Modelling mode I failure at crack tip with verifications using digital image correlation

Introduction

It is widely accepted that crack tip shape and crack width have significant effects on the stress intensity factor (SIF). For a crack with finite width (i.e. blunt crack), researchers are working on correction factors to account for the deviations that result from linear elastic fracture mechanics (LEFM) predictions.¹ Although literature is abundant on the use of finite element method to predict SIF,² most work focused on comparing finite element predictions and those from LEFM on sharp crack, and often not complemented by experimental observations. Also, some modelling techniques require manual node adjustment (i.e. collapsed elements or quarter-node approach) and/or complex mesh morphing strategy,³ and therefore not feasible for routine use by simulation analysts. SIFs calculated in finite element commercial software, such as ANSYS, often differ from contour to contour. Experimental verifications are indeed important in cases of a crack with finite width.

In recent years, digital image correlation (DIC) has enabled researchers and engineers to measure strains at minute locales and view full-field data without requiring any physical contact, or installing relatively large and expensive strain gauges.⁴ It is convenient, simple, and accurate in measuring deformation and strains in large-scale structures.⁵ Open-source programme such as GOM correlate makes 2D DIC feasible and cost-effective. Some examples of DIC usage for monitoring structural behaviour and long term reliability of materials under different loading conditions in both static and dynamic situations⁵ are as follows: Abshirini et al.’s investigation on mode I failure of Brazilian disc using DIC method,⁶ Huang et al.’s⁷ study on the flexural behaviour and crack formation in cementitious composite, UK National Physical Laboratory’s (NPL) measuring of hard-to-detect micro-crack opening in concrete structures.⁸ Recently, digital image correlation was used to evaluate fracture mode-mixity.⁹

In this work, experiments and simulations were conducted on brittle Poly (methyl methacrylate) (PMMA) plastic to compare the actual and predicted strain fields around crack tip with finite width, and the critical force at which unstable crack growth initiates. The effects of mesh size and crack width on the simulated critical SIF were investigated. The aim of this work is to determine if SIFs computed using commercial finite element method have experimentally verifiable advantages over the traditional stress-based modelling approaches in predicting Mode I brittle failure at blunt crack tip.

Methods

Specimen

Centrally straight crack Brazilian discs (CSCBD), commonly used to investigate the fracture toughness in various materials¹⁰ were prepared according to dimensions in Figure 1. The PMMA discs were machined to a diameter of 60 mm and 15 mm thick. The centre notch was of 2 × 15.5 mm in length and 1 mm in width. PMMA had been chosen in this study due to its brittle nature and the vast amount of related publications. Specimen dimensions were specified in compliance with ASTM D3967-95a¹¹ requiring a thickness to diameter ratio $\left(t/d\right)$ within the range of 0.2-0.75 and ASTM E 399-90¹² requiring the specimen to be sufficiently large compared to the crack length and the plastic zone size. The benefit of using a Brazilian Disc specimen is that it is much easier to prepare than other specimens because of its simple geometry, and the uniaxial compression test is relatively easy to set up.

Figure 1.

(a) Brazilian disc and crack tip polar coordinate system¹⁵; (b) Actual disc dimensions.

The quality of DIC results depends mostly on the resolution of the speckle pattern.¹³ Fine speckle pattern, a good focusing lens and a high resolution (10 MP) camera are essential for DIC to track the random speckle pattern with accuracy.¹⁴ The specimen was first sprayed with a layer of plastic primer followed by white spray paint before the black speckle pattern was applied. Spray painting was used to apply random speckle patterns, as shown in Figure 2. Natural drying under sunlight took place before subsequent application of paints.

Figure 2. Fine speckle pattern on a 60 mm Brazilian disc.

Compression test and digital image collection

Figure 3 shows the test setup. Compression was applied to CSCBD until fracture using Instron 3367 universal testing machine, at speeds of 1 mm/minute and 5 mm/minutes, respectively. During the compression test, the camera was mounted onto the tripod and calibrated using a built-in spirit level to ensure the image sensor was aligned perpendicular to the region of interest. In the setup, a high brightness LED light was used to illuminate the surface of the specimen. The camera was set to capture one image per second in black and white. Both the image acquisition and compression testing were carried out simultaneously. Image acquisition was started one to three-seconds before load application. Images and force data were synchronized during post-processing. GOM Correlate was used to perform the DIC.

Figure 3. Setup for compression test and digital image collection.

LEFM and numerical analysis

According to,¹⁰ critical SIF of Mode I fracture, K_I, can be computed using equation (1).

(1)

$$K_I=\frac{P}{\mathit{RB}}\sqrt{\frac{a}{\pi }}{Y}_I\left(a/R,\mathrm{\theta }\right)$$

where $P$ is the critical load at fracture, $R$ is the radius of the disc, $a$ is the semi crack length,$B$ is the thickness of the disc and $\mathrm{\theta }$ is the angle of crack relative to load, which is zero in this study. $Y_I\text{is}$ a geometry factor and is a function of crack length ratio $(a/R$) and the crack angle $\mathrm{\theta }.$

Ansys Mechanical Workbench was used to simulate the compression test. Plane stress model was used, as the finite element results do not show significant differences from those of plane strain model. This is expected as the specimen dimensions meet the requirement of small plastic zone for plain strain consideration (i.e. ASTM E 399-90¹²), and will not be further discussed in this paper. The material properties of PMMA are summarized in Table 1.¹⁵

Table 1. Material properties of PMMA.

Young’s Modulus (MPa)	Poisson’s Ratio	Density (g/cm3)	Tensile Yield Strength (MPa)	Compressive Yield Strength (MPa)
1800	0.37	1.18	62	104

A force was applied through a rigid top plate in contact with the specimen, which was supported by yet another rigid plate at the bottom. Pre-meshed crack approach and linear elements were adopted. Ansys fracture tool, which calculates SIF using contour integration,¹⁶ was used to compute SIFs on 6 contours around the crack tip.

As the quality of the mesh directly affects the accuracy and speed of the solution, a finer mesh was used around the crack tip since it was the region of interest. In this study, the effects of element type (i.e. triangles vs. quadrilaterals), minimum element size at crack tip (i.e. 50 μm, vs. 2.5 μm) and crack width (i.e. 1.0 mm and 0 mm) on stress-strain contour and SIF were investigated. Figure 4 shows the different meshes on the CSCBD model. Quadratic elements reported lower SIF values, and will not be presented in this study.

Figure 4.

Equivalent strain fields in different meshes (a) Triangle mesh, crack width = 1.0 mm; (b) Quadrilateral mesh, crack width = 1.0 mm; (c) Zoom in at crack of (b); (d) Quadrilateral fine mesh, zoomed in at crack; (e) Triangle mesh, crack width = 0 mm, zoomed in at crack; The scales in (c), (d) and (e) are comparable.

Results

Experiments on CSCBD show consistent critical load ranging from 4212 N to 4502 N, with no significant differences due to the two different speeds. With the dimensions in Figure 1, using equation (1), K_I turns out to be between 0.91 to 0.98 $\mathrm{MPa}\times \surd \mathrm{m}$. Figure 5 shows the DIC of two samples at the initiation of unstable crack.

Figure 5. DIC on sample specimens displaying strains normal to crack path.

It is well-known that finer mesh tends to increase the predicted stresses at the singularity sites of crack tips, even well before the critical load. Therefore, predicting the critical load by studying the magnitude of stresses at crack tip had not been a viable option. The conventional practice in predicting unstable sharp crack growth is to obtain the SIF using LEFM at the crack front and compare it with the critical stress intensity factor (i.e. fracture toughness) of the material. However, in this FEM study simulating both a blunt crack and a sharp crack, the SIFs predicted by the model differ vastly from one contour to the next. These values are presented in Figure 6 for different meshes (i.e. Tri for triangles, Qua for quadrilaterals, Qua_Fine for fine quadrilaterals) and crack widths (CW). Results for blunt crack models of triangle and quadrilateral meshes almost overlapped in Figure 6.

Figure 6.

SIFs (in Pa × √m) of 5 contours around crack tip for different meshes.

Nevertheless, it was found that triangle mesh at the 1.0 mm-wide rounded crack was less sensitive towards stress concentration at crack tip. The onset of unstable crack in the experiment coincided with the onset of yielding at crack tip in the mesh, i.e. when the Mises stress based factor of safety is less than unity, as shown in Figure 7. The factor of safety in this case is the ratio of yield strength to Mises stress. And PMMA, being, a brittle material, fractures close to its yield strength.

Figure 7. Mises-stress based factor of safety around crack tip for different models.

Factor of safety in the red regions is less than unity (a) Triangle Mesh; (b) Quadrilateral Mesh; (c) Quadrilateral Fine Mesh.

Figure 8 shows the stress fields at crack tips for both blunt and sharp cracks in different meshes. They were mostly comparable for the blunt cracks, but only the quadrilateral fine mesh captured the stress singularity well at the sharp crack tip.

Figure 8.

Mises stress (in Pa) around sharp and blunt crack tip in (a) triangle meshes (b) quadrilateral meshes (c) quadrilateral fine meshes.

The Mises strain field recorded in DIC at impending crack growth is now compared with that predicted in finite element models, namely the triangle and quadrilateral meshes, both with 1.0 mm crack width (See Figure 9). Comparison of the Mises strain values as recorded in the DIC and that predicted by the models were made along two different paths. Path A crosses the stress concentration at crack tip along the horizontal axis, whereas path B is parallel to path A and crosses the specimen centroid.

Figure 9.

Comparison of the strain values as recorded in the DIC on two specimens (Ex1, Ex2) and those predicted by models using triangle mesh (FE-Tri) and quadrilateral mesh (FE-Rec) along paths A and B, as depicted in the insert.

Discussion

The critical SIFs obtained from the experiment, in the range of 0.91 to 0.98 $\mathrm{MPa}\times \surd \mathrm{m}$, are consistent with 0.87-1.20 $\mathrm{MPa}\times \surd \mathrm{m}$ reported by Choi et al.,¹⁷ 1.02 $\mathrm{MPa}\times \surd \mathrm{m}$ reported by Zhou et al.,¹⁸ and 1.17$\mathrm{MPa}\times \surd \mathrm{m}$ ($37.1\mathrm{MPa}\times \surd \mathrm{mm}$) by Lerch et al.¹⁹ This implies the applicability of LEFM for the blunt “notch root-radius”²⁰ of 0.5 mm of PMMA in this study and validates the experimental procedures in the present study.

The SIFs obtained from the finite element models were significantly lower from the actual values ranging from 0.87 to 1.17 $\mathrm{Pa}\times \surd \mathrm{m}$ reported elsewhere.^17-19 The sharp crack models showed poorer predictions—their SIFs in the first contours being closest to the actual values. This correlates with the sharp drop of peak stresses away from the singularity of the sharp crack tips, in contrast with the more gradual drop in stress values at the blunt crack tip without singularity²⁰ (see Figure 8). Although all models captured the rise of stress or strain amplitude at crack tips (see Figures 4, 7 and 8), the different attempts made using different element types and sizes did not improve the SIF predictions. This finding implies that the practice of predicting unstable crack growth using SIFs computed by commercial FEM, such as the pre-meshed crack approach of Ansys fracture tool in this case, begs a closer examination, and experimental verifications.²¹ There is evidence that the interaction integral approach used in the pre-meshed crack approach to determine SIFs may not be as accurate as that using the J-integral approach or the modified Virtual Crack Closure Technique.²²

Figure 7 shows that both the coarse and fine meshes of quadrilateral elements are able to capture noticeable yielding zones at crack tip equally well, whereas the triangle mesh displays yielding only at a single node. The insensitivity of triangle elements towards stress concentration at the blunt crack tip actually helps failure prediction in the present scenario. Yielding at a single node provides a clear-cut indicator for FEM analysts to determine the onset of failure. On the contrary, the larger yielding zones in quadrilateral element models obscure the precise moment of impending fracture, since the question of how large a yielding zone needs to be to signify failure cannot be answered easily. Further case studies to take advantage of the triangle mesh insensitivity towards stress concentration may lead to a simple yet experimentally verifiable practice in predicting crack tip failures on brittle materials such as PMMA.

It can be seen from Figure 9 that the equivalent strains captured by the DIC matched reasonably well (i.e. to the same order of magnitude) with those predicted by the finite element models. DIC managed to capture the stress concentration around the crack tip even better than the models. However, the small strains at far field regions seemed to chatter. The general agreement between DIC and the model predictions verifies the models.

Conclusions

The following are the conclusions:

Data availability