Blatz-Ko 圓球波方程李群性質及其應用

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陳聲偉(Sheng-Wei Chen) 查詢紙本館藏

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土木工程學系

論文名稱

Blatz-Ko 圓球波方程李群性質及其應用
(Symmetries and Application of the wave equations governing radial motion of the Blatz-Ko materials)

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摘要(中)

主要研究由Blatz-Ko材料所導出的圓球對稱動態變形波方程式，並利用李群理論導出此波方程式的李群與求得三個不變解，其中兩個不變解為可分離變數解，另一個則為不可分離變數不變解，利用不可分離變數不變解對原Blatz-Ko波方程做降維後導出的是一個含有奇異性(singular)的二階non-autonomous常微分方程式。我們利用變數變換將此常微分方程式轉換成autonomous 微分方程組，並且成功的分析其相平面(phase plane)上的軌跡。根據上面所述可以完整的分析方程式的解在全域上的行為模式，而這組特殊解的分析可以明確的指出在什麼樣的問題以及初始條件下可能發生解的奇異性。另外也分析多參數子群在群轉換作用下，所得到新的不變解在相平面上的初位置與時間彼此間參數變化所對應的新的關係。本論文也研究Blatz-Ko波方程的一種對稱性-倒數時間平移對稱，此一特殊的對稱性質能讓將不同的波動方程初始邊界條件彼此連結起來，根據這個連結轉換可以從已知的解轉換到其它的解。在實驗及計算方面根據這個倒數時間轉換也可以減少Blatz-Ko材料在實驗或數值計算的時間。最後分析在逆時間轉換下的解的極限行為與導數性質。

摘要(英)

This dissertation investigates the symmetry properties of the spherical wave equation for Blatz-Ko materials. Three invariant solutions are obtained by symmetry reduction. Two of them are in separable forms. The non-separable one is governed by a singular, non-autonomous second order ordinary differential equation. A variable transform is applied to turn this non-autonomous equation into an autonomous equation. By analyzing the phase plane of this autonomous equation we find out the condition for the solutions to develop singularity. Similar analyses are also carried out for the governing equation of an invariant solution derived from a multi-parameter symmetry. A special symmetry, the inverse time translational symmetry, of the Blatz-Ko wave equation is also studied. A correspondence between initial/boundary value problems is derived using this special symmetry. And this correspondence can be applied to reduce the duration of the experiments or numerical calculations for the dynamic behavior of the Blatz-Ko spheres. The correspondence can also be used to analyze the limiting dynamic behavior of the Blatz-Ko sphere.

關鍵字(中)

★ 奇異性
★ Blatz-Ko材料
★ 波方程
★ 逆時間轉換

關鍵字(英)

★ Blatz-Ko materials
★ singularity
★ inverse time translation
★ wave equation

論文目次

摘要………………………………………………………………….….Ⅰ
英文摘要…………………………………………………………….….Ⅲ
誌謝…..…………………………………………………………………Ⅴ
目錄………………………………………………………………..........Ⅶ
圖目錄……………………………………………………………..........Ⅹ
表目錄……………………………………………………………........
第一章緒論……………………………………………………………01
第二章Blatz-Ko 圓球波動方程的群與不變解………………………04
2.1 前言...........................................................................................04
2.2 Blatz-Ko 材料圓球體徑向運動方程式 ..............................04
2.3李群理論簡介.............................................................................08
2.3.1 微分方程的李群................................................................08
2.3.2 不變解................................................................................13
2.4 Blatz-Ko 材料圓球波動方程的李群轉換................................14
2.4.1 Blatz-Ko 材料圓球波動方程的對稱群............................15
2.4.2 Blatz-Ko 材料圓球波動方程的不變解............................22
2.5 Blatz-Ko 材料圓球波動方程的解析解...................................24
第三章解的奇異性分析.................................................26
3.1 前言…………………………………………………………...26
3.2 (2.4.27)解的相平面分析...........................................................26
3.3 方程式(2.4.22)的奇異解之分析..............................................29
3.3.1 跳躍條件分析....................................................................30
3.3.2 漸近解逼近法....................................................................33
3.3.3解的奇異性與初始邊界條件之關連..................................36
第四章多參數子群不變解解的奇異性分析........................................67
4.1 前言…………………………………………………………...67
4.2 多參數子群不變解...................................................................67
4.3 分析(4.2.5)的相平面................................................................69
4.3.1 case(1) 與 ....................................................70
4.3.2 case(2) 與 ....................................................72
4.3.3 case(3) 與 ...................................................74
4.3.4 case(4) 與 ....................................................77
4.4 對Blatz-Ko材料作數值解分析...............................................79
4.4.1分析case(1)的物理特性......................................................80
4.4.2分析case(2)的物理特性.....................................................82
4.4.3分析case(3)的物理特性......................................................84
4.4.4分析case(4)的物理特性.....................................................85
第五章 Blatz-Ko 圓柱方程的倒數時間轉換之對稱性…………….110
5.1 前言.........................................................................................110
5.2用連結(2.2.12)式的初始邊界值問題…………………….111
5.3 ITTS轉換群解分析………………………………………….114
5.3.1 ITTS轉換群計算解分析………………………………115
5.3.2 case(1)當時(5.3.3)式數值解…………………….115
5.3.3 case(2)當時(5.3.3)式數值解…………………...117
5.3.4 ITTS作用在解析解的分析............................................117
第六章總結…………………………………………………………..131
參考文獻………………………………………………………………134

參考文獻

1. P.J. Blatz and W.L. Ko, Application of finite elastic theory to the deformation of rubbery materials. Trans.Soc. Rheol., 6, 223-251 (1962).
2. J.K. Knowles and E. Sternberg, On the ellipticity of non-linear elastostatics for a special material. J. Elasticity, 5, 341-361 (1975).
3. C.O. Horgan, Remarks on ellipticity for the generalized Blatz-Ko constitutive model for compressible nonlinearly elastic solid. J. Elasticity, 42, 165-176 (1996).
4. M.F. Beatty and D.O. Stalnacker, The Poisson function of finite elasticity. ASME J. Appl. Mech., 53, 807-813 (1986).
5. M. Cheref, M. Zidi and C. Oddou, Analytical modelling of vascular prostheses mechanics. Intra and extracorporeal cardiovascular fluid dynamics. Comput. Mech. Pub., 1, 191-202 (1998).
6. M. Zidi, Finite torsional and anti-plane shear of a compressible hyperelastic and transversely isotropic tube. Int. J. Engrg. Sci., 38, 1481-1496 (2000).
7. R. Abeyaratne and C.O. Horgan, The pressurized hollow sphere problem in finite elastostatics for a class of compressible elastic materials, Int. J. Solids Structures, 20, 715-723 (1984).
8. D.-T. Chung, C.O. Horgan and R. Abeyaratne, The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials, Int. J. Solids Structures, 22, 1557-1570 (1986).
9. R. Abeyaratne and C.O. Horgan, Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinear elastic medium. J. Elasticity, 15, 243-256 (1985).
10. C.O.Horgan and R.Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void. J.Elasticity, 16, 189-200 (1986).
11. S.Biwa, Critical stretch for formation of a cylindrical void in a compressible hyperelastic material. Int. J. Non-Linear Mech., 30, 899-914 (1995).
12. H.C. Lei (李顯智) and H.W. Chang, Void formation and growth in a class of compressible solids. J. Engrg. Math., 30, 693-706 (1996).
13.李顯智,唐又新,孔洞承受的極限壓力,中華民國力學學會期刊, 33, 205-212 (1997).
14. A. Mioduchowski and J.B. Haddow, Combined torsional and telescopic shear of a compressible hyperelastic tube. J. Appl. Mech., 46, 223-226 (1979)s.
15. M. Zidi, Circular shearing and torsion of a compressible hyperelastic and prestressed tube. Int. J. Non-Linear Mech., 35, 201-209 (2000).
16. M. Zidi, Torsion and telescopic shearing of a compressible hyperelastic tube. Mech. Res. Comm., 26, 245-252 (1999).
17. M. Destrade and G. Saccomandi, Finite amplitude elastic waves propagating in compressible solids. Phys. Rev. E, 72, 016620 (2005).
18. H.A. Erbay and V.H. Tuzel, Dynamic extension of a compressible nonlinearly elastic membrane tube. IMA J. Appl. Math., 70, 25-38 (2005).
19. E.R. Ferreira and Ph. Boulanger, Finite-amplitude damped inhomogeneous waves in a deformed Blatz-Ko material. Math. Mech. Solids, 10, 377-387 (2005).
20. E.R. Ferreira and Ph. Boulanger, Superposition of transverse and longitudinal finite-amplitude waves in a deformed Blatz-Ko material. Math. Mech. Solids, 12, 543-558 (2007).
21. F. Peyraut, Orientation preserving and Newton-Raphson convergence in the case of an hyperelastic sphere subjected to an hydrostatic pressure. Comput. Methods Appl. Mech. Engrg., 192, 1107-1117 (2003).
22. F. Peyraut, Loading restrictions for the Blatz-Ko hyperelastic model---application to a finite element analysis. Int. J. Non-Linear Mech., 39, 969-976 (2003).
23. P.A. Du Bois, S. Kolling and W. Fassnacht, Modelling of safety glass for crash simulation. Comp. Materials Sci., 28, 675-683 (2003).
24. Z.Q. Feng, B. Magnain and J.M. Cros, Solution of large deformation contact problems with friction between Blatz-Ko hyperelastic bodies. Int. J. Engrg. Sci., 41, 2213-2225 (2003).
25. Z.Q. Feng, B. Magnain and J.M. Cros, Solution of large deformation impact problems with friction between Blatz-Ko hyperelastic bodies. Int. J. Engrg. Sci., 44, 113-126 (2006).
26. J. B. Haddow and L. Jiang, Finite amplitude azimuthal shear waves in a compressible hyperelastic solid. J. Appl. Mech., 68, 145-152 (2001).
27. F.A.McClintock, A criterion for ductile fracture by the growth of holes. J.Appl. Mech., 35, 363-371 (1968).
28. A.Needleman, Void growth in an elastic-plastic medium. J.Appl. Mech., 39, 964-970 (1972).
29. A.L.Gurson, Continuum theory of ductile rupture by void nucleation and growth : Part Ⅰ- yield criteria and flow rules for porous ductile media. J.Energ.Matl.Tech.,Trans.ASME, 2-15 (1977).
30. U.Stigh, Effects of interacting cavities on damage parameter. J.Appl. Mech, 53, 485-490 (1986).
31. H.S.Hou and R.Abeyarante, Cavitation in elastic and elastic-plastic solids, J.Mech.Phys.Solids, 40, 571-592 (1992).
32. A.N.Gent,Cavitation in rubber: a cautionary tale. Rubber Chem.Tech., 63, G49-G53 (1990).
33. C.O.Horgan and D.A.Polignone,Cavitation in nonlinearly elastic solids: a review. Appl.Mech.Rev., 48, 471-485 (1995).
34. J.M.Ball, Discontinous equilibrium solutions and cavitation in nonlinear elasticity. Phil.Trans.R.Soc.Lond, A306, 557-610 (1982).
35. C.A.Stuart, Radially symmetric cavitation for hyperelastic materials, Ann.Inst.Henri Poincare-Analyse non lineare, 2, 33-66 (1985).
36. F.Meynard, Existence and nonexistence results on the radially symmetric cavitation problem. Quart.Appl.Math. 50, 201-226 (1992).
37.S.Biwa, E.Matsumoto and T.Shibata, Effect of constitutive parameters on formation of a spherical void in a compressible non-linear elastic material. J.Appl.Mech. 61, 395-401 (1994).
38.X.-C. Shang and C.-J. Cheng, Exact solution for cavitated bifurcation for compressible hyperelastic materials. Int.J.Engrg.Sci., 39 1101-1117 (2001).
39. G.B. Whitham, Linear and Nonlinear Waves. Weily, New York (1974).
40. H.C. Lei and H.W. Chang, A list of hodograph transformations and exactly linearizable systems. Int. J. Non-Linear Mech., 31, 117-127 (1996).
41. H.C. Lei, Linearity of the steady state of a nonlinear anisotropic diffusion process. J. Chinese Inst. Engineers, 18, 461-470 (1995).
42. H.C. Lei, Study of a hodograph transformation and its applications. J. Inst. Chinese Engineers, 25, 707-714 (2002).
43. J. Weiss, M. Tabor and G. Carnevale, The Painleve property for partial differential equations. J. Math. Phys., 24, 522-526 (1983).
44. A.A. Alexeyev, Some notes on singular manifold method – several manifolds and constraints. J. Phys. A- Math. Gen., 33, 1873-1894 (2000).
45. L.V. Ovsiannikov, Group Analysis of Differential Equations (W.F. Ames, trans.). Academic Press, New York (1982).
46. P.J. Olver, Applications of Lie Groups to Differential Equations. Springer-Verlag, New York (1986).
47. H.C. Lei, Group splitting and linearization mapping of a solvable nonlinear wave equation. Int. J. Non-Linear Mech., 33, 461-471 (1998).
48. L.V. Ovsiannikov, Group Analysis of Differential Equations (W. F. Ames, trans.). Academic Press, New York (1982).
49. N.H. Ibragimov, Tramsformation groups applied to mathematical physics. Reidel, Boston( 1985).
50. G.W. Bluman and S. Kumei , Symmetries and Differential Equations. Springer-Verlag , New York (1989).
51. P.J. Olver, Applications of Lie Groups to Differential Equations. Springer-Verlag, New York (1986).
52. J.M. Hill, Some Partial Solutions of Finite Elasticity. (1972) Ph.D. Thesis, University of Queensland, Brisbane, Australia.
53. J.M. Hill, On static similarity deformations for isotropic materials. Q. Appl. Math., 40, 287-291 (1982).
54. K.A. Ames and W.F. Ames, On group analysis of the von Karman equations. Int. J. Nonlinear Anal.: Theory, Meth. Appl., 6845-853 (1982).
55. K.A. Ames and W.F. Ames, Analysis of the von Karman equations by group methods. Int. J. Non-linear Mech., 20, 201-209 (1985).
56. D. Levi and C. Rogers, Group invariance of a neo-Hookean system: incorporation of stretch change. J. Elasticity, 24, 295-300 (1990).
57. H.C. Lei and J.A. Blume, Lie group and invariant solution of the plane-strain equation of motion of a neo-Hookean solid. Int.J. Non-Linear Mech , 31, 465-482 (1996).
58. H.C. Lei (李顯智) and M.J. Hung, Linearity of waves in some systems of non-linear elastodynamics . Int.J. Non-Linear Mech., 32, 353-360 (1997).
59. C.O. Horgan and J.G. Murphy, Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials. IMA J. Appl. Math.,70, 80-91 (2005).
60. C.O. Horgan and J.G. Murphy, A Lie group analysis of the axisymmetric equations of finite elastostatics for compressible materials. Math. Mech. Solids, 10, 311-333 (2005).
61. W.A. Strauss, Nonlinear wave equations. American Mathematical Society, Rhode Island, 1989.
62. S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string. Communication on Pare and Applied Mathematics, 33, 241-263 (1980)
63. C. H. Hsu, and S. S. Lin and T. Makino, Smooth solutions to a class of quasilinear wave equations. J. Diff. Eq., 224, 229-257 (2006).
64. J. Kevorkian, Partial Differential Equations: Analytical Solution Techniques. Wadsworth & Brooks/Cole, Belmont, California, 1990.

指導教授

李顯智(Hin-Chi Li)

審核日期

2008-1-15

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