A Scaling Approach to Modeling Indentation Measurements
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A Scaling Approach to Modeling Indentation Measurements Yang-Tse Cheng and Zhiyong Li* Materials and Processes Laboratory, General Motors Research and Development Center, Warren, Michigan 48090, U.S.A. Che-Min Cheng Laboratory for Non-Linear Mechanics of Continuous Media, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China. ABSTRACT Using dimensional analysis and finite element calculations, the relationships between hardness, elastic modulus, final contact depth, and the work of indentation are extended to conical indentation in elastic-plastic solids with various cone angles. These relationships provide new insights into indentation measurements. They may also be useful to the interpretation of results obtained from instrumented indentation experiments. 1. INTRODUCTION For over one hundred years, indentation experiments have been performed to obtain the hardness of materials [1]. Recent years have seen significant improvements in indentation equipment and a growing need to measure the mechanical properties of materials at small length scales. It is now possible to monitor, with high precision and accuracy, both the load and displacement of an indenter during indentation experiments [2-4]. However, questions remain, including what properties can be measured using instrumented indentation techniques and what is hardness? Many authors have addressed these basic questions [5-18]. Rather than an exhaustive literature review, this paper summarizes our recent results [19-29], obtained using a scaling approach to indentation modeling, that may be useful to the interpretation of results obtained from instrumented indentation experiments. We consider a three dimensional, rigid, conical indenter of a given half angle, θ , indenting normally into a homogeneous solid. The friction coefficient at the contact surface between the indenter and the solid is assumed zero. The quantities of interest from the loading portion of indentation measurements include the force ( F ) and the contact depth ( hc ) or the projected contact area ( Ac = πa 2 ) (Fig. 1a), from which the hardness under load, H = F / Ac , can be evaluated. For unloading, the maximum indentation depth, hm , and the residual final depth, h f , at which the indenter detaches from the surface of materials during unloading (Fig. 1b), have been used to characterize the properties of materials [6]. Finally, the total- and irreversiblework of indentation, Wtot and W p , defined respectively as the area under the loading curve and that between the loading and unloading curves, have also been used for materials characterization [12-17, 24]. It is desirable, therefore, to investigate the relationships between these seemingly different measures of material properties obtained from indentation experiments. Q1.1.1
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Figure 1. Illustration of (a) conical indentation, and (b) loading and unloading curves, where Wtot is the total work, W p is the irreversib
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