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### Spherical Indentation Stress-Strain and Compressive Yield Stress

**Strength of Materials and Engineering **

Extracting Stress-Strain and Compressive Yield Stress Information From Spherical Indentation

21 pages

Thomas F. Juliano,

Mark R. VanLandingham,

Tusit Weerasooriya, and

Paul Moy

Weapons and Materials Research Directorate, ARL

Analytical determination of vibrations
and friction torque due to rotation,
taking into account the hydrodynamic
action of a lubricating oil film, and
determination of the elastic and damping
characteristics of bearings and bearing
assemblies are some of the problems
considered in this book. The methodology
and techniques of measuring the
dynamic characteristics of bearings are
presented. Experimental data and the
methodology of statistical analysis are
also given.

The book is intended for scientists and
engineers engaged in the instrument
manufacturing and machine building
industries.

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Open and view: Spherical Indentation Stress-Strain and Compressive Yield Stress

Introduction:

Depth-sensing indentation, also referred to as nanoindentation, is a technique in which a hard indenter tool (usually made of diamond or crystalline Al_{2}O_{3}) is pressed into a sample surface while load and displacement are measured continuously. It is commonly employed to find elastic modulus and indentation hardness properties of various materials on the microscale (1–3). Because indentation crudely resembles a uniaxial compression test, it may seem that stress-strain relationships may be readily discerned from load-displacement data for any tool geometry. However, it has been shown that such a relationship is unique to nonself-similar indenter geometries (i.e., a sphere) but not to pyramidal or conical indenters (4).

Pioneering work to determine expressions for indentation stress and strain was done by Tabor and focused primarily on the response of metals to contact loading (5). Since that time, a number of studies have attempted to describe indentation stress-strain curves with some promising success (6–12). These works have concentrated on a limited modulus range or specific class of material. However, no general study exists that proposes an easily implemented, general framework to estimate the elastic modulus and yield stress of a material for spherical indentation, exclusively using the loading-curve data. Such a procedure is developed in this work, and a number of materials important for military applications are used to exemplify the approach. This work is especially useful in measuring mechanical stress-strain curves on materials that cannot be measured through bulk measurement or materials that are heterogeneous at the microscale. Examples include single material grains, thin films, and microelectromechanical system components.

Figure 1. Schematic of indenter contact with a sample surface. (A number of variables from equations 1–5 are depicted.)

Content:

List of Figures iv

List of Tables iv

Acknowledgments v 1.

Introduction 1 2.

Theory 1 3.

Experimental 5 4.

Results and Discussion 6 5.

Conclusions 11 6.

References 12

Distribution List 14

Figure 1. Schematic of indenter contact with a sample surface. (A number of variables from equations 1–5 are depicted.) .2

Figure 2. Curves showing true stress and true strain for compression data (solid lines) and 20-µm-radius indenter data (dotted lines). Curves are shown for (a) Ti-6-4, (b) RHA steel, (c) WC-Co, (d) PMMA, and (e) PC 9

Table 1. Compressive yield stress values for Ti-6-4, RHA steel, WC-Co, PMMA, and PC measured using compression testing and elastic modulus values measured using ultrasonic testing (Ti-6-4, RHA steel, and WC-Co) and DMA (PMMA and PC)..7

Table 2. Elastic modulus and compressive yield stress values estimated from indentation testing of Ti-6-4, RHA steel, WC-Co, PMMA, and PC using a 20-μm-radius indentation tip.7

Table 3. Elastic modulus and compressive yield stress values estimated from indentation testing of Ti-6-4, RHA steel, WC-Co, PMMA, and PC using a 50-μm-radius indentation tip..7

Table 4. Elastic modulus and compressive yield stress values estimated from indentation testing of Ti-6-4, RHA steel, WC-Co, PMMA, and PC using a 500-μm-radius indentation tip.

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