FSI of viscoelastic plate

1. OUTLINE
1
3.
Results
2.
Study of
viscoelastic
splitter plate
behind the
cylinder
4.
Conclusions
and Future
Work
1.
Work
Presented in
APS I and II

2. 2
Work Presented in
APS-I and II
1

3. 3
Study of vortex-induced-vibration(VIV) of
a viscoelastic splitter plate behind a
cylinder
WORK PRESENTED IN APS-I

4.  Linear Viscoelastic models
4
WORK PRESENTED IN APS-I I
 Maxwell Body:
 Voigt Body:
 Standard Linear Solid:
η E
1 0 1
d d
p q q
dt dt
 
 
  
1 0 1
p , 0,
q q
E


  
1 0 1, 1
p 0,q E q 
  
1
1 0 1, 1
2 2
p , (1 )
E
q E q
E E


   

5.  Applications
 Literature Review
 Objectives of the present work
 Finite Strain Viscoelastic
5
Study of Viscoelastic splitter plate behind the cylinder
2

6. Snoring
Printing process
6
Auregan et al. 1995
Watanabe , 2002
FPCL, JHU
Phonation
Piezoelectric Energy
Harvesting
InTechOpen

7. • Chen et al. (2011). Analyzed the flutter . Either increasing the
structure-to-fluid ratio or decreasing dimensionless bending
stiffness ( )causes the system transits from periodic to
chaotic.
• Chen et al.(2014). Increase of either elastic or viscous component
of flag stabilizes the system.
• Tang et al. (2007) considered Kelvin-Voigt model , flutter amplitude
decreases with the increase of the material damping.
• Purohit et al. (2016). At lower flexibility forcing field of flowing field
dominates the vibration plate, as the plate stiffer plate response is
effected by elastic inertial force.
7
2 3
EI
u L


8. Based on the multiplicative decomposition of deformation gradient into elastic and inelastic part
In thermodynamic equilibrium spring of maxwell element is relaxed , ,hence
Stress
For equilibrium and non equilibrium parts of material neo-Hookean model has been implemented
8
Second Piola-Kirchho stress tensor, 2
is strain energy density function, E Lagrangian strain tensor, C right Cauchy-Green deforformation tensor
ij
ij ij
ij ij
W W
S
E C
W
 
 
 
2
2 /3 2 2 2 2 2 2 2 2 2 2 2 2
1 1 2 3 2 1 2 2 3 1 3 3 1 2 3
2 2 2
1 2 3
W= ( 3) ( 1 1ln( )
4
, shear and bulk modulus, , invarients as I , ,
, , is Eigen values (principal stretch ratios) of Green deform
i
i i
k
I J J
k I J I I I I

            
  

   
       
2 2 2
1 2 3
ation tensor Cij, J=  
2 2
1 2
1 1
Split of free energy, ( ) ( )
2 2
e eq neq e
E E
   
      
0
e
  ( )
eq 
  
2
The ansatz, ( ) ( ), elastic right Cauchy-Green tensor (loosly speaking strain E )
eq neq e e
C C C
    
1
2 +2F F = , F deformation gradient of dashpot in fig
eq neq T
eq neq i i i
e
S S S
C C C
 
  
  
  

9. 9
Results
3
 Free vibration of cantilever beam under time varying sinusoidal l
 Vortex induced vibration of viscoelastic splitter plate behind a cy

10. 10
 P1=7.125 N, P2=6.84 ( 50 nodes with 0.285 per node, p1 25, p2 24 nodes)
 L=10 m , f(t) = sin(0.2t), density 10 kg/m3
 Viscoelastic parameters
 Kneq, Keq are bulk modulus corresponding to E2 and E1
 Sneq, Seq are shear modulus for E2, E1 in N/m2
 Relaxation time (sec)
 A: tip (point A) vibration amplitude
 Standard linear solid
2
/ E
 


11. cas
e
Keq Kneq Seq Sneq R.
time
A
1 2333.33 2333.33 500 500 0.1 9
2 2333.33 2333.33 500 500 1 8
3 2333.33 2333.33 500 500 2 6.6
4 2333.33 2333.33 500 500 10 4.7
5 2333.33 2333.33 500 500 100 4.5
6 2333.33 233333.33 500 50000 0.1 2.85
7 2333.33 233333.33 500 50000 1 0.3
8 2333.33 233333.33 500 50000 10 .07
9 233333.33 2333.33 50000 500 1 0.5
Variation of amplitude with relaxation time

12. Variation of amplitude with relaxation time

13. cas
e
Keq Kneq Seq Sneq R.
time
A
1 2333.33 2333.33 500 500 0.1 9
2 2333.33 2333.33 500 500 1 8
3 2333.33 2333.33 500 500 2 6.6
4 2333.33 2333.33 500 500 10 4.7
5 2333.33 2333.33 500 500 100 4.5
6 2333.33 233333.33 500 50000 0.1 2.85
7 2333.33 233333.33 500 50000 1 0.3
8 2333.33 233333.33 500 50000 10 .07
9 233333.33 2333.33 50000 500 1 0.5
× 100
Effect of Kneq and Sneq on amplitude
Amplitude decreases with increase of
relaxation time

14. cas
e
Keq Kneq Seq Sneq R.
time
A
1 2333.33 2333.33 500 500 0.1 9
2 2333.33 2333.33 500 500 1 8
3 2333.33 2333.33 500 500 2 6.6
4 2333.33 2333.33 500 500 10 4.7
5 2333.33 2333.33 500 500 100 4.5
6 2333.33 233333.33 500 50000 0.1 2.85
7 2333.33 233333.33 500 50000 1 0.3
8 2333.33 233333.33 500 50000 10 .07
9 233333.33 2333.33 50000 500 1 0.5
× 100
Effect of Keq and Seq on amplitude

15.  Amplitude decreases with increase of modulus
 Structure vibrates with the forcing frequency
15

16. 16
case Keq Kneq Seq Seq R. time A
1 2333.33 2333.33 500 500 0.1 2.17
2 2333.33 2333.33 500 500 1 2.25
3 2333.33 2333.33 500 500 2 2.98
4 2333.33 2333.33 500 500 10 2.89
5 2333.33 2333.33 500 500 20 2.96
6 2333.33 2333.33 500 500 100 2.97
 Amplitude almost constant
 Structure vibrates with forcing frequency
 (w=0.2, f=.05)

17.  Time to achieve steady state decreases with increase of Relaxation time.
 ( relaxation time =100 has different behavior)
17

18.  Re=100,

18
case Keq Kneq Seq Seq Relaxation
time
A
1 2333.33 2333.33 500 500 0.1 1.02
2 2333.33 2333.33 500 500 1 0.18
3 2333.33 2333.33 500 500 2 0.53
4 2333.33 2333.33 500 500 10 0.6
5 2333.33 2333.33 500 500 100 0.9
6 233.33 233.33 50 50 1 chaotic
7 2333.33 233333.33 500 50000 1 No vibration
8 2333.33 23333.33 500 5000 1 No vibration
/ 10, 1400, 0.4
s f E
  
  
Amplitude is not monotonic with Modulus as here fluid and
structure both play role

19. 19
tau=2
tau=100
tau=1
tau=0.1
Time for steady state is not
monotonic, increases and then
tau=10

20. 20

21. Symmetry breaking bifurcation which oscillates in upper or lower part of cylinder wake
21
Case 6, chaotic Case 7,

22.  Amplitude increases with increase of relaxation time
22
case Keq Kneq Seq Sneq R. time A
1 2333.33 2333.33 500 500 0.1 .017
2 2333.33 2333.33 500 500 0.5 0.00005
3 2333.33 2333.33 500 500 1 0.19
4 2333.33 2333.33 500 500 2 .47
5 2333.33 2333.33 500 500 10 0.00005
6 2333.33 2333.33 500 500 20 0.6
7 2333.33 2333.33 500 500 100 0.7
8 233.33 233.33 50 50 1 chaotic
9 233.33 23333.33 50 5000 1 No vibration
10 2333.33 233333.33 500 50000 1 No vibration
11 4666.66 4666.66 1000 1000 1 No vibration
12 2333.33 23333.33 500 5000 1 No vibration
13 2333.33 4666.66 500 1000 1 0.48

23. 23
Steady state time increases with relaxation then further increasing the
relaxation time steady state time decreases

24. 24
Damped to small amplitude vibration

25. 25
Conclusions and Future Work
5

26. For constant modulus
 amplitude decreases with increase of viscosity
 Steady state time decreases with decrease of viscosity
 Steady state time increases with relaxation then further increasing the
relaxation time steady state time decreases
 Amplitude increases with increase of relaxation time
26