Experimental assessment of toughness in ceramic matrix composites using the J-integral with digital image correlation part II: application to ceramic matrix composites

Abstract

This paper is the second of a two-part series assessing the feasibility of using the J-integral with full-field deformation data from digital image correlation (DIC) to characterize toughness in continuous fiber, SiC_f/SiC_c ceramic matrix composites (CMCs). In part I, line and area integral adaptations of the J-integral were validated with analytical and experimental deformation fields. It was found that variability in experimental data and contour truncation at the crack wake introduced error and path-dependency, but this could be mitigated through data filtering. In this paper, the line and area integral methods were evaluated for cross-ply and longitudinal CMC tensile specimens. It is demonstrated that these approaches cannot be used for quantitative characterization, but can be successfully used to examine qualitative trends. For both architectures, area integral measurements of the potential energy release rate, J, were larger (and arguably more accurate) than line integral measurements. Resistance curves were characteristic of tough materials, but the stress intensity magnitudes were larger in the longitudinal laminates than the cross-ply (for equivalent crack extension). Experimentally derived bridging laws showed that fiber bridging tractions were largest in the cross-ply laminates, which was attributed to a lower volume fraction of longitudinal fibers. Quantitative, single-value measurements of fracture toughness could not be established since (1) noise in the DIC data masked strain signals indicative of first matrix cracking, and (2) widespread matrix cracking rendered the J-integral invalid (regardless of size and position, the boundaries of the line and area contours inevitably intercepted microcracks). Rather, gross estimates of toughness and stress intensity factors were made under the assumption that the limitations in the spatial resolution of the DIC data essentially transforms regions with fine microcracking along the contour into continuous regions of elastic deformation. When evaluated with the J-integral, the smoothed deformation fields manifest as resistance curves indicative of a tough material. Thus, the J-integral was valuable for qualitative assessments of toughening mechanisms in the CMCs.

Appendices

Appendix 1

In addition to the J-integral measurements, stress intensity factors were also calculated using the following closed form analytical expressions for single edge-notch tensile specimens per [17]:

$$ K = \sigma \sqrt {\pi a} \times f\left( \frac{a}{W} \right) $$

(5)

$$ f\left( \frac{a}{W} \right) = \frac{{\sqrt {\frac{2W}{\pi a}\tan \frac{\pi a}{2W}} }}{{\cos \frac{\pi a}{2W}}}\left[ {0.752 + 2.02\left( \frac{a}{W} \right) + 0.37\left( {1 - \sin \frac{\pi a}{2W}} \right)^{3} } \right] $$

(6)

where a is the total crack length, W is the width of the tensile coupon, B is the thickness of the coupon, and P is the applied axial load. To account for composite orthotropy, Eq. (5) is multiplied by the following correction factor [18]:

$$ Y\left( \rho \right) = 1 + 0.01\left( {\rho - 1} \right) - 0.016\left( {\rho - 1} \right)^{2} - 0.002\left( {\rho - 1} \right)^{3} $$

(7)

where c is the notch half-width of the notches minor axis, b is the notch length; and ρ and λ are material parameters defined as

$$ \lambda = \frac{{E_{y} }}{{E_{x} }} $$

(8)

$$ \rho = \frac{{\left( {E_{x} E_{y} } \right)^{1/2} }}{{2G_{xy} }} - \left( {\nu_{xy} \nu_{yx} } \right)^{1/2} $$

(9)

E _y , E _x , G _xy , $\nu_{xy}$, and $\nu_{yx}$ are the longitudinal and transverse tensile moduli, shear moduli, and Poisson’s ratios of the composite laminate, respectively. The accuracy of Eq. (7) is ≤2 % and is valid for the range of $\rho$ within 0 ≤ $\rho$ ≤ 4.

Laminate properties were derived from the reduced transformed stiffness matrix for a symmetric cross-ply laminate [11]. The stiffness $\left( \left[ {\overline{Q}^{\theta } } \right]\right)$ in each lamina oriented an angle (θ) counterclockwise from the transverse direction in the material coordinate system is given by

$$ \left[ {Q^{\theta } } \right] = \left[ {\begin{array}{*{20}c} {\frac{{E_{1} }}{{1 - \nu_{12} \nu_{21} }}} & {\frac{{\nu_{12} E_{2} }}{{1 - \nu_{12} \nu_{21} }}} & 0 \\ {\frac{{\nu_{12} E_{2} }}{{1 - \nu_{12} \nu_{21} }}} & {\frac{{E_{2} }}{{1 - \nu_{12} \nu_{21} }}} & 0 \\ 0 & 0 & {G_{12} } \\ \end{array} } \right] $$

(10)

$$ \left[ {T_{1}^{\theta } } \right] = \left[ {\begin{array}{*{20}c} {{ \cos }^{2} } & {{ \sin }^{2} } & {2 \times { \cos } \times { \sin }} \\ {{ \sin }^{2} } & {{ \cos }^{2} } & { - 2 \times { \cos } \times { \sin }} \\ { - { \cos } \times { \sin }} & {{ \cos } \times { \sin }} & {m^{2} - n^{2} } \\ \end{array} } \right] $$

(11)

$$ \left[ {T_{2}^{\theta } } \right] = \left[ {\begin{array}{*{20}c} {{ \cos }^{2} } & {{ \sin }^{2} } & {{ \cos } \times { \sin }} \\ {{ \sin }^{2} } & {{ \cos }^{2} } & { - { \cos } \times { \sin }} \\ { - 2 \times { \cos } \times { \sin }} & {2 \times { \cos } \times { \sin }} & {m^{2} - n^{2} } \\ \end{array} } \right] $$

(12)

$$ \left[ {\overline{Q}^{\theta } } \right] = \left[ {T_{1} } \right]^{ - 1} \left[ Q \right]\left[ {T_{2} } \right]. $$

(13)

The lamina engineering properties were derived via the concentric cylinder model [12] using known properties of the composite’s monolithic constituents.

The stiffness of the symmetric laminate was computed as

$$ \left[ {\overline{Q} } \right] = \frac{1}{2}\left( {\left[ {\overline{Q}^{0} } \right] + \left[ {\overline{Q}^{90} } \right]} \right). $$

(14)

The laminate compliance was then

$$ \left[ {q^{*} } \right] = \left[ {\overline{Q} } \right]^{ - 1} $$

(15)

such that the laminate engineering constants in (8) and (9) become

$$ \begin{array}{*{20}c} {E_{y} = \frac{1}{{q_{11}^{*} }}} \\ {E_{x} = \frac{1}{{q_{22}^{*} }}} \\ {G_{xy} = \frac{1}{{q_{66}^{*} }}} \\ \end{array} \quad \begin{array}{*{20}c} {\nu_{xy} = - \frac{{q_{12}^{*} }}{{q_{22}^{*} }}} \\ {\nu_{yx} = - \frac{{q_{12}^{*} }}{{q_{11}^{*} }}} \\ \end{array}, $$

(16)

J was evaluated along the contour length for each image and converted to the plane-stress stress intensity factor, K, using the expressions [19]:

$$ K^{2} = \frac{{J\sqrt {E_{x} E_{y} } }}{{\sqrt {{{\left( {\alpha + \beta } \right)} \mathord{\left/ {\vphantom {{\left( {\alpha + \beta } \right)} 2}} \right. \kern-0pt} 2}} }} $$

(17)

$$ \alpha = \sqrt {\frac{{E_{x} }}{{E_{y} }}} ,\quad \beta = \frac{{E_{x} }}{{2G_{xy} }} - \nu_{xy} $$

(18)

where x and y correspond to principle directions (1 and 2), the elastic constants are laminate constants, and K is mode-I.

Appendix 2

Fracture characterization: unidirectional CMC

Using the line and area integrals, resistance curves were computed for two unidirectional SiC/SiC CMCs loaded in uniaxial tension under ambient environment. The edge-notched tensile coupons each consisted of a double layer of longitudinally oriented fibers. DIC data were collected using a Grasshopper 5 megapixel CCD camera fitted with a 75-mm Tamron CCTV lens over a 6.5-mm (2048 pixel) FOV. For both datasets, a DIC step size of 2 pixels was used. Smoothing was conducted using 35-node strain filters (line integral only), and 35-node displacement filters. A 45-pixel subset was used for DIC on the first dataset and a 41 pixel dataset for the second. During both tests, two cracks initiated and propagated from the notch. Consequently, stress intensities were plotted against crack opening displacement rather than crack extension and data were not transformed into a crack coordinate system. Resistance curves for the first dataset are shown in Fig. 8; inset in the figure is longitudinal strain fields corresponding to various lengths of crack opening displacement. The first cracks that can be detected by DIC occur at a crack opening displacement of 0.4 µm. At this COD, the stress intensity factors range between 0.9K (line integral) and 1.1K (area integral). The plane stress fracture toughness of the composite is estimated to be at or below this range of stress intensities.

Fig. 8

Line and area integral derived resistance curves for the first of two uniaxial CMC laminate tension tests. Fibers are oriented in the loading direction. Two cracks initiated simultaneously from the notch tip, with final fracture ultimately dominated by the upper crack. Plane stress fracture toughness is attained at or below the range 0.9K–1.1K. Note All stress intensity factors for CMC tests are normalized to a reference value, K. Red squares line; blue circles area (Color figure online)

Up to about 2 µm of crack opening displacement, the area and line integrals measured similar stress intensities. The area integral measurements were higher than line integral for all loading thereafter. This was also observed in the cross-ply tests, and is attributed to the difference in filtering for the two integration methods. Although both displacements and strains are filtered in the line integral, a majority of the J computation incorporates filtered strain data. Filtering strains is preferable to filtering displacements, but this is not possible for the area integral; strains are computed it each Gauss point individually, then J immediately thereafter.

Resistance curves for both tensile tests are plotted in Fig. 9. Inset in 9 is longitudinal strain fields for both tests which correspond to a point at which cracks have initiated from the notch tips. Stress intensities for the first test were generally less than those of the first tests. Both tests exhibited toughening behavior; the rate of increase in stress intensities decreased as crack opening displacement increased. As expected, area integral measurements were larger than line integral measurements. In the second test, crack initiation was detectable in DIC data at a crack opening displacement of 0.55 µm. This corresponds to a plane stress fracture toughness value at or below the range 1.3K (line integral) to 1.5K (area integral). Combining the results from both tests, estimates of plane stress fracture toughness as computed by the J-integral fall within or below the range 0.9K–1.5K.

Fig. 9

Resistance curves derived from J-integral for longitudinal unidirectional laminate CMCs loaded in uniaxial tension. Inset is longitudinal strain fields corresponding to earliest detection of crack initiation at edge-notch. J-integral measures of fracture toughness were at or below the range 0.9K–1.5K. Note All stress intensity factors for CMC tests are normalized to a reference value, K (Color figure online)

Comparison of results from cross-ply and unidirectional laminates

For both the high-temperature cross-ply and ambient temperature unidirectional CMC tests, the range of critical stress intensity factors measured by the J-integral was close to published values of fracture toughness for monolithic reaction bonded silicon carbide. Again, fracture toughness estimates for the composite were defined as being at or below the J derived stress intensity factor corresponding to the first crack initiation at the notch, as detected from DIC strain fields. However, fracture toughness may not be an appropriate metric for characterizing fracture behavior in the CMCs that were tested. Cracks initiated in the matrix at low stress intensities because the matrix is inherently stiff and brittle. The existence of these cracks, however, did not result in a change in slope in the stress–strain curves of the test coupons. Unlike their monolithic counterparts, the CMC cracks did not grow unstably under sustained loading after crack initiation. As evidenced in the R-curves in Figs. 4 and 9, the stress intensities grew larger as the cracks propagated, but the rate of growth decreased with crack propagation. This is attributed to fiber bridging in the crack wake. Fibers bridging the crack apply a traction that closes the crack faces. The applied stress intensity must increase to overcome these tractions [15]. Although the estimated range of fracture toughness in the composites was near that of the monolithic matrix, the cracks that initiated grew stably under continued loading. The SIF magnitudes were larger in the unidirectional resistance curves than in the cross-ply curves. This is because the unidirectional layups contained only longitudinal fibers, which are oriented in a more favorable direction for toughening than the transverse fibers in the cross-ply.

The J-integral approach was adopted to characterize fracture toughness at the constituent level and is therefore not expected to generate results that are equivalent to a macroscale closed form solution of the stress intensity factor. One of the challenges of this new approach toward constituent level characterization of fracture toughness is the lack of a baseline to determine how much the constituent scale measurement should deviate from the macroscale. It is expected that the macroscale measurement will be incorrect, as it assumes a single crack in a material that ultimately fails through the formation of multiple widespread matrix cracks. However, the J-integral measurements at the constituent scale, while a promising new approach that provide more realistic depictions of the local fracture toughness, currently incorporate a significant amount of error. While displacement and strain error measurements [such as root mean square (RMS) and standard deviation] can be assigned to an area of interest over which DIC data have been collected [20], it cannot be assigned to each node of data individually. For example, displacement and strain errors for a set of test data can be determined by measuring the RMS error in DIC measurements generated for a static image pair. However, adding the RMS error bounds to each data point then using these modified data to evaluate J has no effect, the added error is uniform across the FOV and equates to nothing more than a rigid body translation.