Indentation size effects in polymers and related rotation gradients
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Svetoslav Nikolov Max-Planck-Institut für Eisenforschung, 40237 Düsseldorf, Germany (Received 13 December 2006; accepted 6 March 2006)
Similar to metals, the hardness of many polymers increases with decreasing indentation depths at depth ranges from several microns down to several nanometers. While for metals such phenomena are commonly attributed to geometrically necessary dislocation densities, such an explanation cannot be applied to polymers. To provide a micromechanically motivated model for the indentation size effect in polymers, here we propose an elasto-plastic extension of the higher order elasticity model recently developed by the authors. In this model, size effects in polymers (as well as in nematic liquid crystals) are related to Frank elasticity arising from bending distortions of the polymer chains and their interactions. On the basis of this theory, we derive a simple model for indentation size effects in polymers. Unlike other models, our model includes only elastic size effects due to rotational gradients. It is shown that the proposed model can explain the experimentally observed size effects in polymers. Together with the existing experimental data mentioned here, new experimental data for silicon rubber are also presented and discussed.
I. INTRODUCTION
Indentation experiments have been very useful for characterizing size-dependent deformation at the micron scale and below, where the size-dependence manifests itself by an indentation-depth-dependent hardness. For metals at the micron scale, the size-dependent deformation mechanisms are usually related to plastic deformation and the notion of geometrically necessary dislocations increasing flow stress and hardness values.1,2 At smaller indentation depths below 200 nm, the roughness of the surface3,4 and other surface effects may, however, alter the deformation mechanisms and consequently influence the hardness.3,4 Apart from metals, size effects have also been observed in polymeric materials5–9 where the notion of geometrically necessary dislocations cannot be applied. Some authors attributed these size effects to structural differences in depth5,7 while for plastic deformation in glassy polymers, Lam and Chong10 proposed a kink pair model based on geometrically necessary kinks with density proportional to the induced strain gradients. The model is similar to other developments suggested in
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Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2007.0197 1662 J. Mater. Res., Vol. 22, No. 6, Jun 2007 http://journals.cambridge.org Downloaded: 13 Nov 2014
strain gradient plasticity of metals and the notion of geometrically necessary dislocations. Unlike metals, however, where elastic strains are small and often negligible, in polymers the elastic strain and elastic deformation energy is not small compared to the inelastic deformation energy. Recent microbeam experiments have also shown an increased (normalized) bending stiffness with decreasing thickness,11,12 which indicate that the sizedependent defor
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