Extracting Crack-tip Field Parameters in Anisotropic Elastic Solids From Full-field Measurements Using Least-squares Method and Conservation Integrals (Experimental and Applied Mechanics) Part 2

EXPERIMENT

Experiment procedure

The composite material investigated in this study is Cyply 1002, a glass-fiber reinforced epoxy laminate, manufactured by Cytec Engineered Materials. A cured composite panel of that has 0/90 cross-ply laminates are used. Each composite panel has 13 plies of 3.3mm thickness. The composite laminate has 13 plies in the arrangement of cross-ply [(0/90)6/0] lay-ups. The elastic properties of the composite panel were measured in this study following ASTM D3518 and Mill handbook MIL-HDBK-17-1F. The lay-up sequences and measured material properties of the specimen are shown in the following table.

Table 1. Elastic properties of the composite material

Lay-up	Transverse Young’s modulus Ex	Longitudinal Young’s modulus Ey	Shear modulus Gxy	Poission ratio
[(0/90)6/0]t	22.39Gpa	22.14Gpa	3.172Gpa	0.127

Translaminar fracture tests were conducted using the extended compact tension (ECT), also known as eccentrically loaded single-edge-notch specimen (ESET) following ASTM E1922-04 [9]. The specimen geometry is shown in Fig.1 (a). The specimens with an initial notch of 0.20 mm width were machined using an abrasive water jet cutter. The crack length-to-width ratio (a/W) was 0.5. In order to minimize delamination of surface plies, the fiber direction in the surface plies was parallel to the initial notch direction. This led to self-similar crack growth in the composite laminates without extensive delamination as shown in Fig.1 (c). The fracture tests were conducted at an extension rate of 1 mm/sec using MTS servo-hydraulic mechanical test frame. The applied load and loading-point displacement were recorded with 10 Hz sampling rate.

Fig.1 (a) Geometry of specimen, (b) Speckle patterns, data field used in calculation (red area) and integral paths (blue contours). (c) Self-similar crack growth.

To achieve full-field measurements using DIC, random speckle patterns were spray-painted on the composite specimens. During the fracture tests, a series of digital images (1280 x 960 pixels) were acquired with a frame rate of 1 Hz. The size of the field-of-view was 19.1 mm x 13.5 mm (22.5 um/pixel) centered on the initial notch. A typical image of the speckle pattern is shown in Fig.1 (b). The displacement fields near the notch tip were obtained from the images using a custom-built DIC software. The parameters for the DIC analysis included a subset-size of 41 x 41 pixels and a subset spacing of 10 pixels.

Experiment result and discussion

A typical load-displacement relation is plotted in Fig. 2. It shows that the linear part is up to more than 80% of the peak load, excluding the small nonlinear part at the beginning when the load is small. It is also observed that AV/ V is less than 10%, which represents that relatively small damage zone occurs around the crack tip during fracture. So the formula provided in ASTM E1922-04 can be used to get the applied stress intensity factor as well as the fracture toughness.

Fig. 3 shows a typical result of the contour maps of the displacement fields around the crack tip obtained by digital image correlation at the 70th second. A length of 1 millimeter corresponds to about 44.5 pixels in the images processed by digital image correlation. It is observed that the obtained displacement fields are smooth. The fluctuation happens only near the crack face. Actually, the displacement data in this area are invalid. Since the subset used in DIC overlap the crack face, the displacements are determined by areas on both sides of the crack surface, which have opposite displacement. Therefore, the data near the crack face should be eliminated in the post processing. In the present study, data within 30 pixels near the crack face, which is equal to the summation of the half length of the subset and the width of the notch (crack), are excluded. In addition, these nearly symmetric displacement fields reflect the mode I loading in the fracture test.

Fig.3 Example of the displacement fields around the crack tip obtained by digital image correlation.

Fig. 4 Comparison of mode I and mode II stress intensity factors obtained by different methods during the fracture test.

Fig.5 Variation of rack length during the test.

Fig.6 R curve.

Fig. 4 shows the results of mode I and mode II stress intensity factors extracted by different methods during the whole fracture test. Five different methods are used to get K1 in the present study. The terms " K1 by ASTM" and " K1 by ASTM modified" in the legend represent the results obtained by the formula in ASTM E 1922-04 which is derived based on the isotropic elasticity and its modified version which brings a constant corrected factor considering the influence of the orthotropy [10], respectively. The formula provided in ASTM 1922-04 is shown as:

where

P is the applied load at the pins, B is the specimen thickness, W is the width of the specimen, a is the original crack length, and a = a / W . If the influence of material orthotropy is considered, a corrected factor Y(p) is introduced into the previous formula as follows:

where

p is defined as a constant related to the orthotopic material properties

The terms " K1 by NLLS" and " K1 by Interactional Integral" represent the results obtained by the non-linear least squares methods and interactional integral descried in "Methodologies" section, respectively. Since mode I loading is dominant in the test, K1 can also be obtained directly by the relation between J andK1, assuming K11 is zero. The result is shown by term " K1 by J Integral".

Firstly, it is observed that the difference between the results of " K1 by ASTM" and " K1 by ASTM modified" is about 16% at the maximal load, which means the influence of the orthotopic material properties is relatively large. Secondly, the results of terms " K1 by ASTM modified", " K1 by NLLS", " K1 by Interactional Integral" and " K1 by J Integral" are very close to each other before reaching the maximal load, which can inversely verify the validity of the parameter-extraction methods for the present study. In addition, it is noteworthy that the values of K1 of the terms " K1 by NLLS", " K1 by Interactional Integral" and " K1 by J Integral" keep increasing for a little while after reaching the maximal load, opposite to the immediately decrease of that of " K1 by ASTM modified". It is due to the fact that the formula provided in ASTM (both original and modified versions) are based on two assumptions that the damage zone (process zone) occurs in front of the crack tip is negligible, and crack length is constant during the whole test. For the first assumption, the damage zone size before crack propagation in the present study has been pointed out to be relatively small and thus the assumption of constant crack length before crack propagation exists. It explains the consistence of " K1 by NLLS", " K1 by Interactional Integral", " K1 by J Integral" and " K1 by ASTM modified" before reaching the maximal load. However, the crack length as well as the damage zone size will increase after reaching the maximal load. So the ASTM formula is not valid any more after the maximal load. Generally, K, is a combination of the applied load and the crack length, with a general form as follows:

Here Y andF are both corrected factors only related to orthotopic material properties and specimen geometry, respectively. At the beginning of the crack propagation, the effective crack length aeff, which is defined as the sum of the actual crack size and a process zone correction, keep increasing while the applied stress o decreases a little. The combination of the two variables will cause the applied stress intensity factors to keep increasing for a little while until the crack propagate unstably. So the least-squares method and conservation integrals which can reveal this phenomenon show superiority over ASTM in accurately measuring the fracture toughness K1C in the present study for fiber-reinforced composite. The terms " K11 by NLLS" and " K11 by Interactional Integral" in Fig. 4 represent the results of K11 obtained by non-linear least squares methods and interactional integral. Comparing to KI , K II is negligible. It is consistent with the fact that KI is dominant in this test.

Fig.5 shows the variation of the crack length in the fracture test. The results obtained by two different methods show good agreement with each other, except the beginning part where the displacements are very small and the accuracy of the calculation is influenced significantly by the noise in the data. It can be seen that the crack length starts to significantly increase at the 70th second. It is also the time when the plot of " KI by ASTM" and " KI by ASTM modified" starts to deviate from the linear part, corresponding to the time when the increase of load deviates from linearity. There is a little increasing jump of the crack length which is consistent with the decreasing jump of " K1 by ASTM" and the load around 80 second. It is observed from the captured images that the bridging fibers in the damage zone suddenly break at this moment. After the crack reaches about 14 millimeters at the 80 second, the crack grows rapidly.

Fig.6 shows the extracted R curve, which reflects the material resistance to crack extension. Strictly speaking, the present fiber-reinforced composite has a rising R curve rather than a flat one. The rising R curve reflects the increase of the resistance. To understand the shape of the R curve, it is better to study the fracture process of the studied fiber-reinforced composite, which can be described briefly as follows [11]. The matrix of the composite ruptures at first, then the load is carried by the fibers. It will then cause a mechanism called fiber bridging, where the propagating crack leaves fibers intact. The fibers do not fail at the same time, because the fiber strength is subject to statistical variability. Consequently, the material exhibits quasi ductility, where damage accumulates gradually until final failure. In other words, a damage zone at the crack tip increases in size as the crack grows. Thus the driving force must increase to maintain the crack growth. After the crack reaches a relatively long length, the R curve reaches steady-state with further growth. The critical crack length in the present study when the crack grows unstably is nearly 14 millimeters, corresponding to the 80th second.

CONCLUDING REMARKS

Until recently, research efforts in fracture mechanics of composite materials have been mainly based on a continuum analysis for homogeneous anisotropic linear elastic materials that contains a flaw of known length. This is an extension of conventional LEFM concepts to account for anisotropic constitutive responses. Nevertheless, the applications of LEFM principles to fiber reinforced composite have been less successful due to the complexity of the crack-extension process in micromechanical point of view. On the other hand, even after extensive studies on micromechanical models to account for various failure mechanisms, such as fiber pull-out, matrix cracking, fiber/matrix debonding and interlaminar delamination, it is still illusive that this bottom-up approach will provide a capability to predict macroscopic crack growth in fiber-reinforced composite materials.

In this study, an experimental investigation method based on full-field optical deformation measurements are presented as a way to characterize subcritical and steady-state crack advances as well as stress intensity factors during fracture tests. The experimental analysis method still relies on the fracture mechanics theories in anisotropic elasticity under the assumption of homogeneous approximation. Nonetheless, for some limited cases, self-similar and collinear crack advances in composite materials can be experimentally investigated to account for the crack-tip damage zone in a similar way to Irwin’s plastic-zone correction factor and R-curve. Furthermore, the method is expected to provide a foundation for top-down approach to multiscale analysis of fracture in heterogeneous solids.