Indentation and imprint mapping for the identification of constitutive parameters of thin layers on substrate: Perfectly bonded interfaces

Abstract

Indentation tests are at present frequently employed for the identification of material parameters at different scales. The indentation curve provides experimental data for the calibration of mechanical models through traditional semi-empirical formulae or through simulation of the test and inverse analysis. A recently proposed innovative inverse analysis technique combines the traditional indentation test with the mapping of the residual deformations (imprint), thus providing experimental data apt to be used to identify isotropic and anisotropic material parameters in more accurate fashion and in larger number. In this paper, such new methodology is employed for the identification of material properties in film–substrate systems. The film and a significant portion of the underlying bulk material are incorporated in the finite element models built up to simulate the indentation test in finite strain regime. The substrate material properties are considered among the unknown parameters to be identified through inverse analysis, carried out by a batch, deterministic approach, using conventional optimisation algorithms for the minimisation of a suitably defined discrepancy norm. Several numerical examples are discussed in order to test the performance of the proposed methodology in terms of result accuracy and computing effort.

Introduction

Research in the field of surface engineering is fast growing since the life-time of industrial products and the performance of materials for advanced mechanical applications are more and more often enhanced by the application of coatings, e.g., see [1]: in MEMS technology, thin gold layers are deposited on silicon to allow electrical connections with other devices; the surfaces of metal-working tools, precision ball bearings and computer magnetic storage systems are coated by hard ceramic and diamond like materials in order to improve wear and corrosive resistance; friction and wear are alternatively lowered by boundary lubrication provided by the surface deposition of soft, low shear strength material, such as MoS₂ or graphite.

The most commonly used coating technique consists of vapor deposition in a variety of modes, see [1]. Parameters like deposition rate, temperature, gas pressure, impurity level, affect the characteristics and the mechanical properties of the coating and of the interface between the underlying material. The mechanical properties of thin films can be strongly different from those of the corresponding bulk materials also because of their unique microstructure, large surface to volume ratio and reduced dimensions.

This paper presents a methodology for the determination of the elastic–plastic characteristics of these layered systems, based on the instrumented indentation test (see, e.g. [2]). This popular technique is frequently used to evaluate the local mechanical properties of thin films due to its capability to deform materials on a very small scale. However, in order to obtain accurate estimates, it is crucial to distinguish between film and substrate contributions to the overall specimen response.

In the approach most widely used, penetration depth is kept smaller than some critical value that depends on film–substrate properties. The presence of the substrate and its influence on the test results are hence neglected, and formulae or methods developed for homogenous materials are applied, see, e.g. [3], [4], [5]. Significant research work has been devoted to the determination of this critical penetration depth, say $h_{\text{c}}$. A well known empirical rule for hardness measurement equals $h_{\text{c}}$ to about one-tenth of the film thickness. However, it has been proven that $h_{\text{c}}$ depends on the mechanical properties of both film and substrate (described, e.g., by the ratio between the yield strengths and elastic moduli of coating and substrate: $a_{\text{y}}\equiv {\sigma }_{\text{yc}}/{\sigma }_{\text{ys}}$ and $a_{\text{E}}\equiv E_{\text{c}}/E_{\text{s}}$, respectively, with clear meaning of the introduced symbols) as well as on the indenter geometry. The results of recent finite element (FE) parametric studies of film–substrate systems (see, e.g. [6], [7] and references therein) permit to point out some general qualitative rules, for instance: the critical indentation depth increases with decreasing $a_{\text{y}}$ and decreases with increasing indenter tip radius; for a hard film on a soft substrate ($a_{\text{y}}>10$ and $a_{\text{E}}>0.1$) indentation experiments should be carried out up to a maximum depth of 5% of the film thickness to avoid any influence from the substrate; viceversa, for a soft film on a hard substrate ($a_{\text{y}}<1$) the one-tenth rule is too severe and the coating hardness can be correctly measured for a penetration depth up to 30% of the film thickness. In any case, it could be rather difficult to get reliable parameter estimates for very thin films resting on these rules, since the suggested indentation depth could be too small to consider the indentation response representative of the macroscopic material behavior (for instance, $h_{\text{c}}$ could result of the same order of magnitude of the material microstructure). Sophisticated material models accounting for size effects and microscopic-to-macroscopic transition procedures should be eventually adopted to overcome this limitation, see [8], [9], but with a clear and sometimes practically unaffordable increase of the problem complexity.

Alternatively, the mechanical parameters characterizing the film can be derived, once the substrate properties are known, through empirical and analytical models developed for the film–substrate system interpreted as a layered composite material, see [10], [11]. Beyond the elastic range, the film properties can be estimated through techniques based on FE analyses and either the rule of mixture [12], [13] or dimensional analysis [14], [15].

Huber et al., see [16], [17], developed a neural network methodology for the identification of material parameters entering elastic–plastic constitutive laws of film–substrate systems, considering indentation depth up to twice the film thickness. Their procedures, really effective in terms of computing costs, basically rest on the assumption that the same constitutive description can be ascribed to both the coating material and its support, which is not always the case.

An alternative model calibration approach based on indentation test and imprint mapping (through atomic force microscope or laser profilometer, depending on the imprint size) has been proposed by Bolzon et al. [18] and Bocciarelli et al. [19]. These authors have shown that the use of information consisting of the geometry of the residual imprint, besides the traditional indentation curve, can enhance the effectiveness of the calibration process by increasing both the number of the identifiable parameters and the reliability of their estimate. This procedure is applied here for the identification of material properties of ductile films on hard substrates (as found, e.g., in MEMS technology) and of hard protective ceramic coatings on metals, in the case of perfect adhesion between the layers. The case of weak debonding interfaces is the subject of current research.

Strain hardening is neglected for the present demostrative purposes, even if it could be dealt with as in [18], but friction between the indentation tool and the specimen surface (which might affect significantly the response of piling-up materials like mild metals) and the internal friction angle for pressure-sensitive materials like ceramics are taken into account.

The substrate material properties are considered among the unknown parameters to be identified through inverse analysis, carried out by a batch, deterministic approach that uses conventional optimization algorithms for the minimization of a suitably defined discrepancy norm, expressing the difference between measured and computed quantities, as in [20], [21].

The performance of the proposed procedure is verified by means of some numerical exercises employing pseudo-experimental data, i.e. data generated by a direct analysis assuming a certain value for the set of the sought parameters, which should then be found as solution of the inverse problem. The significance of experimental and modelling noise on measurable quantities is addressed by introducing random errors in the computer generated data to be input in the inverse analysis.