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Atomistic simulations of shock waves in cubic silicon carbide
Computational Materials Science 45 (2009) 419–422
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Atomistic simulations of shock waves in cubic silicon carbide Q. Cheng, H.A. Wu *, Y. Wang, X.X. Wang CAS Key Laboratory of Materials Behavior and Design, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
a r t i c l e
i n f o
Article history: Received 23 September 2008 Received in revised form 29 October 2008 Accepted 31 October 2008 Available online 16 December 2008 PACS: 62.50.Ef 61.43.Bn 62.20.x
a b s t r a c t Molecular dynamics simulations were performed to investigate the mechanical properties of shock waves in cubic silicon carbide (3C–SiC, b–SiC). Shock wave was produced and distinctly demonstrated when a moving sample material impacted to the static sample material on one side. At rather small impact velocity, compressive shock wave traveled with a velocity equal to the longitudinal sound speed. A linear Hugoniot relationship for normalized particle velocity was verified from our atomistic simulation results, which is comparable with the recent shock experiments on SiC powders. Part of cubic lattice atoms transforms into amorphous state when the impact velocity exceeds the critical value of 4.91 km/s. The size of the amorphous region is well proportional to the particle velocity. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Molecular dynamics Hugoniot relationship Shock wave Silicon carbide
1. Introduction Silicon carbide (SiC) is a kind of promising and interesting engineering materials due to its excellent chemical stability, high stiffness, high hardness, high thermal conductivity and excellent micromechanical characteristics. It has been widely used in the applications of optoelectronic and tribological devices, such as heat exchanger, ceramics fans, semiconductor coating [1–5]. Li [6] predicted absolute conductivity values for a perfect cubic silicon carbide, which were in satisfactory agreement with experimental data, except in low-temperature region (below 400 K). Gao [7] simulated high-energy displacement cascades in b-SiC by using molecular dynamics methods with modified Tersoff potential. Devanathan [8] successfully proved that carbon sublattice displacements of SiC could result in amorphization, in which chemical short-range disorder played a significant role. Ishimaru [9] and colleagues found that the ratio of heteronuclear (Si–C) to homonuclear (Si–Si and C–C) bonds was changing upon annealing which implied the relaxation of amorphous SiC. Atomistic simulations of indentation test on b-SiC were performed in Szlufarska [10,11] group. They found that indentation-induced amorphization in SiC was mainly driven by coalescence of dislocation loops under the
* Corresponding author. E-mail address: [email protected] (H.A. Wu). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.10.020
indenter. However, very few studies have been carried out on the physical responses of shocked SiC materials because of difficulties in the preparation of the test specimen, and also because of the issues associated with measuring physical properties at such small scale. Therefore, special investigations about the characteristics of shock waves in silicon carbide may be significantly important. There exists a large number of polytypes of SiC just like carbon, which originate from different stacking sequences of silicon–carbon pair layer. The most common type in polytypes is the zinc– blende structure. In this work characteristic of wave propagation in shocked b-SiC was studied by molecular dynamics simulation with a modified Tersoff many-body potential. The results demonstrate that the speed of wave propagation is consistent well with that of elastic longitudinal wave under lower shock velocity and is approximately proportional to shock velocity under higher shock velocity. 2. Modeling and simulation Molecular dynamics method [12] is based on the classical Newton mechanics and modern statistical mechanics, which can be extended to simulate many equilibrium and nonequilibrium phenomenon at ultra-small scales. It is a good approach to investigate the physical properties of nano-scale materials. Lagrange equation of motion can be written as
d oL oL ¼ 0; dt oq_ i oq_ i
i ¼ 1; . . . ; N
ð1Þ
420
Q. Cheng et al. / Computational Materials Science 45 (2009) 419–422
where L is the Lagrangian of a conservative system and qi is a set of N independent generalized coordinates. The interaction between atoms in the 3C–SiC system is described by the Tersoff covalent many-body potential [13]. Tang and Yip [14] pointed out that the original Tersoff potential could not be used for the lack of physical meaning when material systems took a large compression. We adopted the Tersoff potential parameters from T94 [15].
E¼
1X V ij 2 i–j
ð2Þ
V ij ¼ fC ðr ij Þ½fR ðr ij Þ þ bij fA ðr ij Þ Here, fC is the cutoff function and fR, fA are the repulsive and attractive energy functions, respectively. All of them are functions of rij, which is the distance between atom i and j. bij is the bond order number. The lattice constant of 3C–SiC is 4.2791 Å. In the simulation, we choose [1 0 0] [0 1 0] [0 0 1] crystal orientations to coincide with coordinate directions X, Y, Z, respectively. The size of the simulation cell is 427.91 Å 42.791 Å 42.791 Å, in which 80,000 atoms are evolved. Periodic boundary conditions are applied in y and z directions, while free surface in x direction. The timestep of simulation is 1.0 femtosecond. The temperature is kept at almost 0 K and thermal oscillations are ignored. In the first part of our simulations, uniform extension and compression deformations were performed to compute the elastic constants. The strain rate in these deformations is approximatively 106 s1, which can be considered as quasi-static processes from our previous work [16]. The positions of atoms in x direction were scaled 0.1% and relaxed 5 picoseconds incrementally. In the second part, a strip of atoms in the left side of sample in x direction is assigned a uniform velocity, which is defined as piston impact velocity. Assigning different velocities can simulate different piston impact velocities. The shock strip is 150 Å, approximately 1/3 of the whole length. The simulation case is schematically demonstrated in Fig. 1. In the theory of elastic wave [17], the speed of elastic longitudinal wave can be evaluated from the following generic formula
co ¼ vxx ¼
sffiffiffiffiffiffiffi C xx
q
ð3Þ
where Cxx and q are the elastic modulus in x direction and density of the materials, respectively. The following linear Hugoniot relationship holds for many materials [18]
U S ¼ U o þ s1 U P
ð4Þ
where s1 is a constant in the range 0.5–2.5, and Uo is approximatively equal to co, US and UP represent shock velocity and particle velocity, respectively. 3. Results and discussions In the static deformations, only minor difference could be found between tensile and compressive modulus in the [1 0 0] direction. The corresponding static tensile and compressive curves are plotted in Fig. 2, in which the slope of the curves represents elastic modulus Cxx. The calculated elastic moduli are 445.9 GPa and 447.2 GPa for tensile and compressive deformation, respectively. The corresponding calculated velocities of elastic longitudinal waves are 11.45 km/s and 11.47 km/s, respectively. At the beginning of impact simulations, the atoms in the piston region were assigned a uniform velocity of 0.02 km/s. The stress
Fig. 2. Stress–strain curves for static homogeneous deformations of 3C–SiC.
Fig. 1. (a) Geometry view of the simulations, (b) Atomistic view of the simulations.
Q. Cheng et al. / Computational Materials Science 45 (2009) 419–422
421
Fig. 3. Stress and velocity profile of elastic shock waves (a) stress profile (b) velocity profile.
and velocity profiles in the shock direction are depicted in Fig. 3. The shock wave travels through the solid and brings about instantaneous changes of physical properties, such as stress, particle velocity, temperature, etc. In our simulations, the wave front can be identified from the velocity/stress profile where velocities/stresses abruptly change. It is obvious that the distances between adjacent waves are almost equal at the same time intervals. The velocity of shock wave calculated from the profiles is 11.56 km/s, which is very close to the theoretical value of 11.47 km/s obtained above. The calculated wave velocity is appreciably higher than the experimental value of of 9.5 km/s [19]. Single crystalline model is employed in our simulation study, while polycrystalline samples were used in experiments. Thus, the difference may be acceptable.
The relationship between shock wave velocity and assigned particle velocity in the piston region is plotted in Fig. 4. Both the shock velocity and particle velocity are normalized by the longitu-
Fig. 6. Atomistic details of amorphous region in Fig. 5b.
Fig. 4. Particle velocity dependence of shock velocity. The velocities are normalized by longitudinal sound velocity.
Fig. 7. Particle velocity dependence of the number of atomic layers of the amorphous region.
Fig. 5. Atomistic configurations at different impact velocities. (a) 4.91 km/s, (b) 5.00 km/s.
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Q. Cheng et al. / Computational Materials Science 45 (2009) 419–422
dinal sound velocity of 3C–SiC. It can be seen that normalized shock velocity is approximately proportional to normalized impact/particle velocity from Fig. 4. The linear relationship of the corresponding velocities for metals has been proved by experiments [20] and simulations [21]. Shock tests on SiC powder with different ratio of initial porosity were conducted by Gu [22]. Particle velocities and shock velocities in the experiments were in the range 0.4– 1.1 km/s and 1.1–2.0 km/s, respectively. The linear curves of Hugoniot relationship were clearly observed and in nearly the same slope for different samples. Due to limitations of experiments, the impact velocity in experiments could not reach velocity as high as in our atomistic simulations. However the same tendency was observed. At different impact velocities, the magnitude of stresses is in different levels. Atomistic configurations at two different impact velocities are compared in Fig. 5. The cubic lattice transformed into amorphous state when an impact velocity above the critical value of 4.91 km/s is assigned. Amorphization will lead to release of the high impact energy to the lattices, and it is not reversible. Details of amorphous region at the critical velocity are shown in Fig. 6. The number of atomic layers of the amorphous region is well proportional to the normalized particle velocity as depicted in Fig. 7. 4. Summary The mechanical properties of shock waves in cubic silicon carbide is investigated with molecular dynamics simulation at rather small impact velocities, the velocity of shock wave propagation is consistent well with analytical elastic longitudinal sound speed. Normalized velocity of shock waves is proportional to normalized particle velocity, and it approaches to 1.0 as normalized particle velocity decreases. The linear Hugoniot relationship is compared with experimental results on SiC powder with different ratio of initial porosity, which shows the same tendency. When the impact velocity reaches above the critical value of 4.91 km/s, part of the cubic lattice atoms transforms into amorphous state. The number
of atomic layers of the amorphous region has a linear dependence of the particle velocity. Acknowledgements This research was supported by the National Natural Science Foundation of China under the Grant No. 10632080, the National Basic Research Program of China under the Grant No. 2006CB300404. References [1] M.E. Levinshtein, S.L. Rumyantsev, M. Shur, Properties of Advanced Semiconductor Materials: GaN, AlN, InN, BN, SiC, SiGe, Wiley, New York, 2001. [2] D. Emin, T.L. Aselage, C. Wood, Materials Research Society, Novel Refractory Semiconductors: Symposium held April 21–23, 1987, Materials Research Society, Pittsburgh, Anaheim California, USA, 1987. [3] J. R. O’Connor, J. Smiltens, Silicon Carbide, a High Temperature Semiconductor, in: Proceedings Pergamon, Oxford, 1960. [4] A. Pechenik, R.K. Kalia, P. Vashishta, Computer-Aided Design of HighTemperature Materials, Oxford University Press, New York, 1999. [5] M.F. Thorpe, M.I. Mitkova, North Atlantic Treaty Organization, Cientific Affairs Division Amorphous Insulators and Semiconductors, Kluwer Academic, Dordrecht, 1997. [6] J. Li, L. Porter, S. Yip, J. Nucl. Mater. 255 (1998) 139–152. [7] F. Gao, W.J. Weber, Phys. Rev. B 63 (2004) 054101. [8] R. Devanathan, F. Gao, W.J. Weber, Appl. Phys. Lett. 84 (2004) 3909. [9] M. Ishimaru, Phys. Rev. Lett. 89 (2002) 055502. [10] Izabela Szlufarska, Appl. Phys. Lett. 86 (2005) 021915. [11] Izabela Szlufarska, Phys. Rev. B 71 (2005) 174113. [12] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Press, 1987. [13] J. Tersoff, Phys. Rev. B 39 (1989) 5566. [14] M.J. Tang, S. Yip, Phys. Rev. B 52 (2005) 1515. [15] J. Tersoff, Phys. Rev. B 49 (1994) 16349. [16] H.A. Wu, European J. Mech. A/Solids 25 (2006) 370–377. [17] D. Royer, Elastic Waves in Solids, Springer, 1996. [18] E.M. Bringa, J. Appl. Phys. 96 (2004) 3797–3799. [19] http://www.ioffe.ru/SVA/NSM/Semicond/SiC/mechanic.html. [20] Y. Chinone, S. Ezaki, F. Fuijita, R. Matsumoto, Springer Proc. Phys. 43 (1989) 198–206. [21] M.D. Furnish, Shock Compression of Condensed Matter, 483–488, New York, 2000. [22] Y.B. Gu, Int. J. Impact Eng. 32 (2006) 1768–1785.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Atomistic simulations of shock waves in cubic silicon carbide Q. Cheng, H.A. Wu *, Y. Wang, X.X. Wang CAS Key Laboratory of Materials Behavior and Design, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
a r t i c l e
i n f o
Article history: Received 23 September 2008 Received in revised form 29 October 2008 Accepted 31 October 2008 Available online 16 December 2008 PACS: 62.50.Ef 61.43.Bn 62.20.x
a b s t r a c t Molecular dynamics simulations were performed to investigate the mechanical properties of shock waves in cubic silicon carbide (3C–SiC, b–SiC). Shock wave was produced and distinctly demonstrated when a moving sample material impacted to the static sample material on one side. At rather small impact velocity, compressive shock wave traveled with a velocity equal to the longitudinal sound speed. A linear Hugoniot relationship for normalized particle velocity was verified from our atomistic simulation results, which is comparable with the recent shock experiments on SiC powders. Part of cubic lattice atoms transforms into amorphous state when the impact velocity exceeds the critical value of 4.91 km/s. The size of the amorphous region is well proportional to the particle velocity. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Molecular dynamics Hugoniot relationship Shock wave Silicon carbide
1. Introduction Silicon carbide (SiC) is a kind of promising and interesting engineering materials due to its excellent chemical stability, high stiffness, high hardness, high thermal conductivity and excellent micromechanical characteristics. It has been widely used in the applications of optoelectronic and tribological devices, such as heat exchanger, ceramics fans, semiconductor coating [1–5]. Li [6] predicted absolute conductivity values for a perfect cubic silicon carbide, which were in satisfactory agreement with experimental data, except in low-temperature region (below 400 K). Gao [7] simulated high-energy displacement cascades in b-SiC by using molecular dynamics methods with modified Tersoff potential. Devanathan [8] successfully proved that carbon sublattice displacements of SiC could result in amorphization, in which chemical short-range disorder played a significant role. Ishimaru [9] and colleagues found that the ratio of heteronuclear (Si–C) to homonuclear (Si–Si and C–C) bonds was changing upon annealing which implied the relaxation of amorphous SiC. Atomistic simulations of indentation test on b-SiC were performed in Szlufarska [10,11] group. They found that indentation-induced amorphization in SiC was mainly driven by coalescence of dislocation loops under the
* Corresponding author. E-mail address: [email protected] (H.A. Wu). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.10.020
indenter. However, very few studies have been carried out on the physical responses of shocked SiC materials because of difficulties in the preparation of the test specimen, and also because of the issues associated with measuring physical properties at such small scale. Therefore, special investigations about the characteristics of shock waves in silicon carbide may be significantly important. There exists a large number of polytypes of SiC just like carbon, which originate from different stacking sequences of silicon–carbon pair layer. The most common type in polytypes is the zinc– blende structure. In this work characteristic of wave propagation in shocked b-SiC was studied by molecular dynamics simulation with a modified Tersoff many-body potential. The results demonstrate that the speed of wave propagation is consistent well with that of elastic longitudinal wave under lower shock velocity and is approximately proportional to shock velocity under higher shock velocity. 2. Modeling and simulation Molecular dynamics method [12] is based on the classical Newton mechanics and modern statistical mechanics, which can be extended to simulate many equilibrium and nonequilibrium phenomenon at ultra-small scales. It is a good approach to investigate the physical properties of nano-scale materials. Lagrange equation of motion can be written as
d oL oL ¼ 0; dt oq_ i oq_ i
i ¼ 1; . . . ; N
ð1Þ
420
Q. Cheng et al. / Computational Materials Science 45 (2009) 419–422
where L is the Lagrangian of a conservative system and qi is a set of N independent generalized coordinates. The interaction between atoms in the 3C–SiC system is described by the Tersoff covalent many-body potential [13]. Tang and Yip [14] pointed out that the original Tersoff potential could not be used for the lack of physical meaning when material systems took a large compression. We adopted the Tersoff potential parameters from T94 [15].
E¼
1X V ij 2 i–j
ð2Þ
V ij ¼ fC ðr ij Þ½fR ðr ij Þ þ bij fA ðr ij Þ Here, fC is the cutoff function and fR, fA are the repulsive and attractive energy functions, respectively. All of them are functions of rij, which is the distance between atom i and j. bij is the bond order number. The lattice constant of 3C–SiC is 4.2791 Å. In the simulation, we choose [1 0 0] [0 1 0] [0 0 1] crystal orientations to coincide with coordinate directions X, Y, Z, respectively. The size of the simulation cell is 427.91 Å 42.791 Å 42.791 Å, in which 80,000 atoms are evolved. Periodic boundary conditions are applied in y and z directions, while free surface in x direction. The timestep of simulation is 1.0 femtosecond. The temperature is kept at almost 0 K and thermal oscillations are ignored. In the first part of our simulations, uniform extension and compression deformations were performed to compute the elastic constants. The strain rate in these deformations is approximatively 106 s1, which can be considered as quasi-static processes from our previous work [16]. The positions of atoms in x direction were scaled 0.1% and relaxed 5 picoseconds incrementally. In the second part, a strip of atoms in the left side of sample in x direction is assigned a uniform velocity, which is defined as piston impact velocity. Assigning different velocities can simulate different piston impact velocities. The shock strip is 150 Å, approximately 1/3 of the whole length. The simulation case is schematically demonstrated in Fig. 1. In the theory of elastic wave [17], the speed of elastic longitudinal wave can be evaluated from the following generic formula
co ¼ vxx ¼
sffiffiffiffiffiffiffi C xx
q
ð3Þ
where Cxx and q are the elastic modulus in x direction and density of the materials, respectively. The following linear Hugoniot relationship holds for many materials [18]
U S ¼ U o þ s1 U P
ð4Þ
where s1 is a constant in the range 0.5–2.5, and Uo is approximatively equal to co, US and UP represent shock velocity and particle velocity, respectively. 3. Results and discussions In the static deformations, only minor difference could be found between tensile and compressive modulus in the [1 0 0] direction. The corresponding static tensile and compressive curves are plotted in Fig. 2, in which the slope of the curves represents elastic modulus Cxx. The calculated elastic moduli are 445.9 GPa and 447.2 GPa for tensile and compressive deformation, respectively. The corresponding calculated velocities of elastic longitudinal waves are 11.45 km/s and 11.47 km/s, respectively. At the beginning of impact simulations, the atoms in the piston region were assigned a uniform velocity of 0.02 km/s. The stress
Fig. 2. Stress–strain curves for static homogeneous deformations of 3C–SiC.
Fig. 1. (a) Geometry view of the simulations, (b) Atomistic view of the simulations.
Q. Cheng et al. / Computational Materials Science 45 (2009) 419–422
421
Fig. 3. Stress and velocity profile of elastic shock waves (a) stress profile (b) velocity profile.
and velocity profiles in the shock direction are depicted in Fig. 3. The shock wave travels through the solid and brings about instantaneous changes of physical properties, such as stress, particle velocity, temperature, etc. In our simulations, the wave front can be identified from the velocity/stress profile where velocities/stresses abruptly change. It is obvious that the distances between adjacent waves are almost equal at the same time intervals. The velocity of shock wave calculated from the profiles is 11.56 km/s, which is very close to the theoretical value of 11.47 km/s obtained above. The calculated wave velocity is appreciably higher than the experimental value of of 9.5 km/s [19]. Single crystalline model is employed in our simulation study, while polycrystalline samples were used in experiments. Thus, the difference may be acceptable.
The relationship between shock wave velocity and assigned particle velocity in the piston region is plotted in Fig. 4. Both the shock velocity and particle velocity are normalized by the longitu-
Fig. 6. Atomistic details of amorphous region in Fig. 5b.
Fig. 4. Particle velocity dependence of shock velocity. The velocities are normalized by longitudinal sound velocity.
Fig. 7. Particle velocity dependence of the number of atomic layers of the amorphous region.
Fig. 5. Atomistic configurations at different impact velocities. (a) 4.91 km/s, (b) 5.00 km/s.
422
Q. Cheng et al. / Computational Materials Science 45 (2009) 419–422
dinal sound velocity of 3C–SiC. It can be seen that normalized shock velocity is approximately proportional to normalized impact/particle velocity from Fig. 4. The linear relationship of the corresponding velocities for metals has been proved by experiments [20] and simulations [21]. Shock tests on SiC powder with different ratio of initial porosity were conducted by Gu [22]. Particle velocities and shock velocities in the experiments were in the range 0.4– 1.1 km/s and 1.1–2.0 km/s, respectively. The linear curves of Hugoniot relationship were clearly observed and in nearly the same slope for different samples. Due to limitations of experiments, the impact velocity in experiments could not reach velocity as high as in our atomistic simulations. However the same tendency was observed. At different impact velocities, the magnitude of stresses is in different levels. Atomistic configurations at two different impact velocities are compared in Fig. 5. The cubic lattice transformed into amorphous state when an impact velocity above the critical value of 4.91 km/s is assigned. Amorphization will lead to release of the high impact energy to the lattices, and it is not reversible. Details of amorphous region at the critical velocity are shown in Fig. 6. The number of atomic layers of the amorphous region is well proportional to the normalized particle velocity as depicted in Fig. 7. 4. Summary The mechanical properties of shock waves in cubic silicon carbide is investigated with molecular dynamics simulation at rather small impact velocities, the velocity of shock wave propagation is consistent well with analytical elastic longitudinal sound speed. Normalized velocity of shock waves is proportional to normalized particle velocity, and it approaches to 1.0 as normalized particle velocity decreases. The linear Hugoniot relationship is compared with experimental results on SiC powder with different ratio of initial porosity, which shows the same tendency. When the impact velocity reaches above the critical value of 4.91 km/s, part of the cubic lattice atoms transforms into amorphous state. The number
of atomic layers of the amorphous region has a linear dependence of the particle velocity. Acknowledgements This research was supported by the National Natural Science Foundation of China under the Grant No. 10632080, the National Basic Research Program of China under the Grant No. 2006CB300404. References [1] M.E. Levinshtein, S.L. Rumyantsev, M. Shur, Properties of Advanced Semiconductor Materials: GaN, AlN, InN, BN, SiC, SiGe, Wiley, New York, 2001. [2] D. Emin, T.L. Aselage, C. Wood, Materials Research Society, Novel Refractory Semiconductors: Symposium held April 21–23, 1987, Materials Research Society, Pittsburgh, Anaheim California, USA, 1987. [3] J. R. O’Connor, J. Smiltens, Silicon Carbide, a High Temperature Semiconductor, in: Proceedings Pergamon, Oxford, 1960. [4] A. Pechenik, R.K. Kalia, P. Vashishta, Computer-Aided Design of HighTemperature Materials, Oxford University Press, New York, 1999. [5] M.F. Thorpe, M.I. Mitkova, North Atlantic Treaty Organization, Cientific Affairs Division Amorphous Insulators and Semiconductors, Kluwer Academic, Dordrecht, 1997. [6] J. Li, L. Porter, S. Yip, J. Nucl. Mater. 255 (1998) 139–152. [7] F. Gao, W.J. Weber, Phys. Rev. B 63 (2004) 054101. [8] R. Devanathan, F. Gao, W.J. Weber, Appl. Phys. Lett. 84 (2004) 3909. [9] M. Ishimaru, Phys. Rev. Lett. 89 (2002) 055502. [10] Izabela Szlufarska, Appl. Phys. Lett. 86 (2005) 021915. [11] Izabela Szlufarska, Phys. Rev. B 71 (2005) 174113. [12] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Press, 1987. [13] J. Tersoff, Phys. Rev. B 39 (1989) 5566. [14] M.J. Tang, S. Yip, Phys. Rev. B 52 (2005) 1515. [15] J. Tersoff, Phys. Rev. B 49 (1994) 16349. [16] H.A. Wu, European J. Mech. A/Solids 25 (2006) 370–377. [17] D. Royer, Elastic Waves in Solids, Springer, 1996. [18] E.M. Bringa, J. Appl. Phys. 96 (2004) 3797–3799. [19] http://www.ioffe.ru/SVA/NSM/Semicond/SiC/mechanic.html. [20] Y. Chinone, S. Ezaki, F. Fuijita, R. Matsumoto, Springer Proc. Phys. 43 (1989) 198–206. [21] M.D. Furnish, Shock Compression of Condensed Matter, 483–488, New York, 2000. [22] Y.B. Gu, Int. J. Impact Eng. 32 (2006) 1768–1785.