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quiz1_solution

# quiz1_solution - 1.033/1.57 Q#1 Stress Strength Conical...

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z n 1 z 1.033/1.57 Q#1: Stress & Strength Conical Indentation Tests October 8, 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 strengths at very fine scales. We have already studied the ‘ﬂat’ indentation problem in the homework set. Focus of this exercise is a more realistic shape of intenders, that is the conical indentation test, as sketched in the Figure below. The indenter (which is made of diament) is considered a rigid cone of half-apex angle α . The depth of the indenter in the material is h . A vertical force F is exerted on the rigid indenter in the direction of the Oz -axis. The test gives access to the hardness H of the indented material, defined as the average pressure: F H = A where A = πR 2 is the projected contact area at the surface z = 0 (see figure). The aim of this exercise is to relate the hardness to the strength properties of the indented material. Given the ﬂat surface of the indenter, the contact of the rigid indenter with the material (along the cone surface oriented by unit normal n ) can be considered to be without friction. Throughout this exercise we will assume quasi-static conditions (inertia effects neglected), and we will neglect body forces. F r R Rigid Indenter α F r R Rigid Indenter 2 h z 2 n 1 α

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October 8, 2003 page 2 1. Statically Admissible Stress Field : For purpose of analysis, we separate the half- space in two subdomains, noted respectively 1 and 2 (see figure). In these domains, we consider diagonal stress fields in a cylinder
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