My advisor wants me to work on the fracture model with two plates of different properties containing a non-viscous fluid first similar to Korneev et al 2014
So I have this Materials for us that may help
Krauklis, P.,V., 1962. About some low frequency oscillations of a liquid layer in elastic medium,
Prikladnaya matematika i mekhanika, 26, 1111-1115. (in Russian)
So we will derive the general dispersion equation for that and that of Low frequency symmetric mode as in Krauklis wave in trilayer but the mechanical properties of the plates will be different.
The plates should be Aluminum and Lucite
From the dispersion equation, we will (1)estimate the phase velocity and attenuation
(2) Vary the thickness of elastic plates as indicated by Korneev et al 2014 ( Krauklis waves in a trilayer) with repect to phase velocity and attenuation
(3) Vary the thickness of the fluid layer ( Keeping that of the elastic layer constant) with respect to the phase velocity and attnuation
( 3) Compare our Phase velocity results with those of Korneev et al 2014 my superimposing his phase velocity solution with our at a particular plate thickness and fluid layer thickness.
The second paper summary
[8/24/2016 10:47:31 AM] Pavel Uglich: Well. this paper contains all dispersion equations and mechanical parameters. Comparison is possible.
The authors compare their results with the previous existing dispersion equations for Krauklus wave
Since symmetry is embedded in the solution, we need to satisfy boundary conditions on
just one interface. The problem has four unknowns Aj , Bj ( j =1,2) , with four equations at
the boundary using two components for both stress and displacement. I obtain the dispersion
equation for symmetric modes by finding values of Vf for which the determinant of the
system is zero. Using z = h / 2 , we can obtain (after some algebra) the equation
[8/24/2016 10:49:22 AM] isa ali: The plots should be in log log scale
[8/24/2016 10:50:44 AM] isa ali: the phase velocity , attenuation and frequency scale should be the same.
[8/24/2016 10:52:02 AM] Pavel Uglich: Plots in log scale are available in Maple, it' not hard.
[8/24/2016 11:15:30 AM] isa ali: Sorry for the delay in my response
[8/24/2016 11:16:53 AM] isa ali: So our first task is to verify the exact solution of the dispersion equation and the numerical solutions for phase velocity and attenuation
[8/24/2016 11:17:36 AM] isa ali: Use the range of frequency the used in the article I attached earlier
[8/24/2016 11:21:57 AM] isa ali: I will also like us to look closely at this article and how the dispersion equation for low frequency symmetric mode was derived for a Thin Elastic Regime
Use this model isa ali: and look also at the boundary conditions
For symmetry purpose
Use some notation like equation 42 and 43 to simplify the dispersion equation
That is the dispersion equation they obttained
isa ali: He used some notation to simply it
so that the determiinant will not have some element like a11, a12.....
Well, It's not hard. They use asymptotic expression for tanh to simplify dispersion equations.
To find the dispersion curve for all the possible modes of propagation in the Fracture model in the presence of a viscous fluid
Then simplify the equation to obtain the dispersion equation of the K-wave or the trapped mode in the fluid layer when the thickness of the two elastic layers are finite
Just as in Krauklis wave in trilayer by korneev et at 2014
isa ali: et al. 2014
: Evaluate the phase velocity and attenuation from the dispersion equation
: Compare their exact and numerical solutions for both phase velocity and attenuation
Compare the results of the phase velocity to that of Korneev et al 2014 , Korneev 2008 and Nikikin et al 2016 at a particular thickness of the elastic layer
Compare our attenuation results with those of Korneev et al 2008 and Nikitin et al. 2016
: Get the phase velocity at various plate thicknss like this
To see how it will be affected in the presence of a viscous fluid
Do same for Attenuation results
: At a particular plate thickness , vary the aperture of the fracture or fluid layer just as is done in this
Consider similar values on x- and y axis for all plots please)
Compare the phase velocity and attenuation of K-waves in the presence various viscous fluid just like we did in the last project at certain plate thickness and at a particular aperture