# Characterization of a cracked specimen with full-field measurements: direct determination of the crack tip and energy release rate calculation

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## Abstract

The crack characterization in a pre-cracked aluminum specimen is investigated in this study using the grid method. The images of this grid are analyzed to provide the crack tip location as well as the displacement and strain fields on the surface of the specimen during a tensile test. Experimental data are used to calculate the energy release rate with the compliance method A fracture analysis is also performed using the invariant M\(\uptheta \) integral in which both real and virtual displacement fields are introduced. This integral is implemented in the finite element software Cast3M. Both approaches give similar results in 2D case.

## Keywords

Aluminum Crack tip identification Finite element analysis Grid method M\(\uptheta \) integral## 1 Introduction

In the mechanical and civil engineering fields, many materials such as wood, asphalt, composites, steel, polymers or aluminum are submitted to complex and long-time loadings (Moutou Pitti et al. 2008; Réthoré et al. 2010; Grédiac and Toussaint 2013). Among several cases, the real serviceability of structures made of these materials is often due to the crack initiation and the crack growth process during their lifetime (Dubois and Moutou 2012). This must be taken into account to enhance the reliability of such structures (Moutou Pitti and Chateauneuf 2012). The characterization of the crack tip location and growth appears therefore to be a key-issue in the communities involved in prediction of the behavior of structures and in machine safety (Chalivendra 2009; Atkinson and Eftaxiopoulos 1992; Suo and Combescure 1992; Parks 1974, 1977). This is due to the singularity of the mechanical fields at the crack tip on the one hand, and to the problems raised by the identification of the crack tip location and propagation on the other hand, even when recent image analysis methods are employed.

An overview of the literature in this field shows that many studies have been undertaken in order to characterize cracking in various types of materials. In the past, McNeill et al. (1987) have been already evaluated stress intensity factor by means of digital image correlation. Recently, Zanganeh et al. (2013) have localized the crack tip using a comparative study based on displacement field data. The digital image correlation (DIC) technique has been used in order to characterize crack tip growth (Yates et al. 2010; Pop et al. 2011). Recently, Méité et al. (2013a, b) have studied mixed-mode fracture by means of full-field optical and finite element calculations. DIC has also been applied to evaluate fatigue crack propagation law with the so-called integrated digital image correlation for which an a priori solution for the displacement/strain fields is necessary to reduce measurement noise (Mathieu et al. 2011). The same technique has been proposed to extract fracture mechanics parameters from kinematic measurements (Réthoré et al. 2012), and to identify crack tip location (Réthoré et al. 2012). Some of these methods are combined with a finite element analysis to update at the same time the actual mechanical fields during the crack propagation process. Réthore et al. (2008) have pointed out the drawback due to the virtual extension field theta that plays a major role when displacement measurements are noisy. Several authors also mention the use of interaction integrals to extract fracture mechanics parameters (Réthore et al. 2005, 2008 for instance).

In the present paper, the grid method is employed to analyze the cracking process in an aluminum specimen. The big advantage is that no a priori closed-form solution for the displacement/strain distribution is necessary to analyze the strain field, the latter being directly measured, within certain limits, at the crack tip despite the high strain gradient that occurs. This is obtained thanks to the very good compromise between strain resolution and spatial resolution of this measurement method.

The paper is organized as follows. First, the main characteristics of the grid method are recalled. The experimental device and the experimental procedure are then briefly described. Obtained results are given and discussed by comparing typical experimental and numerical strains and displacement maps. In the second section the invariant integral \(M\theta \) formulation applied in the isotropic case is recalled and then used to compute the energy release rate in the opening mode case. In the third section, the procedure employed to determine the crack tip localization is described. It is directly based on some features of the strain maps near the crack tip. The crack tip location is identified for different values of the load applied to the specimen. The critical energy release rate is deduced from the measurements using the compliance method, assuming that displacements are imposed. These values are finally compared with the numerical energy release rate given by the \(M\theta \) integral approach.

## 2 Experimental procedure

### 2.1 The grid method

The grid method is one of the white-light techniques available to measure bidimensional displacement and strain fields. It consists first in depositing a crossed grid on the surface of the specimen under investigation in order to track with a camera the slight change in the grid as loading increases. In the current case, a 12-bit/1,040 \(\times \) 1,376 pixel SENSICAM camera connected to its companion software CamWare was employed. The grid was transferred using the procedure described in (Piro and Grédiac 2004). The pitch of the grid was equal here to 0.2 mm along both directions.

With this technique, the in-plane displacement and strain fields are deduced from the images of the grid taken during the test by processing them with the windowed Fourier transform (WFT). This transform actually provides the phases and its derivatives of the quasi-regular marking of the surface. It can be easily demonstrated that the change of the phase between any current grid image and the reference grid image is directly proportional to the in-plane displacement (Surrel 2000). Strain components and local rotations are merely obtained from the spatial derivatives of the phase, using the definition of these quantities (respectively half the sum and half the difference between the displacement gradient and its transpose). In practice the WFT is merely carried out by convolving the grid images by a suitable kernel. Since the window of this transform has a certain width greater than one pixel, estimating the strain components at a given pixel relies on the information contained in the neighboring pixels. The kernel of the WFT used here to perform this calculation is a 2D Gaussian envelope whose standard deviation is equal to sigma \(=\) 5 pixels. Using the classic “3-sigma rule”, the width of this envelope can be estimated to 2 \(\times \) 3 \(\times \) 5 \(=\) 30 pixels. The lowest distance between two independent measurements being equal to the width of the Gaussian envelope, this width is actually the spatial resolution of the technique for strain measurement. Note that sigma \(=\) 5 pixels is the lowest value that can be used here because the frequency of the grid in the image is f \(=\) 1/5 pixels \(-\)1 and because the following inequality holds: f \(\times \) sigma \(\ge \) 1, as recently rigorously demonstrated in Sur and Grédiac (2014).

It has been shown that the metrological performance of this technique could be significantly improved (especially for small strain measurements) by getting rid of most of the grid marking defects that unavoidably occur when grids are printed on their support (Badulescu et al. 2009a, b) . In particular, a very good compromise is obtained between resolution in strain and spatial resolution. For instance, a typical resolution is strain is some hundreds of microstrains for a spatial resolution in strain equal to 30 pixels. These quantities however strongly depend on various parameters such as the quality of the grid and lighting. Another feature of this technique is that the displacement and strain components are calculated pixelwise, thus allowing detecting very localized phenomena. Note however that these measured quantities are not independent from one pixel to each other because measuring the phases and their derivatives is performed at any pixel by relying on the information contained in the neighboring pixels. Full details on strain calculation with this technique can be found in (Badulescu et al. 2009a, b) for unidirectional and crossed grids, respectively.

### 2.2 Experimental device and procedure

A 48.35 \(\times \) 41.6 \(\hbox {mm}^{2}\) grid was also transferred after the fatigue test around the crack in order to measure the displacement and strain fields in this region during the quasi-static test by applying the grid image processing technique described above.

This procedure illustrate the fact that both the location of the crack tip and the strain distribution can be obtained for various crack angles, the orientation of the crack near the tip changing during the quasi-static test to become perpendicular to the applied load.

## 3 Invariant integrals

## 4 Results and discussion

### 4.1 Displacement and strain maps

#### 4.1.1 Comparison between numerical and experimental results

This type of strain map was used to determine the crack tip location and to deduce the crack propagation during the test, as explained in the following section.

### 4.2 Finding the crack tip location from the strain and local rotation maps

Finding the location of the crack tip is a key issue in many problems dealing with cracked specimen characterization. In many recent studies in which DIC is involved, an a priori knowledge of the form of the strain field ahead of the crack is necessary to have an information that the measurement technique cannot directly provide. In this case, the crack tip location is part of the unknowns that are measured. This location is found by performing optimization calculations (Zanganeh et al. 2013 for instance). In addition to these calculations, the problem is that any deviation of the actual strain distribution from the model, as in the example discussed above, can be really detected since no reliable measurement of the strain field in this zone can be made, the resolution and spatial resolution in strain of the measurement method used being not sufficient. In the present study, the fact that strain fields can be measured despite the high strain gradient (beyond a certain strain level which is the resolution of the technique) leads the crack tip location to be determined directly. Finding the crack tip has been performed here using two different routes: the first one relies on the strain fields, the second one on the local rotation fields. They are briefly explained and illustrated below.

The points located in each valley and near the crack tip are then collected (red points in the figure). They are used to define a straight line for each valley by linear regression of the white point cloud. Since there are here three lines, the information is redundant and there is no unique crossing point in practice. The crack tip location is therefore considered to be the point located at the lowest distance from the three crossing points in the least square sense. Figure 8 shows such straight lines superimposed to each valley. Comparing the coordinates of the actual crack tip considered to define the mesh (633,628) (or (25.30, 25.09) in mm) and the identified one (633.50, 627.15) (or (25.28, 25.08) in mm) shows that the distance between them is tiny (0.022 mm in the current example).

A second simulation has been performed with a crack which becomes curved near the tip, as in the current experiment when the crack propagates, thus leading to lobes whose shape is different from the preceding one. Despite this new crack shape, performing the same procedure leads to an error of the same order of magnitude as in the preceding case (results not reported here).

Values of critical forces and crack tip location

Critical \(\hbox {N}_\mathrm{i}\) | Force | Grip displacement | Crack growth | dF (kN) | Crack tip position | |
---|---|---|---|---|---|---|

F (kN) | d (mm) | da (mm) | X (mm) | Y (mm) | ||

0 | 0.000 | 0.000 | ||||

1 | 40.000 | 0.297 | 0.000 | 40.000 | 32.333 | 16.947 |

2 | 51.110 | 0.35 | 0.150 | 11.110 | 32.233 | 16.835 |

3 | 60.000 | 0.477 | 0.082 | 8.890 | 32.178 | 16.774 |

4 | 70.950 | 0.686 | 0.046 | 10.950 | 32.179 | 16.728 |

5 | 80.160 | 0.864 | 0.018 | 9.210 | 32.162 | 16.721 |

6 | 96.340 | 1.441 | 0.071 | 16.180 | 32.097 | 16.692 |

7 | 97.350 | 1.441 | 0.422 | 1.010 | 31.899 | 16.320 |

8 | 98.680 | 2.246 | 0,815 | 1.330 | 31.240 | 15.840 |

9 | 98.260 | 2.245 | 0.963 | \(-\)0.420 | 30.600 | 15.120 |

10 | 85.500 | 3.083 | 8.867 | \(-\)12.720 | 22,000 | 12.960 |

11 | 76.880 | 3.922 | 4.585 | \(-\)8.620 | 17.440 | 12.480 |

As a last remark, noise impairing the current strain and local rotation measurements can be seen in Figs. 9 and 10 (see the blobs superimposed to the smooth actual strain and rotation fields, especially for low values of the applied load). The relative weight of this noise logically decreases as the strain level increases from one figure to each other. It is however worth noting that noise does not seem to impair the crack tip location for the early stages of the load since the three straight lines converge to nearly the same point.

Note that the out-of-plane movement induces fictitious strain whose order of amplitude is equal to \(\Delta l/l\), where l is the distance between the grid and the CCD chip of the camera (Molimard and Surrel 2012), and \(\Delta l\) the amplitude of the movement. This distance being equal to 80 cm and the amplitude of this movement being no greater than 1 mm, this fictitious strain is not greater than 0.0125 %, which is negligible compared to the strain values that are provided by the system in this zone.

### 4.3 Crack tip location

### 4.4 Critical energy release rate

*da*, \(b\) is the thickness of the specimen (equal to 10 mm here), and \(C\) is the compliance given by Eq. (4). \(d\) denotes the displacement of the moving grip. \(\Delta F_{C}\) represents the change in force between two consecutives critical points reported in Table 1.

### 4.5 Numerical energy release rate

The finite element software cast3M has been used in order to implement the M\(\uptheta \) integral and obtain numerical results. The value for the Young’s modulus \(E\) and the Poisson’s ratio \(v\) are those already given above.

## 5 Conclusion

The grid method was employed here to measure the displacement, strain and local rotation fields near the crack tip of a pre-cracked specimen subjected to a tensile test. It has been shown that these fields are very similar to those obtained with a FE analysis in 2D configuration. The strain and rotation fields were employed to deduce the crack tip location in the images and to measure the crack propagation during the test. The crack tip propagation was also employed to obtain the critical release rate at various load levels using the compliance method. Finally, the M\(\uptheta \) integral in the crack growth case has been implemented in a finite element software in order to compute the numerical energy release rate. The obtained values have been compared with the experimental data. This illustrates the efficiency of the proposed identification crack tip technique and the relevancy of the numerical approach. In further work, the model proposed in this paper will be generalized to mixed mode crack growth process and deconvolution will be used to enhance the strain maps near the crack tip (Grédiac et al. 2013).

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