一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

袁学刚; 朱正佑

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

袁学刚^1,2,
朱正佑¹

上海大学数学系,上海 200072;2.烟台大学数学与信息科学系,山东烟台 264005

基金项目: 国家自然科学基金资助项目(10272069);上海市重点学科基金资助项目

详细信息

作者简介:
袁学刚(1971- ),男,吉林市人,副教授,博士(E-mail:mengjn@ytu.edu.cn);朱正佑,男,教授,博士生导师(联系人.Tel:+86-21-56331454;E-mail:chjcheng@mail.shu.edu.cn)

中图分类号: O175；O343
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出版历程
- 收稿日期: 2002-12-20
- 修回日期: 2004-06-25
- 刊出日期: 2005-02-15

Qualitative Study of Cavitated Bifurcation for a Class of Incompressible Generalized neo-Hookean Spheres

YUAN Xue-gang^1,2,
ZHU Zheng-you¹

Shanghai Institute of Applied Mathematics and Mechanics;Department of Mathematics, Shanghai University, Shanghai 200072, P. R. China;

摘要

摘要: 研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题，该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料，得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比，证明了当摄动参数属于某些区域时，从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点，空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了，空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．
- 不可压的广义neo-Hookean材料 /
- 空穴分岔 /
- 正规形 /
- 稳定性和突变
Abstract: The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials,in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations.The condition of void nucleation for this problem was obtained.In contrast to the situation for a homogeneous isotropic neo-Hookean sphere,it is shown that not only there exists a secondary turning bifurcation point on the cavitated bifurcation solution which bifurcates locally to the left from trivial solution,and also the critical load is smaller than that for the material with no perturbations,as the parameters belong to some regions.It is proved that the cavitated bifurcation equation is equivalent to a class of normal forms with singlesided constraints near the critical point by using singularity theory.The stability of solutions and the actual stable equilibrium state were discussed respectively by using the minimal potential energy principle.
- incompressible generalized neo-Hookean material /
- cavitated bifurcation /
- normal form /
- stability and catastrophe

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参考文献(14)

[1]	Ball J M. Discontinuous equilibrium solutions and cavitations in nonlinear elasticity[J].Philos Trans Roy Soc Lond Ser A,1982,306(3)：557—610. doi: 10.1098/rsta.1982.0095
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[5]	Horgan C O, Polignone D A. Cavitation in nonlinear elastic solids: A review[J].Applied Mechanics Review,1995,48(8)：471—485. doi: 10.1115/1.3005108
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[7]	尚新春,程昌钧.超弹性材料中的球形空穴分岔[J].力学学报,1996,28(6)：751—755.
[8]	SHANG Xin-chun, CHENG Chang-jun.Exact Solution for cavitated bifurcation for compressible hyperelastic materials[J].International Journal of Engineering Science,2001,39(11)：1101—1117. doi: 10.1016/S0020-7225(00)00090-2
[9]	REN Jiu-sheng,CHENG Chang-jun,SHANG Xin-chun. The growth of a void in a rubber rectangular plate under uniaxial extension[J].Journal of Shanghai University (English Edition),2001,5(3)：177—182. doi: 10.1007/s11741-996-0020-y
[10]	HOU Hang-sheng. A study of combined asymmetric and cavitated bifurcation in neo-Hookey material under symmetric dead loading[J].Journal of Applied Mechanics,1993,60(1)：1—7. doi: 10.1115/1.2900746
[11]	REN Jiu-sheng,CHENG Chang-jun. Cavitated bifurcation for composed compressible hyperelastic materials[J].Acta Mechanica Solida Sinica,2002,15(3)：208—213.
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[14]	Golubitsky M, Schaeffer D G.Singularities and Groups in Bifurcation Theory[M].New York：Springer-Varlag, 1985,463.

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一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

作者简介:
袁学刚(1971- ),男,吉林市人,副教授,博士(E-mail:mengjn@ytu.edu.cn);朱正佑,男,教授,博士生导师(联系人.Tel:+86-21-56331454;E-mail:chjcheng@mail.shu.edu.cn)

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Qualitative Study of Cavitated Bifurcation for a Class of Incompressible Generalized neo-Hookean Spheres

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一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

作者简介: 袁学刚(1971- ),男,吉林市人,副教授,博士(E-mail:mengjn@ytu.edu.cn);朱正佑,男,教授,博士生导师(联系人.Tel:+86-21-56331454;E-mail:chjcheng@mail.shu.edu.cn)

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出版历程

Qualitative Study of Cavitated Bifurcation for a Class of Incompressible Generalized neo-Hookean Spheres

计量

出版历程

目录

作者简介:
袁学刚(1971- ),男,吉林市人,副教授,博士(E-mail:mengjn@ytu.edu.cn);朱正佑,男,教授,博士生导师(联系人.Tel:+86-21-56331454;E-mail:chjcheng@mail.shu.edu.cn)