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# Module 2: DIC and crack tip fields

This module will focus on the application of digital image correlation (DIC) to the study of crack tip fields.

For this module, we will use an image-based technique to determine the deformation fields in solid media. This method is called Digital Image Correlation (DIC). The method's premise is to embed or emboss a random pattern of high-contrast markers that move with the material, and to compare their current states to a reference state. Once the spatial distribution of marker deformations is determined, kinematic quantities of interest such as strain tensors, etc. can be used to answer mechanics questions.

### Readings

#### Week 1

A general introduction to DIC as a method, with a specific implementation

I strongly encourage you to review the NCORR manual. Section 2.10 will prove particularly useful if you intend to elaborate your analysis, and measure stress fields from the deformation data.

#### J-integral and calculating the first Piola-Kirchoff stress

The J-integral is a powerful tool for analyzing the crack-tip driving force. If the displacement and stress data are known, the J-integral can be directly evaluated around a contour surrounding the crack tip. A detailed explanation of how this can be achieved for neo-Hookean solids using the 1st PK stress is given in this paper, also cited below; see the discussion around equation 25.

If you wish to understand how the J-integral relates to the strain energy release, John Hutchinson has a wonderful explanation in his notes on non-linear fracture mechanics, specifically on pages 29-35. While these notes are concise, they are mechanics-dense. I will discuss them briefly in class to help guide your reading.

### Exercises

#### Week 1

1. Form new groups.

2. Install NCORR on a PC with one of your group member's accounts. Each account will require a separate install, but once installed, it should always work with that sign in

3. Download and process the image samples from NCORR. Use the hole in the plate dataset to familiarize yourself with the software, the speckle pattern, and calculation of strains. You should work with the documentation in-hand, so you have a clear idea of what each of the software `knobs' does in the experiment.

4. Write a script that allows you to calculate the deformation gradient tensor components from the displacement fields for the hole-in-the-plate data. Make sure that this will work with your data, so you can re-use it at a later date.

5. Write a script that allows you to calculate the 1st Piola-Kirchhoff stress fields for the hole in the plate data for a Hookean solid or a neo-Hookean solid (see e.g. Hs or nHs, normalized by a modulus. Make sure that this will work with your data, so you can re-use it at a later date.

#### Week 2

Pattern application:

1. You'll have to apply the pattern that you'll use for DIC to your gel samples. To determine the density and droplet size for your pattern, you will practice on a sheet of paper, and review the pattern on the microscope. This requires a bit of art, because the resulting pattern will depend on how far away the can is, for how long you spray the paint, etc.

**You'll have a 2 MPx camera (1920 x 1080 pixels) to record your images. Details of the camera's specifications can be found here. Given that you'll need about 3 pixels for each speckle in your pattern, decide approximately how your pattern should look to obtain your desired resolution.**

2. You're nearly ready to test! We'll use the same testing machine that we used in the prior module, but this time with a camera. You'll have to align the camera at the right distance and focus to resolve your pattern sufficiently. **Bear in mind, that you'll want good synchronization between the load values and the images, in case you want to compare your data with theoretical curves. **Try to work out a way to do this.

n.b. Testing can be carried out according to the schedule. If you have multiple samples to test, please try to be efficient in gathering your data, so everyone who is ready to test today has a chance to record some data.

#### Week 3

Today's goals are simple - for everyone to get a data set, and to obtain deformation and stress fields from it. A few pointers:

- If the DIC is not working - see if there is a reason why it isn't working. Thus far, we've seen examples of the pattern fragmenting, which will not generate useful deformation data, and likely will cause the DIC algorithm to fail
- The goal is to measure deformation up to the point where the crack propagates. This may require different patterning methods. Can you think of a way to ensure the pattern tracks the material even at very large deformation?
- Once you have DIC data, see if your displacement field calculations from weeks 1 and 2 work with the data.

#### Week 4

Today you should converge on a hypothesis that you wish to test in your experiments. This hypothesis should relate to something you don't currently understand - it could be about constitutive response near the crack tip, or some other unknown behavior of the crack. You are encouraged to get creative!

Some suggestions as you formulate your hypothesis:

- Use your existing data. The measurement you've made might suggest a path forward, or raise a question that you cannot currently answer, and may require additional testing or analysis.
- Consider how you can test your hypothesis. What additional data or analysis would you need?

#### Week 5

Finalize your analysis and write the report. Pay close attention to formatting figures, and writing concisely. The best way to do this is to edit repeatedly, until you find that you cannot improve the text futher.

### Groups:

1. Lepere, de Tournemire, Nauche C = 4.33e-2;

2. Vincent, Denervaud, Herbault; C = 4.33e-2;

3. Zennaro, El Haouat, Vallat; C = 1.87e-2;

4. Zen-Ruffinen, Charmillot, Dhaouadi; C = 1.87e-2;

5. Chalhoub, Antille, Altuntas; C = 1.20e-2;

6. Desaules, Gouttenoire, Vignon; C = 1.20e-2;

7. Halevi, Griffon, Dupille; C = 8.10e-3;

8. Rosset, Mangin, Touzeau, Beyeler; C = 8.10e-3;

### Lecture notes

**Previous years (2023)**

### Reference

2D Neo-Hookean reference paper (see Sec 2.1)

Displacement components for an LEFM crack with stress intensity factor K (Anderson, Fracture Mechanics 4 ed.)

### Student Submissions

**2023**

Group 5 (2023) - A clever way to spot the crack propagation for neo-hookian solids in Mode-I fracture testing using Matlab

**2024**

Group 6 (2022) - 1: How to achieve a good speckle pattern for samples used in DIC.pdf

Group 6 (2022) - 2: How to Model a Hyperelastic Material in Finite Element Analysis.pdf

Group 4 (2022) - 3: Material Model for silicone based on uni-axial traction.pdf

Group 6 (2022) - 4: Data synchronisation for DIC

Group 2 (2022) - 5: A Simplified Guide to Ncorr Installation