=Paper=
{{Paper
|id=Vol-1712/p04
|storemode=property
|title=Nano-indentation Testing for Al-Cu Bulks
|pdfUrl=https://ceur-ws.org/Vol-1712/p04.pdf
|volume=Vol-1712
|authors=Fabio Maugeri,Giuseppe D'Arrigo,Michele Calabretta
}}
==Nano-indentation Testing for Al-Cu Bulks==
https://ceur-ws.org/Vol-1712/p04.pdf
Nano-indentation Testing for Al-Cu Bulks
Fabio Maugeri∗ , Giuseppe D’ Arrigo† , and Michele Calabretta‡
∗ Department of Electrical, Electronics, and Informatics Engineering, University of Catania, Catania Italy
† CNR-IMETEM, Italy, Str. Primosole, Catania Italy
‡ STMicroelectronics, Str. Primosole, Catania Italy
Abstract—The present paper provides a practical method for
rapid evaluation of the mechanical properties of a copper spec-
imen, avoiding long and onerous fatigue analysis. Instrumented
indentation (also known as relative penetration depth) is used to
calibrate and validate a finite element (FE) analysis simulation
model using COMSOL Multiphysics. The selected tool (CSM
Nano Scratch Tester) allows for the validation of the FE model,
providing for the individual characteristics of the contact point
and penetration depth with remarkable accuracy (in the order
of 10−11 m). The validated FE model enables to determine the
stress state originating from the indentation test and the pile up
on the specimen for loads in the range 0 to 300 mN.
Keywords-Nanoplasmonics, FEM, Surface Plasmon Polariton
Fig. 1. Berkovich tip geometrical characteristics.
I. I NTRODUCTION
Recently nanofabrication is gaining a great interest and
several methods have been recently developed for studying • Angle at the vertex between the probe axis and each edge
phenomena and developing tools at nanoscale [1]. 76.9o .
• Angle at the vertex between the probe axis and the
Indentation is a non-destructive method which is widely
used to estimate mechanical properties of small volumes of opposing face 65.3o .
• Triangle at the base is equilateral
material. It requires high forces, thus such diamond penetrators
as Vickers or Berkovich tip are applied. However, precise The dimension of the impression surface once the load is
results can be obtained only if the penetrators geometry is removed is calculated as follows:
well defined, as well as the exact position of the contact point
PM ax
is established. A penetrator can present certain roundness due H= (1)
to the use and the presence of dust, which can significantly A
influence the indentation curve. where P is the test force in mN and A is the surface area in
Often, due to minor geometrical irregularities, the four µm2 . The latter is calculated via optical microscope available
facets of a Vickers tip form a line rather than meet at a in the CSM Nano Scratch Tester. The change in loads and
single point. For this reason, the present study describes the displacement allows us to obtain the indentation curve. In
nano-indentation hardness test of a copper specimen using a particular, displacement depends on the parameters which can
Berkovich indenter tip. A high resolution instrument (CSM be obtained by using Oliver and Pharr method. Specifically,
Nano Scratch Tester) has been employed to continuously in case Berkovich indenter tip is employed, the projection of
register the loads and displacement of the penetrator during the the contact surface is related to the penetration depth via the
loading-holding-unloading cycle with a remarkable accuracy following formula:
of one hundredth of a nanometer (10−11 m).
The resulting indentation curve has been analysed using A = π· tan2 (70.3)· h2C = 24.5· h2C (2)
Oliver and Pharr method, whereas the FE model offers a
load curve with great precision. From the obtained indentation Penetration depth results from:
curves (elastic and plastic field) one can yield different me-
chanical characteristics (e.g. hardness, Youngs modulus, etc.). hC = hm − hs (3)
II. I NDENTATION USING A B ERKOVICH TIP where hm is the maximum penetration and hs is the vertical
displacement of the surface at the perimeter of the contact and
The geometrical characteristics of the diamond penetrator
thus can be obtained from the general Sneddon equations for
Berkovich are shown in Fig. 1.
indenters of any shape, always expressed as:
Copyright c 2016 held by the authors.
Fm
hs = (4)
S
19
Fig. 2. Lateral view of the Indentation curve and Penetration.
The indentation region is very small with respect to the size
of the sample. A high mesh refinement is used for modeling
the large deformation area. In the outer zones multi-point
constraints (MPCs type linear) are introduced lot interfacing
one element with two adjacent elements to reduce the total
number of degrees of freedom.
Finally, semi-infinite elements (CINAX4 clement type) re-
semble the far field domain in the radial direction. The
elements at the interlayer between the film and the substrate
are reduced in height to be able to more accurately detect
the behavior of that zone. The imposed geometrical boundary
conditions are: the nodes along the axis of rotation can move
only along such an axis, i.e. the y axis; and all the nodes on
the bottom of the mesh are fixed, i.e. the displacements along
x and y direction, are constrained [2], [3], [4], [5], [6].
An important feature of COMSOL is its capability to model
Fig. 3. Mesh of the whole sample with the indenter modeled as a movable the contact between two bodies as a sliding through contacting
rigid surface.
surfaces, which arc in our case the indenter surface and the
specimen surface. The damping coefficient was calculated
with the same procedure reported in [5], [6]. From the initial
where is a factor dependent only on the indenter’s geom-
geometry the nodal gaps between the surfaces are defined.
etry.
During the analysis, the program controls the variation of
III. G EOMETRICAL M ODEL the nodal gaps. In particular whenever the distance between
the indenter and the specimen becomes zero gap closure the
The present study offers a 3D Berkovich hardness model contact takes place and an external reaction force is exerted
illustrating one third of the original geometry (see Fig. 3). The on the nodes involved in the contact. The contact constraint is
base of the penetrator is modelled as an isosceles triangle with enforced by tile definition of the master and the slave surfaces,
the base 129.9µm and sides 75µm. The specimen is modelled only the master surface can penetrate into the slave surface and
as a prism with a triangular base similar to the penetrator the contact direction is always taken as being normal to the
base and height 75µm. The contact point (for the sake of master surface. We have chosen the indenter surface as the
convenience) bears the following coordinates (x;y;z)=(0;0;0). master surface due to the larger stiffness of the indenter with
respect to the specimen. In Fig. 2 the indenter is represented
IV. A XISYMMETRIC M ODEL by a rigid movable surface. Since the coating is very hard, we
In order to define an axisymmetric model, we have used have also modeled the tip with axisymmetric elements to take
an equivalent conical indenter with a semi-apical angle of into account the stiffness of the indenter.
θ = 70o 300 . This system has geometric and loading symmetry This mesh is shown in Fig. 3. The friction coefficient
around the axis of the indenter. The specimen is modeled between the tip and the specimen surface is assumed to be
with 2635 4-node axisymmetric reduced integration elements zero, because in the case of a hard coating on a softer substrate,
(CAX4R element type). like TiN/HSS system, the friction has a negligible effect on
20
the nanoindentation process.
Coating and substrate are assumed to be initially stress free
and in perfect contact during the indentation process. Both
the TiN coating and the HSS substrate are assumed to be
homogeneous and isotropic and having a perfect elastic-plastic
behavior [7], [8], [9], [10], [11], [12].
The indentation procedure is simulated by two subsequent
steps: loading and unloading. During loading, the rigid surface
or the modeled tip moves along the y direction (see Fig. 2
and 3) and penetrates the specimen up to the maximum depth;
during unloading, the tip returns to the initial position. At each
depth increment, the program makes many iterations according Fig. 4. Elastoplastic curve for σ ≥ σsn .
to a specified convergence rate to reach an equilibrate and
congruent configuration [13], [14], [15], [16], [17].
Purely elastic deformation takes place only during the
beginning of the indentation process. The Mises yield criterion
is applied for determining the occurrence of the plastic defor-
mation. The equivalent Mises stress is given by the expression:
r
(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2
σM ises = (5)
2
where σ1 , σ1 , σ1 are the three principal stresses. Whenever
σM ises reaches the yield strength σ0 , the material begins to
deform plastically.
V. E LASTOPLASTIC MODEL OF C OPPER S PECIMEN
The specimen to be indented is copper, while the Berkovich
indenter tip is diamond (extremely hard material) so as not
to suffer significant deformation during indentation process. Fig. 5. Differential mesching in Berkovich.
Diamond indenter operates only in elastic field and is char-
acterised by Young’s modulus, density and Poisson’s ratio:
Diamond E =1141 GPa; ρ = 3530kg/m3 ; ν = 0.07. 3) Kinematical constraint in z direction of the indenter tip
The copper specimen will be subjected to stresses lead- (null displacement in x and y directions);
ing to permanent deformation, thus, elements characterizing 4) Kinematical constraint in z direction of the axis inter-
elastoplastic behaviour of materials must be added to the secting the contact point, belonging to both the indenter
three previously mentioned characteristic dimensions. Copper and the specimen;
E =125 GPa; ρ = 8920kg/m3 ; ν = 0.34. In order to do that, 5) All other boundaries are considered to be unconstrained.
Ramberg-Osgood law must be applied:
A. Mesh Generation
σ = Knp , f or σ ≥ σsn (6) Being exclusively composed of tetrahedrons, mesh is de-
fined with greater attention in the area surrounding the contact
•σp plastic strain. point. The three dimensional domains are organized in pairs
•σ stress in the corresponding direction. [18], [19] in the following way:
• K material constant which corresponds to the value of σ
• every pair includes equally dimensioned domains in x
at σp = 1.
and y direction (except for the pair of domains located
• n hardening behaviour of the material.
furthest from the contact point);
• σsn yield strength.
• the pairs are numbered from 1 to 3 (ranging from smallest
For copper, such values have been obtained in literature: to largest dimensions);
K = 530, n = 0.44, σsn = 70M P a. • Maximum dimension of the elements 2µm;
• Maximum dimension of the elements 6µm;
VI. B OUNDARY C ONDITION
• Maximum dimension of the elements 9µm;
The following boundary conditions have been introduced:
1) Symmetry in compliance with the symmetrical plane of VII. S OLUTION S TRATEGY
the model; The geometrical model is based on the load that acts
2) Fixed constraint τ at the base of the model; vertically on the base of the Berkovich tip using stationary
21
Fig. 6. Load versus Time. Fig. 8. Load versus penetration.
Fig. 7. Indentation test on copper specimen.
Fig. 9. Graphic on the verge.
analysis and parametric solver from the Structural Mechanics
module in Comsol Multiphysics 5.1. This option is set up to Examining the case of maximum load being 300 mN, the
facilitate dynamic analysis that depends on the variation of following curve is obtained (see Fig. 10 and 11).
load in time. The final solution is sought interactively from
a series of multistage elastoplastic static solutions generated X. R ESULTS
under the control of variable displacement parameter. In every
phase, the geometrical configuration is calculated using the The numerical analysis on the model at 51330 degrees of
previous one. The load is measured within the range of 30 freedom (plus 436590 internal DOF) provides the following
seconds (during which the maximum load of 300 mN is curves. Estimated indentation curves indicate a mutual agree-
achieved) with a step being 1s. ment if compared to their experimental counterparts. A prior
validation test versus the experimental values is assessed by
VIII. C ONTACT BETWEEN PARTS juxtaposition of the load curves.
In COMSOL Multiphysics the problem of contact between Fig. 10 (left) shows Von Mises stress state originating from
parts is solved by determining the initial contact pair, i.e. the point of maximum contact from a lateral perspective, while
source and destination. In this case, penetrator is considered Fig. 10 (right) indicates the stress state on a copper specimen
to be the source, whereas the specimen is the destination. (maximum value being the contact point between the edge of
The problem of contact between parts is thus solved by the tip and the specimen).
using Penalty method with the offset null penalty function. Moreover, the shape of the impression (Fig. 11 (right)) can
Examining the case of maximum load being 300 mN, the be evaluated from a lateral perspective (in a section along the
following curve is obtained (see Fig. 8 and 9). contact point) in the direction perpendicular to the direction of
the applied load, passing by the vertex of the triangular base
IX. C OPPER S PECIMEN of the pyramid as shown in Fig. 11 (left), with the maximum
Prior to carrying out the test, it is necessary to choose the pile up value being 35 mN and load 150 mN.
solid specimen surface (by using electronic microscope) that Moreover, the shape of the impression (Fig. 12 (right)) can
does not have visible depressed areas or ridges which could be evaluated from a lateral perspective (in a section along the
influence the test result. Several indentation tests are then contact point) in the direction perpendicular to the direction of
carried out on a copper specimen in seven different points the applied load, passing by the midpoint of the triangular base
with the maximum load ranging from 50 mN to 350 mN (i.e. of the pyramid as shown in Fig. 12 (left), with the maximum
a step being 50 mN) and dwell time 10 seconds. pile up value being 64 mN and load 120 mN.
22
Fig. 10. Simulations of Von Mises stress.
Fig. 11. Shape of the impression from a lateral perspective (passing by the vertex).
Fig. 12. Shape of the impression from a lateral perspective (passing by the the midpoint).
XI. C ONCLUSION By means of suitable parameters and modelling strategies, the
developed model allows a more accurate representation of the
The present study offers a numerical model able to manage indenter tip geometry and materials in contact with it [20],
and predict strong singularities on the contact point between [21], [22], [23], [24].
the indenter tip and the specimen in a nano-indentation test.
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