A Finite Element Method for the Determination of Optimal Viscoelastic Material Properties from Indentation Tests of Polymer Film and Wire with Polymer Insulation

Abstract

The time-dependent behavior of bulk polymer film and wire with polymer insulation is studied using indentation. The indenter is displaced into the material at a constant rate and then held at a fixed indentation depth to monitor load relaxation. A finite element simulation of the experiment is performed; this analysis is parameterized in terms of the unknown shear compliance modeled as a Prony series. An optimization method is then presented to determine the unknown material parameters by minimizing the RMS error between the model and the experimental data. The method is demonstrated with poly (vinyl chloride) (PVC) films after thermal aging and pristine polyethylene sheet; excellent agreement between the model and the data is demonstrated. The method is also demonstrated to successfully characterize the material properties for the compression of a wire with PVC insulation; the resulting properties are then shown to adequately predict the crossed-cylinder indentation behavior of the same wire using a 3D finite element model. The chief benefit of the method is that an analytical solution method is not required for its implementation; as such, the optimization approach can be readily applied to the determination of material properties from arbitrarily complex experimental geometries.

Appendix

Appendix—Correction of Raw Data

Before performing material property optimization for the indentation data for PVC films, corrections to the experimental data set must be performed in certain cases. This is typically true for materials at long aging times that are relatively stiff. In these cases, it is difficult to precisely define the moment of zero indentation during the experimental setup. As such, there can be a region of the data in which indenter displacement occurs but the force changes only slightly; after a sufficient amount of displacement, the force increases rapidly as expected. If the finite element optimization method is applied to the data as collected, the obtained model results often will not fit the data very well. An example of this phenomenon is shown in Fig. 16 for the load portion of PVC film aged for 3 weeks at 100°C. Clearly, minimal contact between the indenter and the film was achieved for the first 0.6–0.8 s of the experiment.

Fig. 16

Comparison of data and ANSYS best fit for indentation force versus time for 0.019 in. PVC film aged 3 weeks at 100°C

In order to correct this feature, the classical film indentation model relationship between indenter force F and indentation depth δ is used [equation (6)]. This relationship is parabolic in nature and is valid for any value of δ; as such, this expression can be differentiated to find:

$$ F = {\left( {\frac{{\pi ER}} {h}} \right)}\delta ^{2} \to \frac{{dF}} {{d\delta }} = {\left( {\frac{{2\pi ER}} {h}} \right)}\delta $$

(12)

This equation describes a line that has a value of 0 when the indentation depth δ is 0. For the experiment using the DMA, the indenter is moved at a constant rate. Calling this constant rate as v = dδ/dt, the equation above can be rewritten in a more useful form in terms of time using the chain rule as:

$$ \frac{{{\text{d}}F}} {{{\text{d}}\delta }} = \frac{{{\text{d}}F}} {{{\text{d}}t}}\frac{{{\text{d}}t}} {{{\text{d}}\delta }} = {\left( {\frac{{2\,\pi \,E\,R}} {h}} \right)}\,\delta \quad \to \quad \frac{{{\text{d}}F}} {{{\text{d}}t}} = v^{2} \,{\left( {\frac{{2\,\pi \,E\,R}} {h}} \right)}\,t $$

(13)

where the substitution δ = vt has been made recognizing the constant indentation rate. Thus, the rate of change in indentation force with time (dF/dt) is linear and is equal to 0 at the moment the indentation begins (t = 0).

Based on this result, the time derivative of the force (dF/dt) is approximated as the ratio of the change in force (ΔF) for each step to the change in time (Δt) for each step and considered. The resulting data for ΔF/Δt for the PVC film aged 3 weeks is shown in Fig. 17. The data clearly becomes linear only in the later part of the experiment; a straight line is passed through the linear portion as shown and an effective start point of the experiment is identified. The values of time, force and indentation depth at this point are t _S  = 0.80 s, F _S  = −0.0701 N and δ _S  = 0.007 mm, respectively. The data is then adjusted to account for the new effective starting point as:

$$ t_{{{\text{new}}}} = t - t_{S} ;F_{{{\text{new}}}} = F - F_{S} ;\delta _{{{\text{new}}}} = \delta - \delta _{S} $$

(14)

where the values with the “new” subscript value are used in the ANSYS optimization. The ANSYS fit that is obtained for this data is shown in Fig. 18; it is clearly more representative of the data set. Modifications in this manner were performed for the PVC film data aged for 3, 4, 5 and 6 weeks. The PVC film data aged for shorter periods did not exhibit such difficulties and was used as is.

Fig. 17

Plot of the derivative of indentation force with respect to time for PVC film aged 3 weeks at 100°C

Fig. 18

Comparison of modified experimental data with the optimal finite element model result for PVC film aged 3 weeks at 100°C