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New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials

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Abstract

New expanding spherical cavity model (ECM) for conical indentation is proposed. For polymeric materials description, the model incorporates isotropic non-monotonic strain hardening. For capturing the indentation size effect (ISE), the model incorporates the strain gradient dependence in yield strength based on lower-order strain gradient plasticity assumptions. Specifically, the forward gradient of the equivalent (accumulated) plastic strain is utilized as a non-local part of the yield strength. To predict the indentation depth-dependent hardness based on the proposed model, it is sufficient to numerically integrate one nonlinear ODE of the first order, and then calculate the definite integral. For the local perfect plasticity model, the hardness is obtained as an analytical expression that differs from known ECMs. The hardness estimate obtained numerically using the proposed model is compared with the experimental ISE data for polycarbonate (PC) and polymethyl methacrylate (PMMA). For the local perfect plasticity model, the formula obtained in the study is compared with the experimental data on the hardness of preliminary work-hardened materials. In both cases, the model shows good agreement with the experimental data. Fitting the experimental data on ISE, we found that intrinsic length scale of PMMA should be near 3 microns and near 9 microns for PC.

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Notes

  1. Most of the data are given for metallic materials; however, the leftmost experimental points correspond to polymeric materials. For them, it is not clear what is meant by the state of full-hardening, which is indicated in [46] (see, for example, stress–strain curve for PMMA on Fig. 8). In [34] it is indicated that the pre-strain of the samples was in limits 1.1 and 1.5.

  2. In contrast to viscoplastic models, in which the plastic strain rate vanishes at the elastic–plastic boundary and increases rapidly with deepening into the plastic region up to values comparable to those predicted by rate-independent plasticity.

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Acknowledgements

This research was funded by the Russian Science Foundation (Project No 22-11-00163).

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  1. Institute of Machinery and Metallurgy, Khabarovsk Federal Research Center FEB RAS, Komsomolsk-on-Amur, Russia

    Georgiy M. Sevastyanov

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Appendix 1

Appendix 1

In [46] (Fig. 5 on p. 425), the abscissa axis displays the complex \(B\ln z = {3 \mathord{\left/ {\vphantom {3 {\left( {3 - \lambda } \right)}}} \right. \kern-0pt} {\left( {3 - \lambda } \right)}}\ln \left[ {{3 \mathord{\left/ {\vphantom {3 {\left( {\lambda + 3\mu - \lambda \mu } \right)}}} \right. \kern-0pt} {\left( {\lambda + 3\mu - \lambda \mu } \right)}}} \right]\) for various materials. Here, \(\lambda = \left( {1 - 2\nu } \right){{\sigma_{y} } \mathord{\left/ {\vphantom {{\sigma_{y} } E}} \right. \kern-0pt} E}\), \(\mu = \left( {1 + \nu } \right){{\sigma_{y} } \mathord{\left/ {\vphantom {{\sigma_{y} } E}} \right. \kern-0pt} E}\). In our study in Fig. 6 (hexagram markers), we use the values \(\lg \left( {{E \mathord{\left/ {\vphantom {E {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}} \right)\) extracted from the Marsh data (see table below; the first line of the table corresponds to the rightmost hexagram on Fig. 6, further from right to left). Poisson’s ratio values are taken from publicly available sources.

Material

Poisson’s ratio \(\nu\)

\(B\ln z\)(extracted)

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\)(extracted)

\(\lg \left( {{E \mathord{\left/ {\vphantom {E {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}} \right)\)(calculated)

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\), calculated by Eq. (2) (Johnson model, incompressible elasticity), with relative error

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\), calculated by Eq. (36) (Johnson model, compressible elasticity), with relative error

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\), calculated by Eq. (37) (Durban–Masri model, compressible elasticity), with relative error

Tellurium-lead

0.42

6.2

2.95

2.86

4.31 (+ 46%)

4.21 (+ 43%)

4.21 (+ 43%)

Aluminum

0.33

5.5

3.05

2.55

3.83 (+ 26%)

3.64 (+ 19%)

3.64 (+ 19%)

Copper

0.36

5.1

3.00

2.38

3.57 (+ 19%)

3.41 (+ 14%)

3.42 (+ 14%)

Mild steel

0.3

4.8

2.95

2.24

3.36 (+ 14%)

3.15 (+ 7%)

3.16 (+ 7%)

Steel 1

0.28

4.5

3.10

2.11

3.16 (+ 2%)

2.94 (− 5%)

2.96 (− 5%)

Steel 2

0.28

4.3

2.95

2.02

3.02 (+ 2%)

2.81 (− 5%)

2.83 (− 4%)

Steel 3

0.28

4.1

2.75

1.93

2.88 (+ 5%)

2.67 (− 3%)

2.61 (− 5%)

Steel 4

0.28

3.9

2.60

1.84

2.74 (+ 5%)

2.54 (− 2%)

2.60 (0%)

Steel 5

0.28

3.7

2.50

1.76

2.62 (+ 5%)

2.43 (− 3%)

2.49 (0%)

Copper-beryllium

0.285

3.5

2.35

1.67

2.48 (+ 6%)

2.31 (− 2%)

2.38 (+ 1%)

PMMA 1

0.38

3.2

2.25

1.55

2.30 (+ 2%)

2.20 (− 2%)

2.26 (0%)

Epoxy resin

0.35

3.1

2.10

1.50

2.22 (+ 6%)

2.11 (0%)

2.18 (+ 4%)

Polystyrene

0.34

3.0

2.30

1.46

2.16 (− 6%)

2.05 (− 11%)

2.13 (− 7%)

PMMA 2

0.38

2.8

1.95

1.38

2.04 (+ 5%)

1.96 (0%)

2.03 (+ 4%)

Polyacetal resin

0.44

1.3

1.15

0.73

1.04 (− 10%)

1.11 (− 3%)

1.05 (− 9%)

Experimental data [34], Table 1 on p. 432) (the first line of the table corresponds to the rightmost pentagram in Fig. 6 in the presented study, further from right to left). Poisson’s ratio values are taken from publicly available sources.

Material

Poisson’s ratio \(\nu\)

\({E \mathord{\left/ {\vphantom {E {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\)

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\)

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\), calculated by Eq. (2) (Johnson model, incompressible elasticity), with relative error

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\), calculated by Eq. (36) (Johnson model, compressible elasticity), with relative error

\({H \mathord{\left/ {\vphantom {H {\sigma_{y} }}} \right. \kern-0pt} {\sigma_{y} }}\), calculated by Eq. (37) (Durban – Masri model, compressible elasticity), with relative error

Lead alloy

0.4

828

3.16

4.40 (+ 39%)

4.276 (+ 35%)

4.272 (+ 35%)

Aluminum

0.33

459

3.35

4.00 (+ 19%)

3.813 (+ 14%)

3.803 (+ 14%)

Copper

0.36

318

3.22

3.76 (+ 17%)

3.6 (+ 12%)

3.6 (+ 12%)

Mild steel

0.3

268

3.27

3.64 (+ 11%)

3.43 (+ 5%)

3.43 (+ 5%)

Beryllium copper

0.285

102.6

2.64

3.00 (+ 14%)

2.80 (+ 6%)

2.83 (+ 7%)

PTFE

0.46

22.6

1.73

2.00 (+ 16%)

1.97 (+ 14%)

1.99 (+ 15%)

PCTFE

0.38

15.3

1.28

1.74 (+ 36%)

1.70 (+ 33%)

1.76 (+ 38%)

Nylon

0.4

13.1

1.25

1.63 (+ 30%)

1.62 (+ 30%)

1.66 (+ 33%)

Perspex

0.38

11.2

1.07

1.53 (+ 43%)

1.53 (+ 43%)

1.57 (+ 47%)

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Sevastyanov, G.M. New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials. Acta Mech (2024). https://doi.org/10.1007/s00707-024-03921-2

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  • DOI: https://doi.org/10.1007/s00707-024-03921-2

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