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Transient creep in mild steel and copper at room temperature
OOOI-6160/85 $3.00 f0.00
Acra merall. Vol. 33, No. 5, pp. 835-840, 1985
Copyright 0 1985Pergamon Press Ltd
Printed in Great Britain. All rights reserved
TRANSIENT COPPER
CREEP IN MILD STEEL AND AT ROOM TEMPERATURE H. D. CHANDLER
School of Mechanical Engineering, University of the Witwatersrand, 200 1 Johannesburg, South Africa
1 Jan Smuts Avenue,
(Received 20 April 1984; in revised form 15 October 1984)
Abstract-Transient creep curves were obtained from mild steel and copper specimens which had been work hardened by cycling. Results are presented in terms of an “overstress” model in which strain rates are related to differences between the actual stress and the corresponding stress-strain point on a quasi-static stress-strain curve. It is further shown that a semi-empirical equation used to describe steady state plastic deformation can be adapted to describe transient behaviour when re-written in terms of the ‘overstress’ rather than the actual stress.
R&mm&-Nousavons obtenu des courbes de fluge transitoire dans l’acier doux et dans le cuivre durcis par des cycles de d&formation. Nous prksentons les rksultats en terme d’un modele de “surcontrainte” dans lequel les vitesses de d&formation sont likes B la diffkrence entre les v&tables contraintes et le point correspondant de la contrainte4formation sur une courbe contrainte4bformation quasi-statique. Nous montrons d’autre part que I’on peut adapter une kquations semi-empirique utilisie dans la description de la dbfoxmation plastique stationnaire pour dircrire le comportement transitoire en l’itcrivant & nouveau en fonction de la “surcontrainte” plut6t que de la contrainte rbelle. Zusammenfassung-An
zyklisch verfestigten Kupfer- und kohlenstoffarmen Stahlproben wurden transiente Kriechkurven gemessen. Die Ergebnisse werden anhand des “uberspannungs”-Modells dargestellt, bei dem die Dehnungsraten mit Unterschieden zwischen der anliegenden Spannung und der entsprechenden Spannung auf einer quasi-statischen Spannungs-Dehnungskurve zusammenhlngen. AuDerdem wird gezeigt, da8 eine fiir die Beschreibung der stationlren plastischen Verformung benutzte Gleichung auf die Beschriebung des ubergangsverhaltens angepal3t werden kann, wenn statt der tatsiichlichen Spannung die “tiberspannung” verwendet wird.
INTRODUCTION
The majority of theories describing plastic deformation behaviour in metals are empirical. This is because the variety and complexity of the microstructural events involved preclude strict formal descriptions of macroscopic behaviour. However, considerable success has been achieved in the formulation of mechanical constitutive equations using semi-empirical approaches in which deformation mechanisms are considered [l, 21. Results are often presented in the form of deformation- mechanism maps in which curves of constant strain rate are plotted in stress-temperature space [2]. These maps are mainly applicable to steady state deformation. Maps including transient deformation have been presented in which curves of total strain after a given time are plotted on stress-temperature coordinates [2], but these are limited to describing monotonic loading and not cyclic loading conditions. Transient deformation, including cycling effects, has been described in which the metal is represented as a composite material [3-61. When metals are plastically deformed, dislocation cell structures are developed and, in the composite model approach, different mechanical properties are assigned to cell wall and cell interior components. In descriptions of 835
creep behaviour [3-61, different steady state power law creep equations are followed by each structural component and, with the requirement of deformation compatibility, different stresses are built up in each. Thus, during cycling, internal residual stresses appear in the model and such representations can, at least qualitatively, describe phenomena such as the Bauschinger effect [7] and cyclic creep acceleration or retardation [4]. These models lead to time hardening type constitutive relationships for creep in which the shear strain rate 1’is a function of the shear stress CT,, time t and absolute temperature T: ?i =f(o,,
t, T).
(1)
The assumption of a relatively stable composite structure is, perhaps, not altogether justified. Recent microstructural investigations [8-lo] indicate that, during cycling, cell walls may break down producing dislocation ‘avalanches’ and that movement of the relatively mobile avalanche dislocations and their incorporation into new wall structures contributes significantly to the overall deformation producing, for example, “strain bursts” in single crystals [8]. Most empirical deformation theories take the form of equation (1). Alternatives are viscoplastic theories of the “Overstress” type [ 1l-131 which specifically
836
CHANDLER:
TRANSIENT CREEP IN MILD STEEL AND COPPER
Before carrying out creep tests, specimens were work hardened by cycling to steady state conditions i.e. when successive mechanical hysteresis loops were identical. Fully reversed cycling was carried out on mild steel between fixed force limits at equivalent shear stress rates varying from 0.016 to 160 MPa s-’ and on copper at rates of 0.005 and 5MPas-‘. Hysteresis curves from the slowest loading rate were taken to approximate “equilibrium” stress-strain curves for each material as far as determining tranY’ sient effects was concerned. The minimum strain rates Strain Y attained during cyclic testing were much larger than Fig. 1. Schematic equilibrium stress strain curve together steady strain rates calculated for copper and pure with a curve tested at a faster rate showing a harder iron using equations 2.9, 2.12 and data from Ref. [2], response. In terms of overstress models, the strain rate is a to try to eliminate steady state effects. function of &J,. The creep tests were performed by initially cycling specimens to steady state conditions between fixed describe transient behaviour. Such theories assume force limits at a rate of 16 MPa s-’ and then holding the existence of a steady state stress-strain curve the force constant during the increasing tensile phase which may be determined experimentally by testing of the cycle either at the maximum or at a smaller the material at a very low loading rate. At higher force for 3&60 min. For mild steel, equivalent cyclic loading rates, the stress-strain curves show a harder shear stress limits were from f 150 to f 210 MPa and response and lie above the steady state curve. If a creep tests were carried out at stresses between 150 material is loaded to a certain shear stress a,! and the and 210 MPa. For copper, cycling was carried out at stress held constant, the shear strain rate i is exf66 and k75 MPa with creep test stresses between pressed as a function of the difference in stress 6a,? 55 and 75 MPa. Eighteen creep curves were deterbetween the actual stress point (Q,“,y’) and the corremined from mild steel and six from copper specimens. Creep strain rates recorded ranged over four orders sponding stress point (B’, y’) on the steady state curve as illustrated in Fig. 1: of magnitude from 10m3to lo-’ s-‘, the slowest rate for each material being much larger than the steady (2) it =f@J state creep rates calculated from Ref [2], again, to Since the stress difference 6a,”is a function of the total reduce steady state effects. Strain rates together with total creep strains were strain, such theories lead to strain hardening rather measured at various time intervals from the creep than time hardening constitutive relationships. curves and total creep strains used to determine a~,~, The purpose of the present paper is to describe overstress, values from mechanical hysteresis loops. ambient temperature, constant stress creep behaviour Results are presented below in terms of equivalent of mild steel and copper which have been subjected shear stress and strain by dividing measured axial to cycling in the plastic region and relate behaviour to an overstress deformation model. The results are stresses and multiplying axial strains by,/% further analysed in terms of a semi-empirical approach [1,2] and it is shown that, together with the RESULTS notion of a steady state stress-strain curve as used Figure 2 shows steady state semi-hysteresis loops, by overstress models, the transient creep behaviour may be represented by equations which are very superimposed at zero stress, for mild steel cycled similar to those normally used to describe steady state between f 210 MPa at different loading rates. As the loading rates decrease, the corresponding shear strain deformation. amplitudes increase although these increases become less with decreasing loading rate. This indicates an EXPERIMENTAL Materials tested were mild steel 070M20 (En3) which was normalised prior to machining and commercially pure copper which was annealed at 500°C following machining. Button head type specimens having a parallel section of length 27 mm and diameter 7.5 mm were used. Push-pull cyclic and constant force creep tests were performed on an ESH servo-hydraulic testing machine of 250 kN maximum capacity operated in force control mode and strains were measured using an Instron strain gauge extensometer of 1Omm gauge length.
200
2 z 150 I b 100 ul j 50 rn 0 0
2
4
6
6 10 12 14 16 Shear Strain x IO3
18
20
22
Fig. 2. Steady state semi-hysteresis loops for mild steel tested at an equivalent shear stress amplitude of &-210 MPa at different loading rates.
CHANDLER:
TRANSIENT CREEP IN MILD STEEL AND COPPER
For mild steel, tl = 9.5, and C = 2.5 x lo-l5 MPa-9.5; for copper a = 4.3 and C = 10e6 MPa-4,3. In terms of this purely empirical approach, the behaviour of the two materials appears to be very different.
200 B 2 150 2 L 130 v) 4
50 0
837
SEMI-EMPIRICAL FORMULATION 0
2
4
6
8 10 12 14 16 18 Shear Straw x lo3
20
22
Fig. 3. Steady state semi-hysteresis curves for mild steel tested at a loading rate of 0.016 MPa s-’ at various stress amplitudes.
To try to rationalise the behaviour of the two materials, a semi-empirical approach following the same lines as that previously used for steady state creep [1,2] was attempted in which equations are formulated in terms of the overstress da,? rather than the current stress state u,~. The strain rate for a system in which mobile dislocations of density pm with Burgers vectors of magnitudes b are moving with an average velocity 6 is given by [14]:
approach to a quasi-static steady state stress-strain curve. Figure 3 shows superimposed steady state cyclic curves for mild steel loaded at rates of 0.016 MPa s-’ at different stress amplitudes. Results indicate that a single quasistatic stress-strain curve can be used for a range of loading conditions although curves appear to become amplitude dependent at lower cyclic stresses. Creep test results are shown in Fig. 4 plotted as strain rate i against overstress 60,~for both steel and copper. Results show considerable scatter, most of which can probably be attributed to measurement error arising from the large spread of creep rates (over four orders of magnitude) together with the small values and limited ranges of 64,. Points from each creep curve have been allocated a particular symbol in Fig. 4 and data from the six copper curves and nine of the mild steel curves have been presented. Within the limits of experimental error, results appear to be independent of the loading conditions for each creep curve and to follow a simple power law relationship over the range investigated:
Structural studies referred to above [8-IO] suggest that during cyclic deformation cell walls break down into dislocation avalanches which form new cell wall structures. When the stress is held constant during a cycle, it is probable that avalanche dislocations are unstable with respect to the applied stress and move, forming new walls which are stable, and this movement of avalanche dislocations is one of the major contributors to the observed transient creep deformation. The density of avalanche dislocations would decrease during creep to the equilibrium value and it might therefore be expected that the transient creep rate r’, would be a function of the excess mobile dislocation density 6p, above the equilibrium number. Equation (4) could thus be written as:
d = C(6u.J”.
i, a 6p, b6.
y’ = p,,,bC.
(3)
10-3 r
10-7c 0 0.5
I
1.0
2.0 3.0 4.0 Overstress 60, (MPa)
10.0
,
15.0 20.0
Fig. 4. Creep strain rates i plotted against overstress 60,.
(4)
(5)
838
CHANDLER:
TRANSIENT
CREEP IN MILD STEEL AND COPPER
For steady state deformation, the mobile dislocation density is expressed as a function of the applied stress o,, Burgers vector, magnitude b and the shear modulus of elasticity p [2]: Pm K (uJ@)2.
(6)
Similarly the excess dislocation density 6p, can probably be written in the same way with overstress 60, replacing the applied stress: +,
0~ (&l@)2.
In the steady state formulation, equation average velocity of the mobile dislocations by:
(7) (4), the is given
fi cc b exp - [AG(a,)/kT]
(8)
Where G(a,) is the Gibbs free energy to overcome obstacles in the lattice, T is the absolute temperature and k is Boltzmann’s constant. The Gibbs free energy depends on the applied stress and the strength and shape of the lattice obstacles through the relationship L21: P4 AG(a,y) = AF 1 - > (9) [ where AF is the activation energy required to overcome obstacles and z^is a shear stress at which such barriers cease to impede dislocation movement. As well as being a measure of obstacle strength, Q also represents the arrangement of lattice barriers and is proportional to pb/l where 1 is the obstacle spacing. It therefore serves to characterise the internal structure, e.g. degree of work-hardening of the material. The constants p and q specify the obstacle shape. For example, values ofp = 1 and q = 1 represent an array of rectangular shaped barriers. To describe more realistically shaped barrier hills, values of p = 3/4 and q = 4/3 are often used [l, 21. Combining equations (5), (7), (8) and (9) does not describe the transient creep results shown in Fig. 4. Creep rates so determined would be functions of 6af and the applied stress Q,~whereas experimental creep
01
0
rates are proportional to higher powers of 6~9 and appear to be independent of a, over the conditions examined. It would appear, therefore, that equation (9) needs to be replaced by an expression in which the Gibbs free energy term is a function of the overstress aa,*rather than the current stress a,. By analogy with equation (Y), the form suggested is:
where AF, p and q represent the activation energy and obstacle shape and would be expected to have about the same values as those in the steady state creep equations. The material property 6z^which represents an overstress value at which obstacles cease to block dislocation movement can probably be interpreted in terms of the residual stress state associated with the dislocation sub-structure and may be a function of the difference between the flow stress Q and the residual or back stresses in the material. Thus, as with steady state creep, 6f can probably be regarded as a measure of work hardening. For the range of experimental conditions tested in which the shapes of the mechanical hysteresis loops are similar as shown in Fig. 3, 6z^ is probably constant. However, if the hysteresis loop shapes are not similar, indicating a different degree of structural hardening, 6r^may well be a function of stress-strain curve shape. Combining equations (5), (7), (8) and (10) leads to the relationship:
where Ilti is a constant. With p = 314 and q = 413, equation (11) is the same as equation (2.12) from Ref. (2) with a~,~, the overstress, replacing G,~,the actual stress and 62^replacing the material constant z^. In Ref. [2], equation (2.12) was used to determine steady state strain rates for deformation-mechanism map regions where plasticity is limited by lattice resistance (Pierls-Nabarro forces).
2
1
3
60. MPa3” Fig.
5.
Creep strain rates from Fig. 4 plotted against overstress in terms of equation (12) for copper.
CHANDLER:
71 3
TRANSIENT
4
CREEP IN MILD STEEL AND COPPER
5
6
7
839
8
60s PAPa”
Fig. 6. Creep strain rates from Fig. 4 plotted against overstress in terms of equation (12) for mild steel. Equation
(11) can be re-arranged
[1n~($$l’;‘=(~)II[(!$).-
to give: 11.
(12)
If applicable, plotting the left-hand side of equation (12) against @a,?)”should result in a straight line with intercept at (6~,~)f’= 0 of -(AF/kT)“g and slope -(AF/kT)“@. 65^-Pwhich would allow values for the material parameters AF and 6z^ to be determined. Creep results from Fig. 4 are plotted in this way in Fig. 5 for copper and in Fig. 6 for mild steel. The constant &, has been taken as 10” which is the same value as that used in Ref. [2]. Results indicate a reasonable fit to straight line curves. Values of AF and ST/~ from Figs 5 and 6 are shown in Table 1 together with values of b and /* from Ref. 123. Values of activation energy AF are comparable to those used to fit steady state data [2] (AF = 0.5 yb3 for obstacle controlled plasticity of copper and pure iron and AF = 0.1 pb3 for lattice resistance controlled deformation of pure iron). Obstacle strengths may be classified according to activation energy [2] and values between 0.2 pb3 and 1.Op’b3 indicate medium strength obstacles such as forest dislocations and values less than 0.2 pb3 indicate weak barriers to flow such as lattice friction. For cyclically hardened materials such as those tested in the present investigation, it would be expected that dislocation forests would be the primary barrier to plastic flow. However the activation energies tend towards the low end of the medium obstacle range. This is perhaps consistent with deformation being largely due to movement of Table 1. Values of material parameters AF and Sr^determined from Fies 5 and 6 Mild steel Shear modulus ~1 Burgers vector b AF ST/u
6.40 x
10’ MPa 2.48 x 10~“m 0.12pb’ 1.11 x 10-1
Copper 4.21 x IO4 MPa 2.56 x lo-” m 0.13pb3 4.66 x lO-4
avalanche dislocations from disrupted cell wall structures since it would be expected that the broken down wall structures would act as weaker obstacles than fully formed cell walls and that avalanche dislocations are more mobile than heavily pinned wall dislocations. CONCLUSIONS
Transient creep of cyclically hardened mild steel and copper can be described in terms of semiempirical equations which are normally used to describe steady state behaviour if, instead of expressing the strain rate as a function of the current stress, it is expressed as a function of the difference between the current stress and the corresponding stress-strain point on a quasi-static stress-strain curve. Values of activation energies are similar to those for steady state creep and are consistent with the deformation mechanisms involved. Using this approach it may be possible, at least for certain transient effects, to construct deformation-mechanism maps similar to those used to describe steady state deformation. REFERENCES 1. U. F. Kocks, A. S. Argon and M. F. Ashby, Prog.
Mater. Sci. 19 (1975). 2. H. J. Frost and M. F. Ashby, DeformationMechanism Maps. Pergamon Press, Oxford (1982). 3. D. K. Shetty and M. Meshii, MetaN. Trans. 6A, 349 (1975). 4. T. Mura, A. Novakovic and M. Meshii, Murer. Sci. Engng 17, 221 (1975). 5. D. K. Shetty, T. Mura and M. Meshii, Mater. Sci. Engng 20, 261 (1975). 6. M. Meshii, M. Ueki and H. D. Chion, in Proc. Fifrh Int. Conf. on Strength of Metals and Alloys, Aachen, 1979 (edited by P. Haasen, V. Gerald and G. Kostorz), Vol. 1, p. 245. Pergamon Press, Oxford (1979). 7. 0. B. Pedersen, L. M. Brown and W. M. Stobbs, Acta metall. 29, 1843 (1981). 8. F. Lourenzo and C. Laird, Acta metall. 32, 671 (1984).
840
CHANDLER:
TRANSIENT
CREEP IN MILD STEEL AND COPPER
9. F. Lourenzo and C. Laird, Acta metall. 32, 681 (1984). 10. H. D. Chandler and J. V. Bee. To be published. 11. M. A. Eisenberg, C. W. Lee and A. Phillips, Inr. J. Solidr Struct. 13, 1239 (1977).
12. E. Krempl, J. Mech. Phys. Solids 27, 363 (1979). 13. D. Kujawski, V. Kallianpur and E. Krempl, J. Mech. Phys. Solids 28, 129 (1980). 14. E. Orowan, Proc. Phys. Sot. 52, 8 (1940).
Acra merall. Vol. 33, No. 5, pp. 835-840, 1985
Copyright 0 1985Pergamon Press Ltd
Printed in Great Britain. All rights reserved
TRANSIENT COPPER
CREEP IN MILD STEEL AND AT ROOM TEMPERATURE H. D. CHANDLER
School of Mechanical Engineering, University of the Witwatersrand, 200 1 Johannesburg, South Africa
1 Jan Smuts Avenue,
(Received 20 April 1984; in revised form 15 October 1984)
Abstract-Transient creep curves were obtained from mild steel and copper specimens which had been work hardened by cycling. Results are presented in terms of an “overstress” model in which strain rates are related to differences between the actual stress and the corresponding stress-strain point on a quasi-static stress-strain curve. It is further shown that a semi-empirical equation used to describe steady state plastic deformation can be adapted to describe transient behaviour when re-written in terms of the ‘overstress’ rather than the actual stress.
R&mm&-Nousavons obtenu des courbes de fluge transitoire dans l’acier doux et dans le cuivre durcis par des cycles de d&formation. Nous prksentons les rksultats en terme d’un modele de “surcontrainte” dans lequel les vitesses de d&formation sont likes B la diffkrence entre les v&tables contraintes et le point correspondant de la contrainte4formation sur une courbe contrainte4bformation quasi-statique. Nous montrons d’autre part que I’on peut adapter une kquations semi-empirique utilisie dans la description de la dbfoxmation plastique stationnaire pour dircrire le comportement transitoire en l’itcrivant & nouveau en fonction de la “surcontrainte” plut6t que de la contrainte rbelle. Zusammenfassung-An
zyklisch verfestigten Kupfer- und kohlenstoffarmen Stahlproben wurden transiente Kriechkurven gemessen. Die Ergebnisse werden anhand des “uberspannungs”-Modells dargestellt, bei dem die Dehnungsraten mit Unterschieden zwischen der anliegenden Spannung und der entsprechenden Spannung auf einer quasi-statischen Spannungs-Dehnungskurve zusammenhlngen. AuDerdem wird gezeigt, da8 eine fiir die Beschreibung der stationlren plastischen Verformung benutzte Gleichung auf die Beschriebung des ubergangsverhaltens angepal3t werden kann, wenn statt der tatsiichlichen Spannung die “tiberspannung” verwendet wird.
INTRODUCTION
The majority of theories describing plastic deformation behaviour in metals are empirical. This is because the variety and complexity of the microstructural events involved preclude strict formal descriptions of macroscopic behaviour. However, considerable success has been achieved in the formulation of mechanical constitutive equations using semi-empirical approaches in which deformation mechanisms are considered [l, 21. Results are often presented in the form of deformation- mechanism maps in which curves of constant strain rate are plotted in stress-temperature space [2]. These maps are mainly applicable to steady state deformation. Maps including transient deformation have been presented in which curves of total strain after a given time are plotted on stress-temperature coordinates [2], but these are limited to describing monotonic loading and not cyclic loading conditions. Transient deformation, including cycling effects, has been described in which the metal is represented as a composite material [3-61. When metals are plastically deformed, dislocation cell structures are developed and, in the composite model approach, different mechanical properties are assigned to cell wall and cell interior components. In descriptions of 835
creep behaviour [3-61, different steady state power law creep equations are followed by each structural component and, with the requirement of deformation compatibility, different stresses are built up in each. Thus, during cycling, internal residual stresses appear in the model and such representations can, at least qualitatively, describe phenomena such as the Bauschinger effect [7] and cyclic creep acceleration or retardation [4]. These models lead to time hardening type constitutive relationships for creep in which the shear strain rate 1’is a function of the shear stress CT,, time t and absolute temperature T: ?i =f(o,,
t, T).
(1)
The assumption of a relatively stable composite structure is, perhaps, not altogether justified. Recent microstructural investigations [8-lo] indicate that, during cycling, cell walls may break down producing dislocation ‘avalanches’ and that movement of the relatively mobile avalanche dislocations and their incorporation into new wall structures contributes significantly to the overall deformation producing, for example, “strain bursts” in single crystals [8]. Most empirical deformation theories take the form of equation (1). Alternatives are viscoplastic theories of the “Overstress” type [ 1l-131 which specifically
836
CHANDLER:
TRANSIENT CREEP IN MILD STEEL AND COPPER
Before carrying out creep tests, specimens were work hardened by cycling to steady state conditions i.e. when successive mechanical hysteresis loops were identical. Fully reversed cycling was carried out on mild steel between fixed force limits at equivalent shear stress rates varying from 0.016 to 160 MPa s-’ and on copper at rates of 0.005 and 5MPas-‘. Hysteresis curves from the slowest loading rate were taken to approximate “equilibrium” stress-strain curves for each material as far as determining tranY’ sient effects was concerned. The minimum strain rates Strain Y attained during cyclic testing were much larger than Fig. 1. Schematic equilibrium stress strain curve together steady strain rates calculated for copper and pure with a curve tested at a faster rate showing a harder iron using equations 2.9, 2.12 and data from Ref. [2], response. In terms of overstress models, the strain rate is a to try to eliminate steady state effects. function of &J,. The creep tests were performed by initially cycling specimens to steady state conditions between fixed describe transient behaviour. Such theories assume force limits at a rate of 16 MPa s-’ and then holding the existence of a steady state stress-strain curve the force constant during the increasing tensile phase which may be determined experimentally by testing of the cycle either at the maximum or at a smaller the material at a very low loading rate. At higher force for 3&60 min. For mild steel, equivalent cyclic loading rates, the stress-strain curves show a harder shear stress limits were from f 150 to f 210 MPa and response and lie above the steady state curve. If a creep tests were carried out at stresses between 150 material is loaded to a certain shear stress a,! and the and 210 MPa. For copper, cycling was carried out at stress held constant, the shear strain rate i is exf66 and k75 MPa with creep test stresses between pressed as a function of the difference in stress 6a,? 55 and 75 MPa. Eighteen creep curves were deterbetween the actual stress point (Q,“,y’) and the corremined from mild steel and six from copper specimens. Creep strain rates recorded ranged over four orders sponding stress point (B’, y’) on the steady state curve as illustrated in Fig. 1: of magnitude from 10m3to lo-’ s-‘, the slowest rate for each material being much larger than the steady (2) it =f@J state creep rates calculated from Ref [2], again, to Since the stress difference 6a,”is a function of the total reduce steady state effects. Strain rates together with total creep strains were strain, such theories lead to strain hardening rather measured at various time intervals from the creep than time hardening constitutive relationships. curves and total creep strains used to determine a~,~, The purpose of the present paper is to describe overstress, values from mechanical hysteresis loops. ambient temperature, constant stress creep behaviour Results are presented below in terms of equivalent of mild steel and copper which have been subjected shear stress and strain by dividing measured axial to cycling in the plastic region and relate behaviour to an overstress deformation model. The results are stresses and multiplying axial strains by,/% further analysed in terms of a semi-empirical approach [1,2] and it is shown that, together with the RESULTS notion of a steady state stress-strain curve as used Figure 2 shows steady state semi-hysteresis loops, by overstress models, the transient creep behaviour may be represented by equations which are very superimposed at zero stress, for mild steel cycled similar to those normally used to describe steady state between f 210 MPa at different loading rates. As the loading rates decrease, the corresponding shear strain deformation. amplitudes increase although these increases become less with decreasing loading rate. This indicates an EXPERIMENTAL Materials tested were mild steel 070M20 (En3) which was normalised prior to machining and commercially pure copper which was annealed at 500°C following machining. Button head type specimens having a parallel section of length 27 mm and diameter 7.5 mm were used. Push-pull cyclic and constant force creep tests were performed on an ESH servo-hydraulic testing machine of 250 kN maximum capacity operated in force control mode and strains were measured using an Instron strain gauge extensometer of 1Omm gauge length.
200
2 z 150 I b 100 ul j 50 rn 0 0
2
4
6
6 10 12 14 16 Shear Strain x IO3
18
20
22
Fig. 2. Steady state semi-hysteresis loops for mild steel tested at an equivalent shear stress amplitude of &-210 MPa at different loading rates.
CHANDLER:
TRANSIENT CREEP IN MILD STEEL AND COPPER
For mild steel, tl = 9.5, and C = 2.5 x lo-l5 MPa-9.5; for copper a = 4.3 and C = 10e6 MPa-4,3. In terms of this purely empirical approach, the behaviour of the two materials appears to be very different.
200 B 2 150 2 L 130 v) 4
50 0
837
SEMI-EMPIRICAL FORMULATION 0
2
4
6
8 10 12 14 16 18 Shear Straw x lo3
20
22
Fig. 3. Steady state semi-hysteresis curves for mild steel tested at a loading rate of 0.016 MPa s-’ at various stress amplitudes.
To try to rationalise the behaviour of the two materials, a semi-empirical approach following the same lines as that previously used for steady state creep [1,2] was attempted in which equations are formulated in terms of the overstress da,? rather than the current stress state u,~. The strain rate for a system in which mobile dislocations of density pm with Burgers vectors of magnitudes b are moving with an average velocity 6 is given by [14]:
approach to a quasi-static steady state stress-strain curve. Figure 3 shows superimposed steady state cyclic curves for mild steel loaded at rates of 0.016 MPa s-’ at different stress amplitudes. Results indicate that a single quasistatic stress-strain curve can be used for a range of loading conditions although curves appear to become amplitude dependent at lower cyclic stresses. Creep test results are shown in Fig. 4 plotted as strain rate i against overstress 60,~for both steel and copper. Results show considerable scatter, most of which can probably be attributed to measurement error arising from the large spread of creep rates (over four orders of magnitude) together with the small values and limited ranges of 64,. Points from each creep curve have been allocated a particular symbol in Fig. 4 and data from the six copper curves and nine of the mild steel curves have been presented. Within the limits of experimental error, results appear to be independent of the loading conditions for each creep curve and to follow a simple power law relationship over the range investigated:
Structural studies referred to above [8-IO] suggest that during cyclic deformation cell walls break down into dislocation avalanches which form new cell wall structures. When the stress is held constant during a cycle, it is probable that avalanche dislocations are unstable with respect to the applied stress and move, forming new walls which are stable, and this movement of avalanche dislocations is one of the major contributors to the observed transient creep deformation. The density of avalanche dislocations would decrease during creep to the equilibrium value and it might therefore be expected that the transient creep rate r’, would be a function of the excess mobile dislocation density 6p, above the equilibrium number. Equation (4) could thus be written as:
d = C(6u.J”.
i, a 6p, b6.
y’ = p,,,bC.
(3)
10-3 r
10-7c 0 0.5
I
1.0
2.0 3.0 4.0 Overstress 60, (MPa)
10.0
,
15.0 20.0
Fig. 4. Creep strain rates i plotted against overstress 60,.
(4)
(5)
838
CHANDLER:
TRANSIENT
CREEP IN MILD STEEL AND COPPER
For steady state deformation, the mobile dislocation density is expressed as a function of the applied stress o,, Burgers vector, magnitude b and the shear modulus of elasticity p [2]: Pm K (uJ@)2.
(6)
Similarly the excess dislocation density 6p, can probably be written in the same way with overstress 60, replacing the applied stress: +,
0~ (&l@)2.
In the steady state formulation, equation average velocity of the mobile dislocations by:
(7) (4), the is given
fi cc b exp - [AG(a,)/kT]
(8)
Where G(a,) is the Gibbs free energy to overcome obstacles in the lattice, T is the absolute temperature and k is Boltzmann’s constant. The Gibbs free energy depends on the applied stress and the strength and shape of the lattice obstacles through the relationship L21: P4 AG(a,y) = AF 1 - > (9) [ where AF is the activation energy required to overcome obstacles and z^is a shear stress at which such barriers cease to impede dislocation movement. As well as being a measure of obstacle strength, Q also represents the arrangement of lattice barriers and is proportional to pb/l where 1 is the obstacle spacing. It therefore serves to characterise the internal structure, e.g. degree of work-hardening of the material. The constants p and q specify the obstacle shape. For example, values ofp = 1 and q = 1 represent an array of rectangular shaped barriers. To describe more realistically shaped barrier hills, values of p = 3/4 and q = 4/3 are often used [l, 21. Combining equations (5), (7), (8) and (9) does not describe the transient creep results shown in Fig. 4. Creep rates so determined would be functions of 6af and the applied stress Q,~whereas experimental creep
01
0
rates are proportional to higher powers of 6~9 and appear to be independent of a, over the conditions examined. It would appear, therefore, that equation (9) needs to be replaced by an expression in which the Gibbs free energy term is a function of the overstress aa,*rather than the current stress a,. By analogy with equation (Y), the form suggested is:
where AF, p and q represent the activation energy and obstacle shape and would be expected to have about the same values as those in the steady state creep equations. The material property 6z^which represents an overstress value at which obstacles cease to block dislocation movement can probably be interpreted in terms of the residual stress state associated with the dislocation sub-structure and may be a function of the difference between the flow stress Q and the residual or back stresses in the material. Thus, as with steady state creep, 6f can probably be regarded as a measure of work hardening. For the range of experimental conditions tested in which the shapes of the mechanical hysteresis loops are similar as shown in Fig. 3, 6z^ is probably constant. However, if the hysteresis loop shapes are not similar, indicating a different degree of structural hardening, 6r^may well be a function of stress-strain curve shape. Combining equations (5), (7), (8) and (10) leads to the relationship:
where Ilti is a constant. With p = 314 and q = 413, equation (11) is the same as equation (2.12) from Ref. (2) with a~,~, the overstress, replacing G,~,the actual stress and 62^replacing the material constant z^. In Ref. [2], equation (2.12) was used to determine steady state strain rates for deformation-mechanism map regions where plasticity is limited by lattice resistance (Pierls-Nabarro forces).
2
1
3
60. MPa3” Fig.
5.
Creep strain rates from Fig. 4 plotted against overstress in terms of equation (12) for copper.
CHANDLER:
71 3
TRANSIENT
4
CREEP IN MILD STEEL AND COPPER
5
6
7
839
8
60s PAPa”
Fig. 6. Creep strain rates from Fig. 4 plotted against overstress in terms of equation (12) for mild steel. Equation
(11) can be re-arranged
[1n~($$l’;‘=(~)II[(!$).-
to give: 11.
(12)
If applicable, plotting the left-hand side of equation (12) against @a,?)”should result in a straight line with intercept at (6~,~)f’= 0 of -(AF/kT)“g and slope -(AF/kT)“@. 65^-Pwhich would allow values for the material parameters AF and 6z^ to be determined. Creep results from Fig. 4 are plotted in this way in Fig. 5 for copper and in Fig. 6 for mild steel. The constant &, has been taken as 10” which is the same value as that used in Ref. [2]. Results indicate a reasonable fit to straight line curves. Values of AF and ST/~ from Figs 5 and 6 are shown in Table 1 together with values of b and /* from Ref. 123. Values of activation energy AF are comparable to those used to fit steady state data [2] (AF = 0.5 yb3 for obstacle controlled plasticity of copper and pure iron and AF = 0.1 pb3 for lattice resistance controlled deformation of pure iron). Obstacle strengths may be classified according to activation energy [2] and values between 0.2 pb3 and 1.Op’b3 indicate medium strength obstacles such as forest dislocations and values less than 0.2 pb3 indicate weak barriers to flow such as lattice friction. For cyclically hardened materials such as those tested in the present investigation, it would be expected that dislocation forests would be the primary barrier to plastic flow. However the activation energies tend towards the low end of the medium obstacle range. This is perhaps consistent with deformation being largely due to movement of Table 1. Values of material parameters AF and Sr^determined from Fies 5 and 6 Mild steel Shear modulus ~1 Burgers vector b AF ST/u
6.40 x
10’ MPa 2.48 x 10~“m 0.12pb’ 1.11 x 10-1
Copper 4.21 x IO4 MPa 2.56 x lo-” m 0.13pb3 4.66 x lO-4
avalanche dislocations from disrupted cell wall structures since it would be expected that the broken down wall structures would act as weaker obstacles than fully formed cell walls and that avalanche dislocations are more mobile than heavily pinned wall dislocations. CONCLUSIONS
Transient creep of cyclically hardened mild steel and copper can be described in terms of semiempirical equations which are normally used to describe steady state behaviour if, instead of expressing the strain rate as a function of the current stress, it is expressed as a function of the difference between the current stress and the corresponding stress-strain point on a quasi-static stress-strain curve. Values of activation energies are similar to those for steady state creep and are consistent with the deformation mechanisms involved. Using this approach it may be possible, at least for certain transient effects, to construct deformation-mechanism maps similar to those used to describe steady state deformation. REFERENCES 1. U. F. Kocks, A. S. Argon and M. F. Ashby, Prog.
Mater. Sci. 19 (1975). 2. H. J. Frost and M. F. Ashby, DeformationMechanism Maps. Pergamon Press, Oxford (1982). 3. D. K. Shetty and M. Meshii, MetaN. Trans. 6A, 349 (1975). 4. T. Mura, A. Novakovic and M. Meshii, Murer. Sci. Engng 17, 221 (1975). 5. D. K. Shetty, T. Mura and M. Meshii, Mater. Sci. Engng 20, 261 (1975). 6. M. Meshii, M. Ueki and H. D. Chion, in Proc. Fifrh Int. Conf. on Strength of Metals and Alloys, Aachen, 1979 (edited by P. Haasen, V. Gerald and G. Kostorz), Vol. 1, p. 245. Pergamon Press, Oxford (1979). 7. 0. B. Pedersen, L. M. Brown and W. M. Stobbs, Acta metall. 29, 1843 (1981). 8. F. Lourenzo and C. Laird, Acta metall. 32, 671 (1984).
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TRANSIENT
CREEP IN MILD STEEL AND COPPER
9. F. Lourenzo and C. Laird, Acta metall. 32, 681 (1984). 10. H. D. Chandler and J. V. Bee. To be published. 11. M. A. Eisenberg, C. W. Lee and A. Phillips, Inr. J. Solidr Struct. 13, 1239 (1977).
12. E. Krempl, J. Mech. Phys. Solids 27, 363 (1979). 13. D. Kujawski, V. Kallianpur and E. Krempl, J. Mech. Phys. Solids 28, 129 (1980). 14. E. Orowan, Proc. Phys. Sot. 52, 8 (1940).