Nonlocal Gradient Plasticity Theory for Predicting Size Effects in Metallic Micro/Nano Pillars

Abstract

There are numerous experimental results that indicate a size-dependent mechanical behavior of materials at the micron or submicron scales, in the sense that “smaller is stronger”. As no material length scale exits in the conventional (local) constitutive theories, they fail to capture this size dependency of material behavior at small length scales. A number of theories for generalizing the classical plasticity theories in order to account for the observed size effects have been proposed and still motivates researchers in this area. Among them the nonlocal continuum theories of either integral or gradient type have been introduced to model material behavior size dependency in a computationally effective way [1, 2]. The concept behind these theories is that; plastic strain gradients lead to the enhancement in the density of geometrically necessary dislocations (GNDs) and thereby to an elevation in the material’s strength [3]. Higher-order gradient dependent plasticity theory which enforces microscopic boundary conditions at interfaces and free surfaces has been introduced to improve first-order gradient theories as they fail to predict size effects. Recently, micro/nano compression tests have attracted several researchers to investigate the small-scale mechanical behavior of micro/nano-pillars with sizes range from few micrometers down to a few hundred of nanometers (see [4, 5] and reference quoted therein). Typically focused ion beam (FIB) machining from a thin film or a bulk specimen is used to produce columnar structure which then subjected to compression using a nanoindenter outfitted with a flat punch indenter (e.g. [4, 5]). Usually extrinsic defects like high initial dislocation density in the order of 15 2 10 /m , vacancy clusters, intermetallic components, and near surface amorphous layers are formed at a layer adjacent to the milled surface [5, 6]. Deformation in micro/nano-pillar’s compression is macroscopically uniform, so one may doubt how gradient enhanced theories could be able to capture size effect as gradient terms have minor effects. However, the authors believe that higher-order gradient plasticity theories can capture this size effect even for macroscopically uniform deformation as higher-order boundary conditions play an important role once the size of specimen decreases [7, 10]. For example, initial defects or amorphous layers or surface roughness at external surfaces of the micro/nanopillars may increase the level of surface energy for embedding dislocation annihilation through the surface [10]. Moreover, as mentioned above, the fabrication process of these pillars cause initial GNDs formation within the specimen with higher density at milled surfaces so one may claim that this causes initial plastic strain distribution which is related to the initial GND density distribution. In the current paper, different initial effective plastic strain distributions that correspond to constant plastic strain gradients within a micropillar sample have been assumed and the ability of gradient enhanced plasticity theory to capture size effect for different pillar sizes is presented.