Determination of the elastic properties of multi-layered foils by the four-point micro-bending test

Abstract

A new test method for the determination of the elastic properties of thin multi-layered foils is introduced, where the geometry of the test is scaled down from the macroscopic four-point bending test. The mechanical test makes use of a novel, multi-purpose mechanical tester with nanometre displacement capability in three orthogonal directions, which allows the elimination of uncertainties associated with misalignment or twist of the jig. The particularly small dimensions of the test specimens prevent the use of the classical expressions of beam theory. To account for deflections of the same magnitude as the thickness of the beam, the analysis proceeds in three distinct steps. Firstly, the limits of applicability of the elementary beam theory are shown, the emphasis being put on large deformations. Secondly, a novel approach is introduced that deals properly with large deformations, making use of elliptic integrals to calculate the deflection of the foil. Thirdly, it is shown that the effects of frictional forces at the fulcrums must be included to describe experimental data properly while the range of experimental displacements allowed is extended. The analytical model is shown to compare favourably with the results of a 2D finite element model. As a result, a set of master curves are calculated and used to deduce the composite Young's modulus of the foil. Experimental data collected on a range of titanium/titanium nitride nano-composite films are further exploited following these methods, and shown to agree well with theoretical results derived by the classical laminate theory, using macroscopic bulk properties of the constituents. Thereby, it is demonstrated that small length scale metrology does not necessarily imply small deformation regimes.

Acknowledgements

This research was carried out as part of the ‘Materials Metrology’ programme funded by the National Measurement Systems Directorate of the UK Department of Trade and Industry, and was also supported by NPL's Strategic Research Programme. The authors wish to acknowledge R. Morrell and L. Neil McCartney for fruitful discussions, and M.G. Brookes and P.J. Kelly, University of Salford (UK), for providing the materials used in this study.