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Method of identification of viscoelastic materials with a stress relaxation function Rudnitsky V. A., Kren A. P., Tsarik S. V Institute of Applied Physics, National Academy of Sciences, Minsk, Belarus Contact

INTRODUCTION

The present paper is concerned with the use of the dynamic indentation for the determination of polymeric meterial's time-dependent function, which characterizes its viscoelastic properties. The method includes the impact loading of the material tested by a rigid indenter, recording of the indenter's velocity during impact, converting the analog signal to a digital one and computer processing of the data. The apparatus of the contact dynamic method of indentation described [1] consists in recording in digital form identer's velocity V(t) while active stage of test impact by indenter dropping a height, and next processing the data V(t) to obtain conventional characteristics of material tested or its rheological dependencies. Using low impact energy the dynamic indentation method can be used to test both a material and small details and products from polymers and rubbers, i.e. to carry out incoming and outgoing inspection.
The material's viscoelastic properties can be determined from the solution of equation obtained on the basis of phenomenological models, consisting of combination of mass, springs and dashpots [2]. In practice using of these models leads to complicated differential equation with constant coefficients, which sometimes has no physical explanations.
The validity of the determination of viscoelastic properties and identification procedure depends on the correct choice of influencing factors, and its measurement accuracy. The hardness and viscous factor are the most convention characteristic of the viscoelastic material. The durometers commonly used for testing the viscoelastic materials (polymers or rubber) are Shore hardness meters. These ones determine the indentation depth in the material caused by a spring loaded rigid indenter. The rigidity is measured in Shore's units. Though this process of material properties determination is simple, it doesn't allow to evaluate adequately quality of the material due to the absence of the viscous data. More informative and precise are the dynamic methods. The contact dynamic method of indentation that is discussed in this paper belongs to this category.

EXPERIMENTAL APPARATUS AND PROCEDURE

The schematic drawing of the experimental arrangement of the dynamic indentation test is shown in Fig.1. The apparatus consists of impact device including mainly the tungsten carbide indenter with spherical tip attached to rotating lever, permanent magnet mounted in the indenter and stationary inductive coil.

Fig 1: Experimental arrangement


The initial data about the motion of the indenter during the impact are the analog voltage signal produced by inductive coil. Taking into account the proportionality of inductive coil voltage to indenter's velocity during indentation into material tested one can obtain the velocity-time dependence V(t). Using this curve one can measure material resilience as follows [3]

(1)

where V_imp and V_reb are indenter's velocities at the impact and rebound accordingly.
Moreover the V(t) curve allows to evaluate material's stifness by measuring impact active stage time. Though it is rather difficult to determine exactly the beginning of the impact due to flatness of the curve V(t) in this range.
Numerical differentiation and integration of the curve V(t) gives the current contact force F(t) and depth of the intrusion a(t) correspondingly. The dependence of the contact force versus depth of indentation F(a) can be extracted from F(t) and a(t) curves. The typical curves V(t), a(t) and F(t) are presented in Fig. 2.

Fig 2: Typical depth indentation (curve #1), indenter velocity (curve #2), contact force (curve #3) curves obtained on rubber sample.

The dependence of F(a) is shown in Fig.3.

Fig 3: The dependence of contact force versus depth of indentation.

Firstly the force - displacement curve allows to obtain the dynamic stiffness as the slope angle of the curve at its beginning

(2)

Moreover our investigation shows that the curve F(a) gives more information about materials viscoelastic properties and their changes during impact loading. It is useful to consider the dynamic force-displacement dependence in more detail.
The mechanical contact between a rigid spherical indenter and elastic half-space (known as the Hertz elastic problem) is described as follows[4]

(3)

where E^* is reduced elastic modulus, which is the constant for a purely elastic materials, R is radius of the indenter tip.
A tentative solution of the viscoelastic problem is given by replacing the elastic modulus in Eq 3 by the corresponding integral viscoelastic operator, as it was made in [5]

(4)

where the equation kernel E'(t) is relaxation function of the tested material, which characterizes both elastic and viscous material properties.
The numerical determination of the relaxation function E'(t) is rather complex and difficult problem because it depends upon the approximation functions applied. To overcome this problem it is propose to determine the section modulus E(t) defined as the ratio of the current contact force F(t) to the product of and a ^3/2(t).
Taking into account the Eq 3 the section modulus E(t) is written as

(5)

Section modulus E(t) is calculated for all pairs F(t) and for each time point with sampling 2 ms during the active stage of the impact indentation test.

EXPERIMENTAL

The procedure described above was used to determine the section modulus E(t) of polyizoprene rubber samples with different percentage of plastisizers. The experiments were carried out using rubber samples with Shore hardness 76 and 52 HS with different resiliences R 20%, 40% and 30%, 60%, respectively. The samples were disk shaped with diameter 60 mm and thickness 6 mm. Indenter was made from tungsten carbide. The tip of the indenter was spherical with radius R=1.25 mm. The impact energy is approximately equal to 1 mJ.
Fig. 4 shows the value of elastic modulus E₀ (curves #5, 6) under static test and change of the section modulus E (curves #1, 2, 3, 4) at dynamic loading versus normalized depth of the indentation. In this figure straight line is elastic modulus E₀ under static loading. The section modulus E is characterized by the raise at the beginning of the impact and gradual decrease to the end of the active stage of the impact.

Fig 4: The dependencies of elastic modulus E0 under static test and change of the section modulus E at dynamic loading versus normalized depth of the indentation. (1- (76HS, R20%), 2- (76HS, R40%), 3- (52HS, R30%) and 4- (52HS, R60%) Shore's units and rebound resilience, respectively, 5- static modulus for rubber samples with 76HS, 6- for 52HS)

As can see in Fig. 4 section modulus value and its change considerably differs from the static one. For the tested rubber samples the static elastic modulus practically is constant value, but the dynamic section one is decreased very strong approaching to the static one. The difference in values between elastic modulus and static one can be explained by considerably influence of strain rate history while the impact test. As a result the dynamic section modulus allows to identify the mechanical properties of the polymeric products during its manufacture and operation.

CONCLUSION

Results obtained by the present study can be established as follows:
1. The dynamic indentation apparatus for testing of properties viscoelastic material and products from this material was established.
2. The method of evaluation of viscoelastic properties by section dynamic modulus was proposed.
3. It is shown experimentally that section modulus determined as a current slope of curve F versus product 3R^1/2a ^3/2/4 exceed static modulus by gradually decrease to the end of active stage of the impact on whole its way.

REFERENCES

1. V. A. Rudnitsky and S. V. Tsarik, The proceedings of the 2nd International Conference on Computer Methods and Inverse Problems in Nondestructive Testing and Diagnostics, Minsk (1998).
2. J. D. Ferry, Viscoelastic Properties of Polymers, New York (1961).
3. M. M. Reznikovsky and A. I. Lukchomskaya, The mechanical testing of rubbers, Moscow (1968). (In Russian)
4. K. L. Johnson, Contact mechanics, Cambridge University Press (1987).
5. E. H. Lee, J. R. M. Radok, Trans. ASME, 82, (1960).

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