类石墨烯二维原子晶体的微态理论模型
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摘要
基于微态(Micromorphic)连续介质理论,提出了针对类石墨烯二维原子晶体的新力学模型.该模型以有限大小的布拉维单胞为基元体,考虑基元粒子的宏观位移和微观变形,依据微态理论基本方程,推导了全局坐标系下模型的主导方程.然后针对布拉维单胞中含有两个原子的类石墨烯晶体,通过分析单胞中声子振动模式与基元体自由度的关系,获得了微态形式下声子色散关系的久期方程,并根据二维晶体声子色散特性对久期方程进行了简化,进而确定了类石墨烯晶体模型的本构方程.最后,以石墨烯和单层六方氮化硼为例,利用简化的表达式拟合了它们面内声子色散关系数据,计算了模型材料的常数,石墨烯模型的等效杨氏模量、泊松比分别为1.05 TPa和0.197,氮化硼分别为0.766TPa和0.225,均与已有的实验值相符合.
A novel mechanical model is proposed for graphene-like two-dimensional(2D) atomic crystals based on micromorphic continuum theory,in which the macro-displacement and micro-deformation of a basal element in the Bravais cell of finite size are considered.The governing equations of the model are derived from the basic equations of micromorphic theory in global coordinates.For the Bravais cell of graphene-like crystals containing two atoms,the secular equations of phonon dispersions are then obtained in micromorphic form by analyzing the relations between the vibrational modes of phonons and the independent degrees of freedom of the basal elements,and are further simplified according to the properties of phonon dispersion of 2D crystals,thus the constitutive equations of the model are conformed.Finally,the material constants are determined respectively by fitting the data of the in-plane phonon dispersion relations of graphene and monolayer hexagonal boron nitride with the simplified expressions.The obtained equivalent Young's modulus and Possion' ratio of graphene are 1.05 TPa and 0.197 respectively,and those of monolayer hexagonal boron nitride are 0.766 TPa and 0.225 respectively;both show good agreements with available experimental values.
A novel mechanical model is proposed for graphene-like two-dimensional(2D) atomic crystals based on micromorphic continuum theory,in which the macro-displacement and micro-deformation of a basal element in the Bravais cell of finite size are considered.The governing equations of the model are derived from the basic equations of micromorphic theory in global coordinates.For the Bravais cell of graphene-like crystals containing two atoms,the secular equations of phonon dispersions are then obtained in micromorphic form by analyzing the relations between the vibrational modes of phonons and the independent degrees of freedom of the basal elements,and are further simplified according to the properties of phonon dispersion of 2D crystals,thus the constitutive equations of the model are conformed.Finally,the material constants are determined respectively by fitting the data of the in-plane phonon dispersion relations of graphene and monolayer hexagonal boron nitride with the simplified expressions.The obtained equivalent Young's modulus and Possion' ratio of graphene are 1.05 TPa and 0.197 respectively,and those of monolayer hexagonal boron nitride are 0.766 TPa and 0.225 respectively;both show good agreements with available experimental values.
引文
1 Lee C,Wei X,Kysar JW,et al.Measurement of the elastic properties and intrinsic strength of monolayer graphene.Science,2008,321(5887):385-388
2 Bosak A,Serrano J,Krisch M,.et al.Elasticity of hexagonal boron nitride:Inelastic x-ray scattering measurements.Physical Review B,2006,73(4):041402
3 Zhang B,Mei L,Xiao H.Nanofracture in graphene under complex mechanical stresses.Applied Physics Letters,2012,101(12):121915-121915.
4 Zhang B.Layered graphene structure of a hexagonal carbon.Physica B:Condensed Matter,2013,413:73-75
5 Zhang B,Yang G,Xu H.Instability of supersonic crack in graphene.Physica B:Condensed Matter,2014,434:145-148
6韩同伟,贺鹏飞,王建等.单层石墨烯薄膜拉伸变形的分子动力学模拟.新型碳材料,2010,25(4):261-266(Han Tongwei,He Pengfei,Wang Jian,et al.Molecular dynam ics simulation of a single graphene sheet under tension.New Carbon Materials,2010,25(4):261-266(in Chinese))
7韩同伟,贺鹏飞,骆英等.石墨烯力学性能研究进展.力学进展,2011,41(3):279-293(Han Tongwei,He Pengfei,Luo Ying,et al.Research progress in the mechanical properties of graphene.Advances in Mechanics,2011,41(3):279-293(in Chinese))
8 Alzebdeh KI.An atomistic-based continuum approach for calculation of elastic properties of single-layered graphene sheet Solid State Communications,2014,177:25-28
9 Sakhaee-Pour A.Elastic properties of single-layered graphene sheet.Solid State Communications,2009,149(1):91-95
10 Li C,Chou TW.A structural mechanics approach for the analysis of carbon nanotubes.International Journal of Solids and Structures,2003,40(10):2487-2499
11 Scarpa F,Adhikari S,Phani AS.Effective elastic mechanical properties of single layer graphene sheets.Nanotechnology,2009,20(6):065709
12 Boldrin L,Scarpa F,Chowdhury R,et al.Effective mechanical properties of hexagonal boron nitride nanosheets.Nanotechnology,2011,22(50):505702
13王飞,庄守兵,虞吉林.用均匀化理论分析蜂窝结构的等效弹性参数.力学学报,2002,34(6):914-923(Wang Fei,Zhuang Shoubing,Yu Jilin.Application of homogenization FEM to the equivalent elastic constants of honeycomb structures.Acta Mechanica Sinica,2002,34(6):914-923(in Chinese))
14唐文来,彭倚天,倪中华.基于有限元分析的石墨烯弹性性能和振动特性.东南大学学报,自然科学版,2013,43(2):345-349(Tang Wenlai,Peng Yitian,Ni Zhonghua.Analytical and numerical investigations for thin-film with constraint under uniaxial tension.Journal of Southeast University(Natural Science Edition),2013,43(2):345-349(in Chinese))
15 Eringen AC.Microcontinuum Field Theories I:Foundations and Solids,1999.New York:Springer
16 Chen Y,Lee JD.Determining material constants in micromorphic theory through phonon dispersion relations.International Journal of Engineering Science,2003,41(8):871-886
17张朝晖,高原,聂君锋等.含约束薄膜的单轴拉伸的理论与数值研究.工程力学,2011,28(11):31-37(Zhang Zhaohui,Gao Yuan,Nie Junfeng,et al.Analytical and numerical investigations for thinfilm with constraint under uniaxial tension.Engineeing Mechanics,2011,28(11):31-37(in Chinese))
18 Berinskii IE,Borodich FM.On the isotropic elastic properties of graphene crystal lattice.Surface Effects in Solid Mechanics.Berlin Heidelberg:Springer,2013:33-42
19 Zhang B,Yang G.A micromorphic model for monolayer hexagonal boron nitride with determined constitutive constants by phonon dispersions.Physica B:Condensed Matter,2014,451:48-52
20 Chen Y,Lee JD,Eskandarian,A.Atomistic viewpoint of the applicability of microcontinuum theories.International Journal of Solids and Structures,2004,41(8):2085-2097
21 Miro P,Audiffred M,Heine T.An atlas of two-dimensional materials.Chemical Society Reviews,2014,43:6537-6554
22 Maultzsch J,Reich S,Thomsen C,et al.Phonon dispersion in graphite.Physical Review Letters,2004,92(7):075501
23 Mohr M,Maultzsch J,Dobardεzic E,et al.Phonon dispersion of graphite by inelastic x-ray scattering.Physical Review B,2007,76(3):035439
24 Dubay O,Kresse G.Accurate density functional calculations for the phonon dispersion relations of graphite layer and carbon nanotubes.Physical Review B,2003,67(3):035401
25 Serrano J,Bosak A,Arenal R,et al.Vibrational properties of hexagonal boron nitride:inelastic X-Ray scattering and ab initio calculations.Physical Review Letters,2007,98(9):095503
26 Wirtz L,Rubio A,de La Concha RA,et al.Ab initio calculations of the lattice dynamics of boron nitride nanotubes.Physical Review B,2003,68(4):045425.
27 Geick R,Perry CH,Rupprecht G.Normal modes in hexagonal boron nitride.Physical Review,1966,146:543-547
28 Al-Jishi R,Dresselhaus G.Lattice-dynamical model for graphite.Physical Review B,1982,26(8):4514-4522
29 Blakslee OL,Proctor DG,Seldin EJ,et al.Elastic constants of compression-annealed pyrolytic graphite.Journal of Applied Physics,1970,41(8):3373-3382
30 Baεzant ZP,Christensen M.Analogy between micropolar continuum and grid frameworks under initial stress.International Journal of Solids and Structures,1972,8(3):327-346
31 Kumar RS,McDowell DL.Generalized continuum modeling of 2-D periodic cellular solids.International Journal of Solids and Structures,2004,41(26):7399-7422
32 Akg(o|¨)z B,Civalek(O|¨).Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory.Materials&Design,2012,42:164-171
33徐巍,王立峰,蒋经农.基于应变梯度有限元的单层石墨烯振动研究.固体力学学报,2014,35(5):441-450(Xu Wei,WangLifeng,Jiang Jingnong.Finite element analysis of strain gradient on the vibration of single-layered graphene sheets.Chinese Journal of Solid Mechanics,2014,35(5):441-450(in Chinese))
34王晓明,王飞,赵学增等.基于Cosserat理论的四边简支自由振动微平板尺度效应研究.固体力学学报,2012,33(1):63-68(Wang Xiaoming,Wang Fei,Zhao Xuezeng,et al.On the size effects in a freely-vibrating micro-plate with the four edges simply-supported based on the cosserat theory.Chinese Journal of Solid Mechanics,2012,33(1):63-68(in Chinese))
2 Bosak A,Serrano J,Krisch M,.et al.Elasticity of hexagonal boron nitride:Inelastic x-ray scattering measurements.Physical Review B,2006,73(4):041402
3 Zhang B,Mei L,Xiao H.Nanofracture in graphene under complex mechanical stresses.Applied Physics Letters,2012,101(12):121915-121915.
4 Zhang B.Layered graphene structure of a hexagonal carbon.Physica B:Condensed Matter,2013,413:73-75
5 Zhang B,Yang G,Xu H.Instability of supersonic crack in graphene.Physica B:Condensed Matter,2014,434:145-148
6韩同伟,贺鹏飞,王建等.单层石墨烯薄膜拉伸变形的分子动力学模拟.新型碳材料,2010,25(4):261-266(Han Tongwei,He Pengfei,Wang Jian,et al.Molecular dynam ics simulation of a single graphene sheet under tension.New Carbon Materials,2010,25(4):261-266(in Chinese))
7韩同伟,贺鹏飞,骆英等.石墨烯力学性能研究进展.力学进展,2011,41(3):279-293(Han Tongwei,He Pengfei,Luo Ying,et al.Research progress in the mechanical properties of graphene.Advances in Mechanics,2011,41(3):279-293(in Chinese))
8 Alzebdeh KI.An atomistic-based continuum approach for calculation of elastic properties of single-layered graphene sheet Solid State Communications,2014,177:25-28
9 Sakhaee-Pour A.Elastic properties of single-layered graphene sheet.Solid State Communications,2009,149(1):91-95
10 Li C,Chou TW.A structural mechanics approach for the analysis of carbon nanotubes.International Journal of Solids and Structures,2003,40(10):2487-2499
11 Scarpa F,Adhikari S,Phani AS.Effective elastic mechanical properties of single layer graphene sheets.Nanotechnology,2009,20(6):065709
12 Boldrin L,Scarpa F,Chowdhury R,et al.Effective mechanical properties of hexagonal boron nitride nanosheets.Nanotechnology,2011,22(50):505702
13王飞,庄守兵,虞吉林.用均匀化理论分析蜂窝结构的等效弹性参数.力学学报,2002,34(6):914-923(Wang Fei,Zhuang Shoubing,Yu Jilin.Application of homogenization FEM to the equivalent elastic constants of honeycomb structures.Acta Mechanica Sinica,2002,34(6):914-923(in Chinese))
14唐文来,彭倚天,倪中华.基于有限元分析的石墨烯弹性性能和振动特性.东南大学学报,自然科学版,2013,43(2):345-349(Tang Wenlai,Peng Yitian,Ni Zhonghua.Analytical and numerical investigations for thin-film with constraint under uniaxial tension.Journal of Southeast University(Natural Science Edition),2013,43(2):345-349(in Chinese))
15 Eringen AC.Microcontinuum Field Theories I:Foundations and Solids,1999.New York:Springer
16 Chen Y,Lee JD.Determining material constants in micromorphic theory through phonon dispersion relations.International Journal of Engineering Science,2003,41(8):871-886
17张朝晖,高原,聂君锋等.含约束薄膜的单轴拉伸的理论与数值研究.工程力学,2011,28(11):31-37(Zhang Zhaohui,Gao Yuan,Nie Junfeng,et al.Analytical and numerical investigations for thinfilm with constraint under uniaxial tension.Engineeing Mechanics,2011,28(11):31-37(in Chinese))
18 Berinskii IE,Borodich FM.On the isotropic elastic properties of graphene crystal lattice.Surface Effects in Solid Mechanics.Berlin Heidelberg:Springer,2013:33-42
19 Zhang B,Yang G.A micromorphic model for monolayer hexagonal boron nitride with determined constitutive constants by phonon dispersions.Physica B:Condensed Matter,2014,451:48-52
20 Chen Y,Lee JD,Eskandarian,A.Atomistic viewpoint of the applicability of microcontinuum theories.International Journal of Solids and Structures,2004,41(8):2085-2097
21 Miro P,Audiffred M,Heine T.An atlas of two-dimensional materials.Chemical Society Reviews,2014,43:6537-6554
22 Maultzsch J,Reich S,Thomsen C,et al.Phonon dispersion in graphite.Physical Review Letters,2004,92(7):075501
23 Mohr M,Maultzsch J,Dobardεzic E,et al.Phonon dispersion of graphite by inelastic x-ray scattering.Physical Review B,2007,76(3):035439
24 Dubay O,Kresse G.Accurate density functional calculations for the phonon dispersion relations of graphite layer and carbon nanotubes.Physical Review B,2003,67(3):035401
25 Serrano J,Bosak A,Arenal R,et al.Vibrational properties of hexagonal boron nitride:inelastic X-Ray scattering and ab initio calculations.Physical Review Letters,2007,98(9):095503
26 Wirtz L,Rubio A,de La Concha RA,et al.Ab initio calculations of the lattice dynamics of boron nitride nanotubes.Physical Review B,2003,68(4):045425.
27 Geick R,Perry CH,Rupprecht G.Normal modes in hexagonal boron nitride.Physical Review,1966,146:543-547
28 Al-Jishi R,Dresselhaus G.Lattice-dynamical model for graphite.Physical Review B,1982,26(8):4514-4522
29 Blakslee OL,Proctor DG,Seldin EJ,et al.Elastic constants of compression-annealed pyrolytic graphite.Journal of Applied Physics,1970,41(8):3373-3382
30 Baεzant ZP,Christensen M.Analogy between micropolar continuum and grid frameworks under initial stress.International Journal of Solids and Structures,1972,8(3):327-346
31 Kumar RS,McDowell DL.Generalized continuum modeling of 2-D periodic cellular solids.International Journal of Solids and Structures,2004,41(26):7399-7422
32 Akg(o|¨)z B,Civalek(O|¨).Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory.Materials&Design,2012,42:164-171
33徐巍,王立峰,蒋经农.基于应变梯度有限元的单层石墨烯振动研究.固体力学学报,2014,35(5):441-450(Xu Wei,WangLifeng,Jiang Jingnong.Finite element analysis of strain gradient on the vibration of single-layered graphene sheets.Chinese Journal of Solid Mechanics,2014,35(5):441-450(in Chinese))
34王晓明,王飞,赵学增等.基于Cosserat理论的四边简支自由振动微平板尺度效应研究.固体力学学报,2012,33(1):63-68(Wang Xiaoming,Wang Fei,Zhao Xuezeng,et al.On the size effects in a freely-vibrating micro-plate with the four edges simply-supported based on the cosserat theory.Chinese Journal of Solid Mechanics,2012,33(1):63-68(in Chinese))
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