Modeling for Dynamic Length Metrology in Accurate Manufacture

Accuracy is a prerequisite for modern manufacture, and length metrology (also commonly referred to as dimensional or geometrical metrology) is the tool to control accuracy. Length measurements must always refer to 20 °C, zero force and 50 %RH (in case of polymer parts). Polymers are relatively less stable materials as compared to metals and ambient conditions highly affect their dimensions and hence polymer parts show shrinkage and creep behavior at room temperature due to their low melting point and weak atomic bonding compared to metals.

Consequently, the traditional method of dimensional validation of polymer parts with micrometer level accuracy is performed by measurement of the so-called reference dimension in a laboratory with controlled environment long time after production because the parts need time to “settle” (which for some polymer materials takes weeks).

However, in the manufacturing industry there is an increasing push for measurements being performed already soon after production in a non-controlled environment in order to reduce the waiting time for quality control and the cost for equipment in a control laboratory, - but without compromising the level of accuracy.

The target of the present PhD study is thus to provide a procedure to measure the dimension of specific polymer parts in the production with an uncertainty less than 10 μm. This is done by firstly classifying the influencing parameters on dimension of polymer parts and not surprisingly the performed literature review shows that temperature, moisture, shrinkage and probe force are the main factors affecting dimensions and consequently also the uncertainty of the measurements.

The relation between dimension and temperature is described by the Thermal Expansion Coefficient (CTE). The wide range of the CTE values often encountered in polymer manufacturers’ datasheets as well as the 3D nature of the temperature field in the part right after production leads to an increase in the uncertainty beyond the above mentioned aimed value of 10 μm.

This means that simple 1D based compensation for the thermal effect on dimension is typically not possible in a real production environment. Depending on the case at hand, the actual 3D temperature field in production must be reconstructed based on a very limited number of temperature measurements at the surface of the part and subsequently be used in a 3D thermomechanical calculation of the dimensional changes when cooling the part down to the reference state of a uniform temperature field of 20 °C.

This however calls for relatively heavy 3D FE-analyses, which make this procedure not suited for in-line corrections in production. If however a number of these 3D FE analyses are made a priori with proper variations of boundary conditions, simple correction formulae suitable for in-line corrections can be constructed.

Two examples of this procedure are presented in the thesis. In the first, a steel gauge block is measured (a limited number of temperatures at the surface and one measurement of the length) and it turns out that this case is quite close to 1D such that the classical compensation procedure yields almost the same accuracy as the 3D thermomechanical analysis.

In the second case, however, a LEGO brick is measured while sitting in a fixture and this case is by no means 1D so here the full-blown 3D thermomechanical analysis really shows its potential. Apart from the thermal effect, the dimension of the polymer part is also influenced by the variation in the relative humidity of the ambient.

The water molecules diffuse to the molecular structure of the polymer and this disorders the atomic bonds and consequently exhibits an effect on physical properties and dimension. To quantify this, an experimental study was carried out to obtain the moisture expansion coefficient (CME) which (similarly to the CTE) provides a measure of the corresponding relative length variation due to change in concentration content.

The obtained values for the CME were then subsequently used in a numerical simulation of the effect of moisture on final dimensions. Moreover, as mentioned before the injection moulded polymer parts experience long-term shrinkage from the time of production and several weeks onwards. This phenomenon which most likely is attributed creep was described numerically by a simple Kelvin-Voigt model based on measurements over a period of 60 days from the time of production.

The measurement equipment (force probes) furthermore can deform the polymer part and hence influence the measured length - even though the forces are very small they still have to be considered since their contribution to the measured length might be comparable to the desired accuracy. Hence, classical Hertzian contact theory was used to suggest an analytical solution for the compensation of the effect of probe forces on dimension.

To support this, corresponding numerical modelling as well as experimental investigations were carried out. As a final part of the PhD project and to make the picture complete, the thermal stability of tools used in e. g. injection moulding or subsequent machining of metallic parts was analyzed and a model for predicting and visualizing the various heat fluxes in an applied tool was proposed.

The developed simple expression for compensating for all the effects combined was finally applied to two cases in order to predict the reference dimension for final validation: 1) Measurement of polymer parts 4 days after production and 2) Measurement of polymer parts right after production. The results showed that the proposed Dynamic Length Metrology (DLM) algorithm was capable of predicting the reference dimension with an expanded uncertainty of 10 μm.