CRITICAL ANALYSIS OF THE EVALUATION OF PLASTIC
MATERIAL PROPERTIES OBTAINED FROM STANDARD
ROUND TENSILE SPECIMENS
Magdalena Gromada1, Gennady Mishuris2, Andreas Öchsner3,4,
1
Faculty of Mechanical Engineering and Aeronautics
2
Department of Mathematics
Rzeszow University of Technology, W. Pola 2,
35-959 Rzeszow, Poland
3
Centre for Mechanical Technology and Automation
4
Department of Mechanical Engineering
University of Aveiro, Campus Universitário de Santiago,
3810-193 Aveiro, Portugal
ABSTRACT: Since the well-known papers of Bridgman (1) and DavidenkovSpiridonova (2) have been published in the fifties, researchers applied their simple
formulae to determine the yield stress for hardening plastic materials. In the years
1970-1980 several attempts based on computer simulation have been done to verify
the assumptions of the approximation formulae, to recognize the possible error or
even to improve the formulae (3-7). It was stated that the error connected with
application of the simple formulae can be estimated as 10 % in comparison with the
numerical simulations, which was considered as an acceptable accuracy. Recently,
researchers have applied the same simple formulae to analyze such complicated
effects arising from damage and fracture of the material in the last stage of plastic
deformation (8,9) and for notched specimens (8). This arises in these new
circumstances once again the questions for accuracy of the simple formulae and for
their justification for notched specimens. Moreover, new generation of powerful
computers and software enables us to believe in success. In this paper, the finite
element method is applied for the high accurate simulation of the tensile tests. The
obtained results are discussed in the context of the approximation formulae and
previously known results.
1. INTRODUCTION
The tensile test is an important standard engineering procedure useful to characterize
some important elastic and plastic variables related to the mechanical behavior of
materials. Due to the non-uniform stress and strain distributions existing at the neck
for high levels of axial deformation, it has been long recognized that significant
changes in the geometric configuration of the specimen have to be considered in order
to properly describe the material response during the whole deformation process up to
the fracture stage. Although in many engineering applications the design of structural
parts is restricted to the elastic response of the materials involved, the knowledge of
their behavior beyond the elastic limit is relevant since plastic effects with usually
large deformations take place in many manufacturing procedures such as metal
forming. Other important applications of elastoplastic models for metals are:
crashworthiness, impact problems, inelastic buckling of thin-walled structures,
superplastic forming, etc. (see, e.g. (10–12)).
Analytically derived formulae serve in practical use to evaluate the complex stress
and strain state in the necking region of tensile specimens. Among others, the
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formulae proposed by Bridman and Davidenkov-Spiridonova are well known and
considered in the following.
2. THEORETICAL BACKGROUND
Under the assumptions that the stress components  r and   are identical and that
the yield stress k  k ( eq , eq ) is a constant for each increment in the symmetry plane
( z  0 ), Bridgman has derived from the equilibrium equation the following formula:
 z
k
( r,0) 
 0,
r
 (r)
(1)
where    r  is the curvature radius of the longitudinal trajectory and the radial
distance of the point, which lies both on this trajectory and in the minimum section of
the specimen (see Fig. 1).
Figure 1: Principle configuration for a necking specimen.
He and Davidenkov-Spiridonova suggested calculating the curvature due to differing
formulae, respectively:

a 2  2aR  r 2
,
2r

Ra
.
r
(2)
Let us note that both of the presented formulae are only approximations and might be
improved. For example, the expressions (2) can be rewritten in a general way by
introducing the auxiliary function:

Ra
,
rG  r 2
 
(3)
 
which has to satisfy at the free boundary r  a the condition G  a 2  1 and Gt   0 .
Additionally, from the physical point of view one can conclude that the function  r 
should strongly decrease such that:  r    , when  r   0 and  a  R . So that
 r   0 for all 0  r  a and the following restriction for the introduced function
can be derived:
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G t  
a
,
t
0  t  a2 .
(4)
Let us note that condition (4) is fulfilled for the two above mentioned formulae (2).
Substituting representation (3) into the equilibrium equation (1) one can determine the
stress component  z by taking into account the boundary condition  z  k at the
free surface as:
z  
k
G r 2  ,
2aR
(5)
where G a 2   2aR was chosen.
The average longitudinal stress  z across the minimum section is given then by:


2
2 k
k
 z  2   z rdr   2
G r 2 rdr   3 F (a 2 )  F (0) ,

a 0
a 2aR 0
2a R
a
a
(6)
where F t   Gt . Let us consider the following simple function:

 a 
Gt       1    ,
 t
     1.
(7)
Under the additional assumptions:
0    1, 0   1
  0 ,      0,
or
(8)
function (7) fulfils all mentioned restrictions and one can calculate:
Gt   t 
F t  
t 2
2
2a 1   
t
2 

2
2
2a 2 1   
 F t  ,
2 

2a 2 1    
t  c1 .
  2aR  a 2 

2




 2aR  a 2 
4a 1   
t
2   4   
4
2
(9)
.(10)
Finally, the equation for the average stress for such a function takes the form:

a  a1    
 z  k 1 
.

4R
4 R4   


(11)

It is easy to see if   0 or   1 holds then formula (3) reduces to that suggested by
Davidenkov-Spiridonova. On the other hand, if one takes any possible combination
from (9) such that
1   
4 

4 R  2 R  
a 
a 
,
1 
 ln 1 
 1 

a 
a   2R 
4 R 
(12)
the Bridgman average stress will be obtainable (but not the Bridgman solution itself).
Moreover, one can show that it is possible to take such a combination of the
parameters  and  from (8) that the magnitude of the curvature (3) and the
corresponding average value (11) will differ significantly from those of Davidenkov3 - 80
Spiridonova or Bridgman (cf. Fig. 2 and 3). Let us note the formula (11) can be also
non-linear with respect to the ratio a / R by choosing  as a function of this ratio. For
example, one curve in the Fig. 3 has been drawn with the
factor  (a / R)  (12 R  3a) /(16 R  a) and is close to Bridgman’s solution.
Figure 2: Influence of parameters in function (7) on the ratio /R.
Figure 3: Influence of parameters in function (7) on the normalized average stress.
Since the choice of function G t  in (3) has a drastic influence on the final result of
the stress a stronger analysis should be performed in order to approximate better the
real material properties. As an example, we are going to show in the following that in
the case under consideration one can take more appropriate functions of the class (7)
given better results than those proposed by Bridgman and Davidenkov-Spiridonova.
Simultaneously, we will show that the Davidenkov-Spiridonova formula reveals in all
cases under consideration better results than the Bridgman approximation while this is
in literature controversially discussed.
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The main strategy of the approach is to assign different flow curves into a FE code
and to perform for each of them accurate simulations of the necking behavior. Based
on these calculations, we are going to extract the geometric shape of the necking zone,
i.e. radius of curvature and minimum diameter, and the applied force. Based on these
quantities which can be considered as experimental measurements, the approximation
formulae will be applied in order to calculate the effective stress and effective plastic
strain. These approximated values are then compared with the flow curve which has
been assigned before to the finite element code and which can be considered as the
exact solution. Additionally, we are going to investigate some of the important
assumptions of the approximation formulae and compare them with known results and
those based on our finite element analysis.
3. FINITE ELEMENT SIMULATION
Because of symmetry, only one-half of the length of the tensile specimen is modeled
where the axial coordinate z ranges from 0 to 10 mm and the radial coordinate r
ranges from 0 to 3 mm. All nodes along the z = 10 axis will continue to have their zdisplacements increased by a constant step/increment. The whole mesh comprises
15600 four-node, isoparametric elements written for axisymmetric applications with
bilinear interpolation functions. The constructed mesh density is a significant
improvement in comparison with those used in papers mentioned in the introduction.
The material for all elements is treated as an elastic-plastic material, with Young’s
modulus of 210000 MPa, Poisson’s ratio of 0.3, and initial yield strength of 200 MPa.
Different flow curves are investigated: i.e. linear hardening with a plastic modulus of
300 MPa, non-linear hardening with k( ε p ) = 100 + 100(1+51 ε p )0.5 and ideal
plasticity. All analyses were performed with the option for large displacements and
finite strains and the convergence checking is done on residuals with a control
tolerance of 0.01 for the relative forces. It turned out during the preliminary
investigations that the necking simulation based on axisymmetric FE models is critical
to a loss of accuracy. In order to ensure the accuracy and the convergence of the
simulation, it is not enough to use a thin and refined mesh but also to check several
criteria during the whole simulation. A first test is the equilibrium of forces between
the sum of the reaction forces in the symmetry plane (z = 0) and the displaced nodes.
Furthermore, the boundary conditions at the symmetry plane, i.e. rz = 0 for z = 0, and
the free surface, i.e. rz = r = 0 for r = a0, should be checked for each increment. It
should be mentioned here that it is necessary to introduce a local coordinate system
which is aligned to a principle material axis, e.g. z-axis, in order to follow the
significant deformation of the material in the necking region and to display the
stresses in a correct way.
4. RESULTS
First of all, Fig. 4 shows the initial and final shape of the tensile specimen for linear
hardening material. It can be seen that a deep necking region is developed.
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Figure 4: Initial and final shape of the tensile specimen for linear hardening.
One of the crucial assumptions of the Bridgman approximation is that the
circumferential stress is equal to the radial stress, i.e.  r    . As it can be seen in
Fig. 5 and as mentioned in even in the textbooks (13,14), this assumption does not
hold near the free surface. For comparison, also the distribution of the axial stress z
is drawn.
Figure 5: Distribution of the stress components  r ,  , z in the minimum section
for the last increment (linear hardening).
Figure 6 illustrates the distribution of the equivalent stress and plastic strain in the
minimum section for the final stage extracted from the FEM solution. Additionally,
one can observe a linear relationship between these two curves which is a simple
consequence of the linear hardening flow. However, these values would be not
available from real experiments. Here, only an average equivalent plastic strain is
calculated instead of the unknown distribution due to the standard engineering
approximation  p  2 ln( a0 / a) which has been derived under the assumption that the
elastic deformation can be neglected and that the material is plastic incompressible.
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The respective value is drawn in Fig. 6 together with the real distribution and the
accurate average based on the finite element method.
Figure 6: Distribution of equivalent stress and equivalent plastic strain in the
minimum section for the last increment (linear hardening).
Furthermore, it can be seen that the difference between these both average values is
visible. However, the error is less than 2 % as shown in Tab. 1 (cf. row “Eq. stress”).
Figure 7: Comparison of equivalent plastic strain obtained from FE analysis and
approximation formula in the minimum section for the last increment
(linear hardening).
Since the real distribution of the equivalent stress is also not obtainable from
experimental investigations, the same procedure was applied the calculation of
average values due to the Bridgman and the Davidenkov-Spiridonova formulae where
the average axial stress is calculated as the ratio between applied force and
instantaneous minimum cross section area. It is obvious from that figure that the
approximation based on the Davidenkov-Spiridonova formula gives a much better
prediction of the exact average value calculated due to the FE calculation than the
Bridgman formula. Accurate values for the error are summarized in Tab. 1.
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Figure 8: Comparison of equivalent stresses obtained from FE analysis and
approximation formulae in the minimum section for the last increment
(linear hardening).
Based on these average values, it is then possible to calculate the distribution of the
axial stress in the minimum section of the specimen. Figure 9 compares the exact
distribution with different approximation formulae. It turns out that also here the
Davidenkov-Spiridonova formula gives a better approximation than that derived by
Bridgman.
Figure 9: Distribution of the normalized axial stress obtained from FE calculation
(linear hardening) and approximation formula for different values of the
parameters  and , cf. Eq. (11).
However, a better approximation in this particular case is possible based on the
optimized values of  and  as indicated in the figure. Unfortunately, such
optimization requires the knowledge about the real stress distribution. On this stage,
these values can not be considered as a general rule and further investigations are
required.
The radius of curvature necessary for the approximation formulae is an extremely
difficult variable to be measured in practice (e.g. (3,15)). Based on our FE
simulations, we were able to accurately determine the necking curvature. Thus, the
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additional error due to a possible inaccurate measurement of this radius ( 5 or 10 %
of R) can be extracted from the results presented in Tab. 1.
Table 1: Errors connected with different analytical evaluation procedures.
Error %
Bridg. Dav./Spir. Bridg. Dav./Spir.
Bridg.
Dav./Spir.
hardening
linear
nonlinear
ideal
a
1.19
1.11
1.20
a/R
0.72
1.07
1.72
Eq. stress
-1.913
-0.315 -4.098
-0.995 -11.052
-4.236
R + 5%
-2.530
-1.051 -4.910
-2.017 -12.168
-5.750
R - 5%
-1.245
0.486 -3.226
0.112
-9.863
-2.612
R + 10%
-3.103
-1.730 -5.668
-2.965 -13.220
-7.164
R + 10%
-0.521
1.362 -2.287
1.314
-8.595
-0.867
Eq. strain
1.372
1.372
1.271
1.271
1.060
1.060
It can be seen from Tab. 1 that both approximation formulae, i.e. Bridgman and
Davidenkov-Spiridonova, reveal the largest error for the case of ideal plasticity. This
fact has never been reported in literature. Therefore, the reconstructed flow curves
based on the approximation formulae are compared with the exact FE curve for ideal
plasticity in the following Fig. 10.
Figure 10: Exact and reconstructed flow curves for ideal plasticity.
A clear deviation is possible to observe in both cases. As just mentioned above, the
Davidenkov-Spiridonova formula is quite closer to the exact solution.
5. CONCLUSION AND FURTHER WORK
Classical approximation formulae for the equivalent plastic strain and stress were
analyzed and compared with numerical simulations. These investigations based on
high-accurate FE simulation reveal that the engineering formulae give unexpectable
good accuracy in comparison with some earlier reported results. It turned out that the
approximation formula proposed by Davidenkov-Spiridonova results in quite better
results than the Bridgman one. This should be of special interest because many
investigators still use the formula proposed by Bridgman despite the fact that the
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Davidenkov-Spiridonova formula is mentioned in the classical textbooks by
Kachanov (13) and Hill (14) and in several scientific publications. In the case of
nearly ideal plasticity or a behavior with a plateau region, the formula still can be
improved and will be subject of our future investigations. Further important aim will
be the development of approximation formulae for elasto-plastic materials with
damage evolution and clarifying the situation for notched specimens.
ACKNOWLEDGMENTS
This research has been conducted during the Visiting Fellowship of G. Mishuris at
LSBU granted by the Leverhulme Trust. A. Öchsner is grateful to Portuguese
Foundation of Science and Technology for financial support.
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