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An Anisotropic Elastic-plastic Model for the
Optimization of a Press Machine’s Auxiliary
Worktable Plate Thickness
W. Rajhi
College of Engineering
University of Hail
Hail, Kingdom of Saudi
Arabia
B. Ayadi
College of Engineering
University of Hail
Hail, Kingdom of Saudi
Arabia
A. Alghamdi
College of Engineering
University of Hail
Hail, Kingdom of Saudi
Arabia
N. Messaoudene
College of Engineering
University of Hail
Hail, Kingdom of Saudi
Arabia
Abstract—In this work an anisotropic elastic–plastic finite
element model strongly coupled with ductile damage is applied to
determine the suitable thickness of added auxiliary worktable
plate to a 100 ton maximum capacity press machine. The major
focus of this study is to minimize the amount of stress transmitted
to the optional worktable plate and the press machine body while
allowing them to withstand plastic deformation and damage
during compression testing until final sample fracture. The
worktable plates and the press machine body are made of
TRIP800 grade steel. AISI 316L stainless steel is chosen as test
material for the cylindrical billets. The proposed model is based
on a non-associative plasticity theory and the “Hill 1948”
quadratic (equivalent) stress norm is considered to describe the
large plastic anisotropic flow accounting for mixed isotropic and
kinematic hardening with isotropic damage effect. For each
material the model uses an experimental data base obtained from
a set of tensile tests conducted until the final fracture in three
directions, the rolling direction (RD) or 0, the transverse
direction (TD) or 90, and the 45 direction. After several
numerical simulations of compression testing using
ABAQUS/Explicit FE® software, thanks to the user’s developed
VUMAT subroutine, varying cylindrical billet diameters and
material, worktable plates number and thicknesses and spatial
plate configurations the solution of 100mm thick worktable plate
is selected since in that case the cylinder specimen is totally
damaged and the stress state inside the worktable plates and the
press machine body remains admissible.
Keywords-compression testing; hardening; plastic anisotropy;
ductile damage; iIdentification; numerical simulation
I. INTRODUCTION
The proposed elastic–plastic model coupled with isotropic
ductile damage has been used in several researches to find
solutions to a set of problems related especially to the
occurrence of the ductile damage, which arises in metal
working processes, in order to either delay the damage
occurrence during the metal forming processes [1-5] or to
stimulate cavitation growth and totally ductile failure in the
case of metal cutting processes [5-8]. The extension to the
anisotropic damage case of the proposed model in order to
simulate the directional effect of the ductile damage in metal
forming process has been deeply discussed in [5, 9, 10]. In this
work, an isotropic version of the model presented in [9] is used
to solve a mechanical engineering problem which consists to
determine the suitable thickness of auxiliary worktable plates
of a 100ton maximum capacity press machine, their number
and their spatial positions in relation to each other, in order to
carry out compression testing until final sample fracture while
ensuring the protection of the worktable plates and the body of
the press machine against plastic deformation and damage
(Section II). Section III discusses the deduction of the elastic–
plastic constitutive equations fully coupled with isotropic
damage from the anisotropic damage model case. Numerical
aspects are well described in the case of the isotropic damage
model in [1, 5, 11] and extended to the anisotropic damage case
in [5, 9, 12]. Section IV is devoted to the identification of the
model parameters for the TRIP800 and the AISI 316L grade
steels using experimental results from the tensile tests
conducted until the final fracture of specimens loaded in three
directions i.e., the rolling direction (RD) or 0, the transverse
direction (TD) or 90, and the 45 direction. Thus, the
methodology proposed to determine the suitable optional
worktable plate thickness is presented and discussed.
II. PROBLEM FORMULATION
Press Master (see Figure 1) is a testing press machine
which is used in our mechanical laboratory for static three point
bending flexural test of steels, aluminum alloys and rock
materials under different control parameters. Because the press
machine is not equipped with auxiliary worktable, the
operating cannot include compression testing. An auxiliary
worktable is required to carry out compression testing of
different cross sectional area parts. During compression testing
the cylindrical specimens are placed between the piston rod and
the worktable which is positioned upon the work platform
beam. The auxiliary worktable may include one or more
metallic plates. These plates and the work platform beam both
are made of steel grade that may be assumed to behave
elastoplastically with mixed isotropic and kinematic hardening.
The TRIP800 steel which constitutes the material of the press
Engineering, Technology & Applied Science Research Vol. 8, No. 2, 2018, 2764-2769 2765
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machine body is selected in this work for the auxiliary
worktable plate’s material because TRIP steels offer an
outstanding combination of strength and ductility being thus
suitable for structural and reinforcement parts of complex
shape.
Fig. 1. Illustration of the critical press machine parts.
The typical true stress–strain curves for the TRIP800,
resulting from uniaxial tensile tests conducted along the rolling
direction (RD) or 0, the transverse direction (TD) or 90, and the
45 direction are given in Figure 2. The TRIP800 flat tensile
specimen is shown in Figure 3 with all dimensions in
millimeters. In Figure 2, one can see that the TRIP800 steel
exhibits slight plastic anisotropy with an overall ultimate stress
around 1040.0MPa reached for an overall total strain value of
about 24.5%, while the maximum total strain at the final
fracture is observed for the rolling direction around 27.0 % i.e.,
about 26.0% of plastic strain at fracture. Furthermore, the
stress–strain curves obtained for the 45 and (TD) directions
have roughly the same elastic–plastic trend before reaching
their necking points. The level for which the stress-strain
responses bifurcate for each direction is shown in Figure 2
where the 45 stress-strain curve reaches the ultimate stress
before that of the (TD) direction. Hence, for many metallic
orthotropic sheets, the ductile damage is highly anisotropic
even if the anisotropy of the plastic flow is small. Thus, it is
worth noting that three tensile tests are performed for each
loading direction in order to identify the tensile specimens
containing the lower defects density so as to use the
corresponding stress–strain curves i.e., revealing the greatest
ductility during the model parameters identification.
Present work’s purpose is to allow the press machine,
within its limit capacity, able to ensure compression testing
until final fracture of metallic cylindrical specimens using
optional worktable plates made of TRIP800 steel while
optimizing the plates’ thickness, their number and their
configurations according to various spatial arrangements. The
problem may be modeled as illustrated in Figure 4. The
worktable plates shape is assumed to be square
(400mm*400mm) of variable thickness. To achieve the above
target, this study aims to predict through a set of numerical
simulations of compression testing the limiting diameter of the
cylindrical specimens, for a given material, allowing the
continuity of the compression process until final fracture while
remaining within the press machine capacity range. In addition,
the number of the auxiliary plates, their thickness and spatial
positions in relation to each other can be optimized so that the
maximum values of the equivalent plastic strain and damage
reached during the compression process inside the body of the
press machine and the worktable plates remain negligible and
the stress should have the lower criticality even if it is elastic.
Fig. 2. TRIP800 uniaxial tensile tests conducted along the rolling
direction (RD) or 0, the transverse direction (TD) or 90, and the 45 direction.
Fig. 3. TRIP800 flat tensile test specimen dimensions.
Fig. 4. Schematic representation of the critical press machine parts.
III. CONSTITUTIVE EQUATIONS
During compression loading test, the cylindrical sample
materials as well as the TRIP800 steel assigned to both the
work platform beam and the worktable plates are assumed to
behave elastoplastically with mixed isotropic and kinematic
hardening according to an anisotropic elastic–plastic model
fully coupled to the ductile damage. In this work, only the
piston rod will be considered as a rigid part. The proposed
model is based on a non–associative plasticity theory and the
“Hill 1948” quadratic (equivalent) stress norm is considered to
describe the large plastic anisotropic flow accounting for mixed
isotropic and kinematic hardening with isotropic damage effect.
However, in the context of CDM, when the anisotropic effect
of the ductile damage is taken into account (see [10]), a fourth-
rank damage–effect tensor is used [14-16] to define the
effective state variables needed to describe the stress–strain
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behavior accounting for the full effects of the anisotropic
ductile damage. In this case, the damage variable may be
modeled using a symmetric forth or second rank tensor in order
to represent the ductile damage of orthotropic metallic
materials. Following the concept of effective stress with the
hypothesis of total energy equivalence, the matrix
representations of the effective stress and the kinematic stress
tensors are given by:
-1
: M
(1)
-1
: X M X
(2)
where M is the fourth rank damage effect tensor. In the CDM
literature, several forms of damage effect tensor M are
formulated [14]. The effective isotropic hardening stress is
given by:
1-
R
R
(3)
And M defines an Euclidean norm of the second order
damage tensor . According to the damage effect tensor
forms presented in [14], it is clear that when the ductile damage
is assumed to be isotropic the damage effect tensor M can be
written under the following form:
1 1 1
, , ,
1 1 1
, ,
iso
f f f
M diag
f f f
(4)
where is the isotropic damage variable and f is a
degreasing function of Ω called the damage effect function. To
make the link between the constitutive equations of the
anisotropic damage model and those of the isotropic damage
case, the following damage effect function is chosen to
describe the stress–strain behavior accounting for the full
effects of the isotropic ductile damage:
1-f (5)
The constitutive equations for the isotropic damage case
may be deduced readily from those of the anisotropic case by
substituting the damage effect tensor M by the isotropic
damage effect tensor given by:
1- 1
iso M (6)
Where 1 is the fourth order unit tensor. Thus, in the
isotropic damage case, the Euclidean norm of the second
order damage tensor transforms to the scalar damage variable
Ω. In this work the plastic anisotropy is taken into account via
the fourth rank symmetric and positive definite operator
H which defines the anisotropy of plastic flow. It is expressed
in terms of Hill’s anisotropic parameters F, G, H, L, M and N:
.
-
- -
0 0 0 2
0 0
G H Sym
H H F
G F F G
H
L
0 0 2
0 0 0 0 0 2
M
N
(7)
Using the non-associative framework, the anisotropic
plastic flow is modeled using a yield function fP :
, , ; - - -p yHf X R X R
(8)
The parameter σy is the initial size of the plastic yield
surface. The norm
H
X defines the “Hill 1948” quadratic
equivalent stress in the effective stress space. The fully coupled
anisotropic plasticity–isotropic damage model constitutive
equations accounting for the kinematic and isotropic hardening
are given by:
(1- )
eE (9)
(1- )X C (10)
(1 )R Q r (11)
where E, C and Q are the elasticity, kinematic and isotropic
hardening modules. The application of the generalized
normality rule leads to the evolution relations for plastic flow
with hardening and anisotropic damage effect:
p
n
(12)
1
1
r br
(13)
p
a
(14)
where n represents the outward unit normal to the plastic yield
surface in the damaged state. The material constants a and b are
the non-linearity parameters for kinematic and isotropic
hardening respectively.
is the plastic (Lagrange) multiplier.
The effect of the damage is crucially governed by (6) and (15):
•
•
0-
1-
sY Y
S
(15)
where Y is the total energy release rate density tensor. In
conclusion, noting that both the anisotropic and isotropic
damage models have the same number of material parameters
which are the elastic proprieties E and υ to describe the
isotropic elastic behavior, eleven plastic parameters σy, F, G, H,
L, M, N, C, Q, a and b for the plastic anisotropy and four
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damage parameters S, s, β and Y0 to control the damage
evolution. Finally, the related numerical aspects are deeply
described in [1, 5, 11].
IV. RESULTS
This section is devoted to the numerical simulation of
compression testing of cylinder specimens made of different
materials according to the schematic representation illustrated
in Figures 3-4. The identification of the elastic–plastic and
damage parameters of the isotropic damage model presented
above are required to perform the numerical simulation of
compression testing. Two grade steel materials are considered
for each compression testing simulation. The TRIP800 steel for
the press machine parts (the work platform beam and the
worktable plates) and the AISI 316L stainless steel assigned
beforehand to the cylindrical specimen. The identification
procedure is based on two main steps. The first step consists on
the determination of the elastic–plastic parameters through a set
of uncoupled (without damage effect) tensile test calculations
under Abaqus/Explicit® using the user subroutine VUMAT
while respecting the specimen geometry (see Figure 4). The
tensile specimen given in Figure 3 is meshed using hexahedral
tri–linear elements (C3D8R) with a meshing refinement
(minimum mesh size of 1.0mm3) located on the central region
or gage length of the specimen. The numerical tensile test was
carried out with a constant velocity of 3.0mm/s. Anisotropic
parameters G and H and the set of hardening parameters are
varied in order to get the best fit between the numerical force–
displacement curve and the one obtained experimentally along
the rolling direction while respecting two following conditions:
1) G+H=1 and 2) the retained plasticity parameters should
provide a numerical response comparable with experience, up
to a plastic strain range between 15% and 20%, from which the
two responses bifurcate due to the emphasized effect of ductile
void growth on experimental response. The TRIP800 (TD) 90
and 45 force–displacement experimental responses are
explored later to determine the remaining parameters F and N
respectively taking into account
that 90 /y F H and 45 2 / 2y F g N . In the
second identification step, fully coupled calculations are
performed taking into account the damage effect in order to
recover the optimum damage parameters which provide the
best fit between the end of numerical and experimental force–
displacement curves. In conclusion, the same VUMAT is used
with a flag parameter (NCD) allowing to perform an uncoupled
analysis when NCD=0 and a fully coupled analysis when
NCD=1. Τhe superposed experimental, numerical uncoupled
and fully coupled force–displacement curves for the three
loading directions RD, 45 and TD for the TRIP800 steel are not
presented here (see [9] for more details concerning the AISI
316L material parameters identification) and we limit ourselves
only to report in Τable 1 the best values of the TRIP800 and
AISI 316L steels parameters resulting from the identification
procedure described above. The parts shown in Figure 5,
excluding the piston rod, are meshed using hexahedral tri–
linear elements (C3D8R) with an overall meshing size of about
2.0mm3 for the 50mm length cylinder specimen, 7mm3 for the
100mm thick worktable plate and 20mm3 for the encastred
work platform beam. Τhe piston rod is considered as a rigid
part meshed using bilinear rigid quadrilateral elements (R3D4)
with meshing size of about 5.0mm3. If a modification of the
meshing size defined during the fully coupled analysis of the
identification procedure is intended, before performing the
compression testing simulations with the new meshing size, the
damage parameters values given in Table I particularly the
parameter S should be adjusted by supplementary set of
identification calculations for each material as the proposed
model is written in local approach where the damage
parameters depend strongly on the meshing size [18]. The
numerical compression testing was carried out with a constant
piston velocity of 100.0mm/s. The friction coefficient defined
between the different parts is about 0.3. In order to reduce the
CPU time in the followed dynamic explicit algorithm, a mass
scaling value of 10-6 is defined through all the compression
testing simulations. According to Figure 6, 36mm is the
limiting diameter value of the AISI 316L cylinder specimens
which the press machine can totally damage without exceeding
its maximum loading capacity range. Otherwise, beyond this
limiting diameter, the loading amount provided by the press
machine is unable to ensure the total failure as revealed by the
numerical simulation of the compression testing of the AISI
316L cylinder specimens given by Figure 6, where the damage
doesn’t exceed 5% and 10% for 44 and 40mm cylinder
diameters respectively when the loading reaches the maximum
capacity force of the press machine.
TABLE I. TRIP800 AND AISI 316L MODEL PARAMETERS
Model
Parameters
Grade Steel
TRIP800 AISI 316L [9]
Elasticity
E 195000MPa 200000MPa
υ 0.3 0.3
Anisotropic Plasticity
σy 406MPa 225MPa
F 0.45 0.8
G 0.47 0.76
H 0.53 0.24
L 1.5 1.5
M 1.5 1.5
N 1.45 1.55
C 49000MPa 15000MPa
Q 4000MPa 2000MPa
a 380 300
b 4.2 0.6
Damage
S 102MPa 140MPa
s 1 1
β 1.5 1.5
Y0 0 0
The first set of compression testing simulations checks the
suitable AISI 316L cylinder specimen diameter which the press
machine can deform until total failure occurs without
exceeding the capacity specified by the manufacturer. In Figure
5 initial geometries, mesh and boundary conditions of the
compression testing simulation are schematized. The worktable
is assumed to be made from one plate 100mm thick. The
distribution of the Von Mises stress, the damage and the
equivalent plastic strain inside the 36mm diameter cylinder at
the end of the compression testing process is shown in Figure
7. The mechanical field distribution within the worktable plate
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and the body of the press machine analysis will be discussed
later. Furthermore, the compression testing simulation was
successfully accomplished as the maximum values of the
damage and the equivalent plastic strain were located at the
center of the AISI 316L cylinder correctly at the intersection of
the two well-known shear bands. This result is marked by the
fast drop of the force–displacement curve (not presented here)
at the end of the compression testing. According to Figure 7(a),
the maximum VOM stress obtained at the end of the
compression process within the cylinder σend (σend=860MPa) is
considerably greater than the ultimate stress of the AISI 316L
[9] as significant dissimilarity between the tensile and the
compression behaviors at large deformations is often expected
in ductile materials.
Fig. 5. Geometry, boundary conditions, and meshing of the parts.
Fig. 6. Numerical force–damage curves of the AISI 316 L compression
testing obtained for different diameters
A second set of compression testing simulations was
carried out altering TRIP800 worktable plate’s number, plate
thickness and configuration according to various spatial
arrangements of plates from one calculation to another until
obtaining the optimal configuration which ensures the
protection of the worktable plates and the body of the press
machine against plastic deformation and damage. Furthermore,
the solutions exhibiting low criticality were collected. It is
worth noting that the appropriate configuration means the
numerical solution allowing the compression testing until total
failure of the AISI 316L cylindrical specimen without
exceeding the press machine capacity and giving the lowest
elastic stress value inside both the TRIP800 worktable plates
and the press machine body. After several numerical
simulations of compression testing, the solution of 100mm
maximum thickness is selected for a well-defined number of
worktable plates. Accordingly, eleven configurations based on
how worktable plates with different number and thicknesses
are arranged spatially, were analyzed. In proposed
configurations, the plates are positioned from the piston rod to
the work platform beam across the plates thickness direction
according to the rank reported in Table II.
(a) (b)
(c)
Fig. 7. (a) Distribution of the Von Mises stress, (b) the damage and (c) the
equivalent plastic strain, at the end of the compression testing of 36mm
diameter cylindrical AISI 316L billet.
TABLE II. CONFIGURATIONS AND RELATED MAXIMUM VOM STRESS
STATES OBTAINED DURING COMPRESSION TESTING OF THE AISI 316L SAMPLES
Configuration
VOM Stress
Maxi in
plates σP
(MPa)
VOM Stress Maxi
in machine body σB
(MPa)
A 1 plate of 50mm 462 200
B 1 plate of 75mm 429 110
C 3 plates of 25mm 556 319
D 2 plates of 37.5mm 463 208
E 2 plates of 50mm 430 129
F 1 plate of 100mm 421 81
G 2 plates of
2550mm
481 181
H 2 plates of
5025mm
439 166
I 3 plates of
255025mm
478 156
J 3 plates of
502525mm
445 152
K 2 plates of
7525mm
432 99
Maximum VOM stress values reported in Table II are
reached immediately before the total rupture of the first
meshing element generally located at the specimen center.
Configurations reported in Table II are judged to be of low
criticality as the maximum VOM stress within the body of the
press machine is less than the yield stress of the TRIP800 steel.
The above configurations are chosen among harshest stress
state configurations causing a significant problem within the
worktable plates and the press machine body. In fact, as
illustrated in Figure 8(a), the use of two 20mm thick plates
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leads to the excessive bending distortion of the TRIP800
worktable plates before the total fracture of the AISI 316L
specimen occurs. Likewise, as illustrated in Figure 8(b), high
stress i.e., 98% of the TRIP800 yield stress, is detected within
the press machine body with a slight bending distortion of the
inferior plates owing to the use of five 20mm thick plates. The
results reported in Table II prove the TRIP800 steel's validity
while giving the worktable plates excellent combination of
strength and ductility under extreme loading conditions since
almost the maximum VOM stress values inside the plates σP
for various configurations are not very far from the yield stress
value of the TRIP800 steel. The slight decrease of the billet
diameter carries the greater part of the configurations below the
elastic barrier. Nevertheless, the maximum VOM stress values
within the press machine body σB remain below the elastic
limit for all configurations. According to the configurations
sets (B, C, D) and (F, K, I or J), the higher the number of
plates, the higher the maximum VOM stress within the plates
and the press machine body. The couple of values defined by
the gaps between σend and the maximum VOM stress within
both the worktable plates and the press machine body (σend-σP,
σend-σB) is used as criterion for the selection of the suitable
configuration. Accordingly, the F and K configuration provides
the maximum pair gap values of about (440, 780MPa) and
(428, 761MPa) for the plates and the press machine body
respectively and may be chosen as the appropriate
configurations.
(a) (b)
Fig. 8. (a) Excessive bending distortion of two 20mm thick plates,
(b) high stress in the machine press body due to the use of five 20mm thick
plates
V. CONCLUSION
In this work, a mechanical engineering problem related to
the optimization of the press machine optional worktable
thickness was solved using an anisotropic elastic–plastic finite
element model strongly coupled with ductile damage. The
numerical simulation of compression testing of different
materials varying worktable plates’ thickness, their number and
their spatial positions in relation to each other is crucial in order
to find the best solution allowing the worktable plates and the
press machine body to withstand against plastic deformation
and damage. The use of 100mm worktable plate reduces
greatly the amount of stress transmitted to the press machine
parts. Eventually, the above solution will be considered
practically to perform compression testing of cylindrical
samples made of any material.
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