Thematic areas “Physico-chemical nanomaterials science”

Applications of local gradient elasticity to the description 
of the size dependence of shear modulus in nanofilms
Hrytsyna O. R.
Department for mathematical methods of computing experiment, Center of Mathematical Modeling of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Dudajev str., 15, Lviv, 79005, Ukraine; Department of mechanics, Institute of Construction and Architecture, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 45, 84503, Slovak Republic 
E-mail: hrytsyna.olha@gmail.com

The classical theory of elasticity does not describe experimentally observed phenomenon of change in the mechanical characteristics of material in nanoscale systems [1]. We show that a local gradient theory of elasticity, taking the local mass displacement into account [2], describes the size effect of the shear modulus. To this end, we study the stress strain state of an infinite isotropic layer  which surfaces are subjected to a shear displacements . For this boundary problem, the classical elasticity predicts the linear dependence of the displacement on space coordinate: . Such a displacement causes uniform shear strain  and shear stress . The last one is proportional to the shear modulus , which does not depend on the layer thickness. Within the framework of the local gradient elasticity the following relation  is obtained for shear stress. Here,  ,  is characteristic length,  is the dimensionless coupling factor between the mechanical fields and the local mass displacement [2]. Above formula describes the size effect of shear modulus. The size effect of shear modulus is manifested only in thin films. It is shown that the effect of the film thickness on the shear modulus becomes more important in case of parameter  increase.

1.	Zhang B. Physical fundamentals of nanomaterials. ‑ Micro and Nano Technologies. – Elsevier, 2018. ‑ 486 p. 
2.	Hrytsyna О. R. Electrothermomechanics of nonferromagnetic polarizable solid bodies with regard for the tensor nature of the local displacements of mass // J. Math. Sci. ‑ 2017. ‑ 226, No 2. ‑ 139–151.