### What happens in a state of uniform dilatation

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Multiaxial States

The computing project Multiaxial States concerns the analysis of strain and stress at a point in a solid body. The goal is to write a MATLAB program that will allow the computation of the strain associated with a prescribed motion and the stress associated with that strain through Hooke's Law. We will compute principal values, maxima, and graphically represent the state by mapping the motion both in the (x,y)  coordinates and in the principal coordinates.

Finally, we will produce Mohr's circle to represent the state.

The theory needed to execute this project is contained in the Course Notes entitled Multiaxial Strain.

The general steps are as follows:

1. Specify the components of the deformation tensor (matrix) F and use it to compute strain tensor E. The tensor F is the mapping matrix. We can use it to compute the deformed location of points that we identify in the undeformed configuration (e.g., the vertices of the initial square of material).

2. Develop code to plot both the undeformed square and the deformed version of it. Establish the vertices of the square with the points (0,0), (1,0), (0,1), and (1,1).

Compute the deformed positions of those points using F.

3. Compute the angle for the principal directions along with the principal values of strain. Develop code to plot the undeformed and deformed square defined in the coordinate system associated with the principal directions (i.e., take the initial square to have its sides along those directions and map the vertices to find the deformed body.

4. Compute the stresses associated with the strains by virtue of Hooke's law for plane strain.

5. Solve the problem of finding principal values using the eigenvalue solver (the "eig" command) in MATLAB. Output the principal values of stress, the maximum shear stress, and plot Mohr's circle for the stress state. Do the same for the strain.

6. Examine the implications of different states of strain. For example:

a. What happens in a state of uniform dilatation (elongation without shearing)?

b. Verify that pure shear has principal directions at 45 degrees from the original orientation of the block.

c. Show that the principal directions for stress and strain are the same. Is this due to the nature of Hooke's Law? Do you expect it to be true for all material models?

d. Explore any other feature of the problem that you find interesting.

e. Extend the code to do three dimensional states of strain for an extra nice project. Note that this step is not required but you are encouraged to do it to earn top marks. The visualization plots (especially the deformed and undeformed cube need to be rendered in 3D).

Write a report documenting your work and the results (in accord with the specification given in the document Guidelines for Doing Computing Projects). Post it to the Critviz website prior to the deadline.

Please consult the document Evaluation of Computing Projects to see how your project will be evaluated to make sure that you can get full marks. Note that there is no peer review process for reports in this course.

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