List of Symbols
Step 3: Step 3: Derive the element characteristic (stiffness and mass) matrices and characteristic (load) vector. The strain can be expressed as
3.5 Computation of Young’s Modulus and Hardness from Load-Displacement Plot This section presents the methodology to compute the Young’s modulus and hardness
from the load-displacement plots that were extracted from nanoindentation process. To calculate Young’s modulus and hardness, Oliver and Pharr [Oliver and Pharr (1992)]
analytical method has been used. It is described as follows.
3.5.1 Computation of Elastic Modulus
During the nanoindentation method, a small indentation in the specimen is made and the indentation load P, and displacement h, during one complete cycle of loading and unloading are recorded. It is important to determine the stiffness of the contact between the indenter and the material being tested to obtain the mechanical properties of interest. Assuming the case of the elastic contact for a cone and a flat specimen, the force and the indenter displacement are given by Sneddon’s equations [Sneddon (1948, 1965)]:
𝑃 =2𝜋𝐸𝑟tan 𝛼 ℎ2 (3.37)
where, Er is the reduced modulus, α is the indenter cone half-angle (Figure 3.3 (a)).
Taking the derivative of P w.r.t. h to obtain:
𝑑𝑃
𝑑ℎ= 2 [𝜋2𝐸𝑟tan 𝛼] ℎ (3.38) where, 𝑑𝑃 𝑑ℎ⁄ is defined as the contact stiffness (S), which is a very important quantity in the analysis of nanoindentation test data. Substituting the term [2𝜋𝐸𝑟tan 𝛼] back to the equation (3.37), we can write
𝑃 =12𝑑𝑃𝑑ℎℎ (3.39)
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From the Figure 3.6 (a), the displacement h of the indenter and the radius of the circle of the contact ‘a’ is related by:
ℎ =𝜋2𝑎 cot 𝛼 (3.40)
Substituting the value of h in equation (3.38), we get 𝑑𝑃
𝑑ℎ = 2 [ 2
𝜋 𝐸𝑟tan 𝛼] × [𝜋
2 𝑎𝑐𝑜𝑡𝛼]
⇒ 𝑑𝑃𝑑ℎ= 2𝐸𝑟𝑎 (3.41)
The projected contact area, 𝐴 = 𝜋𝑎2. Thus, from equation (3.38), we get
𝐸𝑟 = 12𝑑𝑃𝑑ℎ√𝜋√𝐴 (3.42)
The equation (3.42) applies to the elastic contact of any axis-symmetric indenter (such as a sphere, cone) and can be used to calculate the combined elastic modulus of the indenter and the specimen. The contact stiffness 𝑆 = 𝑑𝑃 𝑑ℎ⁄ can be measured from the initial slope of the unloading curve. The contact radius can be found from the depth of the circle of contact hc
and the geometry of the indenter. Once the stiffness S is measured, reduced modulus Er, which accounts for the measured elastic displacement contributing from both the sample and the indenter tip can be computed by:
𝐸𝑟 = 12𝛽𝑆√𝜋√𝐴 (3.43)
where, 𝛽 is a constant that depends on the geometry of the indenter. It is used to determine the elastic modulus and hardness values accurately as it affects contact stiffness and projected area which are important for calculating elastic modulus and hardness respectively.
The standard values of 𝛽 suggested by Oliver and Pharr and other researchers are listed in Table 3.5.
Table 3.5 List of indenter parameters for spherical indenter [Králík and Němeček (2014)]
Indenter type
Projected area (A)
Intercept factor (k)
Geometry correction factor (β)
Sphere 𝜋𝑎2 0.75 1
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is given by:
1
𝐸𝑟 =1−𝐸𝑖2
𝑖 +1−𝐸𝑠2
𝑠 (3.44)
where Ei = 1141 GPa and i= 0.07 are the elastic modulus and Poisson’s ratio respectively for diamond indenter as per Goel et al. (2014b) and Nawaz et al. (2017). From this equation we can calculate the Esample (elastic modulus) for the given sample.
3.5.2 Computation of Hardness
Hardness is the property of material which shows the resistance to the plastic deformation. It is given by the ratio of the applied force (Pmax) to the corresponding area of contact (A):
Hardness, 𝐻 = 𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑙𝑜𝑎𝑑
𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝑎𝑟𝑒𝑎=𝑃𝑚𝑎𝑥𝐴 (3.45) where Pmax is the maximum load and A is the projected area of contact or hardness impression. The effect of indentation depth on hardness measurement has been a real area of concern. When low loads are applied, the resultant area of contact might be very small or sometimes it recovers elastically with no residual impression left behind, due to which, the contact area of impression approaches to zero. This gives an exaggerated hardness value as the contact area becomes nearly zero in equation (3.45). The most common method to determine the hardness of a material is by static indentation.
3.5.3 A Case Study
After the development of the numerical model, simulations were carried out. For a typical process condition of 5 nm depth of indentation with 300 nm radius spherical indenter and 0.1 mm/s loading speed, a case study on computing elastic modulus and hardness is presented below.
A. Elastic Modulus
Figure 3.11 shows the load-displacement plot that was extracted from ODB field output.
The step by step procedure to compute the elastic modulus is as follows.
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Figure 3.11 Load-displacement plot
From the extracted load-displacement plot, maximum load, corresponding maximum displacement and final displacement after the unloading are recorded. In the present case, maximum load was 121.89 μN for the maximum displacement of 5 nm. The displacement after unloading was 0.28 nm.
From the unloading curve, 1/3rd data points (7 points) of the total unloading data points (21 points) are linearly fitted. Usually between 25% and 100% of the unload curve data are commonly used for the fitting function depending on the quality of the unloading data [Oliver & Pharr (2004), Shuman et al. (2006)]. In the present case we have found that 1/3th of the data points give good results while calculating Young’s modulus and hardness. The fitted line is extended to meet the x-axis for the ease of finding the slops from the linearly fitted line as shown in Figure 3.12.
Figure 3.12 Fitting the unloading curve and finding the slope TH-2306_10610325
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From the linearly fitted line, the slope is calculated. This is regarded as the stiffness, S = 40059.72 N/m2.
From the P-h plot, the fixed and variable parameters for indentation parameters were obtained and listed in Table 3.6.
Table 3.6 Indentation parameters (fixed and variable parameters)
Parameters Value Unit
Indentation depth at max load (h) 4.9993× 10−8 m Indentation load at max load (Pmax) 0.000122 N
Unloading displacement 4.72 × 10−8 m
Depending on the indenter tip used, the contact radius was calculated. For spherical indenter, the contact radius was computed as
Contact radius, 𝑎 = √𝑅2− (𝑅 − ℎ)2 = √ℎ(𝐷 − ℎ) =√4.9993(600 − 4.9993)
=54.54 nm (3.46)
where, Ris the radius of the indenter, D is the diameter of the spherical indenter, and h is the max depth at max load.
Corresponding contact area between the indenter tip and workpiece at maximum displacement was calculated as
Contact area, 𝐴 = π𝑎2 = 9.34879 × 10−15 m2 (3.47)
After finding out the stiffness (S), contact area (A) and taking indenter geometry correction factor = 1, the reduced modulus (Er) was calculated as
𝑆 = 𝛽2√𝐴√𝜋 𝐸𝑟
𝐸𝑟 =2𝛽1 √𝜋×40059.72
√9.34879×10−15
𝐸𝑟 = 3.67253 × 1011 N/m2 (3.48)
In the present simulation, the diamond indenter was considered as perfectly rigid body.
However, in the physical experiments, there may be a slight deformation in the nanometric tip which might have absorbed some load. Therefore, it is quite appropriate to assume an TH-2306_10610325
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infinite value of Young’s modulus for the diamond as well. Thus, the second term in the Equation (3.44) will become zero [Yu et al. (2003)]. Then by using Equation (3.44), the elastic modulus of SiC sample was computed as
1
𝐸𝑟
=
1−𝐸𝑖2𝑖
+
1−𝐸𝑠2𝑠
1
3.67253×1011
= 0 +
1−(0.23)𝐸 2𝑠
𝐸𝑠 = 348 GPa (3.49)
Thus, the Young’s modulus (Es) for the silicon carbide sample is found as 348 GPa.
B. Hardness
The hardness value of silicon carbide can be calculated by using Equation (3.45). It is given as,
Hardness, 𝐻 =𝑃𝑚𝑎𝑥𝐴 =9.348790.00012210−15 = 13.04 𝐺𝑃𝑎 (3.50) After determining the required output parameters, the results were validated with the published experimental results. These are presented in the following section.