Experimental Mechanics DOI 10.1007/s11340-011-9561-5
Extraction of Mechanical Properties with Second Harmonic Detection for Dynamic Nanoindentation Testing G. Guillonneau & G. Kermouche & S. Bec & J.-L. Loubet
Received: 11 February 2011 / Accepted: 22 September 2011 # Society for Experimental Mechanics 2011
Abstract In this article, a new method based on the detection of the second harmonic of the displacement signal to determine mechanical properties of materials from dynamic nanoindentation testing, is presented. With this technique, the Young’s modulus and hardness of homogeneous materials can be obtained at small penetration depths from the measurement of the second harmonic amplitude. With this innovative method, the measurement of the normal displacement is indirectly used, avoiding the need for very precise contact detection. Moreover, the influence of the tip defect and thermal drift on the measurements are reduced. This method was used for dynamic nanoindentation tests performed on fused silica and on an amorphous polymer (PMMA) because these materials are supposed not to exhibit an indentation size effect at small penetration depths. The amplitude of the second harmonic of the displacement signal was correctly measured at small depths, allowing to calculate the Young’s modulus and the hardness of the tested materials. The mechanical properties calculated with this method are in good agreement with values obtained from classical nanoindentation tests. Keywords Nanoindentation . Second harmonic detection . Dynamic measurements . Tip defect . Hardness G. Guillonneau (*) : S. Bec : J.-L. Loubet Ecole Centrale de Lyon, Laboratoire de Tribologie et Dynamique des Systèmes, Université de Lyon, UMR 5513 CNRS/ECL/ENISE, 69134 Ecully, France e-mail:
[email protected] G. Kermouche Ecole Nationale d’Ingénieurs de Saint-Etienne, Laboratoire de Tribologie et Dynamique des Systèmes, Université de Lyon, UMR 5513 CNRS/ECL/ENISE, 42000 Saint-Etienne, France
Introduction Indentation tests were developed at the beginning of the twentieth century, with hardness tests performed by Brinell [1], using a hard steel sphere as an indenter. The technique was industrialized and the Brinell test became a standard test to determine the hardness of materials. Other standard hardness tests have been developed afterwards and are routinely used, like the Vickers or the Knoop tests [2], which used indenters with different geometries to determine materials hardness. For these tests, the hardness is calculated from the measurement of the projected contact area from optical imaging of the residual indent. This clearly imposes a lower limit on the indentation length scale. The Rockwell [2] hardness test was the first standard test which does not use optical observation but uses measurement of the tip displacement to determine the hardness of materials. Since the 1970s, micro and nano-hardness techniques have been developed and are currently widely used [3–6]. They are based on the simultaneous and continuous measurement of the load and the penetration depth of an indenter which is pressed normally to the sample surface. The measured penetration depth is used to calculate the projected contact area, which is necessary to calculate the hardness of the tested material. Moreover, in the 1970s, Bulyshev [3] showed that it is possible to relate the load-displacement curve to the elastic modulus of the indented material. Doerner and Nix [7], Oliver and Pharr [8] and Loubet and al. [9, 10] have greatly contributed to the understanding of nanoindentation tests and to the development of methods to analyze the measured data. In addition to Young’s modulus and hardness, other information can be extracted from the load-displacement curves. For instance, it is possible to estimate the fracture toughness of brittle materials. Using a tangential force
Exp Mech
sensor, nanoscratch tests can also be performed [11]. More recently, in the 90s, new tests have been developed using dynamic measurement. Named CSM (Continuous Stiffness Measurement) [8–13], it offers significant improvements in nanoindentation techniques. CSM consists in superimposing an oscillation of small amplitude to the force signal. From the analysis of the dynamic response of the system, it is possible to calculate the dynamic contact stiffness and the dynamic contact damping coefficient. So, in addition to elastic properties, viscoelastic properties can be measured using this technique, which is useful to characterize polymeric materials for example. This dynamic method makes possible the measurement of mechanical properties continuously during the penetration, which is very interesting for heterogeneous materials like multilayered or graded materials where properties vary with penetration depth. Stress relaxation tests can also be performed. With nanoindentation experiments, the main difficulty in interpreting the data comes from the precise determination of the projected contact area, especially at a small scale. In literature, different formulas and procedures are proposed to calculate this area from the displacement measurement [8, 9]. However, for very small penetrations (lower than 100 nm), the identification of material properties is particularly difficult to obtain. The aim of this paper is to propose a new method based on second harmonic detection to determine the material’s mechanical properties from dynamic nanoindentation tests. With this technique, elastic modulus and hardness can be calculated at low depths from equations which use the measurement of the second harmonic amplitude. With this method, the precise contact detection is not needed and the influence of the tip defect for small penetrations is reduced. In this paper, the classical methods used to determine the local elastoplastic properties of materials from indentation curves will be first shortly described. The uncertainties due to the measurement of the penetration depth will be discussed. Secondly, the proposed second harmonic detection method will be explained. In the third part, experimental details will be given and in the last part, results obtained on fused silica and on PMMA will be given and compared to results obtained from classical nanoindentation tests.
About Nanoindentation Measurement
Fig.1 (a) Load-displacement curve in nanoindentation test (b) Schematic illustration of indentation with a non-perfectly sharp indenter (zj is the uncertain height due to the tip defect)
both elastic and plastic deformations occur, resulting in the formation of a plastic impression conforming to the indenter shape. Nanoindentation hardness is defined as the indentation load divided by the residual projected contact area, Ac. It corresponds to the mean contact pressure. From the load-displacement curve (Fig. 1(a)), the hardness H can be obtained at maximum load Fmax by the following equation: H¼
Fmax Ac
ð1Þ
Classical Indentation Measurement Nanoindentation techniques are widely used to measure the Young’s modulus E and the hardness H of materials. Figure 1(a) shows a typical load-displacement curve obtained during indentation on an elastic-plastic sample. When the indenter penetrates normally into the sample,
The equivalent contact radius a can be defined from the projected contact area Ac by: Ac ¼ pa2
ð2Þ
The contact radius a is determined from the loaddisplacement curve, as will be discussed in the next
Exp Mech
paragraph. For non viscous material, the contact stiffness S is obtained from the slope of the tangent to the unloading curve at maximum load: dF S¼ dz
Ac ¼ pa2 ¼ 24:56z2c
»
S 2ba
ð4Þ
Where β is a correction coefficient used to account for the non axis-symmetry of the indenter (β=1 for axissymmetric indenters) [14]. The reduced contact modulus Ec* includes elastic deformation that occurred both in the sample and the indenter. The reduced Young’s modulus E of the tested material can be calculated using the following equation: 1 1 n2 1 1 n 2i ¼ » » ¼ E Ei E Ec
ð7Þ
ð3Þ
Where F is the normal load and z is the measured tip displacement. Based on Sneddon’s relationship [24], for a solid of revolution, a geometry-independent relation involving the contact stiffness, contact radius and reduced contact modulus Ec* can be derived: Ec ¼
the projected contact area is expressed by the following geometrical equation:
ð5Þ
Where Ei and νi, E and ν are the Young’s modulus and Poisson’s ratio of the indenter and the sample respectively. To calculate the contact stiffness S from equation (3), Oliver and Pharr [8] proposed to describe the unloading curve not with a linear function as suggested by Doerner and Nix [7], but with a power law expression: m F ¼ B z zf ð6Þ Where zf is the residual depth (see Fig. 1(a)), m and B are two coefficients empirically determined by fitting the unloading curve. The contact stiffness S is then calculated by differentiating equation (6) at the maximum displacement z = zmax. The contact stiffness can also be continuously measured during the whole test using the dynamic CSM method (Continuous Stiffness Measurement) [8, 12, 13]. In such dynamic tests, S is thus named dynamic contact stiffness. The CSM method will be explained in details in paragraph III. To determine hardness and elastic modulus, the next step is to determine the projected contact area from the displacement measurement. This will be explained in the following paragraph. Determination of the Projected Contact Area Under Load The projected contact area Ac can be calculated as a function of the contact depth. For example, with a Berkovich indenter,
With zc the contact depth (see Fig. 1(b)). Thus the main difficulty is first to relate the contact depth to the measured tip displacement. It is well known that it is not possible to determine exactly the contact depth with classical indentation measurements, i.e., load, tip displacement and also contact stiffness measurements [15]. Nevertheless, a fairly good approximation of the contact depth can be obtained using models. The most frequently used is the Oliver and Pharr method [8] which states that the contact depth zc is linked to the measured tip displacement z, by the following equation: zc ¼ z "
F S
ð8Þ
Where ε is a parameter which depends on the indenter geometry (for a Berkovich indenter, ε=0.75) and F is the normal load. The contact area Ac is then obtained through a polynomial function of the contact depth determined from calibration tests [8, 16]: 1=128 Ac ¼ C0 z2c þ C1 zc þ C2 z1=2 c þ ::: þ C8 zc
ð9Þ
Using this method, equation (8) implies that the contact depth is lower than the measured total displacement (zc
