Strain Gradient Effect in Cone Indentation
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Strain Gradient Effect in Cone Indentation Anthony A. DiCarlo1, Henry T. Y. Yang1 and Srinivasan Chandrasekar2 1 Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106 2 School of Industrial Engineering, Purdue University, West Lafayette, IN 47907-1287 ABSTRACT A size effect is known to exist in the strength and hardness of metals at small length scales. For example, the hardness of a pyramid indentation at the macro-scale is constant and typically independent of indentation size due to the self-similar feature of the indentation. But, at the meso-scale, this hardness has been observed to increase with decreasing indentation size. This increase has been attributed to the influence of strain gradient on the flow stress. At this point, the contribution of a rotational gradient to the hardness is unclear. This study investigates the sensitivity of the hardness to rotational strain gradients through a Cosserat continuum. Finite element simulation of cone indentation is employed to conduct this investigation. The effect of varying indentation strain fields is modeled using indentation with cones of varying angles. The results demonstrate the role of rotational gradients in indentation.
INTRODUCTION Due to self-similarity, the hardness of an indentation created using sharp tool geometry, such as cone, wedge, or pyramid, should be independent of penetration depth. With the emergence of nano-indentation equipment, the hardness has been found to be higher at small penetration depths. Pethica et al. [1] used nano-indentation to measure the hardness over a wide range of penetration depths in gold, nickel, and silicon and found that the relationship between hardness and indentation depth varied at different depth regimes. This effect has also been observed in a variety of materials including iron, aluminum, fused silica [2], and tungsten single crystal [3]. The reason for this size effect is uncertain. Some investigators have attributed this phenomenon to indenter tip shape [3, 4] or strain rate [5]. However, the size effect has still been observed in well-calibrated equipment. Consequently, some researchers [6, 7, 8] proclaim that the size effect is a result of an intrinsic material property due to geometrically necessary dislocations. This idea is the foundation for strain gradient plasticity developed by Fleck et al. [9]. Li et al. [10] incorporated the strain gradient plasticity into a dimensional analysis model to demonstrate that the hardness is no longer independent of indentation size, but dependent upon a material length parameter. Strain gradient plasticity adopts the notion that the stress is not only a function of strain, but also the strain gradient. This notion lends itself well to finite element analysis. Shu and Fleck [11] have incorporated strain gradient plasticity into finite element analysis through the rotational gradient continuum developed by the Cosserat brothers [12]. This study incorporates a similar model to investigate the effect of rotatio
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