M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
463–471
LOAD DETERMINATION BY ANALYSING THE STRESS STATE
FOR THE OPEN-DIE FORGING OF THE ALUMINIUM ALLOY
AlMgSi0.5
DOLO^ANJE OBREMENITVE Z ANALIZO NAPETOSTNEGA
STANJA PRI PROSTEM KOVANJU ALUMINIJEVE ZLITINE
AlMgSi0,5
Mileta Janji}, Milan Vuk~evi}, Nikola [ibali}, Sreten Savi}evi}
University of Montenegro, Faculty of Mechanical Engineering, Cetinjska 2, 81000 Podgorica, Montenegro
mileta@ac.me
Prejem rokopisa – received: 2016-05-19; sprejem za objavo – accepted for publication: 2016-06-03
doi:10.17222/mit.2016.092
This paper deals with the stress state for the open-die forging of a staggered axial-symmetric specimen of the aluminium alloy
AlMgSi0.5. The experimental-theoretical method of visioplasticity, adjusted for the process of open-die forging, consisted of
five steps. The basic equations of the plasticity theory, i.e., balance equation, conditions of a plastic flow and Levy-Mises
equations are applied. The basic partial differential equations of visioplasticity are obtained theoretically, while the displace-
ment of points of the meridian cross-section, which represents the basis for its solution, can be experimentally obtained. The
deformation force is calculated on the basis of the obtained results and is compared to the experimental values.
Keywords: visioplasticity, strain rate, stress, deformation force, displacement, tensor
^lanek obravnava napetostno stanje pri prostem kovanju stopni~astega osnosimetri~nega vzorca iz aluminijeve zlitine
AlMgSi0,5. Eksperimentalno-teoreti~na metoda, prilagojena postopku prostega kovanja, je setavljena iz petih korakov.
Uporabljena je bila osnovna ena~ba teorije plasti~nosti: ravnote`na ena~ba, pogoji plasti~nega te~enja in Levy-Misesove ena~be.
Osnovna parcialna diferencialna ena~ba vizioplasti~nosti je dobljena teoreti~no, medtem ko se premik meridiana pre~nega
preseka, kar predstavlja osnovo za njeno re{itev, lahko dobi eksperimentalno. Sila deformacije je izra~unana na osnovi dobljenih
rezultatov in je primerjana z eksperimentalno dobljenimi vrednostmi.
Klju~ne besede: vizioplasti~nost, raztezek, trgalna hitrost, napetost, sila deformacije, premik, tenzor
1 INTRODUCTION
The use of numerical methods for the deformation
parameters’ calculation is constantly being enhanced by
the increment of simplicity of usage, the development of
advanced software, reducing the time for the preparation
of input data and obtaining of a wide spectrum of output
information. Beside this, the questions asking whether
the entry data were correctly taken and are the obtained
results valid always are being raised.1,2 For this purpose,
it is necessary to develop or adapt the experimen-
tal–theoretical methods serving to verify the numerically
obtained results.1,3 There were many attempts by several
authors to determine the stress state for a volumetric
deformation, such as setting of a mesh at the specimen’s
cross-section before the deformation and a comparison
with a deformed mesh,4–6 determination of a deformed
image of the cross-section after the deformation
process,7,8 use of the Upper Bound Elemental Technique
(UBET),9 etc.
One of the methods for the determination of the
stress state is the visioplasticity method.10,11 This is an
experimental–theoretical method based on the known
kinematic parameters of the deformation process. This
method enables the analysis of plane and axial-sym-
metric problems in the processing by deformation. The
method uses the basic equations of the theory of
plasticity: balance equations, conditions of a plastic flow
and equations by Levy-Mises.
The procedure of the calculation of the tensor stress
components by the visioplasticity method can be divided
into five steps:
• Deformation of the specimen and preparation for the
reading of point displacements of the meridian
cross-section,
• Determination of the strain tensor components,
• Determination of the strain rate tensor components,
• Solution of the basic equation for visioplasticity,
• Determinations of stress tensor components.
There are three different variants of this method. The
differences are conditioned by the deformation processes
and the manner of the experimental determination of
displacements of points of a specimen’s cross-section.
This paper presents the originally adjusted method of
visioplasticity, adjusted to open-die forging.
Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471 463
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
UDK 621.73.042:669.715:620.172.21 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 51(3)463(2017)
2 THEORETICAL BASICS
2.1 Stresses
The stress tensor for an axial-symmetrical problem
has the following shape: (r = z = 0):12
T
r rz
rz z
 
 
  
  
=
0
(1)
and the balance equations are:
∂
∂
∂
∂
∂
∂
∂
∂
   
  
r rz r
rz z rz
r z r
r z r
+ +
−
=
+ + =
⎫
⎬
⎪
⎭
⎪
0
0
(2)
The effective stress 42,67 is:
       


 
e r r r z rz= − + − + − +
1
2
6 2( ) ( ) ( ) (3)
and the average normal stress is equal to:
( )   a r z= + +13 (4)
The stress intensity has major importance in the
theory of plastic flow because on the basis of the biggest
deformation energy spent for the shape change, an
elementary part changes from the elastic into plastic state
when the stress intensity reaches the boundary value.
From (3), it follows that:
( ) ( ) ( )       


 
r r r z rz e− + − + − + =6 2
2 2 (5)
which represents the condition of plastic flow for an
axial-symmetric stress state.
2.2 Strain
There is no displacement in the tangential direction
for the axial-symmetric state and the changes of two
other displacements in this direction are:12
u
u ur z
  
= = =0 0 0,
∂
∂
∂
∂
, (6)
The stress tensor for the axial-symmetric deformation
state is:
T
r rz
rz z








 
 
 
  

=
0
(7)
The components of the strain tensor are:

 

 

 

 
r
r
r
r
z
z
z
rz
u
r
u
r
u
z
u
= =
= =
= =
∂
∂
∂
∂
∂
0
0
r z
z
u
r∂
∂
∂
+
⎫
⎬
⎪
⎪
⎭
⎪
⎪
(8)
The effective strain is:

 
 
 
 
 
 
 


 
e r z r z rz/= − + − + − +
2
3
3 2 2( ) ( ) ( ) (9)
2.3 Strain rates
Analogously with the previous tensor, the tensor of
strain rates for the axial-symmetric state is:12
T
x xy xz
xy y yz
xz yz z

  
  
  




















= (10)
The components of the strain-rate tensor in a
cylindrical coordinate system, on the basis of 8, are:
 
 
 







 
r
r
r
r
z
z
z
v
r
v
r
v
z
= =
= =
=
∂
∂
∂
∂
0
0
zr
r zv
z
v
r
= +
⎫
⎬
⎪
⎪
⎭
⎪
⎪∂
∂
∂
∂
(11)
The effective strain rate is:
 (  ) (  ) (  ) 
 
 
 
 
 
 



 
e r z r z rz/= − + − + − +
2
3
3 2 2 (12)
In the cylindrical coordinate system for the axial-
symmetric state, the equations by Levy-Mises are as
follows:
.
 ' ( )
 ' ( )
 ' (

   

   

   
 
z z sr
z
sr
r
r r s
v
z
v
r
= − =
= − =
= −
∂
∂
r
r
rz rz
r z
v
r
v
z
v
r
)
 '
=
= = +
⎫
⎬
⎪
⎪
⎪⎪
⎭
⎪
⎪
⎪
⎪
∂
∂
∂
∂
∂
∂
 2
(13)
The coefficient of proportionality of  ' in 13 is:
 



'

=
3
2
e
e
(14)
3 EQUATION OF VISIOPLASTICITY
By combining of first two equations from Equation
(13) the following can be written:
  ' ( )
 
   z r z r− = − (15)
which gives the following:


 

 
z
z r
r=
−
+
 
'
(16)
By differentiating the above equation with respect to
r for constant z and  we obtain the following:
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
464 Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
∂
∂
∂
∂
∂
∂
  
 

 
z r z r
r r r
= + ⋅
−⎛
⎝
⎜
⎞
⎠
⎟
 
'
(17)
From the first system balance equation (2) we obtain
the following:
∂
∂
∂
∂
   r r zr
r r z
= −
−
− (18)
while from the second and third equations the relation
of the stress and strain rates of the system can be
obtained (13):



 




 


r
r
sr
sr
= −
= −

'

'
(19)
Substituting (19) into (17) we obtain:
∂
∂
∂
∂
 
 

 
r r sr
r r z
=
−⎛
⎝
⎜ ⎞
⎠
⎟ −
 
'
(20)
If the above expression is replaced into equation (17)
we obtain the following expression:
∂
∂
∂
∂
∂
∂
 
 

 
 
 

 
z z zr z r
r z r
= −
−
− +
−⎛
⎝
⎜
⎞
⎠
⎟
 
'
 
'
(21)
Since from the fourth equation of the Levy-Mises
system (13):


 
= −

'
rz
(22)
then equation (21) takes the following shape:
∂
∂
∂
∂
∂
∂
 
 

 

 

 

 
z z r z r
r r r z
=
−⎛
⎝
⎜
⎞
⎠
⎟ −
−
−
⎛
⎝
⎜ ⎞
⎠
 
'
 
'

'
⎟ (23)
Substituting Equation (14) for  ' into equation (23)
we obtain:
∂
∂
∂
∂
∂
∂



 




 




z
e
z r
e
r
e
z
r r r z
=
−⎛
⎝
⎜
⎞
⎠
⎟ −
−
−
2
3
1
2
 

 


r
e


⎛
⎝
⎜
⎞
⎠
⎟ (24)
Equation (24) represents the gradient of an axial
stress, which depends on the strain rates and the effective
stress at every point of the deformation zone. This is the
basic equation of visioplasticity.
The components of the strain rates and the effective
stress are determined experimentally so as to, using the
defined mathematical operations via Equation (24),
obtain the value of an axial stress in the relevant points
of a deformable body volume. The only point where the
axial stress cannot be determined is for r = 0, because
Equation (24) becomes undefined. However, because of
the process symmetry at this point, the following applies:
∂
∂
 z
r
= 0 (25)
From the above it can be seen that the problem of
determining the axial component of stress in the process
of the axial-symmetric deformation is reduced to the
problem of determining the components of the strain
rates, the effective stress and the effective strain in the
deformation zone.
With knowledge of the axial stress at each point of
body volume, it is easy to obtain other components of the
stress using the Levy-Mises equations (13):
  

 



  

 




r z e
r z
e
z e
z
e
= +
−⎛
⎝
⎜
⎞
⎠
⎟
= +
−⎛
⎝
⎜
⎞
2
3
2
3
 

 
 ⎠
⎟
=
 

rs
e rz
e3


(26)
4 STUDY TOPIC
The research was carried out in laboratory conditions,
which were adjusted to be as similar as possible to the
manufacturing conditions.7,8
The staggered axial-symmetric processing body with
two levels of height from upper and one level on the
lower partition plane is adopted (Figure 1).
Aluminium alloy AlMgSi0.5, the chemical compo-
sition of which is given in Table 1, is used as a test
material.
Table 1: Chemical composition AlMgSi0.5 in mass fractions (w/%)
Tabela 1: Kemijska sestava AlMgSi0,5, v masnih odstotkih (w/%)
Fe Si Ti Cu Zn V Cr Mn Mg Ni
0.207 0.477 0.01 0.09 0.068 0.004 0.01 0.1 0.493 0.02
The testing is carried out at the hot processing tem-
perature: T = 440 °C. The deformation is accomplished
with a constant velocity: v = 2 mm/s. Graphite grease is
used for the lubrication.
The experiment is carried out on the machine for the
static testing with a hydraulic drive.
The upper and lower dies are used as tools (Figure
1), located in a slideway, which ensures their coaxiality
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471 465
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 1: System: tools workpiece
Slika 1: Sistem: orodje-obdelovanec
and has the role of a chamber for the maintenance of a
constant temperature. The slideway is heated together
with the die and the specimens.
The dies are made of tool steel for work in hot state
designated by X38CrMoV-5-1.13
The specimen dimensions are: ! d0 × h0 = 33×33,94
mm. The specimen’s height is calculated from the
condition of volume constancy.
5 DETERMINATION OF THE STRESS STATE
5.1 Determination of kinematic field using the speci-
men volume
The basis for the determination of a stress state using
the visioplasticity method is the familiarity of the kine-
matic field. Two kinds of process can be differentiated:10
• stationary (drawing, milling, direct extrusion),
• non-stationary (opposite extrusion, squeezing in open
dies).
For stationary processes, it is characteristic that the
elements of the kinematic and stress fields in one
unmovable point of space where the process is carried
out are independent of time. Therefore, the kinematic
field can be determined experimentally in any time
(phase). Also, the duration of the phase is also not im-
portant under the assumption of the constant strain rate.
For the non-stationary processes, the kinematic and
stress fields in one unmovable space point where the
deformation is carried out, are changed over time. In
such a processes, the analyses of one defined increment
of deformation cannot give an image about the kinematic
and stress field for the entire process time, but only for
the observed increment. Therefore, it is necessary to
determine the interval (phase) at the end of which the
stress state can be determined in the specimen volume.
The determination of the kinematic field necessary
for solving the basic equation of visioplasticity (24) is
carried out experimentally. It is possible to determine the
components of the displacement velocities on the basis
of the measured values of the displacements of the
meridian cross-section points according to 7, 8:
v
u
t
v
u
t
r
r
z
z
=
=
⎫
⎬
⎪
⎭
⎪




(27)
The constant deformation state is the assumption for
such a method of determination.
The displacements of the points in the radial and
axial directions are determined on the basis of known
numerical value coordinates of the mesh nodes (Figure
2):
u r r
u z z
r
z
= −
= −
⎫
⎬
⎭
0
0
(28)
where:
ur, uz are the displacements in radial and axial directions
at the end of the process,
r, z are the coordinates of node points at the end of the
process,
r0, z0 are the coordinates of node points at the start of
the process.
In this way it is possible to determine the partial
derivations of displacements using the radius, serving to
obtain the deformation values.8,9
The determination of the displacement velocity
represents the basis of the kinematic analysis. Since the
deformation process in an open die is a non-stationary
process, the displacement velocities are determined on
the basis of the displacements of points at the start and at
the end of a finite interval of the deformation at the end
of the deformation process, under the assumption of a
constant deformation velocity. This interval must be
small enough to represent the true state of the
displacement velocities, but not too small so as to avoid
the impact of the anisotropy of deformation. Based on
the preliminary experiments, it was proved that the
optimal interval of deformation at the end of the process
for the increment of a tool step of z = 2 mm.14
At the beginning of the adopted interval, it was
necessary to determine the deformed mesh of the
specimen’s radial cross-section. The deformation process
is stopped at the flash height of hf = 3 mm. The process
of preparation of the specimen’s cross-section, and the
measurement of the lines of the deformed mesh is the
same as for the determination of the deformations.
On the basis of the known values of the nodal point
coordinates in the deformed and non-deformed mesh at
the start of the observed interval, the displacements of
the points in the radial and axial directions are
determined:
u r r
u z z
ra a
za a
= −
= −
⎫
⎬
⎭
0
0
(29)
where:
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
466 Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 2: Displacements of cross-section points of the deformed mesh
Slika 2: Premiki to~k preseka v deformirani mre`i
ura, uza are the displacements in the radial and axial
directions at the start of the observed deformation
interval, and ra, za are the coordinates of the nodal points
at the start of the observed deformation interval.
The increase of the displacements in the nodal points
of the deformed mesh is calculated as the difference in
the displacements at the end and the start of the observed
interval:
 
 
u u u r r r
u u u z z z
ra r ra a
z z za a
= − = − =
= − = − =
⎫
⎬
⎭
(30)
where: ur, uz are the increments of displacements in
radial and axial direction, and r, z are the increments
of the coordinates.
Having in mind that the deformation velocity is con-
stant at v = 2 mm/s, and for the adopted tool step
increment z = 2 mm the time increment is:


t
z
v
s= (31)
It is possible to determine the components of the
displacement velocity (27) on the basis of the values of
the displacements of nodal points in the deformed mesh
of the meridian cross-section (Figure 3):
v
u
t
r
t
v
u
t
z
t
r
r
z
z
= =
= =
⎫
⎬
⎪
⎭
⎪








(32)
where: vr, vz are the displacement velocities in the radial
and axial directions.
Using the knowledge of the kinematic field, i.e., the
distribution of displacement velocities, it is possible to
determine the displacement velocities for the points of
the meridian cross-section according to 11, while the
effective strain rate is determined using Equation (12).
The partial derivations are determined for the small
enough values r, z and t on the basis of the
following Equations (33), (34) and (35):
∂
∂
v
r
v
r
r r=


(33)
∂
∂
v
z
v
z
z z=


(34)
∂
∂
∂
∂
v
z
v
r
v
z
v
z
r z r z+ =




(35)
5.2 Determination of the stress components
The distribution of an axial stress z in planes z = const.
is obtained by the integration of the basic visioplasticity
Equation (24):
 

 




 




z
r
e
z r
e
r
er r
=
−⎛
⎝
⎜
⎞
⎠
⎟ −
−⎛
⎝
⎜
⎞
⎠
⎟ −∫
2
3
1
20
∂
∂
 

 

∂
∂z
r C
zr
e



⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣⎢
⎤
⎦⎥
+d 1
(36)
To obtain the absolute values of the axial stresses, it
is necessary to determine the integral constant C1. From
the first equation of the system (26) it is necessary to
obtain an expression for an axial stress at each point of
the deformation zone as in the following Equation (37):
  

 


z r e
r z
e
= −
−⎛
⎝
⎜
⎞
⎠
⎟
2
3
 

(37)
The radial stress r is not known for the complete
deformation zone, but in the free area in the flash
cross-section it is equal to zero, which means that the
axial stress can be determined in that cross-section. If the
first term from Equation (36) for a axial stress is
designated with z, the following Equation (38) can be
written:
 z z C= + 1 (38)
The integral constant at the point of the free area at
the end of the flash can be obtained in the following
way:
C z z z
r r
z z z
r r
f
f
f
f
1 = −=
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
=
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
  (39)
On the basis of known values z, the values of other
components of the stress tensor can be obtained from
Equation (26).
5.3 Solving the basic visioplasticity equation
The method is based on a calculation of the axial
stress component z by solving the basic visioplasticity
Equation (24), where the main problem is the determi-
nation of integral constant C1 (Equation (36)). Having in
mind that the integration of the visioplasticity equation is
done with a radius, the value of the axial stress com-
ponent at one point of radius r for a particular value of
the height y must be known in order to determine the
integral constant C1. The only points where it is possible
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471 467
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 3: Vectors of the displacement velocities at certain points of
the cross-section
Slika 3: Vektorji hitrosti premika dolo~enih to~k na preseku
to determine the value of the axial stress component are
the points for the maximum values of the radius on the
flash level (Figure 1). These values are determined on
the basis (37) form the condition that the radial stress
component in these points is equal to zero: r = 0. Other
deformation kinematic parameters are known, where the
effective stress is determined from the corresponding
curved strengthening for the value of the effective strain.
The value of the sub-integral function of Equation
(36) is known because all the strain and kinematic
parameters and the effective stress are known, and they
are presented in Figure 4. Using the integration of this
function on r, the indefinite integral given in Figure 5 is
obtained, and the integral constant is determined in a
previously described way.
The integral constant is impossible to determine
because of the unknown radial stress at other points. For
this reason, on the basis of the system balance Equation
(2), it follows that:

 
z
rz rzz
r r
z C= +
⎛
⎝
⎜
⎞
⎠
⎟ +∫
∂
∂0 2
d (40)
If the first term of the previous expression is marked
with z0, then the following can be written:
 z z z
r
z z z
r
f f C=
=
⎛
⎝
⎜
⎞
⎠
⎟
=
=
⎛
⎝
⎜
⎞
⎠
⎟
= +
0 0
2 (41)
and so:
C z z z
r
z z z
r
f f2
0
0
0
= −=
=
⎛
⎝
⎜
⎞
⎠
⎟
=
=
⎛
⎝
⎜
⎞
⎠
⎟
  (42)
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
468 Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 6: Sub-integral function of the second equation for the speci-
men’s axis
Slika 6: Podintegralna funkcija druge ena~be za os vzorca
Figure 4: Sub-integral function of the visioplasticity equation for the
cross-section of the partition plane
Slika 4: Podintegralna funkcija ena~be vizioplasti~nosti na preseku
delilne ravnine
Figure 7: Indefinite integral of the second balance equation z0 + C2
in the specimen’s axis
Slika 7: Nedolo~ni integral druge ravnote`ne ena~be z0 + C2 na osi
vzorca
Figure 5: Undefined integral of the visioplasticity equation z + C1
at the cross-section of the partition plane
Slika 5: Nedolo~eni integral ena~be vizioplasti~nosti z + C1 na pre-
seku delilne ravnine
In this way, all the values of the axial stress com-
ponent in the axis of the specimen’s symmetry are
obtained, i.e., r = 0 mm, which actually represents the
possibility for a determination of the integral constant for
all the values of the height z.
The values of the sub-integral function (40) as a
function of the height are given at Figure 6, and the
values of indefinite integral for r = 0 mm in Figure 7.
The integral constant is determined on the basis of:
 z z
r
z z
r
C
=
⎛
⎝
⎜
⎞
⎠
⎟
=
⎛
⎝
⎜
⎞
⎠
⎟
= +
0 0
(43)
and so:
C z z
r
z z
r
= −
=
⎛
⎝
⎜
⎞
⎠
⎟
=
⎛
⎝
⎜
⎞
⎠
⎟
 
0
0
0
(44)
which is graphically represented in Figure 8 as a
function of the height z.
The values of the normal stress at all the points of the
specimen’s meridian cross-section are obtained in the
previously described way, and other tensor components
are determined on the basis of relations (26). The values
of the axial and shear stress using the upper and lower
contour of the specimen’s contour, and in the cross-sec-
tion of the partition plane, are given in Figure 9 and
Figure 10.
MATLAB has developed the program for a determi-
nation of all the stress parameters for the specimen’s
cross-section using the visioplasticity method, all
numerical integrations and for the determination of the
integral constants. As the entry data, besides the curve of
strengthening and the geometrical parameters, the prog-
ram uses the output results from the strain and kinematic
analyses.
6 DEFORMATION FORCE
The familiarity with the stress state in the specimen’s
meridian plane enables the determination of deformation
force. Also, this value, in a relatively simple way, can be
experimentally measured with a high accuracy and can
be used for an estimation of the accuracy of the stress
deformation analysis. In this case it is the z-axis direc-
tion and the stress components that are the axial and
shear stress. The deformation force is calculated as a
sum of the integrals using the area of change of the
mentioned stress components:12
F A Am z
A
rz
A
= ∫ ∫ d + d (45)
Figure 11 presents the experimentally obtained
deformation forces during the process and the values of
the forces at the end of the deformation process cal-
culated using the visioplasticity method on the basis of
the determined stress state for the upper contour, the
lower contour and the cross-section of the partition
plane. On the basis of the law of static balance of a body,
the forces obtained for the upper and lower die should be
the same; however, certain deflections emerge because of
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471 469
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 9: Axial stress z
Slika 9: Napetost v osi z
Figure 10: Shear stress rz
Slika 10: Stri`na napetost rz
Figure 8: Integral constant C as a function of the height z
Slika 8: Konstanta integrala C v funkciji vi{ine z
the calculation errors. These deflections are relatively
small, which indicates that the forces are determined
with a satisfactory accuracy.
7 CONCLUSION
This paper deals with the stress state for the open-die
forging process for an axial-symmetric specimen made
of the aluminium alloy AlMgSi0.5. The originally
adjusted, experimental–theoretical visioplasticity method
for the open-die forging.
The visioplasticity method consists of five steps.
Using the basic equations of the plasticity theory, i.e.,
balance equation, condition of plastic flow, and Levy-
Mises equation, enables the obtaining of the basic
visioplasticity equation. The starting basis is made of
experimentally determined displacements of the points
of the meridian cross-section.
When solving the basic visioplasticity equation, the
determination of the integral constant represents the
main problem. It is calculated for the cross-section of the
partition plane because an axial stress is known at the
free end of the flash and it is equal to the flow stress.
Then the axial stress along the specimen axis and
integral constant on the basis of the known axial stress in
the cross-section of partition plane and z axis. After this,
an axial stress in all cross-sections is determined, and the
integral constants on the basis of known values of the
stress along the z axis.
The deformation force, as the sum of the integrals for
the area of normal and shear stress for the upper contour,
the lower contour and the cross-section of the partition
plane, is obtained on the basis of the obtained results.
The obtained values are: Fu = 529 kN, Fl = 555 kN, Fp =
576 kN. The experimentally measured values at the end
of the deformation process are: FI = 500.66 kN in FII =
600.35 kN. The deflections of the experimental values
are in range of 16.61 %, and for the calculated values on
the basis of the stress state obtained by the visioplasticity
method 8.16 %, and they are between the minimum and
maximum experimental values of the experimental
values of the deformation force at the end of the process.
It can be concluded that the adapted visioplasticity
model for the determination of the stress state for the
open-die forging gives the results that overlap well with
the experimental results, so that it can be applied to other
families of axial-symmetry specimens made of different
materials.
Acknowledgements
This work was carried out as a part of the research on
the project: RTPPIMS&NS, number: 01-376, funded by
The Ministry of Science of Montenegro.
APPENDIX
T stress tensor
r radial normal stress
z axial normal stress
 tangential normal stress
rz radial-axial shear stress
e axial normal stress
a average normal stress
ur radial displacement
uz axial displacement
u tangential displacement
T
 strain tensor

r radial normal strain

z axial normal strain

 tangential normal strain
rz radial-axial shear strain

e effective strain
T

 strain rate tensor

 r radial strain rate

 z axial strain rate

 tangential strain rate

rz radial-axial strain rate

 effective strain rate
 ' coefficient of proportionality
T temperature
v deformation velocity
d0 diameter of specimen
h0 height of specimen
D basic diameter of workpiece
D1 diameter of second degree of workpiece
Df diameter of flash
hf height of flash
H0 height of lower degree under the partition plane of
workpiece
H1 height of first degree above the partition plane of
workpiece
H2 height of second degree above the partition plane of
workpiece
vr radial displacement velocity
M. JANJI] et al.: LOAD DETERMINATION BY ANALYSING THE STRESS STATE FOR THE OPEN-DIE FORGING ...
470 Materiali in tehnologije / Materials and technology 51 (2017) 3, 463–471
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 11: Deformation forces as a function of motion
Slika 11: Sile deformacije v odvisnosti od premika
vz axial displacement velocity
ur radial displacement
uz axial displacement
r radial coordinate
z axial coordinate
r0 radial coordinate at the start of the deformation
process
z0 axial coordinate at the start of the deformation
process
r radial increment
z axial increment
t time increment
ura radial displacement at the start of the observed
deformation interval
uza axial displacement at the start of the observed
deformation interval
ra radial coordinate at the start of the observed
deformation interval
za axial coordinate at the start of the observed
deformation interval
ur radial increment of displacement
uz axial increment of displacement
C, C1, C2 integral constants
Fm maximum deformation force
Fu deformation force for the upper contour of the
workpiece
Fl deformation force for the lower contour of the
workpiece
Fp deformation force for the partition plane of work-
piece
FI experimental measured deformation force for work-
piece I
FII experimental measured deformation force for work-
piece II
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