Modeling Tensile Damage and Fracture Behavior of Fiber-Reinforced Ceramic-Matrix Minicomposites

Evolution of damage and fracture behavior of fiber-reinforced mini ceramic-matrix composites (mini-CMCs) under tensile load are related to internal multiple damage mechanisms, i.e., fragmentation of the brittle matrix, crack defection, and fibers fracture and pullout. In this paper, considering multiple micro internal damage mechanisms and related models, a micromechanical constitutive stress–strain relationship model is developed to predict the nonlinear mechanical behavior of mini-CMCs under tensile load corresponding to different damage domains. Relationships between multiple micro internal damage mechanisms mentioned above and tensile micromechanical multiple damage parameters are established. Experimental tensile nonlinear behavior, internal damage evolution, and micromechanical tensile damage parameters corresponding to different damage domains of two different types of mini-CMCs are predicted. The effects of constitutive properties and damage-related parameters on nonlinear behavior of mini-CMCs are discussed.


Introduction
With the development of thermodynamics, the advancement of component integrated design technology, the weight reduction brought by structural simplification and the comprehensive development of material technology, the thrust-to-weight ratio of aeroengines has gradually increased. However, on the premise of maintaining the engine layout and not changing the conventional metal materials, the improvement of aerodynamic, thermal, component design and structural weight reduction techniques can only increase the ratio between the thrust and weight of the aeroengines to approximately 14. For aeroengines with ratios between the thrust and a weight of approximately [12][13][14][15] or higher, more breakthroughs must be made in new materials, new process applications and new structural design, such as polymer-matrix composites (PMCs) or metal-matrix composites (MMCs) for cold section components in aeroengine and ceramic-matrix composites (CMCs) for hot section components in aeroengines. Two main types of CMCs are used in aeroengines, including, C/SiC and SiC/SiC. For aeroengines, the operating temperatures of C/SiC and SiC/SiC are approximately 1650 • C and 1450 • C, respectively. Increasing the operating temperature of SiC fibers can raise the operating temperature of SiC/SiC to 1650 • C. Results show that the application of CMCs in the hot section components of the combustion chamber, turbine, and nozzle can increase the operating temperature of the aeroengine by 300-500 • C, reduce the weight by 50-70% and increase the thrust by 30-100% [1][2][3][4]. (5) where E f is the elastic modulus of the fiber, η is the ratio between debonding length l d and half of fragmentation length of the brittle matrix l c , α f and α c are the axial thermal expansion coefficient of the fiber and the composite, and ∆T is the temperature difference between testing and fabricated temperature.

Domain III
Upon approaching saturation of fragmentation of the brittle matrix, some fibers gradually fracture and pullout inside of the minicomposite. Considering intact and broken fiber in the different damage regions, the axial stress of the fiber can be determined by Equation (6) [4].
where Φ is the stress of unbroken fiber, which can be determined by Equation (7) [24].
where L is the average pullout length of a broken fiber, and P the probability of fiber being broken. Considering fibers fracture inside of mini-CMCs, the nonlinear stress-strain relationship at damage Domain III is acquired by Equation (5).  Figure 1 shows the experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density of brittle matrix versus applied stress curves of Hi-Nicalon TM SiC/SiC mini-CMC. The predicted results of tensile curves and fragmentation density of brittle matrix evolution curves both agreed well with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including: (1) Domain I, the linear domain. The domain starts from the initial loading to the first matrix fragmentation stress σ mc = 200 MPa, and the first fragmentation density of the matrix is approximately λ mc = 0.02/mm, and the corresponding composite tensile strain is approximately ε c = 0.06%.
(2) Domain II, the nonlinear region due to matrix fragmentation. The domain starts from σ mc = 200 MPa to the saturation matrix fragmentation stress σ sat = 560 MPa, and the saturation fragmentation density of the matrix is approximately λ sat = 1.76/mm, and the composite tensile strain is approximately ε c = 0.34%.     Figure 2 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density of the brittle matrix versus applied stress curves of Hi-Nicalon TM Type S SiC/SiC minicomposite. The predicted results of tensile curves and fragmentation density of brittle matrix evolution curves agree with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including:     Figure 3 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density evolution of brittle matrix curves versus applied stress of Tyranno TM ZMI SiC/SiC minicomposite. Predicted results of tensile curves and fragmentation density of brittle matrix evolution curves agree well with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including: (1) Domain I, the linear region. The domain starts from initial loading to σ mc = 150 MPa with λ mc = 0.01/mm and ε c = 0.05%.
(2) Domain II, the nonlinear region due to matrix fragmentation. The domain starts from σ mc = 150 MPa to σ sat = 350 MPa with λ mc = 1.49/mm and ε c = 0.198%.     Figure 4 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density evolution of the brittle matrix versus applied stress of SiC/SiC minicomposite without heat treatment. The predicted results of tensile curves and fragmentation density evolution of the brittle matrix curves agree well with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including: (1) Domain I, the linear region. The domain starts from initial loading to σ mc = 35 MPa with λ mc = 0.04/mm and ε c = 0.013%.
(3) Domain III, the secondary linear and final fracture domain. The domain starts from σ sat = 340 MPa to σ uts = 399 MPa.  Figure 4 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density evolution of the brittle matrix versus applied stress of SiC/SiC minicomposite without heat treatment. The predicted results of tensile curves and fragmentation density evolution of the brittle matrix curves agree well with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including: (1) Domain I, the linear region. The domain starts from initial loading to σmc = 35 MPa with λmc = 0.04/mm and εc = 0.013%.     (1) Domain I, the linear region. The domain starts from initial loading to σ mc = 50 MPa with λ mc = 0.02/mm and ε c = 0.012%.
(2) Domain II, the nonlinear region due to matrix fragmentation. The domain starts from σ mc = 50 MPa to σ sat = 210 MPa with λ sat = 0.28/mm and ε c = 0.13%.   Figure 6 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density evolution of brittle matrix versus applied stress of SiC/SiC minicomposite after heat treatment at 1500 °C. The predicted results of tensile curves and fragmentation density of brittle matrix versus applied stress curves both agree well with experimental data. Three main  Figure 6 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density evolution of brittle matrix versus applied stress of SiC/SiC minicomposite after heat treatment at 1500 • C. The predicted results of tensile curves and fragmentation density of brittle matrix versus applied stress curves both agree well with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including: (1) Domain I, the linear region. The domain starts from initial loading to σ mc = 50 MPa with λ mc = 0.005/mm and ε c = 0.017%.
(3) Domain III, the secondary linear and final fracture domain. The domain starts from σ sat = 220 MPa to σ uts = 246 MPa.  Figure 6 shows experimental data and theoretical predicted results of tensile nonlinear curves and fragmentation density evolution of brittle matrix versus applied stress of SiC/SiC minicomposite after heat treatment at 1500 °C. The predicted results of tensile curves and fragmentation density of brittle matrix versus applied stress curves both agree well with experimental data. Three main damage domains are obtained from the nonlinear tensile curve, including: (1) Domain I, the linear region. The domain starts from initial loading to σmc = 50 MPa with λmc = 0.005/mm and εc = 0.017%.

Discussion
In this section, the constitutive properties and damage-related parameters on tensile nonlinear stress-strain curves of SiC/SiC mini-CMC are discussed.

Discussion
In this section, the constitutive properties and damage-related parameters on tensile nonlinear stress-strain curves of SiC/SiC mini-CMC are discussed. Figure 8 shows the tensile nonlinear curves and the evolution of interface debonding fraction for different fiber volumes. For low fiber volume V f = 25%, the debonding fraction increases from η (σ d = 166 MPa) = 0 to η (σ uts = 521 MPa) = 0.33, and the failure strain of the composite is ε f (σ uts = 521 MPa) = 0.6%. For high fiber volume V f = 30%, the debonding fraction increases from η (σ d = 204 MPa) = 0 to η (σ uts = 626 MPa) = 0.32, and the failure strain of the composite is ε f = 0.64%. At high fiber volume, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the composite tensile strength and failure strain increase.  Figure 9 shows the tensile nonlinear curves and the evolution of the interface debonding fraction for different interface shear stresses. For low interface shear stress τi = 20 MPa, the debonding fraction increases from η (σd = 166 MPa) = 0 to η (σuts = 521 MPa) = 0.5, and the failure strain of the composite is εf = 0.73%. For high interface shear stress τi = 40 MPa, the debonding fraction increases from η (σd = 168 MPa) = 0 to η (σuts = 521 MPa) = 0.25; and the failure strain of the composite is εf = 0.25%. At high interface shear stress, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the composite failure strain decreases.    Figure 9 shows the tensile nonlinear curves and the evolution of the interface debonding fraction for different interface shear stresses. For low interface shear stress τ i = 20 MPa, the debonding fraction increases from η (σ d = 166 MPa) = 0 to η (σ uts = 521 MPa) = 0.5, and the failure strain of the composite is ε f = 0.73%. For high interface shear stress τ i = 40 MPa, the debonding fraction increases from η (σ d = 168 MPa) = 0 to η (σ uts = 521 MPa) = 0.25; and the failure strain of the composite is ε f = 0.25%. At high interface shear stress, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the composite failure strain decreases.  Figure 9 shows the tensile nonlinear curves and the evolution of the interface debonding fraction for different interface shear stresses. For low interface shear stress τi = 20 MPa, the debonding fraction increases from η (σd = 166 MPa) = 0 to η (σuts = 521 MPa) = 0.5, and the failure strain of the composite is εf = 0.73%. For high interface shear stress τi = 40 MPa, the debonding fraction increases from η (σd = 168 MPa) = 0 to η (σuts = 521 MPa) = 0.25; and the failure strain of the composite is εf = 0.25%. At high interface shear stress, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the composite failure strain decreases.    Figure 10 shows the tensile nonlinear curves and the evolution of the interface debonding fraction for different interface debonding energy. For low interface debonding energy ζ d = 4 J/m 2 , the debonding fraction increases from η (σ d = 192 MPa) = 0 to η (σ uts = 521 MPa) = 0.31, and the failure strain of the composite is ε f = 0.59%. For high interface debonding energy ζ d = 6 J/m 2 , the debonding fraction increases from η (σ d = 234 MPa) = 0 to η (σ uts = 521 MPa) = 0.57, and the failure strain of the composite is ε f = 0.57%. At high interface debonding energy, the interface debonding stress increases, the debonding fraction at the same applied stress decreases, and the composite strain during nonlinear Domain II decreases, and the failure strain of the composite decreases.

Conclusions
In this paper, the tensile damage and fracture behavior of two different mini-CMCs are investigated. Multiple microdamage mechanisms of matrix fragmentation, fibers failure and pullout are considered in the analysis of tensile nonlinear curves. Experimental tensile nonlinear curves and internal damage evolution of SiC/SiC and C/SiC mini-CMCs are predicted. The effects of material properties and damage-related parameters on tensile damage and fracture at different domains are analyzed.
(1) Predicted tensile nonlinear curves and matrix fragmentation density evolution curves agree with experimental data, and the matrix fragmentation approaches saturation before tensile fracture, and the interface partial debonding remains till tensile fracture. (2) Microdamage parameters of first matrix fragmentation stress, saturation matrix fragmentation stress and density, composite tensile strength and failure strain are obtained from tensile stressstrain curves and can be used to characterize tensile nonlinear behavior of mini-CMCs. (3) At higher fiber volume, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the composite tensile strength and failure strain increase. (4) At higher interface shear stress and interface debonding energy, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the failure strain of the composite decreases. Acknowledgments: The authors also wish to thank four anonymous reviewers and editors for their helpful comments on an earlier version of the paper.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Conclusions
In this paper, the tensile damage and fracture behavior of two different mini-CMCs are investigated. Multiple microdamage mechanisms of matrix fragmentation, fibers failure and pullout are considered in the analysis of tensile nonlinear curves. Experimental tensile nonlinear curves and internal damage evolution of SiC/SiC and C/SiC mini-CMCs are predicted. The effects of material properties and damage-related parameters on tensile damage and fracture at different domains are analyzed.
(1) Predicted tensile nonlinear curves and matrix fragmentation density evolution curves agree with experimental data, and the matrix fragmentation approaches saturation before tensile fracture, and the interface partial debonding remains till tensile fracture. (2) Microdamage parameters of first matrix fragmentation stress, saturation matrix fragmentation stress and density, composite tensile strength and failure strain are obtained from tensile stress-strain curves and can be used to characterize tensile nonlinear behavior of mini-CMCs. (3) At higher fiber volume, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the composite tensile strength and failure strain increase. (4) At higher interface shear stress and interface debonding energy, the debonding fraction at the same applied stress decreases, and the composite strain at nonlinear Domain II decreases, and the failure strain of the composite decreases.