Pseudoelastic fatigue of CuZnAl single crystals: the effect of concomitant diffusional processes

Materials Science and Engineering A290 (2000) 108 – 121 www.elsevier.com/locate/msea

Pseudoelastic fatigue of CuZnAl single crystals: the effect of concomitant diffusional processes A. Yawny, F.C. Lovey, M. Sade * Centro Ato´mico Bariloche, Comisio´n Nacional de Energı´a Ato´mica, 8400 San Carlos de Bariloche, Argentina Received 24 December 1999; received in revised form 28 March 2000

Abstract Experiments have been performed in CuZnAl single crystals in order to further understand the evolution of material properties after continuous pseudoelastic cycling. In a fatigue experiment of this type, permanent and recoverable effects are observed for temperatures above 273 K. Experiments in the stress – strain – time space have been designed in order to separate both contributions. The occurrence of recoverable changes can be related to the ordering of the beta-phase and to the stabilization of the martensite, both depending in turn on the atomic diffusion phenomena in these alloys above 273 K. The kinetics of these processes show a considerable increase of the associated time constant after the pseudoelastic cycling procedure. The relation of these changes with the microstructural evolution of the material is analyzed and a discussion is offered on the role that each mechanism, either diffusive or due to microstructural changes, plays on fatigue. The relevance of defining a reference state in order to determine the consequences of pseudoelastic cycling is shown. A model which considers the stabilization of martensite and the beta recovery has been used to reproduce the stress deformation behavior after cycling. An enhancement of stabilization kinetics during fatigue has allowed us to obtain a closer fit with the experimental results. © 2000 Elsevier Science S.A. All rights reserved. Keywords: CuZnAl; Pseudoelastic cycling; Stabilization; Diffusion; Fatigue

1. Introduction Results related to pseudoelastic fatigue in CuZnAl single crystals have been recently presented [1–3]. In the work of Malarrı´a et al. [3] attention was paid to the mechanical behavior at low temperatures (liquid nitrogen). A strong correlation was found between the mechanical and the microstructural evolution. An increase in the pseudoelastic hysteresis and a strong hardening associated with the slope of the transformation were found. These effects were explained by the formation of bulk defects parallel to the basal plane of the martensite, named basal plane defects (BPD). It should be noticed that at the reported low temperature tests diffusion is greatly inhibited and no mechanical recovery was observed in these experiments. However, a partial recovery in the pseudoelastic behavior at low temperature was measured after an up-aging at 323 K [3]. * Corresponding author. Tel.: +54-2944-445265; fax: 54-2944445299. E-mail address: [email protected] (M. Sade).

Pseudoelastic experiments performed at 305 K [2] have shown that the stress to transform decreases with cycling and tends to recover the original value after aging several hours at the test temperature. However, the amount of the recovery remains unprecisely characterized as can be seen in figure 1 in Ref. [4]. In that work, the stress to start the transformation (t b − M) increases during aging at the experimental temperature (Texp) after the pseudoelastic cycling experiments performed between Texp = 301 and 357 K. The samples studied in Ref. [4] were subjected to a thermal treatment consisting in holding them at 1123 K for 20 min, followed by air quenching to room temperature (RT), where they were kept several days before taking them to the test temperature Texp. The magnitude of the changes in the transformation stresses, the magnitude and sign of the hysteresis evolution during cycling and the subsequent recovery are strongly dependent on the previous history of the sample: heat treatment, time elapsed after heat treatment and temperature of aging before the fatigue experiment. Additionally, those changes are not constant throughout the whole trans-

0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 0 9 2 5 - 4

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

109

Table 1 Samples used in this worka Sample

Cu (wt.%)

Zn (wt.%)

Al (wt.%)

Ms (K)

f (mm)

L (mm)

m

a1 a2 b1 b2 c1 c2

76.02 76.02 75.53 75.53 75.85 75.85

16.66 16.06 16.70 16.70 16.27 16.27

7.93 7.93 7.77 7.77 7.89 7.89

293 293 278 278 288 288

3.00 2.88 2.96 3.04 2.80 2.80

12 12 11.8 21.3 8.0 8.0

0.33 0.33 0.434 0.434 0.48 0.48

a

Samples with the same letter and different number belong to the same single crystal. Nominal composition in wt.%, nominal Ms, diameter, central length and Schmid factor corresponding to the stress induced variant are shown.

formation cycle, i.e. although a decrease in the stress to start the transformation was always present, a hardening in the subsequent transformation curve was sometimes observed, thus leading to an increase of the stress to transform at higher transformed fractions [4]. For the sake of clarity in this paper we distinguish between permanent and recoverable effects. Several examples of permanent defects have been detected and shown in the literature, such as the BPD already mentioned and surface defects parallel to the habit plane of the transformation, named habit plane defects (HPD) [5]. The HPD evolve leading to microcrak formation, playing a significant role on the fatigue life of the material [5]. The recoverable effect is relevant in the temperature range at which diffusion is present. CuZnAl alloys are particularly suitable for this analysis since diffusion is present at temperatures as low as 273 K [6,7]. The interaction of the diffusive mechanisms with permanent defects is a significant question if fatigue behavior is to be understood. The aim of this work is to go a step further in the analysis of the different phenomena involved in pseudoelastic fatigue at temperatures at which diffusion is present both in the beta phase and in the martensite. CuZnAl single crystals with an electron concentration e/a = 1.48 have been selected for this study.

the transformation shear direction are shown in Table 1. The thermal treatment used in this work was as follows: samples were kept at 1123 K for 20 min, air quenched to room temperature (RT) and immediately taken to the experimental temperature Texp which was constant during the different stages of the experiment as will be explained below. In all the experiments the specimens reached the desired temperature, Texp, in less than 120 min from the air quenching. Special care was taken to avoid the samples to transform to martensite during this procedure. A Universal Testing Machine Instron 1123 with an Environmental Chamber Instron 3110 was used for the testing experiments. The temperature of the sample was continuously monitored by a thermocouple in good thermal contact with the sample or spot welded to it. The working temperature Texp was kept constant within a fluctuation smaller than 0.25 K during each experiment, which lasted between 1 and 2 weeks. In order to select the crosshead speed to be used, so as to follow the mechanical evolution of the samples, tests were performed at successive decreasing crosshead speeds. The hysteresis of the pseudoelastic transforma-

2. Experimental CuZnAl alloys were melted from pure metals (99.999%). The composition of the alloys used in this work is shown in Table 1. The samples are identified with a letter indicating the single crystal from which they were prepared and numbers are used to distinguish different samples from the same crystal. Single crystals were prepared by a modified Bridgman technique. Cylindrical tensile samples were prepared by spark-erosion. The orientation of the samples was determined by the Laue Technique and the tensile axes are shown in Fig. 1. The diameter of the samples, the gauge length and the Schmid factors relative to the habit plane and

Fig. 1. Orientation of the tensile axes of the used samples. The name of the samples are indicated in Table 1. Samples with the same starting letter belong to the same single crystal. a ( ); b (); c ( + ).

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

110

complete experiment is presented, although more details will be given together with the obtained results.

2.1. Stage 1: reaching the asymptotic state stage

Fig. 2. Diagram of the experimental sequence followed for the samples in this work. After keeping the samples at 1123 K for 20 min they were air cooled and put into the temperature chamber of the testing machine. For further reference, the origin of the time axis was taken at the moment of taking the sample out from the thermal treatment furnace. The lowest reached temperature was in between room temperature (TR) and the experimental temperature (Texp). Reaching a stable Texp value took less than 120 min in most of the cases. After that, the samples were subjected to the different stages described in the text. Table 2 The temperature for all the experimental stages is showna Sample

a1 a2 a2r b1 b1r b2 c1 c2

Texp (K)

334.0 334.0 333.5 333.5 333.5 333.5 332.0 332.5

Stages 1

2

3

4

5

Y Y Y Y Y Y Y Y

Y Y – Y Y Y Y –

Y Y – Y Y Y Y –

– – – Y – – – Y

– – – – – – Y

a A complete sequence includes five stages. A Y indicates which stage was followed by each sample. Samples a2r and b1r are the same a2 and b1 which were used in a second experimental sequence with an intermediate annealing in between (20 min at 1073 K and air cooling).

tion cycle decreases if the crosshead speed decreases until a critical value is reached. For lower values no change in the hysteresis width due to the latent heat release is detected. Thus, the results to be obtained are free from thermal effects. This is an important experimental fact in order to study the actual changes behind the hysteresis evolution during the pseudoelastic cycling experiments. In this way thermal effects do not affect the obtained results. In our case 0.1 mm min − 1 was the chosen crosshead speed value to follow the mechanical evolution. For convenience, the experimental sequence can be divided in several stages as schematically shown in Fig. 2. This does not necessarily mean that all samples were subjected to the whole sequence. Table 2 shows the sequence followed by each sample at the constant temperature Texp. A short summary of the sequence for a

Once the sample reaches Texp the evolution of the transformation and retransformation stresses are followed just to obtain an asymptotic condition which could be considered a reference point for further experiments. This was done by performing a single pseudoelastic cycle at several time intervals. This is the so called ‘reaching the asymptotic state stage’. Point A in Fig. 2 shows the end of this stage. To describe the evolution, the start of the time axis was considered at the moment of the air quench.

2.2. Stage 2: static stabilization stage Samples are partially transformed, and stabilization of the transformed martensite is allowed during a time interval tst. After this, the transformation is continued until 100% of the sample gauge length is transformed and then unloaded as in a normal cycle. The increase in the hysteresis of the stabilized martensite quantitatively indicates the amount of stabilization. This stage is named ‘static stabilization’ to distinguish it from a possible stabilization effect during cycling performed at higher frequencies, which could show different features. More details about the procedure to analyze stabilization effects can be found in Ref. [6].

2.3. Stage 3: the beta phase reco6ery after static stabilization The beta phase recovers after the static stabilization of the martensite. This process can be quantified by doing single cycles as was mentioned in Stage 1. This stage ends when the material reaches again the reference state.

2.4. Stage 4: pseudoelastic fatigue stage Pseudoelastic fatigue is now performed up to a definite number of cycles. Parameters of the cycling experiments are shown in Table 3. In this stage a higher crosshead speed than the one used to follow the evolution was used.

2.5. Stage 5: beta phase reco6ery after fatigue Following the same procedure as in Stages 1 and 3, the recovery of the beta phase after fatigue is followed and the kinetics of the process analyzed. An additional stabilization experiment with the subsequent recovery was performed in a sample after Stage 5. In what follows, a constant ratio r=(dt/dT)$1

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

111

Table 3 Experimental parameters of the cycling stage: crosshead speed, frequency n, number of cycles, resolved shear stress to transform from beta to 18R for cycle 1, maximum resolved shear stress during the whole cycling and maximum strain during each cycle Sample

Crosshead speed (mm min−1)

n (Hz)

N

tb “m (MPa)

tM (MPa)

o b−m (%)

b1 c2

10 10

0.12 0.12

5180 5000

42.1 49.0

44.2 50.0

5.2 6.4

MPa K − 1 (Clausius – Clapeyron) is considered for the relation between the resolved shear stress and the temperature to transform from the b-phase to the martensite for CuZnAl single crystals at the temperature range of the present experiments [8]. The structure of the stress induced martensite is 18R for all samples used in this work.

3. Results The results are presented in the sequence of stages previously described. The variation of certain characteristics of the pseudoelastic cycles are used to describe the evolution along the different stages. Typical descriptors used here are the resolved shear stress to transform and retransform from the b-phase to the 18 R-martensite phase t b − M and t M − b, the hysteresis of the transformation Dt and the maximum resolved shear stress at the end of the transformation t max. The first three values can be determined at different transformed fractions and can be converted to temperature values by using the r-value mentioned above. Resolved shear stresses are obtained as usual as t=m·s, where s is the applied stress and m is the corresponding Schmid factor of the stress induced variant for each sample.

transformation used in this experiment. This elongation value is indicated with dashed lines in the Fig. 3(a). It can be noticed that for Texp = 333 K and crosshead speed 0. 1 mm min − 1 a decrease in t b − M and an increase in t M − b are detected for the successive cycles. Both shifts show different magnitudes being the decrease of the stress to transform smaller than the increase in the stress for retransforming for the mentioned experimental conditions. A final resolved shear stress hysteresis Dt = 0.8 MPa, equivalent to a thermal hysteresis of DT = 0.8 K is obtained, corresponding to cycle R of Fig. 3(b). It must be taken into account that t b − M does not depend on the crosshead speed, if this is low enough. However, t M − b and conse-

3.1. Stage 1 As an example Fig. 3 shows results from sample a2 (Texp = 333 K). Pseudoelastic cycles recorded at different time intervals after the sample has reached Texp are represented. The resolved shear stress t versus the strain o referred to the initial length of the samples is shown and only three cycles are plotted for the sake of clarity. Cycles E, F and R correspond to t = 60, 140 and 3128 min, respectively. Several features can be considered from Fig. 3(a). The hysteresis corresponding to the pseudoelastic transition strongly decreases as time evolves, being the shape of the hysteresis cycle markedly triangular. The widest hysteresis always corresponds to the low-displacement zone of the curve which corresponds with the first transforming-last retransforming part of the sample. The evolution of the critical resolved shear stress as a function of time is shown in Fig. 3(b). The plotted t values correspond to an elongation of 10% of the total strain due to the

Fig. 3. Mechanical evolution during Stage 1 for sample a2, crosshead speed =0. 1 mm min − 1 and Texp =333 K. (a) t – o curves obtained for three different time intervals after the air quench: cycle E (60 min), cycle F (140 min) and cycle R (3128 min). Cycle R corresponds to the asymptotic state; (b) The evolution of the critical resolved shear stresses as a function of time for the same sample as Fig. 3 (a), values corresponding to cycles E, F and R also shown in this figure.

112

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

3.2. Stage 2

Fig. 4. Dt=t b − M − t M − b versus time at Texp $ 333 K are plotted for different samples. Values of t b − M and t M − b obtained at 10% of o b − M. Using values of sample a2 a double exponential fit of the curve is drawn for guiding the eye, the corresponding time constants are 198 and 942 min. The exact experimental temperature for each specimen is shown in Table 2. Sample a2r is the same one as sample a2, with a new performed thermal treatment and experimental sequence. Points corresponding to samples a2r and b1 were shifted in the time axis to emphasize the kinetics of the mechanisms.

quently Dt are dependent on the crosshead speed. An analysis of the detected behavior will be postponed to the discussion. Similar experiments for several samples at nearly the same test temperature show the same characteristics for the stresses evolution. The points Dt versus t for the different samples at Texp =333 K are plotted in Fig. 4. Because of some differences among the time involved in the heat treatments of the samples, the whole points do not fall on the same line. However by applying a shift in the time axis, the evolution for all the samples can be superimposed and this is an evidence of a similar kinetics for all the cases, as it is noticeable from the figure. Using as an example the time evolution of Dt obtained for sample a2 (see Fig. 4), a single exponential fit analysis gives a time constant of t1 =281 min. However a fit with a double exponential function as plotted in Fig. 4 better adjusts to the experimental points. The corresponding time constants are 198 and 942 min. It is here emphasized that, within the experimental resolution, no additional changes were detected in t b − M or t M − b after 2 d (2880 min) at Texp = 333 K for the samples used in this work. This fact is important in order to further distinguish among different contributions to the observed mechanical evolution after fatigue. As an example the asymptotic values measured at 10% of the transformed fraction are t b − M = 34.8 and t M − b =34.0 MPa for sample a2 at Texp =333 K. At the end of the first stage, the sample is at point A of Fig. 2(a). This state is considered as a reference state from which a further evolution is measured. This will be strongly justified taking into account the experimental evidence that follows for the next stages.

A typical result is shown in Fig. 5(a) for sample b1. In this figure, two cycles are shown: one corresponds to the reference state previously defined and the other shows the path followed by the sample during the static stabilization experiment. The sample was stressed from the unloaded condition up to point b and the crosshead position was kept constant during 1060 min. A decrease in the stress with time was observed. Then the sample was loaded to go on transforming (point d) up to the end (point e) and then completely retransformed by going back to the unloaded condition. Two features are noticeable from the figure: 1. The obtained decrease in the retransforming stress for the stabilized part of the samples was Dt M − b = + 4 MPa which corresponds to a stress free transformation temperature shift (DAs) of approximately 4 K. This has been found to be the maximum obtained value in all the samples analyzed at Texp = 333 K. The amount of the stabilization was determined at a deformation corresponding to 10% of transformation. 2. After the stabilization was stopped and further transformation of the sample reinitiated (from 50 to 100%), a decrease between 0. 5 and 1 MPa in t b − M, in comparison with the asymptotic cycle of Stage 1, is detected in the region of the curve corresponding to the portion of the sample which had been kept in the beta phase during the experiment. The decrease in t b − M for the in b-phase aged part of the sample was observed in all the samples in which this stage was performed.

3.3. Stage 3 The recovery of the specimen b1 after the stabilization stage has been followed by performing complete pseudoelastic cycles, immediately after the sample was unloaded. In the first cycle of this recovery stage no distinguishable step was observed at the point in which the static stabilization had been performed. Fig. 5(b) shows the recovery of the stresses to transform and retransform as a function of time for further cycles, once the stabilization experiment was finished. The start of the time axis was taken after the sample was unloaded, point f in Fig. 5(a), which is considered the start of the recovery. Negative values in the t-axis indicate the time elapsed during the static stabilization procedure. From the plotted data a time constant of approximately 15 min has been determined for Texp = 333 K for sample b1. We notice however that this time constant is only an estimate because: the kinetics of the b-phase recovery is fast compared with the imposed crosshead speed for measuring the pseudoelastic behavior evolution. In this way the obtained 15 min can be

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

considered as an upper limit. It is emphasized that at the end of this stage the stresses t b − M and t M − b went back to the same values they had before the stabilization experiment. However, the hysteresis already reached the reference state value from the start of the recovery stage. Fig. 5(b) also shows the results for Stages 2 and 3 for the other samples tested at the same temperature Texp = 333 K. The magnitude of the static stabilization for every sample can be seen by following the arrows. Time constants for the b recovery are of the same magnitude than the 15 min mentioned before. The samples are now at point B in Fig. 2. The maximum obtained effect from a static stabilization, during the time intervals of the same order of magnitude as those used for fatigue experiments in this work is then known.

113

The fact that the transformation and retransformation stresses recover up to the levels previously determined at the end of the Stage 1 reinforce the definition of the reference state. The resolved shear stresses corresponding to the non-stabilized part of the sample also recovered the values of the reference state. In order to obtain a first estimate of the kinetics of the stabilization mechanism at the same experimental temperature, the same procedure was performed for different stabilization time intervals. The sample was allowed to recover to its reference state after each stabilization test. Sample b1r was used for this and the obtained amounts of stabilization versus the stabilization time are shown in Fig. 5(c). A relaxation time of 160 min was obtained, by an exponential fit of the experimental points.

Fig. 5. Example of the stabilization stage and post stabilization recovery for sample bl. (a) two cycles are shown: one corresponds to the reference state (solid curve) and the other shows the path followed by the sample during the static stabilization experiment. The sample was stressed from the unloaded condition up to point b and the crosshead position was kept constant during 1060 min. Then the sample was loaded to go on transforming (point d) up to the end (point e) and then completely retransformed by complete unloading. The magnitude Dt of the stabilization is indicated. (b) recovery of the stresses to transform and retransform for several samples as a function of time. The decrease in the stress to retransform in the stabilization experiment is shown and both stresses to transform and retransform for further cycles are presented. (c) Kinetics of the stabilization mechanism measured for sample b1r: a deformation amount corresponding to approximately 60% of transformation was kept constant during the stabilization period. The measurement of the increased hysteresis was performed at a deformation corresponding to 36%.

114

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

Fig. 6(a). An important experimental fact is that the evolution of the behavior of the samples during a pseudoelastic cycling procedure saturates from a certain number of cycles on. This fact had been previously observed [5]. The cycle N=5153 plotted in Fig. 6(a) corresponds to this state.

3.5. Stage 5 The cycle N= 5157 in Fig. 6(a) corresponds to the end of the recovery after fatigue. The evolution of the stresses to transform in this recovery stage can be seen in Fig. 6(b) for the sample b1. We have observed that the slope associated to the transformation, except for the very initial transformed fractions, completely recovers after 1440 min at Texp = 333 K. The hysteresis, which had increased during the previous cycling, shows only a partial recovery leading to a permanent effect. Within the resolution of the experiment t b − M recovers the value it had before cycling and t M − b remains at a lower value. It should be remarked that for the very beginning portion of the transforming cycle (below 15%) the stress remains below the values corresponding to the reference state.

3.6. Stabilization after fatigue Fig. 6. (a) Mechanical evolution after pseudoelastic cycling for sample b1. N = 1 corresponds to the reference state before cycling, N =5153 after cycling, and N= 5157 after the post cycling recovery of the beta phase; (b) kinetics of the post cycling beta phase recovery. The plotted stresses were measured at a deformation corresponding to 15% of transformed sample as indicated by the straight line in Fig. 6(a).

3.4. Stage 4 This stage describes the results of a fatigue experiment, although emphasis will be put on the evolution far from the end of the fatigue life of the sample. Fig. 6(a) shows cycle N =1 and cycle N=5153, at the beginning and at the end of the continuous pseudoelastic fatigue, respectively, for sample b1 cycled at Texp = 333 K. It is easily observed that a softening is associated to the whole transformation cycle. The stress to transform near the 100% transformed fraction remains nearly constant while the stress level to start the transformation decreases. Thus, an increase of the mean value of the slope associated to the pseudoelastic transformation is observed. The down-shift in t b − M measured at transformed fraction of 15% is 4 MPa which corresponds to an up-shift of 4 K in the stress-free temperature of transformation Ms(0). In addition, an increase in the hysteresis is also observed along the whole cycle. Both shifts are dependent on the transformed fraction as can be seen in

A significant experimental result was obtained with samples c1 and c2. As it is indicated in Table 2, sample c1 was used to have a reference behavior of the stabilization stage before fatigue. Sample c2, obtained from the same single crystal was used to analyze the stabilization behavior after cycling, i.e. after Stage 5. The results of these experiments are shown in Fig. 7. Fig. 7(a) shows the t− o, curves corresponding to the asymptotic stage before the fatigue experiment and to the stabilization procedure. A decrease of 2.3 MPa was measured from the last curve at a deformation corresponding to 10% of transformed sample. However a decrease of 0.8 MPa in the stresses to transform and retransform at the right side of the curve has been measured. If we consider that this decrease might be also present at the left side of the curve, a stabilization of 1.5 MPa (corresponding to 1.5 K) can be considered as the magnitude of the obtained stabilization. On the other hand if the same procedure is applied to the obtained curves for sample c2 (stabilization procedure after fatigue), an stabilization effect of only 0.1 K is obtained, at least one order of magnitude smaller than for sample c1 (see Fig. 7(b)). It is clearly visible from Fig. 7(a) and (b) that nearly no step is present in the retransformation curve after fatigue if compared with the precycling condition.

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

4. Discussion

4.1. Precycling beha6ior 4.1.1. Reaching the asymptotic stage Several different phenomena are to be considered to understand the behavior corresponding to Stage 1: amount of retained vacancies and the kinetics of the system to reach thermal equilibrium, stabilization of martensite, and retained disorder in the b phase [8]. The first issue to analyze is the asymptotic or reference stage reached at the end of Stage 1, which has been shown to play a significant role to distinguish among different contributions in the fatigue behavior. It has been reported that after quenching CuZnAl single crystals from temperatures up to 1073 K, extra vacancies are retained [9,10]. Romero et al. measured the variation of the concentration of vacancies at room temperature, being the relaxation time of this evolution dependent on the quenching temperature [9]. No measurement of the concentration of vacancies has been reported for the same heat treatment and alloy composition as the used in the present experiments. However, air cooling from 1123 K is not slow enough to allow the concentration of vacancies to reach its equilibrium value at each intermediate temperature during cooling. Moreover, if migration of vacancies is present at room temperature [9], it is surely present at higher temperatures. It is then reasonable to consider that after the air cooling from 1123 K an extra amount of vacancies is retained and after the samples reach the test temperature (333 K) vacancies migrate, leading to a decrease in their concentration. Somoza et al. [11] have found that after quenching CuZnAl b samples from 620 K and aging at different temperatures (318, 343 and 373 K), no extra vacancies are found after 25 ×104 s (4166 min). Al-

115

though the composition of the alloy and thermal treatment used in that work differs from ours, their result gives support to consider that the asymptotic stage reached in Stage 1 corresponds to the concentration of vacancies at thermal equilibrium C EV. At Texp =333 K a value of C EV = 6.1× 10 − 8 can be obtained from the reported vacancy energy and entropy formation value (E FV = 0.54 eV and S Fv /k=2.2) [12,13]. CuZnAl alloys with electron concentration 1.48 shows two ordering processes during cooling from high temperatures [8,14]. It has been also reported that an incomplete L21 ordering is retained in CuZnAl alloys quenched from different temperatures [8,10,15]. Rapacioli et al. explained the effect of the quenching temperature on the measured Ms (a decrease in CuZnAl alloys) by an amount of disordered CuZn atom pairs, which are retained after quenching [15]. In fact for alloys of lower Al concentration it has been shown that the L21 order can be suppressed for some quenching temperatures [16]. Man˜osa et al. have recently discussed the relation between retained disorder, extra vacancies and martensitic transition temperatures in CuZnAl and CuAlBe alloys [10]. They found a correlation between the evolution of transition temperatures and further ordering at room temperature after quenching CuZnAl single crystals from different temperatures. Being diffusion present at the used Texp in this work, the movement of the excess of vacancies enable further ordering of the beta phase, and this explains our reported decrease in t b − M for Stage 1. The amount of this decrease ranges from 2 to 4 MPa of resolved shear stress. It has been reported [9] that 10 h is a sufficient time interval to reach complete order in CuZnAl alloys at RT, while in our experiments, performed at higher temperatures, t b − M reaches an asymptotic value at higher time intervals. This point may be understood

Fig. 7. (a) t – o, curves corresponding to the asymptotic stage (reference) before the fatigue experiment and the corresponding one for the stabilization procedure for sample c1; a decrease of 2.3 MPa was measured from the last curve at a deformation corresponding to 10% of transformed sample; (b) same stabilization experiment as in Fig. 7(a), after fatigue. Sample c2 was used.

116

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

considering the different thermal treatments carried out in each case: samples quenched from high temperatures like those used by Romero et al. [9] leave a higher density of retained vacancies which enhance the ordering kinetics. In addition the measurement of the stresses to transform might interfere with the ordering mechanism. However, this possibility is counterbalanced by the fact that the kinetics of b recovery after static stabilization of martensite is very rapid as it will be shown below. The stronger effect observed for t M − b during Stage 1, which shows an increase of approximately 12 MPa in 2880 min (2d) can be understood if stabilization of martensite is enhanced by retained vacancies. This has been observed in CuZnAl alloys [6,8], for higher amounts of retained vacancies. During the annealing of the excess of vacancies and while CV tends to C EV, the amount of the stabilization effect in cycle N decreases if compared with cycle N − 1. This can be observed in Fig. 4 at which the hysteresis of each transformation is measured as a function of time. It is reasonable to consider the value of the determined hysteresis as a measurement of the stabilization. Point R in Fig. 3(b) shows the asymptotic obtained state. If it is considered that the amount of vacancies does not change during the time used for each cycle, a direct correlation exists between each point and the amount of vacancies. The kinetics of the stabilization mechanism is fast enough to enable different amounts of stabilization for different parts of the sample, which remain in martensite a different time interval ranging from approximately 960 s for the first part to transform to extremely small values for the last part of the sample transforming to martensite. It is clear then that the narrowest part of the hysteresis cycle corresponds to the last part of the sample to transform to martensite. Macchi et al. obtained a relaxation time for the annealing of extra vacancies as a function of temperature [11]. Using their relationship a relaxation time of 133 min is obtained for Texp = 333 K as used in the present work. From the Dt values versus time shown in Fig. 4, a relaxation time of 198 min is obtained as the first time constant, if a double exponential fit is used. It should be considered that the evolution of Dt given in Fig. 4 is mainly associated to the stabilization process. However the relation between stabilization and amount of vacancies is rather more complicated and it should not be expected the same relaxation time for both evolutions [6]. A clear and important fact shown in this work is that in order to analyze a mechanical evolution during fatigue, a precise reference behavior must be obtained. This role is played by the asymptotic behavior reached in Stage 1 (point A in Fig. 2). From this asymptotic cycle the following information can be obtained: 1. The value of t b − M or its corresponding MS which depends on the particular state of order of the b

phase, reaches a constant value within the resolution of the equipment after 2 days at the present test temperatures (333 K). 2. An asymptotic hysteresis width of 1 MPa (or correspondingly 1 K) has been obtained giving an estimate value of the involved friction. However some contribution to this asymptotic value coming from the stabilization of martensite cannot be disregarded considering the stabilization effect which is discussed below.

4.1.2. Static stabilization of martensite The static stabilization stage, which shows a decrease of t M − b between 1.3 and 4 MPa after approximately 1100 min depending on the specimen, although of smaller absolute value if compared with reported stabilization effects, plays a significant role in the fatigue experiment. The main point here is that this effect does not need an extra amount of vacancies but the obtained asymptotic CV of the b phase is enough to provide for this mechanism. This should impose an unavoidable effect during the fatigue stage if the beta phase is not allowed to recover during these dynamic experiments. It is not possible to assure that the saturation of the stabilization mechanism is reached. Changes in the formation stresses during fatigue of the same order of magnitude as the obtained during static stabilization have been found, probably indicating that this mechanism is a significant one during fatigue at the test temperatures here considered. We have noticed that different saturation values of the static stabilization amount have been obtained for different samples or sometimes for the same sample after different thermomechanical treatments, being 4 MPa the maximum obtained amount. The extrapolation of the plotted curve presented in Fig. 5(c) for sample b1r leads to a stabilization amount of approximately 1.3 MPa, which is lower than the maximum obtained value of 4 MPa. One possible explanation for this is that the thermal treatment performed to sample b1r after fatigue was not enough to recover the initial state and the kinetics of stabilization remained slower than before fatigue. As it is discussed below, the microstructure affects the diffusive mechanisms like stabilization and recovery of the b phase. Hence a difference in the density of defects could explain the different values of stabilization. An important point of the stabilization mechanism is that although the beta phase recovers after annealing at the test temperatures used in this work (333 K), this recovery takes some time, as it has been clearly shown in Fig. 5 (b). The asymptotic point in this figure is recovered after 400 min for sample b1 at T=333 K, although the main recovery is obtained after approximately 15 min. Results of the same order had been previously reported [6,17]. Within the resolution of the

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

experiment, no macroscopic, changes in the state of the material, after the stabilization and subsequent recovery, are observed, as it is emphasized by the reference state which is regained after these stages. The obtained relaxation time of this recovery is higher than the time interval used in a fatigue experiment between two subsequent cycles, implying an accumulative effect during cycling as it will be discussed below. A decrease of the stress to transform in the non stabilized part of the sample was detected in this and a previous work [6]. Abu Arab et al. [6] considered that this effect was the consequence of further ordering in the beta phase. The results here presented show that the stress to transform goes back to its original value, corresponding to the cycle before the stabilization test. This fact suggests that an incomplete state of order is not responsible for this decrease during stabilization This effect is most probably related to the coexistence of b and martensite in a partially ( 50% transformed specimen). A tendency to increase the amount of transformation has been observed in a similar experiment but where the applied stress was kept constant and the displacement was free to move [17]. It is convenient to keep in mind that stabilization is not the only effect taking place in martensite. An example of this has been reported by Coluzzi et al. [18] showing changes in the elastic modulus and in the elastic energy dissipation coefficient Q − 1 as a function of time at TBMf. The beta phase recovers to its original state after retransformation and no changes of As, were detected. Another possible phenomenon taking place in martensite is the rubber effect. This is associated to diffusion in martensite [19]. This effect is induced in the martensite at the same time as stabilization takes place. This makes it rather difficult to separate the contribution of both effects on the martensitic transition temperatures. Moreover different activation energies have been found for both mechanisms, indicating that different diffusion processes originate both effects in martensite [19]. At the moment no direct correlation between rubber effect and the transition temperatures As and Ms, between b and martensite has been found, although some contribution cannot be completely disregarded. In this work we will consider that the rubber effect does not produce significant changes in As or Ms in comparison with the stabilization effect.

4.2. Post cycling beha6ior A remarkable result is related to the behavior of the material after fatigue. Fig. 6 shows the recovery of t b − M and t M − b after fatigue. This behavior cannot be fitted by a single exponential curve. It is clearly observed that the recovery of the b phase shows a slower kinetics after the sample has been pseudoelastically

117

cycled if compared with recovery after static stabilization prior to cycling (see Fig. 5(b) and Fig. 6(b)). An increase of one order of magnitude of the relaxation time after the cycling experiment was measured. The same fact had been reported in a previous work with samples prepared with a slightly different thermal treatment [4]. It is proposed that the generation of structural defects induced by cycling [20] significantly affect the kinetics of diffusive phenomena in the beta phase. For example higher dislocation densities retained after quenching from high temperatures give lower relaxation times for the decay of the concentration of extra vacancies [21]. One way to explain the change in kinetics is that vacancies will decrease the number of jumps before being trapped in a sink, making longer the time constant for diffusional processes. Sinks are associated in this work with the defects introduced during cycling. However it cannot be disregarded that vacancies are not efficient vehicles for the ordering mechanism due to the interaction with stress fields originated in the dislocation arrays retained after cycling. One of the main results of the present work is the inhibition of the stabilization of martensite after pseudoelastic cycling of single crystals of the b phase. The inhibition of the martensitic stabilization after different thermomechanical treatments has been the subject of several works [22–24]. One way to make stabilization more difficult is using thermal treatments with an intermediate step quench at a temperature higher than Ms [24]. In this way, the amount of vacancies in excess is reduced approaching the equilibrium concentration. Results obtained for Stage 2 in the present work show the presence of stabilization for a small amount of vacancies, which cannot be avoided. Other methods to inhibit stabilization include mechanical treatments as well. Duan et al. [22] found that previous rolling of the b phase at high temperatures inhibit the further stabilization process. These authors suggested that the formed dislocations would act as sinks and high mobility pathways for vacancies. In addition they mentioned that the dislocations introduced by rolling were more efficient concerning the inhibition of stabilization than the dislocations introduced after fatigue at low temperatures. However no details are there reported about these fatigue experiments. Similar hot rolling experiments were performed by Wolska et al. [23] at different temperatures and for the same temperatures, reaching different amounts of deformation. They concluded that the inhibition of the stabilization increases for lower temperatures of rolling and for the same temperature, at higher amounts of deformation. Although in the mentioned papers no details about the formed dislocations was available, it is reasonable to consider that they correspond to plastic deformation of the beta phase. Romero et al. [25] analyzed dislocations in the beta phase after deformation of CuZnAl single crys-

118

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

tals and found that they are screw dislocations (b= Ž111 and u=Ž111). Dislocations formed after pseudoelastic cycling are mixed ones, with their glide plane parallel to the basal plane of the martensite and Ž100 type Burgers vectors [20]. The present results indicate that the stabilization of martensite is inhibited here due to the microstructural changes present after fatigue. It is not possible to compare the magnitude of the inhibition effect for different microstructures with the available experimental results. The present experiments indicate that the dislocation arrays formed after cycling affect the efficiency of vacancies as vehicles for diffusion mechanisms both in b and in martensite.

4.3. Modeling of the cycling beha6ior In order to understand the behavior of the material during the pseudoelastic fatigue test it is convenient to analyze first some results of the different stages described above. Fig. 6(a) shows that the decrease of the transformation stress during cycling is almost the same as the measured one during the static stabilization. The nearly complete recovery of the t b − M corresponding to the reference state after fatigue strongly suggests that these phenomena are related with the two diffusive processes mentioned before. On the other hand, the observed hysteresis changes during cycling do not completely recover afterwards. However, it is worthwhile at this point to compare the hysteresis evolution in the present experiments with the previous reported studies made at lower temperatures. Malarrı´a et al. [1] found a correlation between the density of complex dislocation arrays parallel to the basal plane of the martensite, with retained martensite inside, and the increment in the hysteresis value in a fatigue experiment. In the same work [1] the density of these defects as a function of temperature for a fixed amount of cycles were measured showing that the effect on the hysteresis increases as the temperature decreases. This explains the smaller hysteresis increment effect found in the present work for fatigue at Texp around 333 K. The evolution of the mechanical curves during fatigue can then be decomposed into recoverable and unrecoverable changes. The stabilization of martensite and the posterior recovery of the b phase are considered to be the origin of the recoverable component of the mechanical evolution during cycling. The introduction of dislocations in complex arrays gives the non recoverable component of the evolution. Both contributions are non-independent as we can see from the comparison of the kinetics of the static stabilization procedures before and after cycling and from the comparison of the kinetics of the corresponding recovery stages. We conclude that the introduction of permanent defects during cycling give an important decrease in the kinetics of both diffusion related processes.

In the experiments reported in the present work the magnitude of the changes associated with the recoverable part of the evolution are stronger than those related with the non recoverable effects (i.e. mainly an increment from 1 to 2 K in the hysteresis and an ill defined beginning of the transformation). This has to do with the temperature levels involved as was stated before. Taking into account this experimental features, in what follows it will be considered as a simplifying assumption that the mechanical evolution at the present test temperature is only due to the diffusive mechanisms. The evolution of the permanent defects is disregarded and as a first step in modeling the evolutions, the kinetics of the diffusive mechanisms are considered constant during cycling. The effect of changes in the time constants will be analyzed afterwards. In addition, we will consider that the sample transforms following the same sequence in each cycle, leading to a direct relation between each small piece of material and the spent time interval in beta or martensite. This implies that the first part to transform in cycle N, will be the first one in cycle N+ 1. The last one should not show a stabilization effect and in fact no decrease in t b − M was detected at the end of the transformation path in the reported experiments. Fig. 8 schematically shows a typical pseudoelastic cycle (above) and the corresponding time–elongation plot. The period of each cycle is named P. The parameter f is the fraction of the period P at which each part of the sample is in martensite in one cycle ( f·P). The value (1−f )·P gives the corresponding time the same part of the sample remains in the b phase in each cycle. As an example, f= 0 corresponds to a slice of the sample which never transforms to martensite (end of the cycle) and f= 1 to a portion of the sample which might not retransform to the b phase during cycling. The mechanical evolution of the sample can be analyzed using the change in the resolved shear stress to transform from b to martensite for every portion of the sample. The obtained change after N cycles is named SN = − DtN and is defined in the following way: − SN = DtN = t b − M − t0

(1)

where t0 and t bN− M are the resolved shear stresses to transform from b to martensite before cycling and after cycle N, respectively and S0 = DtN = 0 = 0 describes the reference state before cycling. In order to obtain DtN a single exponential kinetics is assumed for both the stabilization of martensite and the b recovery, each process with its corresponding relaxation time as was determined before. This simplifying assumption is considered reasonable taking into account the experimental results and being both mechanisms activated through the migration of vacancies. DtN can be obtained as a function of the change DtN − 1 corresponding to the previous cycle.





A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

DtN = DtN − 1 + (DtSat − DtN − 1) 1− e

−

tS

n

tS

119

e

−

tb tb

(2)

Here ts, is the relaxation time of the stabilization of martensite, tb the relaxation time of the b recovery process, ts the time in martensite (in one cycle), ts =f·P, tb the time in beta (in one cycle), tb = (1− f )·P, DtSat the change obtained in the resolved shear transformation stress after saturation has been reached in the static stabilization stage. Using an induction method, one obtains from Eq. (2)



DtN = DtSat 1− e



t − S tS

e

t N−1 − b tb

   n

n

% e

t t − S + b tS tb

(3)

n=0

After some algebraic manipulation, DtN can be written as



DtN = Dt N 1− e where



−

1− e

n

NP teff

−

(4)



Pf ts

e

Dt N = DtSat 1− e

−

−

P

P(1 − f) tb

(5)

teff

and Fig. 8. Schematic picture of a displacement controlled pseudoelastic cycle: the elements that define the model for the evolution during cycling are indicated. The element DX in the original b phase configuration transforms forth and back to martensite in each cycle, standing a time length ts in martensite and tb in the b phase. In every cycle an incremental change in the transformation shear stress occurs because of the competing effects of the martensite stabilization phenomena and the recovery of the b phase after martensite stabilization. The different elements DX stay a different fraction f of the cycle period P in each phase thus giving to a nonuniform evolution of the transformation stress. The way in which the evolution is obtained for a specific element is represented in the right bottom of the figure and expressed mathematically in Eq. (2) where SN = − DtN.

Fig. 9. Evolution of SN = DtN as defined in Eq. (1) for the case SSAT = −4 MPa, P = 10 s, ts = 9000 s and tb = 900 s. The evolution is plotted by considering f as a parameter.

1 f (1− f) = + tb tef tS

(6)

From the definitions given above it is clear that DtN, DtN − 1, tS and tb, depend on the selected part of the sample to be analyzed, i.e. on f. Before starting the cycling test the sample is in its reference state and DtN = Dt0 = 0. From Eq. (4) it is observed that DtN can be written as the product between a factor which does not depend on the number of cycles Dt N , and a second factor which depends on the total time of the performed cycling (P·N). Dt N corresponds to the saturation state and depends on the considered portion of the sample ( f value). The exponential factor describes the kinetics of the DtN evolution, with an effective relaxation time tef. An example of the application of the presented model is shown in Fig. 9 where SN = − DtN is plotted for different values of f. DtSat has been taken as the maximum obtained value in the static stabilization experiment, i.e. DtSat = − 4 MPa, the period P=10 s, ts = 9000 s and tb = 900 s. Each part of the sample reaches a different saturation value of the change in the stress to transform. We will call it Dt N to distinguish it from the saturation value of the static stabilization stage DtSat. The obtained Dt N values depend on f, as it is clearly observed in Fig. 9. Considering this dependence, the form of the final stress to transform versus (1−f ) curve for the saturation stage can be obtained, and this is shown in Fig. 10 for different set of parameters.

120

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

The effect of modifying ts, and tb has been considered. We have obtained from the applied model that the effect of modifying ts or tb is negligible for small values of f but increases for higher values of this parameter. It is interesting to notice that if ts and tb are multiplied by a same factor h, the same Dt N is obtained for a given value of f in the range 1Bh B100. However, if (tef/P) versus h is calculated, a strong effect is observed on the kinetics of the process for each f. The ratio tef/P gives the amount of cycles performed till t =tef. This analysis indicates that in the case that both kinetics were affected in the same way, the final shape of the saturation cycle would be the same but the number of cycles to reach the saturation state would be different. From our experiments it is clear that the kinetics of both, the stabilization of martensite and the recovery of the b phase, change after fatigue to a certain number of cycles N if they are measured in static condition, as they were measured previously to be cycled. The proposed simplified model considers a constant kinetics for the individual processes along the whole experiment and we adopted the time constants measured in the Stages 2 and 3 described above. Let us consider with some detail the comparison between the present model and the experimental curves (Fig. 10). Data from sample b1 has been considered. The experimental t versus o curves for the pseudoelastic b “M transformation for cycle N = 0 (reference state) and for cycle N = 5180 are shown in full lines. The solid circles represent the result of the model when the same set of parameters used in Fig. 9 are introduced in the calculation. As it can be observed the model reasonably reproduces the main characteristics of the experimental behavior, although some differences arise. A decrease in the ratio ts/tb gives an increase in the Dt N for a certain

Fig. 10. t– o curves for the b–M transformation at the start of cycling and at the saturation stage for samples b1. Black circles represent the result of the model when the same set of parameters as used in Fig. 9 are introduced in the calculation. Open circles and triangles indicate the results obtained if DtSat and ts, are changed.

f at fixed DtSat. An increment in the Dt N values could also be obtained by increasing DtSat for a certain f at constant ts/tb. Results of the model for DtSat = −8 MPa and for ts = 2000 s are shown in Fig. 10 (see empty circles and triangles, respectively). In this way real physical mechanisms have been considered in order to model and predict mechanical evolutions during pseudoelastic fatigue. The obtained results indicate that the actual kinetics of the involved processes could be strongly affected during cycling in a way that the kinetics of the martensite stabilization is favored in comparison with the kinetics of the b phase recovery. This effect is not permanent because the kinetics after cycling, measured in static condition, is slower than before cycling. One way to understand this behavior is to consider the point defect generation during cycling and the possible enhancement of the kinetics due to the moving interfaces between martensite and the b phase during fatigue. In fact, the introduction of point defects during pseudoelastic cycling has been proposed recently for experiments performed at low temperatures [3]. The point defects are expected to be created from dislocation annihilation [3,26]. The dislocations produced during the transformation cycling are due to localized plastic deformation in the 18R martensite [20]. These dislocations are progressively accumulated in bands until saturation is reached after a few thousands of cycles. As occurs in conventional fatigue [27,28], some annihilation can take place within the densely packed dislocations arrays. Dislocations gliding in the same basal plane that forms the bands lead, in this way, to a high density of point defects. Since the point defects are created in the 18R phase, they would act preferentially in the martensite before being trapped by the neighboring dislocations. They would be no longer available for the b recovery or any other subsequent diffusional process. This suggestion is consistent with the enhancement of the kinetics of the martensite stabilization, as compared to b recovery, needed to explain the experimental results in Fig. 10. The fact that the experimental saturation cycle in Fig. 10 was obtained after more than 5000 of cycles, also agrees, qualitatively, with the fact that the creation of dislocation arrays reach a saturation state at a similar number of cycles [5]. Hence, during cycling, DtSat can be higher in absolute value than after the static stabilization. A higher value of DtSat and a lower value of ts, can make the model to reasonably fit with the experimental curve in Fig. 10. On the other hand the interface movement during cycling also enhances the diffusional processes [29]. It has been found that the martensite stabilization can be favored during cycling due to the difference between the equilibrium vacancy concentration in b and martensite [17]. However this mechanism does not provide a clear difference between the kinetics of the martensite

A. Yawny et al. / Materials Science and Engineering A290 (2000) 108–121

stabilization and the b recovery. Therefore the creation of point defects in the martensite remains as the most plausible explanation of the recoverable behavior during fatigue cycling.

5. Conclusions (1) CuZnAl alloys have been shown in this work to be an extremely interesting example where microstructural changes associated with martensitic transformations are clearly related to the kinetics of diffusive mechanisms. Particularly the stabilization of martensite is present during fatigue and both the beta phase recovery and the stabilization mechanism change their kinetics after fatigue. (2) The relevance of defining a reference state for the experimental temperature has been pointed out in order to measure mechanical changes induced by pseudoelastic fatigue. (3) A static stabilization effect before cycling is present also for the equilibrium concentration of vacancies at the used working temperature and plays a significant role on the fatigue behavior. (4) A model which describes the mechanical evolution during cycling has been presented. It takes into account only two diffusive mechanisms: the stabilization of martensite and the b phase recovery. This is reasonable for the experimental temperatures used in this work and direct extrapolations are not possible either to lower temperatures or to different compositions. A reasonable agreement between the model and the experimental results has been found. A closer fit is possible by considering an enhancement of the stabilization kinetics due to creation of point defects in the martensite due to dislocation annihilation. (5) The kinetics of both diffusive mechanisms is altered after cycling: relaxation times increase after cycling at least one order of magnitude if compared with the measured values before cycling. This is associated with the dislocation arrays, which are retained during fatigue.

Acknowledgements The authors thank C. Go´mez and T. Carrasco for the preparation of alloys, single crystals and samples. This

.

121

work has been supported by Comisio´n Nacional de Energı´a Ato´mica, by Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas of the Repu´blica Argentina under grant PIP 4131/97 and by Agencia Nacional de Promocio´n Cientı´fica y Tecnolo´gica (grant PICT97, number 03-688). We also wish to thank Professor V. Torra for many enlightening discussions.

References [1] J. Malarrı´a, M. Sade, Scripta Metall. Mater. 30 (1994) 241. [2] J. Malarrı´a, M. Sade, F.C. Lovey, J. Phys. IV (France) 5 (1995) C8-889. [3] J. Malarrı´a, M. Sade, F.C. Lovey, Z. Metallkd. 87 (1996) 953. [4] A. Yawny, J. Malarrı´a, F.C. Lovey, M. Sade, J. Phys. IV (France) 7 (1997) C5-531. [5] M. Sade, R. Rapacioli, M. Ahlers, Acta Metall. 33 (1985) 487. [6] A. Abu Arab, M. Ahlers, Acta Metall. 36 (1988) 2627. [7] A. Tolley, M. Ahlers, J. Nucl. Mater. 205 (1993) 339. [8] M. Ahlers, Prog. Mater. Sci. 30 (1986) 135. [9] R. Romero, W. Salgueiro, A. Somoza, Phys. State Sol. (a) 133 (1992) 277. [10] L. Man˜osa, M. Jurado, A. Gonza´lez-Comas, E. Obrado´, A. Planes, J. Zarestky, C. Stassis, R. Romero, A. Somoza, M. Morin, Acta Metall. Mater. 46 (1998) 1045. [11] A. Somoza, C. Macchi, R. Romero, Mater. Sci. Forum 587 (1997) 255. [12] A. Ghilarducci, M. Ahlers, J. Phys. F 13 (1983) 1757. [13] H.J. Wollenberger, in: R.W. Cahn, P. Haasen (Eds.), Physical Metallurgy, Elsevier, Amsterdam, 1983, p. 1142. [14] M. Ahlers, J. Phys. Condens. Mater. 5 (1993) 8129. [15] R. Rapacioli, M. Ahlers, Acta Metall. 27 (1979) 777. [16] A. Planes, R. Romero, M. Ahlers, Acta Metall. Mater. 38 (1990) 757. [17] J.L. Pelegrina, M. Rodriguez de Rivera, V. Torra, F.C. Lovey, Acta Metall. Mater. 43 (1995) 933. [18] B. Coluzzi, A. Biscarini, F.M. Mazzolai, J. Phys. IV (France) 5 (1995) C8-823. [19] J.A. Giampaoli, J.L. Pelegrina, M. Ahlers, Acta Metall. Mater. 46 (1998) 3333. [20] M. Sade, A. Uribarri, F.C. Lovey, Phil. Mag. A 55 (1987) 445. [21] W. Salgueiro, R. Romero, A. Somoza, M. Ahlers, Phys. State Sol. (a) 133 (1992) 277. [22] X. Duan, W.M. Stobbs, Scripta Metall. 23 (1989) 441. [23] J. Wolska, M. Chandrasekaran, E. Cesari, Scripta Metall. Mater. 28 (1993) 779. [24] M. Chandrasekaran, E. Cesari, J. Wolska, I. Hurtado, R. Stalmans, J. Dutkiewicz, J. Phys. IV (France) 5 (1995) C2-143. [25] R. Romero, F.C. Lovey, M. Ahlers, Phil. Mag. A 58 (1988) 881. [26] J. Pola´k, Scripta Metall. 4 (1970) 761. [27] S. Basinski, Z. Basinski, A. Howie, Phil. Mag. 19 (1969) 899. [28] P. Charsley, F. Shimmin, Mater. Sci. Eng. 38 (1979) 23. [29] A. Amengual, F.C. Lovey, V. Torra, Scripta Metall. Mater. 24 (1990) 2241.