# A study on the determination of mechanical properties of a power law material by its indentation force–depth curve

## Summary (3 min read)

### 1 Introduction

- Compressive forces are normally used to form micro-components through micro-forming.
- Sharp indenters widely used for micro- and nano- indentation tests are of Berkovich or Vickers types.
- A lot of efforts have been made in the recent few years to derive approaches to extract these mechanical properties from a single or multi-set of P-h curves, and will be briefl reviewed in the following sections.
- It has been shown by many numerical simulations [8, 9, 11, 14, 18-22] that Eq.(1) is a good approximation for elastic, elastic–perfect plastic and elastic–plastic materials.

### 1.2 Unloading curve and Young’s modulus

- The Young’s modulus E can be estimated from the unloading curve which is assumed to be purely elastic.
- In order to resolve this issue, Oliver and Pharr [6,25] introduced another method, taking into account the large elastic recovery during the unloading process of hard materials.
- After the parameters B and m are determined, the unloading slope at the maximum load can be evaluated to be, 1( )m m r S mB h h −= − (12) Another major difference between Doerner and Nix’s and Oliver and Pharr’s methods is the approaches to determine the contact area at the maximum load.
- Since hc obtained by Doerner and Nix’s, Eq.(10), and Oliver and Pharr’s methods, Eq.(14), is always smaller than hm, they can only explain the sink-in expressions.

### 1.3 Dimensional analysis and universal functions

- Cheng et al [9, 8] and Tunvisut et al. [27] have used dimensional analysis to propose a number of dimensionless universal functions, with the aid of computational data points calculated via the FE method.
- For this approach, one of the most complete studies has been published recently by Dao et al. [1].
- Using these functions, the relationships between characteristic parameters, C, S and hr/hm, of a P-h curve and the mechanical properties of a material, E, σy and n, have been set up and then were used to extract mechanical properties of power-law materials by co-equation solving.
- On the other hand, their analysis is less precise for determining the strain hardening coefficient n.
- Even if the mean value gives a good estimate of the expected value, the errors are high and at least six experimental curves on the same material were required.

### 2.1 Numerical model

- Axisymmetric FE models were constructed to simulate the indentation response of elastic-plastic solids using the commercial FE code ABAQUS.
- Fig. 3(a) shows the FE model for axisymmetric calculations.
- The semi-infinite substrate of the indented solid was modelled using 9600 four-noded, bilinear axisymmetric quadrilateral elements, where a fine mesh near the contact region and a gradually coarser mesh further from the contact region were designed to ensure numerical accuracy.
- At the maximum load, the minimum number of contact elements in the contact zone was no less than 35 in each FEM computation.
- An example of the self- adaptive mesh and the Mises stress contour at the maximum load is shown in Fig. 3(b).

### 2.2.1 The loading curve

- 2.1.1 Loading curves for elastic and elastic-perfect plastic materials.
- It was found that this weighting scheme worked very well.

### 2.2.2 The unloading curve

- Zeng and Chiu’s weighting scheme as described in section 1.4 is firstly used in an attempt to describe the indentation-unloading curve (in fact, the upper 50% of the unloading curve) of an elastic-plastic material by weighting the unloading curves of the corresponding elastic and elastic-perfect plastic materials.
- As shown in Fig. 9, if the contact area does not change during unloading, the unloading curve would be a straight line, Pfc, with a slope at the maximum load (note this is different from the straight line for the corresponding elastic-perfect plastic material as proposed in Zeng and Chiu’s method).
- The unloading response of the elastic-plastic material should be a combination of the full contact straight line and the purely elastic curve.
- It was found that excellent fitting can be obtained for at least the upper 50% of the unloading curve by the proposed weighting scheme.

### 2.2.3 The residual depth

- An important characteristic parameter of the P-h curve is the residual depth, hr, after complete unloading (also shown in Table 1 for each simulation case).
- Following the same least square fitting approach used in previous sections.
- With the loading and unloading parts of a P-h curve being related to the mechanical properties of a material by Eq.(21) and (26), respectively, the mechanical properties of the matereial are determined by minimising the errors between the FE and predicted (using Eq.(21) and (26)) P-h curves, as shown in Fig. 14.
- The first term in the equation defines the errors in the loading phase and the second in the unloading.
- The Bates and Watts’ optimisation method improves the efficiency of optimisation by normalising optimised parameters in situations where several parameters, which have different magnitude but are confined within their own individual boundaries, need to be optimised.

### 3 Case studies-effectiveness of the proposed optimisation approach

- To use The Bates and Watts’ optimisation method [29], the lower and upper boundaries of the parameters to be optimised have to be defined.
- Here the optimisation parameters are the mechanical properties of the material of concern.
- In most cases, some background information of the material of concern should be given.
- The number of the matched sets of material properties whose residuals are not more than 10% of the global minimum is given in Table 2.
- Using the Young’s modulus predicted by Oliver and Pharr’s method in the two parameter optimisation seems not to be a good approach, since Oliver and Pharr’s method often overestimates the Young’s modulus.

### 4 Discussion

- This study proposes a novel optimisation approach to extract mechanical properties of a power law material from its given experimental (or FE simulation) indentation P-h curve.
- The present study shows that the non-uniqueness problem is a commonplace for nearly all P-h curves and the matched materials can be found in a deterministic way through optimisation algorithm.
- For three parameter (i.e. E, σy and n) optimisation, although the best matched material is identified in this study (Table 2), it could be difficult to identify it in reality since the material properties are not exactly known, particularly when there is large number of matched materials to choose from.
- Further investigation is carried out in this direction.
- The difficulty in relating the unloading P-h curve to mechanical properties is the main factor that affects the prediction accuracy of the proposed optimisation scheme.

### 5 Conclusions

- A novel optimisation approach to extract mechanical properties of a power law material from its given experimental (or FE simulation) indentation P-h curve is proposed.
- It was found that the prediction accuracy of material properties can be improved by this approach since the entire P-h curve data, except the lower 50% of the unloading curve, are used.
- Using the proposed optimisation approach, it was found that mechanical properties of an elastic-plastic material usually cannot be determined uniquely by using only a single indentation P-h curve of the material.
- This is because in general a few matched set of mechanical properties were found to produce a given P-h curve.
- If the best matched material is identified, the predictions of mechanical properties are quite accurate.

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##### Citations

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### Cites background from "A study on the determination of mec..."

...Due to the self-similarity (Cheng and Cheng, 2004), there can be multiple combinations of mechanical properties ðE; Y; and nÞ, that give rise to almost indistinguishable indentation load–depth curves of a single conical/pyramid indenter (Cheng and Cheng, 1999; Capehart and Cheng, 2003; Alkorta et al., 2005; Tho et al., 2005; Luo et al., 2006)....

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...1 for notations), see the recent review by Oliver and Pharr (2004): S ¼ b 2ffiffiffi p p E ffiffiffiffiffiffi Am p ð1Þ where b is a correction factor and E is determined by the elastic modulus E and Ei, and Poisson’s ratios m and mi of the indented material and the indenter, respectively, as below: E ¼ 1 m 2 E þ 1 m 2 i Ei 1 ð2Þ Due to the self-similarity (Cheng and Cheng, 2004), there can be multiple combinations of mechanical properties ðE; Y; and nÞ, that give rise to almost indistinguishable indentation load–depth curves of a single conical/pyramid indenter (Cheng and Cheng, 1999; Capehart and Cheng, 2003; Alkorta et al., 2005; Tho et al., 2005; Luo et al., 2006)....

[...]

...…there can be multiple combinations of mechanical properties ðE; Y; and nÞ, that give rise to almost indistinguishable indentation load–depth curves of a single conical/pyramid indenter (Cheng and Cheng, 1999; Capehart and Cheng, 2003; Alkorta et al., 2005; Tho et al., 2005; Luo et al., 2006)....

[...]

35 citations

### Cites background from "A study on the determination of mec..."

...However, Luo et al.(124,125) have highlighted the difficulties of calculating the coefficients in these expressions from indents of a single geometry and noted that the use of more than one indenter geometry may be necessary to obtain a unique solution to the fitting of a suitable constitutive expression for uniaxial deformation behaviour....

[...]

...The trend towards the calculation of uniaxial constitutive mechanical response (either stress–strain or creep) from depth sensing data was noted.4,123–128 However, Luo et al.124,125 have highlighted the difficulties of calculating the coefficients in these expressions from indents of a single geometry and noted that the use of more than one indenter geometry may be necessary to obtain a unique solution to the fitting of a suitable constitutive expression for uniaxial deformation behaviour....

[...]

^{1}

35 citations

##### References

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### "A study on the determination of mec..." refers methods in this paper

...To use the Bates and Watts’ optimization method [29], the lower and upper boundaries of the parameters to be optimized have to be defined....

[...]

...For a particular set of P–h data (obtained experimentally or by FE simulation as is the case in this study), the Bates and Watts’ optimization method [29], is used to determine the mechanical properties of the material by minimizing the residual defined in equation (28)....

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