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Crystal plasticity study of β−Sn by nano indentation simulations
Srihari Sundar1,*
Aritra Chakraborty 2
Philip Eisenlohr 2
1 Undergraduate student, Metallurgical and Materials Engineering, Indian Institute of Technology Madras
2 Chemical Engineering and Materials Science, Michigan State University
* sriharisundar95@gmail.com
Crystal plasticity study of β−Sn by nano indentation simulations
Srihari Sundar1,*
Aritra Chakraborty 2
Philip Eisenlohr 2
1 Undergraduate student, Metallurgical and Materials Engineering, Indian Institute of Technology Madras
2 Chemical Engineering and Materials Science, Michigan State University
* sriharisundar95@gmail.com
Abstract
The poisonous nature of lead-based solders has resulted in the electronics industry
shifting to lead-free solders, based on tin and alloys of tin. In this work we aim to
study the deformation behavior of tin and the associated crystal plasticity (CP) parame-
ters by considering at different orientations of crystals. Instrumented nanoindentation
has proved to be an important technique to validate constitutive parameters depict-
ing single crystal deformation behavior, hence crystal plasticity Finite Element Method
(CPFEM) simulations are performed to study the topography variation with changing
crystal orientations.
Properties of β-Sn
The most important allotrope of tin at room temperature is β-Sn, with a squashed di-
amond cubic structure as seen in Fig. 1. This particular structure is highly anisotropic
with large variation of properties in the ‘a’ and ‘c’ directions and a c/a ratio of 0.5456.
Bieler and Telang [1] have proposed the relative slip system activity for β-Sn shown in
Fig. 2.
Fig. 1: Crystal structure of β-tin Fig. 2: Slip system information of β-tin
DAMASK: material point crystal plasticity
Boundary value solvers (ABAQUS in this case) usually lack the capability to solve con-
tinuum mechanics problems incorporating CP. Thus, to model such a constitutive re-
sponse, a hierarchy of solvers at the homogenization level, crystallite level, and the con-
stitutive level is employed [3]. For single crystal simulations, there is no requirement of
homogenization. At the crystallite level (Fig. 3), the elasto-plastic problem is solved with
the plastic velocity gradient (Lp), coming from the constitutive level, used as a predictor
in a Newton–Raphson scheme giving us the stress at that point. A phenomenological
power-law formulation is employed for the calculation of Lp.
Fig. 3: Calculations at the crystallite level
Calculation of Lp [2]:
Lp =
n
α=1
˙γα
mα
⊗ nα
(1)
˙τα
c =
n
β=1
hαβ| ˙γβ
| (2)
˙γα
= ˙γ0
τα
τα
c
n
sgn(τα
) (3)
hαβ = qαβ h0(1 −
τβ
c
τs
)a
(4)
Material parameters: ˙γ0 = 10−3
s−1
, n = 25, a = 2, h0 = 75 MPa
Stiffness / GPa: C11 = 72.3, C33 = 88.4, C44 = 22.0, C66 = 24.0, C12 = 59.4, C13 = 35.8
Simulation setup
The indenter is modeled as a sphero-conical rigid body. The single crystalline sample
is discretized by 6381 8-noded hexahedral finite elements. The sample is indented to a
depth of 8.5 % of the indenter radius over a time of 20 seconds.
Fig. 4: Indentation geometry Fig. 5: Finite element mesh on sample surface
Results
Displacement topography:
Fig. 6: Indentation on [001] Fig. 7: Indentation on [100]
Accumulated shear in slip systems:
22
4444
6666
9999
10-3
10-2
10-1
0 600 1200 1800 2400
increment
accumulatedshear
10-3
10-2
10-1
0 600 1200 1800 2400
increment
accumulatedshear
Fig. 10: Slip system activity
The rows represent slip systems from slip families 2,4,6,9 in Fig. 2.
Conclusions
The simulated topographies reflect the strong anisotropy that is connected to the crys-
tal structure of tin. While there is a four-fold symmetry observed for indentation along
the [0 0 1] direction due to equivalence of ‘a’ and ‘b’ axes, relatively lesser deformation is
observed along the ‘c’ axis (parallel to Y ) as seen from the 1 0 0 direction indentation
results. The slip system activity gives us an idea of the slip systems that are active under
different loading conditions.
Acknowledgements
College of Engineering, MSU for organizing the inGEAR program. Tias Maiti from the
CMM group, MSU for discussions during this project. Associated financial support
through NSF-DMR grant 1411102 is gratefully acknowledged. Also, IITMAANA for fund-
ing of travel to MSU and the MME department at IITM for funding presentation at NMD.
References
[1] T. R. Bieler and A. U. Telang. Analysis of Slip Behavior in a Single Shear Lap Lead-Free Solder Joint During Simple Shear at
25°C and 0.1/s. Journal of Electronic Materials, 38(12):2694–2701, 2009. doi:10.1007/s11664-009-0909-x.
[2] F. Roters, P. Eisenlohr, L. Hantcherli, D. D. Tjahjanto, T. R. Bieler, and D. Raabe. Overview of constitutive laws, kinematics,
homogenization, and multiscale methods in crystal plasticity finite element modeling: Theory, experiments, applications.
Acta Materialia, 58:1152–1211, 2010. doi:10.1016/j.actamat.2009.10.058.
[3] F. Roters, P. Eisenlohr, C. Kords, D. D. Tjahjanto, M. Diehl, and D. Raabe. DAMASK: the Düsseldorf Advanced MAterial Sim-
ulation Kit for studying crystal plasticity using an FE based or a spectral numerical solver. In O. Cazacu, editor, Procedia
IUTAM: IUTAM Symposium on Linking Scales in Computation: From Microstructure to Macroscale Properties, volume 3, pages
3–10, Amsterdam, 2012. Elsevier. doi:10.1016/j.piutam.2012.03.001.