Effect of temperature on matrix multicracking evolution of C/SiC fiber-reinforced ceramic-matrix composites

1 Introduction

Ceramic-matrix composites (CMCs) possess low density, high strength and modulus, and wear and corrosion resistance at elevated temperatures and have already been applied on hot-section components of aeroengines [1,2]. The CMC exhaust cone demonstrator that was designed, built, and tested by SAFRAN was certified for use on commercial aircraft by European Aviation Safety Agency and completed the first commercial flight on A320 jetliner on 2015. The application of CMCs on aeroengines has the following advantages: (1) as the specific strength of CMCs is higher than that of nickel-based alloys, the weight of turbine components made of nickel-based alloys can be reduced by 61% if CMCs are used; (2) due to the high-temperature resistance of CMCs, the turbine temperature can be increased up to 1,650°C, which is beneficial to increase the thrust–weight ratio of the aeroengine; (3) compared with ceramics, the strain tolerance of CMCs is greatly improved, which would not cause catastrophic damage, and it is possible to detect the mechanical degradation before materials failure and improve the reliability of life prediction; and (4) replacing the existing high-temperature metal materials with CMCs that can reduce the weight, pollution emission, and noise level of the aeroengine.

At elevated temperatures, the occurrence of the evolution of matrix multicracking affects the mechanical behavior of fiber-reinforced CMCs [3,4,5,6,7]. When the applied stress increases, the evolution of matrix multicracking and debonding of the interface dissipate the energy entered into the composite and the tensile behavior of CMCs exhibits obvious nonlinear behavior [8,9,10,11,12]. Multiple matrix fracture and debonding of the interface and pullout of the fracture fibers at the matrix crack plane are the main damage mechanisms for the nonlinear behavior of CMCs [13,14,15,16,17,18,19,20,21]. The evolution rate of matrix multicracking, debonding fraction at the interface, and the broken fraction of the fibers affect the deformation characteristics of CMCs. For the problem of first matrix cracking, the energy balance relationship before and after steady-state matrix cracking can be established, the first matrix cracking stress for long-steady-state matrix cracking can be solved [22,23,24,25,26,27,28], and the stress intensity factor is used for determining propagation of short matrix cracking stress [29,30,31,32,33,34]. The stress for short matrix cracking increases with flaw size inside the matrix and approaches the long matrix cracking stress [35,36]. Under tensile loading, short matrix cracking appears first at the linear region of the tensile curve; with increasing applied stress, the short matrix cracking propagates and evolves into long matrix cracking, leading to the decrease of composite modulus and the nonlinear behavior of tensile curves [37]. For the problem of matrix multicracking, the maximum stress criterion is first used to predict the generation of matrix multicracking; however, the predicted matrix multicracking approaches the saturation at single applied stress, leading to the step behavior on the predicted tensile stress–strain curve [22,38]; the interaction matrix cracking model calculates the steady-state strain energy release rate considering the change of energy before and after steady-state matrix cracking and matrix flaw density [39]; the energy balance approach is also used to predict the matrix multicracking by establishing the energy balance relationship between two different matrix cracking spaces or configurations [40,41,42,43]; the critical matrix strain energy (CMSE) criterion presumes that the matrix strain energy exists a critical value, when the matrix strain energy is higher than the critical value, the additional matrix strain energy is dissipated through forming new crackings and debonding propagation of the interface [44,45,46,47,48]; the matrix statistical cracking model considers the matrix flaw distribution using the Weibull model and divides the matrix cracking space into three cases, i.e., long matrix crack spacing, medium matrix crack spacing, and short matrix crack spacing; with increasing applied stress, the long and medium matrix crack spacing continually transfers into the short matrix crack spacing, due to the interaction effect between neighboring cracks, when all the matrix crack spacings are less than debonding length at the interface and the evolution of matrix cracking approaches the saturation [49,50]. However, there are few theoretical and experimental investigations on matrix multiple fracture of CMCs at elevated temperatures [51,52,53,54]. The temperature-dependent material properties of the interface, matrix, and fiber affect the evolution of matrix multicracking in fiber-reinforced CMCs [55]. Li [56] investigated matrix multiple fractures of SiC/SiC composites at elevated temperatures. Li [57] developed a micromechanical approach to predict the fatigue life of fiber-reinforced CMCs with different fiber preforms at room and elevated temperatures. Under cyclic fatigue loading, multiple fatigue damage mechanisms degrade the interface performance and fiber strength. The broken fiber fraction is determined considering fatigue damage mechanisms using Global Load Sharing criterion. The composite fatigue fracture occurs when the broken fiber fraction approaches the critical value. However, for C/SiC composites, the temperature-dependent matrix multifracture behavior is much different from that of SiC/SiC composites. The principal objective of the present study is to develop a temperature-dependent matrix multicracking model for C/SiC composites.

In this paper, the temperature-dependent evolution of the matrix multicracking of C/SiC composites is investigated using the CMSE criterion. The temperature-dependent material properties of the interface, fiber, and matrix are considered in the analysis of the microstress field and damage models. The temperature-dependent matrix strain energy and its critical values are determined. The effects of the interface properties, matrix properties, and environmental temperature on the temperature-dependent evolution of the matrix multicracking and the debonding at the interface of C/SiC composites are discussed. The relationships among the evolution of matrix cracking, debonding of the interface, and environmental temperature are established. The experimental evolution of the matrix multicracking and debonding at the fiber/matrix interface of unidirectional C/SiC composites at elevated temperatures are predicted.

2 Theoretical analysis

The ceramic composite system of C/SiC is used for the case study, and its material properties are given by V_f = 30%, r_f = 3.5 µm, γ_m = 6 J/m² (at room temperature), γ_d = 0.4 J/m² (at room temperature), and τ_i = 20 MPa (at room temperature).

The temperature-dependent elastic modulus of the carbon fiber is given by [58]

The temperature-dependent elastic modulus of the SiC matrix is obtained as follows: [59]

The temperature-dependent axial and radial thermal expansion coefficients of carbon fiber are [60]

The temperature-dependent axial and radial thermal expansion coefficient of SiC matrix is [59]

The temperature-dependent interface debonding energy γ_d(T) and matrix fracture energy γ_m(T) are [61]

where T₀ denotes the reference temperature, which is set as room temperature [61], T_m denotes the fabricated temperature, γd0 and γm0 denote the interface debonding energy and matrix fracture energy at the reference temperature of T₀, and C_P(T) is

The temperature-dependent matrix strain energy is

where A_m is the cross-sectional area of the matrix in the unit cell, l_c(T) is the temperature-dependent matrix crack spacing, and σ_m(T) is the temperature-dependent matrix axial stress.

The temperature-dependent fiber and matrix axial stress and interface shear stress are given in Appendix A, and the temperature-dependent interface debonding length is given in Appendix B.

Substituting the temperature-dependent matrix axial stresses in equation (A.2) into equation (9), the matrix strain energy U_m(T) for the condition of partial interface debonding is given by

where A_m is the cross-sectional area of the matrix in the unit cell, E_m(T) is the temperature-dependent matrix elastic modulus, which can be determined using equation (2), V_f and V_m are the fiber and matrix volume, r_f is the fiber radius, l_d(T) is the temperature-dependent interface debonding length, which can be determined by equation (B.6); σ_mo(T) is the temperature-dependent matrix axial stress in the bonding region, which can be determined using equation (5), ρ is the shear-lag model parameter, and τ_i(T) is the temperature-dependent interface shear stress, which is given by [62]

where τ₀ is the steady-state interface shear stress, µ is the interface frictional coefficient, and A is a constant depending on the elastic properties of the matrix and fibers.

For the condition of complete debonding at the interface, the matrix strain energy is given by

The CMSE U_mcr at the critical cracking stress of the matrix σ_cr is determined as

where k (k ∈ [0, 1]) is the CMSE parameter, l₀ is the initial matrix crack spacing, and σ_mocr(T) is given by

where σ_cr(T) is the temperature-dependent critical stress for matrix cracking using the ACK model [22].

where γ_m(T) is the temperature-dependent matrix fracture energy.

The evolution of matrix multicracking in CMCs can be determined as

3 Results and discussion

In the present analysis, the effects of interface properties, matrix properties, and environmental temperature on the temperature-dependent evolution of matrix multicracking and debonding propagation at the interface of unidirectional C/SiC composites are discussed.

3.1 Effect of interface shear stress on matrix multicracking of C/SiC composites

The effect of interface shear stress (i.e., τ₀ = 30 and 35 MPa) on temperature-dependent evolution of the matrix multicracking and debonding fraction at the interface of C/SiC composites at elevated temperatures of T = 773, 873, 973, and 1,073 K is shown in Figure 1. When the interface shear stress increases, the debonding resistance at the interface increases, leading to the decrease of the debonding fraction at the interface; the stress transfer capacity between the fiber and the matrix increases, leading to the higher first matrix cracking stress, saturation matrix cracking stress, and saturation matrix cracking density.

Figure 1

(a) The matrix multicracking density versus applied stress curves when τ₀ = 30 MPa; (b) the interface debonding fraction versus applied stress curves when τ₀ = 30 MPa; (c) the matrix multicracking density versus applied stress curves when τ₀ = 35 MPa; and (d) the interface debonding fraction versus applied stress curves when τ₀ = 35 MPa of C/SiC composites at elevated temperatures of T = 773, 873, 973, and 1,073 K.

When τ₀ = 30 MPa at an elevated temperature of T = 773 K, the density of matrix multicracking increases from λ = 0.08/mm at σ_cr = 53 MPa to λ = 1.2/mm at σ_sat = 107 MPa; the fraction of the interface debonding increases from η = 0.9% to η = 30.2%. At an elevated temperature of T = 873 K, the density of the matrix multicracking increases from λ = 0.07/mm at σ_cr = 77 MPa to λ = 1.5/mm at σ_sat = 154 MPa and the fraction of the interface debonding increases from η = 0.9% to η = 40.6%. At an elevated temperature of T = 973 K, the density of the matrix multicracking increases from λ = 0.07/mm at σ_cr = 104 MPa to λ = 1.9/mm at σ_sat = 209 MPa and the fraction of the interface debonding increases from η = 0.98% to η = 51.9%. At an elevated temperature of T = 1,073 K, the density of the matrix multicracking increases from λ = 0.08/mm at σ_cr = 134 MPa to λ = 2.2/mm at σ_sat = 268 MPa and the fraction of the interface debonding increases from η = 1% to η = 62.8%.

When τ₀ = 35 MPa at an elevated temperature of T = 773 K, the density of the matrix multicracking increases from λ = 0.09/mm at σ_cr = 73 MPa to λ = 1.7/mm at σ_sat = 146 MPa and the fraction of the interface debonding increases from η = 0.9% to η = 36.6%. At an elevated temperature of T = 873 K, the density of the matrix multicracking increases from λ = 0.09/mm at σ_cr = 95 MPa to λ = 2.0/mm at σ_sat = 189 MPa and the fraction of the interface debonding increases from η = 0.9% to η = 45.3%. At an elevated temperature of T = 973 K, the density of the matrix multicracking increases from λ = 0.09/mm at σ_cr = 121 MPa to λ = 2.3/mm at σ_sat = 242 MPa and the fraction of the interface debonding increases from η = 0.98% to η = 55.3%. At an elevated temperature of T = 1,073 K, the density of the matrix multicracking increases from λ = 0.09/mm at σ_cr = 149 MPa to λ = 2.7/mm at σ_sat = 298 MPa and the fraction of the interface debonding increases from η = 1% to η = 65.3%.

3.2 Effect of interface debonding energy on matrix multicracking of C/SiC composites

The effect of interface debonding energy (i.e., γ_d = 0.5 and 1.0 J/m²) on the temperature-dependent evolution of the matrix multicracking and debonding fraction at the interface of C/SiC composites at elevated temperatures of T = 773, 873, 973, and 1,073 K are shown in Figure 2. When the interface debonding energy increases, the debonding resistance at the interface increases, leading to the decrease of the debonding fraction at the interface and the increase of the evolution rate during matrix multicracking.

Figure 2

(a) The matrix multicracking density versus applied stress curves when γ_d = 0.5 J/m²; (b) the interface debonding fraction versus applied stress curves when γ_d = 0.5 J/m²; (c) the matrix multicracking density versus applied stress curves when γ_d = 1.0 J/m²; and (d) the interface debonding fraction versus applied stress curves when γ_d = 1.0 J/m² of C/SiC composites at elevated temperatures of T = 773, 873, 973, and 1,073 K.

When γ_d = 0.5 J/m² at an elevated temperature of T = 773 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 88 MPa to λ = 2.2/mm at σ_sat = 176 MPa and fraction of the interface debonding increases from η = 0.93% to η = 40.5%. At an elevated temperature of T = 873 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 109 MPa to λ = 2.5/mm at σ_sat = 218 MPa and the fraction of the interface debonding increases from η = 0.95% to η = 48.4%. At an elevated temperature of T = 973 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 134 MPa to λ = 2.8/mm at σ_sat = 269 MPa and the fraction of the interface debonding increases from η = 0.98% to η = 57.5%. At an elevated temperature of T = 1,073 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 162 MPa to λ = 3.1/mm at σ_sat = 324 MPa and the debonding fraction at the interface increases from η = 1% to η = 66.7%.

When γ_d = 1.0 J/m² at an elevated temperature of T = 773 K, the density of the matrix multicracking increases from λ = 0.12/mm at σ_cr = 88.4 MPa to λ = 2.3/mm at σ_sat = 176.8 MPa and the debonding fraction at the interface increases from η = 0.9% to η = 40.1%. At an elevated temperature of T = 873 K, the density of the matrix multicracking increases from λ = 0.12/mm at σ_cr = 109 MPa to λ = 2.6/mm at σ_sat = 218 MPa and the debonding fraction at the interface increases from η = 0.93% to η = 47.8%. At an elevated temperature of T = 973 K, the density of the matrix multicracking increases from λ = 0.11/mm at σ_cr = 134 MPa to λ = 2.9/mm at σ_sat = 269 MPa and the debonding fraction at the interface increases from η = 0.95% to η = 56.4%. At an elevated temperature of T = 1,073 K, the evolution of the matrix multicracking increases from λ = 0.12/mm at σ_cr = 162 MPa to λ = 3.3/mm at σ_sat = 324 MPa and the debonding fraction of the interface increases from η = 0.97% to η = 65%.

3.3 Effect of matrix fracture energy on matrix multicracking of C/SiC composites

The effect of matrix fracture energy (i.e., γ_m = 20 and 30 J/m²) on the temperature-dependent evolution of the matrix multicracking and the debonding fraction at the interface of C/SiC composites at elevated temperatures of T = 773, 873, 973, and 1,073 K are shown in Figure 3. When the matrix fracture energy increases, the energy needed for the matrix multicracking increases, leading to the higher first matrix cracking stress, saturation matrix cracking stress, and saturation matrix cracking density.

Figure 3

(a) The matrix multicracking density versus applied stress curves when γ_m = 20 J/m²; (b) the interface debonding fraction versus applied stress curves when γ_m = 20 J/m²; (c) the matrix multicracking density versus applied stress curves when γ_m = 30 J/m²; and (d) the interface debonding fraction versus applied stress curves when γ_m = 30 J/m² of C/SiC composites at elevated temperatures of T = 773, 873, 973, and 1,073 K.

When γ_m = 20 J/m² at an elevated temperature of T = 773 K, the density of the matrix multicracking increases from λ = 0.14/mm at σ_cr = 78 MPa to λ = 2.4/mm at σ_sat = 156 MPa and the debonding fraction at the interface increases from η = 0.9% to η = 37.2%. At an elevated temperature of T = 873 K, the density of the matrix multicracking increases from λ = 0.13/mm at σ_cr = 97 MPa to λ = 2.8/mm at σ_sat = 195 MPa and the debonding fraction at the interface increases from η = 0.9% to η = 44.9%. At an elevated temperature of T = 973 K, the density of the matrix multicracking increases from λ = 0.13/mm at σ_cr = 122 MPa to λ = 3.2/mm at σ_sat = 243 MPa and the debonding fraction at the interface increases from η = 0.93% to η = 53.8%. At an elevated temperature of T = 1,073 K, the density of the matrix multicracking increases from λ = 0.13/mm at σ_cr = 148 MPa to λ = 3.5/mm at σ_sat = 296 MPa and the debonding fraction at the interface increases from η = 0.95% to η = 62.6%.

When γ_m = 30 J/m² at an elevated temperature of T = 773 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 97.3 MPa to λ = 2.2/mm at σ_sat = 194 MPa and the debonding fraction at the interface increases from η = 0.93% to η = 42.5%. At an elevated temperature of T = 873 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 119 MPa to λ = 2.5/mm at σ_sat = 238 MPa and the debonding fraction at the interface increases from η = 0.95% to η = 50.1%. At an elevated temperature of T = 973 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 146 MPa to λ = 2.8/mm at σ_sat = 292 MPa and the debonding fraction at the interface increases from η = 0.96% to η = 58.5%. At an elevated temperature of T = 1,073 K, the density of the matrix multicracking increases from λ = 0.1/mm at σ_cr = 174 MPa to λ = 3.1/mm at σ_sat = 349 MPa and the debonding fraction at the interface increases from η = 0.98% to η = 66.9%.

4 Experimental comparisons

Li et al. [63] investigated the tensile behavior of unidirectional T-700™ C/SiC composites at room temperature. C/SiC composites were manufactured using the hot-pressing method at ∼1,000°C, which offered the ability to fabricate dense composites via a liquid-phase sintering method at a low temperature. Low-pressure chemical vapor infiltration was employed to deposit approximately 5–20 layer PyC/SiC with the mean thickness of 0.2 µm to enhance the desired nonlinear/noncatastrophic tensile behavior. The nano-SiC powder and sintering additives were ball milled for 4 h using SiC balls. After drying, the powders were dispersed in xylene with polycarbonsilane to form the slurry. Carbon fiber tows were infiltrated by the slurry and wound to form aligned unidirectional composite sheets. The tensile experiments were conducted on an MTS Model 809 servo-hydraulic loadframe (MTS Systems Corp., Minneapolis, MN, USA) equipped with edge-loaded grips, operated at a loading rate of 10 MPa/s. Gage-section strains were measured using a clip-on extensometer (Model No. 634.12F-24, MTS systems Corp.; modified for a 25 mm gage-length). Direct observations of matrix cracking were made using HiROX optical microscope. Matrix crack densities were determined by counting the number of cracks at a length of about 15 mm.

The experimental and theoretical densities of the matrix multicracking and the debonding fraction at the interface of unidirectional C/SiC composites at room temperature and elevated temperatures of T = 773 and 873 K are predicted, as shown in Figure 4.

Figure 4

(a) The experimental and theoretical matrix multicracking density versus applied stress curves and (b) the interface debonding fraction versus applied stress curves of unidirectional C/SiC composites.

At room temperature, the evolution of the matrix multicracking starts from σ_cr = 100 MPa and approaches saturation at σ_sat = 125 MPa and the density of the matrix multicracking increases from λ = 4.2/mm to λ = 9.4/mm. At an elevated temperature of T = 773 K, the density of the matrix multicracking increases from λ = 1.3/mm at σ_cr = 140 MPa to λ = 8.7/mm at σ_sat = 220 MPa and the debonding fraction at the interface increases from η = 0.5% to η = 43%; at an elevated temperature of T = 873 K, the density of matrix multicracking increases from λ = 1.1/mm at σ_cr = 182 MPa to λ = 9.4/mm at σ_sat = 280 MPa and the debonding fraction at the interface increases from η = 0.6% to η = 52.2%.

For the C/SiC composites, when environmental temperature increases, both the first matrix cracking stress and saturation matrix cracking stress increase and the saturation matrix cracking density also increases.

5 Conclusion

In this paper, the temperature-dependent evolution of the matrix multicracking and debonding fraction at the interface of C/SiC composites is investigated. The relationships among applied stress, multicracking density, debonding fraction at the interface, interface/fiber/matrix properties, and environmental temperature are established. The effects of interface properties, matrix properties, and environmental temperature on temperature-dependent evolution of the matrix multicracking and debonding fraction at the interface of C/SiC composites are discussed. The experimental evolution of the matrix multicracking and the debonding fraction at the interface curves of unidirectional C/SiC composites at elevated temperatures are predicted.

For the SiC/SiC composites, when environmental temperature increases, both the first matrix cracking stress and saturation matrix cracking stress decrease and the saturation matrix cracking density also decreases. However, for the C/SiC composites, when environmental temperature increases, both the first matrix cracking stress and saturation matrix cracking stress increase and the saturation matrix cracking density also increases.

6 Data availability

The data used to support the findings of this study are available from the paper.