3.4 Finite Element Method
3.4.2 Software implementation
Chapter 3. Computational Methodology 41 the structural stiffness matrix, {d} is an array containing the unknown nodal displace- ments, and{R} is the structural force vector.
It can be shown that (Reddy,1993) the [K]{d}={R}matrix form as follows.
Kij11 Kij12 Kij13 Kij21 Kij22 Kij23 Kij31 Kij32 Kij33
ui vi wi
=
Rri Rθi Rzi
(3.46)
Where, {R}can be expressed as
Rri =−2π Z
Γ
∂φi
∂r
C11ur
a −C13uz h
zdrdz (3.47)
Rθi=−2π Z
Γ
∂φi
∂r
C16ur
a −C36uz h
zdrdz (3.48)
Rzi=−2π Z
Γ
∂φi
∂z
C13ur
a −C33uz h
+φi z
C12ur
a −C23uz h
zdrdz (3.49)
Then the shape functions for the four node quadrilateral standard element are assumed and the matrix is solved.
Chapter 3. Computational Methodology 42 Table 3.3: Unit chosen for implementation in ABAQUS.
Parameter Unit
Length nm
Force nN
Stress GPa
3.4.2.1 Preprocessing
The following steps are used for the modeling of the nanoindentation in the ABAQUS software:
• Step 1: Defining the unit system for the modeling of the nanoindentation process in the nanolength scales. Table 3.3shows the units used.
• Step2: Building up the model. In this step, the indenter and the substrate are modeled. The substrate is a rectangular box and the indenter is a semicircle. The arc point of the indenter is considered as a reference point. The dimensions of the substrate are 15 nm in height and 10 nm in width.
• Step 3: Defining the properties of the substrate materials. The indenter is modeled as an analytical rigid body and the substrate is of Al or its alloys materials.
For implementing the elastic-plastic behavior of the materials, the stress-strain diagrams obtained from the MD tensile simulations have been given as input to the FEM. The elastic properties are the elastic modulus and the poisons ratio and the plastic properties are the yield stress and plastic stress-strain data which are calculated from the true stress-strain curves. These properties are assigned to the part by section assignment.
• Step 4: In the assembly step, the indenter and the substrate are assembled to- gether. The indenter is placed over the substrate such that the indenter’s edge is just connected and there is no overlap between the edge of indenter and substrate.
• Step 5: The interactions among the surfaces are defined. For the contact problem, two surfaces are defined: one as master surface and the other one as slave surface.
The indenter surface is the master surface and its normal is along the outside radial direction. The substrate is considered as the slave surface. The interactions between the surfaces are assumed as frictionless.
Chapter 3. Computational Methodology 43 Table 3.4: Boundary conditions chosen to model the axi-symmetry nanoindentation
problem in ABAQUS.
Boundary Condition Applied at Steps
BC1 (ENCASTRE,
U1=U2=U3=UR1=UR2=UR3=0)
Bottom edge Loading and unloading
BC2 (U1=UR3=0) Symmetry
boundary
Loading and unloading BC3 (U1=UR3=0) Reference point of
the indenter
Loading and unloading BC4 (U2=-1.5) Reference point of
the indenter
Loading
BC5 (U2=0) Reference point of
the indenter
Unloading
• Step 6: Defining the steps of the simulation. Two steps of the simulations are defined . In the first step the indenter is pushed through the material and in the second step the indenter returns to its initial position. The unloading process automatically starts when the loading is completed up to the specified depth.
Finally, the boundary conditions are specified which are shown in Table3.4.
• Step 7: The mesh generation is done in this step. The indenter surface is considered as analytical rigid body and single element is considered for this. The substrate on the other hand is meshed finely. Two regions for this mesh purpose are defined.
One region includes the place where the indentation actually takes place. The first region has been taken as 5nm by 5 nm and a very fine mesh is applied as this is the main contact region of the indenter and the substrate. Other region is meshed with coarser elements. The mesh generated for the present problem is shown in Fig. 3.11. The element considered for the current problem is CA4XR which is a standard 4 node quadrilateral element generally used for the axi-symmetry modeling (Khan et al.,2010).
• Step 8: In this step necessary output parameters are set up to obtain the output data. For indentation problem, the desired output is the load and displacement.
So, the data outputs are comprised of the resultant force and displacement of the reference point of the indenter which will be recorded during the simulation.
Chapter 3. Computational Methodology 44
Figure 3.11: Mesh generation for the axi-symmetry nanoindentation problem. A zoom view of the fine mesh region is shown in the right side.
3.4.2.2 Simulation
For the simulations, at first, all the preprocessing data are written as an input file. After that, the basic parameters such as step size increment and nonlinearity consideration are fixed with desired solver. For a non-linear problem, ABAQUS-Standard uses the Newton-Raphson method for the solution of governing general equations. Finally, the number of processors that will be used for the simulation are selected and the solution is performed.
3.4.2.3 Post processing of the results
When the indentation simulation is completed, post-processing is performed to obtain the desired output data. A Python script is prepared to obtain all the necessary data.
At first, the nodal forces and displacements for the reference point of the indenter are calculated. These data are exported for further analysis of the hardness and reduced modulus of the materials. The von Mises stress distribution and the effective plastic strain are also visualized from the output database of the simulation.