Name:_______________________________

Name:_______________________________28 May 2014

2014 Final ExamOpen Book, Open NotesDue: 4 June 2014

Feel free to use http://www.continuummechanics.org/cm/interactivecalcs.html

1. What does represent? And what numerical value does it always equal?

2. What is wrong with this tensor equation?

3. What does it mean when the deformation gradient is symmetric and has nonzero values in the off-diagonal terms?

4. What does it mean when the deformation gradient is NOT symmetric?

5. We talked about -0.5 being a key value in Green strains. What is key about it?

6. For an incompressible material, will the trace of a true strain tensor always equal 0?

7. Under what conditions will Hookes Law accurately reproduce the behavior of a nonlinear hyperelastic material that would normally be modeled with Mooney-Rivlin coefficients?

8. What type of stress does a von Mises stress include, and what does it exclude?

9. If you knew that a material had Mooney-Rivlin coefficients of C10 = 0.6 MPa and C20 = 0.2 MPa (with all other Cijs being zero) and you want to use Hookes Law to model it (perhaps because your program cant use M-R coefficients), then what value of elastic modulus, E, would you use?

10. Which stress tensor is the proper one to use in the equilibrium equation, ? 11. Start with the following equation from the Navier-Stokes webpage

and use tensor notation to show how to get to

12. Suppose you have a Green strain tensor and a Second Piola-Kirchhoff stress tensor that are

You can tell right away that this must be an anisotropic (not isotropic) material because the global X,Y,Z coordinate system corresponds to the principal orientation of the strain tensor, but not the stress tensor. What is the angle (in degrees) between the principal directions of the two tensors?

13. Given the principal values of a Green strain tensor:

What must ??? be equal to if the material is incompressible?

14. Given the constitutive law: where C is a constant, figure out what the engineering stress-strain equation would be for uniaxial tension. Make a graph of it for C = 1.0 to see how straight it is. (the graph doesnt have to be elaborate)

15. Given that the data below came from a 10Hz test, estimate value of tan() at 60C and 1,000Hz?

0.0000.1000.2000.3000.4000.5000.6000.7000.8000.9001.000-60-40-20020406080100)Temperature ( C)tan()