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A microfriction constitutive theory for cyclic plasticity of metals
Inr. J. Non-Linear .Wechmicr, Vol. 21. No. I. pp. 51-60. 1986 Printed in Grsat Bnlain
CCO-7462/86/S3.00 + .oO Pergamon Press L:d.
A MICROFRICTION CONSTITUTIVE THEORY FOR CYCLIC PLASTICITY OF METALS J. W. SMITH* and D. S. ZHANGt Department of Civil Engineering, University of Bristol, Bristol BS8 ITR, U.K. and t Department of Process Equipment, East China Institute of Chemical Technology, 130 Meilong Road, Shanghai, 201107, China
l
(Receired
7 January
1985; receiced for publication
3 September
1985)
constitutive theory of plasticity is presented which models the gradual strain softening in the initial plastic range that is exhibited by most real metals. The theory is an attempt to model the successive movements of many dislocations in a crystal lattice. Some forms of the final equations correspond closely to early empirical expressions. The loading-unloading behaviour of the model is compared with the results of a cyclic loading experiment on a sample of high strength steel at elevated temperature. It is shown that the model simulates thecyclic plastic behaviour ofthe metal very accurately so that it is possible to represent the phenomenon of strain softening in repeated cyclic plasticity.
Abstract-A
INTRODUCTION
Most metals, especially high strength steels, exhibit a gradual strain softening in the early part of their plastic strain response when loaded axially. This is in contrast to the sharp discontinuity between elastic and plastic strain that is frequently assumed in elastic-plastic analysis. The unloading and cyclic behaviour of high strength metals in the plastic range is of considerable practical importance in pressure vessel technology, chemical plant, the nuclear industry and other industrial applications. A comprehensive survey of stress-strain relations in the plastic range was carried out by Drucker [l]. In this he discussed the incremental and deformation theories, these being the two most important fundamental types of theory. He also evaluated theories which attempted to describe the slip of crystal planes. Zienkiewicz and Cormeau [2] showed how visco-plastic rheological models can be used to unify the numerical treatment of plasticity and creep in non-linear solids. A method developed by Owen et al. [3] relied on the notion of a material consisting of ideal elastoplastic overlays. The overall response of the system was capable of exhibiting strain softening and Bauschinger’s effect. The endochronic theory of material behaviour was introduced by Valanis [4] and was used to describe certain phenomena including strain hardening, unloading behaviour and continuous cyclic straining. This method did not require a yield or loading condition but instead made use of a quantity called “intrinsic time” replacing real time in the equations of viscoelasticity. In effect this was a memory function which was a measure of the previous deformation history of the material. It was shown that this theory could describe certain behaviour of real metals with considerable accuracy. However, Sadler [5] showed that in some instances the simple endochronic model is fundamentally unstable and requires the introduction of “internal barriers” [6] which is similar to the use of a yield surface. In this paper a constitutive theory is developed which is an attempt to model the microscopic deformation behaviour of a real metal. It is referred to as the “microfriction component” and is essentially a development of classical rheological models. It is, however, similar in concept to the overlay system [3] and to other methods described by Drucker [l].
THE
MICROFRICTION
COMPONENT
The rigid plastic constitutive equation may be obtained by considering the simple element consisting of a friction slider and elastic spring in parallel as shown in Fig. l(a). The friction 51
52
J. W.
a
SMITH
and D. S. ZHANG
a
b:.,-_ a0 o.
E
I
(a)
E
Oh-7
Fig. I. The simple friction element.
slider is not active until the applied stress is greater than its “active stress” o,,. The relationship between stress and strain is 0
-
00
EC---
E
(1)
and is shown in Fig. l(b), If two such elements are connected in series and it is assumed that the active stress CT~> cr, , then the stress-strain relationship will be
and is shown in Fig. 2. It can be seen that the additional friction element makes it possible to model the curve of real material behaviour more closely. Yielding in metals is a macroscopic result of the movement of dislocations in a crystal lattice. The gradual yielding which is observed in many metals may be attributed to the random orientations of the many crystals in the material. In a single direction therefore it would be easy to conceive of there being numerous “active stress” values at which slip could take place in individual crystals. A natural extension of the above models would be to consider a model of unit length consisting of a large number of simple friction elements in series as in Fig. 3. The threshold “active stresses” e, , cr2,. . . , on are all different and randomly distributed in the unit length. This model will be called the “microfriction component”. The threshold stress is shown as a function of x in Fig. 4(a). However, let us assume that the thresholds could be sorted in ascending order such that there is a relationship g= f
l-u)
(3)
where cr(x) = threshold stress in random spatial order o(g) = threshold stress in monotonic ascending order. This is shown in Fig. 4(b). The form of the function is irrelevant, in fact, because elements will yield in order of threshold stress.
Fig. 2. Two simple friction elements connected in series.
Microfriction
constitutive
theory
53
%+$J-$J___~.._~
eDn
cTN
Fig. 3. The ~~of~ction
-I
component.
Suppose that the applied external stress equals a(G). Thus all elements below this value yield while those above remain rigid. Elongation of one element is ’ - ‘(8) = -dg.
&
E(g)
Thus elongation of the component, and hence the strain, is given by
&=
J
cc7- 4s)
dg.
E(g)
o
In equation (5), G is the “active length” there being no elongation of elements with threshold stresses greater than o(G). We shall now consider the integration of equation (5) in the case of some specific forms of functions o(g) and E(g). 1. Suppose that the microfriction component has a uniform stiffness, i.e.
E(g) = E
(6)
and the function o(g) takes the form a(g) =
ag”
Go +
(7)
where E, oo, n and a are material constants, a always being positive as in Fig. 5. According to the definition of active length we have G
6-U
a
i
l/n
0
zz-.
(8)
i
Substituting equations (6), (7) and (8) in equation (5) we obtain (T)’
&=
J
in [o,+ag”]
B-
E
0
n
_
(CT - ~0)~
n+l
The stress-strain
,dg
Eal/n
(9)
.
curves for different values of n are shown in Fig. 6.
(a) Random order Fig. 4. Distribution
(b) Ascending of threshold stresses.
order
J. W. SMITH and D. S. ZHANG
log
OGGG
Fig. 5. Functions
for threshold
stress.
2. Suppose that the function E(g) takes the form
4g) = Eo + Pg”
(10)
and the function o(g) takes the form a(g) = co + ag”
where E,,, p, m, co,a and n are material constants. a must be positive as in Fig. 5 but p may be positive or negative. Now substituting equations (lo), (7) and (8) in equation (5) we obtain
&=
s
Ga - (00 + ug”) dg 0 Eo -t Bg”
= I1 + I2
(11)
where G 0-00
I1 =
I2 =
s 0 Eo + Pg”
I
G
0
(12)
dg
G” dg. Eo -t-Bg”
(13)
Some explicit results of the integrations of I1 and 12 are listed in Table 1 where it should be noted that
and
L
E
0
Fig. 6. Stress-strain
curves for constant
E(g).
Microfriction constitutive theory Table I. Explicit solutions to integrals in equation (11) m=l
In=2
In=3 (A”’ + G)*
II
A2i3 - GA”’
arctan Ll) G
a
I2 n = 2 F
1
fGZ
+ G* I
s E0
- E,$G + E;
no solution
C
s arctan Ll G IF0)I
n=3
no solution
Two empirical formulae reported by Hill [7] are worth referring to at this point. The first is Ludwik’s power law cr = a + bc’
(14)
which leads to e=
o-__a r/c i b 1
which is similar in form to equation (9). The special case (c = 1) gives the straight line strain hardening law in Fig. 6. The second empirical formula is the exponential law of Vote and Palm [8,9] 0 = a + (b - a)(1 - eece) which leads to 1
,q= --ln c
CT- b (a-b 1
which is similar in form to equation (11) with m = n = 1.
J. W. SMITH and D. S. ZHANG
56 CALIBRATION
AND
SIMULATION
Case 1 of the microfriction component axial tension test. In this case we have
OF
AXIAL
TENSION
TEST
will now be calibrated and used to illustrate the
E(g) = E,
o(g) =
60
ag”
+
giving equation (9) thus n
E_
(~7-0~)~
(9)
n + 1 Ea””
It is understood that this refers to the plastic component ofstrain only. Thus the stress may be expressed as
(15)
cT=fJo+
=
fJ()
+
AEB
where E’ = Ecc’!” which is identical to equation 14. The constants may be determined in the usual way by plotting the tension test data in logarithmic coordinates as shown in Fig. 7. Hence
and
E’ =
Ea*“’ = LA n+l
(II+
l)‘n
.
It can be seen that E and a are not independent for this model and therefore the value E’ is used instead. Some examples of the simulation of real metals subjected to axial tension are shown in Fig. 8. The simple microfriction component of equation (9) is used and the elastic part has been added to the total strain. The experimental curves were obtained from Templin and Sturm [lo]. It may be seen that the microfriction component models the plastic strain hardening behaviour reasonably accurately except in the case of structural steel. This was because the carbon atoms in steel pin the dislocations until the stress is large enough to overcome their effect resulting in the characteristic yield plateau which is a difficult discontinuity to model. CYCLIC
LOADING
WITH
STRAIN
SOFTENING
In order to simulate the behaviour of a real material the model should be capable of responding to cyclic loading. The microfriction component as illustrated in Fig. 3 is satisfactory for this purpose and requires to be in series with a notional spring to represent the elastic response.
Fig. 7. The determination
of material
constants.
Microfriction constitutive theory
57
5IlIcon steel ,’ Sttuctural
-
E i 19J.100
Expenment
Brass E : 103LOO uyz 103 L 7=ore E’. 266567&l 9
Htgh
purity
alummium
E = 372Ll dy.21 6 J-073 E’: 10033 9
-annealed
1
0
005
0.15
0 IO
0 20
o-25
o-30 035
O-LO
Strain [a)
STRESS-STRAIN
CURVES
,-v-_;;
-
Exponment
-----
Theory
Brass High purity
alumlntum-annealed
r 0
ib)
0002 EXPANDED
0 00.4 STRAIN
0.006 stram
0.008
0 010
0 012
SCALE
Fig. 8. The microfriction model compared with experiments in axial tension.
On releasing the external stress the elasticcomponent ofstrain reduces to zero. The friction elements will then be pre-loaded in compression because of the remaining stress left in their parallel elastic elements due to the plastic deformation. Thus, on compression the threshold element will yield at an external compressive stress less than go. The model, therefore, exhibits Bauschinger’s effect. Some high strength engineering steels exhibit strain softening when loaded cyclically beyond the yield point. Figure 9 shows experimental stress-strain loops for a cylindrical sample of 1CrMoV steel. The diameter was 7.69 mm, the gauge length 15 mm and the sample was strained in reversed loading at a temperature of 500 “C. At this temperature the Young’s modulus is 160GN/m2. The experiment was strain controlled with a maximum strain of &-1.38 ;O, except for the first few cycles. The stress-strain curves form symmetrical hysteresis loops with a gradually reducing stress amplitude as shown in Fig. 10. The strain softening phenomenon can be incorporated into the microfriction model by considering that the stress amplitude is a function of the total (summed) plastic strain, which can be regarded as one equivalent quantity of the cycle number. Thus
J. W.
SMITH
and D. S. ZHANC
Cycle n -I
6 I
U
,I”,
lcmz
Fig. 9. Experimental stress-strain
50
lOO.LZ
MN/d
loops.
This introduces the idea of adopting the “endochronic” material memory concept by giving the microfriction model parameter values that change with summed plastic strain. In this case it was considered physically realistic to modify the threshold stress of the microfriction model so this would represent gradually increasing size and number of dislocations. The function $ that best fitted the experimental results was given by
cl
II/=
c, +JqTl
+ Cz~l~,l- G-npl)C~.
The other parameters were obtained from the loop ofthe 16th cycle and were calculated to be: C-J,, = 150 MN/m2 E = 160GN/m2 n = 0.3194
E’ = 8.864 x 10” (Nm units).
Using these values the microfriction model was used in incremental form to obtain the theoretical stress-strain loops shown in Fig. 11. The strain amplitude softening is compared with the experiment in Fig. 10. 1100 1 1000
; . f
I -
900-
E
2 :
aoo-
7
Fig. IO. The reduction of stress amplitude.
Microfriction
constitutive
theory
59
800 Stress MN/m2 I
s=
-600
150.00 MN/mi
N = 0.32 E’.&186.Lx10g
Fig. 11. Microfriction
model stress-strain
loops.
CONCLUSIONS
A constitutive theory of plasticity has been presented which accurately models the gradual transition from elastic to plastic behaviour that is exhibited by most metals. The theory is an attempt to model the deformation of dislocations in a crystal lattice at microscopic scale. In effect, it is a development of classical rheological models incorporating frictional sliders and spring elements. Hence, it is referred to as the “microfrictional component”. It has been shown that some forms of the final equations correspond closely to early empirical formulae. The theory was calibrated and compared with results of axial tension tests and found to model the strain hardening behaviour accurately except in the case of structural steel with its characteristic.yield plateau. The cyclic behaviour of the model was compared with the results of an experiment on a sample of high strength steel at elevated temperature. The sample was cyclically loaded both in tension and compression under strain control. The symmetrical hysteresis loops exhibited a gradually reducing stress amplitude in successive cycles until failure. This strain softening phenomenon was modelled in the “microfriction component” by subjecting the threshold stress to a decay that increased with previous summed strain history. Good agreement with the experimental stress-strain loops was obtained. The “microfriction component” has been shown to model material yielding behaviour accurately in both axial and cyclic loading. The equations are capable of being incorporated into a fully 3-dimensional finite element scheme. Acknowledgemenrs-Experimental data from a cyclic loading test, provided by Prof. E.G. Ellison ofthe Department of Mechanical Engineering, University of Bristol, is gratefully acknowledged.
REFERENCES 1. D. C. Drucker, Stress-strain relations in the plastic range -a survey of theory and experiment. Office of Naval Research NR-041-032, (1950). 2. 0. C. Zienkiewia and 0. C. Cormeau, Visco-plasticity -plasticity and creep in elastic solids-a unified numerical solution approach. Int. J. Num. .Clerh. Engng. 8, 821-845 (1974).
60
J. W. SMITH and D. S. ZHANG
3. D. R. Owen, A. Prakesh and 0. C. Zienkiewicz, Finite element analysis ofnon-linear composite materials by use of overlay systems. Compur. Strucr. 4, 1251-1267 (1974). 4. K. C. Valanis, A theory of viscoplasticity without a yield surface. Archiwum Mechaniki Srosowanej, 23, (4). 517-551 (1971). 5. 1. S. Sadler, On the uniqueness and stability ofendochronic theories ofmaterial behavior. J. appl. Mech. 45,263 (1978). 6. K. C. Valanis, Some recent developments in the endochronic theory of plasticity-the concept of internal barriers. Symposium on Constitutive Equations in Viscoelasticity: Phenomenological and Physical Aspects, ASME Winter Annual Meeting, New York City, 5-10 December (1976). 7. R. Hill, The Marhematical Theory oJ Plasticiry. Oxford University Press. London (1950). 8. E. Vote. The relationship between stress and strain for homogeneous deformation. J. Insf. Merals 74,537 (1948). 9. J. H. Palm. Stress-strain relations for uniform monotonic deformation under triaxial loading. Appl. Sci. Res. AZ, 54192 (1949). 10. R. L. Templin and R. G. Sturm, Some stress-strain studies of metals. J. Aeronauf. Sci. 7, 189-198, (1940).
CCO-7462/86/S3.00 + .oO Pergamon Press L:d.
A MICROFRICTION CONSTITUTIVE THEORY FOR CYCLIC PLASTICITY OF METALS J. W. SMITH* and D. S. ZHANGt Department of Civil Engineering, University of Bristol, Bristol BS8 ITR, U.K. and t Department of Process Equipment, East China Institute of Chemical Technology, 130 Meilong Road, Shanghai, 201107, China
l
(Receired
7 January
1985; receiced for publication
3 September
1985)
constitutive theory of plasticity is presented which models the gradual strain softening in the initial plastic range that is exhibited by most real metals. The theory is an attempt to model the successive movements of many dislocations in a crystal lattice. Some forms of the final equations correspond closely to early empirical expressions. The loading-unloading behaviour of the model is compared with the results of a cyclic loading experiment on a sample of high strength steel at elevated temperature. It is shown that the model simulates thecyclic plastic behaviour ofthe metal very accurately so that it is possible to represent the phenomenon of strain softening in repeated cyclic plasticity.
Abstract-A
INTRODUCTION
Most metals, especially high strength steels, exhibit a gradual strain softening in the early part of their plastic strain response when loaded axially. This is in contrast to the sharp discontinuity between elastic and plastic strain that is frequently assumed in elastic-plastic analysis. The unloading and cyclic behaviour of high strength metals in the plastic range is of considerable practical importance in pressure vessel technology, chemical plant, the nuclear industry and other industrial applications. A comprehensive survey of stress-strain relations in the plastic range was carried out by Drucker [l]. In this he discussed the incremental and deformation theories, these being the two most important fundamental types of theory. He also evaluated theories which attempted to describe the slip of crystal planes. Zienkiewicz and Cormeau [2] showed how visco-plastic rheological models can be used to unify the numerical treatment of plasticity and creep in non-linear solids. A method developed by Owen et al. [3] relied on the notion of a material consisting of ideal elastoplastic overlays. The overall response of the system was capable of exhibiting strain softening and Bauschinger’s effect. The endochronic theory of material behaviour was introduced by Valanis [4] and was used to describe certain phenomena including strain hardening, unloading behaviour and continuous cyclic straining. This method did not require a yield or loading condition but instead made use of a quantity called “intrinsic time” replacing real time in the equations of viscoelasticity. In effect this was a memory function which was a measure of the previous deformation history of the material. It was shown that this theory could describe certain behaviour of real metals with considerable accuracy. However, Sadler [5] showed that in some instances the simple endochronic model is fundamentally unstable and requires the introduction of “internal barriers” [6] which is similar to the use of a yield surface. In this paper a constitutive theory is developed which is an attempt to model the microscopic deformation behaviour of a real metal. It is referred to as the “microfriction component” and is essentially a development of classical rheological models. It is, however, similar in concept to the overlay system [3] and to other methods described by Drucker [l].
THE
MICROFRICTION
COMPONENT
The rigid plastic constitutive equation may be obtained by considering the simple element consisting of a friction slider and elastic spring in parallel as shown in Fig. l(a). The friction 51
52
J. W.
a
SMITH
and D. S. ZHANG
a
b:.,-_ a0 o.
E
I
(a)
E
Oh-7
Fig. I. The simple friction element.
slider is not active until the applied stress is greater than its “active stress” o,,. The relationship between stress and strain is 0
-
00
EC---
E
(1)
and is shown in Fig. l(b), If two such elements are connected in series and it is assumed that the active stress CT~> cr, , then the stress-strain relationship will be
and is shown in Fig. 2. It can be seen that the additional friction element makes it possible to model the curve of real material behaviour more closely. Yielding in metals is a macroscopic result of the movement of dislocations in a crystal lattice. The gradual yielding which is observed in many metals may be attributed to the random orientations of the many crystals in the material. In a single direction therefore it would be easy to conceive of there being numerous “active stress” values at which slip could take place in individual crystals. A natural extension of the above models would be to consider a model of unit length consisting of a large number of simple friction elements in series as in Fig. 3. The threshold “active stresses” e, , cr2,. . . , on are all different and randomly distributed in the unit length. This model will be called the “microfriction component”. The threshold stress is shown as a function of x in Fig. 4(a). However, let us assume that the thresholds could be sorted in ascending order such that there is a relationship g= f
l-u)
(3)
where cr(x) = threshold stress in random spatial order o(g) = threshold stress in monotonic ascending order. This is shown in Fig. 4(b). The form of the function is irrelevant, in fact, because elements will yield in order of threshold stress.
Fig. 2. Two simple friction elements connected in series.
Microfriction
constitutive
theory
53
%+$J-$J___~.._~
eDn
cTN
Fig. 3. The ~~of~ction
-I
component.
Suppose that the applied external stress equals a(G). Thus all elements below this value yield while those above remain rigid. Elongation of one element is ’ - ‘(8) = -dg.
&
E(g)
Thus elongation of the component, and hence the strain, is given by
&=
J
cc7- 4s)
dg.
E(g)
o
In equation (5), G is the “active length” there being no elongation of elements with threshold stresses greater than o(G). We shall now consider the integration of equation (5) in the case of some specific forms of functions o(g) and E(g). 1. Suppose that the microfriction component has a uniform stiffness, i.e.
E(g) = E
(6)
and the function o(g) takes the form a(g) =
ag”
Go +
(7)
where E, oo, n and a are material constants, a always being positive as in Fig. 5. According to the definition of active length we have G
6-U
a
i
l/n
0
zz-.
(8)
i
Substituting equations (6), (7) and (8) in equation (5) we obtain (T)’
&=
J
in [o,+ag”]
B-
E
0
n
_
(CT - ~0)~
n+l
The stress-strain
,dg
Eal/n
(9)
.
curves for different values of n are shown in Fig. 6.
(a) Random order Fig. 4. Distribution
(b) Ascending of threshold stresses.
order
J. W. SMITH and D. S. ZHANG
log
OGGG
Fig. 5. Functions
for threshold
stress.
2. Suppose that the function E(g) takes the form
4g) = Eo + Pg”
(10)
and the function o(g) takes the form a(g) = co + ag”
where E,,, p, m, co,a and n are material constants. a must be positive as in Fig. 5 but p may be positive or negative. Now substituting equations (lo), (7) and (8) in equation (5) we obtain
&=
s
Ga - (00 + ug”) dg 0 Eo -t Bg”
= I1 + I2
(11)
where G 0-00
I1 =
I2 =
s 0 Eo + Pg”
I
G
0
(12)
dg
G” dg. Eo -t-Bg”
(13)
Some explicit results of the integrations of I1 and 12 are listed in Table 1 where it should be noted that
and
L
E
0
Fig. 6. Stress-strain
curves for constant
E(g).
Microfriction constitutive theory Table I. Explicit solutions to integrals in equation (11) m=l
In=2
In=3 (A”’ + G)*
II
A2i3 - GA”’
arctan Ll) G
a
I2 n = 2 F
1
fGZ
+ G* I
s E0
- E,$G + E;
no solution
C
s arctan Ll G IF0)I
n=3
no solution
Two empirical formulae reported by Hill [7] are worth referring to at this point. The first is Ludwik’s power law cr = a + bc’
(14)
which leads to e=
o-__a r/c i b 1
which is similar in form to equation (9). The special case (c = 1) gives the straight line strain hardening law in Fig. 6. The second empirical formula is the exponential law of Vote and Palm [8,9] 0 = a + (b - a)(1 - eece) which leads to 1
,q= --ln c
CT- b (a-b 1
which is similar in form to equation (11) with m = n = 1.
J. W. SMITH and D. S. ZHANG
56 CALIBRATION
AND
SIMULATION
Case 1 of the microfriction component axial tension test. In this case we have
OF
AXIAL
TENSION
TEST
will now be calibrated and used to illustrate the
E(g) = E,
o(g) =
60
ag”
+
giving equation (9) thus n
E_
(~7-0~)~
(9)
n + 1 Ea””
It is understood that this refers to the plastic component ofstrain only. Thus the stress may be expressed as
(15)
cT=fJo+
=
fJ()
+
AEB
where E’ = Ecc’!” which is identical to equation 14. The constants may be determined in the usual way by plotting the tension test data in logarithmic coordinates as shown in Fig. 7. Hence
and
E’ =
Ea*“’ = LA n+l
(II+
l)‘n
.
It can be seen that E and a are not independent for this model and therefore the value E’ is used instead. Some examples of the simulation of real metals subjected to axial tension are shown in Fig. 8. The simple microfriction component of equation (9) is used and the elastic part has been added to the total strain. The experimental curves were obtained from Templin and Sturm [lo]. It may be seen that the microfriction component models the plastic strain hardening behaviour reasonably accurately except in the case of structural steel. This was because the carbon atoms in steel pin the dislocations until the stress is large enough to overcome their effect resulting in the characteristic yield plateau which is a difficult discontinuity to model. CYCLIC
LOADING
WITH
STRAIN
SOFTENING
In order to simulate the behaviour of a real material the model should be capable of responding to cyclic loading. The microfriction component as illustrated in Fig. 3 is satisfactory for this purpose and requires to be in series with a notional spring to represent the elastic response.
Fig. 7. The determination
of material
constants.
Microfriction constitutive theory
57
5IlIcon steel ,’ Sttuctural
-
E i 19J.100
Expenment
Brass E : 103LOO uyz 103 L 7=ore E’. 266567&l 9
Htgh
purity
alummium
E = 372Ll dy.21 6 J-073 E’: 10033 9
-annealed
1
0
005
0.15
0 IO
0 20
o-25
o-30 035
O-LO
Strain [a)
STRESS-STRAIN
CURVES
,-v-_;;
-
Exponment
-----
Theory
Brass High purity
alumlntum-annealed
r 0
ib)
0002 EXPANDED
0 00.4 STRAIN
0.006 stram
0.008
0 010
0 012
SCALE
Fig. 8. The microfriction model compared with experiments in axial tension.
On releasing the external stress the elasticcomponent ofstrain reduces to zero. The friction elements will then be pre-loaded in compression because of the remaining stress left in their parallel elastic elements due to the plastic deformation. Thus, on compression the threshold element will yield at an external compressive stress less than go. The model, therefore, exhibits Bauschinger’s effect. Some high strength engineering steels exhibit strain softening when loaded cyclically beyond the yield point. Figure 9 shows experimental stress-strain loops for a cylindrical sample of 1CrMoV steel. The diameter was 7.69 mm, the gauge length 15 mm and the sample was strained in reversed loading at a temperature of 500 “C. At this temperature the Young’s modulus is 160GN/m2. The experiment was strain controlled with a maximum strain of &-1.38 ;O, except for the first few cycles. The stress-strain curves form symmetrical hysteresis loops with a gradually reducing stress amplitude as shown in Fig. 10. The strain softening phenomenon can be incorporated into the microfriction model by considering that the stress amplitude is a function of the total (summed) plastic strain, which can be regarded as one equivalent quantity of the cycle number. Thus
J. W.
SMITH
and D. S. ZHANC
Cycle n -I
6 I
U
,I”,
lcmz
Fig. 9. Experimental stress-strain
50
lOO.LZ
MN/d
loops.
This introduces the idea of adopting the “endochronic” material memory concept by giving the microfriction model parameter values that change with summed plastic strain. In this case it was considered physically realistic to modify the threshold stress of the microfriction model so this would represent gradually increasing size and number of dislocations. The function $ that best fitted the experimental results was given by
cl
II/=
c, +JqTl
+ Cz~l~,l- G-npl)C~.
The other parameters were obtained from the loop ofthe 16th cycle and were calculated to be: C-J,, = 150 MN/m2 E = 160GN/m2 n = 0.3194
E’ = 8.864 x 10” (Nm units).
Using these values the microfriction model was used in incremental form to obtain the theoretical stress-strain loops shown in Fig. 11. The strain amplitude softening is compared with the experiment in Fig. 10. 1100 1 1000
; . f
I -
900-
E
2 :
aoo-
7
Fig. IO. The reduction of stress amplitude.
Microfriction
constitutive
theory
59
800 Stress MN/m2 I
s=
-600
150.00 MN/mi
N = 0.32 E’.&186.Lx10g
Fig. 11. Microfriction
model stress-strain
loops.
CONCLUSIONS
A constitutive theory of plasticity has been presented which accurately models the gradual transition from elastic to plastic behaviour that is exhibited by most metals. The theory is an attempt to model the deformation of dislocations in a crystal lattice at microscopic scale. In effect, it is a development of classical rheological models incorporating frictional sliders and spring elements. Hence, it is referred to as the “microfrictional component”. It has been shown that some forms of the final equations correspond closely to early empirical formulae. The theory was calibrated and compared with results of axial tension tests and found to model the strain hardening behaviour accurately except in the case of structural steel with its characteristic.yield plateau. The cyclic behaviour of the model was compared with the results of an experiment on a sample of high strength steel at elevated temperature. The sample was cyclically loaded both in tension and compression under strain control. The symmetrical hysteresis loops exhibited a gradually reducing stress amplitude in successive cycles until failure. This strain softening phenomenon was modelled in the “microfriction component” by subjecting the threshold stress to a decay that increased with previous summed strain history. Good agreement with the experimental stress-strain loops was obtained. The “microfriction component” has been shown to model material yielding behaviour accurately in both axial and cyclic loading. The equations are capable of being incorporated into a fully 3-dimensional finite element scheme. Acknowledgemenrs-Experimental data from a cyclic loading test, provided by Prof. E.G. Ellison ofthe Department of Mechanical Engineering, University of Bristol, is gratefully acknowledged.
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J. W. SMITH and D. S. ZHANG
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