A computationally efficient approach for predicting toughness enhancement in ceramic composites with tailored inclusion arrangements

Abstract

Advanced manufacturing techniques such as extrusion-based methods have enabled the fabrication of ceramic composites with ordered inclusion phases (i.e. the size and position of the inclusion can be precisely controlled) to improve their overall strength and toughness. Conventional theories, simulation approaches, and experimental methods for analyzing fracture in composites with randomly dispersed inclusion phases (resulting in homogeneous, isotropic effective properties) become inadequate at understanding and designing composites with ordered inclusions for enhancing effective properties such as toughness. In addition, existing methods for analyzing fracture in composites can be computationally expensive and pose challenges in accurately capturing experimentally observed fracture growth. For example, extended finite element and phase-field methods are computationally expensive in evaluating the large design space of possible inclusion arrangements enabled by the new manufacturing techniques. In this work, a closed-form analytical model for the mixed-mode stress intensity factor in a composite with selected inclusion arrangements is presented, which expedites the analysis for various composite designs. Moreover, the fracture initiation calculation is adapted to approximate crack propagation with computational efficiency. The accuracy of this model for predicting fracture initiation is validated by linear elastic fracture mechanics analysis using the finite element method. The prediction of fracture propagation is validated using a phase-field model, as well as a 4-point bending experiment. Finally, the model is applied to analyze three different composite inclusion arrangements to study the effect of various material combinations and geometries on the overall toughness of the composite; a complete sampling of (and optimization) over the entire design space, however, is beyond the scope of this work. The relative increase in crack length (compared to a homogeneous material) is used as a metric to compare the relative toughness of three different composite designs. Within these designs, using the fast-running approximate method, the effect of the ratio of inclusion radius to inclusion spacing, and the elastic mismatch on the resulting crack length are compared to determine the composite arrangements that result in the greatest toughness enhancement for selected material properties. In particular, a multi-phase cubic array resulted in the greatest toughness enhancement of the designs considered.

Appendices

Appendix A: Approximate stress intensity factor of a kinked crack in a 4-point bending test

The parameters c and d, used in (2) are functions of the kink angle, $\omega$, with an infinitesimal kink length. This approximation is asymptotically accurate for small kink angles (in a homogeneous material) and is given by:

$$\begin{aligned} \begin{aligned} c=&\,\frac{1}{2}\left( \exp \left( -j\frac{\omega }{2}\right) +\exp \left( -j\frac{3\omega }{2}\right) \right) , \\ d=&\,\frac{1}{4}\left( \exp \left( -j\frac{\omega }{2}\right) -\exp \left( j\frac{3\omega }{2}\right) \right) , \end{aligned} \end{aligned}$$

(10)

where $j=\sqrt{-1}$ (Cotterell 1965; He and Hutchinson 1989; Williams 1956).

Appendix B: Influence of a nearby inclusion on a straight crack

The perturbation in stress intensity factor in mode I and II fracture due to the difference in material properties of the inclusions and the matrix is dictated by the following coefficients appearing in (3):

$$\begin{aligned} \begin{aligned} C_1 =&\,\frac{(1-\alpha )(1-2\nu )}{1+\alpha -2\nu }, \\ C_2 =&\,\frac{3(1-\alpha )}{2(1+3\alpha -4\nu \alpha )}, \\ C_3 =&\,\frac{(1-\alpha )(11+19\alpha +32\nu ^2\alpha -22\nu -40\nu \alpha )}{16(1+\alpha -2\nu )(1+3\alpha -4\nu \alpha )}, \\ C_4 =&\,\frac{-(1-\alpha )(1-2\nu )}{4(1+\alpha -2\nu )}, \\ C_5 =&\,\frac{9(1-\alpha )}{16(1+3\alpha -4\nu \alpha )}, \end{aligned} \end{aligned}$$

(11)

where $\nu$ is Poisson’s ratio (assumed to be same for the inclusions and the matrix) and

$$\begin{aligned} \alpha =\frac{E_\text {inc}}{E_\text {mat}} \end{aligned}$$

(12)

is the ratio of the Young modulus of the inclusion, $E_\text {inc}$, to the Young modulus of the matrix, $E_\text {mat}$. Note that in most cases, the Poisson ratio of the composite materials will not be identical. Hence, the results predicted from this model will be most accurate when the Poisson ratios of the constituents are nearly the same value. Despite this assumption, the model is still able to accurately capture experimental results.