Material model calibration by indentation, imprint mapping and inverse analysis

Dedicated to Professor Giannantonio Sacchi Landriani
https://doi.org/10.1016/j.ijsolstr.2004.01.025Get rights and content

Abstract

The identification of elastic–plastic material parameters by means of indentation tests and their finite element simulation is considered in this paper with the innovative provision of measuring the imprint geometry besides the indentation curves. The inverse analysis is carried out by a deterministic approach using conventional algorithms. The proposed methodology is validated using “pseudo-experimental” (computer generated) data with and without noise. Also friction between the indenter tool and the indented specimen is dealt with by inverse analysis and investigated through a parametric study. Sensitivity with respect to the sought parameters is examined for measurable quantities, including residual displacements on the specimen surface.

Introduction

Since a long time (see, e.g., Tabor, 1951; Mott, 1957) indentation tests have represented a practical methodology for material characterisation. A comprehensive survey of the subject can be found in Bhushan (1999). Not only hardness, like in the remote origin of this experimental methodology, but also material properties such as Young modulus and yield stress are inferred from the indentation curves (imposed force versus indenter penetration depth) derived from laboratory tests on various materials in many engineering situations. At present, indentation procedures are widely used in micro- and nano-technologies, in industries dealing with thin film coatings and in those producing micro- and nano-electro-mechanical systems (MEMS and NEMS) for a large and still growing number of applications.

Experiments have demonstrated that complex size effects arising at the nano-scale can be evidenced and fruitfully investigated by indentation tests (see, e.g., Begley and Hutchinson, 1998; Swadener et al., 2002). Recently, theories have been proposed in order to explain strength enhancement as the structural size diminishes and to interpret this phenomenon within the size effect framework. Micro- and nano-indentation might provide tools to validate such theories (Bhushan et al., 1996; Shu and Fleck, 1998; Guo et al., 2001; Yuan and Chen, 2001; Bazant and Guo, 2002; Vodenitcharova and Zhang, 2003).

Clearly, the main purpose of experimental tests on materials is the quantitative assessment of the parameters included in the constitutive models to be used for structural analysis and design. As the realism and the interpretative capacity of these models grow, the material parameters become more numerous and less amenable to direct separate measurement. Therefore, the combination of experiments, their simulation (by computer or, sometimes, by analytical solutions) and inverse analysis represents a spreading approach to material characterisation with roots in elastic–plastic structural mechanics (see, e.g., Maier et al., 1982; Bittanti et al., 1983; Bui, 1993). Such an approach has been applied to indentation tests in the recent literature, by exploiting the traditional indentation curves as source of experimental data (see, among others: Giannakopoulos and Suresh, 1999; Venkatesh et al., 2000; Nakamura et al., 2000; Tardieu and Constantinescu, 2000; Dao et al., 2001; Kucharski and Mróz, 2001; Huber et al., 2002; Gu et al., 2003). However, Capehart and Cheng (2003) showed that 1% noise level precludes the accurate identification of strain hardening parameters when the only experimental information consists of loading–unloading curves in conical indentation. This result agrees with the remarks by Venkatesh et al. (2000) and by Dao et al. (2001), according to whom plastic properties extracted from indentation curves can be strongly influenced by small variations of input data.

At present, instruments apt to measure the local profile of the indented specimen are fairly frequently available in laboratories to different purposes. In particular, e.g., atomic force microscopes (AFM) can provide an accurate mapping of the geometrical consequences of micro- and nano-indentation tests (see, e.g., Bhushan et al., 1996). Consideration of imprint profiles for accurate assessment of the contact area has been proposed by Taljat et al. (1998); Matsuda (2002) and Tunvisut et al. (2002) with reference to indentation tests on metals. Fairly satisfactory results were reported by Taljat et al. (1998) for various material properties in the case of spherical indentation. However, the formulae on which this analytical method is based are unable to accurately determine the strain hardening exponent (unless the yield stress or, at least, the ratio of yield stress over Young's modulus are known), and the load–depth curves can significantly be affected by friction, i.e. by a parameter which is usually difficult to evaluate.

The method proposed by Tunvisut et al. (2002) for conical and pyramidal indentation relies on the accurate measurement of the final contact area, besides on maximum load and initial unloading slope of the indentation curves; it turns out that 5% error in the slope of the unloading branch may give rise to errors up to 30% in yield stress estimates and up to 50% in the estimated hardening exponent.

In this paper, geometrical data concerning the imprint generated in the specimen by the indentation test are considered an important experimental information, supplementary to the usual one consisting of measured indentation depth at various instants of the force cycle imposed on the indenter. Both kinds of data are employed as input in the subsequent inverse analysis for parameter identification. Information about the whole geometry of the residual imprint is exploited, not only the contact area like in the above cited papers. The potentialities and limitations of this methodology are preliminarily investigated herein with reference to a conventional isotropic elasto-plastic material model, namely Chaboche (1986) model with non-linear kinematic hardening.

In Section 2, a brief description is provided of the conventional finite element (FE) discretisation generated and of the material model adopted in order to simulate the indentation tests and their consequences on the specimens in terms of residual elastic–plastic deformations.

The inverse analysis employed for the parametric identification, outlined in Section 3, consists of a classical least-square procedure.

Any identification method is affected by an error which is intrinsically connected with the model approximations and with the noise associated to the experimental tests. However, the potentialities of an experimentally novel approach to inverse analysis, like the present one, can preliminarily be investigated by means of “pseudo-experimental” data, i.e. by giving to measurable quantities values which are generated through a direct (or “forward”) analysis performed by introducing an a priori known set of material parameters into a simulation of the experiment.

Section 4 gathers, in comparative critical terms, the results of various numerical exercises carried out by inverse analysis. The purpose is to elucidate the main features and some pros and cons of the enriched indentation methodology proposed herein.

In view of the importance of sensitivity analysis in the design of laboratory tests, see e.g. Kleiber et al. (1997), Section 5 is devoted to the sensitivity of measurable quantities with respect to unknown parameters, with emphasis on the novel data achievable by mapping the imprint and the surrounding surface of the specimen.

The interpretation and the quantitative appraisal of the role of friction on the indenter–specimen interface may be crucial to the effectiveness of the extended identification method. Hence, these problems are focussed in Section 6.

The closing remarks (Section 7) are intended as a tentative assessment of the present preliminary results and of their future prospects on the use of imprint mapping (besides the traditional indentation curves) for material parameter identification through indentation tests at various scales.

Section snippets

Computer simulations

The material model considered herein is the popular associative elastic–plastic–hardening isotropic constitutive model proposed by Chaboche (1986). In its formulation specialised to kinematic hardening only, it can be expressed by the following rate relationships:ε̇ij=ε̇ije+ε̇ijpε̇ije=1+νEσ̇ij+1−2νEṗδijε̇ijp=λ̇fσijf=32ij−Xij)(σij−Xij)−σ0Ẋij=23cε̇ijp−γXij23ε̇rspε̇rspf⩽0λ̇⩾0fλ̇=0In the above formulae: εij represent the strain tensor components, decomposed into their elastic (reversible) ε

Inverse analysis

A conventional deterministic batch (non-sequential) approach is adopted herein for the parametric identification, namely the available experimental information are exploited all together without processing uncertainties which may affect both the measurements and the system modelling. Sequential stochastic approaches such as Kalman filtering (see, e.g., Bittanti et al., 1983; Stavroulakis et al., 2003) will be employed in planned further studies.

The inverse analysis problem is formulated below

Comparative numerical results

Some inverse analyses are presented and discussed in this Section in order to elucidate in computational terms potentialities and limitations of imprint mapping, as here proposed supplement to the indentation curves in the considered methodology.

Sensitivity

The design of experiments to be combined with parameter identification procedures may be enhanced by preliminary sensitivity analyses (see, e.g., Kleiber et al., 1997). In fact, these are intended to quantify the influence of each sought parameter on measurable quantities and, hence, to corroborate the conjectures on its identifiability.

In the present context, sensitivity matrices have been repeatedly computed by forward finite difference, namely:qi(xk)xkxk=x̄kΔqi(x̄k+Δxk)−Δqi(x̄k)Δxkwhere qi

On the role of interface friction

Friction between the indenter and the specimen is often neglected in the literature on hardness tests, in view of the following circumstances (see, e.g., the comprehensive monograph by Bhushan, 1999): the tip angle of the conical or pyramidal indenter can be selected in order to minimise frictional effects; the influence of friction is usually weak on the traditional experimental data consisting of force versus indentation depth relationship. In the preceding computational exercises (Section 4)

Conclusions and future prospects

The objective pursued in this paper is a preliminary assessment of the potentialities of an indentation method which provides, besides the usual experimental data concerning force versus penetration depth, also systematic measurements concerning the residual deformed configuration of the specimen, namely the imprint and the surrounding area. In the light of the present study, such measurements (which can be carried out at the nano- and micro-scale, e.g., by an atomic force microscope) turn out

Acknowledgements

The authors are grateful to Prof. G. Coccia (Department of Chemistry, Materials and Chemical Engineering `Giulio Natta', Politecnico di Milano) and to Ing. E.J. Chiarullo for information and discussions concerning indentation tests at the micro-scale and related use of atomic force microscope.

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