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Numerical Assignment: Investigating Material Heterogeneity with Numerical Homogenization Method

## Problem Layout

Introduction

This individual assignment is designed for you to investigate the topic of material heterogeneity and the so-called ‘numerical homogenisation’ method for bi-phasic materials. You will be given the problem layout and your task is to make all engineering calculations and assessments required. This should be done via Finite Element (FE) analysis.
A comprehensive write-up of the methodology should be included. Not only would your report be judged by the results you present, it will also be marked on its engineering quality. Hence the critical thinking involved when analysing the results should be clearly presented along with the critical discussion and conclusions.
You are encouraged to make use of these references, and others available in the library/www, to demonstrate your academic knowledge of this topic. However please be aware of the plagiarism and copyright issues when preparing your report.

1. Let us assume two elastic engineering materials, with Young’s modulus of 1GPa (material in red) and 200MPa (material in blue), both having Poisson’s ratio of 0.3. Two constituents are distributed in a certain way (red material always at the ‘centre’, of a square shape) – see below

2. Use ABAQUS to carry out numerical homogenisation, using simplified KUBC boundary conditions (see below in Fig. 2), to derive the apparent bulk and shear moduli for up to 7 different volume fractions (i.e. 0%, 10%, 20%, 30%....100%) of the material with a Young’s modulus of 1GPa, respectively. Plot the resulting apparent properties of all cases using the diagram of ‘apparent elastic properties v.s. volume fraction’. Discuss your findings based on the ‘homogenisation theory’ that was introduced in our lecture session.

3. Let us assume two elastic engineering materials, with Young’s modulus of 1GPa and 200MPa, both having Poisson’s ratio of 0.3. The distribution of two constituents could be random.

4. Use ABAQUS to carry out numerical homogenisation, using simplified KUBC boundary conditions (see below in Fig. 2), to demonstrate the idea of ‘possible solution space’ between upper and lower Hashin-Shtrikman bounds. In other words, for a bi-phasic material system, at a given volume fraction ratio between those two materials, there are many (infinite) possible apparent properties due to different microstructures.

5. Create two ‘bi-phasic material systems’, which have different geometries/topologies, as above. For each system, you should create up to 5 cases of varying volume fractions (e.g. 0%, 10%, 20%.... 100%) of the material with a Young’s modulus of 1GPa (thus 100%, 90%, 80% ….. 0% for the other material of 200MPa), respectively. Plot the resulting apparent properties of all cases for both systems, using the diagram of ‘apparent elastic properties v.s. volume fraction’. Discuss your findings based on the ‘homogenisation theory’ that was introduced in our lecture session.

## Assignment Submission:

Hint: It is important to choose the microstructure of the bi-phasic material systems wisely, so you can demonstrate that those two scenarios have different relationships between apparent properties and volume fractions

1. The maximum size of the report is limited to 4 pages for BEng and 5 pages for MEng, including all references. An extra cover/declaration page is permitted, however no appendices should be included. The report should be submitted by 3:59pm (UK time), 7:59pm (Dubai time) and 11:59pm (Malaysia time) Friday 10th December through CANVAS. Late submission will lead to mark reduction according to school policy. Abaqus .cae files should also be saved and you may be asked to provide them later. Do not wait until the last minute/hour to submit your report. Reasons such as ‘late submission due to canvas taking a few minutes to submit’ may not be accepted.

2. The quality of your engineering analysis and discussion in your report is key. Therefore, you should not try to fill the report with screenshots. We are looking for high quality figures/tables with critical analysis and discussion.

3. You are encouraged to research into the topics relevant to this assignment, such as homogenisation method in relation to the numerical methods and so-called KUBC/SUBC/MBC/PBC variations in boundary conditions, although this assignment only asks you to use simplified KUBC (i.e. averaging the force/stress based on a displacement/strain input). Demonstrating your understanding and efforts in researching relevant topics, in both ‘introduction and mini literature review’ and ‘discussion’ sessions of your report, is particularly encouraged.

4 Notes on Plagiarism  Full details of the university plagiarism policy can be accessed via the link: http://www.hw.ac.uk/registry/resources/PlagiarismGuide.pdf. According to the university guidelines, “Plagiarism involves the act of taking the ideas, writings or inventions of another person and using these as if they were one’s own, whether intentionally or not. Plagiarism occurs where there is no acknowledgement that the writings or ideas belong to or have come from another source”.
Students must pay particular attention to the following examples of common plagiarism mistakes made by other students when reflecting on their own work:

1.“I thought it would be okay as long as I included the source in my bibliography” [without indicating a quotation had been used in the text]

2.“I made lots of notes for my essay and couldn't remember where I found the information”

3. “I thought it would be okay to use material that I had purchased online”

4. “I thought it would be okay to copy the text if I changed some of the words into my own”

5 “I thought that plagiarism only applied to essays, I didn't know that it also applies to oral presentations/group projects etc”

6. “I thought it would be okay just to use my tutor's notes”

7. “I didn't think that you needed to reference material found on the web”

8. “I left it too late and just didn't have time to reference my sources”