THE APPLICATION OF AN
ANISOTROPIC YIELD FUNCTION OF
THE SIXTH DEGREE TO
ORTHOTROPIC MATERIAL
T. TAKEDA
Faculty of Engineering, Yamagata Uniuersity, Japan
By combining Drucker’s yield function with Hill’s quadratic yield function, an anisotropic yield function of the sixth
degree is proposed. The effects of the third deviatoric stress invariant and initial anisotropy are also included.
Experimental evaluation is made on 1050 aluminium tubes under multiaxial stress states. The tubes in the as-received
condition are subjected to progressive reductions in the hot extruding and cold drawing processes. They are annealed
by heating at 200°C for 1 h. By applying proportionally combined loadings of axial load, internal pressure, and
torsion to the specimens, a change of yield stress with a rotation of the principal stress axes and a difference between
the directions of the principal stress and principal strain increment are examined. In the tension-internal pressure
stress field, it is found that this aluminum tube exhibits orthotropic anisotropy of high strength in the tangential
direction. The yield surface in the tension-torsion stress field lies outside von Mises’ yield surface. The torsional yield
stress deviates considerably from von Mises criterion. Such behavioural characteristics can be expressed precisely by
the proposed yield function. In addition, it is experimentally verified that the normality rule is obeyed in strain
behaviour.
1
2 EXPERIMENTAL PROCEDURE
INTRODUCTION
It is only rarely that engineering materials have isotropic
properties. Metallic bars and tubes in the as-received
condition have usually been subjected to plastic deformation such as extruding, drawing, or stretching. Work
hardening and anisotropy resulting from deformation
processing cannot be completely removed by low temperature annealing. Furthermore, there are some cases
where metals have anisotropic textures even after full
annealing. For such materials, it is advisable that yield
behaviour is discussed on the premise of initial anisotropy.
The present author (1)-(3)t introduced the concept of
a modified stress deviator derived by Ohashi et al. (4x5)
into an anisotropic yield function. This anisotropic yield
function can be expressed by means of combining
Drucker’s yield function (6) with Hill’s quadratic yield
function (7).It is a sixth power equation in the stress
components. It has been verified experimentally that this
proposed yield function describes exactly the yield
surface of materials with axisymmetric anisotropy
(carbon steel and aluminum alloys), i.e., transverse isotropy, and it plays the role of plastic potential in defining
the direction of the plastic strain increment vector (1)-(3).
In this paper, experiments are carried out on 1050
aluminum tubes with orthotropic anisotropy under the
combined loadings of axial load, internal pressure, and
torsion. Yield surfaces and strain behaviour are determined and then the applicability of the anisotropic yield
function is examined.
The MS. of this paper was received at the Institution on I5 October 1990 and
accepted for publication on 19 February 1991
t References are given in the Appendix.
J O U R N A L OF STRAIN ANALYSIS VOL 26 NO 3 1991
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2.1 Specimens
The material used was a 1050 aluminium supplied in the
form of tubes of 40 mm outer diameter, 2 mm wall thickness, and 2 m long. The chemical compositions are given
in Table 1 . The tubes had been subjected to progressive
deformation in the industrial forming process. First,
tubing of 54 mm outer diameter x 5 mm wall thickness
was produced by hot extrusion from a hollow billet. Secondly, it was plug-drawn at room temperature through
two dies to the dimensions of 43.4 mm outer
diameter x 2.55 mm wall thickness. After process
annealing, it was again plug-drawn to its final dimensions. Finally, it was set straight by a roller leveller.
Tubes in the as-received condition often show variations in strength characteristics. Thus, they were cut into
lengths of 110 mm and were annealed in our laboratory.
Figure 1 shows the variation of yield stress as a function
of annealing temperature. The holding time at these temperatures was 1 h. The yield stress was defined by the
proof stress at 0.05 percent offset plastic strain. With
increasing temperature, yield stress decreased slightly up
to 200°C and then droped rapidly in the range of 200°C400°C. It appears that the first gradual decrease in yield
stress was due to recovery and the subsequent drop was
due to recrystallization.
Table 1. Chemical compositions of 1050aluminium (wt%)
Si
Fe
Cu
Mn
0.10
0.30
0.01
0.00 0.00
1991
0309-3247/91/07o(ro201
Mg
Zn
Ti
Others
Each
Al
0.00
0.02
<0.03
99.56
20 1
$02.00
+ 0.05
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T. TAKEDA
Thin-walled
cylindrical specimen
4- Experimental
a
Hoop stress
f
-$==
TXY
o,
of
Magnification
part A
Axial stress
4
Shear stress
Annealing temperature T “C
Fig. 1. Variation of tensile yield stress as a function of annealing temperature
Fig. 2. Coordinate system of stress components and stress state under
combined loadings
(1) nX:ay:txy= constant.
The purpose of this investigation was to determine the
anisotropic metal plasticity. Hence, the author chose
200°C for the annealing condition where recovery occurs
but recrystallization does not. The average grain size in
this annealed condition was about 65 pm, although
grains became elongated in the longitudinal direction of
the tube.
2.2 Equipment
A hydraulic-type multiaxial stress testing machine was
used for this investigation. It enabled a tabular specimen
to be loaded axially, twisted, and inflated by internal
pressure. These three types of loadings can be produced
simultaneously in any desired proportion. The maximum
capacities of this machine are 98 kN axial load, 196 MPa
internal pressure, and 490 Nm torque. A pair of special
chucks was devised to carry out experiments on the
tubes, having a uniform length of 110 mm. In order to
measure the axial load and torque to which the specimen
was subjected, a load cell with strain gauges fixed to its
surface was bolted to the end of one of the chucks. A
pressure transducer was used to measure internal pressure. Strains were measured by 45 degree-rosette gauges
fixed directly to the surface of the specimen itself. The
rosette consisted of three strain gauges, each of which has
a gauge length of 5 mm, mounted on a single backing.
Strain gauges were aligned axially, circumferentially, and
at an inclination of 45 degree to the specimen axis.
(2) ay/ax= constant in the tension-internal pressure
stress field.
(3) J3t x y / ~ x = constant in the tension-torsion stress
field.
The flow curves must be expressed by some kind of
stress-strain relationship in order to determine the
parameter values of anisotropic yield function. In this
paper, the von Mises-type equivalent stress-strain
relationship is used as the first approximation, i.e.
a = (a:
- a,ay + 0; + 32:y)1”
+
dip = [(4/3){(d&I)’ (d.$)(d&!)
+ (dc;)’} + (dyIY)’/3]‘ I 2
p =
s
dip
(1)
where dE:, de:, and dy:y are axial, circumferential and
engineering shear plastic strain increments, respectively.
The equivalent plastic strain p is obtained by integrating
along the strain path.
Reasonable definitions of equivalent stress and equivalent plastic strain for anisotropic material are given after
the yield function has been determined.
3 EXPERIMENTAL RESULTS AND DISCUSSION
2.3 Multiaxial stress test
Figure 2 shows a coordinate system of stress components
and the combined stresses acting within the wall of a
thin-walled cylindrical specimen subjected to axial load,
internal pressure, and torque. Here, radial stress is very
small and is assumed to be zero. The specimens were
deformed along three different loading paths, maintaining the following stress ratios.
202
3.1 Combined axial load-internal pressure-torsion
loading
When subjecting a specimen to loading with a constant
stress ratio 6,: ay: T , =
~ 1: tan’ 8 : tan 8 under a combination of axial load, internal pressure and torsion, the
principal stresses become a1 = ox (1 + tan2 0) and o2 =
a3 = 0, i.e., only the major principal stress crl acts on the
plane inclined at angle 8 to the specimen axis.
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APPLICATION OF AN ANISOTROPIC YIELD FUNCTION
e=45,6o
\
e = 75
Experimental
: Calculated by equations
(2) and (3b)
Stress state
I .o
e percent
0.5
0
I .5
Fig. 3. Equivalent stress-strain curves obtained from proportional loading tests of uI ;u,, ; T,,, = 1 : tan2 0 : tan 0
Figure 3 shows the equivalent stress-strain curves for
principal stress direction 8. The flow curves were taken at
intervals of 15 degrees. The curve for 8 = 0 degrees is
lowest and the curves for 8 = 45 degrees and 60 degrees
are highest. Thus, this aluminium shows anisotropy. The
flow curve in uniaxial tension, i.e., the a,( = a,) - p relation for 8 = 0 degrees, can be expressed by Hollomon's
nth power hardening law (8)
a, = Fp"
(2)
The values of the plastic coefficient F and the work
hardening exponent n are given in Table 2. Figure 4
shows the relation between flow stress and angle 8. Here,
-
1.5 -
.
L
3.2 Combined tensiowinternal pressure loading
The principal stress axes d o not rotate under any combination of tension and internal pressure. Figure 7 shows
the yield surface in the ax - ay plane. The flow stress
ratios as a function of equivalent plastic strain were
obtained from each flow curve in much the same way as
0
0
-
0
.-
-
-(I
1.00
y1
?!
0
y1
: Experimental
- : Calculated by equation (3b)
3
z
0
15
30
45
8 degrees
60
15
90
Fig. 4. Relation between flow stress and principal stress direction 0 (uf:
tensile flow stress given in equation (2))
JOURNAL OF STRAIN ANALYSIS VOL 26 NO 3 1991
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the flow stress ratio a,/q was taken as the ordinate,
because the ratios were almost equal at various values of
offset strain, ranging from a minimum equivalent plastic
strain of 0.05 per cent to a maximum of 2.0 percent in
each flow curve of Fig. 3. As angle 8 increases, the ratio
a,/a, rises to a maximum value and then decreases gradually.
Figures 5(a) and (b) show the plastic strain paths in the
E: - E; and E: - y:,
planes. Letting the plastic strain
increment ratios be M = ds;/de: and N = dy:y/de:, the
principal strain increment direction 8' is computed by
the equation 8' = (1/2) tan-' { N / ( 1 - M)}. Figure 6
shows the relation between the angles 8' - 8 and 8. The
values of M and N were obtained by the least squares
method, assuming the plastic strain path to be straight.
The angle 8' - 8 increases in a negative direction with
angle 8 and changes sinusoidally.
Table 2. Values of plastic coefficient and work
hardening exponent in Hollomon's equation
1050 Al
F MPa
n
Uniaxial tension
124.2
6.39 x lo-*
1991
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203
T. TAKEDA
2
.
5
1
a"
I
$
Experimental
m
I
o
I
: Experimental
&
8
I
30
I
45
60
;P
."t
I
-8
0 degrees
Fig. 6. Variation of principal strain increment direction 8' with principal stress direction 0
e=
-1.0
15
equivalent to compression in the z direction when
neglecting the hydrostatic stress components, this aluminium shows orthotropic anisotropy for which the
strength differs in the x, y, and z directions.
Figure 8 shows the plastic strain paths in the ef) - e!
plane. The plastic strain increment ratio de:/de:
was
obtained by the least squares method, assuming a linear
strain path. The strain increment vectors corresponding
to angle 4 = tan-'(de:/de3 are indicated by the arrows
on the yield surface in Fig. 7.
I
3.3 Combined tensiomtorsion loading
Rotation of the principal stress axes is possible by combinations of tension and torsion. Figure 9 shows the yield
surface in the a, - ,/3t,, plane. Again, a dimensionless
coordinate system can be adopted. The experimental
points lie outside von Mises' circle. The experimental
point on the ordinate, i.e., shear stress, is farthest from
von Mises' circle. Figure 1qa) shows the plastic strain
paths in the e: - y:,/,/3 plane under combined loadings.
The directions of the plastic strain increment vectors,
i.e. I(/ = tan-' {(dy:,/,/3)/de:},
were determined by
assuming linear strain paths. They are indicated by the
arrows on the yield surface in Fig. 9. Figure l q b ) shows
the plastic strain paths in the e: - E: plane. The plastic
3.0
1
-1.2
vonMises
Tresca
c f Percent
-0.5-
(b)
: Calculated by
<<
Fig. 5. Plastic strain paths in - E: and E: - y& planes (a) in ~ l-) E:
plane (b) in - y:y plane
in the preceding paragraph, and were almost equal. This
means that the equi-strain surfaces described by the
proof stress at various offset strains have similar forms.
Hence, the surface expressed with the dimensionless
coordinate system of adof and a,/a, is considered to be
the yield surface. The value of o,/a, on the loading path
ay/cx= co (tension in the y direction) is larger than unity,
while the experimental point on the loading path
oy/ox= 1 (balanced biaxial tension) lies inside von Mises'
and Tresca's yield surfaces. As balanced biaxial tension is
204
Drucker /
(C= 2.25)
equation (4)
.
0.6
0
"
-
0
: Experimental
-: Calculated by equation (3b)
L
0
0.4
---- : Calculated by f = J:
0.2
I
-
GI:
de!
I
0.2
-0.2
c
I
I
I
0.4
0.6
0.8
'
-0.2
Fig. 7. Yield surface in
8,
- uy plane (of:tensile flow stress given in
equation (2))
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APPLICATION O F AN ANISOTROPIC YIELD FUNCTION
2.5
-: Calculated by
Experimental
Experima
I-pKq I
equation (4)
2.c
1.5
E
2
E
>
m
1.0
a2
A
0.5
ay/a, = 0
percent
E:
-1.0
(a)
EP
,
Fig. 8. Plastic stain paths in E: - E: plane
0
strain increment ratio M = ds:/dc,P increases negatively
as the ratio ,/3 zxy/axincreases. Figure 11 shows this M
value as a function of loading direction a = tan-'(J3
percent
1 .o
0.5
2.0
1.5
-0.5
TXY/~J.
4 APPLICATION OF ANISOTROPIC YIELD
FUNCTION
I
\/TxylO,
The anisotropic yield function of the sixth degree is able
to include the effects of the third deviatoric stress invari-
= I
Experimental
I
. Calculated by
equation (4)
t
2
-2.0
(b)
Fig. 10. Plastic strain paths in E: - y:/J3
and E: - E: planes (a) in
E: - yfY/,/3 plane; (b) in 4 - E: plane
(I
0.6
0:
-
I
= {= tan-'(\/3TXy/o, )}degrees
15
I
I
45
60
I
I
I
75
'
I
90
I
I
0
: Experimental
0
30
I
0.5
0
0.2
I
I
I
0.4
0.6
0.8
ITO
1.2
-
1.o
OJOI
Fig. 9. Yield surface in ur - J3 txyplane (ar:tensile flow stress given
in equation (2))
JOURNAL OF STRAIN ANALYSIS VOL 26 NO 3 1991
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0 : Experimental
-. Calculated by equation (4)
Fig. 11. Relation between plastic strain increment ratio M and loading
direction a
1991
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205
T. TAKEDA
Table 3. Values of parameters in equation (3b) for numerical calculation
2.25
-2.038 x lo-'
-5.893 x
ant and initial anisotropy (1H3).It is written as
f = Ji(J2 + A i j k l c i j
(34
where J 2 = (1/2) oij a;, and J , = (1/3) aij O i k b i i are the
second and third deviatoric stress invariants, '4ijkl is the
anisotropy parameter, and C is a parameter allowing for
the effect of J , . It is assumed that there is no coupling
effect between the parameters Aijk1 and C. If A i j k l = 0,
this equation reduces to Drucker's yield function (6).For
plane stress loading, equation (3a) is expressed as
{(1/3)(0: - O, O,
x
{(1/3)(0:
+e
y
Oil)
+ + T$}'
- o,by + o:)
0:)
- Axxyy(Or - OYl2
- AyyzzO: - A z Z x x ~ :
- C{(1/27)(20,
x
(0,
- CJ;
-
+ 4Axyxy~:,)
o ~ ) ( o, 20,)
+ oY)+ (1/3k, + ~,)~:,>'
+
= (427){1 - 4CP7 - 3(AXXYY
A,,,,))
(3b)
In this equation uf is the tensile yield stress; however,
the tensile flow stress given in equation (2) is used in this
investigation. Although equation (3b) apparently
involves five parameters, the number of independent
-4.649 x lo-'
-7.330 x
variables is four. Thus, we can take parameter C as being
defined arbitrarily and relate it to the parameter C of
Drucker's yield function. Here, the author tries to
separate the effect of J , from the presence of initial anisotropy. The attempt is made by setting C = 2.25 which is
the upper limit when adding the convexity restrictions of
yield surfaces to Drucker's yield function (9KlO). In Fig.
7, the experimental points on the loading paths oY/o, = 0
and 1/2 are compared. The loading path oy/ux= 1/2 is
independent of J , , i.e. 5, = 0, whereas the loading path
o,/o, = 0 is dependent. But the directions of the major
principal stress and specimen axis are the same, and the
space between these two paths is narrow. Assuming that
the results obtained from these two loading tests make
no difference in the appearance of anisotropy, the dimensionless stress value on the loading path oY/o, = 1/2
reveals only the effect of J , . The broken curve in Fig. 7
represents the result calculated from Drucker's yield
function with C = 2.25. The experimental point on the
loading path O ~ / O ,= 1/2 almost lies on the broken curve,
and we can see that the above-mentioned consideration
is valid. The similar examination can be made by comparing the experimental points on the loading paths
o,/o, = 2 and infinity where the direction of the major
principal stress is the same as that of the circumference of
specimen. In order to determine the parameters Axxyy,
I50
6
E
B
Experimental
I0
Stress state
0
0.5
1 .o
I .5
2.0
e percent
Fig. 12. Equivalent stress-strain curves (Zea:equivalent stress defined by equation (5))
206
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APPLICATION OF AN ANISOTROPIC YIELD FUNCTION
Ayyzz,A,,,,, and Axyxy,we need four relationships involving them. These are commonly obtained by substituting
into equation (3b) the dimensionless stress values from
four kinds of loading tests. However, in this investigation, the parameter values were determined in the following manner. First, the dimensionless stresses in balanced
biaxial tension, tension in the y direction and simple
shear were chosen as characteristic values, and three
relationships were obtained.
Another relationship was obtained by the least
squares method where the dimensionless stresses in the
loading paths 8 = 15,30,45,60, and 75 degrees in section
3.1 were used as characteristic values, because this
resulted in high accuracy in determining the yield function. The parameter values of equation (3b) used for the
numerical calculation are given in Table 3.
Using the associated normality flow rule and assuming
that the yield function and plastic potential are indentical, the plastic strain increment de; is given as
dsc
= (df/daij)
dl
(4)
where d l is a scalar constant.
In Figs 3-1 1, the results calculated by equations (2),
(3b), and (4) agree well with the stress-strain curves, yield
surfaces and strain behaviour obtained from the experiments. It can also be seen that the directions of the
plastic strain increment vectors are almost normal to the
yield surface in Figs 7 and 9.
The last stage is an attempt at introducing the anisotropic yield function into an equivalent stress-strain
relationship. The equivalent stress ifeq is derived directly
from equation (3a) by setting ifeq = afin equation (3b).
The equivalent plastic strain p is defined on the basis
of the strain hardening hypothesis. They become
and
raiir
p =J
{(2/3) dE{ de{}”’
Equation (6) is the general expression of equivalent
plastic strain defined in equation (1). The author gave
this definition for the equivalent plastic strain, because
the length of the plastic strain trajectory was used in
determining the parameter values of yield function. The
experimental data for 8 = 30,60, and 90 degrees in Fig. 3
are calculated by equations (5) and (6) as examples, and
they are plotted in Fig. 12. It can be seen that the experimental data are correlated with a single stress-strain
curve, i.e., ifeq( =af)- p curve.
5 CONCLUSIONS
Multiaxial stress tests were carried out on 1050 aluminium tubes with orthotropic anisotropy. The anisotropic yield function of the sixth degree was applied to
the experimental results and its validity was examined.
JOURNAL OF STRAIN ANALYSIS VOL 26 NO 3 1991
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The results are summarized as follows.
(1) Under proportionally combined loadings of axial
load, internal pressure, and torsion, flow stress rises
to a maximum value with a rotation of the principal
stress axes and then decreases gradually. The difference between the directions of the principal stress
and principal strain increment changes sinusoidally
with a rotation of the principal stress axes.
(2) The yield surface in the tension-internal pressure
stress field reveals orthotropy of high strength in the
tangential direction.
(3) The yield surface in the tension-torsion stress field
lies outside von Mises’ yield surface. It becomes
inflated in the direction of the torsion axis. The
plastic strain increment ratio of circumferential strain
to axial strain increases negatively with an increase in
the ratio of shear stress to tensile stress.
(4) All of these results can be expressed precisely by the
proposed yield function and the associated flow rule.
(5) The proposed yield function was introduced into the
equivalent stress-strain relation. This relation provides a satisfactory equivalence correlation for the
multiaxial deformation behaviour of orthotropic
material.
ACKNOWLEDGEMENT
The author would like to thank Kyowa Electronic
Instruments Co. Ltd for providing the rosette strain
gauges.
APPENDIX
REFERENCES
( 1 ) TAKEDA, T. and NASU, Y., ‘Multliaxial yield behavior of mild
steel with axisymmetric anisotropy’,J . Japan SOC.Tech. Plasticity,
1987,28,1282-1288 (in Japanese).
(2) TAKEDA, T., TAKAHASHI, Y., KIKUCHI, S., and NASU, Y.,
‘Yield surfaces and stress-strain relations of 20240 and T6 aluminum alloys’, J . Japan Soc. Tech. Plasticity, 1989, 30, 1324-1329
(in Japanese).
(3) TAKEDA, T. and NASU, Y., ‘Evaluation of yield function including effects of third stress invariant and initial anisotropy’,J . Strain
Analysis, 1991,26,47-53.
(4) OHASHI, Y., OHNO, N., SUGIYAMA, T., AND KANAYAMA,
S., ‘Strain-rate effect on strain anisotropy of aluminum alloy under
combined loading at elevated temperature’, Trans J S M E , 1980,
A46,459-461 (in Japanese).
(5) OHASHI, Y. and OHNO, N., ‘Inelastic stress-responses of an
aluminum alloy in non-proportional deformations at elevated
temperature’,J . Mech. Phys Solids, 1982,30,287-304.
(6) DRUCKER, D. C., ‘Relation of experiments to mathematical
theories of plasticity’, J . Appl. Mech., 1949,16,349-357.
(7) HILL, R. The mathematical theory of plasticity, 1950, (Clarendon
press, Oxford).
(8) HOLLOMON, J. H., ‘Tensile deformation’, Trans AIME, 1945,
162,268-290.
(9) BETTEN, J., ‘Plastische Anisotropie und Bauschinger-Effekt;allgemeine Formulierung und Vergleich mit experimentell ermittelten FleiDortkurven’, Acta Mech., 1976,25,7%94.
(10) TAKEDA, T., SHIRATORI, E., KARASHIMA, S. and NASU,
Y., ‘A study on the anisotorpicyield condition containing the third
invariant of deviatoric stresses’, J . Japan Soc. Tech. Plasticity,
1983,24,44248(in Japanese).
1991
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