated with 28 layers while samples #l and #2 have only 8 layer). 2.3. Accuracy of the measured effective interphasial moduli This section addresses the accuracy of the measured effective interphasial moduli. The quality of the measured value depends on the composite homogeneity and measurement error. The constituent inhomogeneity is minimized in the measurements since the irradiated ultrasonic velocity represents a statistical average over more than 100 fibers insonified by ultrasonic beam. Some local variation in matrix and composite properties can be viewed as error in their moduli measurements. Below we discuss the effect of these errors on the accuracy of the measured interphasial moduli. To assess error in the measured interphasial moduli, one must consider two important aspects. One is the experimental error in the measured velocity and constituent (fiber and matrix) modulus error. The other is the propagation of these errors during the inversion process. The experimental scatter in the velocity measurements was estimated to be less than 0.5% (Chu and Rokhlin, 1994b) and as a result, error in the composite and constituent moduli is within one percent (Chu and Rokhlin, 1994a; Chu et al., 1994). While the experimental error seems reasonably small, the propagation of error during inversion could have significant error amplification. This amplification is related to the stiffness ratios between composite constituents. To illustrate error propagation during inversion, it is convenient to define an error magnification factor Sj as

where M’ is the interphasial modulus and P’ is a given composite modulus or constituent property. When the interphasial modulus is expressed explicitly in terms of the composite moduli and constituent properties (M’ = M’(p)), the error magnification factor is given by s-PaM1 I MI@’

(6)

As an example, Fig. 8 shows the calculated error magnification factors versus interphasial axial shear modulus (normalized by the matrix shear modulus) for a typical ceramic matrix composite. When Gi/G” > 0.1 the magnification factors increase as the interphasial modulus increases. On the other hand when the interphasial modulus approaches zero (G: e G”), the magnification factors 8: and 8: become infinite due to the presence of G,’in the denominator. In general, error in measurement of the composite moduli has the largest effect on the precision of the interphasial modulus determination, while error in the fiber modulus has the least effect. It is important to note that for typical ceramic and metal matrix composites the ratio Gi/G” is between 0.02 and 0.15 (see the vertical dashed lines in Fig. 8), which is in the vicinity of the error magnification factor minima. Thus, assuming that the measured composite and constituent moduli have one percent error, error in the measured interphasial moduli is expected to be about 5-10%.

721

Fiber-matrix interphase elastic moduli

.’ --

I

L -

I-

_

_

---_

--__

--__Sf

---_

---___

--_ -_

---__ -_

I

-30.0

CERAMIC

MATRIX

--__

COMPOSITE

~,,,,,~~~~,‘l”,ll~rllllll,~~ll,~~~~,~~~ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 INTERPHASIAL MODULUS

--_

--------. S, 0.7 0.8 G.‘/G”

0.9

1 0

Fig, 8. The error magnification factors Sj for determination of interphasial axial shear modulus versus the normalized interphasial modulus. 3. INTERPHASE CHARACTERIZATION

USING ULTRASONIC ATTENUATION DATA

In this section interphase characterization by ultrasonic scattering (attenuation) will be discussed. The use of ultrasonic attenuation data to monitor fiber-matrix interphase condition in composites was suggested by Huang et al. (1994) and has been implemented in the study of fatigue damage in a Ti-based metal matrix composite (Rokhlin et al., 1994). This approach is appropriate when the fiber diameters are large as in the case of Sic fiber composites, and scattering by matrix microstructures (porosity, grains, etc.) is negligible compared with fiber scattering as for a titanium-based matrix. In this case the fiber-matrix interphase condition affects greatly the wave propagation in composites at higher frequencies, and by measuring the frequency dependency of the wave attenuation one can infer information about the fiber-matrix interphase. Appropriate models are needed to establish the quantitative relationship between the attenuation in composite and the averaged fiber-matrix interphase conditions. This is usually done by taking into account the scattering from multiple fibers using mathematical models based on simplified physical assumptions (see for example True11 et al., 1969; Waterman and Truell, 1961; Ma1 and Yang, 1995; Huang and Rokhlin, 1995a). Here we start with the effect of the fiber-matrix interphase on single-fiber scattering. Then we give an example of comparison of the attenuation data with theoretical predictions for a unidirectional titanium alloy composite reinforced with Sic fibers. Finally, in Section 4, we end with an example of the use of attenuation data to monitor fatigue damage in crossply titanium-based SIC composites. 3.1. Ultrasonic wave scattering by a single multi-phase fiber Assume a longitudinal wave incident normally on an interphase between the matrix and a fiber with unit potentials. v =

i i”J,(kf”r)exp(in6), “= -co

(7)

where J,(e) are the nth order Bessel functions of the first kind, and 8 and r are the polar angle and radius, respectively. The time-dependent term e-‘“’ is suppressed throughout. The general solutions for scattered longitudinal and shear waves in the matrix are represented as outgoing waves: f$ =

t A,H,(k;“f) n= --oD

exp(inO),

v/’ =

i B,H,(@V) exp(inO), n= -co

(8)

722

S. I. Rokhlin et al.

where H,(x) = J,,(x) + iN,(x), N,,(x) is a Bessel function of the second kind or a Neumann function, and H,(x) is a Hankel function of the first kind. h$”and K: are the wave numbers of the longitudinal and transverse modes in the matrix material. For a multi-phase fiber which consists of a fiber core and shell plus a homogeneous interphase between the fiber and the matrix, the incident and scattered waves in the matrix and inside the fibers must satisfy boundary conditions on each of the three boundaries: core/shell, shell/interphase and interphase/matrix. The scattering coefficients A, and B, can be found by solving a 12th order linear system of equations directly (Huang et al., 1995) or solving a 4th order equivalent boundary condition system using a transfer matrix approach. The scattering cross-section, which is defined as the total power scattered per unit length divided by the incident wave intensity (power per unit area), is found to be (Pao and Mow, 1973): (9) where Qi and Q, are the non-dimensional scattering cross-sections for longitudinal and transverse waves, respectively. The scattering cross-sections in the equations are normalized by the geometric limit 4r,(r,: the fiber radius). Although the problem of single 3-phase fiber scattering can be solved exactly, the effect of the fiber-matrix interphase on the scattering is hidden in the complicated structure of the solution. To better understand the role of the interphase on scattering and with the aim of developing effective inversion schemes for interphase characterization from scattering data, we study simplified equivalent models in the low frequency range. First we replace the fiber core and shell with an equivalent homogeneous fiber, then we replace the thin fiber-matrix interphase by interfacial springs (Huang et al., 1995; Huang and Rokhlin, 1995b). As an example we calculate scattering cross-sections for a longitudinal wave incident on a Sic fiber (SCS-6) in titanium alloy matrix. Attenuation measurements for a unidirectional composite (fiber fraction 24%) with such a fiber-matrix interphase system will be discussed in the next section. The elastic properties of each phase in this composite are listed in Table 2. The unusually high radial-to-shear modulus ratio of the interphase was discussed in Section 2, and the moduli were obtained from bulk wave velocity measurements using a 5 MHz transducer. The matrix properties were obtained from a slice of pure matrix material cut from the same composite sample. Figure 9 shows the calculated total scattering cross-section Q, + Qt from such a fiber with a 3pm carbon interphasial layer. The results are shown versus frequency (bottom axis) and wave number (kf”rr-the top axis). The fiber radius rf is 71 pm (including the 3 pm interphasial layer). The solid line in the figure represents results when the actual 2-phase Sic fiber with the 3 pm interphase is used, the coarse dashed line is for an equivalent fiber with the 3 pm interphase and the fine dashed line is for an equivalent fiber with the interfacial springs (K, = C,‘,/h = 1.O x lOi N/m3 and Kt = G:/h = 1.3 x 10” N/m3). One sees from the figure that the solutions obtained using the equivalent fiber with interfacial springs agree well with the exact solutions for the 3-phase fiber model in the low frequency range f > 20 MHz (k;“r, c 1.5). To illustrate the effect of the fiber-matrix interphase on ultrasonic wave scattering, we show in Fig. 10 the dependence of the scattering cross-section on the interphase stiffness at a fixed frequency f = 13 MHz (k;“r, = 1.0). In the figure, the top axis is the Table 2 Properties of each phase in a SCS-6/titanium Phase Fiber core (carbon) Fiber shell (Sic) Interphase (carbon-rich coating) Matrix (Ti)

alloy composite

GAGPa)

GAGPa)

P (g/cc)

r Olm)

49 446 31 193

16 177 4.6 45

1.7 3.2 2.1 5.4

18 68 71 -

723

Fiber-matrix interphase elastic moduli

2

1

0

3

6

5

4

2~o~E&----

+ SPRINGS + INTERPHASE CARBON CORE

f

I

K

2 ”

l.O-

z z K 2 0.5 a sb

0.0

WAVE

LONGITUDINAL INCIDENCE

z

0

10

20

30

40

FREQUENCY

50

70

60

ID

f MHz

Fig. 9. Single fiber scattering cross-section for a Sic fiber with a carbon-rich interphase embedded in a titanium ahoy matrix. The coarse dashed line represents the results when the actual fiber is replaced by an equivalent fiber, and the fine dashed line when the interphase is further replaced by distributed springs.

_--

EQUIVALENT FIBER+SPRINGS ACTUAL FIBER+SPRINGS

klm rf = 1.0 f = 13 MHz

NORMAL

STIFFNESS

Kn

(N/m3)

Fig. 10. Single fiber scattering cross-section from Sic/titanium interface with varying interfacial stiffnesses for a longitudinal wave incidence. The dashed line represents results when the Sic fiber is replaced by an equivalent fiber.

124

S. I. Rokhlin et al.

interphase radial modulus C’i’,whose value varies from 100 GPa to 0.1 GPa, simulating the weakening of the interphase. The bottom axis is the corresponding normal K,, interfacial stiffness. The ratio of the interphase radial-to-transverse shear modulus and the thickness and density of the interphase are kept constant. The calculation is done for an incident longitudinal wave using models of an equivalent fiber with interphase layer (the solid line) and an equivalent fiber with interfacial springs (the dashed line). One sees from the figure that the interphase stiffness reduction first leads to a slight scattering crosssection decrease to a minimum, then a sharp increase with further interphase stiffness reduction. Finally it reaches the limit defined by the scattering cross-section from a cylindrical cavity of the same diameter (debonded fiber). The fiber impedance (37 g/cc - Km/s) is higher than that of the matrix (32 g/cc - Km/s). The initial crosssection reduction with decrease of interphase stiffness is due to reduction of the impedance of effective fiber (including the interphase), which leads to impedance matching between the effective fiber and the matrix. The further decrease of the effective fiber impedance to a value lower than that of the matrix results in a sharp increase in the scattering cross-section. One sees that there is a significant effect of interphasial stiffnesses on the scattering cross-section suggesting that the scattered wave or scattering-induced attenuation may be used for interface characterization. One also notes that once again the spring model gives a close approximation to the interphase layer model. 3.2. Experimental measurement of attenuation spectrum in a unidirectional composite To confirm experimentally the feasibility of using wave scattering data for interface characterization in composites, we have performed ultrasonic attenuation measurements on a SCSd/titanium alloy unidirectional composite with fiber fraction of c = 24%. The fibers are distributed randomly as shown by the scanning electron micrographs (SEM) in Fig. 1l(a). As one can easily see from the figure there is a 3 pm carbon-rich interphasial layer between the fiber and the matrix. There are two layers of Sic deposited by the CVD process on the carbon core, one with coarse grains, one with fine grains; also there exists a pyrolytic carbon coating between the carbon core and the inner layer of the Sic shell. These microstructures do not have a noticeable effect on elastic wave scattering and their effect is neglected in calculation. Immersion ultrasonic attenuation measurements were performed in the throughthickness direction at normal incidence. The through-transmitted signals are measured using 15 and 20 MHz broadband transducers. The frequency used for attenuation measurements is higher than that used for velocity measurements, thus wave interaction with the fibers is more local and may vary more spatially. The samples are mounted on a rotation table which is interfaced with a host computer via a DC motor controller. Alignment is achieved by maximizing the reflection echo received by each transducer in the pulse-echo mode. Excitation and reception of the signals is done using a Panametrics 5052PR pulserreceiver and a LeCroy 9400 digital oscilloscope. The amplitude spectrum of the received signal is obtained from an FFT of the time-domain signal after digitization and averaging. The attenuation spectrum is obtained by deconvolving the signal transmitted through the sample with the signal transmitted through water without the sample. The transmission loss is also compensated for beam diffraction and radiation losses on the water-sample interfaces. The longitudinal wave attenuation (in Np/cm) versus frequency is shown in Fig. 1l(b) by open circles for one sample with thickness h = 2.7 mm. The calculated attenuation spectrum using the independent scattering model is shown by a solid line. In the model the scatterers (fibers) act independently of each other. The attenuation a! of stress waves in this case is (True11 et al., 1969) a=

4rf

yn,(Q~

+ QJ,

n,=C-

number of scatterers per unit area. ~~

(10)

The properties of each phase used in the calculations are the same as those in Table 2. The interphase moduli were obtained independently by bulk wave velocity measurements with a 5 MHz transducer as described in Section 2. One sees from Fig. 1l(b) that agreement

Fiber-matrix

725

interphase elastic moduli

jYzer--

INDEPENDENT

SCATTERING

MODEL

4

2

0 0

5

10

15

i0

FREQUENCY f (MHZ) (b) Fig. 11. (a) SEM photograph of a XX-B/titanium alloy unidirectional composite sample. (b) Measured attenuation spectrum (open circles). The theoretical predictions are given by the solid line.

127

Fiber-matrix interphase elastic moduli

between theory and experiment is very good in the frequency range measured, since the impedance difference between the fiber and the matrix is small the fibers can be considered as weak scatterers, rendering the effect of multiple scattering negligible. Recall that from Fig. 10 the dependence of the scattering cross-section on the interphase moduli is not monotonic and has a minimum. Also for higher interphase moduli the sensitivity of the scattering cross-section to the change of interphase moduli is small, see for example, at C,‘, > 10 GPa in Fig. 10. Explanation of this is given by Huang et al. (1995). For CA smaller than 10 GPa the scattering cross-section depends strongly on interphase moduli CC,‘,)and thus the attenuation technique can be used effectively to evaluate interphase damage since it leads to significant reduction of the moduli (see for an example measurement of fatigue-induced interphase damage in Section 4.2). 4. APPLICATION

TO INTERPHASIAL

DAMAGE ASSESSMENT

4.1. Oxidation damage in ceramic matrix composites Oxidation damage is a major concern for ceramic matrix composites and has been a critical area for composite development. Successful prevention of oxidation damage requires identification of dominant damage mechanisms. It is known (Bhatt, 1989; Chu et al., 1993) that one major damage mechanism is oxidation of the fiber-matrix interphases. Thus one may use the interphase characterization techniques described above to quantify the damage severity and identify the transition from the nonoxidized to the oxidized state.

THEORY BEFORE OXIDATION * * * + c AFTER OXIDATION (6OO’C. 1 HR)

__

2

ooooo

1

0

15

30 REFRACTION

60 45 ANGLE (deg.)

75

(b)

* + t * * AFTER (6OO’C.

1

o+0

/

1

OXIDATION 1 HR)

,

15 REFR%ON4iNGLE

,

--,7-R-,

60

75

91

(deg.)

Fig. 12. The angular dependence of the phase velocities in (a) l-3 and (b) 2-3 planes for SiC/RBSN composites before and after one hour oxidation at 600°C. The points are experimental data and the solid lines are theoretical calculations from the reconstructed composite moduli.

S. I. Rokhlin et al.

728 1.2 :: O”l.0

z w $

1400

.-,------q-r \

600

0.8

‘&

0.6

/

“C

O ‘a> \

5 a 0.4

“C

R \ .\,/

\ \

>v

(4

-

\ \

\

*0,

._

-_*----‘ 11 ---_ \\

900 OC .

0

Cl

0

1,

0

:

(b)

Fig. 13. Reduction of composite moduli (a) C,, and (b) C,, as a function of oxidation exposure time for different temperatures. The moduli are normalized with those of the as-received composites.

Figure 12 shows the measured ultrasonic phase velocities versus refraction angle for composites before and after oxidation at 600°C for one hour. It is clear that the velocities are significantly reduced due to oxidation damage. The composite moduli calculated from the measured velocity data decrease as the oxidation exposure time increases (shown in Fig. 13). The transition from the nonoxidized to the oxidized state can be quantified by finding the effective interphasial moduli for different exposure times. Table 3 shows the radial and axial shear stiffnesses of the interphase determined for composites before and after oxidation at 1400°C for 6, 18 and 60 min. As one can see, the interphasial elastic moduli for different composites before oxidation are similar. They become very different Table 3 Elastic moduli (in GPa) of the fiber-matrix interphasial layers obtained from ultrasonic measurements for composites before and after oxidation at 1400°C for different exposure times Exposure times (min)

Interphasial radial stiffness Before

After

Reduction (olo)

Before

After

6 ia 60

5.14 6.85 4.02

3.62 2.50 0.0

30.6 64.5 100

1.81 1.78 1.23

1.17 0.67 0.0

Interphasial axial shear stiffness Reduction (a7’0) 36.3 62.4 100

129

Fiber-matrix interphase elastic moduli

(a)

0

15

30 REFRACTION

45 60 ANGLE (deg.)

75

9

i

(b)

5

4 -

oo300 * * * **

BEFORE OXIDATION AFTER OXIDATION

1 31,,,,,,,,,,,,,,,,,,,,,,,,,,,,r 15 30 0 REFRACTION4tNGLE

60

75

!

(deg.)

Fig. 14. The measured phase velocities in (a) l-3 and (b) 2-3 planes for an SCS-O/RBSN composite (no interphasial layer) before and after oxidation at 900°C for two hours. The velocity does not change with the oxidation treatment, thus indicating that oxidation-induced change for composites with KS-6 fibers is due to the interphase oxidation.

after oxidation. The reduction of the interphasial moduli due to oxidation is also shown in the table. It is clear that during the transition, the elastic moduli of the interphasial layer are significantly reduced and these reductions characterize the severity of interface oxidation. To verify interface oxidation as a dominant damage mechanism, ultrasonic measurements on a composite without the fiber-matrix interphasial layers were made. Figure 14 shows the measured phase velocities for the SCS-O/RBSN composite before and after oxidation at 900°C (the most damaging oxidation temperature) for two hours. It is clear that the measured velocity data before and after oxidation are essentially identical, thus showing that the composite is not affected by the oxidation. This verifies the point that the dominant damage mechanism for the reduction of composite moduli (or phase velocities) is oxidation of the fiber-matrix interphasial layers. 4.2. Fatigue damage in metal matrix composites The samples used in this work are [OEX&, SiC/Ti-15V-3Cr-3Al-3Sn metal matrix composites. The composite has an B-ply symmetry layup made by hot isostatic pressing of a foil/fiber/foil layup with thickness about 1.7 mm (Soboyejo et a,.). The matrix is metastable /? titanium alloy Ti-15V-3Cr-3Al-3Sn (Ti-15-3, weight ratio) and the Sic fiber is

730

S. 1. Rokhlin et al.

6-

FREQUENCY

f (MHZ)

TRANSDUCER

(b)

SAMPLE (815’c, 10 h~=O.,.u)

1/ /

3

WATER

c=JI FATIGUE ?? oomo

00ooO

LOAD

ATTENUATION VALUES AT MEASURED BY A 20 MHz C,, VALUES OBTAINED MEASURED BY A 5MHz

f=13 MHz TRANSDUCER

FROM VELOCITIES TRANSDUCER

160

0~,,,,,,,,,,,,,,,,,,~,,,,,_,,,,,,,,,,,,,,,.,,,~140 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0

FATIGUE CYCLE N/N, Fig. IS. Ultrasonic results for a [O/90] cross-ply SCS-6/Ti-15-3 fatigue tensile sample (heat treated at 815” for 10 h) measured in the through thickness direction. (a) The attenuation spectra of the sample at different stages of fatigue life; (b) the attenuation dependence on fatigue life cycles at a fixed frequency f = 13 MHz (solid circles). The dependence of the through-thickness modulus Cs, on the fatigue life cycle is also plotted here for comparison (open circles).

SCS-6 by Textron. The fiber volume fraction of the composite is 35% and the composite density is 4.18 g/cm3. To simulate the effect of heat treatment during material processing, as-received samples were heat-treated at temperatures above (8lYC) and below (540°C) the matrix /I transus. The heat treated samples were first fatigued to failure under different stress ranges to obtain their S/N curves. Based on the S/N curves, two stress-controlled

Fiber-matrix interphase elastic moduli

731

fatigue tests (Aa equals 50 or 70% of the ultimate strength, R = C,in/Cmax = 0.1) were selected for ultrasonic damage assessment using samples with different heat treatments. The fatigue cycling frequency is 10 Hz. Ultrasonic bulk wave velocity and attenuation measurements were performed on samples prior to fatigue, as well as at different stages of fatigue (with step size 0.1 of the fatigue life) (Rokhlin et al., 1994). As an example the ultrasonically measured attenuation spectra (in Np/mm) of the same sample measured at different fatigue stages are shown in Fig. 15(a) (fatigue stress level at 70% of ultimate stress). One sees from the figure that at frequencies below the high frequency dynamic noise limit there is a significant increase of attenuation as fatigue progresses. The high frequency dynamic noise is mostly due to wave interference between neighboring fibers and laminae. Figure 15(b) shows the dependence of the attenuation on the fatigue life cycle at a fixed frequency f = 13 MHz. One sees that the attenuation increases linearly at a significant rate before 50% of the fatigue life cycle due to the extension of partial debonding in the 90” laminae (with fibers perpendicular to the loading direction). Considering the increase of attenuation as fatigue progresses, one concludes that attenuation is sensitive to fatigue-induced damage on the fiber-matrix interface at the early stages of fatigue life cycle. Then the attenuation increases steadily but at a smaller rate until 70% of the fatigue life cycle, mainly due to fiber bonding occurring in 0” laminae. Finally the attenuation increases rapidly again, probably due to matrix cracking. To compare with the phase velocity approach, we also plot in Fig. 15(b) the dependence of C,, of the composite on the fatigue life cycle which was determined from velocity measurements at 5 MHz as described in the previous section. One sees that the rate of increase of attenuation and that of the drop in C,, are similar. The strong dependence of measured attenuation spectra on fatigue damage and the calculated scattering cross-section from a single fiber with an interface of variable stiffness indicate a possibility of using attenuation data for quantitative assessment of interfacial damage. Multiple scattering must be considered in this composite to calculate the attenuation due to the high fiber fraction (35%). Various procedures for determination of the wave propagation characteristics in a multi-scatterer medium have been proposed, for example Waterman and True11 (1961), Ma1 and Yang 1995) and Huang and Rokhlin (1995a). Significant additional effort is needed to account for the partial debonding of the fibers, inhomogeneous distributions of damage among different fiber/matrix interfaces, the effect of O/90” fiber orientation in cross-ply composites and the periodical iaminae layup and fiber distribution in each laminate.

REFERENCES Auld, B. A. (1990), Acousfic Fields and Waves in Solids. Krieger Publishing, Malabar, FL. Bhatt, R. T. (1989). Oxidation effect on the mechanical properties of Sic fiber-reinforced reaction-bonded silicon nitride matrix composites. NASA TM-102360. Bhatt, H., Donaldson, K. Y., Hasselman, D. P. H. and Bhatt, R. T. (1992). Effect of HIPing on the effective thermal conductivity/diffusivity and the interfacial thermal conductance of uniaxial Sic fiber-reinforced RBSN. J. Mater. Sci. 27, 6653-6661. Blatt, D., Karpur, P., Stubbs, D. A. and Matikas, T. E. (1993). Observation of interfacial damage in the fiber bridged zone of a titanium matrix composite. Scripta Met&. et Muter. 29, 851-856. Chu, Y. C. and Rokhlin, S. I. (1992). Determination of macro- and micromechanical and interfacial elastic properties of composites from ultrasonic data. J. Acoust. Sot. Am. 92, 920-931. Chu, Y. C. and Rokhlin, S. I. (1993). Inverse determination of fiber-matrix interphase properties in composites from ultrasonic data. In Ultrasonic Characterization and Mechanics of Interfaces (Edited by S. I. Rokhlin, S. K. Datta and Y. D. S. Rajapakse), AMD-Vol. 117, pp. 113-124. ASME, New York. Chu, Y. C. and Rokhlin, S. I. (1994a). Stability of determination of composite moduli from velocitv data in planes of symmetry for weak and strong anisotropies. J. Acoust. Sot. &n. 95, 213-225. Chu, Y. C. and Rokhlin, S. I. (1994b). Comparative analvsis of throunh-transmission ultrasonic bulk wave methods for phase velocity measurements in anisotropic materials. J. &oust. Sot. Am. 95, 3204-3212. Chu, Y. C. and Rokhlin, S. I. (1994c), Fiber-matrix interphase characterization in composites using ultrasonic velocity data. J. Appt. Phys. 76, 4121-4129. Chu, Y. C. and Rokhlin, S. I. (1995). Determination of fiber-matrix interphase moduli from experimental moduli of composites with multi-layered fibers. Mech. Mater. (in press). Chu, Y. C., Rokhlin, S. I. and Baaklini, G. Y. (1993). Ultrasonic assessment of interfacial oxidation damage in ceramic matrix composites. Transactions of ASME, J. Engng Muter. Technol. 115, 237-243.

132

S. I. Rokhlin et 01.

Gosz. M. and Achenbach, J. D. (1993). Interface effect on bulk wave velocities in fiber composites. In C&asonic Characterization and Mechonics of Interfaces (Edited by S. I. Rokhlin, S. K. Datta and Y. D. S. Rajapakse) AMD-Vol. 177. pp, 125-133. ASME. New York. Huang; W., Brisuda, S. and Rokhlin, S. I. (1994). -Fiber/matrix interface characterization by ultrasonic wave scattering. In Review of Progress in QNDE, Vol. 13, 1367-1374 (Edited by D. 0. Thompson and D. E. Chimenti). Plenum Press, New York. Huang, W.. Brisuda, S. and Rokhlin, S. I. (1995). Ultrasonic wave scattering from fiber-matrix interphases. J. Acoust. Sot. Am. 97, 807-811. Huang, W. and Rokhlin, S. I. (1995a). Frequency dependences of ultrasonic wave velocity and attenuation in fiber composites. Theory and experiments. In Review of Progress in QNDE. Vol. 14, 1233-1240 (Edited by D. 0. Thompson and D. E. Chimenti). Plenum Press, New York. Huang, W. and Rokhlin, S. I. (1995b). Effect of fiber/matrix interphase in low frequency ultrasonic wave scattering: Spring boundary condition approach. In Review of Progress in QNDE, Vol. 14 (Edited by D. 0. Thompson and D. E. Chimenti). Plenum Press, New York. Mal, A. K. and Yang, R. B. (1995) Characterization of fiber-matrix interface degradation in a metal matrix composite. In Review of Progress in QNDE, Vol. 14 (Edited by D. 0. Thompson and D. E. Chimenti). Plenum Press, New York. Matikas, T. E. and Karpur, P. (1993). Ultrasonic reflectivity technique for the characterization of fiber-matrix interface in metal matrix composites. J. Appt. Phys. 74, 228-236. Nagy, P. B. (1994). Leaky guided wave propagation along imperfectly bonded fibers in composite materials. J. Nondes. Eval. 13, 137-145. Ning, X. J., Pirouz, P., Lagerlof, K. P. D. and DiCarlo, J. (1990). The structure of carbon in chemically vapor deposited BC Monofilaments. J. Muter. Res. 5, 2865-2876. Pao, Y. H. and Mow, C. C. (1973). Diffraction of Elastic Waves and Dynamic Stress Concentrations. Crane, Russack. New York. Rokhlin, S. I. and Wang, W. (1989). Ultrasonic evaluation of in-plane and out-of-plane elastic properties of composite materials. In Review of Progress in QNDE, Vol. 8B (Edited by D. 0. Thompson and D. E. Chimenti), pp. 1489-1496. Plenum Press, New York. Rokhlin, S. I. and Wang, W. (1992). Double through-transmission bulk wave method for ultrasonic phase velocity measurement and determination of elastic constants of composite materials. J. Acoust. Sot. Am. 91, 3303-3312. Rokhlin, S. I., Chu, Y. C. and Huang, W. (1994). Ultrasonic evaluation of fatigue damage in metal matrix composites. Symposium on Wuve Propogation ond Emerging Technologies (Edited by V. K. Kinra, R. J. Clifton and G. C. Johnson), ASME AMD-Vol. 188, pp. 29-46. Soboyejo, W. 0. Rabeeh, B. M., Chu, Y. C., Laverentyev, A. and Rokhlin, S. I. (1995). An investigation of fatigue damage in a symmetric [O/90],, silicon carbide fiber-reinforced titanium matrix composite. Metafl. Trans. (submitted). Truell, R., Elbaum, C. and Chick, B. B. (1969). Ultrmonic Methods in Solid Stote Physics. Academic Press, New York. Waterman, P. C. and Truell, R. (l%l). Multiple scattering of elastic waves. J. Moth. Phys. 2, 512-537.

APPENDIX A The parameters N and D in eqn (4) are given by N = (2 + 3~ + &)qr + 2910 + rrnrl,, + 2&&t/s D = -&VI

+ 2(c4 - 1X1 + r&l

+ (1 + 2P + 2c, + POls

-

(Al)

- &Xl + g, + 2g,&,) + 2pVz + 2&V, - 2g,<,%

+ P&ls

+ Yfbl,

- 4g&*

+ 2g,pGl,

W&t1 + a@- 1)11

W)

where ‘I,

=

&c - 1)” - (c + 1)’ + 4cg#c

- l)(c - 1) + 12c*

q2 = (c - l)(c’ - 3cz + 3c + 1) + 2cg,(l - 6c + 4c4 - &c - I)@’ + 5c2 - c + 1)

(A4)

qs = (c - l)(c’ - c* + 5c + 1) + 4g,c(l - 3c + c2) - &c - l)(c’ + 3cz - 3c + 1)

(A5)

n, = (c’ + 2c3 + 1) + gc(c - l)(c’ - 3c2 - 3c - 1) - 2&r qs = (2g,&&(C +$[(c

+ c + l)(c - 1)s

(A6)

+ 2c + 1) - (cd - 2c + 1) + 2g&c]

+ 1)” - 12c2] - 4cg,(2c - l)(c - 1) - (c - 1)4(/gr

(A7)

rl6 = (c - l)(c’ - 3c2 + 3c + 3) - 4g,(l + c + 3cs - 2cr - cd) - gf(c - 1)(5c3 + 9cr + 3c + 7) (A8) rl, = (c’ - 1) + g&2 - 2c - l)(c2 + 1) - $(c - I)@’ + 2)

(A9)

‘I* = cr + g,(c - I)@’ - cs - c - 1) + g&r + c + l)(c - 1)2

(Al@

99 = (c’ - 3) - g=(c - l)(c’ + c2 + c + 3)

(All)

Fiber-matrix interphase elastic moduli rlICI-- -2g,c(2cZ

733

- 1) + (c - l)(c + 1)’ - gf(c + l)(c - 1)’

(A12)

q,, = (c - 1)[(c3 + 9c2 + 3c + 3) - gf(5c3 - 3c2 + 3c + 7)] + 4gJ - 1 - c + 3cz - 4~’ + c4) (A13) &,, =

G;/Km,

g, = G:/G;,

&

=

‘$/K’,

p = ZhG:/(aC:,),

g,

=

G:/G’,

c = /Jr zz

1 - vm

Note that the interphasial redial stiffness C:, in the above formulation must be obtained first using (2).

(A14) (A15)