1. Measurement of Dynamic Properties of Viscoelastic Materials
Victor M. Kulik1, Boris N. Semenov1, Andrey V. Boiko2,
Basel M. Seoudi 3, H. H. Chun4 and Inwon Lee4
1 Institute of Thermophysics,
Russian Academy of Sciences, Novosibirsk, , Russia
2 Institute of Theoretical and Applied Mechanics,
3 Department of Naval Architecture and Ocean Engineering,
Pusan National University, Busan , Korea
4 Advanced Ship Engineering Research Center (ASERC),

2. Field of application Frequency: 10 – 10000 Hz Relative deformation
MPa
T, 0C 10
0.1
- 50
100
Frequency: 10 – Hz
Relative deformation
(%) = (10-4 – 5)%
Temperature range is determined by
specification of piezoelectric
accelerometers
Modulus of elasticity range is determined by
possibilities of sample preparation

3. Schematic diagram of measurement setup
Exciter Br ü el
& Kj æ r
4808 H y
Function
Generator
Agilent
33220A
Power
Amplifier B K j
2721 4
Vibration Amplitude
Digital
Multimeter
34401A
Phase Difference
Universal Counter
53131A
Accelerometer
Conditioning
2693 3 5 1 2 –
ase P
late ; S
ample L
oad M
ass (M)
Reference Accelerometer; 5
Load Accelerometer
PC &
LabVIEW
S/W
GPIB
Interface

4. Photos of the measurement setup; (a) whole system, (b) various samples, (c) reference and load accelerometers, (d) close-up of a sample

5. Mathematical model z Governing equations r m, , E,tg,  Displacement r
m, , E,tg, 
aexp(wt)
Zaexp(w-)
Нагрузка М
sample
Displacement
(r, z) - radial displacement
(r,z) - axial displacement

6. Mathematical model Boundary conditions
a) Absence of radial displacements at bonded surfaces
 = 0 at z = 0,  = 0 at z = H.
b) Axial displacements at the lower and upper edges
 = aet at z = 0
 = Zae(t -) at z = H.
c) Absence of strains at the side surface (at r = R) rz = 0, i.e.
and rr = 0, i.e.
Internal strain at the upper edge is equal to pressure developed by the movement of the finite load mass M

7. Solution procedure
A pseudospectral approximation of the wave equations was used.
A grid was set up based on Chebyshev Gauss–Lobatto knots independently in z and r, thereby producing so-called tensor product grid. Then, a linear coordinate transformation mapped the problem from the polynomial domain to the physical domain
Our tests showed that as the number of knots in each direction is enlarged from 12 to 20; the approximated values of E and m change less than 1% in the region of interest that is within an estimated experimental error in measuring the amplitudes and phase shifts of the axial displacement.
Then, the original system of equations is approximated by the following matrix equation
Boundary conditions were applied explicitly by changing the corresponding rows in the left and right-hand sides of the equations to make inhomogeneous problems, which are then easily solved by the left matrix division. Final approximations of E and tg were obtained by Gauss-Newton iterations of the obtained solutions to satisfy the compatibility condition.

8. Sample preparation Silicon rubber S2 (Dow Corning Inc
Sample preparation Silicon rubber S2 (Dow Corning Inc.) = Polydimethylsiloxane : catalyzing additive = 9 : 1

9. Gluing of samples

10. Results for various mass ratio
Ratio of amplitudes
Phase lag
M/m ● ♦ ■ ▲
Modulus of elasticity
Loss tangent

11. Results for various sample sizes
Phase lag
Ratio of amplitudes
D=H, mm ● ■ ▼  
Modulus of elasticity
Loss tangent

12. Dependencies of viscoelastic properties on relative deformation
f, Hz
● - 45
♦ - 180
■ - 800
Modulus of elasticity
Loss tangent
~ U/f2
For example, from 30 Hz to 3 kHz
frequency changes in 102 →  changes in 104
It is not possible to change Uvibr in 104 times
3 modes of operation
- constant displacement, U/f2
constant velocity, U/f
- constant acceleration, U

13. Dynamic deformation of cylindrical sample M/m = 5, D/H =1 ,  = 0
Dynamic deformation of cylindrical sample M/m = 5, D/H =1 ,  = 0.495, tg = 0.1
Parametrical resonance f=143 Hz
Natural resonance f = 794 Hz

14. CONCLUSIONS
An improved method to measure the dynamic viscoelastic properties of elastomers is proposed.
No special equipment other than the ordinary vibration measurement apparatus is required.
The recorded magnitude ratio and the phase difference between the load disc and the base plate vibrations represent the axial, dynamic deformation of the sample. The data are sufficient to obtain the dynamic properties of the sample, oscillation properties of vibration exciter, sensitivity of gauges having no effect on the calculation results.
For accurate calculation of the properties, a two-dimensional model of cylindrical sample deformation was used. Therefore, a form factor, which takes into account the sample sizes in one-dimensional models, is not required in this method.
Typical measurement of the viscoelastic properties of a silicone rubber Silastic® S2 were measured over the frequency range from 10 Hz to 3 kHz under the action of wide region of deformation (ratio of vibration magnitude to sample thickness was from 10-4 % to 5%).
Modulus of elasticity and loss tangent fall on single curves when the ratio of load mass to sample mass and dimensions of the sample changed in a very wide region.