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quiz2_solution (1)

quiz2_solution (1) - 1.033/1.57 Q#2 Elasticity Bounds...

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1.033/1.57 Q#2: Elasticity Bounds Conical Indentation Test November 14, 2003 MIT 1.033/1.57 Fall 2003 Instructor: Franz-Josef ULM Instrumented nano-indentation is a new technique in materials science and engineering to determine material properties at very fine scales. A typical indentation test is composed of a loading and an unloading part. The loading part is used to extract strength properties, the unloading portion is used to extract the elasticity properties of the indented material. We have already studied the link between strength properties and hardness measurements for a ‘flat’ indenter (Homework Set #2) and for a conical indenter (Quiz #1). This exercise deals with the link between elastic properties and the indentation result upon unloading. An indentation test is a surface test, but its effect is felt in a bulk volume of characteristic size L around the indentation cone. Within this bulk zone the material undergoes deformation as a consequence of the indentiation, while the material situated outside this zone will ‘not feel’ the localized indentation (no deformation). The focus of this exercise is to estimate this characteristic size L associated with the elastic unloading by means of the upper bound displacement approach. To this end, we consider a rigid conical indenter of half-apex angle α , which —during the loading phase— has penetrated into the material to an indentation depth h (see figure 1 TOP). At this stage, we consider an infinitesimal unloading | s | << h . The slope of the unloading, dF /ds > 0 (see figure 1. BOTTOM), is related to the stiffness properties of the indented material by: dF 2 = AM (1) ds π where A = πR 2 is the projected contact area at the surface z = 0 (see figure), and M is the indentation modulus. In this exercise, we will first establish an upper bound for the relation between M and the elasticity constants of a homogeneous, linear elastic isotropic material, char- acterized by the Lamé constants λ, G (= const ) . Then, we will use this solution to evaluate the characteristic length scale L . Throughout this exercise we will assume quasi-static conditions (inertia effects neglected), and we will neglect body forces.
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