Journal of Applied Mathematics and Physics, 2014, 2, 425-430
Published Online May 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.26051
How to cite this paper: Zinovieva, O., Romanova, V., Balokhonov, R., Zinoviev, A. and Kovalevskaya, Zh. (2014) Numerical
Study of the Influence of Grain Size and Loading Conditions on the Deformation of a Polycrystalline Aluminum Alloy.
Journal of Applied Mathematics and Physics, 2, 425-430. http://dx.doi.org/10.4236/jamp.2014.26051
Numerical Study of the Influence of Grain
Size and Loading Conditions on the
Deformation of a Polycrystalline Aluminum
Alloy
O. Zinovieva1,2, V. Romanova2, R. Balokhonov2, A. Zinoviev2, Zh. Kovalevskaya2,3
1National Research Tomsk State University, Tomsk, Russia
2Institute of Strength Physics and Materials Science of the Siberian Branch of the Russian Academy of Sciences,
Tomsk, Russia
3National Research Tomsk Polytechnic University, Tomsk, Russia
Received January 2014
Abstract
This work is concerned with numerical simulations of surface roughening in a polycrystalline
aluminum alloy. Using 3D finite difference model, high-resolution simulations are conducted. Ef-
fects of loading conditions and grain size on surface roughening and mesoscale deformation
processes in AL6061-T3 aluminum alloy under quasistatic uniaxial tension are investigated.
Keywords
Polycrystalline Materials, Surface Roughening, 3D Model
1. Introduction
One of the behavioral features of aluminum alloys, limiting their possible use as a material for machine parts, is
deformation-induced surface roughness formed on the free surface under wide range of deformation conditions.
Despite the increasing number of experimental and computational studies in this area (see, e.g., [1]-[7]), the
causes and the mechanisms of surface roughening continue to be debated among researchers. The surface
roughening phenomena are shown to be dependent on many microstructural factors such as grain size [1], crys-
tallographic texture [3] [4], etc. Grain anisotropy which causes plastic strain localization and, consequently, sur-
face roughening, has a significant influence on the changes of surface morphology [3].
Real behavior of aluminum alloys is related to the complex multi-level structure of the metal, and the
processes occurring within the bulk and on the surface of material depend on a variety of factors. Investigation
of the influence of individual factors on the deformation and fracture in the experiment is often not possible. In
this regard, numerical simulation is an important addition to the experimental studies. In [8] the evolution of
mesoscale deformation in polycrystalline Al6061-T6 alloy containing 280 grains has been analyzed in detail.

O. Zinovieva et al.
426
Plastic deformation is shown to arise at the grain boundaries which are sources of stress concentration in materi-
al. Localization of stresses and strains is more pronounced near the free surface than in the bulk of material.
This paper continues a series of computational studies of surface roughening phenomena and mesoscale de-
formation processes in polycrystalline materials [5]-[8]. The influence of boundary conditions (BCs) and grain
size on the qualitative characteristics of the surface roughness in polycrystalline AL6061-T3 alloy is analyzed.
Microstructural model proposed in [8] is modified to apply periodic BCs. The role of free surface and grain
boundaries in the evolution of deformation processes at the mesolevel is discussed. Optimal thickness of the
three -dimensional (3D) model for the study of surface roughening is defined.
2. Three-Dimensional Microstructure-Based Simulation
In order to describe the mechanical behavior of a material, we use the mathematical tools of continuum me-
chanics, assuming that a medium retains its continuity at the meso- and macrolevels in plastic deformation. To-
tal system of equations to describe the elasto-plastic behavior of a material includes equations of motion and
continuity, expressions for components of the strain rate tensor and constitutive equations defining the relation-
ship between the stress and strain tensors. The procedure used in the 3D numerical analysis is detailed elsewhere
(see, e.g., [9]).
In this work 3D polycrystalline structures are generated by the step-by-step packing method proposed in [9].
Rectangular volume is discretized by a regular mesh of 200 × 75 × 200 cubic cells with a step of 10 µm. As the
initial conditions, nucleation centers are distributed randomly within the volume. All grains are assumed to grow
according to a spherical law and with the same rate.
Polycrystalline structures, periodic in the X1-direction, are shown in Fig ure 1. In all three cases, the size of
the computational domain is 2000 × 750 × 2000 µm, and the number of grains varies from 200 to 1000. Average
grain diameters are 300, 200 and 170 microns, respectively. In order to obtain a periodic structure in a certain
direction, an additional condition is checked: if the growing grain goes beyond the surface, its growth continues
on the opposite side by parallel transfer. From the equation of spherical volume V, the grain diameter can be ex-
pressed as
33
6π6π,
g mc
VV hN= =
(1)
where Nc is the number of cells approximating the grain, hm is the step of computational mesh.
A rigorous description of elasto-plastic material behavior calls for crystal plasticity models taking into ac-
count explicitly the crystallographic orientation of individual grains and the slip planes, and is based on physics
of dislocation interaction [10]. Such models are of fundamental importance for use in describing the behavior of
materials with a limited number of slip systems and plastically anisotropic materials. They should be also used
when grains rotate significantly because a phenomenological plasticity models based on experimental data and
their approximation do not account for the microscale processes in an explicit form. The macroscopically iso-
tropic AL6061-T3 aluminum alloy under study is characterized by fcc lattice and has 12 slip systems, respec-
tively. For the sake of simplicity, assuming that the crystallographic orientation relative to the applied load has
no significant effect on the yield stress within the grain, we omit an explicit formulation of crystal plasticity
taking an implicit account of the crystallographic orientation of grains through the difference between their elas-
tic and plastic characteristics within ±2 ÷ 5% about average values.
(a) (b) (c)
Figure 1. Polycrystalline models with average grain size of
300 (a); 200 (b); 170 µm (c).

O. Zinovieva et al.
427
In describing the behavior of polycrystalline aluminum alloys, the Hall-Petch effect has special significance.
It should be noted that in this work this effect is intentionally disregarded to separate the impact of the grain size
and mechanical properties.
The elastic-to-plastic transition is defined by the von Mises criterion with allowance made for strain-harden-
ing:
( )
12
ip
eqij ijyeq
SS σ σε = =
, (2)
where
( )
ip
y eq
σε
is the strain-hardening function of the i-th grain,
p
eq
ε
is the accumulated equivalent plastic
strain,
ij
S
is the deviatoric stress tensor. The strain-hardening function for the Al6061-Т3 alloy is given in the
form of
( )
()
065 1exp0.048
ii p
y eq
σσ ε
=+ ⋅−−
[MPa], (3)
Here
0
i
σ
is the initial yield stress of the i-th grain. Average values of the shear modulus and the bulk mod-
ulus defined in the calculations are 28.4 and 82.0 GPa, respectively. Average value of initial yield stress
0
i
σ
is
107 MPa. Properties remain constant within each grain, while varying in passing through the grain boundary.
The system of equation is complemented by initial and boundary conditions and solved numerically using the
finite difference method [11]. In order to describe loading conditions, let us introduce the Cartesian coordinate
system as presented in Fig ure 1. Boundary conditions on x3 = 0 and x3 = L3 sides simulate uniaxial tension along
the X3-a xi s:
3 33
30 3
,
x xL
UU
υυ
= =
=−=
, (4)
where xi are the spatial coordinates,
33
Ux=
is the velocity vector component, Li are the sizes of computational
domain. The bottom surface is a symmetry plane, its displacements are fixed in the vertical direction and are not
constrained in the horizontal plane. On top surface BCs correspond to free surface conditions. On lateral sides
10x=
and
11
xL=
of microvolumes the absence of external forces (free surface) or periodic BCs are applied.
Thus the loading scheme used in calculations is shown in Fig ure 1.
3. Calculation Results
Numerical results obtained in calculations are shown in Fig ures 2-5. Surface roughening is shown to be mostly
affected by microstructure of the subsurface layer [5]. Basing on this observation, we have determined the mini-
(a) (b) (c)
(d) (e) (f)
Figure 2. Distributions of equivalent plastic strains ((a)-(c)), and surface
roughness patterns ((d)-(f)) in polycrystals of 350 ((a), (d)), 200 ((b), (e)), 100
µm in thickness ((c), (f)), ε = 0.3%, arrows indicate the direction of tension.

O. Zinovieva et al.
428
(a) (b)
(c) (d)
Figure 3. Distributions of equivalent plastic strains ((a), (b)) and surface roughness patterns
((c), (d )) in polycrystals with free lateral surfaces ((a), (c)) and with periodic BCs on lateral
surfaces ((b ), (d)), ε = 0.18%.
Figure 4. Roughness profiles along the diagonal line (see Figure 3(c)) in the
polycrystal with free lateral surfaces (a) and along the midline (see Figure
3(d)) in the sample with periodic BCs (b), ε = 0.18% µm.
mum thickness of the model, needed to reproduce the surface roughening with acceptable accuracy. For doing
so, we calculate uniaxial tension of polycrystalline structures of 500 × 350 × 1000, 500 × 200 × 1000, and 500 ×
100 × 1000 µm (Figures 2(a)-(c)). The lateral
10x=
and
11
xL=
sides are free of loading. Average grain
diameter is about 50 µm, so lateral sides in a direction normal to free surface comprises 7, 4, and 2 layers of the
grains, respectively.
Analysis of the calculation results shows that the main influence on the deformation processes on the surface
has a microstructure lying within 1 - 2 grain diameters. The model that contains only 2 grain layers in thickness
fails to adequately describe surface roughness due to influence of the bottom surface (Figure 2(c) and Fig ure
2(f)). Using a model of a thickness more than 4 average grain diameters for the considered loading conditions,

O. Zinovieva et al.
429
(a) (b) (c)
Figure 5. Roughness profiles in polycrystals with average grain size of 300 (a); 200
(b); 170 µm (c) (curves 1); roughness profiles in a homogeneous material (curves 2), ε
= 0.22% µm.
we observed that the surface topography varies slightly (cf. Figure 2(d) and Figur e 2(e)). Thus, to investigate
the effects of surface roughening, it is recommended to use a model of a thickness of 3 - 4 average grain diame-
ters.
In order to analyze the stress-strain state which is responsible for surface roughening, we have compared the
surfaces of a homogeneous isotropic material and a material with internal boundaries. Generally the surface
roughening phenomena is related to the microstructure. Due to polycrystalline structure, all components of the
stress and strain tensors are nonzero at mesoscale, including the stress tensor component
σ
22 acting across the
free surface. In uniaxial tension the stress tensor component
σ
22 exhibits a quasi-periodic distribution of positive
and negative values, i.e., the regions exposed to tensile strains alternate with those experiencing compression,
compensating each other. Therefore from the macroscopic standpoint, the equilibrium conditions are satisfied.
Thus, at mesolevel periodically distributed fields of normal stresses and strains arising in the bulk of the material
act from inside towards the surface, resulting in surface roughening. This conclusion was arrived at in our earlier
work [12] [13] on the basis of 2- and 3D calculations for a metal matrix composite and a coated material.
To analyze the effects of grain size and loading conditions on the surface roughening, a series of calculations
for the microstructures shown in Figure 1 with different B Cs on the lateral sides has been performed. Loading
conditions have a significant impact on the surface roughness characteristics. Using 3D matrix-based statistical
analysis, authors [2] have experimentally investigated the influence of the loading conditions (uniaxial, biaxial,
and plane strain) on the surface roughening in AA5754-O aluminum alloy. In this paper we evaluate the influ-
ence of loading conditions on qualitative characteristics of the deformation-induced surface roughness. We con-
sider two idealized cases of BCs: in the first case the lateral sides are free of loading, in the second one they are
assigned with periodic BCs that simulate conditions of constrained deformation. The calculations are performed
for a model containing 1000 grains (Figure 1(c)).
Figure 3 compares the results of calculations for polycrystals with free boundaries and periodic BCs. In both
cases, we can observe mesoscopic relief folds, consisting of several smaller ones (Figure 4). These computa-
tional results are in agreement with experimental findings on aluminum alloys [3] [14]. Qualitative difference is
in the orientation of folds and areas of plastic strain localization. When lateral sides are free of external forces,
the surface folds and localization bands tend to form at an angle of 45˚ to the axis of tension (Figure 3(a) and
Figure 3(c)), due to the direction of maximum macroscopic shear stress. In the case of periodic BCs, interlacing
folds are oriented perpendicular to the loading direction (Figure 3 (b) and Figure 3(d)). In the case of the spe-
cimen with free lateral sides, the height of relief folds is smaller, and the width is greater than in case of the
sample with periodic BCs (Fig ure 4). In the second case, increase of the surface height data scattering is due to
the restriction of deformations in the lateral direction. For both specimens under study, surface roughness pro-
files demonstrate the presence of several large folds consisting of two or three smaller ones (Figure 4). This al-
lows us to class the roughness evolution as a mesoscale phenomenon.
For the study of the grain size effect, we have performed a series of calculations for specimens with an aver-
age grain size of 300, 200 and 170 µm (Figure 1). It is concluded that the grain refinement gives rise to the de-
crease of roughness fold height relative to the average level corresponding to the surface of a homogeneous
sample, and to the increase of fold number (Figure 5). In all cases, the transverse sizes of the folds formed at the
early stage of the plastic flow are of 3 - 4 grain diameters.
4. Conclusion
In this paper we have investigated the effect of the loading conditions and grain size on the qualitative characte-
1
2
500 µm
0.08
δ
X
2
,
µ
m
0
-0.08

O. Zinovieva et al.
430
ristics of surface roughness and deformation processes in polycrystalline AL6061-T3 aluminum alloy at the
mesoscale under quasi-static tension. It is shown that the increase of grain size leads to the formation of larger
relief folds on the surface of polycrystals loaded. Periodic BCs also cause the formation of higher folds with
lower period by comparison with the loaded specimen with free lateral surfaces. In order to optimize numerical
calculations, we determine the minimum thickness of the specimen for the study of the surface roughness phe-
nomena. It is of 3 - 4 average grain diameters.
Acknowledgem ents
This work is partially supported by Federal Targeted Programme “R & D in Priority Areas of Science and
Technology Sector of Russian Federation in 2014-2015” and Russian Foundation for Basic Research (grant No.
14-08-00277 А).
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