Elastic contact between indenter and specimen

Introduction

Background

 How to compare simulation with experiment Examples:
Compare with rigid punch Al and Cr films on Si TiN on 440C stainless I TiN on 440C stainless II
Silicon on Insulator I Silicon on insulator II Al-W multilayers Cu-Nb multilayers
Simulation Files   RETURN TO HOME
Introduction
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INTRODUCTION

We have developed and implemented an elasticity model for helping to interpret nanoindentation data generated from layered specimens. This webpage includes downloadable files generated from the model simulations.  All downloadable simulations are for single, isotropic films on silicon.  For multiple layers (up to many dozens) or different substrates it is possible to perform simulations.  Please Email me. 

The model is meant to be used for simulating unloading compliance as a function of the size of the indent when the specimen is a thin film or multilayered thin film on top of a substrate. The model employs linear elasticity theory to calculate the average elastic displacement beneath a uniformly loaded area. The contact stiffness estimated in this way is a little lower than estimated using a flat punch (which is also an approximation to a real nanoindentation experiment), but the results obtained are satisfactory if not excellent when compared with experimental data.

We can model any arbitrary stack of isotropic layers (up to dozens of layers) on top of an isotropic. Undoubtedly it should be possible to find a system of layers that is so pathlogical that the computer program crashes. Nevertheless, we haven't come across one of these in any of our research. We can also model a transversely isotropic layer on top of an isotropic substrate, and Hertzian (spherical) contact for alayer on top of a substrate. The Hertzian solution is for a rigid indenter rather than a specified pressure distribution. All of the calculations are done using Mathematica®.

Below are simulations that that you can download if you wish to compare them with your data or model. The simulations are text files (open with notepad, wordpad, Excel, etc.) containing brief headers followed by simulated results in four columns:

  1. h/A1/2 : Total film thickness/square root of area (total film thickness = all layers except substrate)
  2. hC : Unloading compliance multiplied by total film thickness (in micrometers squared/Newton)
  3. A1/2/h : Square root of area/total film thickness
  4. Eeff : Effective modulus in GPa (=unloading stiffness/square root of area)

The information carried in columns 1 and 2 is duplicated in columns 3 and 4. Numbers in column 3 are the inverses of those column 1. Moreover, column 4 is the inverse of the product of columns 1 and 2. Normalization in columns 1-3 by film thickness is necessary so that the simulations can be compared with films of arbitrary thickness. The comparison between simulation and data requires that the actual film thickness be known.

In these simulations the effect of the diamond indenter is taken into account using 1137 GPa for the Young's modulus of diamond and 0.07 for the Poisson's ratio. It should therefore be possible to compare the model with experimental data with minimal effort .

 

Background; Comparison of model with experimental data

BACKGROUND

The contact compliance (1/stiffness) for an indenter with a specimen is

(1)

where A is the projected area of the indent and nd and Ed are Poisson's ratio and Young's modulus of the diamond indenter (0.07 and 1137 GPa). Er is the property of the specimen and depends on the size of the indent in relation to the film thickness. b (beta) is a numerical factor that depends on indenter geometry. It varies a little, and different researchers use different values. Most researchers use b = 1.128 which is consistent with a flat, circular punch. In earlier simulations we employed b = 1.086 corresponding to uniform pressure over a triangular region. More recently we have gone to b = 1.23 based on experimental measurements and finite element analysis simulations [1]. The new value is substantially larger than the old; yet both values were adequate for fitting experimental data. The reason for this apparent discrepancy is that in the previous studies where b = 1.086 was used a different way of measure unloading compliance was also used based on Doerner and Nix [2]. The older way of calculating compliance tended to overestimate the compliance in a way that cancels out the effect of using a lower b value. (Some of the errors did not cancel out: we always obtained a high value for the Youn'g modulus of silicon). These days we determine compliance (contact stiffness) from experimental data in a way that is similar to Oliver-Pharr method [3], and based on this obtain b ~ 1.23.

The product bEr is calculated from by the simulations using elasticity theory and methods described in reference [4].

In the simulation files, the value of b being used is 1.086. To correct this for our new way of analyzing data we multiply compliance calculated from the model by (1.086/1.23) and effective modulus by (1.23/1.086). Most researchers would use the numerical factor (1.128/1.086).

In the simulations the effects of the diamond indenter are already taken into account.

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DIFFERENT WAYS OF COMPARING MODEL SIMULATIONS WITH EXPERIMENTAL DATA

  1. hC vs. h/A1/2. This is a good way of comparing model with experimental data when you know what the areas of the indents are. Be warned, though: the areas determined by the instrument are based on depth, and they aren't always very accurate; accurate area measurement is tedious and requires some experience. This method is good for identifying whether you have properly removed the machine compliance from your data, because an unaccounted for machine compliance will come out as an intercept on the hC axis. Columns 1 and 2 are hC vs. h/A1/2, respectively. Depending on what you use for b, you should multiply hC from the simulation by (1.086/b) before comparing it with your data.
  2. Eeff vs A1/2/h. This offers perhaps the most sensitive comparison between data and theory, but it is also the most likely to suffer from any errors in the data. Be warned: the areas determined by the instrument are based on depth, and they aren't always very accurate; accurate area measurement is often tedious and requires some experience.
  3. CP1/2 vs. P1/2/h from the experimental data with (HA)1/2C vs. (HA)1/2/h where H, the hardness of the film, is used as a fitting parameter. This offers the most robust comparison because it does not rely on having to measure the area. For poly crystalline films the hardness is constant throughout some range of load, so this comparison works remarkably well.

REFERENCES

  1. 1. Jakes, J.E., C.R. Frihart, J.F. Beecher, R.J. Moon, and D.S. Stone, Experimental method to account for structural compliance in nanoindentation measurements. Journal of Materials Research, 2008. 23(4): p. 1113-1127.
  2. Doerner, M.F. and W.D. Nix, A method for interpreting the data from depth-sensing indentation instruments. Journal of Materials Research, 1986. 1(4): p. 601-9.
  3. Oliver, W.C. and G.M. Pharr, Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research, 1992. 7(6): p. 1564-1580.
  4. Stone, D.S., Elastic rebound between an indenter and a layered specimen. I. Model. Journal of Materials Research, 1998. 13(11): p. 3207-13.


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Comparison between our model and rigid punch model of King (Int. Journal of Solids and Structures, 1987. 23(12): p. 1657-64.)

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D. Stone, T. W. Wu, P.-S. Alexopoulos, and W. R. LaFontaine, "Indentation Technique to Investigate Elastic Moduli of Thin Films", Mater. Res. Soc. Symp. Proc. 130 J. C. Bravman, D. M. Barnett, W. D. Nix, and D. A. Smith, Eds. MRS (1989) 105-110.

Films on top of Si

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D. Stone, T. W. Wu, P.-S. Alexopoulos, and W. R. LaFontaine, "Indentation Technique to Investigate Elastic Moduli of Thin Films", Mater. Res. Soc. Symp. Proc. 130 J. C. Bravman, D. M. Barnett, W. D. Nix, and D. A. Smith, Eds. MRS (1989) 105-110.

0.25 mm TiN film on 440C stainless steel substrate I:

hC vs h/A1/2

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D. S. Stone, K. B. Yoder, and W. D. Sproul, Hardness and Elastic Modulus of TiN Based on Continuous Indentation Technique and New Correlation. Journal of Vacuum Science and Technology A , 9(4) July/August (1991) 2543-2547.

TiN film on 440C stainless steel substrate II:

CP1/2 vs. P1/2/h gives hardness of the film ~ 32 GPa.

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D. S. Stone, K. B. Yoder, and W. D. Sproul, Hardness and Elastic Modulus of TiN Based on Continuous Indentation Technique and New Correlation. Journal of Vacuum Science and Technology A , 9(4) July/August (1991) 2543-2547.

Aluminum-Tungsten multilayers, modeled as transversely isotropic layer, on Silicon

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(nanoindentation able to detect composition of composites based on modulus measurement)

M.F. Tambwe, D.S. Stone, J.-P. Hirvonen, I. Suni, and S.-P. Hannula. Nanoindentation as a Composition Microprobe for Nanolayer Composites, Scripta Materialia, 37, November (1997) 1421-1427.

Silicon on Insulator (SOI) I

CP1/2 vs. P1/2/h

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J.E. Jakes, C.R. Frihart, J.F. Beecher, R.J. Moon, P.J. Resto, Z.H. Melgarejo, O.M. Suárez, H. Baumgart, A.A. Elmustafa, and D.S. Stone. Nanoindentation near the edge, Journal of Materials Research 29(3), March 2009, pp. 1016-1031.

 

Cu/Nb nanolayer composites on Si substrate

Eeff vs A1/2/h

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Tambwe, M.F., D.S. Stone, A.J. Griffin, H. Kung, Y.C. Lu, and M. Nastasi, Haasen plot analysis of the Hall-Petch effect in Cu-Nb nanolayer composites. Journal of Materials Research, 1999. 14(2): p. 407-17.

 

Silicon on Insulator (SOI) II

Eeff vs A1/2

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J.E. Jakes, C.R. Frihart, J.F. Beecher, R.J. Moon, P.J. Resto, Z.H. Melgarejo, O.M. Suárez, H. Baumgart, A.A. Elmustafa, and D.S. Stone. Nanoindentation near the edge, Journal of Materials Research 29(3), March 2009, pp. 1016-1031.

   
   
   
   
Simulation files

 

 

 

 

FILM YOUNG'S

MODULUS

(GPa)

 

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FILM YOUNG'S

MODULUS

(GPa)

 

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FILM YOUNG'S

MODULUS

(GPa)

 

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Simulations for thin films on Silicon Substrate

Below is a matrix containing simulations for films with Young's moduli ranging from 10 to 1000 GPa and Poisson's ratios between 0.05 and 0.45. The files are text, and they consist of a header and 4 columns (see Intro). By clicking on the matrix element you can either display the contents of the file or download it to your computer. The Young's modulus of Silicon is 161 GPa with 0.227 Poisson's ratio. We find that this works well for 100 silicon.

 

If you want to model another system not available here including a multilayer thin film or transversely isotropic film on arbitrary substrate, email me.

 

Please let me know if you have trouble downloading a file.

 

POISSON'S RATIO

  0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
10 e10n05 e10n1 e10n15 e10n2 e10n25 e10n3 e10n35 e10n4 e10n45
20 e20n05 e20n1 e20n15 e20n2 e20n25 e20n3 e20n35 e20n4 e20n45
30 e30n05 e30n1 e30n15 e30n2 e30n25 e30n3 e30n35 e30n4 e30n45
40 e40n05 e40n1 e40n15 e40n2 e40n25 e40n3 e40n35 e40n4 e40n45
50 e50n05 e50n1 e50n15 e50n2 e50n25 e50n3 e50n35 e50n4 e50n45
60 e60n05 e60n1 e60n15 e60n2 e60n25 e60n3 e60n35 e60n4 e60n45
70 e70n05 e70n1 e70n15 e70n2 e70n25 e70n3 e70n35 e70n4 e70n45
80 e80n05 e80n1 e80n15 e80n2 e80n25 e80n3 e80n35 e80n4 e80n45
90 e90n05 e90n1 e90n15 e90n2 e90n25 e90n3 e90n35 e90n4 e90n45
100 e100n05 e100n1 e100n15 e100n2 e100n25 e100n3 e100n35 e100n4 e100n45
110 e110n05 e110n1 e110n15 e110n2 e110n25 e110n3 e110n35 e110n4 e110n45
120 e120n05 e120n1 e120n15 e120n2 e120n25 e120n3 e120n35 e120n4 e120n45
130 e130n05 e130n1 e130n15 e130n2 e130n25 e130n3 e130n35 e130n4 e130n45
140 e140n05 e140n1 e140n15 e140n2 e140n25 e140n3 e140n35 e140n4 e140n45
150 e150n05 e150n1 e150n15 e150n2 e150n25 e150n3 e150n35 e150n4 e150n45
160 e160n05 e160n1 e160n15 e160n2 e160n25 e160n3 e160n35 e160n4 e160n45
170 e170n05 e170n1 e170n15 e170n2 e170n25 e170n3 e170n35 e170n4 e170n45
180 e180n05 e180n1 e180n15 e180n2 e180n25 e180n3 e180n35 e180n4 e180n45
190 e190n05 e190n1 e190n15 e190n2 e190n25 e190n3 e190n35 e190n4 e190n45
200 e200n05 e200n1 e200n15 e200n2 e200n25 e200n3 e200n35 e200n4 e200n45
210 e210n05 e210n1 e210n15 e210n2 e210n25 e210n3 e210n35 e210n4 e210n45
220 e220n05 e220n1 e220n15 e220n2 e220n25 e220n3 e220n35 e220n4 e220n45
240 e240n05 e240n1 e240n15 e240n2 e240n25 e240n3 e240n35 e240n4 e240n45
260 e260n05 e260n1 e260n15 e260n2 e260n25 e260n3 e260n35 e260n4 e260n45
280 e280n05 e280n1 e280n15 e280n2 e280n25 e280n3 e280n35 e280n4 e280n45
300 e300n05 e300n1 e300n15 e300n2 e300n25 e300n3 e300n35 e300n4 e300n45
320 e320n05 e320n1 e320n15 e320n2 e320n25 e320n3 e320n35 e320n4 e320n45
340 e340n05 e340n1 e340n15 e340n2 e340n25 e340n3 e340n35 e340n4 e340n45
360 e360n05 e360n1 e360n15 e360n2 e360n25 e360n3 e360n35 e360n4 e360n45
380 e380n05 e380n1 e380n15 e380n2 e380n25 e380n3 e380n35 e380n4 e380n45
400 e400n05 e400n1 e400n15 e400n2 e400n25 e400n3 e400n35 e400n4 e400n45
420 e420n05 e420n1 e420n15 e420n2 e420n25 e420n3 e420n35 e420n4 e420n45
440 e440n05 e440n1 e440n15 e440n2 e440n25 e440n3 e440n35 e440n4 e440n45
460 e460n05 e460n1 e460n15 e460n2 e460n25 e460n3 e460n35 e460n4 e460n45
480 e480n05 e480n1 e480n15 e480n2 e480n25 e480n3 e480n35 e480n4 e480n45
500 e500n05 e500n1 e500n15 e500n2 e500n25 e500n3 e500n35 e500n4 e500n45
520 e520n05 e520n1 e520n15 e520n2 e520n25 e520n3 e520n35 e520n4 e520n45
540 e540n05 e540n1 e540n15 e540n2 e540n25 e540n3 e540n35 e540n4 e540n45
560 e560n05 e560n1 e560n15 e560n2 e560n25 e560n3 e560n35 e560n4 e560n45
580 e580n05 e580n1 e580n15 e580n2 e580n25 e580n3 e580n35 e580n4 e580n45
600 e600n05 e600n1 e600n15 e600n2 e600n25 e600n3 e600n35 e600n4 e600n45
650 e650n05 e650n1 e650n15 e650n2 e650n25 e650n3 e650n35 e650n4 e650n45
700 e700n05 e700n1 e700n15 e700n2 e700n25 e700n3 e700n35 e700n4 e700n45
750 e750n05 e750n1 e750n15 e750n2 e750n25 e750n3 e750n35 e750n4 e750n45
800 e800n05 e800n1 e800n15 e800n2 e800n25 e800n3 e800n35 e800n4 e800n45
850 e850n05 e850n1 e850n15 e850n2 e850n25 e850n3 e850n35 e850n4 e850n45
900 e900n05 e900n1 e900n15 e900n2 e900n25 e900n3 e900n35 e900n4 e900n45
1000 e1000n05 e1000n1 e1000n15 e1000n2 e1000n25 e1000n3 e1000n35 e1000n4 e1000n45