ABSTRACT
A modified form of the two-slope method used for the determination of mechanical properties of a material is presented in this paper. Modified expressions for the determination of slopes of the loading and unloading curves make use of the energy based parameters which are independent of the indentation size. A correction factor is also introduced to account for the inward radial displacement of material’s surface points which has important implications on the accuracy of the mechanical properties. Mechanical properties obtained after these modifications compares well with the experimental results. The elastic modulus and hardness obtained by the proposed method precisely describe the elasto-plastic behavior of the metals considered in this study which further confirms the accuracy of the method described herein. The proposed method enhances our understanding of the behavior of a material at very small scale of length and may be extended to determine the mechanical properties of materials other than metals.
Introduction
Several methods for the analysis of nanoindentation data to determine the mechanical properties of thin films and bulk materials exist in the literature. All these methods may be classified based on: (1) how load-displacement curves are utilized in the determination process; and (2) how the analysis is performed. Based on first criterion nanoindentation data analysis procedure may be divided into three groups: (i) The unloading curve method (Oliver and Pharr method) [1-3]; (ii) the loading curve method (Methods by Hainsworth et al. [4, 5] and Malzbender et al. [6]) and (iii) the hybrid method (Cheng and Cheng [7, 8]; two-slope method [9, 10]; and finite element method [2]). On the basis of second criterion, these methods are classified as tools for either reverse analysis or forward analysis [11]. In the reverse analysis, mechanical properties are extracted from the experimental load displacement curve. On the other hand, mechanical properties are used to model the experimental load-displacement curve in the forward analysis. Loading curve method and finite element method are usually used as forward analysis tools. In all of these methods, the most important thing is to understand the information contained in the load-displacement curves. But, both the presence of non-uniform stress and displacement fields in the vicinity of contact and complex deformation processes that take place during indentation preclude us in gaining such understanding [2]. Therefore, these methods are either incomplete or heavily depend on the empirical observations.
Oliver and Pharr method, also referred to as an area function technique, is the most widely used method for the determination of mechanical properties of a material. Determination of the contact area is a prerequisite of this technique which is generally obtained with large errors when the data from inhomogeneous samples or from materials that show significant pile up in the perimeter of contact during indentation. Second limitation of this conventional technique is related to computational efforts which are extremely important when grid indentation technique [12] is required to be performed on inhomogeneous samples. Attaf [13] has shown that the computational efficiency of the area function technique can be significantly improved if correlations among nanomechanical quantities dictated by the unified correlations diagram [13] are considered. On the other hand, Oliver proposed two-slope method [9] that does not require determination of contact area at all to determine the mechanical properties. Surprisingly, these studies have not received much attention of material scientist and engineers even though their findings may prove to be instrumental in developing an attractive alternative method to the area function technique. In the two-slope method, stiffness of the load displacement curves are determined by differentiating the algebraic expressions used to fit the loading/unloading curves and then evaluating the differential at peak indentation load. It has been shown that unloading curve parameters are dependent on indentation size [15] and thus pose difficulty in characterizing a material based on these parameters. For some brittle material, it is possible that the power law may not be a good choice to describe the unloading curve [15]. On the other hand, equation used to represent the loading curve itself depends on the indentation size. For instance, a parabolic representation for loading curve is suitable at large peak indentation load for conical or Berkovich indenter. But at small peak indentation load, a second degree polynomial is the appropriate choice as bluntness in the tip of the indenter is more pronounced at this level of load [4]. This anomaly in the representation of loading and unloading curves may be overcome if functional analysis based expressions [16] are used to represent them. A Berkovich indenter does not confirm to an axisymmetric condition. Thus, a correction factor is required in order to apply Sneddons’ solution to determine the mechanical properties from the data acquired using this indenter. King [17] suggested that a value of 1.034 may be used for this correction factor. Recent studies by Meza et al [18], however, have shown that this factor is dependent on the penetration depth. Method that is based on Sneddon’s solution ignores one more correction factor that take proper deflected shape of the material surface points during indentation as introduced by Hay et al. [19]. This correction factor has important implication on the accuracy of the computed mechanical properties. The very intent of this study is to ameliorate the existing two-slope method by giving due consideration on the proper representation of loading/unloading curves and on the implementation of the correction factors described above.
Experimental data
For the purpose of verification, nanoindentation data pertaining to aluminum, copper and tungsten is used. These materials may be considered as elastic perfectly plastic material as the percentage elastic recovery of these materials is less than 10%. First set of data are digitized from the literature [1, 4]. These data were obtained using a Berkovich indenter with peak indentation load greater than 100 mN. Second set of data were acquired by carrying out nanoindentation test on commercially available oxygen free copper using same indenter with peak indentation load of 1.0, 1.5, 2.0 and 2.5 mN.
Two-slope method: background
As the name suggests, two-slope method essentially makes use of the slopes of the loading and unloading curves evaluated at the point of maximum depth of penetration. Oliver [9] derived expressions for reduced elastic modulus (Er), contact area (Ac) and hardness (H) of a material in the following forms:
where C is constant that relates the contact area and contact depth by the relation
denotes the geometric factor which are respectively equal to 24.56 and 0.75 for a Berkovich indenter. The factor ft appearing in Eq. (1) accounts for the lack axial symmetry of a Berkovich indenter. Sl and Su respectively denote slopes of the loading and the unloading curves as shown in fig 1. These slopes are usually evaluated by analytically differentiating the algebraic expressions (obtained by curve fitting) used to represent them. Two well known equations are used to derive above expressions: (i) fundamental relation among Su, Er and Ac; and (ii) equation of loading curve capable of describing the elasto-plastic deformation of a material. In mathematical form; these equations may be written as
where
is a parameter supposed to be an indicator of bluntness in the tip of the indenter. The advantage of this method is that it does not require the computation of area function, a key requirement in the conventional Oliver and Pharr method, for the determination of mechanical properties. The essence of this method is that Eq. (5) must precisely predict the experimental loading curve when the values of mechanical properties determined using Eq. (1) and (3) are substituted in it.
Energy dissipation and nanomechanical quantities
Energy dissipated during indentation may be employed to represent the load-displacement curves which, in turn, may be used to evaluate the slopes of these curves. Starting with the basic definition of the energy based parameters; a procedure for the determination of the loading/unloading slopes is described in the following.
Energy constants
Total, elastic and plastic energy dissipated during indentation are found to be proportional to the absolute energy of indentation [20]. Absolute energy (WS) is considered as the maximum possible energy that can be dissipated during indentation of a material. Mathematically, it is given by the area of the triangle OAhmax in the loading displacement diagram as shown in fig 1. The ratio of absolute to total and elastic energy are constant for a given material may be expressed as:
Where vT and vE respectively describe the total and elastic energy constants and are primarily related to the curvature of the loading unloading curves. Large values of these constants simply mean that the loading and unloading curves have more curvatures. The elastic energy constant may also be related to the percentage elastic recovery of a material. The material which recovers less upon withdrawal of load has higher value of this constant.
Fig 1 Typical nanoindentation load-displacement diagram with terminology
Slopes of loading/unloading curves
Attaf [15], using functional analysis, found that the loading and unloading curves may be respectively represented by the following expressions:
Where Pn]ax and hmax are peak indentation load and maximum depth of penetration respectively. The approximating power of the above expressions is shown in fig 2a and 2b for aluminum and tungsten respectively. Excellent agreement between experimental and theoretical curves could be seen in the figures. Note that, for the materials which recover more upon withdrawal, such an excellent agreement is normally not obtained for unloading curves. Knowing the expressions for loading and unloading curves, their slopes may be evaluated by analytically differentiating the above expression and then evaluating the derivative at the point of maximum depth of penetration as
While Eq. (10) gives reasonable estimate of loading slopes, unloading slopes are overestimated if Eq. (11) is used. However, Eq. (11) may be used to determine the contact depth between the indenter and material to be indented in terms of elastic energy constant and maximum depth of penetration. Slope of the unloading curve can then be determined from the known value of contact depth following the procedure used to determine the contact depth from the known value of initial unloading stiffness in the conventional Oliver and Pharr method.
Fig 2 Modeling nanoindentation load displacement curves using functional analysis based expressions: (a) Aluminum; (b) Tungsten
Contact depth and elastic energy constant
As mentioned, the initial unloading stiffness or slope of the unloading curve may be determined from the known contact depth which may be expressed as a fraction of maximum depth of penetration as [13]:
Equation (12) is found to be applicable even for heterogeneous material such as concrete. For copper, an error of less than 3% was obtained. Total depth of penetration may be additively decomposed into material surface deformation and the contact depth. During indentation, material surface deformation is found to be proportional to the ratio of peak indentation load and initial unloading stiffness. Using these conditions one can obtain a relation for slope of the unloading curve in terms of elastic energy constant as
where £ is known as geometric constant and is equal to 0.75 for a Berkovich indenter. Note that it is the geometric constant that makes the difference between Eq. (11) and Eq. (13). Equation (11) overestimates the slope of the unloading curve by 25% for a Berkovich indenter than that given by Eq. (13). In this way, the main ingredients of the two-slope method i.e. loading and unloading slopes may be determined without resorting to curve fitting method.
Modified two-slope method
The issue related to the representation of loading/unloading curves is already discussed in the preceding sections and we concluded that it is advantageous to use energy based expressions to determine their slopes. The second issue is related to the use of correction factors in the Sneddon’s solution and loading curve equation respectively given by Eq. (4) and Eq. (5). Role of these two correction factors have been studied from different perspectives by many researchers. However, no definite conclusions regarding their determination and implementation are found to exist in the literature. It would be reasonable to assume that the correction factors due to lack of axial symmetry and inward radial displacement of materials’ surface points exits simultaneously and may be employed simply by multiplying these two factors if a Berkovich indenter is used to obtain the load displacement curve. This assumption regarding implementation of correction factors is also used by Meza et al. [18]. By substituting Eq. (10) and (13) in Eq. (1) – (3) and by replacing ft with Py , following expressions for elastic modulus, area of contact and hardness is obtained respectively.
In above expressions, factor yis calculated according to the formulae given in Hay et al [19] for a Berkovich indenter. A value of 1.034 for the factor is used to check the validity of the proposed expressions when the peak indentation load is in excess of 100 mN. Knowing all the parameters appearing in above equations, the mechanical properties for aluminum, copper and tungsten were determined. As shown in table 1, the elastic modulus of all these materials compare well with that obtained from the conventional Oliver and Pharr method. But Eq. (15) yields higher values of the contact area for all these material than the conventional method. It is due to the fact that the initial unloading stiffness given by Eq. (13) is still larger than that obtained by the differentiation of the power law used to represent the unloading curve. The contact area determined by the conventional method cannot be questioned as it gives accurate value of contact area at least for materials like aluminum, copper and tungsten [1]. As a result, an area correction factor is used to obtain the accurate value of the contact area and hence the hardness. We assume that this correction factor is equal to (a/3)2. With this assumption the hardness value for each material is calculated (as shown in table 1) which is in reasonable agreement with that obtained by both Oliver and conventional methods. Elastic modulus and hardness are substituted back in Eq. (5) for each material and corresponding loading curve is modeled. Excellent agreement between theoretical and experimental loading curve could be seen in fig 3 which further ensures the accuracy of the mechanical properties determined using this method and validates our assumption concerning area correction factor.
Fig 3 Modeling nanoindentation loading curves using Eq. (4) with values of Er and H determined in this study.
Following the above procedure, elastic modulus and hardness value of copper subject to different peak indentation load are calculated. Here different values for / are assumed successively, but best results were obtained when a value of 1.126 is selected. This is in agreement with the findings of Meza et al. which states that / depend on the maximum depth of penetration: it attains higher values at lower depth and remains constant when the depth of penetration is large. The elastic modulus and hardness values obtained using modified method and Oliver method are compared with those obtained experimentally as shown respectively in fig 4 and fig 5. It can be seen that the values obtained by the proposed method is in reasonable agreement (< ±10%) with experimental values. On the other hand, Oliver method gives error in excess of 10% for both elastic modulus and hardness as shown respectively in fig 4a and fig 5a. From this observation, it may be concluded that the proposed method gives consistent results regardless of the indentation size. Oliver method is more prone to error when indentation size is small. This is very important in case of thin films and coatings where depth of penetration is very small.
Fig 4 Comparison of modulus (in GPa): (a) Oliver method; (b) This study vs. experimental data for copper at loads less than 3 mN
Fig 5 Comparison of hardness (in GPa): (a) Oliver method; (b) This study vs. experimental data for copper at loads less than 3 mN
Conclusions
Energy based parameters/expressions are employed to ameliorate the two-slope method used for the determination of elastic modulus and hardness of a material. The proposed method is validated using the nanoindentation data on elastic perfectly plastic materials such as aluminum, copper and tungsten. Elastic modulus and hardness of these materials compare well with experimental results with a relatives error of less than 10%. Main conclusions of this study may be enumerated as follows:
• Functional analysis based expressions are very accurate as far as the representation of load-displacement curves is of concern for the material which recovers less upon withdrawal of load. Expression for unloading curve, however, overestimates the initial unloading stiffness. Alternatively, it may be calculated from the known value of contact depth expressed in terms of elastic energy constant and maximum depth of penetration. The slope of the loading curve at the peak indentation load may be obtained directly by analytically differentiating the energy based expression.
• Correction factors due to the lack of axial symmetry and inward radial displacement of the material surface points have important implications on the accuracy of the mechanical properties determined by this method. The magnitude of / depends on the indentation size and has higher value at lower depth of penetration. In this study, a constant value equal to 1.034 is used at large peak indentation load. At low load level, a value which gives the least error is selected using hit and trial method. y may be determined using the expression given by Hay et al.
• The proposed expression for the determination of the contact area overestimates its value as compared to the conventional method. For the material considered in this study, a very accurate value of contact area is usually obtained using the conventional Oliver and Pharr method. To correct this, a factor has been introduced in the expression which is found to be approximately equal to (a/)2 for all the materials considered in this study regardless of indentation size.
Table 1 Comparison of mechanical properties in GPa obtained from three different methods.
|
Material |
This Study |
Oliver Method |
Oliver and Pharr Method |
Ref. |
|||
|
Er |
H |
Er |
H |
Er |
H |
||
|
Aluminum |
70.44 |
0.194 |
69.55 |
0.197 |
72.42 |
0.21 |
1 |
|
Copper |
130.00 |
0.894 |
122.74 |
0.804 |
136.50 |
0.970 |
4 |
|
Tungsten |
320.16 |
4.470 |
344.87 |
4.32 |
320.4 |
3.80 |
1 |
