Storage and loss stiffnesses and moduli as determined by dynamic nanoindentation
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W.D. Nix Stanford University, Department of Materials Science and Engineering, Stanford, California 94305 (Received 24 July 2008; accepted 16 October 2008)
The storage and loss stiffnesses for the composite response of the sample, indenter, and load frame during dynamic nanoindentation are derived. In the first part of the analysis, no physical model is assigned to the composite system. It is shown that this case is equivalent to the conventional nanoindentation analysis. In the second part of the analysis, the sample is modeled as a standard linear solid in series with the indenter and load frame. The results for the storage and loss stiffnesses as computed by the two methods differ by at most 3% for the elastomeric system under consideration. Results for the storage and loss moduli are also similar. The relative merits and weaknesses of each analysis are discussed. I. INTRODUCTION
The goal of this work is to provide further understanding of two methods of analysis for dynamic nanoindentation; to substantiate our thesis we compute the storage and loss stiffnesses and moduli for an elastomer using the two methods. First, we consider the equations from the conventional analysis, highlighting the fact that these equations describe the composite response of the sample, indenter, and load frame.1,2 We refer to this as the “black box derivation” to remind the reader that these equations do not assume a physical model for the components contained in the box (i.e., the sample, indenter, and load frame.) The global storage and loss stiffnesses are calculated for the composite response. Next, we consider a recently presented method of analysis in which the sample is modeled as a standard linear solid.3 We report on new progress using this technique. Again, we compute global storage and loss stiffnesses, but we subsequently show that storage and loss stiffnesses can also be extracted for the sample. No assumptions regarding the stiffnesses of the indenter or load frame are made. We compute the storage and loss moduli for the sample using both methods, and we conclude by evaluating the merits and weaknesses of both approaches. II. THE BLACK BOX DERIVATION
We begin with a model of the instrument and sample system during dynamic nanoindentation, as shown in Fig. 1. The sample, indenter, and load frame accommodate a)
Address all correspondence to this author. e-mail: [email protected]. DOI: 10.1557/JMR.2009.0112 J. Mater. Res., Vol. 24, No. 3, Mar 2009
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the total imposed displacement; their composite response is represented by a storage stiffness kstorage and loss stiffness kloss . The details of the response of each of these components will be considered later. As the indenter tip is pressed into the material, the leaf springs that support the indenter column are deflected; the stiffness of these springs is represented by ks . Additionally, the indenter displacement is measured by capacitive plates oriented perpendicular to the indenter column. As the indenter oscillate
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