·Table of Contents ·Materials Characterization and testing | Method of identification of viscoelastic materials with a stress relaxation functionRudnitsky V. A., Kren A. P., Tsarik S. VInstitute of Applied Physics, National Academy of Sciences, Minsk, Belarus Contact |
Fig 1: Experimental arrangement |
(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.
Fig 4: The dependencies of elastic modulus E_{0} 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.
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