Modeling brittle and tough stress–strain behavior in unidirectional ceramic matrix composites
Introduction
The detailed deformation behavior, and the factors which determine that response, are critical to the development of optimized composite systems and to the design of structural components using composites. Ceramic matrix composites (CMCs, ceramic matrices reinforced with ceramic fibers), pose a particular problem because the constituent materials are both brittle. The matrix and fiber failure are controlled by pre-existing or induced flaws that are difficult to control in any processing scheme. The use of fiber/matrix interface coatings to mechanically decouple the fibers and the matrix has been successful in eliminating the most egregious problem of perfectly brittle, monolithic-like behavior in the CMC. But, the overall deformation of the composite still depends on the flaws in the matrix and the fibers. Predicting the stress–strain and failure behavior of unidirectional CMCs as a function of the matrix and fiber flaws is the subject of this paper.
Two fundamental regimes of damage evolution exist in CMCs. First, at lower stresses, flaws in the matrix grow to form matrix cracks extending across the material. Because of debonding at the engineered interface between fibers and matrix, the matrix crack does not penetrate the fibers, which then remain to bridge the matrix crack and restrain the crack opening1, 2, 3. Being inherently a crack propagation problem, matrix cracking above the minimum cracking stress is controlled by flaws and by the micromechanics of matrix crack growth. Price and Sinyth[4] and subsequently He et al.[5] demonstrated that the stress–strain curve in the material could be predicted given a knowledge of the average matrix crack spacing vs stress. Curtin[6], Zok and Spearing[7] and Yang and Knowles[8], considered the statistical evolution of cracking and its dependence on micromechanical parameters but did not predict stress–strain deformation. Recently, Ahn and Curtin presented a fully statistical treatment of the matrix crack evolution and associated stress–strain behavior. All of the works to date have assumed that the fibers do not fail during matrix cracking.
At higher stress, the second regime of fiber damage and ultimate failure occurs. If matrix cracking can reach the fully-saturated state prior to composite failure, the subsequent deformation and failure are controlled entirely by the fiber flaw population, as shown by Curtin[9], Curtin and Zhou[10] and Hui et al.[11]. We refer to this as “tough” behavior. The theory for ultimate strength in this regime has been well-verified by many workers and, in this regime, both the composite strength and failure strain are controlled by the fiber properties (strength and modulus). Tough behavior is highly desirable since the fiber properties are easier to control and/or measure than are the matrix properties.
Composite failure is also possible in the regime of matrix cracking, however, if the fibers are sufficiently weak or the matrix comparatively strong. We subsequently refer to this as “brittle” behavior. The strength of the composite failing around only a single matrix crack was studied first by Thouless and Evans12, 13. Curtin[14] and Phoenix and Raj[15] demonstrated that the failure stress is controlled by the same characteristic fiber strength relevant in the saturated-matrix crack regime. However, the failure strain is controlled by the matrix flaw population and is so generally much smaller than the failure strain of the fiber bundle. There is thus a transition in behavior for the composite failure strain which depends on both the matrix and fiber flaw populations. The two regimes of behavior, and the transition between them, are shown schematically in Fig. 1; the transition occurs when the characteristic matrix cracking strength σR is comparable to the fiber bundle strength. It is important to understand the overall dependence of composite deformation and of the transition from tough to brittle behavior on both matrix and fiber statistical strengths. With a predictive capability for the total deformation behavior, one can than investigate the required matrix properties and/or the required fiber strengths to obtain tough, high failure-strain behavior.
In this paper, we develop a new approach to calculating composite tensile strength as a function of the matrix cracking, spanning the range from a single crack to fully-saturated cracking. In combination with the simplified matrix cracking model of Ahn and Curtin[16], generalized to include a fiber/matrix interface fracture energy, we develop a simple but accurate model for predicting the entire stress–strain and failure behavior of unidirectional composites as a function of the underlying constituent properties. To demonstrate the suitability of the analysis, it is applied to predict properties of SiC/SiC unidirectional minicomposites, using recent data of Lissart and Lamon[17], and very good results are obtained for most materials using measured fiber strength distributions. Furthermore, it is demonstrated that, in spite of some rather different macromechanical responses and flaw populations, nominally similar materials possess the same interfacial sliding resistance τ. The values for τ as derived from the matrix crack evolution are in excellent agreement with the values derived by two other methods, hysteresis loops and fiber pullout analysis. This agreement is non-trivial and demonstrates the accuracy of our analysis. We then show that allowing for some variations in the fiber strength can lead to good predictions for all materials tested, which also highlights the interplay between matrix and fiber flaw populations in determining failure.
The remainder of this paper is organized as follows. In Section 2, we present a simplified model for matrix crack evolution, introduce the controlling matrix flaw parameters and determine the stress–strain behavior due to the matrix cracking. In Section 3, we present the new analyses of fiber damage evolution for arbitrary matrix crack densities and show that in the limits of both saturated and single matrix crack cases previous results are essentially regained. We then determine the additional strain due to fiber damage and hence the full stress–strain evolution law for the composite, including failure. In Section 4, we show that the analysis can predict quite well the various deformations measured in six different samples presented by Lissart and Lamon and derive values for τ by various means. In Section 5we discuss other implications of the model and present examples of trends with changing matrix and fiber flaw parameters.
Section snippets
Matrix cracking
Ahn and Curtin[16] showed that the statistics of matrix crack evolution depends on the micromechanics of bridged matrix flaws, as described by Marshall et al.[18] and Chiang et al.[19] and on the underlying initial flaw population, but that a three-parameter Weibull form for the number of flaws is quite generally appropriate. Specifically, the number of flaws N(σ, L, A) in a length L (and some implicit reference area A) that are able to propagate at applied stress σ can be expressed as
Fiber statistics and failure probability
The ceramic reinforcing fibers are brittle materials and must also be described statistically by a flaw distribution. We use the two-parameter Weibull form for the number of flaws which can fail in length L at stress σ,where σ0 is the characteristic fiber strength at tested gauge length L0, In the composite, where fiber breaks are accompanied by fiber sliding against the matrix with the sliding stress τ, there exists a critical gauge length δc, and associated critical strength σc
Materials and constitutive data
Lissart and Lamon have recently presented considerable data and analysis on the uniaxial mechanical behavior of SiC/SiC minicomposites[17]. These materials consist of single 500-fiber tows of Nicalon fibers, with a thin Carbon coating on the fibers, which have been subsequently infiltrated with a SiC matrix using the chemical vapor infiltration method. Lissart and Lamon investigated two different materials, labeled B and C, and presented force–strain data on three samples of each type and
Discussion
We begin our discussion by pointing out some implications of the model and providing some illustrative examples. First, the model shows that tough behavior is best obtained in materials with relatively weak matrices. Fig. 1 presents a set of baseline calculations for fixed elastic properties (Ef=Em=Ec=200 GPa), fixed fiber strength and fraction (σc=2000 MPa, m=5, f=0.3), fixed matrix Weibull modulus ρ=3, σth=σ*=σdeb=0 and varying matrix strength σR ranging from 100 to 900 MPa. The multiple matrix
Acknowledgements
W. A. C and B. K. A. thank the Air Force Office of Scientific Research for support of this work through Grant F49620-95-1-0158. W. A. C. also thanks the Center for Collaborative Research at the University of Tokyo for support of this work through a Visiting Scientist appointment.
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