Finite size specimens with cracks of icosahedral Al Pd Mn quasicrystals
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摘要
Icosahedral quasicrystals are the most important and thermodynamically stable in all about 200 kinds of quasicrystals currently observed. Beyond the scope of classical elasticity, apart from a phonon displacement field, there is a phason displacement field in the elasticity of the quasicrystal, which induces an important effect on the mechanical properties of the material and makes an analytical solution difficult to obtain. In this paper, a finite element algorithm for the static elasticity of icosahedral quasicrystals is developed by transforming the elastic boundary value problem of the icosahedral quasicrystals into an equivalent variational problem. Analytical and numerical solutions for an icosahedral Al–Pd–Mn quasicrystal cuboid subjected to a uniaxial tension with different phonon–phason coupling parameters are given to verify the validity of the numerical approach. A comparison between the analytical and numerical solutions of the specimen demonstrates the accuracy and efficiency of the present algorithm. Finally, in order to reveal the fracture behavior of the icosahedral Al–Pd–Mn quasicrystal, a cracked specimen with a finite size of matter is investigated, both with and without phonon–phason coupling. Meanwhile, the geometry factors are calculated, including the stress intensity factor and the crack opening displacement for the finite-size specimen. Computational results reveal the importance of phonon–phason coupling effect on the icosahedral Al–Pd–Mn quasicrystal. Furthermore, the finite element procedure can be used to solve more complicated boundary value problems.
Icosahedral quasicrystals are the most important and thermodynamically stable in all about 200 kinds of quasicrystals currently observed. Beyond the scope of classical elasticity, apart from a phonon displacement field, there is a phason displacement field in the elasticity of the quasicrystal, which induces an important effect on the mechanical properties of the material and makes an analytical solution difficult to obtain. In this paper, a finite element algorithm for the static elasticity of icosahedral quasicrystals is developed by transforming the elastic boundary value problem of the icosahedral quasicrystals into an equivalent variational problem. Analytical and numerical solutions for an icosahedral Al–Pd–Mn quasicrystal cuboid subjected to a uniaxial tension with different phonon–phason coupling parameters are given to verify the validity of the numerical approach. A comparison between the analytical and numerical solutions of the specimen demonstrates the accuracy and efficiency of the present algorithm. Finally, in order to reveal the fracture behavior of the icosahedral Al–Pd–Mn quasicrystal, a cracked specimen with a finite size of matter is investigated, both with and without phonon–phason coupling. Meanwhile, the geometry factors are calculated, including the stress intensity factor and the crack opening displacement for the finite-size specimen. Computational results reveal the importance of phonon–phason coupling effect on the icosahedral Al–Pd–Mn quasicrystal. Furthermore, the finite element procedure can be used to solve more complicated boundary value problems.
Icosahedral quasicrystals are the most important and thermodynamically stable in all about 200 kinds of quasicrystals currently observed. Beyond the scope of classical elasticity, apart from a phonon displacement field, there is a phason displacement field in the elasticity of the quasicrystal, which induces an important effect on the mechanical properties of the material and makes an analytical solution difficult to obtain. In this paper, a finite element algorithm for the static elasticity of icosahedral quasicrystals is developed by transforming the elastic boundary value problem of the icosahedral quasicrystals into an equivalent variational problem. Analytical and numerical solutions for an icosahedral Al–Pd–Mn quasicrystal cuboid subjected to a uniaxial tension with different phonon–phason coupling parameters are given to verify the validity of the numerical approach. A comparison between the analytical and numerical solutions of the specimen demonstrates the accuracy and efficiency of the present algorithm. Finally, in order to reveal the fracture behavior of the icosahedral Al–Pd–Mn quasicrystal, a cracked specimen with a finite size of matter is investigated, both with and without phonon–phason coupling. Meanwhile, the geometry factors are calculated, including the stress intensity factor and the crack opening displacement for the finite-size specimen. Computational results reveal the importance of phonon–phason coupling effect on the icosahedral Al–Pd–Mn quasicrystal. Furthermore, the finite element procedure can be used to solve more complicated boundary value problems.
引文
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[2]Levine D and Stcinhardt P J 1984 Phys.Rev.Lett.53 2477
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[6]Ma J Y,Wang J B and Wang R H 2008 J.Phys.D:Appl.Phys.41085413
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[8]Bak P 1985 Phys.Rev.B 32 5764
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[10]Fan T Y 2011 Mathematical Theory of Elasticity of Quasicrystals and Its Applications(Heidelberg:Springer)
[11]Ding D H,Yang W G,Hu C Z and Wang R H 1993 Phys.Rev.B 487003
[12]Letoublon A,de Boissien M,Boudard M,Mancini L,Gastaldi J,Hennion B,Caudron R and Bellissent R 2001 Phil.Mag.Lett.81 273
[13]Capitan M J,Calvayrac Y,Quivy A,Joulaud J L,LeFebvre S and Gratias D 1999 Phys.Rev.B 60 6398
[14]Tanaka K,Mitarai Y and Koiwa M 1996 Phil.Mag.A 76 1715
[15]Edagawa K and So GY 2007 Phil.Mag.87 77
[16]Bachteler J and Trebin H R 1998 Eur.Phys.J.B 4 299
[17]Gao Y and Ricoeur A 2010 Phys.Lett.A 374 4354
[18]Zhu A Y and Fan T Y 2007 Chin.Phys.16 1111
[19]Gao Y,Ricoeur A and Zhang L L 2011 Phys.Lett.A 375 2775
[20]Chen W Q,Ma L and Ding H J 2004 Mech.Res.Commun.31 633
[21]Fan T Y and Peng Y Z 2000 Chin.Phys.B 9 764
[22]Li L H and Fan T Y 2008 Sci.China Ser.G 51 773
[23]Li L H and Fan T Y 2007 Phys.Lett.A 372 510
[24]Schaaf G D,Roth J,Trebin H R and Mikulla R 2000 Phil.Mag.A 801657
[25]Wang X F,Fan T Y and Zhu A Y 2009 Chin.Phys.B 18 709
[26]Sladek J,Sladek V and Pan E 2013 Int.J.Solids Struct.50 3975
[27]Sladek J,Sladek V,Krahulec S,Zhang Ch and W¨unsche M 2013 Int.J.Fract.181 115
[28]Radi E and Mariano P M 2011 Math.Meth.Appl.Sci.34 1
[29]Hu C Z,Wang R H and Ding D H 2000 Rep.Prog.Phys.63 1
[30]Henshell R D and Shaw K G 1975 Int.J.Num.Meth.Eng.9 495
[31]Barsoum R S 1975 Int.J.Num.Meth.Eng.10 25
[32]Ricoeur A and Kuna M 2003 J.Eur.Ceram.Soc.23 1313
[33]Tada H,Paris P C and Irwin G R 1985 The Stress Analysis of Cracks(Del Research Corporation)
[2]Levine D and Stcinhardt P J 1984 Phys.Rev.Lett.53 2477
[3]Ronchetti A M and Tasker P W 1987 Phil.Mag.56 237
[4]Ovid’ko I A 1992 Mat.Sci.Eng.:A 154 29
[5]Wollgarten M,Beyss M,Urban K,Liebertz H and K¨oster U 1993 Phys.Rev.Lett.71 549
[6]Ma J Y,Wang J B and Wang R H 2008 J.Phys.D:Appl.Phys.41085413
[7]Bak P 1985 Phys.Rev.Lett.54 1517
[8]Bak P 1985 Phys.Rev.B 32 5764
[9]Levine D,Lubensky T C,Qstlund S,Ramaswamy S and Steinhardt P J1985 Phys.Rev.Lett.54 1520
[10]Fan T Y 2011 Mathematical Theory of Elasticity of Quasicrystals and Its Applications(Heidelberg:Springer)
[11]Ding D H,Yang W G,Hu C Z and Wang R H 1993 Phys.Rev.B 487003
[12]Letoublon A,de Boissien M,Boudard M,Mancini L,Gastaldi J,Hennion B,Caudron R and Bellissent R 2001 Phil.Mag.Lett.81 273
[13]Capitan M J,Calvayrac Y,Quivy A,Joulaud J L,LeFebvre S and Gratias D 1999 Phys.Rev.B 60 6398
[14]Tanaka K,Mitarai Y and Koiwa M 1996 Phil.Mag.A 76 1715
[15]Edagawa K and So GY 2007 Phil.Mag.87 77
[16]Bachteler J and Trebin H R 1998 Eur.Phys.J.B 4 299
[17]Gao Y and Ricoeur A 2010 Phys.Lett.A 374 4354
[18]Zhu A Y and Fan T Y 2007 Chin.Phys.16 1111
[19]Gao Y,Ricoeur A and Zhang L L 2011 Phys.Lett.A 375 2775
[20]Chen W Q,Ma L and Ding H J 2004 Mech.Res.Commun.31 633
[21]Fan T Y and Peng Y Z 2000 Chin.Phys.B 9 764
[22]Li L H and Fan T Y 2008 Sci.China Ser.G 51 773
[23]Li L H and Fan T Y 2007 Phys.Lett.A 372 510
[24]Schaaf G D,Roth J,Trebin H R and Mikulla R 2000 Phil.Mag.A 801657
[25]Wang X F,Fan T Y and Zhu A Y 2009 Chin.Phys.B 18 709
[26]Sladek J,Sladek V and Pan E 2013 Int.J.Solids Struct.50 3975
[27]Sladek J,Sladek V,Krahulec S,Zhang Ch and W¨unsche M 2013 Int.J.Fract.181 115
[28]Radi E and Mariano P M 2011 Math.Meth.Appl.Sci.34 1
[29]Hu C Z,Wang R H and Ding D H 2000 Rep.Prog.Phys.63 1
[30]Henshell R D and Shaw K G 1975 Int.J.Num.Meth.Eng.9 495
[31]Barsoum R S 1975 Int.J.Num.Meth.Eng.10 25
[32]Ricoeur A and Kuna M 2003 J.Eur.Ceram.Soc.23 1313
[33]Tada H,Paris P C and Irwin G R 1985 The Stress Analysis of Cracks(Del Research Corporation)