Damage accumulation and lifetime prediction of fiber-reinforced ceramic-matrix composites under thermomechanical fatigue loading

Abstract In this paper, the damage accumulation and life prediction in fiber-reinforced ceramic-matrix composites (CMCs) subjected to thermomechanical fatigue (TMF) loading are investigated. The relationships between TMF damage mechanisms, fatigue hysteresis-based damage parameters, fraction of broken fiber, and applied cycles are established. Evolution of fatigue hysteresis dissipated energy, fatigue hysteresis modulus, fatigue peak strain, fatigue broken fiber fraction versus applied cycle curves, and the fatigue life S–N curves is analyzed. Damage accumulation and fatigue life of cross-ply silicon carbide/magnesium aluminosilicate composite under in-phase (IP)- and out-of-phase (OP)-TMF and isothermal fatigue (IF) loading are predicted. Under the same fatigue peak stress, the fatigue lifetime decreases from IF loading at 566°C to IF loading at 1,093°C, IP-TMF and OP-TMF. The TMF loading significantly reduced the fatigue lifetime of CMCs.


Introduction
Ceramic-matrix composites (CMCs) are being designed for hot section components in commercial aero engines. As new materials, it is necessary to develop models, methods, and tools to predict the degradation, damage, and failure mechanisms subjected to cyclic loading at different temperatures and environments [1,2]. For the real life applications, the CMC components are subjected to thermomechanical fatigue (TMF) [3,4].
Many researchers performed experimental and theoretical investigations on TMF behavior of fiber-reinforced CMCs. TMF life was significantly shorter under TMF loading than isothermal fatigue (IF) loading [5][6][7][8][9][10][11][12]. The thermal and mechanical load cycling in oxidizing could cause coating/matrix cracks, interface degradation, and fiber oxidation [7]. In the turbine airfoil applications, thermal stress and intermediate temperature embrittlement could be just as much damage factors in reducing fatigue life of CMCs as the exposure to the harsh combustion environment and mechanical loading [9]. Shojaei et al. [13] developed a multi-scale constitutive model of fiber-reinforced CMCs using continuum damage mechanics. An asymptotic solution of a microscale representative volume element (RVE) including the fiber, the interphase, and the matrix was provided, and the CMD model and the RVE asymptotic solution were used to study the microscale damage mechanisms in CMC systems. Min et al. [14] developed a micromechanical fatigue damage model to analyze the damage progression and fatigue failure of fiber-reinforced CMCs. The three-phase micromechanics, the shear-lag, and the continuum fracture mechanics models were integrated with a statistical model in the repeating unit cell to predict the progressive damage and fatigue life of the composite structures. Skinner et al. [15] developed a multiscale damage model for length scaledependent behavior of fiber-reinforced CMCs. The damage mechanism of scale-specific brittle matrix damage initiation and propagation was considered. The nonlinear behavior of 2D C/silicon carbide (SiC) composite was predicted using the developed model. Yang and Liu [16] developed a new continuum TMF damage model for oxide/oxide CMCs and predicted the damage evolution under coupled cyclic thermal shocks and mechanical loading. Li [17,18] investigated the effects of fiber debonding, fracture, and oxidation on matrix cracking of CMCs at room and elevated temperature in oxidative environment. Li [19] found that the degradation rate of the interface shear stress at 800°C in air condition is much higher than that at room temperature, leading to greatly decreasing fatigue limit stress. The comparisons of the interface shear stress degradation between C/SiC and SiC/SiC composites were also investigated [20]. Under multiple loading stress levels, damage mechanism of the interface wear at room temperature and different loading sequences affect the interface debonding extent and the range of the interface sliding [21,22]; under combination of cyclic fatigue and stress-rupture loading, the interface oxidation and fatigue peak stress levels affect the interface debonding and slip length [23]. Fatigue hysteresis loops can be used as an effective tool to monitor damage evolution in fiber-reinforced CMCs [24]. Li [25][26][27] investigated the effects of temperature and oxidation on damage evolution and fatigue lifetime of CMCs. However, in the researches mentioned above, damage accumulation and life prediction of fiber-reinforced CMCs under IP-and OP-TMF and IF loading in oxidizing atmosphere have not been investigated.
The objective of this paper is to investigate the damage accumulation and life of fiber-reinforced CMCs under TMF loading. Evolution of fatigue hysteresis dissipated energy (FHDE), fatigue hysteresis modulus (FHM), fatigue peak strain (FPS), fatigue broken fiber fraction (FBFF) versus applied cycle number curves, and the fatigue life S-N curves is analyzed. Experimental damage accumulation and life of cross-ply SiC/magnesium aluminosilicate (MAS) composite under TMF loading are predicted.

Materials and experimental procedures
Nicalon™ SiC (Nippon Carbon Co., Ltd, Tokyo, Japan) fiber-reinforced barium-stuffed MAS cordierite matrix composite (SiC/MAS CMCs) was received from Corning (Corning, NY, USA) [6] and manufactured by the hot-pressing method at elevated temperature above 1,200°C. The fiber volume fraction was approximately 40%. The TMF tests were conducted on a MTS servo hydraulic load-frame (MTS Systems Corp., Minneapolis MN, USA) and performed by Allen and Mall [6]. The extensometer was an MTS 632.65B-03, with a gage length of 2.54 cm. The TMF tests were conducted at the temperature range of 566°C and 1,093°C in air condition under the load control at a triangular waveform and the fatigue load ratio of R = 0.1. The stress and strain data obtained during the test were used to determine the modulus degradation, strain history, and hysteresis loops.

Theoretical analysis 3.1 Hysteresis-based damage parameters
Mechanical hysteresis of CMCs is affected by the temperature. Under TMF loading, two cases are considered in the present analysis, i.e., in-phase (IP)-TMF and out-of-phase (OP)-TMF, as shown in Figure 1. Temperature-dependent interface shear stress is given by equation (1) [11].
where τ 0 is the interface shear stress corresponding to steady state, μ is the interface frictional coefficient, α rf and α rm are the fiber and matrix radial thermal expansion coefficient, respectively, and A is a constant depending on the elastic properties of the matrix and fibers. Upon unloading and reloading, interface slip and counter slip occur in different damage regions, as shown in Figure 2. The strain of the composite is equivalent to the strain of the intact fibers, as shown in equation (2).
where V f and V f are the volume of the fiber and the matrix, respectively, E m is the matrix elastic modulus, ρ is the shear-lag model parameter, r f is the fiber radius, l d is the interface debonding length, and ξ d is the interface debonding energy. For the condition of complete debonding at the interface, the unloading stress-strain relationship can be determined by equations (7) and (8).
The stress carried by intact and broken fibers can be determined by equation (13) using the global load sharing criterion [33].
where l f is the slip length, Φ is the intact fiber stress, Φ b 〈 〉 is the broken fiber stress, P f is the total fiber failure probability, and P r is the fiber failure probability in the interface debonding and bonding region.
where P fa , P fb , P fc , and P fd are the fiber failure probability of oxidized fibers in the oxidation region, unoxidized fibers in the oxidation region, fibers in the interface debonding and bonding region, ζ is the fraction of oxidation fibers in the oxidized region, and χ is the fraction of oxidation in the multiple matrix cracks.
where m f is the fiber Weibull modulus, K IC is the critical stress intensity factor, Y is a geometric parameter, k is the parabolic rate constant, b 0 is a coefficient, j is an exponent that determines the rate at which interface shear stress drops with the number of cycle N, and p 1 and p 2 are fiber strength degradation parameters.
With increasing applied cycles, the interface shear stress and the fiber strength decrease because of the interface wear and interface oxidation [34]. The evolution of fiber failure probability versus applied cycle number curves can be obtained. When the fraction of the broken fiber approaches critical value, the composite fatigue fractures. The fatigue limit stress is calculated when the fracture applied cycles approach the maximum cycle number.  Figure 3 shows the tensile stress-strain curves at 566°C and 1,093°C in air atmosphere.

Experimental comparisons
At T = 566°C, the tensile stress-strain curves can be divided into three domains as follows: (1) the linear domain, from the initial loading to the first matrix cracking stress σ mc = 95 MPa; (2) the nonlinear domain because of cracking of the matrix and debonding at the interface, from σ mc = 95 MPa to the saturation matrix cracking stress σ sat = 220 MPa; and (3) the secondary linear domain, from σ sat = 220 MPa to composite strength σ uts = 295 MPa, with the failure strain ε f = 0.76%.

Discussion
Comparative analysis of fatigue damage accumulation and lifetime cross-ply SiC/MAS composite under IP-and OP-TMF and IF loading are investigated. Figure 12 shows the evolution of FHDE versus applied cycle number curves of cross-ply SiC/MAS composite under IP-and OP-TMF and IF loading. Under IP-TMF loading at σ max = 120 MPa, the FHDE increases with applied cycle number; however, under IF loading at 566°C and 1,093°C at σ max = 120 MPa, the FHDE decreases with applied cycle number, and the FHDE increases with temperature. Figure 13 shows the evolution of FHM versus applied cycle number curves of cross-ply SiC/MAS composite under IP-and OP-TMF and IF loading. Under σ max = 120 MPa, the degradation rate of FHM is the highest for IF loading at 1,093°C and the lowest for IF loading at 566°C, and the FHM degradation rate under IP-TMF loading lies between that of IF loading at 1,093°C and 566°C. However, the FHM degradation rate is the highest under OP-TMF loading at σ max = 91 MPa compared with that under testing conditions of IF-TMF loading at σ max = 105 MPa, IF loading at 566°C and σ max = 103 MPa, and IF loading at 1,093°C and σ max = 103 MPa. Figure 14 shows the evolution of FPS versus applied cycle number curves of cross-ply SiC/MAS composite under IP-and OP-TMF and IF loading. Under IP-TMF loading at σ max = 120 MPa, the increasing rate of FPS is the highest, compared with that under the test conditions of OP-TMF loading at σ max = 110 MPa, IF loading at 566°C and 1,093°C with σ max = 137 MPa. Compared with OP-TMF loading, the increasing rate of FPS is higher for IP-TMF loading at the same peak stress. Under IF loading, the increasing rate of FPS increases with peak stress and testing temperature.  decreases from IF at 566°C to IF at 1,093°C, IP-TMF and OP-TMF. The TMF loading significantly reduced the fatigue life of CMCs.

Conclusions
In this paper, TMF damage accumulation and life of crossply SiC/MAS composite were predicted. Damage evolution    (1) Under the same fatigue peak stress, the FHM degradation rate under IP-TMF loading lies between that of IF loading at 1,093°C and 566°C. However, the FHM degradation rate is the highest for OP-TMF loading.
(2) Compared with OP-TMF loading, the increasing rate of FPS is higher for IP-TMF loading at the same peak stress. Under IF loading, the increasing rate of the FPS increases with peak stress and the testing temperature.