DOI: 10.1007/s00707-015-1418-z
Acta Mechanica (2015) Page 1
Variation of cutting forces in machining of
f.c.c. single crystals
S. Abolfazl Zahedi
1
, Anish Roy
2
, Vadim V. Silberschmidt
3
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough
University, LE11 3TU UK
1
S.Zahedi@lboro.ac.uk, +44 (0)141 5345565,
2
A.Roy3@lboro.ac.uk, +44 (0)1509 227504,
3
V.Silberschmidt@lboro.ac.uk, +44 (0)1509 227637
Abstract
In this study, micro-machining of f.c.c single-crystal materials was investigated based on a hybrid
modelling approach combining smoothed particle hydrodynamics and continuum finite element
analysis. The numerical modelling was implemented in a commercial software ABAQUS/Explicit
by employing a user-defined subroutine VUMAT for a crystal-plasticity formulation to gain
insight into the underlying mechanisms that drive a plastic response of materials in high-
deformation processes. The numerical studies demonstrate that cutting-force variations in different
cutting directions are similar for different f.c.c. crystals even though the magnitudes of the cutting
forces are different.
Keyword: Single crystal, f.c.c., Shear strain, Cutting-force variation, SPH
Introduction
In recent years mechanical micro-machining has received much attention in the
manufacture of industrial small-size components with complex geometries,
especially for applications in aerospace, biomedical and automotive industries and
microelectronics [1-2]. A growing demand from various applications to reduce
levels of defects in ultra-precision metal cutting requires a fundamental
understanding of machining mechanisms at the micro scale. These applications
basically involve the machining of single-crystal metals or an aggregate of single
crystals (polycrystalline material) where each crystal may be oriented in a
different crystallographic direction in comparison to its neighbours. Apparently,
machining of polycrystalline materials in the micro-scale is inherently different
from machining single crystals. From a fundamental point of view, it is of interest
to investigate the response of machining single crystal materials in different
crystallographic orientations and directions of cutting. This will ultimately
DOI: 10.1007/s00707-015-1418-z
Acta Mechanica (2015) Page 2
indicate the consequence for machining polycrystalline aggregates, albeit ignoring
the effect of grain boundaries [3-4].
Analysis of any machining technique by means of extensive experimentation is an
expensive and time-consuming process. In addition, complexity of the underlying
physics of single-crystal deformation severely affects the outcomes of machining.
As a result, there has been a significant thrust in the development of analytical and
numerical computation methods for characterisation of micro-machining
processes. For instance, Sato et al. [5] used the Schmid factor to predict active slip
systems during the machining process in their model. They assumed one slip
system active continuously in each orientation setups. Micro-plasticity modelling
of machining proposed by Lee et al. [6-7] is another limited research available in
the literature analysing a mechanism of single-crystal machining. Shirakashi et al.
[8] and Kota and Ozdoganlar [9] used the Bishop and Hill’s crystal plasticity
model to predict shear angles and specific energies for f.c.c. single crystals.
The above techniques used to assess forces and stresses in machining process of
single crystals are usually limited to one active slip system at each incremental
deformation. Deformation processes in real-life machining are more complex and
require the use of a comprehensive modelling framework in analysing the cutting
forces and stresses involved. Some researchers used molecular dynamic (MD)
simulation to study the chip-removal mechanism [10-12] but his approach
requires significant computational power in order to model a cutting process in
physically meaningful volumes. Therefore, many MD simulations were applied in
two-dimensional formulations for a small workpiece with unrealistically high
cutting speeds.
Selecting materials for single-crystal machining studies, mainly copper and
aluminium have been preferred in the literature [3-12]. In this paper a well-
developed computational FE/SPH model was applied for these two mono-
crystalline materials to fully understanding the variation of cutting forces.
Copper and aluminium have both f.c.c. structures. The plastic deformation was a
result of resolved shear stress on 12 possible slip systems, with the Schmid
factor determining the slip-system activation. The crystal-plasticity formulation
presented in the next section was implemented as a VUMAT subroutine for
employment in ABAQUS/Explicit together with the SPH (smoothed particle
DOI: 10.1007/s00707-015-1418-z
Acta Mechanica (2015) Page 3
hydrodynamics) technique to predict the deformations, stresses, plastic strain
distribution in the f.c.c. crystalline structure of the workpiece materials. This
allows the effects of crystallinity parameters on cutting-force variations to be
investigated thoroughly.
2. Crystal plasticity
A crystal-plasticity framework has been widely used in predicting the mechanical
behaviour of, and texture evolution in, f.c.c. materials. In the crystal-plasticity
formulation the stress rate is related to the elastic strain rate
as
(1)
where C is the fourth-order elasticity tensor and and
are the total strain rate
and plastic strain rate, respectively. The f.c.c. metals have cubic symmetry; the
elastic moduli for such crystals are particularly simple, and can be parameterized
by only 3 material constants:
,
and
. The following matrix expresses
the elastic moduli of such materials:
.
(2)
The plastic deformation
represents material’s plastic shear and corresponds to
the amount of deformation that remains in the crystal after the load removal.
According to the flow rule:
.
(3)
The plastic strain rate is assumed to be the sum of the shear strain rates
over
the number of considered slip systems. Therefore,
(4)
with
is the Schmid tensor that is equal to a dyadic product of the slip direction
and the slip plane normal
:
DOI: 10.1007/s00707-015-1418-z
Acta Mechanica (2015) Page 4
.
(5)
In Eqs. 4 and 5 the superscript specifies the slip system and is the total
number of available slip systems.
The shear strain rate
of the th slip system in a rate-dependent crystalline solid
is determined by a visco-plastic flow rule as
(6)
where the constant
is the reference strain rate on the slip system ,
is the
variable, which describes the current strength of that slip system at the current
time,
is the shear stress on slip system , and the non-dimensional function
describes the dependence of strain rate on stress. The simplest flow rule is a visco-
plastic power-law expression proposed by Hutchinson [13] to describe
in the
following form:
(7)
where is the material’s rate sensitivity and is the signum function of . It
is worth mentioning that the reference strain-rate
in this equation is assumed to
be 10
-4
1/s. The strength of material
is equal to a sum of the critical resolved
shear stress (CRSS) and the evolved slip-resistance due to strain hardening:
,
(8)
where
(9)
The hardening moduli
in Eq. (9) are evaluated using the hardening model
proposed by Peirce et al. [14] as follows:
(10)
DOI: 10.1007/s00707-015-1418-z
Acta Mechanica (2015) Page 5
where
is the initial hardening parameter, is the latent hardening ratio and
assumed to be 1, is the Taylor cumulative shear strain on all slip systems and
and
are the shear stresses at the onset of yield and the saturation of
hardening, respectively. Therefore, the shear strain is equal to
.
(11)
The user-defined material subroutine VUMAT, initially developed by Huang [15],
modified by Kysar [16] and Zahedi [17-18] further developed by Demiral [19]
was used to implement this single-crystal plasticity formulation. The eight
parameters -
,
,
,
,
,
,
, - were considered as input material
data. Table 1 lists the material parameters used in present simulations for copper
and aluminium.
Table 1 Material parameters of single-crystal copper [20] and aluminum [21]
Copper
Aluminium
GPa
.2 GPa
GPa
GPa
GPa
GPa
MPa
MPa
MPa
MPa
MPa
MPa
The data for mono-crystalline materials in Table 1 show that apart form
and ,
which are the reference strain rate and material’s rate sensitivity, respectively, the
elastic parameters
,
,
and the plastic parameters
,
,
are different
for copper and aluminium. In the following section the machining model is
presented followed by discussion of deformation mechanisms.
3. Machining model
The basic mechanism of chip formation can be understood with the use of a
simple process of orthogonal cutting. Thus, a 3D workpiece with dimensions of
500 µm × 500 µm × 50 µmwas selected as an appropriate representation of a
![Table 1 Material parameters of single-crystal copper [20] and aluminum [21]](/figures/table-1-material-parameters-of-single-crystal-copper-20-and-2l91963h.webp)
![Figure 4 also demonstrates that the total slip that occurs in all active slip systems for a given state of deformation is nearly the same for any f.c.c. structure. For example, when the aluminium or copper crystal were oriented in the (100) plane and cut in the [100] direction, the chip formation appears](/figures/figure-4-also-demonstrates-that-the-total-slip-that-occurs-gwpk64w5.webp)
