Experimental Determination of the Length-Scale Parameter for the Phase-Field Modeling of Macroscale Fracture in Cr–Al₂O₃ Composites Fabricated by Powder Metallurgy

Abstract

A novel approach is proposed to determine a physically meaningful length-scale parameter for the phase-field modeling of macroscale fracture in metal–ceramic composites on an example of chromium–alumina composite fabricated by powder metallurgy. The approach is based on the fractography analysis by the scanning electron microscopy (SEM) with the aim to measure the process zone size and use that value as the length-scale parameter in the phase-field modeling. Mode I and mixed-mode I/II fracture tests are conducted on Cr–Al₂O₃ composites at different reinforcement volume fractions and particle sizes using single-edge notched beams under four-point bending. The fracture surfaces are analyzed in detail by SEM to determine the size of the process zone where the microscale nonlinear fracture events occur. The model adequately approximates the experimentally measured fracture toughness and the fracture loads. It is shown that the model prediction of the crack initiation direction under the mixed-mode loading is in agreement with the experiments and the generalized maximum tangential stress criterion. These outcomes justify using the process zone size as the scale parameter in the phase-field modeling of macroscale fracture in chromium–alumina and similar metal–ceramic composites.

Appendix: Spectral Decomposition

The strain tensor is decomposed into positive (tensile) and negative (compressive) parts:

$$ \varepsilon = {\varepsilon_+ } + {\varepsilon_- }. $$

(A1)

The positive and negative contributions are defined through the spectral decomposition of the strain tensor

$${\varepsilon_\pm } = {\sum\nolimits_{i = 1}^\delta {\left\langle {\varepsilon^i} \right\rangle }_\pm }{n^i} \otimes {n^i} $$

(A2)

with ${\left\langle X \right\rangle_\pm } = \left( {X \pm \left| X \right|} \right)/2$. The ${\varepsilon^i}$ and nⁱ for i = 1,…,$\delta$ are the principal strains and their directions ($\delta$ = 2, or 3 for two- or three-dimensional problems).

The positive and negative contributions into the energy density in Eq. [4] are defined as:

$$ \psi_0^\pm = \frac{\lambda }{2}{\left\langle {{\text{tr}}\left[ \varepsilon \right]} \right\rangle_\pm }^2 + \mu {\text{tr}}\left[ {{\varepsilon_\pm }^2} \right],$$

(A3)

where λ and µ are the Lame coefficients and ${\text{tr}}\left[ X \right]$ is the trace of X.