Experimental assessment of toughness in ceramic matrix composites using the J-integral with digital image correlation part I: methodology and validation

Abstract

The complex nature of damage in ceramic matrix composites (CMCs) renders conventional methods of measuring quantitative fracture properties impractical. This paper is the first of two-part series that assesses the feasibility of using the J-integral with full-field deformation data from digital image correlation (DIC) to characterize toughness in continuous fiber SiC/SiC CMC laminates. It provides a resource for best practices when incorporating experimentally measured, full-field deformation data into J-integral evaluations of toughness. The techniques discussed are important for researchers attempting to measure fracture properties in advanced materials with damage mechanisms that have yet to be well characterized. Two methods for evaluating potential energy release rates J are presented: (1) numerical integration over a line contour and (2) Gaussian integration over an area contour. Accuracy and path independency for both methods were verified using analytically derived deformation fields for a center-cracked, infinite plate of isotropic material under equi-biaxial tension. Inherent noise in the deformation data and necessary contour truncation at the crack surfaces reduced accuracy and introduced path dependency. However, this was mitigated by careful noise filtering of the deformation data prior to the evaluation of J. Applying the line and area integrals to DIC data from tapered, double-cantilever beam, acrylic compact tension specimens resulted in the findings that (1) both integrals measured fracture toughness within a range of published values; (2) both integrals captured the constant stress intensity factor behavior that is characteristic of the tapered beam geometry; and (3) the area integral measurements were consistently larger than line integral measurements. The area integral is more accurate than the line integral as it samples more data points, reducing its sensitivity to experimental noise. Although variability in experimental data can be minimized, it cannot be eliminated. Measurement error is inevitable; thus, the line and area integrals must be used with caution when characterizing quantitative properties including fracture toughness.

Appendix

A comparison of the two contour truncation techniques is shown in Figs. 11 and 12. In Fig. 11a, a contour encompassing a crack tip in a PMMA CT specimen is truncated by removing nodes of suspect data. Specifically, truncation points are defined from confidence values (the DIC software measures the 1-sigma confidence interval—in image pixels—of each subset match; e.g., the subset match is accurate to within a range of pixels with 95% confidence) plotted along the left leg of the contour; all nodes that fall within the boundaries of these two points are removed from the original contour. The equivalent points are then shown for longitudinal strains (plotted along the left leg) in Fig. 11b. The truncated contour is overlaid on a plot of the longitudinal strain field in Fig. 11c. As shown in Fig. 11b, c, even though nodes with poor confidence values were removed, the terminal ends of the truncated contour still reside within a region of unrealistically large strain. The strains at the upper and lower truncation points, 8000 and 15000 µε, respectively, are well beyond the regime of plastic deformation. The second approach of truncation using longitudinal strains plotted along the left leg of the original contour is shown in Fig. 12. In Fig. 12a, nodes with strains that reside within the Gaussian-shaped region of the contour are removed. The corresponding contour plot is overlaid on a longitudinal strain field in Fig. 12b. The truncation points each terminate at regions in which the longitudinal strains are less than 1000 µε, which are more reasonable values of strain near a crack face.

Fig. 11

Illustration of the confidence interval approach to truncation. In a, truncation points are defined at the boundaries of the Gaussian-shaped region of the confidence interval plotted along the left hand leg of the contour. In b, the equivalent points on a plot of the longitudinal strains along the left hand leg. The truncated contour is overlaid on a plot of longitudinal strains in (c). Although data nodes with poor correlations are removed from the contour, nodes with unrealistic strains remain

Fig. 12

Illustration of the strain approach to truncation. In a, truncation points are defined at the boundaries of the Gaussian-shaped region of the longitudinal strains plotted along the left hand leg of the contour. The truncated contour is overlaid on a plot of longitudinal strains in (b). Longitudinal strains at the termination points are below 1000 µε, which are reasonable values of strain near the crack face of a CT specimen

The effect of truncation can be significant, particularly for materials with low fracture toughness. When evaluated with the J-integral, the contour in Fig. 11 results in measured fracture toughness of 0.42 MPa $\sqrt m,$ whereas that of Fig. 12 is 0.56 MPa $\sqrt m$ (a comprehensive discussion of experimental evaluations of fracture toughness in PMMA is provided in “Case 2: application of line and area integrals to experimental DIC data ” section). Truncation by confidence interval incorporates strains that are not real into the fracture toughness calculation and is not used in further analyses of fracture toughness. Truncating by longitudinal strain resulted in predictions closer to the range of published values. Note that truncation by longitudinal strain, however, has its own shortcomings. Real mechanical behavior exists within the Gaussian-shaped region. For materials such as CMCs that do not have well-defined mechanical properties, establishing a cut-off (such as failure strain) to aid in the selection of truncation points is not possible. Consequently, truncating by removing excessive strain along the contour introduces some additional error in the measurement. However, truncating the contour using longitudinal strain appears to be a reasonable approach that results in fracture toughness within the range of published values for PMMA, and was adopted in this work.

Note that measuring uncertainty in the DIC measurements on a node-by-node basis was not possible. In the calibration phase of testing, displacement and strain error can be found by measuring the root-mean-square errors resulting from rigid body translations. However, this error measurement is an average of the error across the entire FOV, whereas the actual error at each data node will vary due to noise. Unlike rigid body translations, there is no method to compute error for each component of displacement/strain at each node.