1. Tensorial modeling of an oscillating and cavitating microshell used as a contrast agent

2. Objectives Formulate an equation for the shell with tensorial analysis using the Mooney Rivlin hyperelastic model. Determine the parametric relations Solve the equation to predict the behaviour of the system

3. Mathematical model Using the Cauchy Stress equation together with the Navier-Stokes equations with their conditions and taking into account spherical symmetry for a thin microshell.

4. Mathematical model The transient Cauchy Eq. With the stresses

5. Mathematical model For a Mooney Rivlin material we have the elastic potential.

6. Cauchy’s Eq. can be integrated as:

7. Mathematical model At the same time we have the R-P Eq.

8. Mathematical model The stresses at both inside as a gas and outside of the shell as a liquid, must stand equilibrium.

9. Mathematical model With both equations and the balance equations we have:

10. Mathematical model Introducing the nondimensional variables

11. Mathematical model We can rewrite the dimensionless equation as

12. Mathematical model For the last equation the initial conditions are And the dimensionless parameters are

13. Results For typical experimental physical values

14. Results

18. Conclusions We obtained a simple model for a Money Rivlin shell The thin shell approach led to a very close interval in the parameters, which showed two modes of collapse. The violent collapse The ever growing collapse, we suppose an elastic response from the shell deformation

19. Conclusions The main parameters  and  A showed to be the main drivers of the collapse however the elastic parameters can shorten or prolong the collapse The linearized equation shows this competence

20. Conclusions Further studies on the frequency and stability of the equation should be done