Pierre-Emmanuel Mazeran a , Moez Beyaoui b , Maxence Bigerelle a , Michèle Guigon a Determination of mechanical properties by nanoindentation in the case of viscous materials A mechanical model based on a generalized Kelvin – Voigt model has been developed to explain and fit the nanoinden- tation curves realized on three amorphous polymers (PC, PMMA and PS). This model includes the responses of quadratic elastic (spring), viscoelastic (two Kelvin – Voigt elements), plastic (slider) and viscoplastic components (dashpot). It is able to fit nanoindentation curves during loading, unloading and hold time periods. With the values of the model parameters and the value of the contact area calculated with the Oliver and Pharr method, it is possible to calculate the values of the mechanical properties of the polymers. A good agreement is found between these values and those obtained with conventional methods. Keywords: Nanoindentation; Polymer; Elastic modulus; Hardness; Viscosity 1. Introduction The determination of the mechanical properties of bulk ma- terials or thin films by instrumented indentation at a nan- ometer scale was developed and has been widely used dur- ing the last two decades [1, 2]. This technique is now currently used to determine the hardness, the elastic modu- lus or other mechanical properties on sufficiently low vol- umes to obtain a local measurement of the mechanical properties. Methods that allow determining the hardness H and the elastic modulus E by taking into account the elastic return of material during the unloading stage have been de- veloped [3 – 5]. In the case of isotropic elastoplastic materi- als, the elastic modulus and the hardness can be given with excellent precision [3]. Nevertheless, in the case of materi- als whose mechanical response is time-dependent, such as polymers or biological materials, the description of the me- chanical behavior by these two mechanical properties is ob- viously inadequate [6 – 19]. The development of the continuous stiffness measure- ment (CSM) method allows the continuous measurement of three parameters (load, contact stiffness and phase), then the calculation of the storage and loss moduli as well as the hardness of the material. It is an interesting opportunity but it remains limited to models with three parameters which are generally insufficient to describe the mechanical beha- vior of a material whose behavior depends on time [10, 14, 16, 18 – 21]. Another approach consists in fitting load – dis- placement curves using various mechanical models includ- ing viscous behavior [8, 10, 14, 16 – 17]. These models are based on the combination of elastic, viscoelastic, plastic and viscoplastic behaviors, modeled by springs, Kelvin – Voigt elements (spring in parallel to a dashpot), sliders and dashpots respectively [22]. These models allow fitting of experimental curves and de- termination of some mechanical properties with more or less convincing results in the quality of the fits and of the me- chanical values that have been computed. In this work, an elastic viscoelastic plastic viscoplastic model (EVEPVP) (Fig. 1) is proposed to fit the indentation curves and to deter- mine the mechanical properties of the indented materials. Each element of the model has an independent quadratic re- sponse: The root square of the load is proportional to the in- dentation depth and/or the indentation depth rate (Table 1). This model makes it possible to suitably adjust the displace- ment curves during the loading and unloading stage, as well as the hold load periods, after the loading and the unloading stages on three massive amorphous polymers: polycarbonate (PC), polymethyl methacrylate (PMMA) and polystyrene (PS). From the estimate of the contact area, it is possible to determine the relation between the displacements of each element and to go back to the material mechanical properties. Moreover, it shows that the values of the mechanical proper- ties determined by our method are in good agreement with the quasi static values determined by the Oliver and Pharr method [5] and the values obtained from tensile tests. 2. Materials Three amorphous polymers (PC, PMMA, PS) were tested. The amorphous polymers were preferred to semi-crystal- line polymers to prevent any heterogeneity of crystallinity and thus of the mechanical properties. P.-E. Mazeran et al.: Determination of mechanical properties by nanoindentation in the case of viscous materials Int. J. Mat. Res. (formerly Z. Metallkd.) 103 (2012) 6 715 Fig. 1. Kelvin – Voigt generalized model with quadratic elements. The model is composed of a spring (elasticity), two Kelvin – Voigt ele- ments (viscoelasticities), a slider (plasticity) and a dashpot (viscoplasticity). a Laboratoire Roberval, UMR CNRS-UTC 6253, Université de Technologie de Compiègne, Compiègne, France b Unité de Dynamique des Systèmes Mécaniques, Ecole Nationale d’Ingénieurs de Sfax, Tunisie  2012 Carl Hanser Verlag, Munich, Germany www.ijmr.de Not for use in internet or intranet sites. Not for electronic distribution.