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Evolution of the cyclic plastic response of Sanicro 25 steel cycled at ambient and elevated temperatures
International Journal of Fatigue 83 (2016) 75–83
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Evolution of the cyclic plastic response of Sanicro 25 steel cycled at ambient and elevated temperatures Jaroslav Polák a,b,⇑, Roman Petráš a, Milan Heczko a, Tomáš Kruml a,b, Guocai Chai c,d Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic CEITEC, Institute of Physics of Materials Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic c Sandvik Materials Technology, SE-811 81 Sandviken, Sweden d Linköping University, Engineering Materials, SE-581 83 Linköping, Sweden a
b
a r t i c l e
i n f o
Article history: Received 4 November 2014 Received in revised form 12 March 2015 Accepted 17 March 2015 Available online 24 March 2015 Keywords: Cyclic plasticity Hysteresis loop analysis Heat resistant steel Dislocation structure Effect of temperature
a b s t r a c t Cyclic plastic response of the austenitic heat resistant steel Sanicro 25 has been studied during strain controlled low cycle fatigue tests performed at ambient and at elevated temperature. Simultaneously with the cyclic hardening/softening curves hysteresis loops during cyclic loading were analyzed using generalized statistical theory of the hysteresis loop. The probability density distribution function of the internal critical stresses, the effective saturated stress and their evolution during cycling were derived for various strain amplitudes. The internal dislocation structure and the surface relief at room and at elevated temperature were studied and correlated with the cyclic stress–strain response and the evolution of the probability density function of the internal critical stresses. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Cyclic plastic response of materials is generally studied during strain controlled low cycle fatigue tests with various total or plastic strain amplitudes and constant strain rate. Since complete stress– strain history in cyclic straining is difficult to report due to enormous number of data usually only the plots of the stress amplitude vs. number of cycles i.e. cyclic hardening/softening curves are reported [1,2]. Sometimes selected saturated hysteresis loops are plotted. These data are correlated with the internal structure of the material and its evolution during cyclic loading to assess the mechanisms of cyclic plastic straining and to find the sources of the cyclic stress. Another approach how to understand the mechanisms underlying the cyclic stress–strain response of materials is the modelization of the cyclic plastic straining based on the properties of dislocations in a particular lattice and the knowledge of the grain arrangement of the polycrystalline complex. Some attempts in this domain [3–5] were only partially successful since until now the phenomenon of cyclic strain localization, which is substantial for cyclic straining, has yet been properly modelled neither in single crystals nor in polycrystals. ⇑ Corresponding author at: Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic. E-mail address: [email protected] (J. Polák). http://dx.doi.org/10.1016/j.ijfatigue.2015.03.015 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.
Deeper insight in the mechanisms of cyclic plastic straining represents the approach adopted originally by Masing [6] who proposed to represent small deformations encountered in cyclic straining of polycrystals by parallel arrangement of several volumes of material having different critical yield stresses. This approach leads to natural explanation of the Bauschinger effect during unloading and to the shape of the hysteresis loop in cyclic straining. Afanasev [7] introduced continuous distribution of microvolumes with critical yield stresses characterized by the probability density function and named this approach statistical theory of the hysteresis loop. In order to take into account the effect of time and temperature on the motion of dislocations which carry the cyclic plastic strain Polák and Klesnil [8,9] and Burmeister and Holste [10] proposed generalized statistical theory of the hysteresis loop where also effective stress is considered. The theory has been applied to find the characteristic parameters of the saturated hysteresis loops in single phase [11,12] and double phase materials [13,14]. Number of researchers [15–20] applied statistical theory without considering the contribution of the effective stress. Austenitic heat resistant stainless steel Sandvik Sanicro 25 has been developed for applications at elevated temperatures, namely for construction of supercritical boilers [21,22]. Mostly creep behavior and high temperature corrosion and to a certain extent also low cycle fatigue properties were studied [22]. Recently [23] basic low cycle fatigue behavior of Sanicro 25 steel at ambient and high temperature was reported.
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In this paper generalized statistical theory of the hysteresis loop has been applied to the analysis of the experimental data recorded during cyclic plastic straining of Sanicro 25 steel at ambient and at elevated temperatures. The saturated effective stress and the probability density function of the internal critical stresses simultaneously with its evolution in cyclic straining were evaluated. The changes of the probability density function during cyclic loading contribute to the interpretation of cyclic hardening/softening behavior of the steel at both temperatures. The dislocation structures and surface relief of cyclically deformed Sanicro 25 steel were also studied and revealed pronounced cyclic slip localization. 2. Experimental Experimental material was supplied by Sandvik, Sweden in the form of the cylindrical rod of 150 mm in diameter. The chemical composition of the material in wt.% was: 0.1 C, 22.5 Cr, 25.0 Ni, 3.6 W, 1.5 Co, 3.0 Cu, 0.5 Mn, 0.5 Nb, 0.23 N, 0.2 Si and the rest Fe. Cylindrical specimens for cyclic straining at room and at elevated temperatures were machined with the axis parallel to the rod. The gauge length and the diameter of specimens were 14 mm and 8 mm respectively for room temperature testing and 15 mm and 6 mm for elevated temperature testing. Specimens were homogenized at 1200 °C for 1 h and cooled in air. Afterward final grinding selected specimens were mechanically polished and finally electropolished. Cyclic straining was performed in computer-controlled electrohydraulic MTS system. Fully reversed strain controlled cyclic loading with constant strain rate 5 103 s1 at room temperature and 2 103 s1 at temperature 700 °C was applied. In block loading high number of data points was recorded for each hysteresis loop in order to compare the hysteresis loop shapes of one specimen cycled with different strain amplitudes. Hysteresis half-loops in relative coordinates were smoothed and digitally differentiated and the first and second derivatives of both tensile and compressive half-loops were evaluated. The evolution of the loop shape has been studied using data collected in MTS LCF low cycle fatigue test program. Individual segments containing only 250 points were used for the analysis of the loop shape during constant strain amplitude cycling. Surface relief produced during cyclic straining was studied on specimens mechanically and electrolytically polished. TESCAN Lyra3 XMU FESEM with focused ion beam (FIB) and TESCAN Mira3 XM FESEM scanning electron microscopes were used for surface observations. Internal dislocation structures were studied in thin foils using transmission electron microscope Philips CM12. Thin foils were prepared from the gage length of the specimens employing spark erosion cutting and standard electropolishing. The foil plane was inclined 45° to the specimen axis. Individual grains were oriented using electron diffraction and Kikuchi lines. 3. Analysis of the hysteresis loop Analysis of the hysteresis loop shape is based on the generalized statistical theory of the hysteresis loop [1,9]. In agreement with the original Masing hypothesis [6] the whole volume of the specimen consists of individual microvolumes arranged in parallel. When external strain is applied to the specimen each microvolume carries certain stress depending on its critical shear stress. Critical shear stress of an individual microvolume of the material has two components, internal critical tensile (or compressive) stress ric, and saturated effective stress res. Internal critical stress is the stress at which the microvolume starts yielding without hardening provided the strain rate is infinitesimally low (i.e. effective stress is zero). The saturated effective stress res depends on the applied
plastic strain rate and temperature and is the same in all volumes. The distribution of microvolumes according to their internal critical stresses is characterized by the probability density function f(ric). The probability density distribution function f(ric) determines the frequency of the occurrence of a volume with internal critical stress ric. All microvolumes are supposed to be arranged in parallel. The macroscopic internal stress component can be obtained by integration over all microvolumes. The total stress r as well as the macroscopic internal rI and the macroscopic effective stress rE components are calculated in quasi-elastic approximation. In quasi-elastic approximation the effective stress increases linearly with strain with the slope of the effective elastic modulus Eeff until saturated value of the effective stress res is reached and then becomes constant. More detailed information can be found in the original paper [9] or in a book [1]. The hysteresis loop can be described using the second integral function G(x) of the probability density function of the internal critical stresses defined as
GðxÞ ¼
Z
x
Z
0
z
f ðzÞdz dx:
ð1Þ
0
Using relative stress rr and strain rr the shape of the hysteresis half-loops are given [9]
rr ¼ er Eeff for er 6 2res =Eeff
ð2aÞ
¼ er Eeff 2G er Eeff =2 res for 2res =Eeff 6 er 6 2ea :
ð2bÞ
Relative stress and strain for tensile and compression hysteresis-half-loops are
rr ¼ r þ ra ; er ¼ e þ ea for tensile half-loop;
ð3aÞ
rr ¼ ra r; er ¼ ea e for compression half-loop
ð3bÞ
where ra and ea are the stress and strain amplitudes, respectively. By double differentiation of the relation (2b) we can obtain the relation for the probability density function of the internal critical stresses
f
er Eeff 2
res
¼
2 @ 2 rr E2eff @ e2r
for 2res =Eeff er 2ea
ð4Þ
Relation (4) allows us to evaluate the probability density function f(ric) and also the saturated effective stress res by plotting the second derivative of the hysteresis half-loop (multiplied by 2/ E2eff) vs. fictive stress erEeff/2. The first and the second derivatives of the relation (2a), in quasi-elastic approximation are zero. In reality, due to plastic strain relaxation during unloading the first derivative for very small fictive stresses (erEeff < res) decreases from 2Eeff to Eeff. The second derivative is thus initially positive, large, decreases to zero and should become zero for (res < erEeff < 2res) [5]. The total shift of the probability density function in a plot of the second derivative vs. fictive stress erE/2 is thus res. Effective stress is thus equal to the offset of the probability density function relative to the origin. 4. Results 4.1. Loop analysis in block loading Fig. 1 shows the plot of the stress amplitude vs. number of cycles during cycling at room temperature and at temperature 700 °C with several blocks of suddenly increasing constant strain amplitude. The number of cycles at each level Ni was inversely proportional to the applied total strain amplitude eai so that Ni eai = 20. Room temperature cycling results in cyclic softening at all strain
J. Polák et al. / International Journal of Fatigue 83 (2016) 75–83
Fig. 1. Stress amplitude vs. number of cycles in loading with increasing strain amplitudes at two temperatures.
levels. When the strain amplitude suddenly increased, the stress amplitude increased too but later cycling lead again to cyclic softening. Cyclic straining with identical strain amplitudes at temperature 700 °C results in cyclic hardening at all strain levels. In both cases the saturated levels of the stress amplitude were not achieved and thus the final data points cannot be used for the construction of the true cyclic stress–strain curve. Nevertheless the hysteresis loops recorded at the end of each block of strain amplitudes were plotted in relative coordinates in Fig. 2 and the behavior of the material at two temperatures was compared. The non-Masing behavior, as often denoted [4], is apparent in cycling at both temperatures. It means that the shape of the smaller hysteresis loops cannot be derived from the shape of the largest hysteresis loop. At room temperature the small loops are higher than the corresponding part of the largest loop and at high temperature the small loop is well below the corresponding part of the largest loop. The loops have been analyzed using the statistical theory. We have evaluated the first and second derivatives of the hysteresis half-loops. Fig. 3 shows plot of the second derivatives of the hysteresis half-loops at the end of block run with the strain amplitude 2.5 103 at two temperatures. At both temperatures the second derivative first decreases, reaches minimum, which is close to zero, and the main peak corresponding to the probability density function is displayed. Effective elastic modulus Eeff has been evaluated at each temperature from the plot of the first derivative vs. fictive stress at the value of the fictive stress where second derivative
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reached the first minimum (see Fig. 3). Effective modulus at room temperature was 200 GPa and at temperature 700 °C it was only 130 GPa. Provided low amplitudes are applied, only the dislocations of the primary system in individual grains are activated and both the effective saturated stress and the probability density function can be assessed. Comparing the plots at both temperatures (Fig. 3a and b) we can see that since effective elastic modulus at 700 °C (130 GPa) is much lower than that at room temperature (200 GPa) the fictive stress in Fig. 3b (x axis) extends to lower values than in Fig. 3a. The initial part of the plot of the second derivative in both cases steeply decreases and nearly reaches zero. It corresponds to the rapid relaxation of the effective stress during unloading from maximum compression. The major part of the plot corresponds to the probability density function of the internal critical stresses f(ric). In order to evaluate the saturated effective stress res from the shift of the probability density function relative to the origin, in agreement with Eq. (4), we have approximated the shape of the probability density by the translated Weibull distribution in the form
" b1 b # b xd xd f ðxÞ ¼ exp a a a
ð5Þ
where a is the scale parameter, b the shape parameter and d the location parameter. d is equal to the saturated effective stress res (see Eq. (4)). In the evaluation only the data whose y coordinate was higher than one half of the peak were included in the fitting. All three parameters were evaluated for all tensile hysteresis halfloops shown in Fig. 2 using least squares fitting procedure. The shape parameter was always very close to 2 and therefore in further analysis it was fixed and all fitting were done again with b = 2. Table 1 shows the parameter a corresponding to the maximum of the probability density function and parameter d = res for both temperatures. Probability density function is derived from the plot of the second derivative vs. fictive stress (erE/2) by shifting it in x axis by res. We can thus evaluate the internal critical stress with the highest probability ricM (it corresponds to the maximum of the probability density) from the fictive stress corresponding to the maximum of the second derivative (erE/2)M and from the saturated effective stress res. Both values are also shown in Table 1. 4.2. Evolution of the loop shape in constant amplitude loading Fig. 4 shows the cyclic hardening/softening curves in cycling with low and high amplitudes at room temperature and at
Fig. 2. Hysteresis loops at the end of each block of strain amplitude; (a) room temperature cycling, (b) cycling at temperature 700 °C.
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Fig. 3. Second derivative of the tensile hysteresis half-loop in cycling with strain amplitude 2.5 103; (a) room temperature cycling, (b) cycling at temperature 700 °C.
Table 1 Parameters of the Weibull distribution fitted to the upper part (y coordinate higher than 50% of peak value) of the plot of the second derivative versus erE/2 for two temperatures. T (°C)
a (MPa)
b
res (MPa)
(erE/2)M (MPa)
ricM (MPa)
22 700
145 221
2 2
104 96
207 240
103 144
temperature 700 °C. In both cases room temperature cyclic straining is accompanied by cyclic softening and high temperature cyclic straining results in cyclic hardening. The rate of room temperature cyclic softening and primarily the rate of high temperature cyclic hardening are so high that for both strain amplitudes the final (saturated) stress amplitudes at high temperature are higher than at room temperature. The analysis of the hysteresis loop shape allows us to get deeper insight in the mechanisms of cyclic plastic straining by studying the evolution of the second derivative of the hysteresis half-loops during the fatigue life. In order not to confuse the tensile and compressive half-loops we have denoted all half-loops as segments (sgm). All odd segments are compressive and all even segments are tensile segments. Zero segment corresponding to the first quarter-cycle has not been analyzed. Fig 5 shows the evolution of the second derivative of the hysteresis half-loop (multiplied by 2/ (Eeff)2) with the number of cycles at room temperature during cycling with low strain amplitude and Fig. 6 with high strain
amplitude. The results obtained by the analysis of the tensile half-loops (Figs. 5a and 6a) are in good agreement with the analysis of the compression half-loops (Figs. 5b and 6b). In low amplitude room temperature cyclic straining (Fig. 5) the early hysteresis loops are characterized by the presence of two peaks in the second derivative, one at the fictive stress 250 MPa the other at around 440 MPa, having approximately the same height. During cycling the second peak diminishes, the first peak increases and moreover it is shifted to the smaller fictive stress. It shows that microvolumes with smaller internal critical stresses become more active which corresponds to cyclic softening. Finally the second derivative has only one peak centered at the fictive stress 210 MPa. In room temperature cycling with strain amplitude 5 103 (Fig. 6) the changes of the second derivative are not so pronounced as in cycling with the strain amplitude 2.5 103. The second derivative of the first segments is characterized by the large peak centered at fictive stress 240 MPa and a small peak centered at 630 MPa (the average derived from tensile and compression halfloops). During cyclic loading the second peak disappears and the first peak is slightly shifted to lower fictive stresses. Both these changes correspond to cyclic softening. Changes of the second derivative in high temperature cycling differ substantially from room temperature cycling. The evolution of the shape of the hysteresis loop is documented in Figs 7 and 8. In cycling with low strain amplitude (Fig. 7) the second derivative of the first segments of hysteresis loops is characterized by the large single peak centered at fictive stress 130 MPa. During cycling
Fig. 4. Cyclic hardening/softening curves in cycling at room temperature and temperature 700 °C with two strain amplitudes (a) ea = 2.5 103, (b) ea = 5 103.
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Fig. 5. Evolution of the second derivative of the tensile and compression segments in room temperature cycling with strain amplitude ea = 2.5 103 during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
Fig. 6. Evolution of the second derivative of the tensile and compression segments in room temperature cycling with strain amplitude ea = 5 103 during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
Fig. 7. Evolution of the second derivative of the tensile and compression segments in cycling with strain amplitude ea = 2.5 103 at temperature 700 °C during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
it diminishes, slightly shifts to higher fictive stresses and the second peak centered at 320 MPa appears. Finally single broader peak the centered at 320 MPa can be derived from the analysis of both tensile and compressive half-loops. The evolution of the second derivative in high temperature cycling with high strain amplitude (Fig. 8) is similar to the
evolution of the second derivative during cycling with small strain amplitude (Fig. 7). Initially single pronounced peak centered at 140 MPa diminishes, shifts to the higher fictive stress and a tendency for the formation of the second broader and smaller peak is apparent. These changes are pronounced both in tensile and compression half-loops.
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Fig. 8. Evolution of the second derivative of the tensile and compression segments in room temperature cycling with strain amplitude ea = 5 103 at temperature 700 °C during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
4.3. Dislocation structures Dislocation structures were studied in specimens cycled to fracture. Fig. 9 shows the typical dislocation arrangement in the grains of the specimen fatigued at room temperature to fracture. Dislocation arrangement in a foil prepared from specimen cycled with low strain amplitude ea = 3.5 103 (Fig. 9a) shows very thin parallel dislocation rich bands which are immersed in the matrix nearly free of dislocations. The bands run parallel with the trace of (1 1 1) slip plane. The arrangement witnesses high planarity of slip in this material. Though localization of the cyclic strain was documented by the appearance of the pronounced persistent slip markings (PSMs) on the surface (see Section 4.4) typical dislocation structures corresponding to well-developed cyclic strain localization in persistent slip bands (PSBs) as ladder-like structures were not observed though several foils were inspected. Room temperature cyclic straining with high strain amplitude also revealed dislocation rich bands parallel to the trace of the primary slip plane. However, they are wider and in several locations bands corresponding to cyclic slip localization (PSBs) were found (Fig. 9b). These bands have the width of about 0.5 lm, run parallel to the primary slip plane and are characterized by alternating dislocation rich and dislocation poor areas.
Dislocation structure in the specimen cycled at 700 °C (Fig. 10) differs substantially from that at room temperature. Fig. 10a shows dislocation arrangement in specimen cycled with low strain amplitude to fracture. Dislocation bands parallel to the trace of primary slip plane could be seen, however, they are interconnected by numerous dislocations belonging to other systems than primary slip system. Cross-slip of primary dislocations substantially contributes to the high dislocation density. Cyclic straining with high strain amplitude at 700 °C produces very high dislocation density. Fig. 10b shows dislocations arranged in bands parallel to the trace of the primary (1 1 1) plane. The bands are closely spaced and only seldom single dislocation segments can be identified. 4.4. Surface relief Surface relief observations of specimens cycled at room temperature with strain amplitude 3.5 103 to 5% of fatigue life is shown in Fig. 11. Room temperature cyclic straining leads to the early localization of cyclic plastic strain and production of PSMs on the surface already at 0.1% of fatigue life. Later parallel PSMs develop in the majority of grains. In some grains the faint PSMs from the secondary slip system could be distinguished (see the left
Fig. 9. Dislocation structure in specimens cycled at room temperature to fracture with two strain amplitudes (a) ea = 3.5 10,- (b) ea = 7 103.
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Fig. 10. Dislocation structure in specimens cycled at temperature 700 °C to fracture with two strain amplitudes (a) ea = 3.5 10, (b) ea = 5 103.
section of Fig. 11). PSMs consist mostly of extrusions. In the central section of well-developed PSMs also intrusions, parallel to extrusions develop. The inset of Fig. 11 shows the extrusion of a largest PSM in central part of the grain accompanied on one side by a thin intrusion. Most probably the intrusion is present also on the other side of the extrusion but it is shielded by the central extrusion protruding from the grain under the angle close to 45°. Cyclic straining at high temperature does not result in such a strong localization of cyclic strain as room temperature cyclic straining. Fig. 12 shows the surface of the specimen cycled at 700 °C to 10% of fatigue life. The surface is covered by a thin sheet of the oxide with variable thickness. Preferably grain boundaries are oxidized; oxide sticks out and decorates cracked grain boundaries. Already at 10% of fatigue life the majority of grain boundaries are cracked. On the surface of the grains very faint slip markings in the form of oxidized extrusions can be distinguished. White straight lines in Fig. 12 denote the directions of slip markings in some grains. A few grains contain slip markings corresponding to two slip systems.
Fig. 12. Surface of the specimen cycled with ea = 3.5 103 for 250 cycles (10% Nf) at temperature 700 °C. The inset shows the detail of the cracked oxidized grain boundary. The white lines denote the direction of oxidized slip traces.
5. Discussion
Fig. 11. Parallel PSMs on the surface of the specimen cycled at room temperature with ea = 3.5 103 for 1850 cycles (5% Nf). The inset shows the detail of the central part of PSM consisting of extrusion and parallel intrusion on one side of the extrusion. The intrusion on the other side is probably hidden by the inclined extrusion.
The cyclic plastic response of the Sanicro 25 steel at room and at elevated temperature differs substantially. Room temperature cyclic straining leads to cyclic softening while cyclic straining at temperature 700 °C leads to intensive cyclic hardening. The analysis of the hysteresis loop in terms of the second derivative of the individual half-loops brings further important information concerning the mechanism of cyclic straining and the reasons for cyclic hardening/softening behavior. It allows evaluating the effective stress and the probability density distribution of the internal critical stresses of the microvolumes. Effective stress is found from the shift of the probability density distribution function vs. fictive stress relative to the origin. It could be estimated with reasonable precision provided the probability density function could be approximated by some analytical function. The use of the Weibull distribution with the shape parameter b = 2 allowed to determine the effective stress from the plot of the second derivative vs. fictive stress of the nearly saturated hysteresis loop corresponding to low strain amplitude i. e. when cyclic strain is due to activation of one slip system. Effective stress at room
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temperature and at 700 °C was found approximately the same, 100 MPa. The lower strain rate applied during cycling at temperature 700 °C could result in decrease of the effective stress at this temperature compared to room temperature value. More information on the evolution of the cyclic plastic response gives the evolution of the hysteresis loop shape and additional information concerning the internal dislocation structures and surface relief of cyclically deformed samples. Room temperature cyclic straining with low strain amplitude results in the activation of single slip in majority of the grains. The dislocation structure (Fig. 9a) revealed approximately random distribution of dislocations corresponding to the primary slip system. The probability density function (derived from the second derivative of the half-loop) reveals two distinct peaks and the diminution of the second peak during cyclic loading (Fig. 5). The second peak disappears during cycling and in the domain of saturation of the stress amplitude the first peak dominates. This behavior can be explained by the parallel deformation of the matrix and of the bands of localized slip from which later persistent slip bands (PSBs) develop. The second peak corresponds to the deformation of the matrix which contains nanoclusters [22,23]. Taking the average value of the effective stress 100 MPa the second peak corresponding to the deformation of the matrix with nanoclusters has the maximum of the probability density function f(ric) around internal critical stress ricM,matrix = 340 MPa while the first peak corresponding to the deformation in the PSB has this value around ricM,PSB = 150 MPa. It is natural that during cycling the plastic strain is concentrated into PSBs and the matrix is deformed only elastically. The shift of the first peak of the probability density function to still lower fictive stresses can be related also to the decrease of the effective stress in the PSBs due to complete dissolution of nanoclusters in PSBs. Cyclic loading with high strain amplitudes revealed only very small second peak in the second derivative which disappears rapidly (Fig. 6). The second maximum of the probability density function f(ric), corresponding to the deformation of the matrix is higher than in cycling with low strain amplitude ricM,matrix = 500 MPa. It is due to activation of secondary slip systems. The position of the first maximum corresponding to the deformation in PSBs is identical with that in cycling with small strain amplitude. Also the small shift of the first peak to lover fictive stresses during cyclic straining is similar to that in low amplitude cycling. The appearance of the PSBs in the dislocation structure of a specimen cycled with high strain amplitude (Fig. 10a) supports the present interpretation. In high temperature cyclic straining continuous cyclic hardening proceeds until the tendency to saturation is reached. The comparison of the second derivatives of the hysteresis loops with low strain amplitude (Fig. 7) and with high strain amplitude (Fig. 8) reveals initially one peak centered at 170 MPa. Since the effective stress is only 70 MPa the probability density function has the maximum at ricM = 100 MPa (Fig. 8). With increasing number of cycles the peak decreases and becomes wider. Simultaneously the effective stress increases to 100 MPa. The maximum of the probability density function in saturation is at ricM = 200 MPa. Data for straining with low strain amplitude (Fig. 7) has higher scatter but approximately the same position of the peaks of the second derivative in the first and in the last cycle as in cycling with high strain amplitude is apparent. Dislocation structure in specimens cyclically deformed at temperature 700 °C (Fig. 10) reveals much higher dislocation density than in specimens cycled at room temperature. No signs of pronounced cyclic strain localization has been found, however, in specimens cycled with low and high amplitudes the banding of dislocations in thin bands parallel to the trace of the primary slip plane (see Fig. 10) could be distinguished. The similar features
witness the observation of the surface of the specimens cycled at 700 °C (Fig. 12). No pronounced PSMs appear on the surface; only very fine parallel slip steps (slightly oxidized) could be resolved in individual grains. These observations witness that the main source of the high cyclic stress in specimens deformed at temperature 700 °C is the high density of dislocations produced during cyclic loading. Due to high temperature cross-slip it is more frequent than at room temperature and though principal strain is carried by dislocations of the primary slip system the cross-slipped segments represent important obstacles raising the yield stress above that recorded in room temperature cyclic straining. Though the nanoclusters do not represent important obstacles to the dislocations in the initial cycles, they probably contribute to the crossslipping and thus to the growth of dislocation density. The probability density function is thus during cyclic loading shifted to higher critical internal stresses and material cyclically hardens. 6. Conclusions Study of the evolution of the cyclic stress–strain response of the Sanicro 25 steel during cycling at two temperatures using the analysis of the hysteresis loop shape, the study of dislocation structures and surface observations led to the following conclusions: (i) Room temperature cyclic straining leads to cyclic softening due to localization of the cyclic strain in the PSBs which emerge on the surface as pronounced PSMs. (ii) High temperature cyclic straining results in pronounced cyclic hardening due to rapid growth of dislocation density as a result of enhanced cross-slip assisted by high temperature and by the presence of nanoclusters. (iii) Analysis of the hysteresis loop shape led to the separation of the contribution of the effective and internal stresses. (iv) Two peaks of probability density function in the onset of room temperature cyclic straining point to the presence of nanoclusters. The disappearance of the second peak is connected with the localization of the cyclic strain in PSBs and the dissolution of these nanoclusters in PSBs due to high local plastic strain amplitude. (v) High temperature cyclic straining is characterized by one peak of the probability density function and its shift to higher internal critical stresses during the fatigue life. This shift is linked to the increase of dislocation density and results in pronounced fatigue hardening.
Acknowledgements The support by the grant No. 13-23652S of GACR and the projects RVO: 68081723 is gratefully acknowledged. This work was realized in CEITEC with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. References [1] Polák J. Cyclic plasticity and low cycle fatigue life of metals. Mater Sci Monograph 1991:63. [2] Suresh S. Fatigue of materials. Cambridge: Cambridge University Press; 1998. [3] Evrard P, Alvarez-Armas I, Aubin V, Degallaix S. Polycrystalline modeling of the cyclic hardening/softening behavior of an austenitic–ferritic stainless steel. Mech Mater 2010;42:395–404. [4] Lin B, Zhao LG, Tong J, Christ H-J. Crystal plasticity modeling of cyclic deformation for a polycrystalline nickel-based superalloy at high temperature. Mater Sci Eng, A 2010;527:3581–7. [5] Barrett RA, O’Donoghue PE, Leen SB. A dislocation-based model for high temperature cyclic viscoplasticity of 9–12Cr steels. Comput Mater Sci 2014;92:286–97.
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[16] Skelton RP, Maier HJ, Christ H-J. The Bauschinger effect, Masing model and the Ramberg–Osgood relation for cyclic deformation in metals. Mater Sci Eng, A 1997;238:377–90. [17] Heino S, Karlsson B. Cyclic deformation and fatigue behaviour of 7Mo–0.5N superaustenitic stainless steel – stress–strain relations and fatigue life. Acta Mater 2001;49:339–51. [18] Sivaprasad S, Paul SK, Das A, Narasaiah N, Tarafder S. Cyclic plastic behaviour of primary heat transport piping materials: influence of loading schemes on hysteresis loop. Mater Sci Eng, A 2010;527:6858–69. [19] Mayer T, Mazza E, Holdsworth SR. A continuous Masing approach for a physically motivated formulation of temperature and strain-rate dependent plasticity. Int J Press Vessel 2013;102–103:1–12. [20] Mayer T, Mazza E, Holdsworth SR. Parameter evolution in a continuous Masing approach for cyclic plasticity and its physical interpretation. Mech Mater 2013;57:86–96. [21] Rautio R, Bruce S. Alloy for ultrasupercritical coal fired boilers. Adv Mater Process 2008;166:35–7. [22] Chai G, Bostrom M, Olaison M, Forsberg U. Creep and LCF behaviours of newly developed advanced heat resistant austenitic steels for A-USC. Proc Eng 2013;55:232–9. [23] Polák J, Petráš R, Heczko M, Kubeˇna I, Kruml T, Chai G. Low cycle fatigue behavior of Sanicro 25 steel at room and at elevated temperature. Mater Sci Eng, A 2014;615:175–82.
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Evolution of the cyclic plastic response of Sanicro 25 steel cycled at ambient and elevated temperatures Jaroslav Polák a,b,⇑, Roman Petráš a, Milan Heczko a, Tomáš Kruml a,b, Guocai Chai c,d Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic CEITEC, Institute of Physics of Materials Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic c Sandvik Materials Technology, SE-811 81 Sandviken, Sweden d Linköping University, Engineering Materials, SE-581 83 Linköping, Sweden a
b
a r t i c l e
i n f o
Article history: Received 4 November 2014 Received in revised form 12 March 2015 Accepted 17 March 2015 Available online 24 March 2015 Keywords: Cyclic plasticity Hysteresis loop analysis Heat resistant steel Dislocation structure Effect of temperature
a b s t r a c t Cyclic plastic response of the austenitic heat resistant steel Sanicro 25 has been studied during strain controlled low cycle fatigue tests performed at ambient and at elevated temperature. Simultaneously with the cyclic hardening/softening curves hysteresis loops during cyclic loading were analyzed using generalized statistical theory of the hysteresis loop. The probability density distribution function of the internal critical stresses, the effective saturated stress and their evolution during cycling were derived for various strain amplitudes. The internal dislocation structure and the surface relief at room and at elevated temperature were studied and correlated with the cyclic stress–strain response and the evolution of the probability density function of the internal critical stresses. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Cyclic plastic response of materials is generally studied during strain controlled low cycle fatigue tests with various total or plastic strain amplitudes and constant strain rate. Since complete stress– strain history in cyclic straining is difficult to report due to enormous number of data usually only the plots of the stress amplitude vs. number of cycles i.e. cyclic hardening/softening curves are reported [1,2]. Sometimes selected saturated hysteresis loops are plotted. These data are correlated with the internal structure of the material and its evolution during cyclic loading to assess the mechanisms of cyclic plastic straining and to find the sources of the cyclic stress. Another approach how to understand the mechanisms underlying the cyclic stress–strain response of materials is the modelization of the cyclic plastic straining based on the properties of dislocations in a particular lattice and the knowledge of the grain arrangement of the polycrystalline complex. Some attempts in this domain [3–5] were only partially successful since until now the phenomenon of cyclic strain localization, which is substantial for cyclic straining, has yet been properly modelled neither in single crystals nor in polycrystals. ⇑ Corresponding author at: Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic. E-mail address: [email protected] (J. Polák). http://dx.doi.org/10.1016/j.ijfatigue.2015.03.015 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.
Deeper insight in the mechanisms of cyclic plastic straining represents the approach adopted originally by Masing [6] who proposed to represent small deformations encountered in cyclic straining of polycrystals by parallel arrangement of several volumes of material having different critical yield stresses. This approach leads to natural explanation of the Bauschinger effect during unloading and to the shape of the hysteresis loop in cyclic straining. Afanasev [7] introduced continuous distribution of microvolumes with critical yield stresses characterized by the probability density function and named this approach statistical theory of the hysteresis loop. In order to take into account the effect of time and temperature on the motion of dislocations which carry the cyclic plastic strain Polák and Klesnil [8,9] and Burmeister and Holste [10] proposed generalized statistical theory of the hysteresis loop where also effective stress is considered. The theory has been applied to find the characteristic parameters of the saturated hysteresis loops in single phase [11,12] and double phase materials [13,14]. Number of researchers [15–20] applied statistical theory without considering the contribution of the effective stress. Austenitic heat resistant stainless steel Sandvik Sanicro 25 has been developed for applications at elevated temperatures, namely for construction of supercritical boilers [21,22]. Mostly creep behavior and high temperature corrosion and to a certain extent also low cycle fatigue properties were studied [22]. Recently [23] basic low cycle fatigue behavior of Sanicro 25 steel at ambient and high temperature was reported.
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In this paper generalized statistical theory of the hysteresis loop has been applied to the analysis of the experimental data recorded during cyclic plastic straining of Sanicro 25 steel at ambient and at elevated temperatures. The saturated effective stress and the probability density function of the internal critical stresses simultaneously with its evolution in cyclic straining were evaluated. The changes of the probability density function during cyclic loading contribute to the interpretation of cyclic hardening/softening behavior of the steel at both temperatures. The dislocation structures and surface relief of cyclically deformed Sanicro 25 steel were also studied and revealed pronounced cyclic slip localization. 2. Experimental Experimental material was supplied by Sandvik, Sweden in the form of the cylindrical rod of 150 mm in diameter. The chemical composition of the material in wt.% was: 0.1 C, 22.5 Cr, 25.0 Ni, 3.6 W, 1.5 Co, 3.0 Cu, 0.5 Mn, 0.5 Nb, 0.23 N, 0.2 Si and the rest Fe. Cylindrical specimens for cyclic straining at room and at elevated temperatures were machined with the axis parallel to the rod. The gauge length and the diameter of specimens were 14 mm and 8 mm respectively for room temperature testing and 15 mm and 6 mm for elevated temperature testing. Specimens were homogenized at 1200 °C for 1 h and cooled in air. Afterward final grinding selected specimens were mechanically polished and finally electropolished. Cyclic straining was performed in computer-controlled electrohydraulic MTS system. Fully reversed strain controlled cyclic loading with constant strain rate 5 103 s1 at room temperature and 2 103 s1 at temperature 700 °C was applied. In block loading high number of data points was recorded for each hysteresis loop in order to compare the hysteresis loop shapes of one specimen cycled with different strain amplitudes. Hysteresis half-loops in relative coordinates were smoothed and digitally differentiated and the first and second derivatives of both tensile and compressive half-loops were evaluated. The evolution of the loop shape has been studied using data collected in MTS LCF low cycle fatigue test program. Individual segments containing only 250 points were used for the analysis of the loop shape during constant strain amplitude cycling. Surface relief produced during cyclic straining was studied on specimens mechanically and electrolytically polished. TESCAN Lyra3 XMU FESEM with focused ion beam (FIB) and TESCAN Mira3 XM FESEM scanning electron microscopes were used for surface observations. Internal dislocation structures were studied in thin foils using transmission electron microscope Philips CM12. Thin foils were prepared from the gage length of the specimens employing spark erosion cutting and standard electropolishing. The foil plane was inclined 45° to the specimen axis. Individual grains were oriented using electron diffraction and Kikuchi lines. 3. Analysis of the hysteresis loop Analysis of the hysteresis loop shape is based on the generalized statistical theory of the hysteresis loop [1,9]. In agreement with the original Masing hypothesis [6] the whole volume of the specimen consists of individual microvolumes arranged in parallel. When external strain is applied to the specimen each microvolume carries certain stress depending on its critical shear stress. Critical shear stress of an individual microvolume of the material has two components, internal critical tensile (or compressive) stress ric, and saturated effective stress res. Internal critical stress is the stress at which the microvolume starts yielding without hardening provided the strain rate is infinitesimally low (i.e. effective stress is zero). The saturated effective stress res depends on the applied
plastic strain rate and temperature and is the same in all volumes. The distribution of microvolumes according to their internal critical stresses is characterized by the probability density function f(ric). The probability density distribution function f(ric) determines the frequency of the occurrence of a volume with internal critical stress ric. All microvolumes are supposed to be arranged in parallel. The macroscopic internal stress component can be obtained by integration over all microvolumes. The total stress r as well as the macroscopic internal rI and the macroscopic effective stress rE components are calculated in quasi-elastic approximation. In quasi-elastic approximation the effective stress increases linearly with strain with the slope of the effective elastic modulus Eeff until saturated value of the effective stress res is reached and then becomes constant. More detailed information can be found in the original paper [9] or in a book [1]. The hysteresis loop can be described using the second integral function G(x) of the probability density function of the internal critical stresses defined as
GðxÞ ¼
Z
x
Z
0
z
f ðzÞdz dx:
ð1Þ
0
Using relative stress rr and strain rr the shape of the hysteresis half-loops are given [9]
rr ¼ er Eeff for er 6 2res =Eeff
ð2aÞ
¼ er Eeff 2G er Eeff =2 res for 2res =Eeff 6 er 6 2ea :
ð2bÞ
Relative stress and strain for tensile and compression hysteresis-half-loops are
rr ¼ r þ ra ; er ¼ e þ ea for tensile half-loop;
ð3aÞ
rr ¼ ra r; er ¼ ea e for compression half-loop
ð3bÞ
where ra and ea are the stress and strain amplitudes, respectively. By double differentiation of the relation (2b) we can obtain the relation for the probability density function of the internal critical stresses
f
er Eeff 2
res
¼
2 @ 2 rr E2eff @ e2r
for 2res =Eeff er 2ea
ð4Þ
Relation (4) allows us to evaluate the probability density function f(ric) and also the saturated effective stress res by plotting the second derivative of the hysteresis half-loop (multiplied by 2/ E2eff) vs. fictive stress erEeff/2. The first and the second derivatives of the relation (2a), in quasi-elastic approximation are zero. In reality, due to plastic strain relaxation during unloading the first derivative for very small fictive stresses (erEeff < res) decreases from 2Eeff to Eeff. The second derivative is thus initially positive, large, decreases to zero and should become zero for (res < erEeff < 2res) [5]. The total shift of the probability density function in a plot of the second derivative vs. fictive stress erE/2 is thus res. Effective stress is thus equal to the offset of the probability density function relative to the origin. 4. Results 4.1. Loop analysis in block loading Fig. 1 shows the plot of the stress amplitude vs. number of cycles during cycling at room temperature and at temperature 700 °C with several blocks of suddenly increasing constant strain amplitude. The number of cycles at each level Ni was inversely proportional to the applied total strain amplitude eai so that Ni eai = 20. Room temperature cycling results in cyclic softening at all strain
J. Polák et al. / International Journal of Fatigue 83 (2016) 75–83
Fig. 1. Stress amplitude vs. number of cycles in loading with increasing strain amplitudes at two temperatures.
levels. When the strain amplitude suddenly increased, the stress amplitude increased too but later cycling lead again to cyclic softening. Cyclic straining with identical strain amplitudes at temperature 700 °C results in cyclic hardening at all strain levels. In both cases the saturated levels of the stress amplitude were not achieved and thus the final data points cannot be used for the construction of the true cyclic stress–strain curve. Nevertheless the hysteresis loops recorded at the end of each block of strain amplitudes were plotted in relative coordinates in Fig. 2 and the behavior of the material at two temperatures was compared. The non-Masing behavior, as often denoted [4], is apparent in cycling at both temperatures. It means that the shape of the smaller hysteresis loops cannot be derived from the shape of the largest hysteresis loop. At room temperature the small loops are higher than the corresponding part of the largest loop and at high temperature the small loop is well below the corresponding part of the largest loop. The loops have been analyzed using the statistical theory. We have evaluated the first and second derivatives of the hysteresis half-loops. Fig. 3 shows plot of the second derivatives of the hysteresis half-loops at the end of block run with the strain amplitude 2.5 103 at two temperatures. At both temperatures the second derivative first decreases, reaches minimum, which is close to zero, and the main peak corresponding to the probability density function is displayed. Effective elastic modulus Eeff has been evaluated at each temperature from the plot of the first derivative vs. fictive stress at the value of the fictive stress where second derivative
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reached the first minimum (see Fig. 3). Effective modulus at room temperature was 200 GPa and at temperature 700 °C it was only 130 GPa. Provided low amplitudes are applied, only the dislocations of the primary system in individual grains are activated and both the effective saturated stress and the probability density function can be assessed. Comparing the plots at both temperatures (Fig. 3a and b) we can see that since effective elastic modulus at 700 °C (130 GPa) is much lower than that at room temperature (200 GPa) the fictive stress in Fig. 3b (x axis) extends to lower values than in Fig. 3a. The initial part of the plot of the second derivative in both cases steeply decreases and nearly reaches zero. It corresponds to the rapid relaxation of the effective stress during unloading from maximum compression. The major part of the plot corresponds to the probability density function of the internal critical stresses f(ric). In order to evaluate the saturated effective stress res from the shift of the probability density function relative to the origin, in agreement with Eq. (4), we have approximated the shape of the probability density by the translated Weibull distribution in the form
" b1 b # b xd xd f ðxÞ ¼ exp a a a
ð5Þ
where a is the scale parameter, b the shape parameter and d the location parameter. d is equal to the saturated effective stress res (see Eq. (4)). In the evaluation only the data whose y coordinate was higher than one half of the peak were included in the fitting. All three parameters were evaluated for all tensile hysteresis halfloops shown in Fig. 2 using least squares fitting procedure. The shape parameter was always very close to 2 and therefore in further analysis it was fixed and all fitting were done again with b = 2. Table 1 shows the parameter a corresponding to the maximum of the probability density function and parameter d = res for both temperatures. Probability density function is derived from the plot of the second derivative vs. fictive stress (erE/2) by shifting it in x axis by res. We can thus evaluate the internal critical stress with the highest probability ricM (it corresponds to the maximum of the probability density) from the fictive stress corresponding to the maximum of the second derivative (erE/2)M and from the saturated effective stress res. Both values are also shown in Table 1. 4.2. Evolution of the loop shape in constant amplitude loading Fig. 4 shows the cyclic hardening/softening curves in cycling with low and high amplitudes at room temperature and at
Fig. 2. Hysteresis loops at the end of each block of strain amplitude; (a) room temperature cycling, (b) cycling at temperature 700 °C.
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Fig. 3. Second derivative of the tensile hysteresis half-loop in cycling with strain amplitude 2.5 103; (a) room temperature cycling, (b) cycling at temperature 700 °C.
Table 1 Parameters of the Weibull distribution fitted to the upper part (y coordinate higher than 50% of peak value) of the plot of the second derivative versus erE/2 for two temperatures. T (°C)
a (MPa)
b
res (MPa)
(erE/2)M (MPa)
ricM (MPa)
22 700
145 221
2 2
104 96
207 240
103 144
temperature 700 °C. In both cases room temperature cyclic straining is accompanied by cyclic softening and high temperature cyclic straining results in cyclic hardening. The rate of room temperature cyclic softening and primarily the rate of high temperature cyclic hardening are so high that for both strain amplitudes the final (saturated) stress amplitudes at high temperature are higher than at room temperature. The analysis of the hysteresis loop shape allows us to get deeper insight in the mechanisms of cyclic plastic straining by studying the evolution of the second derivative of the hysteresis half-loops during the fatigue life. In order not to confuse the tensile and compressive half-loops we have denoted all half-loops as segments (sgm). All odd segments are compressive and all even segments are tensile segments. Zero segment corresponding to the first quarter-cycle has not been analyzed. Fig 5 shows the evolution of the second derivative of the hysteresis half-loop (multiplied by 2/ (Eeff)2) with the number of cycles at room temperature during cycling with low strain amplitude and Fig. 6 with high strain
amplitude. The results obtained by the analysis of the tensile half-loops (Figs. 5a and 6a) are in good agreement with the analysis of the compression half-loops (Figs. 5b and 6b). In low amplitude room temperature cyclic straining (Fig. 5) the early hysteresis loops are characterized by the presence of two peaks in the second derivative, one at the fictive stress 250 MPa the other at around 440 MPa, having approximately the same height. During cycling the second peak diminishes, the first peak increases and moreover it is shifted to the smaller fictive stress. It shows that microvolumes with smaller internal critical stresses become more active which corresponds to cyclic softening. Finally the second derivative has only one peak centered at the fictive stress 210 MPa. In room temperature cycling with strain amplitude 5 103 (Fig. 6) the changes of the second derivative are not so pronounced as in cycling with the strain amplitude 2.5 103. The second derivative of the first segments is characterized by the large peak centered at fictive stress 240 MPa and a small peak centered at 630 MPa (the average derived from tensile and compression halfloops). During cyclic loading the second peak disappears and the first peak is slightly shifted to lower fictive stresses. Both these changes correspond to cyclic softening. Changes of the second derivative in high temperature cycling differ substantially from room temperature cycling. The evolution of the shape of the hysteresis loop is documented in Figs 7 and 8. In cycling with low strain amplitude (Fig. 7) the second derivative of the first segments of hysteresis loops is characterized by the large single peak centered at fictive stress 130 MPa. During cycling
Fig. 4. Cyclic hardening/softening curves in cycling at room temperature and temperature 700 °C with two strain amplitudes (a) ea = 2.5 103, (b) ea = 5 103.
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Fig. 5. Evolution of the second derivative of the tensile and compression segments in room temperature cycling with strain amplitude ea = 2.5 103 during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
Fig. 6. Evolution of the second derivative of the tensile and compression segments in room temperature cycling with strain amplitude ea = 5 103 during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
Fig. 7. Evolution of the second derivative of the tensile and compression segments in cycling with strain amplitude ea = 2.5 103 at temperature 700 °C during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
it diminishes, slightly shifts to higher fictive stresses and the second peak centered at 320 MPa appears. Finally single broader peak the centered at 320 MPa can be derived from the analysis of both tensile and compressive half-loops. The evolution of the second derivative in high temperature cycling with high strain amplitude (Fig. 8) is similar to the
evolution of the second derivative during cycling with small strain amplitude (Fig. 7). Initially single pronounced peak centered at 140 MPa diminishes, shifts to the higher fictive stress and a tendency for the formation of the second broader and smaller peak is apparent. These changes are pronounced both in tensile and compression half-loops.
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Fig. 8. Evolution of the second derivative of the tensile and compression segments in room temperature cycling with strain amplitude ea = 5 103 at temperature 700 °C during the fatigue life (a) tensile half-loops (segments), (b) compression half-loops (segments).
4.3. Dislocation structures Dislocation structures were studied in specimens cycled to fracture. Fig. 9 shows the typical dislocation arrangement in the grains of the specimen fatigued at room temperature to fracture. Dislocation arrangement in a foil prepared from specimen cycled with low strain amplitude ea = 3.5 103 (Fig. 9a) shows very thin parallel dislocation rich bands which are immersed in the matrix nearly free of dislocations. The bands run parallel with the trace of (1 1 1) slip plane. The arrangement witnesses high planarity of slip in this material. Though localization of the cyclic strain was documented by the appearance of the pronounced persistent slip markings (PSMs) on the surface (see Section 4.4) typical dislocation structures corresponding to well-developed cyclic strain localization in persistent slip bands (PSBs) as ladder-like structures were not observed though several foils were inspected. Room temperature cyclic straining with high strain amplitude also revealed dislocation rich bands parallel to the trace of the primary slip plane. However, they are wider and in several locations bands corresponding to cyclic slip localization (PSBs) were found (Fig. 9b). These bands have the width of about 0.5 lm, run parallel to the primary slip plane and are characterized by alternating dislocation rich and dislocation poor areas.
Dislocation structure in the specimen cycled at 700 °C (Fig. 10) differs substantially from that at room temperature. Fig. 10a shows dislocation arrangement in specimen cycled with low strain amplitude to fracture. Dislocation bands parallel to the trace of primary slip plane could be seen, however, they are interconnected by numerous dislocations belonging to other systems than primary slip system. Cross-slip of primary dislocations substantially contributes to the high dislocation density. Cyclic straining with high strain amplitude at 700 °C produces very high dislocation density. Fig. 10b shows dislocations arranged in bands parallel to the trace of the primary (1 1 1) plane. The bands are closely spaced and only seldom single dislocation segments can be identified. 4.4. Surface relief Surface relief observations of specimens cycled at room temperature with strain amplitude 3.5 103 to 5% of fatigue life is shown in Fig. 11. Room temperature cyclic straining leads to the early localization of cyclic plastic strain and production of PSMs on the surface already at 0.1% of fatigue life. Later parallel PSMs develop in the majority of grains. In some grains the faint PSMs from the secondary slip system could be distinguished (see the left
Fig. 9. Dislocation structure in specimens cycled at room temperature to fracture with two strain amplitudes (a) ea = 3.5 10,- (b) ea = 7 103.
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Fig. 10. Dislocation structure in specimens cycled at temperature 700 °C to fracture with two strain amplitudes (a) ea = 3.5 10, (b) ea = 5 103.
section of Fig. 11). PSMs consist mostly of extrusions. In the central section of well-developed PSMs also intrusions, parallel to extrusions develop. The inset of Fig. 11 shows the extrusion of a largest PSM in central part of the grain accompanied on one side by a thin intrusion. Most probably the intrusion is present also on the other side of the extrusion but it is shielded by the central extrusion protruding from the grain under the angle close to 45°. Cyclic straining at high temperature does not result in such a strong localization of cyclic strain as room temperature cyclic straining. Fig. 12 shows the surface of the specimen cycled at 700 °C to 10% of fatigue life. The surface is covered by a thin sheet of the oxide with variable thickness. Preferably grain boundaries are oxidized; oxide sticks out and decorates cracked grain boundaries. Already at 10% of fatigue life the majority of grain boundaries are cracked. On the surface of the grains very faint slip markings in the form of oxidized extrusions can be distinguished. White straight lines in Fig. 12 denote the directions of slip markings in some grains. A few grains contain slip markings corresponding to two slip systems.
Fig. 12. Surface of the specimen cycled with ea = 3.5 103 for 250 cycles (10% Nf) at temperature 700 °C. The inset shows the detail of the cracked oxidized grain boundary. The white lines denote the direction of oxidized slip traces.
5. Discussion
Fig. 11. Parallel PSMs on the surface of the specimen cycled at room temperature with ea = 3.5 103 for 1850 cycles (5% Nf). The inset shows the detail of the central part of PSM consisting of extrusion and parallel intrusion on one side of the extrusion. The intrusion on the other side is probably hidden by the inclined extrusion.
The cyclic plastic response of the Sanicro 25 steel at room and at elevated temperature differs substantially. Room temperature cyclic straining leads to cyclic softening while cyclic straining at temperature 700 °C leads to intensive cyclic hardening. The analysis of the hysteresis loop in terms of the second derivative of the individual half-loops brings further important information concerning the mechanism of cyclic straining and the reasons for cyclic hardening/softening behavior. It allows evaluating the effective stress and the probability density distribution of the internal critical stresses of the microvolumes. Effective stress is found from the shift of the probability density distribution function vs. fictive stress relative to the origin. It could be estimated with reasonable precision provided the probability density function could be approximated by some analytical function. The use of the Weibull distribution with the shape parameter b = 2 allowed to determine the effective stress from the plot of the second derivative vs. fictive stress of the nearly saturated hysteresis loop corresponding to low strain amplitude i. e. when cyclic strain is due to activation of one slip system. Effective stress at room
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temperature and at 700 °C was found approximately the same, 100 MPa. The lower strain rate applied during cycling at temperature 700 °C could result in decrease of the effective stress at this temperature compared to room temperature value. More information on the evolution of the cyclic plastic response gives the evolution of the hysteresis loop shape and additional information concerning the internal dislocation structures and surface relief of cyclically deformed samples. Room temperature cyclic straining with low strain amplitude results in the activation of single slip in majority of the grains. The dislocation structure (Fig. 9a) revealed approximately random distribution of dislocations corresponding to the primary slip system. The probability density function (derived from the second derivative of the half-loop) reveals two distinct peaks and the diminution of the second peak during cyclic loading (Fig. 5). The second peak disappears during cycling and in the domain of saturation of the stress amplitude the first peak dominates. This behavior can be explained by the parallel deformation of the matrix and of the bands of localized slip from which later persistent slip bands (PSBs) develop. The second peak corresponds to the deformation of the matrix which contains nanoclusters [22,23]. Taking the average value of the effective stress 100 MPa the second peak corresponding to the deformation of the matrix with nanoclusters has the maximum of the probability density function f(ric) around internal critical stress ricM,matrix = 340 MPa while the first peak corresponding to the deformation in the PSB has this value around ricM,PSB = 150 MPa. It is natural that during cycling the plastic strain is concentrated into PSBs and the matrix is deformed only elastically. The shift of the first peak of the probability density function to still lower fictive stresses can be related also to the decrease of the effective stress in the PSBs due to complete dissolution of nanoclusters in PSBs. Cyclic loading with high strain amplitudes revealed only very small second peak in the second derivative which disappears rapidly (Fig. 6). The second maximum of the probability density function f(ric), corresponding to the deformation of the matrix is higher than in cycling with low strain amplitude ricM,matrix = 500 MPa. It is due to activation of secondary slip systems. The position of the first maximum corresponding to the deformation in PSBs is identical with that in cycling with small strain amplitude. Also the small shift of the first peak to lover fictive stresses during cyclic straining is similar to that in low amplitude cycling. The appearance of the PSBs in the dislocation structure of a specimen cycled with high strain amplitude (Fig. 10a) supports the present interpretation. In high temperature cyclic straining continuous cyclic hardening proceeds until the tendency to saturation is reached. The comparison of the second derivatives of the hysteresis loops with low strain amplitude (Fig. 7) and with high strain amplitude (Fig. 8) reveals initially one peak centered at 170 MPa. Since the effective stress is only 70 MPa the probability density function has the maximum at ricM = 100 MPa (Fig. 8). With increasing number of cycles the peak decreases and becomes wider. Simultaneously the effective stress increases to 100 MPa. The maximum of the probability density function in saturation is at ricM = 200 MPa. Data for straining with low strain amplitude (Fig. 7) has higher scatter but approximately the same position of the peaks of the second derivative in the first and in the last cycle as in cycling with high strain amplitude is apparent. Dislocation structure in specimens cyclically deformed at temperature 700 °C (Fig. 10) reveals much higher dislocation density than in specimens cycled at room temperature. No signs of pronounced cyclic strain localization has been found, however, in specimens cycled with low and high amplitudes the banding of dislocations in thin bands parallel to the trace of the primary slip plane (see Fig. 10) could be distinguished. The similar features
witness the observation of the surface of the specimens cycled at 700 °C (Fig. 12). No pronounced PSMs appear on the surface; only very fine parallel slip steps (slightly oxidized) could be resolved in individual grains. These observations witness that the main source of the high cyclic stress in specimens deformed at temperature 700 °C is the high density of dislocations produced during cyclic loading. Due to high temperature cross-slip it is more frequent than at room temperature and though principal strain is carried by dislocations of the primary slip system the cross-slipped segments represent important obstacles raising the yield stress above that recorded in room temperature cyclic straining. Though the nanoclusters do not represent important obstacles to the dislocations in the initial cycles, they probably contribute to the crossslipping and thus to the growth of dislocation density. The probability density function is thus during cyclic loading shifted to higher critical internal stresses and material cyclically hardens. 6. Conclusions Study of the evolution of the cyclic stress–strain response of the Sanicro 25 steel during cycling at two temperatures using the analysis of the hysteresis loop shape, the study of dislocation structures and surface observations led to the following conclusions: (i) Room temperature cyclic straining leads to cyclic softening due to localization of the cyclic strain in the PSBs which emerge on the surface as pronounced PSMs. (ii) High temperature cyclic straining results in pronounced cyclic hardening due to rapid growth of dislocation density as a result of enhanced cross-slip assisted by high temperature and by the presence of nanoclusters. (iii) Analysis of the hysteresis loop shape led to the separation of the contribution of the effective and internal stresses. (iv) Two peaks of probability density function in the onset of room temperature cyclic straining point to the presence of nanoclusters. The disappearance of the second peak is connected with the localization of the cyclic strain in PSBs and the dissolution of these nanoclusters in PSBs due to high local plastic strain amplitude. (v) High temperature cyclic straining is characterized by one peak of the probability density function and its shift to higher internal critical stresses during the fatigue life. This shift is linked to the increase of dislocation density and results in pronounced fatigue hardening.
Acknowledgements The support by the grant No. 13-23652S of GACR and the projects RVO: 68081723 is gratefully acknowledged. This work was realized in CEITEC with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. References [1] Polák J. Cyclic plasticity and low cycle fatigue life of metals. Mater Sci Monograph 1991:63. [2] Suresh S. Fatigue of materials. Cambridge: Cambridge University Press; 1998. [3] Evrard P, Alvarez-Armas I, Aubin V, Degallaix S. Polycrystalline modeling of the cyclic hardening/softening behavior of an austenitic–ferritic stainless steel. Mech Mater 2010;42:395–404. [4] Lin B, Zhao LG, Tong J, Christ H-J. Crystal plasticity modeling of cyclic deformation for a polycrystalline nickel-based superalloy at high temperature. Mater Sci Eng, A 2010;527:3581–7. [5] Barrett RA, O’Donoghue PE, Leen SB. A dislocation-based model for high temperature cyclic viscoplasticity of 9–12Cr steels. Comput Mater Sci 2014;92:286–97.
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