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Tensorial modeling of an
oscillating and cavitating
microshell used as a contrast
agent
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
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.
Mathematical model
• The transient Cauchy Eq. With the stresses
Mathematical model
• For a Mooney Rivlin material we have the
elastic potential.
Cauchy’s Eq. can be integrated as:
Mathematical model
• At the same time we have the R-P Eq.
Mathematical model
• The stresses at both inside as a gas and
outside of the shell as a liquid, must stand
equilibrium.
Mathematical model
• With both equations and the balance
equations we have:
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• Introducing the nondimensional variables
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• We can rewrite the dimensionless equation as
Mathematical model
• For the last equation the initial conditions are
• And the dimensionless parameters are
Results
• For typical experimental physical values
Results
Results
=0.6, =0.4
=6, =4
 0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.1
0.0
0.1
0.2
0.3
0.4

0.5
0.6
0.7
Results
Results
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
Conclusions
• The main parameters P 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
Conclusions
• Further studies on the frequency and stability
of the equation should be done