Open access peer-reviewed chapter

Matrix Cracking of Ceramic-Matrix Composites

Written By

Li Longbiao

Submitted: July 29th, 2019 Reviewed: October 7th, 2019 Published: December 11th, 2019

DOI: 10.5772/intechopen.90045

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Abstract

In this chapter, the matrix cracking of fiber-reinforced ceramic-matrix composites (CMCs) is investigated using the energy balance approach. The relationship between the matrix cracking stress, fiber and interface oxidation, and fiber failure is established. The effects of the fiber volume, interface shear stress and interface debonding energy, fiber failure, and oxidation temperature on the time-dependent matrix cracking stress are analyzed. The experimental matrix cracking stress of different fiber-reinforced CMCs is predicted using the present models.

Keywords

  • ceramic-matrix composites
  • matrix cracking
  • oxidation
  • interface shear stress

1. Introduction

Ceramic-matrix composites (CMCs) possess high specific strength and high specific modulus especially at elevated temperature and have already been applied in hot-section components in aeroengine [1]. However, at elevated temperature, the environment affects the mechanical performance of fiber-reinforced CMCs. The matrix cracking stress decreases with operation time due to the interface oxidation [2]. Many researchers performed investigations on matrix cracking of fiber-reinforced CMCs, i.e., the energy balance approach developed by Aveston et al. [3]; Budiansky et al. [4]; Rajan and Zok [5]; and Li [6, 7], and the fracture mechanics approach proposed by Marshall et al. [8], and McCartney [9]. At elevated temperature, the oxidative environment entered inside of fiber-reinforced CMCs through the microcrackings, and the lifetime is affected by the applied load and temperature [10, 11].

In this chapter, the effects of temperature, oxidation, and applied stress on the time-dependent matrix cracking stress of fiber-reinforced CMCs are investigated. The relationship between the matrix cracking stress, fiber and interface oxidation, and fiber failure is established. The effects of the fiber volume, interface shear stress and interface debonding energy, fiber failure, and oxidation temperature on the time-dependent matrix cracking stress are analyzed. The experimental matrix cracking stress of different fiber-reinforced CMCs are predicted.

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2. Theoretical analysis

At elevated temperature, oxygen reacts with the interphase along the fiber length. The time-dependent interphase oxidation length can be determined using the following equation (11) [12].

ζ = φ 1 1 exp φ 2 t b E1

where φ1 and φ2 are parameters dependent on temperature and described using the Arrhenius type laws and b is a delay factor considering the deceleration of reduced oxygen activity.

The oxidation of fiber is assumed to be controlled by diffusion of oxygen gas through matrix cracks. By assuming that the fracture toughness of the fibers remains constant and that the fiber strength σ0 is related to the mean oxidized layer thickness, the time dependence of the fiber strength can be determined by the following equation (12).

σ 0 t = σ 0 , t 1 k K IC Y σ 0 4 E2
σ 0 t = K IC Y kt 4 , t > 1 k K IC Y σ 0 4 E3

When the fiber breaks, the stress carried by the intact and broken fibers can be determined using the global load sharing (GLS) criterion by Eq. (4) [13].

σ V f = T 1 P T + T b P T E4

where Vf denotes the fiber volume fraction, T denotes the load carried by intact fibers, T b denotes the load carried by broken fibers, and P(T) denotes the fiber failure probability.

P T = η P a T + 1 η P b T + P c T E5

where η denotes the oxidation fiber fraction in the oxidation region and Pa(T), Pb(T), and Pc(T) denote the fracture probability of oxidized fibers and unoxidized fibers in the oxidation region and interface debonding region, respectively.

2.1 Downstream stresses

Figure 1 shows the composite under loading of a remote uniform applied stress of σ with a long matrix cracking. A unit cell is extracted from the composite to analyze the microstress field of the damaged fiber-reinforced CMCs, as shown in Figure 2. The fiber and matrix axial stress in the interface oxidation and debonding region can be determined using the following equation.

Figure 1.

The schematic of crack-tip, interface debonding and oxidation.

Figure 2.

The schematic of shear-lag model considering the interface oxidation and debonding.

σ f D z = T 2 τ f r f z , z 0 ζ T 2 τ f r f ζ 2 τ i r f z ζ , z ζ l d E6
σ m D z = 2 V f V m τ f r f z , z 0 ζ 2 V f V m τ f r f ζ + 2 V f V m τ i r f z ζ , z ζ l d E7

In the fiber/matrix interface bonding region, the fiber and matrix axial stresses can be determined using the following equation.

σ f D = E f E c σ E8
σ m D = E m E c σ E9

2.2 Upstream stresses

The upstream region III as shown in Figure 1 is so far away from the crack tip that the stress and strain fields are also uniform. The axial stress of the fiber and the matrix can be determined using the following equations.

σ f U = E f E c σ E10
σ m U = E m E c σ E11

2.3 Interface debonding

The fracture mechanics approach is used to determine the fiber/matrix interface debonding length [14].

ξ d = F 4 π r f w f 0 l d 1 2 0 l d τ i v z l d dz E12

where F(=πrf2σ/Vf) denotes the fiber load at the matrix cracking plane, wf(0) denotes the fiber axial displacement on the matrix cracking plane, and v(z) denotes the relative displacement between the fiber and the matrix. Substituting wf(z = 0) and v(z) into Eq. (12), it leads to the form of Eq. (13).

E c τ i 2 r f V m E f E m l d ζ 2 + 2 E c τ f τ i r f V m E f E m ζ l d ζ στ i 2 V f E f l d ζ T τ i 2 E f l d ζ + r f σT 4 V f E f r f σ 2 4 V f E c τ f σζ 2 V f E f τ f T 2 E f ζ + E c τ f 2 r f V m E f E m ζ 2 ξ d = 0 E13

Solving Eq. (13), the fiber/matrix interface debonding length is determined by Eq. (14).

l d = 1 τ f τ i ζ + r f V m E m 4 E c τ i σ V f + T r f 2 V m E m T 2 4 E c τ i 2 V m E m 4 E c σ V f T + 1 2 + V f E f E c σ V f T 2 σ V f T + r f V m E f E m E c τ i 2 ξ d 1 2 E14

2.4 Matrix cracking stress

The energy relationship to evaluate the steady-state matrix cracking stress is determined by Eq. (15) [4].

1 2 V f E f σ f U σ f D 2 + V m E m σ m U σ m D 2 dz + 1 2 π R 2 G m l d l d r f R r f τ i z r 2 πrdrdz = V m ξ m + 4 V f l d r f ξ d E15

where ξm is the matrix fracture energy and Gm is the matrix shear modulus. Substituting the fiber and matrix axial stresses in Eqs. (6)(11) and the fiber/matrix interface debonding length of Eq. (14) into Eq. (15), the energy balance equation leads to the form of the following equation.

η 1 σ 2 + η 2 σ + η 3 = 0 E16

where

η 1 = l d E c E17
η 2 = 2 V f Tl d E c E18
η 3 = V f l d T 2 E f V f E f 2 τ f r f ζ 2 l d ζ T V f E f 2 τ i r f l d ζ 2 T + 4 3 V f E c V m E f E m τ i r f 2 l d ζ 3 + τ f τ i 2 ζ 3 + 4 V f E c V m E f E m τ f τ i r f 2 ζ l d ζ × l d 1 τ f τ i ζ V m ξ m 4 V f l d r f ξ d E19
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3. Result and discussion

The effects of fiber volume, fiber/matrix interface debonding energy, interface shear stress, fiber strength, and oxidation temperature on the matrix cracking stress, interface oxidation, and interface debonding are analyzed.

3.1 Effect of the fiber volume on matrix cracking stress

The matrix cracking stress, fiber/matrix interface debonding length, and fiber/matrix interface oxidation length versus the oxidation time curves corresponding to different fiber volume of Vf = 30% and 35% are shown in Figure 3. When the fiber volume increases, the matrix cracking stress and the fiber/matrix interface oxidation length increase, and the fiber/matrix interface debonding length decreases.

Figure 3.

(a) The matrix cracking stress versus the oxidation time, (b) the fiber/matrix interface debonding length versus the oxidation time, and (c) the fiber/matrix interface oxidation length versus the oxidation time corresponding to different fiber volume of Vf = 30 and 35%.

When the fiber volume is Vf = 30%, the matrix cracking stress decreases from σmc = 86 to 44 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 6.9 to 6.7 after t = 2 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 9.0 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.94 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

When the fiber volume is Vf = 35%, the matrix cracking stress decreases from σmc = 97 to 50 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 5.8 to 5.7 after t = 1.7 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 8.4 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 1.0 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

3.2 Effect of the fiber/matrix interface debonding energy on matrix cracking stress

The matrix cracking stress, fiber/matrix interface debonding length, and fiber/matrix interface oxidation length versus the oxidation time curves corresponding to different interface debonding energies of ξd/ξm = 0.1 and 0.2 are shown in Figure 4. When the fiber/matrix interface debonding energy increases, the matrix cracking stress and the interface oxidation length increase, and the interface debonding length decreases.

Figure 4.

(a) The matrix cracking stress versus the oxidation time, (b) the fiber/matrix interface debonding length versus the oxidation time, and (c) the fiber/matrix interface oxidation length versus the oxidation time corresponding to different interface debonding energy of ξdm = 0.1 and 0.2.

When the fiber/matrix interface debonding energy is ξd/ξm = 0.1, the matrix cracking stress decreases from σmc = 86 MPa to 32 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 6.9 to 6.7 after t = 1.7 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 9.6 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.88 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

When the fiber/matrix interface debonding energy is ξd/ξm = 0.2, the matrix cracking stress decreases from σmc = 134 to 87 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 5.5 to 5.4 after t = 1.1 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 8.8 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.96 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

3.3 Effect of the fiber/matrix interface shear stress on the matrix cracking stress

The matrix cracking stress, fiber/matrix interface debonding length, and the fiber/matrix interface oxidation length versus the oxidation time curves corresponding to different fiber/matrix interface shear stress of τi = 20 and 10 MPa are shown in Figure 5. When the fiber/matrix interface shear stress in the slip region increases, the matrix cracking stress and the fiber/matrix interface oxidation length increase, and the fiber/matrix interface debonding length decreases.

Figure 5.

(a) The matrix cracking stress versus the oxidation time, (b) the fiber/matrix interface debonding length versus the oxidation time, and (c) the fiber/matrix interface oxidation length versus the oxidation time corresponding to different interface shear stress of τi = 10 and 20 MPa.

When the fiber/matrix interface shear stress is τi = 10 MPa, the matrix cracking stress decreases from σmc = 67 to 43 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 8.0 to 7.7 after t = 2.6 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 9.1 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.93 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

When the fiber/matrix interface shear stress is τi = 20 MPa, the matrix cracking stress decreases from σmc = 102 to 45 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 6.2 to 6.0 after t = 1.7 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 8.9 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.95 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

The matrix cracking stress, fiber/matrix interface debonding length, and the fiber/matrix interface oxidation length versus the oxidation time curves corresponding to different fiber/matrix interface shear stress of τf = 1 and 5 MPa are shown in Figure 6. When the fiber/matrix interface shear stress in the oxidation region increases, the matrix cracking stress and the fiber/matrix interface oxidation length increase, and the fiber/matrix interface debonding length decreases.

Figure 6.

(a) The matrix cracking stress versus the oxidation time, (b) the fiber/matrix interface debonding length versus oxidation time, and (c) the fiber/matrix interface oxidation length versus the oxidation time corresponding to different interface shear stress of τf = 1 and 5 MPa.

When the fiber/matrix interface shear stress is τf = 1 MPa, the matrix cracking stress decreases from σmc = 86 to 28 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 6.9 to 6.7 after t = 1.6 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 9.8 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.86 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

When the fiber/matrix interface shear stress is τf = 5 MPa, the matrix cracking stress decreases from σmc = 86 to 44 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 6.9 to 6.7 after t = 2 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 9 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.94 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

3.4 Effect of the fiber strength on matrix cracking stress

The matrix cracking stress, fiber/matrix interface debonding length, and fiber/matrix interface oxidation length versus the oxidation time curves corresponding to different fiber strength of σ0 = 1 and 2 GPa are shown in Figure 7. When the fiber strength increases, the matrix cracking stress and the fiber/matrix interface debonding length increase, and the fiber/matrix interface oxidation length decreases.

Figure 7.

(a) The matrix cracking stress versus the oxidation time, (b) the fiber/matrix interface debonding length versus the oxidation time, and (c) the fiber/matrix interface oxidation length versus the oxidation time corresponding to different fiber strength of σ0 = 1 and 2 GPa.

When the fiber strength is σ0 = 1 GPa, the matrix cracking stress decreases from σmc = 38 to 18 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 14.9 to 14.7 after t = 2 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 16.3 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.52 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

When the fiber strength is σ0 = 2 GPa, the matrix cracking stress decreases from σmc = 43 to 21 MPa after t = 10 h oxidation at elevated temperature of Tem = 800°C; the fiber/matrix interface debonding length first decreases from ld/rf = 15.3 to 15.2 after t = 2.3 h oxidation at elevated temperature of Tem = 800°C and then increases to ld/rf = 16.6 after t = 10 h oxidation at elevated temperature of Tem = 800°C, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.51 after t = 10 h oxidation at elevated temperature of Tem = 800°C.

3.5 Effect of the temperature on matrix cracking stress

The matrix cracking stress, fiber/matrix interface debonding length, and fiber/matrix interface oxidation length versus the oxidation time curves corresponding to different oxidation temperature of Tem = 600 and 800°C are shown in Figure 8. When the oxidation temperature increases, the matrix cracking stress decreases, and the fiber/matrix interface oxidation length and the interface debonding length increase.

Figure 8.

(a) The matrix cracking stress versus the oxidation time, (b) the fiber/matrix interface debonding length versus the oxidation time, and (c) the fiber/matrix interface oxidation length versus the oxidation time corresponding to different oxidation temperature of Tem = 600 and 800°C.

When the oxidation temperature is Tem = 600°C, the matrix cracking stress decreases from σmc = 86.3 to 73 MPa after t = 10 h oxidation; the fiber/matrix interface debonding length decreases from ld/rf = 6.9 to 6.7 after t = 10 h oxidation, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.19 after t = 10 h oxidation.

When the oxidation temperature is Tem = 800°C, the matrix cracking stress decreases from σmc = 86 to 44 MPa after t = 10 h oxidation; the fiber/matrix interface debonding length first decreases from ld/rf = 6.9 to 6.7 after t = 2 h oxidation and then increases to ld/rf = 9 after t = 10 h oxidation, and the fiber/matrix interface oxidation length increases from ζ/ld = 0 to 0.94 after t = 10 h oxidation.

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4. Experimental comparison

Barsoum et al. [15] investigated the matrix crack initiation in fiber-reinforced CMCs. The experimental and predicted matrix cracking stress versus the fiber volume corresponding to different fiber/matrix interface debonding energy of SiC/borosilicate, SiC/LAS, and C/borosilicate composites is shown in Figures 911.

Figure 9.

The experimental and predicted matrix cracking stress versus the fiber volume corresponding to different fiber/matrix interface debonding energy of SiC/borosilicate composite.

Figure 10.

The experimental and predicted matrix cracking stress versus the fiber volume corresponding to different fiber/matrix interface debonding energy of SiC/LAS composite.

Figure 11.

The experimental and predicted matrix cracking stress versus the fiber volume corresponding to different fiber/matrix interface debonding energy of C/borosilicate composite.

For the SiC/borosilicate composite, the predicted matrix cracking stress with the fiber/matrix interface debonding energy range of ξdm = 0.05, 0.1, and 0.2 agrees with the experimental data corresponding to the fiber volume changing from 10 to 50%, as shown in Figure 9. When the fiber/matrix interface debonding energy is ξdm = 0.05, the matrix cracking stress increases from σmc = 49.4 MPa at the fiber volume of Vf = 10% to σmc = 334 MPa at the fiber volume of Vf = 50%; when the fiber/matrix interface debonding energy is ξdm = 0.1, the matrix cracking stress increases from σmc = 52.2 MPa at the fiber volume of Vf = 10% to σmc = 371 MPa at the fiber volume Vf = 50%; and when the fiber/matrix interface debonding energy is ξdm = 0.2, the matrix cracking stress increases from σmc = 57.1 MPa at the fiber volume of Vf = 10% to σmc = 428 MPa at the fiber volume of Vf = 50%.

For the SiC/LAS composite, the matrix cracking stress with the interface debonding energy range of ξdm = 0.01, 0.02, and 0.03 agrees with the experimental data corresponding to the fiber volume changing from 30 to 50%, as shown in Figure 10. When the fiber/matrix interface debonding energy is ξdm = 0.01, the matrix cracking stress increases from σmc = 228 MPa at the fiber volume of Vf = 30% to σmc = 423 MPa at the fiber volume of Vf = 50%; when the fiber/matrix interface debonding energy is ξdm = 0.02, the matrix cracking stress increases from σmc = 257 MPa at the fiber volume of Vf = 30% to σmc = 489 MPa at the fiber volume of Vf = 50%; and when the fiber/matrix interface debonding energy is ξdm = 0.03, the matrix cracking stress increases from σmc = 281 MPa at the fiber volume of Vf = 30% to σmc = 542 MPa at the fiber volume of Vf = 50%.

For the C/borosilicate composite, the predicted matrix cracking stress with the fiber/matrix interface debonding energy range of ξdm = 0.01, 0.02, and 0.03 agrees with the experimental data corresponding to the fiber volume changing from 30 to 55%, as shown in Figure 11. When the fiber/matrix interface debonding energy is ξdm = 0.01, the matrix cracking stress increases from σmc = 136 MPa at the fiber volume of Vf = 30% to σmc = 438 MPa at the fiber volume of Vf = 55%; when the fiber/matrix interface debonding energy is ξdm = 0.02, the matrix cracking stress increases from σmc = 156 MPa at the fiber volume of Vf = 30% to σmc = 471 MPa at the fiber volume of Vf = 55%; and when the fiber/matrix interface debonding energy is ξdm = 0.03, the matrix cracking stress increases from σmc = 173 MPa at the fiber volume of Vf = 30% to σmc = 504 MPa at the fiber volume of Vf = 55%.

Yang [16] investigated the mechanical behavior of C/SiC composite after unstressed oxidation at elevated temperature of 700°C in air atmosphere. The composite was divided into two types based on the interface bonding, i.e., strong interface bonding and weak interface bonding. For the C/SiC with the strong interface bonding, the matrix cracking stresses of C/SiC corresponding to the proportional limit stresses in the tensile curves are 37, 30, and 20 MPa corresponding to the cases of without oxidation, 4 h oxidation, and 6 h unstressed oxidation, respectively. For the C/SiC with the weak interface bonding, the matrix cracking stresses of C/SiC corresponding to the proportional limit stresses in the tensile curves are 27, 20, and 13 MPa corresponding to the cases of without oxidation, 2 h oxidation, and 6 h unstressed oxidation, respectively. The experimental and predicted matrix cracking stresses of C/SiC composite with strong and weak interface bonding after unstressed oxidation at elevated temperature of 700°C in air are shown in Figure 12a and b, respectively. The matrix cracking stress decreases 18.9% after oxidation for 4 h, 46% after oxidation for 6 h for C/SiC with strong interface bonding, 25.9% after oxidation for 1 h, and 51.8% after oxidation for 6 h for C/SiC with weak interface bonding. The theoretical predicted results agreed with experimental data. The strong interface bonding can be used for oxidation resistant of C/SiC composite at elevated temperature.

Figure 12.

The experimental and predicted matrix cracking stress versus the oxidation time of C/SiC composite after unstressed oxidation at 700°C in air corresponding to (a) strong interface bonding and (b) weak interface bonding.

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5. Conclusion

In this chapter, the time-dependent matrix cracking of fiber-reinforced CMCs was investigated using the energy balance approach. The relationship between the matrix cracking stress, fiber and interface oxidation, and fiber failure was established. The effects of the fiber volume, interface shear stress and interface debonding energy, fiber failure, and oxidation temperature on the time-dependent matrix cracking stress were analyzed. The experimental matrix cracking stress of different fiber-reinforced CMCs was predicted.

  1. When the fiber volume, fiber/matrix interface debonding energy and interface shear stress increase, the matrix cracking stress and the fiber/matrix interface oxidation length increase, and the fiber/matrix interface debonding length decreases.

  2. When the fiber strength increases, the matrix cracking stress and the fiber/matrix interface debonding length increase, and the fiber/matrix interface oxidation length decreases.

  3. When the oxidation temperature increases, the matrix cracking stress decreases, and the fiber/matrix interface oxidation length and the interface debonding length increase.

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Acknowledgments

The work reported here is supported by the Fundamental Research Funds for the Central Universities (Grant No. NS2019038).

References

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Written By

Li Longbiao

Submitted: July 29th, 2019 Reviewed: October 7th, 2019 Published: December 11th, 2019