## BENG 112A Homework Assignment 5: Stress Analysis of an Artery

**Due: Thursday March 3 ^{rd}** at the beginning of class

The goal of this assignment is to perform a stress analysis of an artery using nonlinear finite deformation elasticity and gain insights into:

- The distributions of stress and strain in the vessel wall
- The effects of nonlinear anisotropic interactions between axial and circumferential strains
- The effects of torsion on vessel stress and strain distributions

Before you start, you need to install a patch to Continuity which you can find at http://www.continuity.ucsd.edu/Continuity/Documentation/Help/InstallingPatch. Follow the instructions to install the patch. If you need help, email Jeff Van Dorn [mailto:jvandorn@ucsd.edu]

Download and run a model of artery mechanics in

*Continuity*. Confirm that the solution works as expected, then describe the problem that was solved.Download artery.cont6, launch

*Continuity*and load the model:File→Load Model... or click

- Select the file named artery.cont6
Click

**OK**.

- Run the model and confirm the solutions look correct:
Mesh→Calculate Mesh... or click

Click

**OK**

Mesh→Render→Elements... or click

Click

**Render**to display mesh lines

Biomechanics→Solve Nonlinear...

Enter the

**25**for`Number of Steps`Click the

**Solve**button, and wait for the solver to finish. While the solution is being computed the window will remain open. There will also be output listed to the Python shell window.

Mesh→Render→Elements... or click

Click the

**deformed**radio buttonClick

**Render**to display mesh lines

View→Show→OpenMesh... or click on

Click on

**2. element lines2**in the list on the left, and enter**1,0,0**in the`R,G,B`entry field to change the mesh lines from blue to red.- Press [return] and close the window

The result should look like the screen shot at the bottom of the page.

- By reviewing the inputs to the model (e.g. forms for Edit Nodes, Material Coordinates, Constitutive Law, Boundary Conditions, etc) describe this model as completely, precisely and quantitatively but concisely as you can. Feel free to use equations to support your description but if you do provide equations define what kind of equation it is and what the variables are.

- Render and list stress and strain fields for this problem, and describe them.
- Render stress or strain fields on cross-sections of the deformed mesh:
Biomechanics→Render Surface...

In the pop-up menu next to

`At Xi`select**2**and enter a normalized`Location`between**0.**and**1.**along the length of the mesh for the cross-sectional surface that the solution will be rendered on. (In subsequent renderings, you could specify a different location so that you could see more than one result at a time).Check the

**deformed**radio button to indicate whether you want to the solution rendered on the undeformed or deformed geometrySelect the output variable from the menu. For vector and tensor variables, a choice of components will be presented. Output will be referred to coordinate frames as determined by equations in the

`Output Variables`folder of the`Constitutive Model Editor`Here the coordinates`(x,y,z)`correspond to`material coordinates`where`x`is circumferential,`y`is longitudinal and`z`is radial.Click

**OK**to create a color-coded surface rendering of the chosen output variable

View→Show→OpenMesh... or click on

Click on a

**Textured Field**in the list on the left, and select the**Colors**tab. The fields next to`Range:`show the actual values of the chosen variable at each end of color spectrum. You can change these if you wish. Enter**return**to see the result.You can also show or hide rendered fields by clicking the

**Show/Hide**checkbox in the`Properties`tab.

- List stress and strain fields:
Biomechanics→List Stress and Strain...

In the

`Variables`tab, use checkboxes to deselect any variables you do not wish to listNote that the

`Output Variables`are calculated using the equations entered in the`Output Variable`folder of the submitted`Constitutive model`Click

**OK**to display a listing of the selected`Output Variables`in the Table Manager. You can also use File→Save... in the Table Manager to export the results in a format that Excel can import.

- Save renderings and listings or plots of all components of at least one stress tensor, one strain tensor and one other informative output. Describe whether these solutions agree with your expectations, exactly, approximately or not at all, keeping in mind that these are numerical solutions on a course two-element mesh, so they might not be correct and they certainly won't be perfect. Remember to comment on simple but important features of the solution (e.g. are they axisymmetric?) and whether you expect them to be or not. Are the numbers reasonable or not? Do you know the units of these results?

- Render stress or strain fields on cross-sections of the deformed mesh:
Rerun the model in steps saving solutions for circumferential stresses and strains at 10 or more increments of load so that you can plot stress versus strain at the middle of the wall as the artery is loaded. Repeat the analysis, this time reducing the

*longitudinal stretching*by half or more and compare the results again. What do you notice. Explain the results. Are they realistic for arteries? If not, how might the model be altered to improve the results?Rerun the model again, this time changing the boundary conditions to alter the effects of

*torsion*on the stresses in the artery. Use your results to comment on how torsion affects the*stress distributions*in the wall.