Surface elasticity revisited in the context of second strain gradient theory

• Extension of Gurtin–Murdoch surface elasticity.
• Extension of Mindlin second strain gradient elasticity.
• Relation between surface parameters and strain gradient parameters.
• Estimate of the effective elastic constants of the films of infinite extension.
• Calculation of modulus of cohesion and expression for surface energy.

Surface/interface stresses, when notable, are closely associated with a surface/interface layer in which the interatomic bond lengths and charge density distribution differ remarkably from those of the bulk. The presence of such topographical defects as edges and corners amplifies the noted phenomena by large amounts. If the principal features of interest are such studies as the physics and mechanics of evolving microscopic-/nanoscopic-interfaces and the behavior of nano-sized structures which have a very large surface-to-volume ratio, traditional continuum theories cease to hold. It is for the treatment of such problems that augmented continuum approaches like second strain gradient and surface elasticity theories have been developed by Mindlin (1965) and Gurtin and Murdoch (1975), respectively. In the mathematical framework of the former theory, the surface effect is explicitly revealed through surface characteristic length and modulus of cohesion, whereas within the latter theory, which views the bulk material and its complementary surface as separate interacting entities, the critical role of surfaces/interfaces is directly incorporated through the introduction of the notions of tangential surface strain tensor, surface stress tensor, and surface elastic modulus tensor into the formulation. In the realm of the experimentations, evaluation of the above-mentioned surface parameters poses serious difficulties. One of the objectives of the current study is to provide a remedy as how to calculate, not only these parameters, but also Mindlin’s bulk characteristic lengths as well as Lame constants with the aids of first principles density functional theory (DFT). To this end, surface elasticity is reformulated by maintaining the first and second gradients of the strain tensor for the bulk; as a result two new key equations are obtained. One of these equations is an expression for the net surface stress, needed to relate the surface parameters in surface elasticity to the Mindlin’s second gradient theory parameters. The other equation is for the total elastic energy which is utilized to find an analytical expression for the surface energy. The available data on surface relaxation obtained experimentally and computationally are in good correspondence with the results of the current theory. Moreover, employing the present theory, an estimate for the effective elastic constants of films with infinite extension is provided.