List of Symbols
3.4 Numerical Simulation of Nanoindentation for the Determination of Mechanical Properties
3.4.1 Selection of Solution Methodology
Today high-speed processers and user-friendly solvers have increased the computational efficiency during numerical simulations. Numerical simulations are mainly used to understand the physics of manufacturing processes. During the literature study, it is learnt that molecular dynamic (MD) simulation is well capable of simulating various mechanical problems at atomic level to get the insight of the material micro-structure and grain boundaries. However, the correlation of the results obtained from MD simulation with actual experimental result is still lagging due to its limitation of requirement of huge computational time and length scales [Goel et al. (2014)]. High computational time and memory requirement during MD simulation do not permit the scaling of simulation parameters to the experimental scale. The length scales of process continuum are restricted to few nanometers (or few millions of atoms) and the cutting and indentation velocities should be in the range of 50 m/s in order to reduce the computation time, whereas, in actual experiment, these values are around 1 m/s for machining and 0.1–10 μm/s for nanoindentation. Because of these variations in values, it might cause unexpected spurious effects during the result analysis [Goel et al. (2014)].
Finite element method (FEM) is mostly used in mechanical analysis for the determination of stresses and displacements. However, nowadays with the development of high end computers, FEM can also be used in the analysis of many disciplines such as heat transfer, contact mechanics, fluid dynamics, and electromagnetism [Zienkiewicz and Taylor (1977)].
In the present work, the nanoindentation simulations have been carried out using a commercial finite element (FE) package AbaqusTM. Quasi-static nanoindentation simulations were carried out on silicon and silicon carbide by using spherical indenter tip. Quasi-static nanoindentation permits systematic examination of the load displacement plot to enable better understanding of deformation mechanisms, evaluation of mechanical properties, and aspects of plasticity of brittle materials such as Si and SiC [Goel et al. (2014)]. Since, it is a very slow process, a minute variation in the load or displacement can easily be captured. These variations occur due to pop-in and pop-out events which are associated with the plastic deformation of the material. Generally, the hardness of a material can be determined by static indentation. Thus, in the present work, load-displacement graphs are extracted from the TH-2306_10610325
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simulation results, and by using these results, Young’s modulus and hardness values have been calculated using Oliver and Pharr analytical model. The methodology developed and the input, output parameters used in the present work are shown in Figure 3.4.
Figure 3.4 Schematic of developed numerical model
The step–by–step procedure followed in the FEM based modeling and simulation is outlined below.
Geometric modeling of workpiece and diamond indenter. Details are given in Section 3.4.2 (C).
Material properties were assigned to the workpiece and the indenter. Details are given in Section 3.4.2 (D).
Workpiece geometry was discretized with 4-node axisymmetric, bilinear plane strain reduced integration elements, CAX4R. Then, the mesh sensitivity analysis has been carried out to fine tune the FEM output parameters such as load and displacement.
The objective was to reduce the computation time and to enhance the prediction accuracy. Details of the discretization and mesh sensitivity analysis are given in Section 3.4.2 (E).
Initial boundary conditions such as mechanical constraints ‘encastre’-a fixed boundary condition to keep the workpiece fixed, symmetric boundary condition to two sides of the workpiece, velocity of the indenter. Details are given in Section 3.4.2 (I).
OUTPUT
Load
Displacement
Stress
Pressure
Residual stress Geometry modeling
Indenter geometry Indenter radius
Workpiece geometry height
length
FEM model
Model formulation
Work-tool contact
Element and meshing
Mechanical analysis
Post processing
Boundary conditions
Displacement BC
Fixed BC
Friction
Material definition
Mechanical properties
Drucker Prager properties
Flow stress
Material failure Process conditions
Loading speed
Feed rate
Depth of indentation
NUMERICAL SIMULATION INPUT
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Dynamic explicit solution has been carried out. The solution scheme is presented in Section 3.4.2 (J) in detail.
Solver computed the stress, pressure, force and displacement at each node of domain at the end of the analysis. The node outputs were used to analyze the process. A case study to demonstrate the simulation by using the developed model is presented in Section 3.5.
These steps are discussed at length in the following sections.
3.4.2 Development of FEM based Nanoindentation Model
During the nanoindentation process, the material is pressed using a diamond indenter.
Three faced pyramid shaped Berkovich indenter is the most commonly used diamond indenter in the nanoindentation process. The tip of the Berkovich indenter has three sides and has spherical shape of few nanometers radius at the joint of edges. Figure 3.5 shows the schematic of three faced pyramid shaped Berkovich diamond indenter.
Figure 3.5 Schematic of Berkovich indenter showing front and top views
The nanoindentation process involves frictional interaction between the workpiece and indenter along with the elastic-plastic deformation. In this work, quasi-static nanoindentation simulations were carried out on silicon and silicon carbide by using spherical indenter tip.
Quasi-static nanoindentation permits systematic examination of the load displacement plot that would be useful for study of deformation mechanisms, plasticity of brittle materials and evaluation of mechanical properties [Goel et al. (2014a, b)]. Quasi-static simulation enables simulation for a very small step. Therefore, in this work, only mechanical analysis of nanoindentation process has been attempted.
Three faceted pyramid Berkovich indenter
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A. Assumptions
In the present work following assumptions have been considered for the development of numerical model of quasi-static nanoindentation process.
Workpiece and diamond indenter materials are assumed to be homogeneous and isotropic in nature.
The diamond indenter is assumed to be perfectly rigid.
To induce the plastic deformation in brittle materials such as silicon and silicon carbide without defining independent slip system, von-Mises criterion is used for the plastic yielding [Lu and Bogy (1995), Bhattacharya and Nix (1988a, b), Zhang and Mahdi (1996)].
The workpiece is considered as flat and free of initial residual stresses.
The contact between the indenter and top surface of the work material is assumed to be frictionless. It is because the effect of friction between the indenter tip and the surfaces of the work material in nanoindentation process is negligible as size of the process continuum is very small (in nm) [Bhattacharya and Nix (1988a, b), Lu and Bogy (1995)].
The tip movement is maintained at 0.1 mm/s to meet the quasi-static condition.
As the indentation process is carried out in a very slow loading speed (0.01 to 0.1 mm/s). The temperature effect in the process can be ignored.
B. Governing Equations
In this work, a commercial finite element (FE) software AbaqusTM is used to simulate the nanoindentation process. However a detail theoretical study of the plastic deformation has been carried out. Moreover, the essential step to obtain the field variable using FEM has been studied. These are presented as follows.
During the process of nanoindentation, the tip of the indenter applies compressive forces on the workpiece. The stress generated during plastic deformation can be computed using von-Mises criterion which is given by
𝜎𝑀𝑖𝑠𝑒𝑠 = √(𝜎1−𝜎2)2+(𝜎2−𝜎23)2+(𝜎3−𝜎1)2 (3.1) where 𝜎1, 𝜎2 and 𝜎3 are the three principle stresses. When the 𝜎𝑀𝑖𝑠𝑒𝑠 reaches the yield strength of material (𝜎𝑌), the specimen starts to deform plastically. In this present simulation, von-Mises stress is used to measure the pressure and residual stresses obtained in the TH-2306_10610325
55 numerical simulation. It is to be noted that, after indentation process, the surface of the workpiece material (i.e., place of interest) is free from the hydrostatic stresses. Thus, it is convenient to use von-Mises stress to determine the various stresses and pressures from the nanoindentation simulation, because it also does not include hydrostatic stresses. Also, it is worth to mention that, the onset pressure at the interface of indenter and workpiece is determined which is still not possible to determine during the actual (physical) experiments.
The calculated pressures are compared with the workpiece material’s hardness (relative to a possible HPPT or stress induced amorphization).
In dynamic problems, the field variables such as displacements, velocities, strains, stresses and load are all time-dependent. The procedure involved in deriving the finite element equations of a dynamic nanoindentation problem can be carried out by the following steps [Wang (2007)]: