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Internal friction in a QE22 hybrid composite
Materials Science and Engineering A 370 (2004) 542–545
Internal friction in a QE22 hybrid composite Z. Trojanová∗ , P. Lukáˇc, A. Rudajevová Department of Metal Physics, Charles University, Praha, Ke Karlovu 5, CZ-121 16 Praha 2, Czech Republic Received 12 July 2002
Abstract This paper describes the influence of thermal loading on the damping behaviour of a composite with QE22 matrix reinforced with short Saffil fibres and SiC particles. The composite was prepared by squeeze casting. The logarithmic decrement was measured after a thermal cycle between room temperature and an upper temperature that varied within the range from 40 to 400 ◦ C. A region of strain amplitude independent damping at low strains is followed by a region with amplitude dependent damping at higher amplitude strains. The values of the logarithmic decrement in the strain amplitude dependent region depend very sensitively on the upper temperature of the cycle. The experimental data can be explained considering creation of new dislocations during thermal cycling and recovery processes at higher upper temperatures. © 2003 Elsevier B.V. All rights reserved. Keywords: Internal friction; Thermal cycling; Dislocation damping; Mg alloys; Thermal expansion coefficient
1. Introduction Several magnesium-based metal matrix composites (MMCs) have been developed over the last decade for potential use as light-weight high-performance materials [1]. It is well established that the microstructures and the mechanical properties of MMCs are strongly affected by the nature of the interfaces between the matrix and the reinforcement. The standard operating conditions for most MMCs will generally include some form of thermal loading. Even minor temperature changes may lead to microstructural changes, plastic deformation within the matrix and significant microstructural damage. The coefficient of thermal expansion (CTE) of a ceramic reinforcement is smaller than that of most metallic matrices. The differential thermal expansion between the reinforcement and the metal matrix induces misfit strain at the interface if the composite is submitted to a temperature change. These strain induce thermal stresses which can generate new dislocations at the interfaces. The higher dislocation density in the matrix, especially near the interface regions, as well as the reinforcement/matrix interfaces can provide high diffusivity path in a composite [2]. The higher dislocation
∗ Corresponding author. Tel.: +420-22191-1357; fax: +420-22191-1490. E-mail address: [email protected] (Z. Trojanov´a).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.08.113
density would also affect the precipitation kinetics in a precipitation hardenable matrix. The aim of the present paper is to investigate the internal friction of a QE22 alloy reinforced with short Saffil fibres and SiC particles submitted to thermal loading and to determine possible physical processes leading to the microstructure changes.
2. Experimental procedure Composites were prepared by the squeeze casting method. Commercial QE22 alloy (in wt.%: 2 Ag-2 mixture of rare earth mainly Nd-0.4 Zr, balance Mg) was reinforced with 5 vol.% of ␦-Al2 O3 short fibres (Saffil) with a mean diameter of 3 m and a mean length about 87 m (measured after squeeze casting) as well as with 20 vol.% SiC equiaxial particles. Test specimens for the damping measurements were machined as bending beams (88 mm long with thickness of 3 mm) with the reinforcement plane perpendicular to the main specimen axis. The damping measurements were carried out in vacuum (about 30 Pa) at room temperature. The specimens fixed at one end were excited into resonance (the frequency ranged from 160 to 170 Hz) by a permanent magnet and a sinusoidal alternating magnetic field. Damping was measured as the logarithmic decrement δ of the free decay of the vibrating beam. The signal amplitude is proportional to
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the strain amplitude ε. A special algorithm using all points was used for calculation of the strain amplitude dependence of the logarithmic decrement. Thermal cycles between room temperature and an increasing upper temperature were performed.
3. Experimental results and discussion Fig. 1 shows the optical photograph showing the Saffil fibres and SiC particles. Fig. 2a shows the plots of the logarithmic decrement against the logarithm of the strain amplitude for the QE22 hybrid composite before and after thermal cycling between room temperature and increasing upper temperature. Fig. 2b shows the results obtained at higher temperatures. It can be seen that the strain dependencies of the logarithmic decrement exhibit two region, as for many metallic materials. The logarithmic decrement can be expressed as a sum of two components. δ = δ0 + δH (ε)
(1)
where δ0 is the amplitude strain independent component, for low strain amplitudes and δH is the strain amplitude dependent component of δ. While the amplitude independent component depends weakly on the upper temperature of the cycle, the values of δH increase very strongly with increasing temperature up to 300 ◦ C and then, above 300 ◦ C, the values of δH decrease with increasing upper temperature of the temperature cycle. The experimental data indicate microstructure changes in the sample. An increase of internal thermal stresses due to the difference in CTEs is very probably responsible for these changes. The internal stresses produced by thermal loading of composites can relax by various mechanisms: creation of dislocations, their glide, by decohesion or sliding of the matrix/reinforcement interface, by diffusion of solute atoms in the matrix. An increase in the dislocation density ρ produced by the thermal stress near a reinforcement can be calculated as [3]
Fig. 2. Amplitude dependence of the logarithmic decrement estimated at: (a) lower temperatures of thermal cycling; (b) higher temperatures of thermal cycling.
ρ =
Bf α T 1 b(1 − f) t
(2)
where f is the volume fraction of the reinforcement, t the smallest dimension, b the magnitude of the Burgers vector of dislocations, B is a geometrical constant, α the absolute value of the difference in the CTEs and T the temperature difference. From Eq. (2) it follows that the density of newly formed dislocations is increasing with the temperature difference. When the thermal stresses reach the yield stress value of the matrix, plastic zones can be formed near the interfaces. The strain amplitude dependence of the logarithmic decrement suggests dislocation unpinning processes. The
Fig. 1. Microstructure of QE22 alloy reinforced with Saffil fibres and SiC particles.
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differences in the damping behaviour of specimens thermally cycled to various upper temperatures can be attributed to the interaction between dislocations and point defects including small clusters of foreign atoms and to changes in the dislocation density. The strong strain dependence of ␦ shown in Fig. 2a and b may be explained using the Granato-Lücke theory of dislocation damping [4]. The dislocation structure is assumed to consist of segments of LN along which weak pinning points are randomly distributed. The mean distance between weak pinning points is with LN . The mean total density of dislocations is ρ. A periodic stress σ =σ0 sin ωt is applied. At T = 0 K and at sufficiently high stress the dislocations are able to break-away from the weak pinning points and remain anchored at the strong pinning points. The stress required for the break-away of dislocations depends strongly on the statistic distribution of the pinning points. With increasing temperature, for T > 0 K, the stress for unpinning decreases because the break-away process is thermally activated [5]. Therefore, the break-away can occur at lower stresses than those for double loop. But higher activation energies are required because the break-away is simultaneous from several neighbouring pinning points. At higher temperatures and low frequencies the stress dependence of the decrement component δH can be expressed as [5] 1/2 ρLN ν 3πkT 1/2 3 σ02 δH = 6 ω 2U0 U0 G 4 U0 U0 G 1/2 1 × exp − 3 kT σ0 3
(3)
where G is the shear modulus, σ 0 the amplitude of the applied stress and ω its frequency, ν the dislocation attempt frequency, U0 the activation energy, kT has its usual meaning. With increasing upper temperature of the cycle the decrement component δH increases, too. The observed behaviour may be explained if we consider that during cooling and also during thermal cycling new dislocations are created due to the difference in the CTEs. Number of free foreign atoms or their small clusters can be modified by precipitation in the matrix. Thus, Svoboda et al. [6], who studied the microstructure of QE22 alloy with SiC particulates after T6 thermal treatment, have reported that Mg3 (Ag, Nd) precipitates, rounded precipitates of ␣Nd, rod-like precipitates of complex chemical composition including Zr and Nd and tiny MgO particles were observed as main secondary phases. Kiehn et al. [7] have estimated that ceramic fibres and the reaction products of the inorganic binder (containing Al2 O3 ) in the preform enhanced the Al concentration due to decomposition in the QE22 matrix. Al atoms take part in the precipitation process. TEM investigations have shown that the population of incoherent particles (containing Zr, Nd and Al) at the fibres does not change during step by step annealing. On the other hand new (Alx Mg1−x )Nd cubic particles appear between 120 and 180 ◦ C. These precipitates may serve as strong pinning points. The main weak pinning points are
Fig. 3. C2 parameter versus upper temperature of thermal cycling.
the solute atoms or its small clusters. The reduction of solute content in the matrix due to precipitation and the increase of the total dislocation density lead to an increase of the mean distance between two weak pinning points . After thermal treatment becomes longer because of the enhanced dislocation density due to temperature cycling and due to the lower number of solute atoms. The thermal activation influences the behaviour of all the cycled specimens by the same manner because all experiments were performed at ambient temperature. Experimental data were analysed using Eq. (3) in the form δH = C1 ε exp (−C2 /ε). The estimated constant C2 is plotted against temperature in Fig. 3. The C2 decreases with temperature. The C2 parameter in Eq. (3) is proportional to −3/2 . The distance between weak pinning points increases with increasing temperature. An increase in results in a decrease in the value of C2 , i.e. C2 should decrease with increasing upper temperature of the cycle, which is observed (Fig. 3). The stress necessary for break-away of dislocation loops σ T at a certain temperature T is given by [5] 2/3 kT (4) lnA σT = σM 1 − U1 with A=
2 υ σM 3 ω σ0
kT U1
2/3 (5)
σ M is the break-away stress in the pure mechanical process. For a double loop with the loop length 1 and 2 the break-away occurs under the stress σM =
2Fm . b( 1 + 2 )
(6)
Here Fm is the maximum binding force between a dislocation and a pinning point. U1 = 4/3(Fm3 /Φ)1/2 , where Φ is a constant. At temperature above 300 ◦ C the decrement component δH decreases, which corresponds to a decrease in . The thermal stresses produced at the interfaces are accommodated by the formation of plastic zones. At higher temperatures the yield stress of the matrix is lower than the
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4. Conclusions
Fig. 4. Temperature dependence of the coefficient of thermal expansion coefficient.
thermal stresses. Dislocations can move in the plastic zones and annihilation of dislocations can occur under appearing compressive internal stresses [8]. The dislocation density decreases. The concentration of the weak pinning points remains constant. Therefore, the distance between weak pinning points effectively increases. Thus, the value of C2 should slightly increases above 300 ◦ C, which is observed (Fig. 3). In this connection it is interesting to show the temperature dependence of the coefficient of thermal expansion of the same composite (Fig. 4). It can be seen that the CTE increases with increasing temperature up to approximately 300 ◦ C, then it decreases with increasing temperature. The change in the slope of the plot of CTE against temperature at about 300 ◦ C can be attributed to the change of the tensile thermal stress to the compressive thermal stress in the matrix. Compressive stresses in QE22 + Saffil composites after a temperature cycle with an upper temperature of 400 ◦ C leading to a residual contraction were measured by Trojanová et al. [8]. The motion of dislocations was detected by in situ acoustic emission measurements [8].
In this work the influence of thermal cycling on the internal friction curves for MMCs with QE22 alloy as the matrix and 20 vol.% of SiC particles as well as 5 vol.% of Saffil fibres has been studied. The strain dependence of the decrement exhibits two regions. At lower strains the decrement is practically constant and in the second region, at higher strain amplitudes, it increases with strain and with the upper temperature of the thermal cycle up to 300 ◦ C. The increase of the strain dependent decrement component is caused by an increase in the dislocation density and length of dislocation segments between weak pinning points due to thermal cycling and precipitation process in the matrix. Thermal treatment at temperatures T ≥ 300 ◦ C may result in motion of dislocations and their annihilation under compressive internal stresses, which leads to a decrease of the decrement. Acknowledgements The authors thank the Grant Agency of the Czech Academy of Sciences for financial support under Grant A2041203. References [1] C. Fritze, H. Berek, K.U. Kainer, S. Mielke, B. Wielage, in: B.L. Mordike, K.U. Kainer (Eds.), Magnesium Alloys and Their Applications, Werkstoff-Informationsgesellschaft, Oberursel, 1998, p. 635. [2] R.J. Arsenault, R.M. Fisher, Scripta Metall. 17 (1983) 67. [3] R.J. Arsenault, N. Shi, Mater. Sci. Eng. A81 (1986) 175. [4] A.V. Granato, K. Lücke, J. Appl. Phys. 27 (1956) 583, 789. [5] A.V. Granato, K. Lücke, J. Appl. Phys. 52 (1981) 7136. [6] A. Svoboda, M. Pahutová, F. Moll, J. Bˇrezina, V. Skleniˇcka, in: K.U. Kainer (Ed.), Magnesium Alloys and their Applications, Wiley, Weinheim, 2000, p. 234. [7] J. Kiehn, B. Smola, P. Vostrý, I. Stul´ıková, K.U. Kainer, Phys. Stat. Sol. (a) 164 (1997) 709. [8] Z. Trojanová, F. Chmel´ık, P. Lukáˇc, A. Rudajevová, J. Alloys Comp. 339 (2002) 327.
Internal friction in a QE22 hybrid composite Z. Trojanová∗ , P. Lukáˇc, A. Rudajevová Department of Metal Physics, Charles University, Praha, Ke Karlovu 5, CZ-121 16 Praha 2, Czech Republic Received 12 July 2002
Abstract This paper describes the influence of thermal loading on the damping behaviour of a composite with QE22 matrix reinforced with short Saffil fibres and SiC particles. The composite was prepared by squeeze casting. The logarithmic decrement was measured after a thermal cycle between room temperature and an upper temperature that varied within the range from 40 to 400 ◦ C. A region of strain amplitude independent damping at low strains is followed by a region with amplitude dependent damping at higher amplitude strains. The values of the logarithmic decrement in the strain amplitude dependent region depend very sensitively on the upper temperature of the cycle. The experimental data can be explained considering creation of new dislocations during thermal cycling and recovery processes at higher upper temperatures. © 2003 Elsevier B.V. All rights reserved. Keywords: Internal friction; Thermal cycling; Dislocation damping; Mg alloys; Thermal expansion coefficient
1. Introduction Several magnesium-based metal matrix composites (MMCs) have been developed over the last decade for potential use as light-weight high-performance materials [1]. It is well established that the microstructures and the mechanical properties of MMCs are strongly affected by the nature of the interfaces between the matrix and the reinforcement. The standard operating conditions for most MMCs will generally include some form of thermal loading. Even minor temperature changes may lead to microstructural changes, plastic deformation within the matrix and significant microstructural damage. The coefficient of thermal expansion (CTE) of a ceramic reinforcement is smaller than that of most metallic matrices. The differential thermal expansion between the reinforcement and the metal matrix induces misfit strain at the interface if the composite is submitted to a temperature change. These strain induce thermal stresses which can generate new dislocations at the interfaces. The higher dislocation density in the matrix, especially near the interface regions, as well as the reinforcement/matrix interfaces can provide high diffusivity path in a composite [2]. The higher dislocation
∗ Corresponding author. Tel.: +420-22191-1357; fax: +420-22191-1490. E-mail address: [email protected] (Z. Trojanov´a).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.08.113
density would also affect the precipitation kinetics in a precipitation hardenable matrix. The aim of the present paper is to investigate the internal friction of a QE22 alloy reinforced with short Saffil fibres and SiC particles submitted to thermal loading and to determine possible physical processes leading to the microstructure changes.
2. Experimental procedure Composites were prepared by the squeeze casting method. Commercial QE22 alloy (in wt.%: 2 Ag-2 mixture of rare earth mainly Nd-0.4 Zr, balance Mg) was reinforced with 5 vol.% of ␦-Al2 O3 short fibres (Saffil) with a mean diameter of 3 m and a mean length about 87 m (measured after squeeze casting) as well as with 20 vol.% SiC equiaxial particles. Test specimens for the damping measurements were machined as bending beams (88 mm long with thickness of 3 mm) with the reinforcement plane perpendicular to the main specimen axis. The damping measurements were carried out in vacuum (about 30 Pa) at room temperature. The specimens fixed at one end were excited into resonance (the frequency ranged from 160 to 170 Hz) by a permanent magnet and a sinusoidal alternating magnetic field. Damping was measured as the logarithmic decrement δ of the free decay of the vibrating beam. The signal amplitude is proportional to
Z. Trojanov´a et al. / Materials Science and Engineering A 370 (2004) 542–545
543
the strain amplitude ε. A special algorithm using all points was used for calculation of the strain amplitude dependence of the logarithmic decrement. Thermal cycles between room temperature and an increasing upper temperature were performed.
3. Experimental results and discussion Fig. 1 shows the optical photograph showing the Saffil fibres and SiC particles. Fig. 2a shows the plots of the logarithmic decrement against the logarithm of the strain amplitude for the QE22 hybrid composite before and after thermal cycling between room temperature and increasing upper temperature. Fig. 2b shows the results obtained at higher temperatures. It can be seen that the strain dependencies of the logarithmic decrement exhibit two region, as for many metallic materials. The logarithmic decrement can be expressed as a sum of two components. δ = δ0 + δH (ε)
(1)
where δ0 is the amplitude strain independent component, for low strain amplitudes and δH is the strain amplitude dependent component of δ. While the amplitude independent component depends weakly on the upper temperature of the cycle, the values of δH increase very strongly with increasing temperature up to 300 ◦ C and then, above 300 ◦ C, the values of δH decrease with increasing upper temperature of the temperature cycle. The experimental data indicate microstructure changes in the sample. An increase of internal thermal stresses due to the difference in CTEs is very probably responsible for these changes. The internal stresses produced by thermal loading of composites can relax by various mechanisms: creation of dislocations, their glide, by decohesion or sliding of the matrix/reinforcement interface, by diffusion of solute atoms in the matrix. An increase in the dislocation density ρ produced by the thermal stress near a reinforcement can be calculated as [3]
Fig. 2. Amplitude dependence of the logarithmic decrement estimated at: (a) lower temperatures of thermal cycling; (b) higher temperatures of thermal cycling.
ρ =
Bf α T 1 b(1 − f) t
(2)
where f is the volume fraction of the reinforcement, t the smallest dimension, b the magnitude of the Burgers vector of dislocations, B is a geometrical constant, α the absolute value of the difference in the CTEs and T the temperature difference. From Eq. (2) it follows that the density of newly formed dislocations is increasing with the temperature difference. When the thermal stresses reach the yield stress value of the matrix, plastic zones can be formed near the interfaces. The strain amplitude dependence of the logarithmic decrement suggests dislocation unpinning processes. The
Fig. 1. Microstructure of QE22 alloy reinforced with Saffil fibres and SiC particles.
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Z. Trojanov´a et al. / Materials Science and Engineering A 370 (2004) 542–545
differences in the damping behaviour of specimens thermally cycled to various upper temperatures can be attributed to the interaction between dislocations and point defects including small clusters of foreign atoms and to changes in the dislocation density. The strong strain dependence of ␦ shown in Fig. 2a and b may be explained using the Granato-Lücke theory of dislocation damping [4]. The dislocation structure is assumed to consist of segments of LN along which weak pinning points are randomly distributed. The mean distance between weak pinning points is with LN . The mean total density of dislocations is ρ. A periodic stress σ =σ0 sin ωt is applied. At T = 0 K and at sufficiently high stress the dislocations are able to break-away from the weak pinning points and remain anchored at the strong pinning points. The stress required for the break-away of dislocations depends strongly on the statistic distribution of the pinning points. With increasing temperature, for T > 0 K, the stress for unpinning decreases because the break-away process is thermally activated [5]. Therefore, the break-away can occur at lower stresses than those for double loop. But higher activation energies are required because the break-away is simultaneous from several neighbouring pinning points. At higher temperatures and low frequencies the stress dependence of the decrement component δH can be expressed as [5] 1/2 ρLN ν 3πkT 1/2 3 σ02 δH = 6 ω 2U0 U0 G 4 U0 U0 G 1/2 1 × exp − 3 kT σ0 3
(3)
where G is the shear modulus, σ 0 the amplitude of the applied stress and ω its frequency, ν the dislocation attempt frequency, U0 the activation energy, kT has its usual meaning. With increasing upper temperature of the cycle the decrement component δH increases, too. The observed behaviour may be explained if we consider that during cooling and also during thermal cycling new dislocations are created due to the difference in the CTEs. Number of free foreign atoms or their small clusters can be modified by precipitation in the matrix. Thus, Svoboda et al. [6], who studied the microstructure of QE22 alloy with SiC particulates after T6 thermal treatment, have reported that Mg3 (Ag, Nd) precipitates, rounded precipitates of ␣Nd, rod-like precipitates of complex chemical composition including Zr and Nd and tiny MgO particles were observed as main secondary phases. Kiehn et al. [7] have estimated that ceramic fibres and the reaction products of the inorganic binder (containing Al2 O3 ) in the preform enhanced the Al concentration due to decomposition in the QE22 matrix. Al atoms take part in the precipitation process. TEM investigations have shown that the population of incoherent particles (containing Zr, Nd and Al) at the fibres does not change during step by step annealing. On the other hand new (Alx Mg1−x )Nd cubic particles appear between 120 and 180 ◦ C. These precipitates may serve as strong pinning points. The main weak pinning points are
Fig. 3. C2 parameter versus upper temperature of thermal cycling.
the solute atoms or its small clusters. The reduction of solute content in the matrix due to precipitation and the increase of the total dislocation density lead to an increase of the mean distance between two weak pinning points . After thermal treatment becomes longer because of the enhanced dislocation density due to temperature cycling and due to the lower number of solute atoms. The thermal activation influences the behaviour of all the cycled specimens by the same manner because all experiments were performed at ambient temperature. Experimental data were analysed using Eq. (3) in the form δH = C1 ε exp (−C2 /ε). The estimated constant C2 is plotted against temperature in Fig. 3. The C2 decreases with temperature. The C2 parameter in Eq. (3) is proportional to −3/2 . The distance between weak pinning points increases with increasing temperature. An increase in results in a decrease in the value of C2 , i.e. C2 should decrease with increasing upper temperature of the cycle, which is observed (Fig. 3). The stress necessary for break-away of dislocation loops σ T at a certain temperature T is given by [5] 2/3 kT (4) lnA σT = σM 1 − U1 with A=
2 υ σM 3 ω σ0
kT U1
2/3 (5)
σ M is the break-away stress in the pure mechanical process. For a double loop with the loop length 1 and 2 the break-away occurs under the stress σM =
2Fm . b( 1 + 2 )
(6)
Here Fm is the maximum binding force between a dislocation and a pinning point. U1 = 4/3(Fm3 /Φ)1/2 , where Φ is a constant. At temperature above 300 ◦ C the decrement component δH decreases, which corresponds to a decrease in . The thermal stresses produced at the interfaces are accommodated by the formation of plastic zones. At higher temperatures the yield stress of the matrix is lower than the
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4. Conclusions
Fig. 4. Temperature dependence of the coefficient of thermal expansion coefficient.
thermal stresses. Dislocations can move in the plastic zones and annihilation of dislocations can occur under appearing compressive internal stresses [8]. The dislocation density decreases. The concentration of the weak pinning points remains constant. Therefore, the distance between weak pinning points effectively increases. Thus, the value of C2 should slightly increases above 300 ◦ C, which is observed (Fig. 3). In this connection it is interesting to show the temperature dependence of the coefficient of thermal expansion of the same composite (Fig. 4). It can be seen that the CTE increases with increasing temperature up to approximately 300 ◦ C, then it decreases with increasing temperature. The change in the slope of the plot of CTE against temperature at about 300 ◦ C can be attributed to the change of the tensile thermal stress to the compressive thermal stress in the matrix. Compressive stresses in QE22 + Saffil composites after a temperature cycle with an upper temperature of 400 ◦ C leading to a residual contraction were measured by Trojanová et al. [8]. The motion of dislocations was detected by in situ acoustic emission measurements [8].
In this work the influence of thermal cycling on the internal friction curves for MMCs with QE22 alloy as the matrix and 20 vol.% of SiC particles as well as 5 vol.% of Saffil fibres has been studied. The strain dependence of the decrement exhibits two regions. At lower strains the decrement is practically constant and in the second region, at higher strain amplitudes, it increases with strain and with the upper temperature of the thermal cycle up to 300 ◦ C. The increase of the strain dependent decrement component is caused by an increase in the dislocation density and length of dislocation segments between weak pinning points due to thermal cycling and precipitation process in the matrix. Thermal treatment at temperatures T ≥ 300 ◦ C may result in motion of dislocations and their annihilation under compressive internal stresses, which leads to a decrease of the decrement. Acknowledgements The authors thank the Grant Agency of the Czech Academy of Sciences for financial support under Grant A2041203. References [1] C. Fritze, H. Berek, K.U. Kainer, S. Mielke, B. Wielage, in: B.L. Mordike, K.U. Kainer (Eds.), Magnesium Alloys and Their Applications, Werkstoff-Informationsgesellschaft, Oberursel, 1998, p. 635. [2] R.J. Arsenault, R.M. Fisher, Scripta Metall. 17 (1983) 67. [3] R.J. Arsenault, N. Shi, Mater. Sci. Eng. A81 (1986) 175. [4] A.V. Granato, K. Lücke, J. Appl. Phys. 27 (1956) 583, 789. [5] A.V. Granato, K. Lücke, J. Appl. Phys. 52 (1981) 7136. [6] A. Svoboda, M. Pahutová, F. Moll, J. Bˇrezina, V. Skleniˇcka, in: K.U. Kainer (Ed.), Magnesium Alloys and their Applications, Wiley, Weinheim, 2000, p. 234. [7] J. Kiehn, B. Smola, P. Vostrý, I. Stul´ıková, K.U. Kainer, Phys. Stat. Sol. (a) 164 (1997) 709. [8] Z. Trojanová, F. Chmel´ık, P. Lukáˇc, A. Rudajevová, J. Alloys Comp. 339 (2002) 327.