## Resonance measurements of viscoelasticity

### Introduction

Resonance principle is used in many physical measurements (e.g. NMR, EPR, torsional oscillations and many others). Measurements based on system resonance are, as a rule, more sensitive and often more precise than direct methods. However, resonance methods are applicable only if the measured systems do resonate. This fact represents a major issue in terms of the applicability of the resonance principle in the measurement of viscoelasticity.

According to the generally accepted definition (http://en.wikipedia.org/wiki/Viscoelasticity) viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. In other words, mechanical behavior of viscoelastic systems is given by the combination of elastic and viscose interactions. Solutions of the corresponding constitutive differential equations are principally aperiodic. Consequently, no resonance occurs in these systems.

A simple way to avoid this limitation is to add to the tested viscoelastic object an additional mass (inertial body). The solution of the constitutive differential equation of the overall system (tested object plus inertial body) thus becomes periodical.

The simple way to avoid this limitation may consist in a connection of tested viscoelastic object with additional mass (inertial body), so as the solution of the constitutive differential equation of overall system (object plus inertial body) is periodical. The system (tested object plus inertial body) resonates if the overall mass of the system is sufficiently high.

This principle is already applied in the measurement of viscoelasticity in torsion loading. However, the methodology is applicable more generally, among others for the measurement of viscoelasticity in tensile or bending loading.

### Theory

#### Kelvin body (Voigt model) with additional inertial mass

Fig.1 Voigt model (a) and Voigt model with additional inertial mass (b), rheological schemes

##### Force equilibrium of Voigt model with additional inertial mass

(1)

##### Equation of motion of Voigt model with additional inertial mass

(2)

where *H *is the Hooke coefficient, *L* is the deformation, *N* is the Newton coefficient, *M* is the mass of inertial body.

In case of a uniform rod in tensile loading it also holds

(3)

where *σ* is the stress, *ε* is the strain, is the stress, ,* E* is the Young modulus, *η _{N} *is the viscosity,

*L*is the length of the rod

_{0}*A*is the cross section area of the rod.

_{0}The following also holds

(4)

where *σ* is the stress, *ε* is the strain, is the stress, ,* E* is the Young modulus, *η _{N} *is the viscosity,

*L*is the length of the rod

_{0}*A*is the cross section area of the rod.

_{0}The following also holds

(5)/(6)

##### Self-oscillations

Laplace transform of (2) leads to the eq. (6)

(7)

Suppose that energy is inserted into the system using a short impulse of force Δ*F*.

For the impulse Δ*F *of the force it holds in Laplace transform

(8)

For deformation in Laplace transform it holds

(9)

Resonance occurs only if this holds:

(10)

For the deformation in time domain it holds in this case

(11)

where

(12)

a

(13)

##### Complex stiffness and complex modulus of Kelvin body

###### Complex stiffness

For the complex stiffness *S(iω)* of Voigt model (Kelvin body) it holds

(14)

where *i* is the complex unit.

###### Complex modulus

Complex modulus *E(iω)* of Voigt model is:

(15)

The real part of complex modulus corresponds to the storage modulus; the imaginary part of complex modulus corresponds to the loss modulus .

##### Loss factor

Loss factor of Voigt´s model is:

(16)

where *φ* is the phase shift between real and imaginary parts.

##### Inverse problem solution

The inverse problem solution consists in the determination of parameters *H* and *N* of the Kelvin body on the basis of resonance measurements. The primary results of resonance measurements are damping oscillation curves according to the equation (10). In the first step of the analysis, the frequency *ω* and coefficient *α *are determined. Then the parameters *H *and *N* are calculated using the formulae (11) and (12).

It is also possible to obtain the stiffness and complex modulus by using the formulae (13) and (14).

Fig.2 Kelvin body with additional inertial mass – tensile loading

#### Real viscoelasic body with additional inertial mass

##### Equation of motion

For any linear system it holds

(17)

where the *y(t)* is output variable, the *x(t)* is input variable, *a* and *b* are constant coefficients, *i *and *j* are degree of derivatives.

For real linearly viscoelastic body with additional inertial mass it holds:

(18)

where *L* is the deformation, *F* is the force.

Complex stiffness is thus

(19)

Resonances occur at local minims of the function (18).

Practically, only one single minimum of equation (18) can be found, since the impact of other minims on deformation is negligible.

It is thus possible to proceed in the similar way as in case of Kelvin body. Nevertheless, the resonance frequency depends on mass, and consequently the coefficients *H* and *N* may also be frequency dependent.

Using different masses of the inertial body in measurements enables to determine the dependency of the coefficients *H* and *N* on frequency.

##### Principle of the arrangement of the apparatuses for tensile measurements

The arrangement of apparatuses according to Fig. 2 is not sufficiently universal. The main limitation is caused by gravity force of the inertial body that “preloaded” sample that is measured.

The arrangement according to Fig. 3 enables to solve this problem.

Fig.3 Example of measurement with compensation of the gravity of the inertial body

In arrangement according to Fig. 3, the inertial body is placed above the sample and it is lifted by a compensation spring. The preloading of the sample is given by the deformation of the spring and the spring constant. This construction enables to independently select the pre-straining of the sample and the frequency of oscillations.

##### Applied formulae

For frequency it holds

(20)

where *H _{C}* is the spring constant (spring stiffness).

For damping of oscillations it holds

(21)

For complex stiffness it holds:

(22)

For preloading it holds:

(23)

kde *F _{0}* je předpětí vzorku,

*L*je klidová deformace pružiny,

_{0}*g*je gravitační zrychlení.

where *F _{0}* is the preloading,

*L*is the spring deformation in steady state,

_{0}*g*is the acceleration of gravity.

### Conclusions

The resonance method of viscoelasticity measurements represents alternative to the direct method. The direct method consists in comparison of amplitude and phase of harmonic courses of stress and strain. The meters based on direct method are usually called dynamic mechanical analyzers (DMA).

Compare resonance meters with DMA, the resonance meters are technically less complicated and also more users friendly. Above it, resonance meters are less expensive. The resonance principle enables more sensitive and more precise measurements.