Fiber Reinforced Ceramic Matrix Composites: A Probabilistic Micromechanics-Based Approach

Abstract

Being damage tolerant, the CMCs exhibit nonlinear deformations as a result of cracks that form in the matrix, in the interfaces, and in the fibers. The sequence of cracking modes displays several features that depend on the arrangement of fibers, the microstructure, and the respective properties of constituents. Being ceramic materials, the constituents are highly sensitive to inherent microstructural flaws generated during processing. The flaw populations govern matrix cracking and fiber failures, so that strengths of constituents exhibit statistical distributions. A bottom-up multiscale approach based on micromechanics must account for the contribution of inherent fracture-inducing flaws, variability of constituent strengths, and associated size effects.

The chapter deals with modeling of the stochastic processes of multiple fracture of the matrix and the fibers that govern damage and failure on fiber-reinforced ceramic matrix composites. The models are based on probabilistic approaches to brittle fracture, including the Weibull phenomenological model and the physics-based elemental strength model that considers the flaws as physical entities. The probabilistic models that are discussed permit determination of stresses at crack initiation from microstructural flaws and resulting crack pattern. Applications to the prediction of tensile behavior of unidirectional or woven composites are then discussed.

Appendix

Stress on fiber in the presence of fragmented matrix:

• In the vicinity of interfacial matrix cracks: u + l_o ≤ x < l_d

  $$ {\sigma}_m(x)={\sigma}_m\frac{x-u-{l}_o}{l_d-{l}_o} $$

  (53)
  $$ {\sigma}_f(x)={\sigma}_f\left( 1+a-a\frac{x-u-{l}_o}{l_d-{l}_o}\right) $$

  (54)

• In the rest of fragment: u + l_d < x < 2 l_i − (u + l_d)

  $$ {\sigma}_m=\frac{\sigma }{V_m}\frac{a}{1+a}+{\sigma}_m^{th} $$

  (55)
  $$ {\sigma}_f=\frac{\sigma }{V_f}\frac{1}{1+a}+{\sigma}_f^{th} $$

  (56)

where σ_m and σ_f are, respectively, the stresses operating on the fiber and the matrix; u, l_d, and l_o are defined in Fig. 8; 2l_i is length of fragment i; σ is the remote stress applied to the specimen; V_f and V_m are the volume fractions of fiber and matrix, respectively; $a=\frac{E_m{V}_m}{E_f{V}_f}$ is the load sharing parameter; E_f and E_m are fiber and matrix Young’s moduli; and ${\sigma}_m^{th}$ and ${\sigma}_f^{th}$ are the residual stresses, respectively, in the matrix and in the fiber.