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Is Strain Gradient Elasticity Relevant for Nanotechnologies?

Determination of Strain Gradient Elasticity Constants for Various Metals, Semiconductors, Silica, Polymers and the (Ir) relevance for Nanotechnologies

Strain gradient elasticity is often considered to be a suitable alternative to size-independent classical elasticity to, at least partially, capture elastic size-effects at the nanoscale. In the attached pre-print, borrowing methods from statistical mechanics, we present mathematical derivations that relate the strain-gradient material constants to atomic displacement correlations in a molecular dynamics computational ensemble. Using the developed relations and numerical atomistic calculations, the dynamic strain gradient constants have been explicitly determined for some representative semiconductor, metallic, amorphous and polymeric materials. This method has the distinct advantage that amorphous materials can be tackled in a straightforward manner. For crystalline materials we also employ and compare results from both empirical and ab-initio based lattice dynamics. Apart from carrying out a systematic tabulation of the relevant material parameters for various materials, we also discuss certain subtleties of strain gradient elasticity, including: the paradox associated with the sign of the strain-gradient constants, physical reasons for low or high characteristic lengths scales associated with the strain-gradient constants, and finally the relevance (or the lack thereof) of strain-gradient elasticity for nanotechnologies.

From the results obtained for the dynamic strain gradient constants and associated length scales for the materials investigated, there seems to be a strong indication that classical elasticity is valid for most materials down to a lattice parameter and only breaks down for materials possessing a non-homogeneous microstructure like amorphous silica and polymers. Covalent semiconductors like Si however possess higher non-local length scales compared to metals which may be attributed to the shorter-ranged nature of inter-atomic forces in metals. The high non-locality in amorphous solids possessing an underlying inhomogenous microstructure possibly stems from a group of strongly bonded atoms behaving as a unit. Under such circumstances, parts of the material system may undergo considerable non-affine deformation and high moment stresses may result. Since crystalline materials are highly ordered, they very possibly undergo negligible non-affine deformations as a consequence of which the nonlocal elastic effects are unimportant for such systems. Finally, as far as crystalline systems are concerned, surface elastic effects are the dominant contributors to breakdown of classical continuum elasticity while nonlocal effects are generally negligible.

Please refer to the attached pre-print (accepted for publication by JMPS) for further details.

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Comments

Konstantin Volokh's picture

Interesting paper. I am curious, however, whether simple experimental evidence exists of the length-dependence in elasticity. I mean simple torsion, shear, indentation etc. The mentioned experiments, to the best of my knowledge, usually exhibit length-independent elastic deformations and length-dependent plastic deformations. It would be interesting to learn about simple mechanical tests showing a pronounced length-dependence of elastic deformations.

Thanks for your interest in the paper. One of the main results of our paper is that size-effects due to nonlocal elasticity are so small in crystalline materials that they will elude experimental detection. On the other hand, while size-effects due to nonlocal effects have indeed been observed experimentally for materials such as polypropylene and epoxy, the data is somewhat conflicting and scattered. For polypropylene Mcfarland and Colton (2005) have reported a length scale of 10 microns while Stafford et al. (2004) report to have observed no size-effects for polystyrene thin-films as thin as 150 nm. For Liquid crystal elastomers, Warner (2003) have reported a length scale value of 10nm. A simple model by Nikolov et al (2006) estimates the length scale for rubbers to be around 4.5nm. Please see section 7 of our paper for references to more experimental works.

Rui Huang's picture

See Stafford et al., Macromolecules 39, 5095-5099, 2006 for experiments and a simple explanation of the size effect, and here is a related discussion.

RH

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