Characterisation of ‘mystical’ materials

Several authors have reported the existence of materials that yield indistinguishable P-h curves regardless of tP-he difference in tP-he constitutive parameters and indenter geometry [58, 66, 73, 75], which implies that the number of possible solutions to the inverse analysis of indentation purely based on experimentally obtained P-h curves is infinite. The mystical materials illustrated in Figure 6.9 exhibit very different elastic and plastic behaviours, yet the coefficient of variation between P-h curves is expected to be less than 0.5% [73]. Details of the constitutive parameters that define the stress-strain curves is included in Table 6.3. Bearing this in mind, the capability of the proposed optimisation model to distinguish between mystical materials has been tested by forcing a condition of best fit between the target and predicted curve such that the initial point is a local minimum. Therefore the inverse analysis is run using a (FE simulated) target P-h curve that corresponds to the mechanical properties of material A, i.e. here the target properties, whereas the initial guess vector corresponds to the properties of material B, as detailed in Table 6.3.

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119

Figure 6.9. Different elastic-plastic materials that yield indistinguishable P-h curves, i.e. mystical materials [73].

Table 6.3. Sets of dissimilar mechanical properties that exhibit identical indentation response, i.e.

mystical materials.

Material Parameter

E σy n

A 200000 2000 0.1

B 212900 1245 0.3

C 192570 2340 0

As shown in Figure 6.10a, a single-objective function (SOF) model was incapable of converging to the target properties since the information provided solely by the P-h curve is limited. On the other hand, the non-uniqueness issue of the inverse analysis of indentation was positively addressed by a multi-objective function (MOF) model as the additional information is linked with the plastic behaviour of the material. The enrichment of the model allowed the optimisation algorithm to circumvent the condition of local minimum and converge to the target properties as seen in Figure 6.10b. The same positive outcome was observed using the properties of material C (Table 6.3) as the initial guess vector. The results of the optimisations concerning the SOF and MOF, presented in Table 6.4, highlights the superior performance of the proposed approach over the more conventional SOF. The indentation response measured by the P-h curve is insensitive to the plastic properties, in particular to the strain-hardening exponent, and hence the SOF model lacks of information to search for a better point, which in turn resulted in optimised parameters carrying forward the error throughout the optimisation, i.e. 5.2, 38.9 and 200%. In contrast, the second

Chapter 6- Development of a multi-objective function optimisation model

120 objective allowed the minimisation of the error to 0.14% in E, 0.15% in σy and 0% in n as summarised in Table 6.4.

a)

b)

Figure 6.10. Iteration history through a a) SOF and b) MOF optimisation model using material B as the initial guess

parameters.

Table 6.4. Set up and results for optimisation of mystical materials

Model Parameter Target

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121

6.6. Concluding remarks

The objective pursued in this chapter was an in-depth study of the performance of the proposed characterisation technique to recover elastic-plastic properties of power-law materials from the inverse analysis of the depth-sensing indentation test. The strong dependency of the residual imprint left by the indenter to the plastic behaviour of the indented material was exploited to complement the information provided by the P-h curve and so address the non-uniqueness issue of the inverse analysis of indentation.

Therefore, the inverse analysis is based on a multi-objective function optimisation model that finds a solution to minimise the error in both the P-h curve and the pile-up profile. The model was implemented in the MATALAB environment in order to invoke the lsqnonlin solver available in the Optimization Toobox^TM and to execute the pre-processing of FE models. Furthermore, MATLAB was linked to ABAQUS through a Python script in order to automate the post-processing of FE data.

The outcome was a robust model that has been proven to be capable of converging the three target parameters, E, σy and n, to within less than 0.5% error, despite the values selected as the initial guess. Furthermore, the methodological assessment of the proposed approach drew the following conclusions:

 The use of scaling coefficients, which sum to unity, can be adopted to equilibrate the global contribution of each objective function. The best compromise was found by reducing the components of the residual of the objective function concerning the data from the P-h curve; it was observed that the solution was approached faster and more accurately when c takes a value of 0.25.

 Reducing the overall error by ‘artificially’ reducing the contribution of the P-h curve, wP-hicP-h otP-herwise was observed to drive most of tP-he optimisation procedure in a ratio of approximately 3:1 when the scaling factors are excluded (c1 = c2 = 1), established a better equilibrium between the contributions of each

Chapter 6- Development of a multi-objective function optimisation model

122 objective function. Consequently, the information from both objective functions were better exploited, resulting in a faster convergence and an improved approximation to the target properties.

 The very strong convergence properties, the robustness and reliability of the trust-region algorithm [129] was reflected in a solution insensitive to the definition of bound constraints and the values of the initial guess. Furthermore, allowed the model to account for experimental errors arising from the metallographic preparation of the specimen such as surface roughness and the possibility of an arbitrary tilt.

 Even accounting for noise and outliers along the pile-up profile, the MOF model approaches the target parameters E, σy and n to within less than 2% error.

 The uneven piling-up arising from the misalignment between the indentation axis and the normal of the surface impinges on the solution as a single indentation may define a variety of constitutive behaviours. It has been shown that although a misalignment of 2° around both the +U1 and +U3 axes may significantly affect the solution due to the variability in the pile-up profiles along the three edges of the indenter, the mean of the optimised parameters E, σy and n approached the target parameters with a maximum error of 3, 13 and 28%, respectively. Limiting the surface tilt to within 1°, as recommended by ISO 14577 [13], reduced the corresponding mean errors to 2, 5.6 and 16%.

 The superior performance of the proposed MOF optimisation model, compared with a single-objective function model, was highlighted by its capability to distinguish between materials showing similar indentation responses, referred to in the literature as ‘mystical’ materials. While the single-objective function model was trapped on a local solution, in excess of 5.2, 38.9 and 66.7% away from the (global) target E, σy and n respectively, the multi-objective function optimisation converged to the target properties within an error of less than 0.15% in all three parameters.